Ductile Fracture after Complex Loading Histories:
Experimental Investigation and Constitutive Modeling
by
ARCHVES
MASSAC-u
OF
Stephane Marcadet
Diplome de l'Ecole Polytechnique (2012)
M.S., Massachusetts Institute of Technology (2012)
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
TT
JUL 3 0 2015
, LIBRARIES
Doctor of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
0 2015 Massachusetts Institute of Technology. All rights reserved.
//,
Signature redacted
Author:
Certified by:
-
Signature redacted
D)epartment of Mechanical Engineering
May 22, 2015
Signature redacted
Tomasz Wierzbicki
Professor of Applied Mechanics
Thesis Supervisor
-
Certified by:
Dirk Mohr
Accepted by:_
Signature redact
ONSTITJTE
rC.HN0LLGY
CNRS Associate Professor
Thesis Supervisor
David E. Hardt
Chairman, Committee on Graduate Students
Department of Mechanical Engineering
-3-
Ductile Fracture after Complex Loading Histories:
Experimental Investigation and Constitutive Modeling
by Stephane Marcadet
Submitted to the Department of Mechanical Engineering
On May 22, 2015, in partial fulfillment of the requirements for the degree of
Doctor of Science in Mechanical Engineering
In engineering practice, sheet metal often fails after complex strain paths that deviate
substantially from the widely studied proportional loading paths. Different from previous
works on the ductile fracture of sheet metal, this thesis research addresses the experimental
and modeling issues related to the crack initiation in advanced high strength steels after
loading direction reversal. The main outcome of the present work is a fracture initiation
model for proportional and non-proportional loading.
The starting point of this thesis is a first chapter on the development of a
micromechanically-motivated ductile fracture initiation model for metals for proportional
loading. Its formulation is based on the assumption that the onset of fracture is imminent
with the formation of a primary or secondary band of localization. Motivated by the results
from a thorough unit cell analysis, it is assumed that fracture initiates after proportional
loading if the linear combination of the Hosford equivalent stress and the normal stress
acting on the plane of maximum shear reaches a critical value. A comprehensive fracture
initiation model is then obtained after transforming the localization criterion from the stress
space to the space of equivalent plastic strain, stress triaxiality and Lode angle parameter
using the material's isotropic hardening law. Experimental results are presented for three
different advanced high strength steels. For each material, the onset of fracture is
characterized for five distinct stress states, including butterfly shear, notched tension,
tension with a central hole, and punch experiments. The comparison of model predictions
with the experimental results demonstrates that the proposed Hosford-Coulomb model can
predict with satisfactory accuracy the instant of ductile fracture initiation in advanced high
strength steels.
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In a subsequent chapter, experimental methods are developed to perform compressiontension experiments. In addition, a finite strain constitutive model is proposed combining
a Swift-Voce isotropic hardening law with two Frederick-Armstrong kinematic hardening
rules and a Yoshida-Uemori type of hardening stagnation approach. The plasticity model
parameters
are
identified from
uniaxial
tension-compression
stress-strain
curve
measurements and finite element simulations of compression-tension experiments on
notched specimens. The model predictions are validated through comparison with
experimentally-measured load-displacement curves up to the onset of fracture, local
surface strain measurements and longitudinal thickness profiles. The extracted loading
paths to fracture show a significant increase in ductility as a function of the compressive
pre-strain. The Hosford-Coulomb model is therefore integrated into a non-linear damage
indicator modeling framework to provide a phenomenological description of the
experimental results for monotonic and reverse loading.
Another extension of the modeling framework is presented in a third chapter inspired
by the results from loss of ellipticity analysis. It is demonstrated that the Hosford-Coulomb
model can also be expressed in terms of a stress-state dependent critical hardening rate.
Moreover, it is shown that the critical hardening rate approach provides accurate
predictions of the instant of fracture initiation for both proportional and non-proportional
loading conditions. Enhancements of the finite strain constitutive model are also proposed
to enable a fast identification of all model parameters. The plasticity model parameters are
identified from stress-strain curve measurements from shear loading reversal on specimens
with a uniform thickness reduced gage section. The model is used to estimate the local
strain and stress fields in fracture experiments after shear reversal. The extracted loading
paths to fracture show a significant increase in ductility as a function of the strain at shear
reversal, a feature that is readily predicted by the prosed critical hardening rate model.
Thesis supervisor: Tomasz Wierzbicki
Title: Professor of Applied Mechanics
Thesis supervisor: Dirk Mohr
Title: CNRS Associate Professor, Ecole Polytechnique
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Acknowledgements
I would like to thank Professor Tomasz Wierzbicki and Professor Dirk Mohr who
provided crucial guidance and support throughout my studies at MIT in the Impact and
Crashworthiness Laboratory. I greatly appreciate the opportunities they afforded me in
order to carry out my thesis work independently.
Further, I am grateful to Professors David Parks and Kenneth Kamrin for agreeing to
serve on my thesis committee, and to discuss my research.
It has been a pleasure for me to work with both the former and current members of the
ICL, all of whom have fostered a friendly and productive work environment: Dr. Allison
Beese, Dr. Meng Luo, Dr. Matthieu Dunand, Dr. Christian Roth, Dr. Kai Wang, Dr. Fabien
Ebnoether, Dr. Jessica Papasidero, Dr. Kirki Kofiani, Dr. Gongyao Gu, Dr. Camille Besse,
Mr. Keunhwan Pack, Mr. Xiaowei Zhang, Mr. Colin Bonatti and Mr. Rami Abi Akl.
Special thanks as well to Barbara Smith for her invaluable assistance.
Additionally, I sincerely thank the members of Industrial Fracture Consortium for the
financial support and their regular feedback on my work.
Finally, I'd like to thank my family and friends for their support.
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Contents
Chapter 1: Introduction ..........................................................................................
21
1.1
Ductile Fracture..............................................................................................
21
1.2
Plasticity for Complex Loading ......................................................................
23
1.3
Strain path dependency of FLD and fracture toughness .................................
27
1.4
Examples of non-linear loading paths............................................................
28
1.4.1
Crash ...........................................................................................................
28
1.4.2
Combined Form ing and Crash ................................................................
29
1.5
Complex Loading on Bulk M aterial................................................................
30
1.6
Particularities of Sheet M etal..........................................................................
31
1.7
Thesis Outline ................................................................................................
34
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture................... 37
2.1
Introduction ........................................................................................................
38
2.2
Prelim inaries...................................................................................................
44
2.2.1
Description of the stress state .................................................................
44
2.2.2
Plasticity M odel .......................................................................................
45
2.3
Fracture Initiation M odel for Proportional Loading ......................................
46
2.3.1
M otivation..............................................................................................
46
2.3.2
Localization Criterion in Stress Space.....................................................
47
2.3.3
Fracture initiation model in m ixed strain-stress space.............................
50
2.3.4
Illustration of the HC model ....................................................................
52
2.3.5
Comments on Model Extension for Non-Proportional Loading.............. 55
-8-
2.3.6
2.4
Com m ent on M odel Consistency............................................................
Fracture Experim ents .....................................................................................
57
59
2.4.1
M aterials ..................................................................................................
59
2.4.2
Uniaxial tension experim ents...................................................................
59
2.4.3
Shear experim ents....................................................................................
64
2.4.4
Punch experim ent....................................................................................
64
2.5
Identification of the loading paths to fracture .................................................
64
2.5.1
Plasticity m odel param eter identification ..............................................
64
2.5.2
Loading paths to fracture .........................................................................
66
2.6
M odel calibration and verification ..................................................................
2.6.1
2.7
M odel application ...................................................................................
Sum m ary ............................................................................................................
68
69
70
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets ............ 73
3.1
Introduction ........................................................................................................
74
3.2
Experim ents........................................................................................................
77
3.2.1
M aterial and specim ens............................................................................
77
3.2.2
Experim ental procedure ...........................................................................
80
3.2.3
Experim ental results................................................................................
82
3.3
Combined Chaboche-Yoshida (CCY) plasticity model ..................................
83
3.3.1
Y ield surface ............................................................................................
84
3.3.2
N on-associated flow rule .........................................................................
84
-9-
3.3.3
Definition of the equivalent plastic strain...............................................
85
3.3.4
Isotropic hardening ..................................................................................
85
3.3.5
Non-linear kinematic hardening ............................................................
86
3.3.6
Work hardening stagnation ......................................................................
87
3.3.7
Summary of model parameters ...............................................................
90
3.3.8
Thermodynamic constraints....................................................................
90
3.4
Plasticity model identification and validation.................................................
92
3.4.1
Identification step I: determination of seed parameters ...........................
93
3.4.2
Identification step II: full inverse parameter identification ....................
94
3.4.3
Model verification for reverse loading ...................................................
96
3.4.4
Model verification for monotonic multi-axial loading .............................
101
Effect of loading reversal on ductile fracture initiation ...................................
102
3.5
3.5.1
Characterization of the stress state............................................................
3.5.2
Effect of pre-compression on results for notched tension ........................ 103
3.5.3
Hosford-Coulomb fracture initiation model .............................................
107
3.5.4
Model identification and verification........................................................
109
3.5.5
Discussion of the effect of pre-strain on ductile fracture..........................
112
3 .6
Sum m ary ..........................................................................................................
103
114
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile
Fractu re ...........................................................................................................................
4 .1
Intro d uction ......................................................................................................
1 15
116
10
-
-
4.2
Plasticity m odel................................................................................................
118
4.2.1
Y ield function and flow rule .....................................................................
119
4.2.2
Hardening evolutions ................................................................................
120
4.2.3
Evolution of the back stress tensor ...........................................................
122
4.2.4
W ork Hardening Stagnation .....................................................................
124
4.2.5
Therm odynam ic constraints......................................................................
124
4.2.6
Illustration for uniaxial loading ................................................................
126
4.3
Critical hardening rate fracture initiation m odel..............................................
127
4.3.1
M echanism -based m odeling .....................................................................
127
4.3.2
Phenom enological m odeling for proportional loading .............................
128
4.3.3
Phenomenological modeling for non-proportional loading......................
130
4.3.4
Extended form ulation................................................................................
133
4.3.5
Com m ent on the m odel sensitivity ...........................................................
134
4.4
Experim ents......................................................................................................
134
4.4.1
Overview on experim ental procedures .....................................................
135
4.4.2
Details on experim ental procedures..........................................................
136
4.4.3
Overview on experim ents perform ed........................................................
141
4.5
A pplication and validation ...............................................................................
142
4.5.1
Plasticity m odel param eter identification .................................................
142
4.5.2
Reverse Loading .......................................................................................
143
4.5.3
Fracture m odel param eter identification...................................................
149
-
- 11
4.5.4
4.6
V alidation and discussion .........................................................................
Sum m ary ..........................................................................................................
Chapter 5: Conclusion.............................................................................................
5.1
Sum m ary of findings........................................................................................
150
152
155
155
5.1.1
Hosford-Coulom b .....................................................................................
155
5.1.2
Ductile fracture of sheets after in-plane compression-tension..................
156
5.1.3
Critical Hardening Rate ............................................................................
157
Ongoing and future work .................................................................................
157
5.2
5.2.1
Orthogonal
n ..................................................................................
157
5.2.2
Validations studies ....................................................................................
158
5.2.3
Strain rate and tem perature effects ...........................................................
160
References...............................................................................................................
161
12
-
-
-
- 13
List of Figures
Figure 1-1 Comparison of the von Mises stress versus equivalent plastic strain curve for
monotonic loading with that after loading reversal at a strain of 0.1 to illustrate different
hardening model assumptions: (a) isotropic hardening, (b) permanent softening, (c) work
hardening stagnation, (d) transient behavior.................................................................
25
Figure 1-2 Loading Paths at points (1) and (2) during the three point bending of the hatsh ap ed p rofile ....................................................................................................................
29
Figure 1-3 Comparison of the fracture observed during experimental three point bending
of a martensitic hat-shaped profile with the numerical prediction using isotropic hardening
and linear damage accumulation indicator. ..................................................................
29
Figure 1-4 Comparison between load-displacement curves under compression-tension for
(a) round bars of aluminum (b) flat specimens of DP780 sheets..................................
32
Figure 1-5 Comparison between stress-strain curves under compression-tension for (a)
round bars of aluminum (b) flat specimens of DP780 sheets .......................................
33
Figure 2-1 Eulerian illustration of the ductile fracture process with coalescence through (a)
internal necking, and (b) void sheet fracture. The mesoscopic primary band of localization
is highlighted in gray color, the microscopic secondary band of localization is highlighted
in red . ................................................................................................................................
39
Figure 2-2 (a) Illustration of the stress triaxiality i and the Lode angle parameter 0 in
principal stress space {
1,
-,, %- }. Selected Lode angle parameter values are only shown
for the 60' segment of a-plane where the principal stresses satisfy the order
0- 1
o-j,
c-r . For the other five 60' segments, the same labeling applies because of the
symmetries of the unordered principal stress space; (b) non-linear relationship between 0
and il for plane stress. The blue, black and red curves shows the relationship for biaxial
compression (two negative principal stresses), biaxial tension-compression (one positive
14
-
-
and one negative principal stress), and biaxial tension (two positive principal stresses),
respectively. Open dots highlight the special cases of cases of a. uniaxial compression, b.
pure shear, c. uniaxial tension, d. plane strain tension, and e. equi-biaxial tension"........ 44
Figure 2-3 Localization analysis results: (a) Relationship between the shear and normal
stress acting on the plane of localization; (b) Macroscopic equivalent plastic strain at the
onset of localization as a function of Lode angle parameter and stress triaxiality; each dot
represents the result from a unit cell analysis for a particular stress state, the solid curves
correspond to the predictions of the Mohr-Coulomb model. Note that both plots have been
prepared using the same MC model parameters (friction c, = 0.13, cohesion c2 = 666MPa
).........................................................................................................................................
49
Figure 2-4 Coordinate transformation for plane stress conditions: (a) initial von Mises yield
envelope (solid black line) with subsequent von Mises stress iso-contours and EMC
localization locus (blue line) for proportional loading; (b) representation of the same
envelopes in the modified Haigh-Westergaard space, and (c) in the mixed strain-stress
sp ace ..................................................................................................................................
52
Figure 2-5 Representation of special cases of the Hosford-Coulomb (HC) model in the
modified Haigh-Westergaard space. The blue lines show the strain to fracture for plane
stress con d itio n s................................................................................................................
54
Figure 2-6 Effect of the parameters of the Extended Mohr-Coulomb (EMC) model on the
fracture envelope for plane stress loading. ...................................................................
55
Figure 2-7 Measured true stress versus logarithmic plastic strain curve for uniaxial tension
along different material directions up to the point of necking for (a) DP590, (b) DP780.
Note that each graph shows the curves for three different specimen orientations (00, 450
and 90'), but they lie exactly on top of each other and thus only one curve is visible..... 61
Figure 2-8 Specimen drawings for flat tension specimens with (a) a central hole, (b)-(c)
different notches, and (d) the butterfly specimen for shear testing...............................
62
-15-
Figure 2-9 Measured and simulated force displacement curves for selected fracture
experiments on the DP590 steel. The star symbols represent the experimental curves, while
the simulation results are shown as solid lines. A contour plot of the equivalent plastic
strain at the onset of fracture is shown below each figure............................................
63
Figure 2-10 Comparison of the Hosford-Coulomb (HC) model predictions (blue dots) with
experimental results (end point of the black lines) after calibration based on the SH, CH
and NT6 experiments. The predictions of the Mohr-Coulomb (MC) model are shown as
red dots. The units of the cohesion b are MPa..............................................................
68
Figure 3-1 Comparison of the true stress strain experimental response with a theoretical
isotropic hardening response after loading reversal from uniaxial tension to uniaxial
compression at strain of 0.08 to illustrate different hardening behaviors: isotropic
hardening, permanent softening, work hardening stagnation, transient behavior. ........... 75
Figure 3-2 Uniaxial stress-strain response of DP780 steel for different directions under
un iax ial ten sion .................................................................................................................
78
Figure 3-3 Specimen geometry for uniaxial compression-tension experiments............ 79
Figure 3-4 Displacement measurement using Digital Image Correlation (DIC)........... 79
Figure 3-5 Notched specimen (NCT) for compression-tension fracture testing........... 80
Figure 3-6 Front view of the experimental set-up for compression-tension testing......... 81
Figure 3-7 Schematic side view of the specimen with anti-buckling device and high
pressure clam p s.................................................................................................................
81
Figure 3-8 Experimental results: engineering stress-strain curves as obtained from uniaxial
tension-com pression experim ents.................................................................................
83
Figure 3-9 Experimental results: force-displacement curves as obtained from notched
com pression-tension experim ents.................................................................................
83
-
16-
Figure 3-10 Illustration of the work hardening stagnation model in the plastic strain space.
The sequence shows (a) the initial values, (b) the evolution during monotonic loading, (c)
the point of loading reversal, (d) the transient stagnation after loading reversal, and (e)-(f)
the evolution after stagnation........................................................................................
89
Figure 3-11 Comparison of CCY model (solid lines) and experiments (solid dots) for (a)
uniaxial tension-compression (UTC), and (b)-(f) notched compression-tension (NCT); the
Chaboche model predictions are depicted as dashed lines. ...........................................
98
Figure 3-12 Thickness profiles at the instant of inset of fracture as measured experimentally
(solid dots) and extracted from numerical simulation (CCY=solid lines, Chaboche=dashed
line) after pre-compression up to a surface strain of (a) 0. (b) 0.08, and (c) 0.12; note that
the results are shown for specimens extracted from a different batch of DP780 sheets. 100
Figure 3-13 Results from monotonic experiments: (a) notched tension (NT6), (b) tension
with a central hole (CH), and (c) butterfly specimen (SH); solid dots = experiments, solid
lines = CCY model, dashed lines = Chaboche model; the contour plots show the
distribution of the equivalent plastic strain at the instant of fracture initiation. ............. 102
Figure 3-14 (a) Equiv. plastic strain distribution in the longitudinal specimen cross-section,
and (b) thickness distribution at the instant of fracture initiation...................................
105
Figure 3-15 Detailed analysis of the NCT-13 results: (a) force-displacement curve and local
surface strain; Evolutions of (b) the CCY hardening terms: isotropic hardening resistance
(solid line), axial component of the back stress tensors
(dashed line) and
(dotted line),
(c) the von Mises stress, (d) the thickness profiles, and (e) the stress triaxiality; the dashed
line in (e) depicts the Chaboche model prediction; the labels (,
4,
® and ®indicate
the point of load reversal, onset of stagnation (also maximum load in tension), end of
stagnation, and onset of fracture.....................................................................................
107
Figure 3-16 Loading paths to fracture as extracted from finite element simulations of all
fracture experiments up to the instant of fracture initiation (end point of solid lines); the
Hosford-Coulomb fracture initiation model predictions are shown as solid dots. ......... 109
-
-17
Figure 3-17 Strain to fracture for proportional loading as a function of the Lode angle
param eter and the stress triaxiality.................................................................................
110
Figure 3-18 evolution of the damage indicator at the location of fracture initiation in
com pression-tension experim ents...................................................................................
111
Figure 3-19 Strain to fracture after compression-tension loading of NCT specimens as a
function of the equivalent plastic strain at the point of loading direction reversal; the star
symbols present the hybrid experimental-numerical results, the solid dots care model
p red ictio n s.......................................................................................................................
1 12
Figure 3-20 Net fracture strain after compression-tension loading of NCT specimens as a
function of the equivalent plastic strain at the point of loading direction reversal; the star
symbols present the hybrid experimental-numerical results, the solid dots care model
pred ictio n s.......................................................................................................................
1 13
Figure 4-1 Representation of the Hosford-Coulomb criterion for a power law material with
plane stress condition in the following spaces (a) first and second in-plane stress
components (b) Haigh-Westergaard (c) mixed stress-strain (d) mixed hardening rate-stress.
The initial Von Mises yield envelope (solid black line), the subsequent Von Mises isocontours (black dotted line) and Hosford-Coulomb fracture locus (solid blue line) are
sh o wn ..............................................................................................................................
13 0
Figure 4-2 Effect of plasticity on the prediction of fracture under reverse loading for the
Hosford-Coulomb model as a function of the choice of critical quantity at fracture..... 133
Figure 4-3 Schematic of the dual actuator system ..........................................................
137
Figure 4-4 Geometry of the Mohr-Oswald specimen.....................................................
138
Figure 4-5 Comparaison of the stress-strain response of DP780 under loading reversal after
10% equivalent plastic strain for shear reversal (solid line) and tension followed by
com pression (dotted line) ...............................................................................................
139
Figure 4-6 Geometry of the Dunand-Mohr butterfly specimen......................................
140
18
-
-
Figure 4-7 Effect of model parameters for reverse loading: (a) parameter 7" (b) parameter
C p (c) param eter h (d) param eter yp. ...............................................................................
144
Figure 4-8 Comparison of the Von Mises stress to equivalent plastic strain after loading
reversal for experimental data (black dotted line) and the prediction of the model after
calibration (red solid line) for (a) Shear reversal on DP590 (b) Tension compression on
D P 7 8 0 . ............................................................................................................................
14 6
Figure 4-9 Comparison of the experimental (dotted line) and predicted (solid line) load
displacement response for (a) monotonic shear (b) shear reversal after 25% equivalent
plastic strain (c) shear reversal after 50% equivalent plastic strain................................
147
Figure 4-10 Comparison of the experimental measurements (dotted line) and the model
prediction (solid line) for load-displacement (black) and local surface strain (blue) for (a)
CTR_0 (b) CTR_05 (c) CTR_10 (d) CTR_15 (e) CTR_20. ..........................................
148
Figure 4-11 Calibration of the critical Hardening rate H-C model using the monotonic data
for (a) D P590 (b) D P780.................................................................................................
150
Figure 4-12 Prediction of the onset of fracture (dots) using the critical hardening rate model
and loading paths to fracture (solid lines) for (a) compression followed by tension
experiments on DP780 (b) Shear reversal experiments on DP590.................................
151
Figure 5-1 Comparison of the experimental load-displacement (dashed lines) with the
numerical prediction (red line) using the plasticity model calibrated in chapter 4 for DP590
on orthogonal tests (a) notched tension at 90 degrees with respect to a 5% tensile pre-strain
(b) shear to 50% equivalent plastic strain followed by plane strain tension to fracture. 158
Figure 5-2 Comparison of (b) the crack during experimental three point bending of a
martensitic hat-shaped profile with (a) the crack predicted by FEA using a linear damage
indicator and (c) the crack predicted by FEA using the non-linear damage indicator. .. 159
-
-19
List of Tables
Table 2-1 Lankford R atios ............................................................................................
60
Table 2-2 Dual steel hardening law parameters............................................................
66
Table 3-1 Plasticity model parameters for the Combined Chaboche-Yoshida (CCY) model
and the C haboche m odel................................................................................................
96
Table 3-2 Net fracture strain (equivalent plastic strain at fracture minus compressive prestrain) as a function of the compressive pre-strain. ........................................................
113
Table 4-1 Sum m ary of all experim ents...........................................................................
142
T able 4-2 L ankford R atios..............................................................................................
143
Table 4-3 Sw ift-V oce Law param eters...........................................................................
143
Table 4-4 Reverse Loading Hardening Parameters ........................................................
145
Table 4-5 Fracture Param eters........................................................................................
149
-20-
-
Chapter 1: Introduction - 21
Chapter 1: Introduction
1.1
Ductile Fracture
The Gurson type of models (Gurson, 1977) have received considerable attention over
the past three decades because of their sound micromechanical basis and ability to predict
fracture across many applications. These models are inspired by McClintock, and propose
the nucleation, growth and coalescence of voids as the main mechanism leading to fracture.
Gurson models, or shear-modified Gurson models, are highly relevant for a material
response considered across a wide range of triaxialities. Instead, the stress state in sheet
materials is usually close to plane stress and hence the void growth driving stress triaxiality
does not exceed a value of 0.67. Even inside a localized neck, where three-dimensional
stress states develop, the stress triaxiality seldom exceeds values of 0.7. Consequently,
there is limited void growth in a statistically homogeneous sense in sheet specimens. This
conclusion is supported by experimental observations for pure copper (Ghahremaninezhad
and Ravi-Chandar, 2011), nodal cast iron (Ghahremaninezhad and Ravi-Chandar, 2012a)
and aluminum 6061-T6 (Ghahremaninezhad and Ravi-Chandar, 2012b). Void growth is
important after the onset of shear localization (e.g. Tekoglu, 2012). This may be concluded
from micrographs of fracture surfaces, which show the dimple signature of voids growth
and coalescence. Gurson models were initially developed for plasticity with success.
However, there is growing evidence that the plastic response of sheet materials in terms of
stress strain relation for proportional loading can be predicted with reasonable accuracy up
to the point of shear localization simply by the isotropic growth of a yield surface, provided
a suitable identification method of the hardening law (e.g. Dunand and Mohr, 2010,
Dunand et al., 2011).
Gurson models are often adapted to fracture prediction by proposing that coalescence of
voids dramatically accelerates at a critical porosity leading to imminent fracture initiation.
However, recent experimental evidence regarding ductile fracture at low stress triaxialities
(Barsoum and Faleskog, 2007, Mohr and Henn, 2007, Dunand and Mohr, 2011 a) is
partially not in good agreement with the trends predicted by conventional Gurson models.
- 22 - Chapter 1: Introduction
An alternative approach to predicting fracture with Gurson models is to assume that
ductile fracture occurs when the governing field equations lose ellipticity. This assumption
goes back to Rice's shear localization analysis (Rice, 1976), and has been explored
extensively thereafter (e.g. Needleman and Tvergaard, 1992). Recent examples in the
context of Gurson models are the works by Nahshon and Hutchinson (2008), as well as
those by Danas and Ponte Castaneda (2012). Nahshon and Hutchinson (2008) added a
shear term to the void volume evolution law of the GTN model (Tvergaard and Needleman,
1984), and demonstrated the importance of this modification in their predictions of shear
localization. Danas and Ponte Castaneda (2012) used non-linear homogenization to come
up with a Gurson-type of model that accounts for void shape changes (that are
characteristic for shear loading). Their analysis of the loss of ellipticity at low stress
triaxialities also led to predictions that are very different from those of the traditional
Gurson model.
However, unless the propagation of cracks is to be modeled, the modeling of the post
shear localization behavior is only of little interest in engineering applications, such as
sheet metal forming and crashworthiness, because the width of shear bands is typically of
the size of a few grains. Instead, it is reasonable to assume that the onset of ductile fracture
coincides with the onset of shear localization. As the results from Nahshon and Hutchinson
(2008), and Danas and Ponte Castaneda (2012), show, the predictions of the loss of
ellipticity are material imperfection sensitive which includes unavoidable inaccuracies in
the plasticity model formulation.
Damage indicator models were introduced as an attempt to predict the onset of shear
localization (and hence fracture in an engineering sense) at low computational cost. This
indicator is a dimensionless scalar variable that evolves as a function of the stress state and
plastic deformation. It is initially zero while fracture is assumed to occur as it reaches a
defined critical value. Bao and Wierzbicki (2004) provide a comprehensive overview on
different stress-state dependent damage indicator models, including weighting functions,
based on the work of McClintock (1968), Rice and Tracey (1969), LeRoy et al. (1981),
Cockcroft and Latham (1968), Oh et al. (1979), Brozzo et al. (1972), and Clift et al. (1990).
The choice of the stress-state weighting function is critical in damage indicator models.
Bai and Wierzbicki (2008) developed the so-called Modified Mohr-Coulomb (MMC)
-
Chapter 1: Introduction - 23
model, which is based on a stress-state dependent weighting function that has been derived
from the Mohr-Coulomb failure model in stress space. The MMC model has been
successfully applied in predicting fracture of aluminum 6061-T6 (Beese et al., 2010) and
advanced high strength steels (e.g. Li et al, 2010, Luo and Wierzbicki, 2010, Dunand and
Mohr, 2011 a). In chapter 2 of this thesis, we propose a limiting envelope in the stress space
to indicate the onset of localization for proportional loading. A weighting function is
derived
so that the damage indicator approach is mathematically
equivalent for
proportional loading. The model gives good results for proportional and close to
proportional loading tests. However, it is shown in chapter 3 that further enhancements are
necessary for highly non-linear loading paths. A concerning limitation of such models is
that there is no evidence that the damage indicator quantifies a specific physical
mechanism. Rather than investing efforts in refining empirically such damage indicators,
it is worth considering that other quantities beyond the stress or strain may be the main
relevant physical measure of the initiation of localization. Models such as the HosfordCoulomb model may be transformed in order to remain accurate for proportional loading
while becoming relevant for non-linear loading conditions.
1.2
Plasticity for Complex Loading
Predicting the onset of ductile fracture has been an active field of research for more
than 50 years. In particular, the fracture initiation after monotonic proportional loading
paths has been investigated intensively (e.g. Brunig et al. (2008), Bai and Wierzbicki
(2008, 2010), Sun et al. (2009), Li et al. (2011), Gruben et al. (2011), Chung et al. (2011),
Lecarme et al. (2011), Khan and Liu (2012), Luo et al. (2012), Huespe et al. (2012),
Malcher et al. (2012), Lou et al. (2014)). In industrial practice, in particular during sheet
metal forming, ductile fracture often initiates after complex non-proportional loading
histories. Among these, reverse loading is an important non-proportional
loading
condition, which prevails for instance when a sheet is bent and unbent as it is drawn over
a die radius.
Simulating the mechanical response of ductile materials up to the point of fracture
initiation requires the accurate modeling and identification of the hardening behavior of
- 24 - Chapter 1: Introduction
the material at large strains. Many plasticity models for reverse loading have been
developed for life-cycle analysis. As a result, most experimental procedures are designed
for characterizing the small strain response only. One of few exceptions are the reverse
shear experiments of Barlat et al. (2003) on 3mm thick 1050-0 aluminum sheets. Using
wide shear specimens with a narrow gage section of reduced thickness, they achieved shear
strains of up to 0.22 prior to loading direction reversal. Yoshida et al. (2002) presented an
experimental study on the kinematic hardening response of sheet materials involving a
finite strain compression phase. They bonded several flat specimens together and inserted
the stack of specimens in an anti-buckling device during testing. Other examples of the use
of anti-buckling devices for testing sheet materials under in-plane compression can be
found in Dietrich and Turski (1978), Kuwabara (1995), Yoshida et al (2002), Boger et al
(2005), Cao et al (2009) and Beese and Mohr (2011).
The large strain compression-tension experiments by Yoshida (2002) show that DP
steels feature a Bauschinger effect, transient behavior, permanent softening and work
hardening stagnation. Recall that the Bauschingereffect corresponds to an early yield after
load reversal (Figs. 1-lb and 1-Id); transientbehavior corresponds to a high hardening rate
in the elasto-plastic transition regime resulting from load reversal (Fig. 1-Id); permanent
softening prevails when the stress level after loading reversal remains below that for
monotonic loading for the same equivalent plastic strain (Fig. 1-lb); work hardening
stagnation causes a significantly reduced hardening rate after the transient hardening
regime (Fig. 1-1c).
-
Chapter 1: Introduction - 25
"ZU,
1000-
Permanent Softening
1000-
C-
Isotropic Hardening
800-
$- 800
Bauschinger
effect
0
600
600-
0
0
400-
400
200
200
iI
I
0.1
0.3
0.2
Equivalent Plastic Strain
0
0.4
0
0.3
0.2
0.1
Equivalent Plastic Strain
0.4
(b)
(a)
1200
'
0..
1000
11000
0-
800
800
600
0
>
(
& 600
Workhardening
Stagnation
400
400
0
>
200
I
OL
______________
0.1
0.2
0.3
Equivalent Plastic Strain
(c)
Bauschinger
effect
0.4
200
,n
''0
Transient Behavior
0.1
0.2
0.3
Equivalent Plastic Strain
0.4
(d)
Figure 1-1 Comparisonofthe von Mises stress versus equivalentplastic straincurvefor monotonic
loading with that after loading reversal at a strain of 0.1 to illustrate different hardeningmodel
assumptions: (a) isotropic hardening, (b) permanent softening, (c) work hardeningstagnation, (d)
transient behavior.
Detailed reviews of kinematic hardening models are found in Chaboche (2008), and
Eggertsen and Mattiasson (2009, 2010, 2011). Prager (1956) type of kinematic hardening,
also referred to as linear kinematic hardening, describes both the Bauschinger effect and
- 26 - Chapter 1: Introduction
permanent softening. The main shortcoming of this model is the intrinsic coupling of both
effects, i.e. a material exhibiting a Bauschinger effect without any permanent softening
cannot be described with Prager's model. Furthermore, it describes neither transient
behavior nor work hardening stagnation. Also, this type of hardening is unbounded and
results in a persistent and often unrealistic rate of hardening at large strains. The
Armstrong-Frederick (1966) kinematic hardening rule, also referred to as non-linear
kinematic hardeningmodel, describes the Bauschinger effect and transient behavior. The
governing differential equation includes a recall term which activates the so-called
dynamic-recovery. The recall term is co-linear to the back stress tensor and is proportional
to the increment in equivalent plastic strain. As a result, the evolution of the back stress is
no longer linear and unbounded, and converges towards a saturation value under monotonic
loading. Two parameters are used: one to control the Bauschinger effect and one for the
transient behavior. However, the Armstrong-Frederick model describes neither permanent
softening nor work hardening stagnation.
For improved approximations, several non-linear kinematics hardening models can be
added with different recall constants characterizing the back stress evolution (Chaboche et
al., 1979; Chaboche and Rousselier, 1983). These models give good predictions in the case
of cyclic loading in the range of small strains, as they are able to describe the Bauschinger
effect with great accuracy. The special case of coupling linear kinematic hardening with
non-linear kinematic hardening provides good predictions in the case of moderate and large
strains, as it describes the permanent softening behavior during reverse deformation,
especially with advanced high strength steels free from work hardening stagnation
(Yoshida et al., 2002). However, the permanent softening effect is only represented through
the linear kinematic term. As a result, an increase of the amount of permanent softening in
the model always results in an increase in the strain hardening at large strains.
Mroz (1967) proposed a multi-surface model framework to describe strain hardening.
This idea was developed further by Dafalias and Popov (1976) by making use of a
bounding surface in addition to the yield surface, with the distance between these two
evolving surfaces defining the rate of strain hardening. Chaboche (2008) argues that a
model featuring a combination of linear and non-linear kinematic hardening terms, in
addition to isotropic hardening, can replicate the performance of a Dafalias-Popov type of
-
Chapter 1: Introduction - 27
model. However, one advantage of the Dafalias-Popov type of formulations is that the
material response to monotonic loading can be identified independently from its response
to reverse loading. The Dafalias-Popov model has been developed further by Geng and
Wagoner (2000) to account for permanent softening. Yoshida and Uemori (2002) enriched
the model even further and incorporated work-hardening stagnation.
To the best of the authors' knowledge, experimental results on the effect of loading
direction reversal on the strain to fracture are scarce and only found for bulk materials in
the open literature. Bao and Treitler (2004) performed reverse loading experiments on
notched axisymmetric bar aluminum 2024-T351 specimens with compression followed by
tension all the way to fracture. They observed a substantial increase in ductility due to precompression. Papasidero et al. (2014) made use of a biaxial testing machine to subject
tubular fracture specimens to non-proportional loading. Their results also demonstrated
that pre-compressing aluminum 2024-T351 increases the strain to fracture for subsequent
loading at higher stress triaxialities.
1.3
Strain path dependency of FLD and fracture toughness
The effect of complex loading histories has been studied for several failures modes of
materials. Surprisingly, the literature is scarce regarding ductile fracture.
In many situations in forming applications, localization of deformations in the thickness of
the sheet occurs prior to ductile fracture. Necking being considered itself a mode of failure,
it is often predicted using the Forming Limit Diagram, since it cannot be predicted with
shell elements. It is well known in the literature that the FLD varies under non-linear
loading histories (Muschenborn and Sonne, 1975; Graf and Hosford, 1994; Stoughton,
2000; Stoughton and Zhu, 2004). The dependence of the FLD on the strain path has been
a very active field of research for decades. (Marciniak and Kuczynski, 1967; Cao et al.,
2000; Chow et al., 2001). Interestingly, Stoughton (2000) suggested that the FLD is not
path dependent when converted to a limit in the stress space.
The effect of pre-strain has also been studied for fracture toughness. Enami (2005) studied
the reduction of ductility to cleavage cracks in the case of pre-compression. Eikrem (2008)
studied the crack resistance curve using single edge notched tension (SENT) specimens.
- 28 - Chapter 1: Introduction
He suggests that in the case of crack after cyclic loading, comparing symmetric and
asymmetric cycles, the amount of compression plays a significant role in the fracture
toughness. All studies suggest that the hardening exponent n plays a role, although the
trends are not in agreement. Eikrem claims that, "It is known that plastic deformation will
not only modify the yield and strain hardening behavior, but will also influence in a
detrimental manner the local failure mechanisms, the initiation toughness and fatigue
properties".
1.4
1.4.1
Examples of non-linear loading paths
Crash
Bai (2006) showed analytically and numerically that there exists strain reversal in the case
of prismatic square aluminum tubes subjected to crash loading. The study suggests that
loading histories play a significant role in fracture initiation depending on the wall
thickness and experimental location of fracture.
Recently, Pack and Marcadet studied the three point bending of a hot formed martensitic
hat section. Using conventional damage accumulation indicator with effect of stress state,
the location of crack initiation is wrongly predicted. Two points have competing fracture.
Point (2) is mostly under plane strain tension, while point (1) undergoes plane strain
compression followed by plane strain tension (Fig. 1-2). The complex loading of point (1)
is due to the folding followed by unfolding during the indentation process of the structure
by the punch. A numerical simulation using isotropic hardening plasticity and a linear
damage accumulation rule function of the stress state predicts fracture at point (1), while
experimental fracture occurs at point (2) (Fig 1-3). This suggests that the ductility of point
(1) is underestimated by the model. Therefore, it appears that there is a need to understand
better the effect of complex loading histories of material ductility.
-
Chapter 1: Introduction - 29
1.2
1
0.8
1& 0.6
S0.4
I
-0
0.2
ni
-0.66
-0.33
0
0.33
0.66
Triaxiality
Figure 1-2 Loading Paths at points (1) and (2) during the three point bending of the hat-shaped
profile
Figure 1-3 Comparison of the fracture observed during experimental three point bending of a
martensitichat-shapedprofile with the numericalprediction using isotropic hardeningand linear
damage accumulation indicator.
1.4.2
CombinedForming and Crash
In another attempt to understand the effect of loading histories on ductile fracture, three
point bending tests of a cold formed hat-shaped profile made in DP980 were performed.
Numerical simulations of the process using several modeling approaches were considered.
One approach consisted in considering the material as virgin after the forming process. It
- 30 - Chapter 1: Introduction
has been found that the damage indicator fails to predict the fracture that was observed
experimentally. This error can be attributed to the fact that the remaining hardening
capacity of the material after forming is overestimated by this approach, and that damage
accumulated during forming was not taken into account. When simulating the multi-step
process from forming to crash with isotropic hardening plasticity and linear damage
accumulation rule, the fracture is predicted too early. Once again, this suggests that the
ductility of the material increases when loaded along complex histories.
1.5
Complex Loading on Bulk Material
Johnson and Cook (1985) found early evidence that the prediction of ductile fracture with
damage indicators would be challenged under complex loading. They performed torsion
followed by tension of OHFC copper, and found that the ductility was underestimated for
such loading conditions.
Tai (1990) proposed an interesting experimental way to perform two stage loading on
round bars. The purpose is to investigate the effect of a change of triaxiality during loading
on the ductility of the material. In the first stage, round bars with a notch of a certain radius
Ri are loaded under tension. As a consequence, the stress state in the center of the specimen
has a certain triaxiality l1. The test is then interrupted somewhere along the loading. A new
radius R2 is machined on the specimen before placed back in the testing machine to resume
loading up to fracture. Because of the change in notch radius, the material is then loaded
with a different triaxiality 112. The Lode angle parameter remains constant in such a test due
to the axisymmetry of the specimens.
The work by Bao and Treitler is a rare example of an investigation of the effect of
compression followed by tension on the ductility of a 2024-T351 aluminum alloy. Round
bars with different notch radii are first loaded under compression. The loading is then
reversed to tension up to fracture. In this case, both the Lode angle parameter and the
triaxiality change sign during reversal. It was concluded from this study that the loading
reversal decreased the ductility of the aluminum.
-
Chapter 1: Introduction - 31
Bai revisited such experimental methods in his PhD thesis. He also introduced complex
loading history tests on butterfly specimens with thickness reduction in the gage section.
Shear followed by tension is a similar loading sequence to torsion followed by tension.
Compression followed by tension was also performed. Finally, he was able to perform
some reverse shear tests. All tests suggest that there is a ductility increase with loading
complexity. He proposed an empirical damage indicator to fit the data.
Papasidero and Mohr (2014) introduced an innovative experimental method to test bulk
materials under a combination of torsion and tension using round tubes with thickness
reduction. In particular, they performed torsion followed by tension tests. It was shown that
a non-linear damage indicator could predict the effect of loading complexity on the
ductility of aluminum.
1.6
Particularities of Sheet Metal
In order to investigate the behavior of material under reverse loading, new testing programs
have to be developed. Different types of tests are used to identify the plastic response of
the material and to measure the loading paths to fracture under specific stress state. A main
challenge of sheet metal is the occurrence of localized necking, the concentration of
deformations through the thickness of the sheet. In figure 1-4, the load-displacement
response is compared for bulk aluminum and DP780 sheet metal in reverse loading. While
almost no necking is observed on bulk, DP780 sheets feature a very important phase of
deformations after necking. Uniaxial specimens are not suitable for measuring the behavior
of the material after maximum load. The load-displacement relation after maximum load
is not experimentally repeatable. The statistical spread may be attributed to imperfections
in the experimental set-up (alignment) and material imperfections (inclusions) that trigger
the instability leading to localization. Therefore, identifying the hardening curve at large
strains using an inverse method with FEA cannot be performed on a dogbone specimen
because there is not a right choice of the response after necking. In case of tension after
compression, due to the effect of loading reversal on the plastic behavior, necking occurs
- 32 - Chapter 1: Introduction
very early (Fig. 1-5). To the knowledge of the author, the plastic behavior of sheet metal
after reverse loading has never before been identified all the way to fracture.
100
W0
fj0
40
U.
20
-40
-ABAQUJ
0 15
1
Displacement (mm)
(a)
10
5
0
-j
0
-C
-
r0
-NCT-3
-5
-
NCT-3
-
NCT-13
-10
-2
1
0
-1
Displacement [mm]
(b)
Figure 1-4 Comparison between load-displacement curves under compression-tensionfor (a)
roundbars of aluminum (b) flat specimens of DP780 sheets.
-
Chapter 1: Introduction - 33
1000500
-
I
/
(2I
________
-1
1
.gW
----- Reverse loading 2
------ Reverse loading 3
0-500-1000-
-0.2
0.2
0.1
0.0
-0.1
Strain
(a)
1000
600
400
LU
0
C
200
0
-200
a)
'U
400
-600
-800
-1000
-0.1
0
-0.05
0.05
0.1
0.15
Engineering Strain
(b)
Figure1-5 Comparisonbetween stress-straincurves under compression-tensionfor(a) roundbars
of aluminum (b) flat specimens of DP780 sheets
- 34 - Chapter 1: Introduction
1.7
Thesis Outline
The Thesis is decomposed in seven chapters. Each chapter, apart from Chapter 1 and
Chapter 5, addresses one specific topic and corresponds to a peer-reviewed journal
publication.
The first chapter is a general introduction to the topic and motivations of the thesis.
Previous work related to the field is briefly reviewed. The general organization of the thesis
is outlined.
The starting point of the original research part of this thesis is a second chapter on the
development of a micromechanically-motivated ductile fracture initiation model for metals
for proportional loading. Its formulation is based on the assumption that the onset of
fracture is imminent with the formation of a primary or secondary band of localization.
Motivated by the results from a thorough unit cell analysis, it is assumed that fracture
initiates after proportional loading if the linear combination of the Hosford equivalent
stress and the normal stress acting on the plane of maximum shear reaches a critical value.
A comprehensive fracture initiation model is then obtained after transforming the
localization criterion from stress space to the space of equivalent plastic strain, stress
triaxiality and Lode angle parameter using the material's isotropic hardening law.
Experimental results are presented for three different advanced high strength steels. For
each material, the onset of fracture is characterized for five distinct stress states, including
butterfly shear, notched tension, tension with a central hole and punch experiments. The
comparison of model predictions with the experimental results demonstrates that the
proposed Hosford-Coulomb model can predict the instant of ductile fracture initiation in
advanced high strength steels with satisfactory accuracy.
In a third chapter, experimental methods are developed to perform compression-tension
experiments. In addition, a finite strain constitutive model is proposed, combining a SwiftVoce isotropic hardening law with two Frederick-Armstrong kinematic hardening rules
and a Yoshida-Uemori type of hardening stagnation approach. The plasticity model
parameters
are identified
from uniaxial tension-compression
stress-strain curve
measurements and finite element simulations of compression-tension experiments on
-
Chapter 1: Introduction - 35
notched specimens. The model predictions are validated through comparison with
experimentally-measured
load-displacement curves up to the onset of fracture, local
surface strain measurements and longitudinal thickness profiles. The extracted loading
paths to fracture show a significant increase in ductility as a function of the compressive
pre-strain. The Hosford-Coulomb model is therefore integrated into a non-linear damage
indicator modeling framework to provide
a phenomenological
description
of the
experimental results for monotonic and reverse loading.
The chapter 4 presents another extension of the modeling framework inspired by the results
from loss of ellipticity analysis. It is demonstrated that the Hosford-Coulomb model can
also be expressed in terms of a stress-state dependent critical hardening rate. Moreover, it
is shown that the critical hardening rate approach provides accurate predictions of the
instant of fracture initiation for both proportional and non-proportional loading conditions.
Enhancements of the finite strain constitutive model are also proposed to enable a fast
identification of all model parameters. The plasticity model parameters are identified from
stress-strain curve measurements from shear loading reversal on specimens with a uniform
thickness reduced gage section. The model is used to estimate the local strain and stress
fields in fracture experiments after shear reversal. The extracted loading paths to fracture
show a significant increase in ductility as a function of the strain at shear reversal, a feature
that is readily predicted by the proposed critical hardening rate model.
In chapter 5, the main findings of the thesis are summarized. Some ongoing work is
described and potential directions for further research are discussed.
- 36 - Chapter 1: Introduction
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 37
Chapter 2: Hosford-Coulomb Model for
Predicting Ductile Fracture
A phenomenological ductile fracture initiation model for metals is developed for
predictingductile fracture in industrialpractice. Itsformulation is basedon the assumption
that the onset offracture is imminent with the formation of a primary or secondary band
of localization. The resultsfrom a unit cell analysis on a Levy-von Mises material with
spherical defects revealed that a Mohr-Coulomb type of model is suitable for predicting
the onset of shear and normal localization. To improve the agreement of the model
predictions with experimental results, an extended Mohr-Coulomb criterion is proposed
which makes use of the Hosford equivalent stress in combination with the normal stress
acting on the plane of maximum shear. A fracture initiation model is obtained by
transformingthe localizationcriterionfrom stress space to the space of equivalent plastic
strain, stress triaxialityandLode angleparameterusing the material'sisotropichardening
law. Experimental results are presentedfor three different advanced high strength steels.
For each material, the onset of fracture is characterizedfor five distinct stress states
including butterfly shear, notched tension, tension with a central hole and punch
experiments. The comparison of model predictions with the experimental results
demonstrates that the proposedHosford-Coulomb model can predict the instant of ductile
fracture initiation in advanced high strength steels with good accuracy.
- 38 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
2.1
Introduction
Ductile fracture is a well-known physical process that leads to the formation of cracks in
metals due to the nucleation, growth and coalescence of voids. In cases where macroscopic
localization precedes void coalescence, the ductile fracture process may be described as
follows: as the material deforms plastically, pre-existing (primary) voids evolve and new
ones nucleate (stage
in Fig. 2-1). Due to the increase in porosity and the decrease in
macroscopic strain hardening, the conditions for a discontinuity in the macroscopic strain
field along a planar interface may be met. The result is the formation of a primary band of
localization
at the mesoscale (stage
@ in Fig. 2-1). As stated by Pardoen and Hutchinson
(2000) with reference to Tvergaard (1981), the width of a primary localization band is
expected to be of the order of the inter-void spacing. Subsequently, the material inside the
band experiences accelerated void growth and nucleation (stage @ in Fig. 2-1). As a result,
the porosity and/or the number of voids within the band increase sharply and the
mechanical fields around individual primary voids begin to interact. The nucleation of
secondary voids (which are often several orders of magnitude smaller than primary voids)
is also possible at this stage. The final coalescence phase sets in when the deformation
begins to localize within secondary bands of localization
at the microscale (stage
in
Fig. 2-1). In other words, inside the primary localization band, there is a transition from
diffuse to localized plastic flow, which ultimately leads to primary void coalescence and
the formation of cracks through internal necking or void sheet fracture of the ligaments
between primary voids (stages
®
and
0
in Fig. 2-1). As discussed by Tekoglu et al.
(2015), polycrystalline materials may also fail due to (i) localized plastic flow only (e.g.
necking up to zero cross-sectional area), (ii) localization of plastic flow after damage-free
deformation, followed by void growth and nucleation inside the primary band of
localization, and (iii) direct coalescence (secondary localization) without any prior
occurrence of primary localization.
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 39
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(b)
Figure 2-1 Eulerian illustration of the ductile fracture process with coalescence through (a)
internal necking, and (b) void sheet fracture. The mesoscopic primary band of localization is
highlightedin gray color, the microscopic secondary band of localization is highlightedin red
Research on ductile fracture has addressed several aspects of the ductile fracture process.
A wealth of literature deals with porous plasticity, i.e. the effective large deformation
behavior of mildly porous metals (e.g. Gurson,
1977, Tvergaard,
1981, Mear and
Hutchinson, 1985, Gologanu et al., 1993, Leblond et al., 1995, Benzerga and Besson, 2001,
Molinari and Mercier, 2001, Monchiet et al., 2008, Nahshon and Hutchinson, 2008 and
Danas and Ponte Castafteda, 2012). The formation of primary bands of localization is
expected to come out naturally when solving boundary value problems with accurate
porous plasticity models. In other words, there is no need to introduce any localization
criterion. However, the computation of the stress and strain fields after the initiation of
localization bands is challenging due to the associated loss of ellipticity of the governing
field equations. The use of non-local formulations (e.g. gradient plasticity) appears to be
- 40 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
necessary to regularize the mathematical problem in the post localization regime (e.g.
Anand et al. (2012)).
As an alternative to using highly accurate porous plasticity models, the formation of
primary bands of localization can also be predicted using conventional non-porous
plasticity models in conjunction with localization criteria. This approach is motivated by
the fact that non-porous plasticity models provide an excellent approximation of the multiaxial stress-strain response of metals up to the point of localization (e.g. Dunand and Mohr,
2011 a and Dunand and Mohr, 2011 b). Furthermore, from an engineering perspective, the
formation of a primary band of localization is considered as the onset of fracture.
Accordingly the engineering literature does not differentiate between fracture initiation
models and criteria predicting the formation of primary bands of localization.
Localization analysis with porous plasticity models provides valuable insight into the effect
of stress state on the formation of primary bands of localization (e.g. Rudnicki and Rice,
1975 and Rice, 1977). Recent examples are the localization analysis with a band-like defect
in a shear-sensitive Gurson solid (Nahshon and Hutchinson, 2008), and the investigation
of the loss of ellipticity and peak load by Danas and Ponte Castafteda (2012) using a
homogenization-based porous plasticity model. The main limitation today is the accuracy
of the advanced porous plasticity models. The above models are able to capture first order
effects, but to the best of the authors' knowledge, the above localization estimates have not
yet been utilized to predict the onset of localization in real structures.
Multi-axial experiments provide the only viable alternative to computational localization
analysis (e.g. Hancock and Mackenzie, 1976, Mohr and Henn, 2007 and Haltom et al.,
2013). Here, the main challenge is the analysis of heterogeneous mechanical fields due to
the localization at the structural level (localized necking) which often precedes the
formation of nrimary bands of localization. In particular, the highest strains within a
specimen are often reached below the specimen surface. Except for rare cases where
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 41
tomography-based 3D digital image correlation is possible (e.g. Morgeneyer et al. (in
press)), the strains can only be estimated through statistical analysis of grain deformation
in
micrographs
(e.g.
Ghahremaninezhad
and
Ravi-Chandar
(2012))
or
hybrid
experimental-numerical analysis (e.g. Dunand and Mohr (2010)).
Bao and Wierzbicki (2004) provide a comprehensive overview on functional relationships
between the equivalent plastic strain and the stress state (derived from the works of
McClintock, 1968, Rice and Tracey, 1969, LeRoy et al., 1981, Cockcroft and Latham,
1968, Oh et al., 1979, Brozzo et al., 1972 and Clift et al., 1990) that may be interpreted as
localization or fracture initiation criteria for proportional loading. To account for the effect
of non-proportional loading, the above functions are typically integrated into a damage
indicator framework. Early examples of damage indicator models for predicting the onset
of fracture are the stress triaxiality dependent model of Johnson and Cook (1985) and the
stress triaxiality and Lode parameter dependent model of Wilkins et al. (1980). Recent
examples are the modified Mohr-Coulomb model proposed by Bai and Wierzbicki (2010)
and a micro-mechanism inspired damage indicator model proposed by Lou et al. (2012)
and Lou and Huh (2013). It is worth noting that there is a significant difference between
damage indicator models and Continuum Damage Mechanics (CDM). In the latter
framework, loss in load carrying capacity is modeled through an internal damage variable
while the constitutive equations are derived from the first and second principles of
thermodynamics for continuous media (e.g. Lemaitre, 1985 and Chaboche, 1988). In
particular, in CDM models, the elasto-plastic material response is affected by damage,
whereas the plastic response remains unaffected by damage evolution in damage indicator
models.
Unit cell analyses provide another means for studying the formation of localization bands
in metals. Since the pioneering works of Needleman and Tvergaard (e.g. Needleman, 1972
and Tvergaard, 1981), numerous unit cell analyses have been performed considering a wide
range of unit cell configurations and loading conditions (Koplik and Needleman, 1988,
- 42 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
Brocks et aL., 1995, Needleman and Tvergaard, 1992, Pardoen and Hutchinson, 2000,
Barsoum and Faleskog, 2007, Tvergaard, 2008, Tvergaard, 2009, Scheyvaerts et al., 2011,
Nielsen et al., 2012, Rahman et al., 2012 and Tekoglu et al., 2012). The most recent studies
now consider 3D unit cell models with general 3D boundary conditions (e.g. Barsoum and
Faleskog, 2011 and Dunand and Mohr, 2014). All of the above unit cell analyses are
performed considering a single void only along with periodic boundary conditions which
limits their capability to predicting secondary localization only.
Coalescence models have been developed to describe the transition from diffuse plastic
flow to localized plastic flow within the inter-void ligament of neighboring primary voids
within the mesoscopic band of localization. Simple mechanical models of a periodic array
of square cuboidal voids have been used by Thomason, 1968 and Thomason, 1985 to
analyze the process of internal necking in an approximate manner, while more advanced
models considering spheroidal voids and shear deformation have been developed later (e.g.
Benzerga, 2002, Pardoen and Hutchinson, 2000 and Tekoglu et al., 2012). Coalescence
models predict the formation of secondary bands of localization. The mechanical system
at this stage of the ductile fracture process is characterized by a strong interaction of
neighboring voids. Consequently, coalescence criteria incorporate information on void size
and spacing. In contrast, the formation of primary bands of localization is the outcome of
an instability of the material response at the macroscopic level.
In the present work, a phenomenological fracture initiation model is proposed for
predicting the onset of ductile fracture in engineering practice. The backbone of the
proposed model is a localization criterion for radial loading in terms of the Hosford
equivalent stress and the normal stress acting on the plane of maximum shear. Using the
isotropic hardening law associated with the material's plastic behavior, the criterion is
transformed from principal stress space to the space of equivalent plastic strain, stress
triaxiality and Lode angle parameter. As a final result, a fracture initiation model is
obtained which preserves the underlying physical meaning of the stress-based localization
criterion. The results from fracture experiments for five different stress states are presented
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 43
for three different advanced high strength steels (DP590, DP780 and TRIP780). It is shown
that the Hosford-Coulomb (HC) fracture initiation model can accurately describe the
experimental data for all materials and experiments including pure shear, notched tension
and equi-biaxial tension.
sill M
\
- Planw
Si
Iy
(a)
05
0
-
-
E
CT
-0. 5
Sbsii
(b)
-033
syT eri
0.33
0
Stress triaxiality [-]
067
06 00
- 44 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
Figure2-2 (a) Illustrationof the stress triaxiality q and the Lode angle parameter 9 in principal
stress space {j ,IC,11 , O7
}.
Selected Lode angle parameter values are only shown for the 600
segment of i7-plane where the principalstresses satisfy the order cr,
CTH
(Til .
For the other
five 60' segments, the same labeling applies because of the symmetries of the unorderedprincipal
stress space; (b) non-linearrelationshipbetween 9 and irforplane stress. The blue, black and red
curves shows the relationshipfor biaxial compression (two negative principal stresses), biaxial
tension-compression (one positive and one negative principal stress), and biaxial tension (two
positive principal stresses), respectively. Open dots highlight the special cases of cases of a.
uniaxialcompression, b. pure shear, c. uniaxialtension, d. plane strain tension, and e. equi-biaxial
tension".
2.2
Preliminaries
2.2.1
Descriptionof the stress state
The stress state is described by the stress triaxiality and the Lode angle parameter. The
stress triaxiality is defined as the ratio of the mean stress u,, and the von Mises stress O,
-'
(2-1)
-
(
117
It may be interpreted as a measure of the ratio of the first and second stress tensor
invariants. The Lode angle parameter on the other hand measures the ratio of the third and
second stress tensor invariants,
-
2
3
S=1 -- arccos --- _
r
L 2
__J
3
(2-2)
(J2)3/2
According to the above definition, the Lode angle parameter varies between -1
(axisymmetric compression) and 1 (axisymmetric tension). Figure 1 a provides a graphical
interpretation of the modified Haigh-Westergaard coordinates {7, 0, b}. For plane stress
conditions, the stress space is reduced from 3D to 2D which results in a functional
relationship between the Lode angle parameter and the stress triaxiality (Fig. 2-2b). Based
coordinates,
on the modified Haigh-Westergaard
o-,
au oft
1.
the ordered principal stresses
(
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 45
) may be reconstructed as
a-, =U(7+fi)
(2-3)
7(+ f2
(2-4)
o-,6=
a111 =a (r+ f 3 )
(2-5)
with the Lode angle parameter dependent trigonometric functions
-
Fr
2
f1 [01=-cos
- (1-0)
3
16
-[] 2
f2 [9]=-cos
3
-
f3 []
2.2.2
2
= --
3
r
-(3+0)
_6
IT
cos -(1+)
L6
(2-6)
(2-7)
_
.
(2-8)
PlasticityModel
Before developing the fracture initiation model, we briefly outline the constitutive
equations of a non-associated quadratic plasticity model with isotropic hardening. The
combination of a von Mises yield surface with a Hill'48 flow rule is chosen as it provides
a good approximation of the large deformation response of advanced high strength steels
(Mohr et al, 2010). The reader is referred to Stoughton (2002) for the proofs of the
uniqueness of the stress distribution, the stability of plastic flow and the uniqueness of the
stress and strain state.
The von Mises yield surface is expressed as
f[a,k]=C--k = 0,
(2-9)
with k denoting a deformation resistance that controls the size of the elastic domain. The
direction of plastic flow is assumed to be aligned with the stress derivative of a flow
potential function g[a],
- 46 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
dE = dA
where d,
(2-10)
,
denotes the plastic strain tensor increment in material coordinates; dA > 0 is a
scalar plastic multiplier. The potential function is defined as an anisotropic quadratic
function in stress space
+ G
g2[]fih
-2(1+
-222 +
(1+2G_ +G
G12 )o1 1o 33-
+G
2 22(G
2
2233
+2G
12 ) 2 2073 3
12122(2-11)
+G 33071
+23o-+ 3-1
with the anisotropy coefficients G2, GQ and G33. The above function corresponds to a
special case of an orthotropic Hill'48 flow potential function which accounts for the planar
anisotropy associated with direction-dependent Lankford ratios. For G22=1, G12= -0.5 and
G 33=3, the above potential reduces to the von Mises potential. Isotropic hardening is
introduced into the model through the function
k = k[iz, ]
(2-12)
with the equivalent plastic strain defined as work-conjugate to the von Mises equivalent
stress,
E,= f
2.3
2.3.1
lldA.
0-
(2-13)
Fracture Initiation Model for Proportional Loading
Motivation
A fracture initiation model is developed to predict the onset of ductile fracture in a
macroscopically defect-free solid. In particular, the goal is to predict the strain to fracture,
i.e. the macroscopic equivalent plastic strain that can be achieved before the formation of
a primary or secondary band of localization. At the macroscopic level (average material
response over several inter-void spacing), there is actually no noticeable difference
between the strains at the onset of localization and those at the instant of void coalescence
as all deformation localizes within a narrow band between these two events. It is therefore
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 47
assumed that the onset of fracture coincides with the formation of a primary or secondary
(whichever occurs first) band of localization. It is emphasized that the modeling of ductile
fracture, i.e. the modeling of the propagation of cracks, requires the careful modeling of
the conversion of bulk to surface energy and is beyond the scope of the present work.
The presence of voids may be neglected when describing the macroscopic elasto-plastic
response of sheet metal at low stress triaxialities (e.g. Dunand and Mohr (2011 a)).
However, voids play an important role in triggering the onset of localization. Unit cell
analyses (e.g. Barsoum and Faleskog (2007)) and stability analyses with advanced porous
plasticity models (e.g. Nahshon and Hutchinson (2008)) have demonstrated that shear and
normal localization at low stress triaxialities can be predicted when taking the effect of
voids (and void shape changes) into account. A mechanism-based model for predicting the
onset of ductile fracture would thus require (i) a void nucleation model, (ii) void volume
fraction and shape evolution equations, and (iii) a void volume fraction and shape
dependent shear/normal localization criterion. In theory, a comprehensive porous plasticity
model would satisfy all these requirements since the onset of localization could be
predicted by analyzing the loss of ellipticity of the incremental moduli associated with the
current state of the material. Both Danas and Ponte Castafieda, 2012 and Nahshon and
Hutchinson, 2008 indirectly pursued this approach. However, given the sensitivity of
localization analysis to small changes in the constitutive model formulation and the number
of approximations necessary during non-linear homogenization (see e.g. Ponte Castafieda
(2002)), it is still questionable whether accurate predictions of the equivalent plastic strain
at the onset of fracture will be obtained in the near future using a porous plasticity approach.
A different approach is pursued here. Instead of predicting the onset of localization by
means of an advanced porous plasticity model, we make use of a conventional non-porous
plasticity model along with a localization criterion. The localization criterion for
proportional loading is transformed from stress space to a mixed stress-strain space, before
inserting the resulting functional into a damage indicator model framework to account for
the effect of non-proportional loading.
2.3.2
Localization Criterionin Stress Space
- 48 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
Dunand and Mohr (2014) subjected a unit cell of a Levy-von Mises material with a
central void of 1.2% porosity to combinations of shear and normal loading to determine
the macroscopic equivalent plastic strain at the onset of localization as a function of the
stress state. They performed this type of analysis for more than 160 different stress states
for stress triaxialities ranging from 0 to I and Lode angle parameters ranging from -I to 1.
Their results demonstrate that a Mohr-Coulomb criterion,
(2-14)
2
.
max[rr+cCr]=c
provides a reasonable prediction of the onset of localization, with r and o, denoting the
shear and normal stress on a plane of normal vector n. In terms of the ordered principal
stresses, the Mohr-Coulomb criterion may be rewritten as
(2-15)
~~ -, +c(a,+ auj) = b
(a,
with
C=
1
1+c1
and
b=
2c2
(2-16)
1+c2
which is fully equivalent to (14).
700
-
600
500
c400 F
300
200
L
100
'
0
0
1000
500
Normal stress [MPa]
1500
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 49
(a)
15=0.2
/0.4
1.5
CO
\
1
CU
*50.5-
C-
.
-
il0.6
.--
.
-7-=1.0-
-1
0
-0.5
=0.8
.-
-
0.5
1
Lode angle parameter
(b)
Figure 2-3 Localization analysis results: (a) Relationship between the shear and normal stress
acting on the plane of localization; (b) Macroscopic equivalent plastic strain at the onset Of
localization as afunction of Lode angle parameterand stress triaxiality;each dot represents the
result from a unit cell analysis for a particularstress state, the solid curves correspond to the
predictions of the Mohr-Coulomb model. Note that both plots have been preparedusing the same
MC model parameters (friction C, = 0. 13, cohesion c2 =
666MPa).
However, the predictions of the Mohr-Coulomb (MC) model do not always agree well
with results from experiments (where primary localization may also precede coalescence).
These deficiencies are partly attributed to the shortcomings of the periodic unit cell model
(which can only capture secondary localization). Furthermore, strong simplifying
assumptions with regards to the shape and the volume fraction of the defects triggering the
localization are expected to play a role. Note that the results of Dunand and Mohr (2014)
were obtained assuming the same volume fraction of spherical voids irrespective of the
stress state.
- 50 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
To improve the agreement of model predictions with experimental data (which will be
discussed in Section 6), we construct a simple phenomenological model based on the above
micromechanical results. In particular, an extension of the MC criterion is proposed by
substituting the Tresca equivalent stress in (2-15) by the Hosford (1972) equivalent stress,
uHF+c(jI+
(2-17)
oJllY-b
with
07
{H (2 7)i +(,>
,H)I~01a
>07+ofl c))
0
II
_0HIEa +(01~I _0ja
(2-18)
And 0<a;2 denoting the Hosford exponentl. The above model is referred to as HosfordCoulomb (HC) model: as postulated by Coulomb (1776), the material's deviatoric strength
is decomposed into a cohesion b and a frictional term that is proportional to the normal
stress (0-> + o,, ) / 2 acting on the plane of maximum shear. The HC model actually reduces
to the MC model for a= 1. However, an important difference between the MC and HC
models becomes apparent for biaxial tension (i.e. plane stress states of loading between
uniaxial tension and equi-biaxial tension): since the MC model does not dependent on the
07
=0), while the HC model remains sensitive to the biaxial stress ratio o-,, /-
2.3.3
.
second principal stress, it reduces to a maximum principal stress criterion (because of
Fractureinitiation model in mixed strain-stressspace.
Recall that the onset of ductile fracture is considered to be imminent with the formation
of a primary or secondary band of localization. The above localization criterion (2-17) is
therefore employed to predict the onset of fracture. The results from ductile fracture
experiments are typically presented in terms of the equivalent plastic strain, the stress
triaxiality and the Lode angle parameter. We transform the localization criterion from the
principal stress space {o-r , 1
10-1Jl } to the mixed strain-stress space {7' 0, E,}. Using Eqs.
(3), (4) and (5), th- criterinn (2-17) ic first rewritten in the modified
stress space {rq,0, 7}, where it takes the form
I-
Wetergaard
..........
IIII II.
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 51
cr =
f|[i,
b
]
(2-19)
{(f -2)" +(f 2 f)" +(f1
+c(2i+f, +f) 3
f)")
Fig. 2-4a and b illustrate this transformation for plane stress conditions, with the
localization criterion shown as a blue envelope. Von Mises stress iso-contours are shown
as dotted lines in both figures. Furthermore, we included the trajectories of constant stress
state as straight dashed lines. Since the stress point is located on the yield surface when
plastic localization initiates, the inverse of the isotropic hardening law (12), -, = k-[a:],
may be used to transform the localization criterion from the modified Haigh-Westergaard
space to the mixed stress-strain space
{r, 0,
,},
(2-20)
E'= k-1 [,[N] .
In Fig. 2-4, this final transformation corresponds to a non-affine mapping of the
ordinate from Fig. 2-4b and c.
=.67
. ....
..
.... ..
..
.. ....
.. ....
.
. .. .
..
.. ..
.. ..
..
.. ... ...
... ... ..
V)
... .. ..
...............................
Localization
0.58
..............
4-J
0000
Yield
-/=
0
.......... ........
................
................................
.......................
First in-plane stress
'7=0
A 121-
.
CL
-
52 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
(a)
1200.
1.6
10001.4
Localization :
U
1.2
L_
4800
YieldLocalization
(0
.400-
0*
LU 0.4
0.2
200
0
-0.33
0.33
0
Stress triaxiality
0.66
0
-0.33
(b)
Yieldl
0.33
0
Stress triaxiality
0.66
(c)
Figure 2-4 Coordinate transformationfor plane stress conditions: (a) initial von Mises yield
envelope (solid black line) with subsequent von Mises stress iso-contours and EMC localization
locus (blue line)for proportionalloading; (b) representationofthe same envelopes in the modified
Haigh-Westergaardspace, and (c) in the mixed strain-stressspace.
2.3.4
Illustrationof the HC model
Fig. 2-5 shows the strain to fracture for proportional loading (2-20) as a function of the
stress triaxiality and the Lode angle parameter for different sets of model parameters
{a,b,c}. In all graphs, the parameter b has been adjusted such that the strain to fracture for
uniaxial tension equals 0.8.
For a=1 (Fig. 2-5a), we obtain a representation of the Mohr-Coulomb criterion in
the mixed strain-stress space. The strain to fracture is a monotonically decreasing
function of the stress triaxiality for c>0 and a convex function of the Lode angle
parameter which exhibits a minimum for generalized shear
(0 = 0).
The
characteristic signature of a normal stress dependent criterion is the asymmetry
with respect to the Lode angle parameter. Note that the use of the pressure instead
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 53
of the normal stress in (2-17) (e.g. a Drucker-Prager type of criterion) would result
in a symmetric Lode angle dependency.
For c=O (Fig. 2-5b), the normal stress term is no longer active. Consequently, the
-
criterion becomes independent of the stress triaxiality and symmetric with respect
to the Lode angle parameter (Hosford model).
For a=2 (Fig. 2-5c), the Lode angle dependence is only due to the normal stress
-
term. In the limiting case of a=2 and c=0 (von Mises model), the criterion becomes
independent of both the stress triaxiality and the Lode angle parameter which
corresponds to a fracture initiation model that depends on the equivalent plastic
strain only.
A typical HC surface is shown in Fig. 2-5d (a=1.5 and c=0.1). It exhibits the same
-
stress triaxiality dependence as the MC model, but as the comparison of Fig. 2-5a
and d shows, the models sensitivity to the Lode angle parameter can be adjusted.
a=1 c=0.1
a=1 c=q
0.81
T05
0.6
0.4-
0.5
0
0
0
0
-0.5
0.66
03 0.5
0.33/7
-0.5
05
0.66
(b)
(a)
.1=2
=
c=0.1
a
2
2-1
0.5
0.5
-0.
-0.33
0
0
-0
033
0.5
0.66
0
033 n7
0
-0.5
0
W
0.5
0.66
- 54 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
(d)
(c)
Figure 2-5 Representation of special cases of the Hosford-Coulomb (HC) model in the modified
Haigh-Westergaardspace. The blue lines show the strain tofracturefor plane stress conditions.
All 3D plots include a blue curve which highlights the model response for plane stress
conditions. As illustrated in Fig. 2-2b, the Lode angle parameter is a non-linear function of
the stress triaxiality for plane stress. To shed more light on the effect of the model
parameters a and c, we plotted the strain to fracture as a function of the stress triaxiality for
plane stress conditions in Fig. 2-6:
-
The effect of the friction coefficient c on the MC model (a = 1) is shown in Fig. 26a. Note that the curves are in hierarchical order for
I
< 1/3, i.e. the higher the
friction coefficient, the higher the strain to fracture. For q > 1/3 (biaxial tension),
the friction coefficient has no effect on the strain to fracture predicted by the MC
model. This is due to the fact that the MC model reduces to a maximum principal
stress criterion for biaxial tension with only one independent parameter.
In case of the HC model (a
1), the model response for biaxial tension can still be
adjusted by the Hosford exponent ( Fig. 2-6b) and the friction parameter ( Fig. 2-6c).
Note that the ordering of the curves with respect to the friction parameter changes
at r=1/3. All
curves exhibit an absolute minimum
at r=1/
3
which
corresponds to transverse plane strain tension for a Levy-von Mises material
(
-
o, =0.5o ). Fig. 2-6d shows the criterion for a=2 which includes the special
case c=0 (strain to fracture independent of the stress state).
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 55
21.8.
1.6
j
2
1.81.6
1.4
1.2
=O.1
.c
a=2
1.4
1.21-
a=1.2
0.8
a =1
c=0.35
Sc=
0.2
c = 0.1
0
> 0.8
o0.6
0.40.2-0.33
a=0.8-
0.33
0
Stress triaxiality
0.4
0.2
0.66
a=1
0- 0.33
(b)
(a)
1.8.
1.6c = 0.
T 1.4
c=0.05
1.2
S1 c=0
0.8
2
1.8
1.6
1.4
o1.2
c = 0.2
0.8
c =0.2
C=
c=
0. 1
0.05
=0
a=1.5
0.33
0
0.33
0
Stress triaxiality
(c)
-
Cr-0.6
c0.6
0.4
0.2
-0.33
0.66
0.33
0
Stress triaxiality
0.66
0.4
0.2
"-0.33
a=2|
0.33
0
Stress traxiality
0.66
(d)
Figure 2-6 Effect of the parametersof the Extended Mohr-Coulomb (EMC) model on the fracture
envelope for plane stress loading.
2.3.5
Comments on Model Extensionfor Non-ProportionalLoading
The focus of the present work is on monotonic proportional loading, i.e. loading
histories throughout which the stress triaxiality and Lode parameter remain constant up to
the point of fracture initiation. A model extension for non-proportional loading is
nonetheless included in this paper because of inevitable stress state variations in many
fracture experiments (in particular due to necking in the case of sheet materials).
- 56 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
When using porous plasticity (e.g. Gurson, 1977, Gologanu et al., 1993, Benzerga and
Besson, 2001, Monchiet et al., 2008 and Danas and Ponte Castafieda, 2012) the evolution
law for the void volume fraction (and other possible microstructural state variables) is
loading path sensitive and failure predictions with microstructurally-informed coalescence
criteria (e.g. Thomason, 1985 and Pardoen and Hutchinson, 2000, Benzeraga (2000),
Tekoglu et al. (2012)) are then "naturally" loading path dependent. This is a key advantage
of the latter over phenomenological fracture criteria which are used in conjunction with
non-porous plasticity models that do not feature any loading path dependent damage
measure as internal variable such as the void volume fraction in porous plasticity.
A first approach to evaluating a phenomenological fracture initiation model for
proportional loading based on experimental data with stress state history variations in the
plastic range would be to postulate that the model holds true for the average stress state
history (e.g. Bai and Wierzbicki (2008)). However, as clearly demonstrated by Benzerga
et al. (2012), this approach is in strong contradiction with the results from unit cell
coalescence analysis for non-radial loading paths. A second approach would be to apply
the stress-based localization criterion (Eq. (2-17)) directly even if the loading path is nonproportional (e.g. Stoughton and Yoon (2011) and Khan and Liu (2012)). This corresponds
to assuming that the strain to fracture is independent of the stress state history and depends
on the current stress state only. As will be shown in Section 6, our experimental data
includes different loading paths which show significantly different fracture strains even
though the stress state at the instant of fracture initiation is very similar (see loading paths
NT6, NT20 and PU in Fig. 2-11 d).
As an alternative, we make use of Fischer's integral extension to evaluate our model
for proportional loading based on experimental data with inevitable necking-induced
loading path variations. The resulting final integral form of the HC fracture initiation model
reads
o
-
-pr[,]
=1
(2-21)
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 57
with
denoting the equivalent plastic strain at the onset of fracture after loading
along a path characterized by the histories 17(e") and O(EP). The assessment of the
general validity of this integral approach is deferred to future research as a comprehensive
series of non-proportional loading path experiments would needed for this. Here, we can
only justify the use of the integral formulation through empirical arguments, i.e. it has been
widely used (without any justification) in engineering practice for 30 years (see Johnson
and Cook (1985)) and is mathematically similar to the well-established Palmgren-Miner
rule in high cycle fatigue (Palmgren, 1924). At the same time, it is noted that the integral
extension
for non-proportional
loading
is conceptually
problematic
as discussed
by Benzerga et al. (2012). We therefore emphasize that even though Eq. (2-2 1) depends on
the stress-state history during plastic loading, i.e.
7(c P) and 6(P ), it is just introduced
as a means to identify the criterion for proportional loading from basic fracture
experiments, while it should not be understood as a general recommendation for predicting
fracture initiation after non-proportional loading.
2.3.6
Comment on Model Consistency
The basis of the proposed fracture initiation model is a shear localization criterion in
stress space for proportional loading (Eq. (2-17)). This criterion is transformed from stress
space to the mixed strain-stress space using the material's plasticity model. This approach
is considered as consistent in the sense that the link with the underlying localization
criterion in stress space is preserved. In other words, the final EMC fracture initiation
model (Eq. (2-21) features only parameters that are associated with the localization
criterion (2-17) in stress space.
The derivation of the so-alled modified Mohr-Coulomb (MMC) model (Bai and
Wierzbicki, 2010) is mathematically similar to the EMC model, but it is usually used as
inconsistentmodel. As opposed to the isotropic hardening law provided by Eq. (2-12), Bai
and Wierzbicki (2010) made use of a rather unconventional stress triaxiality and Lode
angle dependent hardening rule
- 58 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
k= g[ ,O ]h[C,]
(2-22)
to transform the Mohr-Coulomb criterion from stress space
(2-23)
to strain space,
"
-pr[q,0
= h-
'
.
(2-24)
g[ O,]
For most engineering materials, the hardening rule does not dependent on the Lode angle
or the stress triaxiality. Consequently, g[q,]= 1 must be used in (2-22) to describe the
isotropic strain hardening. However, when applying the MMC model, the function g[,0]
is not set to unity in Eq. (2-24) even if g[,] =1 is assumed to model strain hardening (e.g.
Bai and Wierzbicki (2010), Luo and Wierzbicki (2010), Beese et al. (2010), Dunand and
Mohr (2011)). This practice is considered as inconsistent modeling. It is mostly done to
obtain a better fit of the model to experimental data. In addition to the two Mohr-Coulomb
parameters, the parameters describing the function g[q,O] can be adjusted to improve the
predictions of the strain to fracture. The main difference between inconsistent and
consistent modeling is that the latter approach preserves the link with the underlying stressbased fracture initiation model. For proportional loading, the application of the EMC model
(2-2 1) provides the same result as the direct use of the underlying stress-based criterion (217). In the case of the inconsistent MMC model, the application of Eq. (2-24) does not
provide the same result as the Mohr-Coulomb criterion in stress space (Eq. (2-23)). In other
words, the link with the original Mohr-Coulomb criterion is lost when using the
inconsistent MMC model.
Another particular feature of the present work is the use of a non-associated flow rule
model to account for the anisotropy in the plastic flow. It is emphasized that the hardening
law is relates the von Mises stress to its work-conjugate equivalent plastic strain. As a
result, the EMC fracture model (2-21) is an isotropic criterion in both the stress and plastic
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 59
strain space. In other words, the anisotropy of the flow rule does not alter the isotropic
response of the fracture model.
Fracture Experiments
2.4
We make use of the experimental data for three different advanced high strength steels
to calibrate and validate the EMC fracture initiation model.
2.4.1
Materials
Two Dual-Phase (DP) steels and one TRIP-assisted steel are considered:
"
1.4mm thick DP590 steel sheets from ArcelorMittal
"
1.06mm thick DP780 steel sheets from US Steel
*
1.43mm thick TRIP780 steel sheets from POSCO
The experimental data for the TRIP780 steel is taken from the literature (Dunand and Mohr,
2011 a), while new experimental results are reported here for dual phase steels.
2.4.2
Uniaxial tension experiments
Tension experiments are performed on different specimen geometries. Dogbone
specimens featuring a 10mm wide gage section are used to characterize the material
response for uniaxial tension up the onset of necking. Specimens are extracted along three
different sheet directions (rolling, transverse and 45 -direction). All experiments are
performed at an axial strain rate of about 10 3 /s . Throughout the experiments, the in-plane
displacement fields are monitored using planar Digital Image Correlation (DIC). In
particular, the evolution of the width strain is determined as a function of the axial strain
using virtual extensometers of about 9mm and 20mm length for the respective directions.
After computing the logarithmic plastic strains in the width and thickness directions
(assuming plastic incompressibility), the Lankford ratios are determined from the average
slopes,
- 60 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
r
depW_
d,,pth
(2-25)
Table 1 summarizes the measured Lankford ratios for all three materials and material
orientations. The axial true stress versus logarithmic plastic strain curves for the DP steels
are shown in Fig. 2-6. As reported by Mohr et al. (2010) for the TRIP780 steel and another
DP590 steel, the axial stress-strain curves for different material orientations lie
approximately on top of each other which motivates the use of an isotropic yield condition.
ro [-]
r45 [-]
r9o [-]
DP590
0.98
0.84
1.13
DP780
0.78
0.96
0.77
TRIP780
0.89
0.82
1.01
Table 2-1 Lankford Ratios
700
600
0- 500
W 400
w 300
-
200
100
0
C
0.05
01
0.15
Logarithmic plastic strain
(a)
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 61
800-
400
00
200
0
0
0.08
0.06
0.02
0.04
Logarithmic plastic strain
(b)
Figure2-7 Measured true stress versus logarithmicplastic strain curve for uniaxial tension along
different material directions up to the point of neckingfor (a) DP590, (b) DP780. Note that each
graph shows the curves for three different specimen orientations (0', 450 and 90 ), but they lie
exactly on top of each other and thus only one curve is visible.
In addition to uniaxial tension experiments, tension experiments are performed on
specimens with circular cut-outs (notched tension) and a central hole ("CH"-specimen).
We use the labels "NT6" and "NT20" to refer to specimens with notch radii of R=6.67mm
and R=20mm, respectively (Fig. 2-8). All specimens had been extracted from the sheets
using abrasive water-jet cutting, with the specimen tensile axis aligned with the sheet
rolling direction. The hole in the CH-specimens is introduced using CNC machining to
minimize the effect of the cutting technique on the onset of fracture at the hole boundaries
(see Dunand and Mohr (2010)). The specimens are tested in a hydraulic universal testing
machine with custom-made high pressure clamps. All experiments are performed under
displacement control at a constant cross-head speed of about 1mm/min. The relative vertical
displacement of points positioned on the lower and upper specimen boundary is measured
using a DIC-based virtual extensometer. The recorded force-displacement curves are
shown as solid dots in Fig. 2-9 and Fig. 2-10. Note that only one curve is shown per
experiment because of the remarkable repeatability of the experimental measurements.
- 62 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
50
50
50
10
10
R
10
0
R6.67
R20
20
20
420
(a) CH
(c) NT6
(b) NT20
1;-0.50
tin i
78
60.77
Is
CO
C14
('4
f~.
1
56.77
(d)
Figure 2-8 Specimen drawingsfor flat tension specimens with (a) a central hole, (b)-(c) different
notches, and (d) the butterfly specimenfor shear testing.
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 63
14
12-
12
10.
10
8
3
2
1
0)
8
4
0 6
o
LL
LL
6
4
4
2
2
0
S
3
2
1
Displacement [mm]
C
4
1.5
1
0.5
2
Displacement [mm]
(b) CH2
(a) SH
8
8
-6
-6
z
2
00
2
1
2
Displacement [mm]
(c) NT20
3
-0
0.5
1
1.5
2
Displacement [mm]
(d) NT6
Figure 2-9 Measured and simulatedforce displacement curves for selected fracture experiments
on the DP590 steel. The star symbols represent the experimental curves, while the simulation
results are shown as solid lines. A contour plot of the equivalent plastic strain at the onset of
fracture is shown below each figure.
- 64 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
2.4.3
Shear experiments
Butterfly specimens (Fig. 2-14d) are subject to tangential loading to characterize the
fracture response for pure shear (q =0, 0 =0). The latest specimen design as optimized
by Dunand and Mohr (2011 b) is used. It features clothoidally shaped specimen shoulders,
convex lateral boundaries and a gage section of 0.5mm thickness while preserving the
original sheet thickness in the shoulder regions. The specimen is tested in a dual actuator
system using the same high pressure clamps as for the above tension experiments. The
shear experiments are performed under combined displacement/force control, i.e. the
horizontal actuator applies a constant tangential velocity of 0.2mm /min while keeping the
vertical force equal to zero (see Mohr and Oswald (2008) for details on the experimental
procedure). The relative tangential and normal displacement of two points located on the
upper and lower specimen shoulder is measured using DIC. The dotted curves in Figs. 8
and 9 show the measured force-displacement curves for shear loading. All curves increase
monotonically up to the point of fracture. The location of onset of fracture is assumed to
coincide with the location of the highest equivalent plastic strain within the gage section.
2.4.4
Punch experiment
Circular discs are extracted from the sheets for punch testing. A hemispherical punch
of a diameter of 45mm is used to apply the loading, while clamping the specimen on a
127mm diameter die. Four about 0.05mm thick Teflon layers with grease are positioned
between the specimen and the punch to reduce the effect of friction. After clamping the
specimens with sixteen M10 screws, the experiments are performed at a constant punch
velocity. The experiments are stopped as soon as the punch force reaches its maximum.
Subsequently, the specimens are cut in half to be able to measure the final specimen
thickness at the apex of the punched specimens. The measured thickness reductions for the
DP590, DP780 and TRIP780 steels were 67%, 61% and 59%, respectively.
2.5
2.5.1
Identification of the loading paths to fracture
Plasticity model parameteridentification
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 65
The plasticity model requires the identification of the isotropic hardening law and of
the anisotropic flow potential function. The parameters of the latter are uniquely
determined from the three Lankford ratios ro, r4 5
and
r9 o, using the analytical
relationships
G0-
, G 22 - rI
- +r90 and G33 - 1+2r550r-+r90
r~o
l+r'b
1+ro
r90 1+r0
r
(2-26)
The isotropic hardening law for the dual phase steels is identified in a two-step procedure.
Firstly, we approximate the true stress versus logarithmic plastic strain curve up to the point
of necking using an exponential law (Voce, 1948),
k, = ko + Q( - e_
.
(2-27)
Secondly, the same experimental curve is approximated using a power law (Swift, 1952),
ksw = A( p +6} .
(2-28)
As suggested by Sung et al. (2010), the final hardening curve is approximated by the linear
combination of the exponential and power law,
k =(1 - a)k, +cks .
(2-29)
The weighting factor a plays an important role in the post-necking range and needs to be
determined through inverse analysis. The NT20 experiment is chosen to identify a . Unlike
in dogbone specimens, the location of the through-thickness necking zone is predetermined
by the notched specimen geometry. Furthermore, the mechanical system for the NT20
experiment does not lose any symmetry in the post-necking range. Hence, the modeling of
one eighth of the specimen without any artificial imperfections is sufficient for simulating
a notched tension experiment. We follow closely the modeling guidelines given by Dunand
and Mohr (2010): eight elements in thickness direction and at least 10 5 explicit time steps,
while using the user material subroutine developed by Mohr et al. (2010) for the nonassociated quadratic plasticity model.
The identification of a is posed as a minimization problem. For this, we introduce the
residual
- 66 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
[ pF NUM [a_-FE)
[ a' '
1=
(2-30)
FD
to quantify the difference between the simulated and measured force-displacement curve
for notched tension (NT20), with FNUM and FFY
denoting the respective computed and
measured forces corresponding to the same displacement u, of the discrete experimental
force-displacement curve with Np data points. After minimizing V/ using a derivative-free
simplex optimization algorithm, the values a= 0.73 and a
=
0.79 are obtained for the
DP590 and DP780 materials, respectively. A summary of all parameters describing the
isotropic hardening of the DP steels are given in Table 2.
ko
Q
7
A
no
n
n
[MPa]
[MPa]
[-]
[MPa]
[-]
[-]
[-]
DP590
345.9
335.8
24.9
1031.0
0.0013
0.2
0.73
DP780
614.0
270.0
32.2
1170.0
3.1 10-
0.11
0.79
Table 2-2 Dual steel hardeninglaw parameters
The black solid lines in Figs. 8 and 9 show the simulated force-displacement curves
for notched tension (NT20 and NT6), tension with a central hole (CH2 or CH4), and the
butterfly shear specimens (SH). In the latter case, only one half of the butterfly specimen
is modeled with symmetry boundary conditions applied to the specimen mid-plane (about
40,000 first-order solid elements). The good agreement of the simulations (solid lines) and
experiments (dots) partially validates the applicability of the calibrated plasticity model.
2.5.2
Loadingpaths to fracture
The calibration of a fracture initiation model requires knowledge of the stress and
deformation history at the material point within the specimen where fracture initiates:
*
For notched tension (NT6 and NT2O) and shear loading (SH), the loading paths to
fracture are extracted at the integration point of the element with the highest
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 67
equivalent plastic strain. As can be seen from the contour plots in Figs. 8 and 9, this
location corresponds to the very center of the NT specimens.
"
For tension with a central hole (CH), the loading path is extracted from the element
on the specimen mid-plane that is positioned at the root of the hole.
"
The punch experiment (PU) does not exhibit any necking and we therefore assume
that the stress state remains equi-biaxial tension (q = 0.67, j = -1) throughout the
entire deformation history. According to the non-associated flow rule, the
equivalent plastic strain to fracture can be determined from the final thickness
reduction
EP=
-
In[t / tin]
t "'
.
(2-31)
1+ 1G22 +2
22Ga
12
The determined loading paths to fracture are shown as black solid lines in Fig. 2-10. Each
row of figures corresponds to a different material. The left plots show the evolution of the
equivalent plastic strain as a function of the stress triaxiality, while the center plots show
the same evolution as a function of the Lode angle parameter. For tension with a central
hole (CH), the stress triaxiality and Lode angle parameters remained more or less constant
at values of 7 =0.32 and 0
=
0.95, respectively. Under notched tension (NT), the stress
state is typically constant up to the onset of through-thickness necking; thereafter, an outof-plane stress builds up which causes an increase in stress triaxiality and a decrease in the
Lode angle parameter. The deformed meshes at the instant of onset of fracture for notched
tension are shown next to the plots of the loading paths in Figs. 8 and 9. The color contours
are chosen such that the maximum value (red) corresponds to the equivalent plastic strain
at the instant of onset of fracture.
- 68 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
SH
SH
0.9
0 .9
CH
1 0.8
0.0.7
a
CH
.8
0 .7
6
0.5
0.4
3NT
0.2
0.
N6
SRP7o
0
0 .6
0 .5
0 .4
MC
a=1
b=1520
c0 185
'EMC
a=1.16
b=1445
C=0.138
NT20
NT20
.3
.2
0
P416
NT6
0.33
0.66
Stress triaxiality
1*
.9
08
.7
*Pu
CH
NT20
SH
NT6
7a=1
b=1231
c=0.107
EMC
a=1.53
b--1102
I
0
.
.
0..9
0..8
0 .7
0 .6
ji0 .5
.4.3
0 .2
0 .1
CH
PU
NT
T6
0 .2
0
am
Ff
iNT6
NT20
0.5
0.2-0-.
0.33m
01
-0.5
0
0.5
Lode angle parameter
1
(e)
00
.i0
SH
0.33
CH
.6SH
0
(d)
.2-
PU
0-1
0.66
0.33
Stress triaxiality
=.6
(c)
0 .2
.1
,7Sc=.6
o~~=-..~-
I
0 .5
0 .4
0 .3
MC
5
4
3
10.2
1Z
0.5
Lode angle parameter
(b)
1
.0. 1
0
-1-0.5
(a)
0. 9
SO. 8
0. 7
0.
0. 5
0
T1
RIP730
(f)
CH
Pu
PP
.8
.7
.6
MC
.5.a=1
b=1096
C=0.22
0
A
0.33
0
EMC
.3
.2
0.6
(g)
.
.4
a=1.78
b=956
Stress triaxlalty
06
NT20
NT6
SH
.1
0.1
0.4
0.2
0--as
0.3
0
0.5
am
q
-0.5
0
0.5
Lode angle parameter
(h)
(i)
Figure 2-10 Comparison of the Hosford-Coulomb (HC) model predictions (blue dots) with
experimental results (endpoint of the black lines) after calibrationbased on the SH, CH andNT6
experiments. The predictions of the Mohr-Coulomb (MC) model are shown as red dots. The units
of the cohesion b are MPa
2.6
Model calibration and verification
The HC model parameter identification is done through inverse analysis. Denote the
loading history for each calibration experiment i with r7, = rq,[, ] and , = 6i [,,]. For a
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 69
given set of HC model parameters X =
{a, b, c}, the strain to fracture cf = if [Z] for an
experiment i is then determined based on Eq. (2-21). The formal minimization problem
for the parameter identification reads
x = arg min(max ,f[x] with
SEX
with i e SE
Z71
(2-32)
denoting a set of calibration experiments. As for the isotropic hardening law
identification, a derivative-free simplex algorithm is used to solve (32) in an approximate
manner.
2.6.1
Model application
Among the loading paths shown in Fig. 2-10, we chose the results from the experiments
S,
= {SH, CH, NT6} to calibrate the three parameters {a, b, c} of the EMC model. The
instants of onset of fracture predicted by the calibrated EMC model are depicted by blue
dots in Fig. 2-10. The overall comparison of the model predictions with the end points of
the loadings paths (black lines) shows satisfactory agreement for all experiments and for
all three materials. The good agreement for SH, CH and NT6 demonstrates that the
mathematical structure of the EMC model is flexible enough to be fitted to three distinct
loading paths. Differences between the model predictions and the experiments become
apparent for the PU and NT20 experiments. The model error for punch loading is small for
the DP590 steel which features a high Hosford exponent. For smaller Hosford exponents
(see DP780 and TRIP780), the model becomes more sensitive to small variations in stress
triaxiality in the vicinity of uniaxial tension (r7
=
0.33,0
=
1.0). Consequently, small
uncertainties in the loading path to fracture for CH have a strong effect on the identified
model parameters b and c (and hence on the results for PU). A more robust calibration is
obtained when including the result from the punch test in the calibration procedure, i.e.
S
= {SH, CH, NT6, PU}.
The results for the DP780 steel (second row in Fig. 2-10) elucidate the effect of the
(
Lode angle parameter. The calibration reveals only little stress triaxiality dependence
c ~ 0), even though the plot of the loading paths in the equivalent plastic strain versus
- 70 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
triaxiality plane (Fig. 2-1 Gd) shows significant variations in the strain to fracture.
According to our model, these variations are only due to the fact that the loading paths
feature different Lode angle parameters (see Fig. 2-1 Ge). The right-most column in Fig. 210 shows the underlying localization criteria in mixed strain-stress space (Eq. (2-20)).
Observe that their shape varies substantially as a function of the material, with a
"
Mohr-Coulomb type of surface (a
1) for the TRIP780 steel (both stress
triaxiality and Lode angle dependent),
*
Hosford type of surface (c ~0) for the DP780 steel (Lode angle dependent
only)
*
Mises type of surface (a
2, c =0) for the DP590 steel (nearly stress state
independent).
This result illustrates not only the flexibility of the EMC model, but also the importance of
validating fracture initiation models for different materials.
To elucidate the effect of the Hosford extension of the Mohr-Coulomb model, we
repeated the EMC model calibration for the special case of a = 1 using the experimental
database S, = {SH, CH, NT6}. The corresponding predictions of the MC model are
depicted as red solid dots in Fig. 2-10. Its approximation of the calibration experiments is
only satisfactory for the TRIP780 steel. For the DP steels, the MC model significantly
underestimates the strain to fracture for notched tension. The calibration for NT6 does not
improve when using two calibration experiments only (e.g. SFv = {CH, NT6}). This is due
to the fact that the two-parameter MC model reduces to a one-parameter model for biaxial
tension. It is thus concluded that the MC model provides only a poor description of
experimental data for biaxial tension.
2.7
Summary
A fully three-dimensional analysis on a unit cell with initial void suggests that localization
at the mesoscale or microscale corresponds to a Mohr-Coulomb limit envelope in the stress
space. The onset of fracture is assumed to be an imminent consequence of the formation of
a bands of localization. It is proposed to replace the Tresca term by a Hosford term with an
-
Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture - 71
additional parameter in order to fit the experimental data. Using of the material's isotropic
hardening law, a consistent transformation from the principal stress space to the space of
equivalent plastic strain, stress triaxiality and Lode angle parameter is performed to obtain
a fracture initiation model for non-proportional loading. Finally, a fracture initiation model
is proposed to predict the onset of fracture in advanced high strength steels at low stress
triaxialities. A number of experiments have been carried out in order to identify the
ductility of three different advanced high strength steel sheets (DP590, DP780 and
TRIP780). These tests include a shear experiment, tension with a central hole, notched
tension and a punch experiment. The Hosford-Coulomb model was successfully calibrated
with all experiments for each material.
- 72 - Chapter 2: Hosford-Coulomb Model for Predicting Ductile Fracture
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 73
Chapter 3: Compression-Tension to Fracture of
Dual Phase Steel Sheets
The effect of loading direction reversal on the onset of ductile fracture of DP780 steel
sheets is investigatedthrough compression-tensionexperiments on flat notchedspecimens.
A finite strain constitutive model is proposed combining a Swift- Voce isotropic hardening
law with two Frederick-Armstrongkinematic hardeningrules and a Yoshida-Uemori type
of hardening stagnation approach. The plasticity model parameters are identifiedfrom
uniaxial tension-compression stress-strain curve measurements and finite element
simulations of compression-tension experiments on notched specimens. The model
predictions are validated through comparison with experimentally-measured loaddisplacement curves up to the onset offracture, local surface strain measurements and
longitudinal thickness profiles. In addition, the model is used to estimate the local strain
and stress fields in monotonicfracture experiments covering plane stress states ranging
from pure shear to plane strain tension.
The extracted loadingpaths to fracture show a
significant increase in ductility as a function of the compressive pre-strain. A HosfordCoulomb damage indicatormodel is presentedto provide a phenomenologicaldescription
of the experimental resultsfor monotonic and reverse loading.
- 74 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
3.1
Introduction
Predicting the onset of ductile fracture has been an active field of research for more
than 50 years. In particular, the fracture initiation after monotonic proportional loading
paths has been investigated intensively (e.g. Brunig et al. (2008), Bai and Wierzbicki
(2008, 2010), Sun et al. (2009), Li et al. (2011), Gruben et al. (2011), Chung et al. (2011),
Lecarme et al. (2011), Khan and Liu (2012), Luo et al. (2012), Huespe et al. (2012),
Malcher et al. (2012), Lou et al. (2014)). In industrial practice, in particular during sheet
metal forming, ductile fracture often initiates after complex non-proportional loading
histories. Among these, reverse loading is an important non-proportional loading condition
which prevails for instance when a sheet is bent and unbent as it is drawn over a die radius.
Simulating the mechanical response of ductile materials up to the point of fracture
initiation requires the accurate modeling and identification of the hardening behavior of
the material at large strains. Many plasticity models for reverse loading have been
developed for life-cycle analysis. As a result, most experimental procedures are designed
for characterizing the small strain response only. One of few exceptions are the reverse
shear experiments of Barlat et al. (2003) on 3mm thick 1050-0 aluminum sheets. Using
wide shear specimens with a narrow gage section of reduced thickness, they achieved shear
strains of up to 0.22 prior to loading direction reversal. Yoshida et al. (2002) presented an
experimental study on the kinematic hardening response of sheet materials involving a
finite strain compression phase. They bonded several flat specimens together and inserted
the stack of specimens in an anti-buckling device during testing. Other examples of the use
of anti-buckling devices for testing sheet materials under in-plane compression can be
found in Dietrich and Turski (1978), Kuwabara (1995), Yoshida et al (2002), Boger et al
(2005), Cao et al (2009) and Beese and Mohr (2011).
The large strain compression-tension experiments by Yoshida (2002) show that DP
steels feature a Bauschinger effect, transient behavior, permanent softening and work
hardening stagnation. Recall that the Bauschingereffect corresponds to an early yield after
load reversal (Fig. 3-1), transient behavior corresponds to a high hardening rate in the
elasto-plastic transition regime resulting from load reversal (Fig. 3-1); permanentsoftening
prevails when the stress level after loading reversal remains below that for monotonic
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 75
loading for the same equivalent plastic strain (Fig. 3-1); work hardening stagnation causes
a significantly reduced hardening rate after the transient hardening regime (Fig. 3-1).
Experimental True Stress
Strain Relation
600
Theoretical Isotropic
400
Hardening Response
200
0
.&-200
Bauschinger Effect
Transient sofi ening
-400
-600
-800
-0.05
0
.05
0.Transient Hardening
Hardening at large strains
Work Hardening Stagnation
Figure3-1 Comparisonof the true stress strain experimental response with a theoretical isotropic
hardeningresponse after loading reversalfrom uniaxial tension to uniaxial compression at strain
of 0.08 to illustratedifferent hardeningbehaviors: isotropic hardening,permanent softening, work
hardeningstagnation, transientbehavior.
Detailed reviews of kinematic hardening models are found in Chaboche (2008) and
Eggertsen and Mattiasson (2009, 2010, 2011). Prager (1956) type of kinematic hardening,
also referred to as linear kinematic hardening, describes both the Bauschinger effect and
permanent softening. The main shortcoming of this model is the intrinsic coupling of both
effects, i.e. a material exhibiting a Bauschinger effect without any permanent softening
cannot be described with Prager's model. Furthermore, it describes neither transient
behavior nor work hardening stagnation. Also, this type of hardening is unbounded and
results in a persistent and often unrealistic rate of hardening at large strains. The
Armstrong-Frederick (1966) kinematic hardening rule, also referred to as non-linear
kinematic hardeningmodel, describes the Bauschinger effect and transient behavior. The
governing differential equation includes a recall term which activates the so-called
dynamic-recovery. The recall term is co-linear to the back stress tensor and is proportional
- 76 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
to the increment in equivalent plastic strain. As a result, the evolution of the back stress is
no longer linear and unbounded and converges towards a saturation value under monotonic
loading. Two parameters are used, one to control the Bauschinger effect and one for the
transient behavior. However, the Armstrong-Frederick model describes neither permanent
softening nor work hardening stagnation.
For improved approximations, several non-linear kinematics hardening models can be
added with different recall constants characterizing the back stress evolution (Chaboche et
al., 1979; Chaboche and Rousselier, 1983). These models give good predictions in the case
of cyclic loading in the range of small strains, as they are able to describe the Bauschinger
effect with great accuracy. The special case of coupling linear kinematic hardening with
non-linear kinematic hardening provides good predictions in the case of moderate and large
strains as it describes the permanent softening behavior during reverse deformation,
especially with advanced high strength steels free from work hardening stagnation
(Yoshida et al., 2002). However, the permanent softening effect is only represented through
the linear kinematic term. As a result, an increase of the amount of permanent softening in
the model always results in an increase in the strain hardening at large strains.
Mroz (1967) proposed a multi-surface model framework to describe strain hardening.
This idea was developed further by Dafalias and Popov (1976) by making use of a
bounding surface in addition to the yield surface, with the distance between these two
evolving surfaces defining the rate of strain hardening. Chaboche (2008) argues that a
model featuring a combination of linear and non-linear kinematic hardening terms in
addition to isotropic hardening can replicate the performance of a Dafalias-Popov type of
model. However, one advantage of the Dafalias-Popov type of formulations is that the
material response to monotonic loading can be identified independently from its response
to reverse loading. The Dafalias-Popov model has been developed further by Geng and
Wagoner (2000) to account for permanent softening. Yoshida and Uemori (2002) enriched
the model even further and incorporated work-hardening stagnation.
To the best of the authors' knowledge, experimental results on the effect of loading
direction reversal on the strain to fracture are scarce and only found for bulk materials in
the open literature. Bao and Treitler (2004) performed reverse loading experiments on
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 77
notched axisymmetric bar aluminum 2024-T351 specimens with compression followed by
tension all the way to fracture. They observed a substantial increase in ductility due to precompression. Papasidero et al. (2014) made use of a biaxial testing machine to subject
tubular fracture specimens to non-proportional loading. Their results also demonstrated
that pre-compressing aluminum 2024-T351 increases the strain to fracture for subsequent
loading at higher stress triaxialities.
The present paper investigates the effect of loading direction reversal on the strain to
fracture in dual phase steel sheets. Section 2 presents an experimental procedure for the
large strain pre-compression of 1mm thick notched specimens prior to fracture testing. A
finite strain plasticity model is presented in Section 3 combining elements of a nonassociated plasticity model for advanced high strength steels (Mohr et al., 2010) with the
non-linear kinematic hardening models of Chaboche (2008) and a Yoshida-Uemori (2002)
type of work hardening stagnation approach. The plasticity model is then used in Section
4 to simulate all fracture experiments, before characterizing the effect of loading reversal
on the strain at fracture initiation in Section 5. In the latter, a Hosford-Coulomb fracture
initiation model with a non-linear damage accumulation rule is proposed to model the
observed loading path effect.
3.2
Experiments
3.2.1
Materialand specimens
All experiments are performed on specimens extracted from 1.06mm thick DP780
dual-phase steel sheets provided by US Steel. Under uniaxial tension, the material exhibits
an initial yield stress of about 450MPa and an ultimate strength of about 800MPa. The
axial stress-strain response for uniaxial tension is approximately isotropic (since the red,
- 78 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
black and blue curves lie almost on top of each other in Fig. 3-2), while the Lankford ratios
are mildly loading direction-dependent: ro=0.75, r 45=0.77, and r9 o=0.95.
1 00C
80C
0~
0)
(0
U)
60C
L..
U)
40(
-UT0
1~
I-
------ 4U
UT90
----UC
20(
0
0.05
True Strain
0.1
Figure3-2 Uniaxial stress-strainresponse of DP780 steelfor different directions under uniaxial
tension.
Stocky dog-bone shaped specimens (UTC-specimens, Fig. 3-3) are designed for
uniaxial tension and compression testing. A gage section width of 10mm is chosen to allow
for the use of an anti-bucking device with a 5mm wide central window for DIC strain
measurement (Fig. 3-4). As far as the choice of the gage section length (14mm) is
concerned, a compromise is sought between two competing effects: firstly, a large gage
length-to-width ratio is desired to guarantee the validity of the assumption of uniaxial stress
fields; secondly, a short gage section length is preferred to delay out-of-plane buckling
under compressive loading.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 79
50
10
14
Figure 3-3 Specimen geometryfor uniaxialcompression-tensionexperiments.
Figure 3-4 Displacement measurement using DigitalImage Correlation(DIC).
To investigate the plastic material response at very large strains, flat notched specimens
with a 20mm notch radius (NCT-specimens, Fig. 3-5) are employed to subject the material
to a compression-tension loading sequence. Notches guarantee that the localized neck will
form at the specimen center perpendicular to the loading axis. This facilitates the numerical
simulation of the experiments (as compared to conventional uniaxial tension specimens
where the position of the through-thickness neck is not known a priori).
-
80 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
50
Figure3-5 Notched specimen (NCT) for compression-tension fracture testing.
All specimens feature 50mm wide and 10mm long shoulder areas. The specimens are
extracted from sheet metal using water-jet cutting with the specimen axis aligned with the
sheet rolling direction. Note that the edge quality of a water-jet cut is sufficient for the
present experiments since fracture always initiates at the specimen center, i.e. away from
the cut edges.
3.2.2
Experimentalprocedure
A hydraulic testing machine (Instron, Model 8802) is used to perform all experiments.
Custom-made high pressure clamps are employed to attach the specimen to the testing
frame (Figs. 3a and 3b). Unlike conventional wedge grips, the clamps work equally well
under compression and tension. Accurate alignment of the upper and lower specimen grips
is critically important to delay buckling (we recommend a tolerance of less than 0. 1mm for
the parallelism of the top and bottom clamping surfaces). Furthermore, it is important to
tighten the specimen clamps under active force control to avoid any plastic precompression due to the Poisson effect during clamping of the specimen.
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 81-
Figure 3-6 Front view of the experimental set-up for compression-tension testing.
Spacer
Device
High Pressure
Grips
Figure 3- 7 Schematic side view of the specimen with anti-buckling device and high pressure
clamps.
Figure 3a shows a photograph of the assembly of the floating anti-buckling device.
The design is similar to that proposed by Beese and Mohr (2011) except for a change in
type and number of springs and bolts to apply an increased average lateral pressure of about
3MPa. Thin Teflon sheets are placed between the specimen and the device to minimize
friction. Note that the applied lateral stress is very small (about 3MPa) as compared to the
axial stress in the specimen (about 800MPa); its effect on the material response is thus
neglected when processing the experimental results. The extensometer function of the
digital image correlation software VIC2D (Correlated Solutions) is used to determine the
relative vertical displacement of two points positioned on the longitudinal specimen axis
- 82 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
at an initial distance of 12.5mm and 10mm for the uniaxial and notched tension specimens,
respectively (as highlighted by blue dots in Figs. 2b and 3-5). For the notched specimens,
the average axial surface strain is also determined in the area of localized necking using a
DIC extensometer of 2mm length (red dots in Fig. 3-5).
3.2.3
Experimentalresults
The experimental program includes:
"
Tension followed by compression on UTC-specimens for tensile pre-strains of
0.05 and 0.10 (axial engineering strains), as well as
*
compression followed by tension on NCT-specimens for compressive prestrains of -0.026, -0.064, -0.091 and -0.132 (axial engineering strain at
specimen center as measured with the 2mm long virtual surface extensometer).
All experiments are performed at a cross-head velocity of about 1mm/min. Monotonic
tension experiments with and without anti-buckling device yielded identical results. This
partially confirms that the effect of the lateral friction due to the anti-buckling device is
negligible. True compressive strains, as high as -0.2, could be achieved without noticeable
buckling. In Fig. 3-2, the true stress-strain curve for monotonic compression along the
rolling direction (green curve) is shown next to that for tension. The curve is only shown
for strains of up to -0.10 as the effect of barreling makes the assumption of uniaxial stress
fields invalid at larger compressive strains.
A summary of all measured force-displacement curves for experiments performed with
loading direction reversal are shown in Fig. 3-8. For both stocky dogbone specimens (Fig.
3-8) and notched specimens (Fig. 3-9), all curves lie on top of each other during the initial
phase of loading (tension phase for the UTC experiments, compression phase for the NCT
experiments) which demonstrates the repeatability of the experimental procedure.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 83
1000
UTC-0
-
Q_
500 .
UTC-5
UTC-10
(0
0
F-
-500
-1000
-0.1
0.1
0
True Strain
Figure 3-8 Experimentalresults: engineeringstress-straincurves as obtainedfrom uniaxial
tension-compressionexperiments.
10
5
C
-
0
0
-j
-
-5
NCT-3
-NCT-3
-10
-2
-1
-
NCT-6
-
NCT-13
0
1
Displacement [mm]
Figure3-9 Experimental results:force-displacement curves as obtainedfrom notched
compression-tension experiments.
3.3
Combined Chaboche-Yoshida (CCY) plasticity model
A new plasticity model is presented combining elements of the non-associated
plasticity model for advanced high strength steels of Mohr et al. (2010), the non-linear
kinematic hardening models of Chaboche (2008) and a Yoshida-Uemori (2002) type of
- 84 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
work-hardening stagnation approach. The model is embedded into the standard finite strain
framework of the commercial finite element software Abaqus/explicit (Abaqus, 2012).
3.3.1
Yield surface
The center of the yield surface in the deviatoric Cauchy stress space is described
through the back stress tensor X. The tensor 4 is introduced to describe the relative position
of a point in stress space with respect to the yield surface center
4= dev[a] - X .
(3-1)
Note from Fig. 3-2 that the stress-strain curves for uniaxial tension along the rolling,
transverse and diagonal directions lie approximately on top of each other. We therefore use
the isotropic von Mises equivalent stress measure to define the yield surface
f(a,k)=
-k=0,(1)
with
2(2)
Note that the above yield surface corresponds to the isotropic von Mises yield surface if
X =0, whereas it is anisotropic otherwise.
3.3.2
Non-associatedflow rule
Despite the isotropic stress-strain response under uniaxial tension, the measurements of the
Lankford coefficients indicate some in-plane anisotropy in the material. An anisotropic nonassociated flow rule (see Stoughton (2002), Cvitanic et al. (2008), Mohr et al. (2010)) is therefore
chosen to describe the evolution of the plastic strain tensor. It is assumed that the increment in
plastic strains dP is aligned with the derivative of the Hill'48 potential function in the 4 - space,
dfk = dA
with
""
(3)
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 85
l
~
2+G 2
+(1+2G
+G
2){
+2G 2 af2 -2(1+G){11
3
-2(G
+G ){22
33
+G 3 i +3 %+32
(4)
In (4), dA > 0 is the plastic multiplier, while the coefficients G1 2, G22 and G3 3 are directly
linked to the Lankford ratios,
G22
G12 =-1,
1+ro
-
+and
r9o 1+ro
G33 =
1+2r45 rO + rO
ri0
(5)
1+ro
For ease of notation, we rewrite Hill's equivalent stress (5) as quadratic form:
G:
H1 l
(3-7)
with the semi-positive definite matrix G.
3.3.3
Definition of the equivalentplastic strain
The equivalent plastic strain increment is defined as work-conjugate to the von Mises
-
space,
dc, = -
I
4:
.
equivalent stress in 4
(3-8)
Combining Eqs. (4) and (8), the relationship between the equivalent plastic strain and the
plastic multiplier is obtained,
dzp dIm
=
d ./
d1
(3-9)
Note that in the absence of a back stress, the above definition of the equivalent plastic strain
is the same as that proposed in Stoughton (2002), Mohr et al. (2010) and in chapter 2 of
this thesis.
3.3.4
Isotropic hardening
The isotropic hardening law describes the evolution of the deformation resistance k
during plastic loading. Formally, we write
- 86 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
(3-10)
dk = 8H [T, ]dz,, 5
where duP is the increment in the equivalent plastic strain. 0
/3
1 is a multiplier
associated with work hardening stagnation (to be detailed below), and H = H[E,] is the
isotropic hardening modulus. In Chaboche (2008), a Voce law is used for the isotropic
hardening function. A limitation of such an evolution is its saturation at large strains. Here,
we use instead a linear combination of the Voce and Swift hardening laws (Sung et al.,
2010) to parameterize the isotropic hardening function,
- w)Q-XT
H'[z,]10
+ wAn(--O + EP)"n-1
(3-11)
with the Voce parameters {Q, r}, the Swift parameters {A, co, n} , and the weighting factor
0 !w
1. A better control of the rate of isotropic hardening at large strains is achieved
using such a combination.
3.3.5
Non-linear kinematic hardening
The evolution law for the back stress is described through the sum of two non-linear
kinematic hardening rules,
(3-12)
.
dX = da, + d6 2
with the initial conditions a, (t =0) =0, X(t= 0) =X 0 . The corresponding evolution
equations read
(2
C1
G :(
-
- CE dA
(3-13)
Hill
and
d42c/ = AV
f~
2
2C2 G:4
4(}2
(3
aj~2dA
-E2
(3-14)
g Hjll
with the work hardening stagnation multiplier 6, and the material model parameters C
and
Note that the back stress evolution direction is given by the non-associated plastic
flow in the non-linear kinematic hardening law.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 87
The two non-linear kinematic hardening rules serve different purposes.
al
is
introduced to model the Bauschinger effect. According to the Frederick-Armstrong
differential equation, the evolution of the back stress is no longer unbound under
,
monotonic loading; it converges towards a saturation stress instead. The second term, a 2
is introduced to model apparent softening. A linear kinematic hardening term is usually
introduced which results in permanent softening. This has proven as a useful assumption
when modeling the response of metals at moderate strains (Eggertsen and Mattiasson 2011,
Zang et al 2011, Geng and Wagoner 2000, Chun et al 2002). However, according to our
experimental observations, the softening effect appears to fade away for very large strains.
In other words, instead of permanent softening, we introduce transientsoftening through a
non-linear kinematic hardening rule. The key difference between transient softening and
the Bauschinger effect is the strain scale, i.e. the Bauschinger effect fades away rapidly
after a few percent of strain after loading reversal, whereas the strain scale associated with
the transient softening effect is at least one order of magnitude higher. In the model, this is
reflected in the choice of parameters (e.g. Y ~ 1072). By introducingfi in the evolution of
the second backstress term only, the fast recovery from the Bauschinger effect remains
fully active during the work hardening stagnation phase.
3.3.6
Work hardeningstagnation
This part of our model is inspired by the work of Yoshida and Uemori (2002).
Originally developed as a two surface model (Dafalias-Popov framework), we borrow
some ideas from Yoshida and Uemori (2002) to define a constitutive equation for the work
hardening stagnation multiplier
p8.
The activation of work hardening stagnation depends on the loading history. To
separate the effect of work hardening stagnation from the effect of the non-associated
plastic flow, we characterize the loading path in terms of the strain-like path tensor
p =
inds,
0
(3-15)
- 88 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
With n the normal to the yield surface. p is equal to the plastic strain tensor in the case of
associated plastic flow. A sphere of radius r is defined around a central point q in a way
that p lies always inside (Fig. 3-10). Denoting the distance between q and p as 8,
5=
(p-):(p
- q),
(3-16)
we have
8-r 5 0.
(3-17)
The work hardening stagnation multiplier is then defined as
8
r
with 0
p
(3-18)
1. The position and size of the sphere are permitted to change according to
the evolution equations
dq = (1-h)dp
(3-19)
h
dr =-(p - q): dp
(3-20)
S
if 8p=1 (i.e. p is located on the sphere boundary, see Figs. 3-1Ob, 3-1Oe and 3-1Of) and
(p -q): dp >0 (i.e. the loading direction dp points outwards). Note that (3-18) was chosen
such that the consistency condition d8 =0 is readily fulfilled. The case 6 =1 with d,8 = 0
corresponds to loading with no work hardening stagnation (Fig. 3-10), i.e. isotropic and
kinematic hardening are both fully active. 8 = 0 represents the opposite limiting case of
maximum work hardening stagnation. Different from the model proposed by Yoshida and
Uemori (2002), we allow for partial work hardening stagnation, i.e. 0 <B < 1.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 89
#=1
dfl=0
dp
dq
no stagnation
I
p=p
(a)
(b)
dp6<0
("d p
t q161
stagnation
stagnationJ
(d)
p
rddsp
dp
dp
no stagnation
(e)
no stagnation
(f)
Figure 3-10 Illustration of the work hardening stagnation model in the plastic strain space. The
sequence shows (a) the initial values, (b) the evolution during monotonic loading, (c) the point of
loading reversal, (d) the transientstagnationafter loadingreversal, and (e)-(f) the evolution after
stagnation.
- 90 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
3.3.7
Summary of model parameters
The proposed plasticity model includes sixteen material model parameters:
*
Three anisotropy coefficients {G 2,G 22 ,G3 3} defining the non-associated
flow rule
*
Seven isotropic hardening parameters including the initial deformation
resistance ko, the Swift parameters {co, A, n}, the Voce parameters {Q, r},
and the weighting factor w
*
the initial back stress tensor X
0
*
the Bauschinger parameters C, and y
*
the softening parameters
*
for work hardening stagnation parameter h
c2
and
72
It is worth noting that the proposed Combined Chaboche-Yoshida (CCY) model reduces
to a conventional Chaboche model when deactivating the work hardening stagnation option
(h = 0) and using G,2 = -0.5, G2 2 = 1, G 33 = 3, ao = 0.
3.3.8
Thermodynamic constraints
The starting point of our considerations is the free energy imbalance of the form
V :5 G:.
(3-21)
In addition to an elastic part V',, the free energy includes a plastic part V/, associated with
kinematic hardening,
V/ = Ve
+ V/,
(3-22)
which both must be positive, i.e.
V/,
0,
(3-23a)
V/,
0.
(3-23b)
Assuming the elastic strain energy potential
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 91
I
Ve =
2
C : (C - e,)): (F -
e,)
(3-24)
along with the elastic constitutive equation
a =/
= C:(E--e,),
(3-25)
the free energy imbalance may be substituted by the requirement of nonnegative rate of
plastic dissipation:
(3-26)
>~!0 .
d, = (7: , -y)
Next, we assume a plastic free energy of quadratic form
3
,[CtI
IQ21
-
3
47,C,
(3-27)
al :a,
which readily satisfies the non-negative requirement (23b) for y, > 0 and C, > 0. After
application of the back stress evolution equations (13) and (14), the rate of change in the
plastic free energy reads
3Q a2: 5
2/,=C1 a1 :aj+#
:
2C]
a2: a 2i
2C2
(3-28)
Combining Eq. (3-26) and (3-28), yields the rate of plastic dissipation
41
=
_V
2+4) ,-y,=
(1-0)
a2
:G:4
_
Hd/
+
-
( # P} +,+- a :
,dl+-
3A
2C
+#8 3A
2: E2
a]:a]+8
2C
3Z
2C2
a 2 :a,
(3-29)
2C2
The second term is unconditionally nonnegative. For loading outside the work hardening
stagnation regime (/8 = 1), the thermodynamic constraints are readily satisfied even though
a non-associated flow rule is used.
A constraint needs to be imposed on the material model parameters if 83<1 and
a 2 : G: 4 <0. In that case, the non-zero dissipation condition is satisfied if
- 92 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
-CL 2 :G :
(3-30)
2
To satisfy (30), it is sufficient to impose a constraint on the magnitude of the back stress
,
tensor a 2
(2 :G:
(3-31)
<25ui
According to the Frederik-Armstrong law, the evolution of a2 is bound to
2
:G-
2
2
- CG
C2 -C mG
(3-32)
3
with Am
max,G
=
G denoting
the largest eigenvalue of G,
max 3, G,1+ G 2 + G2 2 + V +
4G,22
+2G2G2
+ 2G2 +
G2
-
G2.
(3-33)
The free energy imbalance is thus satisfied if the model parameters respect the constraints
y, >0, CI > 0, and 2 C 2 2mxG
3
3.4
k 0.
(3-34)
Plasticity model identification and validation
The plasticity model parameters are identified in a two-step procedure:
1. A first set of parameters is determined using the experimental data of the UTC
experiments whose domain of validity is limited by in-plane barreling and out-ofplane buckling at large compressive strains.
2.
Subsequently, these parameters are used as seed values for an inverse parameter
identification method based on experimental data for the NCT experiments in the
pre- and post-necking range.
The inverse procedure involves finite element simulations of all experiments on NCT
specimens up to the point of fracture and quantifies the difference between the simulation
and experimental results. An improved second set of parameters is then obtained from the
computational minimization of this difference.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 93
3.4.1
Identificationstep I: determinationof seed parameters
The anisotropy coefficients G12 , G2 2 and G33 of the Hill'48 flow potential are
determined from the measured Lankford coefficients ro, r4 5, r9o,. Application of Eq. (3-6)
yields G 2 =-0.43, G22
=
0.88 and G 33 = 2.59. According to the considerations made in
Beese and Mohr (2011), the initial back stress for the cold-rolled material is assumed to
take the special form
X=XO(e (eD -e 3 0e)
(3-35)
where the 1-direction corresponds to the rolling direction, and the 3-direction to the through
thickness direction.
The initial deformation resistance ko and the initial back stress X0 are determined
from the tension/compression asymmetry of the material response (Fig. 3-2). Denoting the
,
absolute values of the initial yield stresses under tension and compression as Y, and Ye
respectively, we have
k0 = (Y +Yj)
(3-36)
and X 0 = (Y, -Y,).
At 0.2% plastic proof strain, we have Y,
_
450MPa and Y ~ 510MPa, and thus,
ko = 48OMPa and XO = -20MPa.
The hardening parameters a = {A, n, co, Q5 r, C1, Y ,, C 2 , Y 2 , h} are identified through
optimization. Simulations for pure uniaxial tension followed by compression are
performed on a single element. The true stress versus logarithmic strain curve is computed
for each experiment and compared with the corresponding experimental result. The cost
function is expressed as
4
M
F, [a]=
SIM
J
2
F
EA3-7
F:
F~
]J
- 94 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
where the subscript
j
=1,2,..,4 differentiates among the stress-strain curves for different
levels of pre-strain; M
denotes the total of experimental data points used for the
computation of the residual for the experiment j.
The seed model parameters listed in
Table 1 are obtained from minimizing F [a].
3.4.2
3.4.2.1
Identificationstep II: full inverse parameteridentification
Finite element model
A finite element model is built to simulate all NCT experiments. Special attention is
paid to the modeling of the post-necking response. Assuming a symmetric mechanical
system, one eighth of the entire specimen is modeled using first-order solid elements with
reduced integration (element type C3D8R of the Abaqus element library). The mesh
features a total of 34,488 elements with eight elements along the half-thickness of the
specimen. The size of the mesh is chosen such that the predicted equivalent plastic strain
at fracture initiation converged (less than 2% change upon further mesh refinement). The
computed axial displacement is reported for the gage section point that corresponds to the
position of the DIC extensometer in the experiments. All simulations are performed under
displacement control with the displacement prescribed at the top boundary of the specimen
shoulder (of the numerical specimen). The explicit solver of the FE software Abaqus is
used to solve the computational boundary value problem using at least 100,000 time steps
for each simulation run.
3.4.2.2
FEA-basedoptimization of model parameters
The set of seed parameters obtained in step I may not accurately capture the behavior
of the material at large strains. Buckling limits the range of strains after loading reversal
for which the parameters are identified. A strong asymmetry of the material behavior in
tension and compression after loading reversal could also affect the relevance of the set of
parameters: the seed parameters are obtained from tension followed by compression tests,
whereas the fracture experiments feature compression followed by tension. An improved
set of parameters is computed through non-linear unconstrained optimization. Each
iteration includes the following steps:
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 95
1. Definition of the material model input data card for a given set of parameters a.
-
2. Simulation of all five NCT experiments (pre-strain levels 0.0, -0.026, -0.064,
0.091 and -0.132) and extraction of the corresponding force versus DIC
extensometer displacement curves, FFA (u), with the subscript j
=1,2,..,5
differentiating among the results for different pre-strain levels.
3. Evaluation of the global cost function
(3-38)
-LAj
j=1
la,[a] =
with N denoting the number of experiments included in the optimization. For each
experiment, a cost function Aj is defined
A
2
=
(3-39)
3
3
The cost function is a measure of how well the model performs with respect to three criteria:
(1) The first criterion Ai evaluates the relative error in the value of the predicted
maximum load.
max(F )-max(Fj(
max F )(
(2) The second criterion A2 evaluates the relative error in the overall predicted drop of
the load from its maximum value to its value at fracture.
max(F
)- F)
max(F
- (max(F/s )- FA
)-F
EA
f
(3) The third criterion A3 evaluates the relative error in the value of the displacement
at predicted maximum load.
- 96 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
Fx F
F L)
UY
(3-42)
uEYP_
U F,6.F6
This set of criteria has been constructed in an attempt to obtain an accurate prediction of
the macroscopic specimen response in the post necking range.
The optimal set of parameters a, is then determined through the minimization of the
cost function,
a~,, = arg min lu [a].
(3-43)
The minimization is performed using a derivative-free Nelder-Mead algorithm as
implemented in the algorithmic development software Matlab (v.7). The final set of
parameters as obtained after 84 iterations is given below the seed values in Table 1. In the
material chosen for this study, the set of seed parameters already provided satisfying
accuracy, and the optimization of the seed parameters provided only minor improvements.
However, this method was found necessary for other materials featuring early buckling or
pronounced tension-compression asymmetry.
MPa
X0
Pa
A
n
co
Q
T
w
C
Y,
C2
MPa
-
-
MPa
-
-
MPa
-
MPa
-
2
h
-
ko
seed
480.
30.
946.5 0.15
0.023
164.3
22.1
0.5
204.6 51.3
108.9
6.1
0.7
final
512.
30.
997.2 0.19
0.031
143.7 32.2
0.5
235.4 50.8
92.1
5.9
0.6
Chaboche seed
480.
30.
707.2
0.22 0.013
41.8
29.4
0.5
322.7
51.2
497.3
1.1
0.
final
495.
30.
743.9
0.11
0.019
63.2
40.4
0.5
301.2
48.5
453.3
1.0
0.
CCY
Table 3-1 Plasticitymodel parametersfor the Combined Chaboche-Yoshida (CCY) model and the
Chaboche model.
3.4.3
Model verificationfor reverse loading
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 97
The comparisons of the measured and predicted stress-strain curves for uniaxial
tension-compression (UTC) are shown in Fig. 3-11 a. The remainder plots in Fig. 3-11 show
the force-displacement curves extracted from finite element simulations (solid lines) of the
NCT specimens next to the experimental measurements (solid dots). The corresponding
surface strain (axial strain at the specimen center as averaged over a distance of 2mm)
evolutions are shown on the blue secondary axis. The proposed model predicts accurately
the load-displacement relation for all experiments. The magnitude of the maximum load as
well as the corresponding displacement are well captured. The post necking behavior is
well predicted as the load drop corresponds to the experimental measurement for all levels
of pre-compression.
A comparison of the computed and measured surface strain evolutions shows that the
proposed plasticity model is able to provide reasonable estimates of the surface strains in
the area of localized necking for all levels of pre-strain. However, the surface strain history
plots are more sensitive to model inaccuracies. For example, the simulation for loading
reversal after a pre-strain of -0.09 (Fig. 3-11 e) overestimates the surface strain at the instant
of fracture by as much as 14% (0.41 versus 0.36 in the experiment). The predicted forcedisplacement curve on the other hand agrees well with the experiment. With the goal of
identifying the loading paths to fracture for monotonic and reverse loading, the model
performance has only been assessed for these loading conditions in the context of the
present work. The reader is referred to the work by Yoshida and co-workers for a more
comprehensive validation of the main model ingredients.
0.6
1000
8..
0
500
a
0.4
0
0.3-S
LL
0.2
it-500
2
-1000
0
-0.1
0
True Strain
0.1
-0.1
0.5
1
1.5
Displacement [mm]
(b) NCT-0
- 98 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
(a) UTC
- 3
0.4 .5
0.3
0.2 0)
4
4)
0
0
U_
lii
0.6
0.5
2
0.1
0
-0.1
0
Q)
0
U-
-0.2
-4 |
-0.3 0
-0.4 _)
-0.5
-0.6
-6 2
-8-
0.5
0
0.3
0.2
0.1 r
2
0
0
- 0
-0.3
-0. 4 S
-0.5 .9
-8 2
-0.6
-10
-0.7
0.5
1..
1
Displacement [mm]
0.7
0.6
0.5
0.4
0.3
0.2 ~
0.1
0
-0.1
-0.2 0"
-0. 3 UJ
3
10
8
6
4
0.3
*
.
-0.4
-8- 2
10
12
-0.7
-0.8
0
-0.5
0
0.5
Displacement [mm]
(e) NCT-9
1
0
u-
40
0.2 (j)
0.1
0 3
-0. 1 G
2-2
-0.2 C
-0.3-0.4 w
-4
-6
-8
-10
-12
-0.5
-0.6
-0.7
-
-2%
0.6
0.5
0.4 a
.
0
10*
0
U_
0
-0.5
(d) NCT-6
100.
86-4
G
-0.1
-2
-4
-6
1.5
1
Displacement [mm]
(c) NCT-3
0.7
0.6
0.5 :
0.4 'F
8
6
4
-
.-
.
10
8-
-1
-0.5
0
0.5
Displacement [mm]
(f) NCT-13
Figure 3-11 Comparison of CCY model (solid lines) and experiments (soliddots) for (a) uniaxial
tension-compression(UTC), and (b)-() notched compression-tension(NCT); the Chaboche model
predictionsare depicted as dashed lines.
For reference, we also repeated the entire calibration and validation procedure with the
Chaboche model, i.e. for the CCY-model with no work hardening stagnation and associated
plastic flow. The corresponding results are shown as dashed curves in Fig. 3-11. The
Chaboche model also provides good predictions of the macroscopic force-displacement
curves, but it tends to overestimate the strains inside the neck.
Remark. Thickness profiles can also be used for "local" validation. We had prepared
these for specimens extracted from a different batch of sheets that had been tested much
earlier (and hence featured slightly different mechanical properties). We also had calibrated
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 99
both the CCY and Chaboche model for those sheets which resulted in similar agreement
of the simulated and measured force-displacement curves as for the material above.
Selected experimental (solid dots) and simulated thickness profiles (solid lines) are shown
in Fig. 3-12. As for the surface strain measurements, the simulation results agree
reasonably well with the experimental measurements. In particular, good agreement is
observed for the ratio of the strains outside and inside the neck (e.g. 0.55 outside the neck
vs 0.35 inside the neck in the case of monotonic tension). Unfortunately, we could not
repeat the thickness profiles measurements on the current batch of specimens due to the
shortage of material.
- 100 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
E
-I,
0.4
lie
0.3
0.
5
0
-5
Longitudinal Position
(a)
*
E
E
*
'N
.-w--
0'
0. 5
I
I
0.
0.
J9 0.
I
0.:
-5
0
5
Longitudinal Position
(b)
-5
0
5
Longitudinal Position
5
E
E 0.
a.
0.
0.2
(c)
Figure3-12 Thickness profiles at the instant of inset offracture as measuredexperimentally (solid
dots) and extractedfrom numerical simulation (CCY=solid lines, Chaboche=dashedline) after
pre-compression up to a surface strain of (a) 0. (b) 0.08, and (c) 0.12; note that the results are
shown for specimens extractedfrom a different batch of DP780 sheets.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 101
3.4.4
Model verificationfor monotonic multi-axial loading
The stress state sensitivity of the fracture initiation on DP780 steel sheets had been
determined in chapter 2 using
"
a flat 10mm wide specimen with a lateral notch (R=6.67mm) for plane strain
tension (Dunand and Mohr, 2010);
"
a flat 20mm wide specimen with a central hole (R=20mm) for uniaxial tension
(Dunand and Mohr, 2010);
*
a butterfly specimen (Dunand and Mohr, 2011) with reduced thickness gage
section for shear;
Here, all experiments are repeated on the specimens extracted from the new batch of sheets
closely following the experimental procedures outlined in chapter 2. The monotonic
experiments are simulated using the CCY plasticity model as calibrated based on the
reverse-loading experiments. The comparison of the measured force-displacement curves
(dotted curve) with the simulation results (solid curves) shown in Fig. 3-13 confirms the
model's ability to provide good predictions of the overall specimen responses. For
reference, we also performed these simulations with the Chaboche model which yields less
accurate predictions. In particular, the force decrease for notched tension is predicted too
early. As a result, the strain at the specimen center at the instant of onset of fracture is
overestimated.
0.7
10
0.2
*
8.
.
4
*
0
0.5
.3
Upn4
W
0.2
0
0. 3
0.1
2
0
0
0.5
1
0
Displacement [mm]
(a)
- 102 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
0.8
12
0.7
0
)
-
14
0.6 A=
110
p
0.5 E
8
04
.
0
6.
A
0.3W
A'
4
0.2
2
0.1
0
C
0
1
0.5
Displacement [mm]
(b)
18
16
14
Z 12
810
08
6
4
2
0
1
2
Displacement [mm]
(c)
Figure 3-13 Results from monotonic experiments: (a) notched tension (NT6), (b) tension with a
central hole (CH), and (c) butterfly specimen (SH); solid dots = experiments, solid lines = CCY
model, dashed lines = Chaboche model; the contour plots show the distributionof the equivalent
plastic strain at the instant offracture initiation.
3.5
Effect of loading reversal on ductile fracture initiation
The effect of pre-compression is investigated at the specimen level to gain insight into
the different ductility of material points that undergo monotonic stretching as compared to
those that are subject to compression-tension cycles when a sheet is drawn over a radius or
when hinge lines propagate during the folding of box structures. The uniform state of the
sheet after production (rolling, etc.) is therefore considered as the reference state of zero
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 103
strain and the strain introduced thereafter up to the point of failure is reported as fracture
strain.
3.5.1
Characterizationof the stress state
The plasticity model is anisotropic due to the Hill'48 flow potential and the non-zero
back stress. However, for the sake of simplicity and due to the lack of experimental data
for different material orientations, it is assumed that the fracture response is not affected
by the orientation of the stress tensor with respect to the material coordinate system. The
stress state is thus characterized through the stress triaxiality and the Lode angle parameter
(which are both isotropic measures). The stress triaxiality 1/ is proportional to the ratio of
the first invariant of the Cauchy stress tensor, I,, and the second invariant of the deviatoric
stress tensor,
=
CV
J2!
-m=
with
=
and 3UV
3
37
=
jdev[a]: dev[a
(3-44)
3
The dimensionless Lode angle parameter, W, measures the ratio of the second and third
invariants of the deviatoric stress tensor, 1 2 and J 3 . Its mathematical definition reads
-
0
2
=1--arccos
3
J3
j
(3-45)
with
J 3 := det(dev[a4.
3.5.2
(3-46)
Effect ofpre-compressionon resultsfor notched tension
Figure 3-14a shows the computed equivalent plastic strain distribution inside the neck
of the notched tensile specimens at the instant of fracture initiation, while Fig. 3-14b
depicts the corresponding thickness profiles. The red dots on the specimen surface in Fig.
3-14a highlight the position of the
2mm
DIC surface extensometer.
In a first
approximation, both the average axial strain and the thickness reduction at the instant of
fracture are unaffected by the amount of pre-compression. The axial (engineering) strain
- 104 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
measured by the DIC surface extensometer is about 0.35 and the thickness reduction is
about 30%. However, the shape of the neck and the strain distribution inside the neck are
strongly influenced by the amount of pre-compression which results in an increase of the
local equivalent plastic strain to fracture as a function of the pre-strain (Fig. 3-14a), ranging
from
Z. = 0.57 for monotonic tensile loading to
i=
0.77
compression to a local equivalent plastic strain of 0.13.
0.
0.8
ENCT-0
IN-(a)
0.55
E
0.5
0>
= 0.45
0
0.4
-NCT-0
-NCT-3
- 0.35
-NCT-6
-NCT-9
-NCT-13
0.3
-5
5
0
Longitudinal Position
(b)
for fracture after pre-
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 105
Figure 3-14 (a) Equiv. plastic strain distribution in the longitudinalspecimen cross-section, and
(b) thickness distributionat the instant offracture initiation.
In the initial compressive phase of loading, the thickness of the specimen is increased.
At the same time, the isotropic hardening capacity in the tensile loading phase decreases
as a function of the pre-strain; as a result, the localization of deformation inside the neck is
more pronounced which causes a steeper thickness gradient in the vicinity of smallest
cross-section (compare the slopes of the thickness profiles in Fig. 3-14b). A more detailed
description of the evolution of the thickness and state variables at the location of fracture
initiation in the NCT-13 experiment is given in Fig. 3-15. In addition to the forcedisplacement curve (Fig. 3-15a)
*
Fig. 3-15b shows the evolutions of the deformation resistance k (solid line),
the back stress tensor component {cE1
tensor component {
2
III
}
(dashed line), and the back stress
(dotted line);
*
Fig. 3-15c shows the evolution of the von Mises equivalent stress;
*
Fig. 3-15d shows the evolution of the longitudinal thickness profile;
*
Fig. 3-15e shows the evolution of the equivalent plastic strain as a function of
the stress triaxiality (so-called "loading path to fracture");
Stagnation (,Q < 1) takes place between points (D and
(,8 = 0) at point
®. At
@
and reaches its maximum
this point, both isotropic hardening and transient softening are
inactive, while the kinematic hardening term associated with the Bauschinger effect is the
only active hardening mechanism. At the same time, through-thickness necking initiates
during the phase of hardening stagnation (compare the thickness profiles
3-15d). Full hardening resumes beyond point
®
and
®
in Fig.
(Fig. 3-15c). It is well known in the
literature that the strain hardening capacity affects the geometry of the neck (e.g. Pardoen
et al, 2004). Here, the strain hardening capacity is affected in several ways by the precompression: the total isotropic hardening capacity decreases as a function of the pre-strain.
However, this effect is competing with the hardening capacity in the Bauschinger
transition, modelled by the kinematic hardening. During the stagnation phase, the rate of
- 106 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
strain hardening is very low (between point
and
®
in Fig. 3-15c), leading to a more
localized evolution of the neck. The consequence on the evolution of triaxiality is observed
in Fig. 3-15e, where there is a rapid increase of the triaxiality as a function of the plastic
strain between point
® and ®. Following the stagnation phase (after point ®),there is
a short phase of significant increase of the rate of hardening (Fig. 3-15c), leading to a more
diffused evolution of the neck. In Fig. 3-15e, there is temporarily almost no evolution of
the triaxiality with respect to the plastic strain after point
3
10
~
@
8
6
4
2
0
-2
@.
0.6
0.5
0.4
F
C/
.
0.1
0
.
0.3
0.2
-
-0.1
-6
-0.2
-0.3-
-8
-10
-12
-0.6
-0.7
-
-4 0
0
-0.5
-1
-0.5
0
0.5
Displacement [mm]
(a)
23
800
600
aCO
'2
400
200
*
I
I
I
I
I
I
I
*
I
I
I
*
I
*
I
*
I
II
I
I
II
I
I
SI
I
I
SI,-------I
I
I
I
I
I
I
I
I
I
I
I
I
I
Ill
I
Ill
I
*
*
*
*
(a 1 j-~~
I
I
I
I
I
I
I
I
I
I
I
S
I
4-
(U
0
1200
k
4
1000
a-
800
.
600
400
4
{~2}I1
~
200
-200
0.4
0.6
0.2
Equivalent Plastic Strain
(b)
0.8
WO
0.2
0.4
0.6
Equivalent Plastic Strain
(c)
0.8
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 107
0.55
0.8
3
E
0.6
0.5
CCY
(DU
4E
C- 0.45
d 0.4
0.4H
0.2
0.35
-5
R66 -0.33
5
0
Longitudinal Position
0.33
0
triaxiality
0.66
0.99
(e)
(d)
Figure 3-15 Detailed analysis of the NCT-13 results: (a) force-displacement curve and local
surface strain; Evolutions of (b) the CCY hardening terms: isotropic hardeningresistance (solid
line), axial component of the back stress tensors (dashedline) and (dotted line), (c) the von Mises
stress, (d) the thickness profiles, and (e) the stress triaxiality; the dashed line in (e) depicts the
Chaboche model prediction; the labels
9, (, @and (@indicate the point of loadreversal, onset
of stagnation (also maximum load in tension), end of stagnation, and onset offracture.
3.5.3
Hosford-Coulombfracture initiationmodel
Assuming that the onset of fracture is imminent with the onset of localization at the
microscale, the Hosford-Coulomb model has been developed based on the results from 3D
unit cell computations for proportional loading (Dunand and Mohr, 2014). In particular, it
is postulated in chapter 2 that ductile fracture initiates after proportional loading when the
linear combination of the Hosford equivalent stress and the normal stress acting on the
plane of maximum shear exceeds a critical value,
5HF
+ c(
with
5-HF
I a ,H-
a
1 a+(7
+ 7, aa(-8
+..
b0
(3-47)
- 108 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
After transformation of the Hosford-Coulomb criterion into mixed strain-stress state space,
the strain to fracture for proportional loading reads,
"p[1,0]
= b(1
{i(f,
+c){
f)"
+(f2 - f)"
+ c(2q + f,+
+(f, - f)"
(3-49)
with the Lode angle parameter dependent trigonometric functions
cos(1
f1[#]=
-
) f 2[]=
cos[
(3+ )
and f 3 []= -- cos[(i+
(3-50)
with the transformation parameter p = 0.1 (Roth and Mohr, 2014), and the fracture model
parameters {a,b,c}. Note that the Hosford exponent a controls the effect of the Lode angle
parameter, while the friction coefficient c primarily controls the effect of the stress
triaxiality on the strain to fracture. The model parameter b is a multiplier controlling the
overall magnitude of the strain to fracture. It is defined such that it is equal to the strain to
fracture for uniaxial tension (which is the same as that for equi-biaxial tension).
To predict the onset of fracture after non-proportional and non-monotonic loading, the
above criterion is embedded into a damage indicator model framework. Let D e [0,1]
denote a scalar damage indicator, with the initial value D = 0 for the undeformed material,
and
D
=
1 for the deformed material at the instant of fracture initiation. The evolution of
the damage indicator is then related to the evolution of the equivalent plastic strain using a
stress state dependent non-linear damage accumulation rule (Papasidero et al, 2014),
dD=m _P f - "-
(3-51)
Irrespective of the choice of the damage accumulation exponent m > 0, the condition
D = 1 is fully equivalent to the direct application of (33) for proportional loading. In the
case of non-proportional loading, values of m < 1 put more weight on the effect of the stress
state at the early stage of loading, whereas values of m > 1 emphasize the effect of the stress
state right before fracture initiation. In the case of m
=1,
the so-called linear damage
accumulation rule is retrieved (e.g. Bai and Wierzbicki, 2010).
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 109
Model identificationand verification
3.5.4
For each experiment performed, we extract the loading path to fracture, E, = E,[rq,
]
at the location within the specimen, where the highest equivalent plastic strain is achieved
corresponding numerical simulation. The solid lines in Fig. 3-16 depict the loading
in the
paths to fracture in the strain versus stress triaxiality plane for eight different experiments.
The end of each path corresponds to the instant of fracture initiation.
S
1.1
T-13,
SH
.0.8
CH
0.6
0
NCT-6
NCT-3
NCT
NT6
o
U
.
0.6NCT-0
S0.4
)0.4
W0.2
W 0.2
-0.33
01
0
0.33
Triaxiality
0.66
0.99
S
-NCT-
.13
0.
-1
CH
NCT-3
NCT-25
NT6
0.5
0
-0.5
Loide Angle Parameter
1
Figure 3-16 Loadingpaths to fracture as extractedfrom finite element simulations of allfracture
experiments up to the instant offracture initiation(endpoint ofsolid lines); the Hosford-Coulomb
fracture initiation model predictionsare shown as soliddots.
The four fracture initiation model parameters {a, b,c,m} are identified based on the
loading paths using a gradient free inverse identification algorithm (Nelder-Mead
minimization in Matlab). The results for the shear (SH), central hole tension (CH) and the
monotonically loaded NCT-specimen (NCT-0) are included in the calibration procedure
for {a,b,c} to cover a wide range of stress states; in addition, the reverse loading
experiment NCT-9 is included in the data basis to identify the damage accumulation
exponent
m . After launching the identification procedure with the seed values
{1.5,0.7,0.01,0.8}, the "optimal" parameters a = 1.65, b = 0.62, c = 0.05 and m = 0.45
are obtained after 76 iterations. Figure 1 b shows a 3D plot of the identified Hosford-
- 110 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
Coulomb criterion
7'=fJ[r,]
for proportional loading. The characteristic strains to
fracture of pure shear, uniaxial tension and plane strain tension are 0.81, 0.64 and 0.51.
f
1.5
0.5
-0.33
0
0
-0.5
0
0.5
0.66
0.33
)7
Figure 3-17 Strain to fracturefor proportionalloading as afunction of the Lode angle parameter
and the stress triaxiality.
The resulting model predictions of the instants of fracture initiation are shown as solid
dots in Fig. 3-16. The solids dots lie exactly on top of the ends of the loading paths for the
calibration experiments (black lines) which indicates that the model has sufficient
mathematical flexibility to be fitted to the experimental data. The blue dots predict the
instants of fracture for the four experiments that have not been included in the calibration
procedure. As for the calibration experiments, the blue dots (model predictions) lie
approximately on top of the ends of the blue solid lines (hybrid experimental-numerical
data) which is seen as a partial validation of the proposed phenomenological model.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 111
1.2
1
---
0.8
E 0.6
0.4
0.2
0
0
0.8
0.6
04
0.2
Equivalent Plastic Strain
1
Figure 3-18 evolution of the damage indicatorat the location offracture initiationin compressiontension experiments.
Figure 9a allows for a more detailed comparison of the model predictions (solid dots)
with corresponding experimental results (star symbols), revealing a maximum relative
error of about 5% for the NCT- 13 experiment. The evolution of the damage indicator for
the compression-tension experiments is shown in Fig. 3-18. The curves all show an abrupt
increase in the damage accumulation rate dD/ dis at the instant of loading reversal. This
is due to the jump in stress triaxiality (and Lode parameter) from -0.39 (-0.8) to +0.39
(+0.8). The corresponding strains to fracture for proportional loading at these stress states
are 1.43 and 0.60, respectively. According to Eq. (3-52), this jump implies an increase in
the rate of damage accumulation, i.e. at a given equivalent plastic strain, the damage
accumulation is much lower under compression than under tension.
-
112 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
0.850.8
0.75
o1
0.5E
0
.0.6-
0
0 0.55
0
o
0.5
Experiment
HC
0 HC no comp.
* HC NL
0
0.03 0.06 0.09 0.12
Compressive pre-strain
0.15
Figure 3-19 Strain to fracture after compression-tension loading of NCT specimens as afunction
of the equivalentplastic strain at the point of loading direction reversal; the star symbols present
the hybrid experimental-numericalresults, the solid dots care model predictions.
3.5.5
Discussionof the effect ofpre-strainon ductilefracture
The basic mechanism responsible for the apparent ductility increase is that void like
defects (which trigger the shear localization at the microscale) do not nucleate or grow
under compression. Hence the main tendency of an increase of the strain to fracture as a
function of the pre-compression strain. However, even after computing the net fracture
strain (i.e. subtracting the pre-compression strain from the final fracture strain, Table 2 and
Fig. 3-20), we observe an increase in ductility due to pre-compression. This second order
effect is attributed to the local thickening of the specimen in pre-compression phase,
thereby delaying the necking related stress triaxiality increases after loading reversal.
-
Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets - 113
NCT-0
NCT-3
NCT-6
NCT9
NTC-13
0.
0.03
0.06
0.09
0.13
Fracture strain
0.56
0.60
0.73
0.75
0.77
Net fracture strain
0.56
0.66
0.67
0.66
0.64
Pre-compression strain
Table 3-2 Net fracture strain (equivalentplastic strain at fracture minus compressive pre-strain)
as afunction of the compressive pre-strain.
0.750.7
0.65
a)
4
0
0.6
C.)
0.55
LL
0.5-
z 0.45.
0.40 35*0
0
0
o
o
Experiment
HC
0 HC no comp.
*
HC NL
0.03 0.06 0.09 0.12
Compressive pre-strain
0.15
Figure 3-20 Netfracture strain after compression-tension loading ofNCT specimens as afunction
of the equivalentplastic strain at the point of loading direction reversal; the star symbols present
the hybrid experimental-numericalresults, the solid dots care model predictions
From a phenomenological point of view, it is worth noting that the present results
suggest that the stress state at the beginning of the loading history has an important effect
on the final strain to fracture. This becomes apparent when comparing the results obtained
with the proposed nonlinear damage accumulation rule (m=0.45) with those obtained using
a linear damage accumulation rule (m=1). A slight increase of the ductility in terms of
equivalent plastic strain at fracture is predicted using the linear rule (open square dots in
Fig. 3-20), but it is well below the prediction of the non-linear rule (star symbols in Fig. 3-
- 114 - Chapter 3: Compression-Tension to Fracture of Dual Phase Steel Sheets
20). An alternative modeling approach using the linear rule would be to consider the
material after pre-strain as a new virgin material with zero initial value of the damage
indicator. Even though the predictions with this approach (open diamond symbols in Fig.
3-20) lie above those obtained with the conventional linear rule, there is still a significant
gap between the model predictions and the experimental results.
3.6
Summary
An experimental procedure is developed in order to perform compression of sheet material
and delay buckling. A uniaxial specimen geometry is used to identify the plastic response
of the material under reverse loading. A geometry with notches is used to characterize the
hardening at high strains after loading reversal, and to measure the ductility after loading
reversal. A Combined Chaboche-Yoshida (CCY) model is proposed to account for the
observed Bauschinger effect, transient softening and work hardening stagnation. The
parameters are identified through an inverse calibration in order to predict the post necking
behavior of the notch tests. The model is carefully validated using local surface strain
measurements and thickness profiles.
The ductility of the material in terms of equivalent strain at fracture increases with
compressive pre-strain. A Hosford-Coulomb damage indicator model with a non-linear
damage accumulation rule is calibrated and validated based on the experimental data. The
model predictions agree well with all experimental results for the DP780 steel with a
maximum relative error of 5% in the strains to fracture. It is worth noting that the same
phenomenological
damage accumulation rule provided an accurate description of
proportional and non-proportional experiments on aluminum 2024-T351 (Papasidero et al.,
2014).
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 115
Chapter 4: Critical Hardening Rate Model for
Predicting Path Dependent Ductile Fracture
Based on the Combined Chaboche Yoshida model for reverse loading, a new plasticity model is
developed so that parameters obtainedfrom monotonic tests can be directly used, and so that the
parametersfor reverse loading are independently identified. An experimental method to perform
reverse shear on sheet material is introduced. Two specimen geometries areproposed: one with a
uniform gage section with thickness reduction is used to identify the parametersof the plasticity
model after shear loading reversal; another specimen geometry, optimized to concentrate strains
in the center of the gage section, is used to characterizethe effect of shear reversal on the ductility
of the material. Data of Compressionfollowedby tension up tofractureon notch specimens is also
considered.
Based on the assumption that ductile fracture is the imminent consequence of the localization of
deformations in a narrow band, it is proposedto predictfractureinitiationwith a criticalhardening
rate. The model is an equivalent of the Hosford-Coulombfracture criterion in stress space for
proportionalloading. The criticalhardeningrate model is successfully calibratedfora wide range
of stressstates in the case ofmonotonic loading. The model is validatedon thefracture experiments
for reverse loading: compressionfollowed by tension and shearreversal.
- 116 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
4.1
Introduction
Since the pioneering work of Gurson (1977), porous plasticity models have received
considerable attention over the past three decades because of their sound micromechanical
basis and ability to predict fracture in many applications. Inspired by the early work of
McClintock (1968) and Rive and Tracey (1969), Gurson-type of models are developed to
provide a mathematical description of the nucleation, growth and coalescence of voids in
solids. It is undisputed that these mechanisms are highly relevant at high stress triaxialities
such as encountered in front of crack tips (Needleman and Tvergaard, 1987). In crack-free
sheet materials, the stress state is usually close to plane stress and hence the void growth
driving stress triaxiality does not exceed the theoretical value of 2/3. Even inside a localized
neck where three-dimensional stress states develop, the stress triaxiality seldom exceeds
values of 0.8. Consequently, there is limited void growth in a statistically homogeneous
sense in sheet specimens. This conclusion is also supported by experimental observations
of Ghahremaninezhad and Ravi-Chandar (2012) from micrographs taken at different stages
during tension experiments on aluminum alloy 6061 -T6 and ... Morgeneyer et al. (2008).
At the macroscopic level, there is also growing evidence that the stress-strain response
of sheet metal can be predicted with reasonable accuracy up to the point of fracture
initiation using non-porous plasticity models. The decrease in the force level that is
observed in the post-necking range can usually be described without introducing damage
into the material model. However, as shown by Dunand and Mohr (2010), Song et al.
(2010), Tardif and Kyriakides (2012) and in Chapter 2, a careful identification of the large
strain hardening response of non-porous models through inverse procedures is required.
Despite the physically sound formulation of Gurson models, it is very difficult to find
experimental evidence that justifies their application to sheet metal as far as the description
of the elasto-plastic material response is concerned. It is reemphasized that this statement
is made with regard to the plasticity of sheet metal only.
An ad-hoc approach to predicting ductile fracture with porous plasticity models is to
assume that fracture initiates when the computed porosity reaches a critical value. A more
physical approach would be to assume that the porous plasticity model provides an accurate
description of the effect of porosity on the material's load carrying capacity which implies
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 117
that ductile fracture is predicted "naturally". In other words, the solution of a boundary
value problem will feature zones of plastic localization (dilational shear bands) that will
eventually cause the loss of load carrying capacity of the structure at hand (e.g. Needleman
and Tvergaard, 1984). The study of Besson et al. (2003) of slant fracture nicely illustrates
this approach. Such simulation models can be supplemented with coalescence criteria (see
review by Benzerga and Leblond, 2010) to account for localization events at the mesoscale.
Note that the latter may even occur before macroscopic localization (Tekoglu et al., 2014).
Recent experimental evidence regarding ductile fracture at low stress triaxialities
(Barsoum and Faleskog, 2007, Mohr and Henn, 2007, Dunand and Mohr, 2011 a) is not in
good agreement with the trends predicted by conventional Gurson models. The qualitative
differences are mostly due to the fact that conventional Gurson models do not predict shear
localization at low stress triaxialities (at reasonable magnitudes of strain). So-called shearmodified Gurson models have thus been developed to capture the localization at low stress
triaxialities. A recent example is the work by Nahshon and Hutchinson (2008) who added
a shear term to the void volume evolution law of the GTN model (Tvergaard and
Needleman,
1984) and demonstrated the importance of this modification in their
predictions of shear localization. Danas and Ponte Castaneda (2012) used non-linear
homogenization to come up with a porous plasticity model that accounts for void shape
changes (that are characteristic for shear loading). Their analysis also shows the loss of
ellipticity at low stress triaxialities.
Given the limited benefits of Gurson type of models as far as predicting the elastoplastic response of sheet metal is concerned, the combination of non-porous plasticity
models with damage indicator models provides an attractive framework for predicting
fracture in industrial practice. Different from porous plasticity and coalescence models,
damage indicator models often have no physical basis and are at most physics-inspired.
The damage indicator is a dimensionless scalar variable that evolves as a function of the
stress state and plastic deformation,
dD=
dp_
"[ 77,9 ]
(4-6)
- 118 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
It is initially zero while fracture is assumed to occur as it reaches a defined critical value.
The heart of these models is the weighting function
T[q, 0], which provides the strain to
fracture for proportional loading as a function of the stress triaxiality q and the Lode angle
parameter 0 . Bao and Wierzbicki (2004) provide a comprehensive overview on different
stress-state dependent damage indicator models including weighting functions based on
the works of McClintock (1968), Rice and Tracey (1969), LeRoy et al. (1981), Cockcroft
and Latham (1968), Oh et al. (1979), Brozzo et al. (1972), and Clift et al. (1990). More
recent representatives of this class of models are the modified Mohr-Coulomb model by
Bai and Wierzbicki (2010) and the micro-mechanically motivated Hosford-Coulomb
model by Mohr and Marcadet (2015). Even though the latter has been derived from a
localization criterion for proportional loading, it gives satisfactory results for both
proportional and non-proportional loading paths (Bai (2008), Papasidero et al. (2015), and
Chapter 3). The main shortcoming of damage indicator models is the lack of physical
arguments justifying their validity for non-proportional loading paths. Even though the
variable D is often called damage and Eq. (4-1) is referred to as damage accumulation rule,
it is emphasized that D has no direct physical meaning (unlike the damage variable used in
continuum damage mechanics). Instead, it may be more appropriate to view the damage
indicator framework as a heuristic mathematical model for predicting path dependent
fracture initiation.
The main objective of the present paper is to provide a mechanism-inspired model for
predicting ductile fracture initiation under proportional and non-proportional loading. An
important byproduct of this work is an advanced plasticity model which accounts for the
direction dependent Lankford ratios, the Bauschinger effect, work hardening stagnation
and quasi-permanent softening. The proposed model is successfully validated using
experimental data for two advanced high strength steels for proportional monotonic
experiments, compression-tension experiments and reverse shear experiments.
4.2
Plasticity model
In view of predicting the large deformation and fracture response for non-proportional
loading paths, a finite strain plasticity model formulation is presented that accounts for (i)
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 119
loading direction dependent Lankford ratios, (ii) the early yield after load reversal
(Bauschinger effect), (iii) the high hardening rate in the elasto-plastic transition regime
resulting from load reversal (transient hardening), (iv) permanent softening, and (v) work
hardening stagnation. As discussed in the review papers by Chaboche (2008) and Eggertsen
and Mattiasson (2009, 2010, 2011), combinations of linear and non-linear hardening rules
can account for the Bauschinger effect, transient hardening and permanent softening, while
further model enrichments are necessary to account for hardening stagnation (e.g. Yoshida
and Uemori, 2002). To account for all five effects (i)-(v), we proposed in Chapter 3 to
combine the plasticity models of Mohr et al. (2010) with the non-linear hardening models
of Chaboche (2008) and Yoshida-Uemori (2002) type of hardening stagnation.
In the sequel, the model of Chapter 3 is reformulated to simplify the associated material
model parameter identification procedure. In particular, the hardening laws are formulated
such that that the model parameters describing the material's response to monotonic
loading do not need to be readjusted when calibrating the parameters that account for
reverse loading effects.
4.2.1
Yield function andflow rule
To define the pressure-independent yield surface, we introduce the tensor 4 as a
measure of the difference between the deviatoric Cauchy stress and a deviatoric back-stress
tensor X,
= dev(a)-
X.
(4-2)
Applying the von Mises equivalent stress definition,
-
3
, =(4-3)
the yield surface is written as
f = 4p -k,. = 0,
(4-4)
- 120 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
with kis denoting the isotropic deformation resistance. Following the recommendation
of Stoughton (2002), Cvitanic et al. (2008) and Mohr et al. (2010), a non-associated flow
rule with a Hill'48 flow potential function is employed,
=d
4G
(4-s5)
5
with the positive definite fourth order tensor G, here shown in the engineering notation
as a matrix
0
0
0
G4 4
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
3
0
0
+ G12
1+2G] 2 + G2 2
0
0
0
0
G12
12 )
-(G
22
-(G
+G1 2
)
-(1+G
G22
+G2)
22
)
-(1
(4-6)
.
and the plastic multiplier dA
The von Mises definition is adopted to define the equivalent plastic strain, i.e.
dT, =
4.2.2
ddc' :9A
(4-7)
Hardeningevolutions
The hardening laws are chosen based on the Combined Chaboche-Yoshida (CCY)
model proposed in Chapter 3. An attempt is made to simplify the model parameter
identification by introducing a deformation resistance B that describes the material
hardening response for monotonic uniaxial tension along the rolling direction. For this
special case, the yield condition is written as
(-I = ko +kkin = B
(4-8)
with k,., denoting the isotropic deformation resistance as introduced in Eq. (4-4), and
denoting the back stress.
kkn
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 121
In most plasticity models, established analytical forms (Holomon, Swift, Voce, power
law, exponential function, etc) are used to parameterize the isotropic hardening law
Ek,[
,] .(4-9)
kan =
The special feature of the current model is that a parametric form (Swift-Voce) is used to
describe B as a function of the equivalent plastic strain,
(4-70)
+(1 - w)O +Q(1 - e-bp
B= wA(o + V
with the Swift parameters {A,co,n}, the Voce parameters {YO,Q,b},
factor 0
w
and the weighting
1. The repartition of the effective hardening response into isotropic and
kinematic hardening will then be described through additional constitutive equations.
For general 3D settings, kso,
kkin
and B will serve as internal variables of the
constitutive model. For notational convenience, we also introduce the sum of kis and
kkif
as dependent variable,
k = k, +kl
(4-81)
The initial configuration of the material shall then be characterized through the initial
conditions
@ ,=0:
B=k =ko =wAOn+(I-w)Y%
and
kkn
(4-92)
=0.
For arbitrary three-dimensional loading, the stress B defines a bounding limit fork,
k ! B,
(4-103)
while the evolution of k is expressed through the differential equation
dk = SB+y,(B - k)d., j
(4-114)
where S e [0,1] is an additional internal variable associated with hardening stagnation.
For loading paths without any hardening stagnation (e.g. monotonic uniaxial tension), we
have S
=
1 at all times, and consequently dk = dB and k = B. In the case where work
hardening stagnation occurs, the strict inequality k < B holds true. The recovery term
- 122 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
y,,(B-k)dE, allows the variable k to converge towards B whenever the material
deformed outside the hardening stagnation regime (S = 1).
The repartition between isotropic and kinematic hardening is then prescribed through
the equation
dki=
if kkfl<lks
0
dk
.kki
-k
(4-15)
(1+P
where we introduced the dependent model parameter p; due to thermodynamic
considerations, it is related to the anisotropy in the plastic flow and is defined as a function
of the maximum eigenvalue A.
of the matrix G,
3
(4-16)
2A
Initially, kis = Y
and kk,, = 0. This means that the apparent strain hardening under
uniaxial tension (evolution of B) is entirely stored into the kinematic
hardening
contribution, until B = 2YO. It has been found that for low cycle fatigue (Dafalias and
Popov, Yoshida), it is not necessary to introduce an isotropic hardening for the yield
surface. When B > 2Y, the apparent strain hardening under uniaxial tension is equally split
into an isotropic and a kinematic hardening contribution,
in order to verify the
thermodynamic constraints (see section 4.2.5). The deformation resistance k,,0 enters
directly into the definition of the yield surface (see Eq. (4-4)), while the evolution of
imposes a constraint on the evolution of the back stress tensors.
4.2.3
Evolution of the back stress tensor
The back stress tensor is decomposed into two terms,
X=C +
The e'VVolUtorul for+the ack-stress
4Lnso
p.(4-17)
U is defindU thuugl tle Uifferential equation
kkf
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 123
dE + Sy,(k, dP -IdkA).
d= - 2
(4-18)
The proposed evolution rule is a combination of terms inspired by the classic Prager
(1952) and Armstrong-Frederick (1966) formulations. Inspired by White et al (1990), the
traditional coefficients of these evolution rules are replaced by instantaneous moduli and
bounding radii that are allowed to change in time. Different from Prager's linear kinematic
hardening rule which makes use of a constant modulus, the hardening modulus dk, / dE,
of the Prager term (first term of the right hand side) is bounded by the evolution of the
variable
kk,1
. Formally, this non-linearity is introduced through the constraint
dkg = min C dkkn
3
ds,'
' dA
(4-19)
which typically becomes active at large strains and for large values of C,,. Similarly,
different from Armstrong-Frederick's kinematic hardening rule which makes use of a
constant radius, the radius of the bounding surface to the backstress is controlled by the
variable
k, . Note that the Armstrong-Frederick recovery effect on the back-stress
evolution (second term on the right hand side of Eq. 4-18) is interrupted during work
hardening stagnation. This feature represents the transient softening effect of chapter 3.
Analogously to the evolution equations of the back stress Pj, we define the hardening
law for a,
d
= Z{dcdp
d a
3 dE,
_
+ ?4jkadEP - cdA
/ 3
(4-120)
with the model parameter y, and the constraint
dk, = dk,, - dk,6.
(4-131)
As discussed in chapter 3, the evolution a is not affected by hardening stagnation to
model the Bauschinger effect.
- 124 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
4.2.4
Work HardeningStagnation
As for the CCY model of chapter 3, the constitutive equations for the evolution rule of
S are inspired by the work of Yoshida and Uemori (2002). Work hardening stagnation is
activated as a function of the loading history. Firstly, a strain-like measure of the loading
path o is introduced,
do =-
2
33
4
dA.
(4-142)
The distance of i to a point q is then defined as
-
0) =
2
_(co-q):(co- q)
3
(4-153)
0.
(4-164)
and limited to
o-r
When C
=or
and
(w-q): do >0, r and q are
dq =
(1 -
dr = h
-
updated as follows:
h) - dco
(4-175)
do)
(4-186)
We can now define the internal variable S:
S((to,q, r) =enf(4-27)
if (o - q): do < 0
then 0
The internal variable S corresponds to the ratio of the distance of O to q when &- increases.
It is set to zero when it decreases. This means that the hardening is deactivated in case of
reverse loading and progressively reactivated at large strains after reversal.
4.2.5
Thermodynamic constraints
The starting point of our considerations is the free energy imbalance of the form
(4-28)
The free energy is limited to an elastic part V/,
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 125
(4-29)
which must be positive, i.e.
(4-190)
,
V/, ' 0
Assuming the quadratic elastic strain energy potential:
VIe =
2
(C
: ( -E,)):
(r - EP)
(4-201)
along with the elastic constitutive equation:
= C:(E -EP)
(4-212)
the free energy imbalance may be substituted by the requirement of nonnegative rate of
plastic dissipation,
d, =
y
(4-223)
Combining equations (4-1), (4-12) and (4-31), we obtain the rate of plastic dissipation
:
- c(+):G
Hill
+
d, =(a+P+) : P, =
(4-234)
0
The first term is unconditionally nonnegative.
A constraint needs to be imposed on the material model parameters when (a + P): G : 4 <0
. In that case, the non-zero dissipation condition is satisfied if
(4-245)
(a + P): G : 41 _-! 2 j
It is thus sufficient to impose a constraint on the magnitude of the total back stress tensor,
V(c +P): G : (ct+ P) <
Hjjl
(4-256)
According to the evolution laws of a and P, the evolution of a + P is bound to
( +p) G :(.
+
: 0.
3
(4-37)
- 126 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
With A,
the largest eigenvalue of the matrix G
Amax,( =max
,G44 ,l+G
2 +G22 ++4G
2
+2GG22 +2G
+G2
-G
22
(438)
And hence the additional thermodynamic parameter constraint
2
-
3
k
ki A max,G
-
iso
kin
-
iso
(4 39)
The condition (4-39) is verified according to equations (4-11) and (4-15) controlling
the evolution on kiso, and kkin. Note: The condition (4-39) means that the sum of the backstresses is bounded by the current isotropic yield surface. When the back-stresses reach the
isotropic yield surface, it is proposed to distribute the total hardening between the isotropic
radius and the evolution of the back-stress so that (4-39) remains satisfied.
4.2.6
Illustrationfor uniaxial loading
For uniaxial loading with an isotropic flow rule, the yield condition becomes:
|a - a -,8|-
k,, = 0.
(4-40)
= dA -
(4-41)
The flow rule simplifies to
de
Equations (9) to (14) remain unchanged. The distribution of the hardening between its isotropic
and kinematic contribution reads
dkiso =
0
if
kki <
dk
i
kkn =
k,,o
(4-42)
2
We can now re-write the evolution of the kinematic back-stress tensors
dc9 = dk, + Sy,(k, - I:id,
With
(4-43)
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 127
dk, = min(dAC,,dkkin
(4-44)
And finally
da = dka +a(ka - a)de,
(4-45)
Critical hardening rate fracture initiation model
4.3
4.3.1
Mechanism-basedmodeling
There is a consensus in the literature that the localization of deformation within a
narrow band is a main precursor to ductile fracture (eg. Danas and Ponte Castaneda (2012),
Nahson and Hutchinson (2008), Pardoen and Hutchinson (2000), Tekoglu et al (2012),
Rousselier and Quilici (2015), Auliffe and Waisman (2015)). Assuming that the initiation
of ductile fracture, i.e. the formation of macroscopic cracks in metals is imminent with the
onset of localization, the onset of ductile fracture can be predicted through an infinite bandtype of localization analysis (Rice, 1976). For a material obeying the incremental stressstrain relationship
d-= L:dc ,(4-46)
Rice (1976) showed that the condition for localization in a planar band reads
det[Linkn, ]= 0
(4-47)
with n denoting the unit normal vector to the localization band. Rice (1976) has also shown
that the above bifurcation condition describes the loss of ellipticity of the governing field
equation. It is worth noting that Rice's criterion remains valid irrespective of the loading
history. The loading history effect on the onset of shear localization is entirely described
by the plasticity model which provides the evolution of the elasto-plastic tangent matrix L
(fourth order tensor) and the stress tensor a (which enters into the corresponding finite
strain formulation, see Mear and Hutchinson (1985)). In engineering practice, the above
approach is seldom used due to the high computational costs associated with solving
- 128 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
equation (39) and its incompatibility with simple non-porous plasticity models (when
prediction localization at low stress triaxialities).
4.3.2
Phenomenologicalmodelingfor proportionalloading
As an alternative to bifurcation analysis, engineers often use fracture initiation criteria
that provide the equivalent plastic strain to fracture as a function of the stress state,
6P' =
(4-48)
]
'r,
The existence of a fracture envelope can be justified for proportional loading in stress
space. It was shown in chapter 2 that any envelope in stress space can be transformed into
the form of (3) for materials featuring isotropic hardening only. For example, the HosfordCoulomb model in principalstress space {-, u1 1
,.
-7
}
reads
with the Hosford (1972) measure of the stress
7HF
=
{
I
-
0
7H
)a
+ (a,, -
CIII
)' + (a,
(4-50)
- 0-111
and the model parameters {a, b, c}. Using coordinate transformations, the same criterion
may be expressed in terms of the stress triaxiality, Lode parameter and Mises equivalent
stress. In the modified Haigh-Westergaardspace {,7, 0, J}, we have,
-
b
-
(4-51)
.
(f-)" +(f2
f)" +(f-
f)}
+c(2r7+ f, + f 3
)
0= O 1 [ri, 0]=
with the Lode angle parameter dependent functions
-
2
S[,,_
fl[o]=-COs -(
3
6
,
-]2
0) ,[0]=-Cos
3
17
2
-(3+0) , A1[0]=_--Cos -(1+0)
16
13
[6
(4-52)
_
For a Levy-von Mises material with isotropic hardening, a third representation of the
Hosford-Coulomb criterion in the mixed stress-strainspace {q, 0, ,} is readily obtained
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 129
when using the inverse isotropic hardening law,
,
to substitute the von Mises
=
equivalent stress by the equivalent plastic strain,
fpr
=k
(4-53)
[ [ 7,O ].
In the latest coordinate frame transformation, we made use of the bijectivity of the
hardening law b = k[Z,]
The bijectivity of the hardening law allows us not only to substitute the equivalent von
Mises stress through the equivalent plastic strain, but we can even express the fracture
criterion in terms of a critical hardening rate, A forth representation of the HosfordCoulomb model in the mixed hardeningrate & stress state space {r,0,dU/ds,} reads
\pr
S =$
k'
,
,
(4-54)
|
All four representations of the Hosford-Coulomb criterion have been visualized for plane
stress conditions in Fig. 4-1 for a power law material
k[E,]=A(EO +-6
(4-55)
n.
with the Swift parameters A = 1100, E, = 0.36 and n
parameters
a =1.5, b=1000MPa and c
=0.1.
=
0.2, and the Hosford-Coulomb
It is reemphasized that all four
representations are fully equivalent and predict the same instant of fracture initiation for a
given proportional loading path in stress space.
- 130 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
120
100 0
80 0
a.
C/)
0
0
W)
U)
60 0
0-
0
40
20 0
0
433
First In-Plane Stress
0.33
Triaxiality
0.66
(b)
(a)
1.2
.C
1
5150
0.8
t 2490
0.6
1340
790
(U
02
ui
0.4
-4Y.33
-
.....
i490
320
220
0
0.33
Triaxiality
0.66
(c)
-0.33
0
0.33
Triaxiality
0.66
(d)
Figure 4-1 Representationof the Hosford-Coulomb criterionfor apower law materialwith plane
stress condition in thefollowing spaces (a) first and second in-plane stress components (b) HaighWestergaard (c) mixed stress-strain (d) mixed hardeningrate-stress. The initial Von Mises yield
envelope (solidblack line), the subsequent Von Mises iso-contours (black dotted line) and HosfordCoulomb fracture locus (solidblue line) are shown.
4.3.3
Phenomenologicalmodelingfor non-proportionalloading
A discussed in the introduction, we seek for a mechanism-based alternative to the
heuristic damage indicator modeling framework to predict ductile fracture initiation for
proportional and non-proportional loading paths. Aside from the lack of physical
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 131
arguments
supporting the particular
mathematical
form of the damage indicator
framework, we also found a strong counterexample: experiments suggest an increase in the
total strain to fracture after shear reversal (see section 4.4.2.2). This is in contradiction with
the prediction of damage indicator models which are insensitive to the effect of loading
direction reversal for pure shear.
Inspired by the fact that Rice's (1976) localization criterion in terms of the tangent
modulus L naturally incorporates the effect of the loading history, we propose a
phenomenological derivative of Rice's model by postulating that ductile fracture initiates
when the hardening rate d6 /dE reaches a critical value for a given stress state {q, 0}.
= g[d5,:]-
(
(4-56)
Note that different from (43), we omit the superscript 'pr', i.e. criterion (12) is a priori
proposed to predict ductile fracture for both proportional and non-proportional loading
paths. We also note that dE defines the von Mises equivalent strain increment of the total
strain tensor,
dE:d.
(4-57)
di=
3
-
Similarly to Rice's localization criterion, the dependence on loading history of ductile
fracture initiation is inherited from the plasticity model.
For materials exhibiting isotropic hardening only, the proposed model would reduce
to a simple stress based criterion. This simple form for modeling the effect of loading
history has been advocated by Stoughton and Yoon (2011). However, for materials
exhibiting non-linear kinematic hardening (such as Bauschinger effect or transient
softening), the above model will immediately predict a history effect on ductile fracture.
This plasticity model effect is shown schematically in Fig. 2 and will be elaborated
further in the subsequent sections dealing with real materials. The stress-strain relation is
shown for monotonic and reverse loading after 0.45 plastic strain. A constant stress state
is assumed even after loading reversal (eg. shear reversal). Fracture under monotonic
- 132 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
loading is assumed to initiate at an equivalent plastic strain of 1. The figure shows the effect
of plasticity on the strain to fracture depending on the choice of critical quantity at fracture:
equivalent plastic strain, corresponding to equation (4-53), in blue, Von Mises stress,
corresponding to equation (4-51), in green, and hardening rate, corresponding to equation
(4-58), in red.
As far as the parametric form of g[q,0] is concerned, we suggest using (48) as
evaluated for a power law hardening model with n
=
0.1,
n
+(f
f)a
+ c(2r7+ f, + f 3
)
f2)" +(f2 - f)
(f -
g[77,]= Htrr
(4-58)
The proposed critical hardening rate model therefore features three model parameters: the
critical hardening modulus H,,, for uniaxial tension, the friction coefficient c, and the
.
Hosford exponent a
120C
Monotonic
1 000
0~
(0
00
C')
Stress
G)
Strain
1~
(0
'I)
600
Hardening Rate
C')
40C
C
0
20C
r
0
1
1.5
0.5
Equivalent Plastic Strain
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 133
Figure4-2 Effect ofplasticity on the prediction offracture under reverse loadingfor the HosfordCoulomb model as afunction of the choice of criticalquantity atfracture.
4.3.4
Extendedformulation
An important underlying assumption in the proposed ductile fracture model is that the
fracture initiation is imminent with the onset of localization. There are a few exceptions
where this assumption does not hold true. Consider for example a material in which LUders'
bands form (e.g. Hall, 1970). In that case, localization bands occur temporarily, but cease
rapidly as the hardening rate picks up. For such a material, the hardening rate right after
initial yield is close to zero or even negative; the proposed model would predict fracture at
this stage of loading even though significantly higher strains are attained in reality. A
similar situation is encountered when PLC bands form in aluminum alloys (e.g. Benallal
et al., 2006) or during work hardening stagnation after loading reversal in DP steels.
The common feature of these special cases is that the localization is stabilized rapidly
due to the material's remaining hardening potential. Localization corresponds to a
catastrophic (i.e. fracture initiation), if there is no more hardening stabilization possible
under continued monotonic loading along the current loading path. The failure criterion
therefore must state that fracture occurs at an instant tf if the localization criterion Eq. (58)
is satisfied, and provided that the hardening modulus did not increase if the loading
continued along the same strain path. Denoting the hardening modulus H at an instant tf
after loading along a specific strain path c[t] from the initial configuration (t =0) to the
current configuration (t = t1 ) as
[tf]]:= 6
(4-59)
the fracture criterion is formally rewritten as
max H[t
t !tf
1
] + 4t](t -t 1 )] g)[t 1], 0 (t]
(4-60)
- 134 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
It is noted that the loading path assumed in (54) is only possible post-localization scenario.
As for propagating instabilities (e.g. Kyriakides, 2001), the formation of a crack after the
onset of localization also depends on the kinematic restrictions imposed by the surrounding
material. In other words, different from the onset of localization, fracture initiation is
expected to depend on the non-local conditions.
Comment on the model sensitivity
4.3.5
At first sight, the formulation of a fracture initiation model in terms of the hardening
rate to be more sensitive to experimental inaccuracies than models that are directly
formulated in terms of strains. It is therefore worth emphasizing that the conversion from
strains to hardening rates is only done computationally, i.e. the identification of the model
parameters {a, H
is based on the measured strains to fracture for different stress
, c}
states. Possible experimental uncertainties in the measurement of stress-strain curve slopes
therefore do not enter the model parameter identification process.
The same applies to inaccuracies in the calibrated plasticity model. Even if thee
plasticity model provides only a poor approximation of the material's large deformation
response, these plasticity model inaccuracies will not affect the model predictions of the
strain to fracture for proportional loading. However, the model predictions for nonproportional loading depend on the slope accuracy of the plasticity model. In other words,
in order to benefit from the model's ability of providing accurate estimates of the strains to
fracture after complex loading histories, it is necessary to use an adequate plasticity model.
For instance, the increase in ductility after reverse shear loading is only possible if a
plasticity model with non-linear kinematic hardening or hardening stagnation is employed.
4.4
Experiments
The proposed plasticity and fracture
models will be validated based on the
experimental data for two different materials: 1.0mm thick DP780 steel sheets provided by
T
UO
Cee,
O)L~
1 A
aWIU.yrrm
Ice
ii
9r steel sheets provided by Arceior'viittai.
!or1 1-al
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 135
4.4.1
Overview on experimentalprocedures
In an attempt to obtain a comprehensive characterization of the plastic and fracture
response of advanced high strength steels the following experiment types have been
performed:
(a) Monotonic uniaxial tension (UT): tensile specimens with a 10mm wide gage section
(Fig. la) are positioned into a universal testing machine and loaded at a constant
cross-head velocity of 2mm/min. In addition to the axial strain, the width strain is
measured using planar Digital Image Correlation (DIC);
(b) Monotonic notched tension (NT-x): the minimum gage section width of all notched
specimens was 10mm (Figs. lb and Ic), while the notch radii were either 20mm (NT20) or 6.67mm (NT-6). The notched specimens were loaded at constant cross-head
velocity of 2mm/min all the
way to fracture.
The relative
axial shoulder
displacements were measured using 17 mm and 15 mm long DIC extensometers for
the NT-6
and NT-20
specimens, respectively.
In addition,
a local relative
displacement has been measured using a 2mm long virtual extensometer at the
specimen center;
(c) Monotonic central hole tension (CH): the employed tensile specimens were 20mm
wide and featured a 4mm diameter hole at the gage section center; as for the NT
specimens, a global relative displacement has been measured using a 40 mm long
axial virtual extensometer;
(d) Monotonic punch experiments (PU): Using a 127 mm diameter hemispherical punch,
a disc specimen is loaded all the way to fracture at a punch velocity of 5 mm/min.
The surface strains are measured in the apex region of the punched specimen using
stereo DIC.
(e) Uniaxial tension-compression experiments (UTC): the specimens are first loaded up
to an axial strain of 0.1 or 0.2 under uniaxial tension, before loading direction
reversal. A low friction anti-buckling device is employed to apply compression stain
increment of up to 0.15 before buckling failure. The axial strains are monitored in
these experiments using a 12 mm long virtual extensometer;
-
136 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
(f) Notched compression-tension experiments (CTR): notched specimens with the same
gage section dimensions as the NT-20 specimens are first subject to compressive
strains of up to -0.13, before removing the anti-buckling device and loading the
specimen all the way to fracture under tension; the global and local axial DIC
extensometer lengths are 12 mm and 2 mm, respectively.
(g) Reverse shear plasticity experiments (MO): tangential loading is applied onto a
rectangular specimen (Fig. 4-4), while keeping the vertical force zero (Mohr and
Oswald, 2008). The strains are measured within the 5 mm wide and 50 mm long gage
section using planar DIC. To avoid any plastic deformation in the gripping areas and
to reduce the required tangential forces, the gage section thickness is reduced to
0.5mm using conventional milling. Fracture initiates near the gage section edges
which limits the validity of the experiments to equivalent strain levels of about 0.3
for monotonic loading; the experiment are performed in a tangential displacementcontrol mode at a constant velocity of 0.5 mm/min;
(h) Reverse shear fracture experiments (BUT): in close analogy with the shear plasticity
experiments, tangential loads are applied to a butterfly-shaped specimen (Fig 4-6,
Dunand and Mohr (2011)). Different from the MO specimen, fracture initiates near
the specimen center where the stress state is close to pure shear. The relative
tangential and normal displacement of the specimen shoulders is monitored using
planar DIC. All experiments are performed a constant tangential velocity of about 0.5
mm/min, while keeping the normal force zero.
4.4.2
Details on experimentalprocedures
The experimental procedures for experiment types (a)-(f) have been described in detail
in Dunand and Mohr (2010), Chapter 2 and Chapter 3. We therefore limit our detailed
description to the experimental procedures for the reverse shear experiments which have
not been reported previously.
All shear tests are performed on a custom-made Instron dual actuator system (Fig. 43). The boundary conditions and the alignment are well controlled by the rigid high
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 137
pressure grips. The vertical actuator controls the vertical load applied to the top of the
specimen. The horizontal actuator is under displacement control.
verllcal
actuat
clamp
specimen
horizota
ackuator
horizontW
b
ba
elldcll
0
Figure 4-3 Schematic of the dual actuator system
4.4.2.1
Reverse shear plasticity experiments
The high width to height ratio of the MO specimen (Fig. 4-4) ensures that the shear
stress field is approximately within the specimen gage section; it can therefore be
calculated based on the force measurements at the specimen boundaries. After machining
the specimens, a random speckle patter is applied onto the gage section surface. A digital
camera monitored the central part of the gage section at resolution of 20 pm/pixel and at a
frame rate of 1Hz. The tangential displacement is applied at a constant horizontal actuator
speed of 0.5 mm/min. The experiments are aborted as soon as small cracks become visible
by eye in the gage section corners.
- 138 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
VF
T
gag. section
t
-
h*2t -h
trH
4-
4
rim7
Figure4-4 Geometry of the Mohr-Oswaldspecimen.
The shear strain is determined from the relative tangential and normal displacements
Au and Av of two points positioned on the vertical symmetry axis of the specimen at an
initial distance of L = 2mm. Following the developments of Mohr and Oswald (2008), the
logarithmic axial strain is
(4-61)
, =In I+-)
(L
and the logarithmic shear strain
(4-62)
dt
A
L+ Av
L
=
2
The work-conjugate shear component of the stress is evaluated using the approximation
F
-
H
A0
(4-63)
exp [e
with the initial cross section area A0 ,and the tangential force F, acting onto the specimen.
Recall that the vertical force is kept zero throughout the experiments and hence U-j =0.
Denoting the elastic shear modulus as G , we compute the plastic strain components
2e = 2c, -q,/ /G
and c
~ c,,
(4-64)
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 139
and the equivalent plastic strain
=
fJ(('p)2
(4-65)
.p )2d
Figure 4-5 shows an example equivalent stress versus equivalent plastic strain curve for
the DP590 material for an MO experiment with loading reversal at a strain of about 0.09.
In the same figure, we also show the stress-strain curve from a tension-compression
experiment (UTC specimen) with loading reversal at the same equivalent plastic strain
(solid dots). The remarkable agreement of both curves is seen as a partial validation of the
reverse shear testing technique. Furthermore, it is noted that significantly larger strains
could be achieved with the MO specimen (before the formation of corner cracks) than in a
UTC specimen which fails due to buckling.
1000
800-
CO
$
C
>
600400
200
-
0
Tension/Compression
Mohr-Oswald
0.2
0.3
0.1
Equivalent Plastic Strain
0.4
Figure4-5 Comparaisonof the stress-strainresponse of DP780 under loading reversal after 10%
equivalentplastic strainforshear reversal (solidline) and tensionfollowed by compression (dotted
.
line)
4.4.2.2
Reverse shear fracture experiments
As mentioned above, the MO specimen fails because of strain concentrations at the
gage section corners and is thus not suitable for measuring the strain to fracture for pure
- 140 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
shear. Instead, the butterfly (BF) specimen (Fig. 4) introduced by Dunand and Mohr (2011)
is used. It features slightly curved specimen shoulders which generate significantly larger
strains at the specimen center than at the free gage section boundaries. As a result, fracture
initiates near the gage section center where pure shear conditions prevail'.
T
FF
~h1A*S
~*C
rc
-
w
1A
Figure 4-6 Geometry of the Dunand-Mohrbutterfly specimen.
A velocity of 0.5mm/min is applied to the horizontal actuator, while keeping the
vertical force zero. The horizontal and vertical relative displacements Au and Av of the
specimen shoulders is measured at an acquisition frequency of 1Hz using a 12 mm long
DIC extensometer which is initially aligned with the vertical axis of symmetry of the BF
specimen. Due to the heterogeneity of the mechanical fields, the equivalent plastic strain
evolution at the specimen center is extracted from a finite element simulation of the
experiment. Following the modeling guidelines of Mohr and Dunand (2011), we made use
of a solid element mesh with four first-order elements along the half-thickness of the
specimen gage section (i.e. an element size of about 0.06 mm). The instant of loading
reversal in reverse loading experiment is then also determined after computing the strain
evolution for a monotonic experiment.
1This statement holds true for most engineering materials tested so far. However, it is important to verify the
validity of this assumption for each experiment performed.
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 141
Overview on experimentsperformed
4.4.3
The same battery of monotonic experiments has been completed for both materials:
*
Uniaxial tension (UT) for three different loading directions
*
Central hole and Notched tension (CH, NT20 and NT6) along the rolling
direction
"
Punch (PU) experiment
*
Mohr-Oswald (MO) and butterfly (BF) shear with the rolling direction parallel
to the vertical axis
For the DP590 steel, the effect of loading direction reversal has been characterized using
"
Reverse Mohr-Oswald (MO) shear plasticity experiments with loading reversal
at an equivalent plastic strain of 0.1 and 0.2;
"
Reverse butterfly (BF) shear fracture experiments with loading reversal at an
equivalent plastic strain of 0.25 and 0.50;
For the DP780 steel, a more extensive experimental program has been performed to
investigate the effect of loading direction reversal:
*
Uniaxial tension-compression
(UTC) experiments with loading direction
reversal at an equivalent plastic strain of 0.05 and 0.1;
*
Reverse Mohr-Oswald (MO) shear plasticity with loading direction reversal at
0.1 (see Fig. 4-6);
"
Notched compression-tension (CTR)
experiments with loading direction
reversal at an equivalent plastic strain of 0.05, 0.10, 0.15 and 0.20;
Table 4-1 also provides a comprehensive summary on all experiments performed. The main
purpose of the performed experiments is to serve as basis for material model identification
and model validation. We therefore omit a separate discussion of the experimental
observations per se in the present section. Instead, the experimental results are introduced
in the next section on the model application and validation.
- 142 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
DP590
DP780
NT6
X
X
NT20
X
X
CH
X
X
PU
X
X
UTC
X
NCT
X
MO (monotonic)
X
X
MO (reverse)
X
X
BUT (monotonic)
X
X
BUT (reverse)
X
Table 4-1 Summary of all experiments
Application and validation
4.5
4.5.1
Plasticitymodel parameteridentification
The proposed plasticity model features the following material parameters:
- G12, G22, G44 (anisotropic flow potential parameters)
- co, ko, A, n,
Q, b, w (Swift-Voce
parameters)
- Cp, yp, y,, h (kinematic hardening and stagnation parameters)
As mentioned above, the great advantage of the model is that these parameters can be
identified sequentially.
5.1.1 Monotonic Loading
Firstly, the parameters G 12 , G22 and G 44 of the G matrix are determined from the
Lankford coefficients.
G -I/
r
+ro
G
,G2
-"
=-I
r0 1+rr9
r~o I+
ro
,+
a'--4
G
Cn
+2r
=
r0
1 + r
0
(A
(4-6
k..
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 143
Then, the Swift parameters co, A, n and the Voce parameters ko, Q, b are independently
determined to fit the data from the true stress-strain response of the material.
ro [-]
r4s [-]
ro [-]
DP590
0.98
0.84
1.13
DP780
0.78
0.96
0.77
Table 4-2 Lankford Ratios.
Finally, the parameter w is determined through the hybrid experimental and numerical
method of chapter 2. It is optimized so that the predicted load displacement response for
the NT20 specimen matches the experimental result. During this optimization, the
parameters for reverse loading are set to dummy values (e.g. Cp=0, yp=100, y =100, h=0),
since they have negligible effect on the simulation.
ko
Q
E
A
[MPa]
[MPa]
[-]
[MPa]
DP590
345.9
335.8
24.9
DP780
614.0
270.0
32.2
0b
n
E
[-]
[-]
[-]
1031.0
0.0013
0.2
0.73
1170.0
3.1 10-'
0.11
0.79
Table 4-3 Swift- Voce Law parameters.
4.5.2
Reverse Loading
The proposed model presents the advantage that the effects of the parameters for
reverse loading are quite independent, and do not affect the response of the model under
proportional loading conditions. The parameter ya is related to the typical recovery time of
the transient behavior. The parameter h is related to the typical length of the work hardening
stagnation phase. The parameter Cp is related to the amount of transient softening. The
parameter yp is related to the rate of strain hardening after the end of the work hardening
stagnation phase. The effect of each parameter is illustrated in Fig. 4-6.
- 144 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
800
.
.
800
700
700
Al
600
0-600
g 500
$ 500
(n 400
m 400
-T) 300
. 300
0 200
0 200
100
100
0
C
0
C
0.05
0.1
0.15
0.2
Equivalent Plastic Strain
0.05
0.1
0.15
0.2
Equivalent Plastic Strain
(a)
700-
.
.
.
80 0
-
800
(b)
70 0
1.4
600-
0-60 0.
g 500
2 50 04)
m 400
n 40 0
.T 300
.T 30 0
0 200
0 20 0
100
100
0
0
0.05
0.1
0.15 0.2
Equivalent Plastic Strain
(c)
C
''0
_
_
_
_
_
_
0.05
0.1
0.15
0.2
Equivalent Plastic Strain
(d)
Figure 4-7 Effect of model parametersfor reverse loading: (a) parameterya (b) parameterCp (c)
parameterh (d) parametery8.
The Von Mises stress-equivalent plastic strain relation is numerically evaluated using
a single element simulation. The boundary conditions are such that the element is under
simple shear and loading reversal is applied at the corresponding experimental equivalent
plastic strain as obtained with the Mohr-Oswald tests. The predicted stress-strain response
is then compared to the experimental response obtained with the Mohr-Oswald tests. The
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 145
parameters x
=
{C,,3 ,yh}
are optimized in order to minimize the area under the
experimental and numerically predicted curves. The error function is expressed as
M
2
1 [x]
,
where the subscript j
levels of pre-strain;
=
=
0
SIM
E0'
2
(4-67)
,,I=- [: ,Z]U O
1,2 differentiates among the stress-strain curves for different
M, denotes the total of experimental data points used for the
computation of the residual for the experiment j . The model parameters listed in Table 4
are obtained from minimizing F, 1[x].
C6
Y2
YE
h
[MPa]
[-]
[-1
[-]
DP590
51.3
74.8
2.1
0.49
DP780
231.6
65.2
6.0
0.64
Table 4-4 Reverse Loading HardeningParameters
The prediction of the stress-strain response under monotonic and shear reversal loading
is compared to the experimental measurement in Fig. 4-7.
- 146 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
900
-
800
700
'u
500
0 400
300
2500
100
U)
0.2
0.4
0.4
00.2
Equivalent Plastic Strain
0.6
0.6
(a)
1200[
a- 1000CO
C')
a)
1~
800
4-'
C/)
C')
a)
C')
600
400
0
200
-
n
0
_
_
0.2
0.1
Equivalent Plastic Strain
0.3
(b)
Figure 4-8 Comparison of the Von Mises stress to equivalentplastic strainafter loadingreversal
for experimentaldata (black dotted line) and the predictionofthe model after calibration(redsolid
line) for (a) Shear reversalon DP590 (b) Tension compression on DP780.
In addition, the predicted load-displacement response for shear reversal on the clothoid
butterfly specimen is compared to the experimental result in Fig. 4-8.
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 147
20
20
15
10
15
5
0
-j
10
0
-j
5
0
-5
-10
-15
0
-20
0
3
2
1
Displacement (mm)
-1
4
0
1
2
Displacement (mm)
3
(b)
(a)
20
15
10
Z5
0
0
-J
-5
-10
-15
-20
-2
0
2
Displacement (mm)
(c)
Figure 4-9 Comparison of the experimental (dotted line) and predicted (solid line) load
displacement response for (a) monotonic shear (b) shear reversal after 25% equivalent plastic
strain (c) shear reversal after 50% equivalentplastic strain.
- 148 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
0.5
8
0.4 C
0.3
4
0.1
0
LL
0.05
0
0.1
0.3
0.2
0.1
0
0
w
-0.2
-0.4
-0.5
-8
0
0.15
0.05
0.1
0.15
Displacement (mm]
(a)
(b)
-_
4
0.2
0.1
0
0
0
-0.3
C
4
0.3
0.2
0
U-
0
-4
-0.1
-0.2
-0.3
-8
-0.4
L
-0.4
-8
0.5
0.4
0.1 U)
a)
-1 ID
-0.2
-4
8
.
6
0.5
0.4
0.3
8
-0.5
-0.6
-0.05
5
-0.3
Displacement [mm}
z
0
C
-0.1
-4
0
0
4
a)
C
0.2 T
0.5
0.4
-0.5
-0.6
_ 10
0.05
-0.05
Displacement [mm]
0
0.05
0.1
Displacement [mm]
.
0
U-
8
0.1
(d)
(c)
0
0.4
0.3
0.2 -E
0.1 0
0
8
4
0
2)
0
0-
-4
-0.1
-0.2
-8
-0.3 5
-0.4
-12
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
Displacement [mm]
(e)
Figure 4-10 Comparisonofthe experimental measurements (dottedline) and the model prediction
(solid line) for load-displacement(black) and local surface strain (blue)for (a) CTRO (b) CTR_05
(c) CTR_10 (d) CTR_15 (e) CTR_20.
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 149
4.5.3
Fracturemodel parameteridentification
The critical hardening rate model set of parameters X = {a, H, c} is calibrated for
each material through an inverse analysis. The minimization problem is:
x = arg min(maX
Hi,
[]
--
)HR[iI)
{a,H,c}
With
Hybri(i)
the equivalent plastic strain at the onset of experimental fracture
determined using the hybrid method, and FiR (i) the equivalent plastic strain at fracture
predicted by the critical hardening rate for the given loading paths and parameters {a,b,c}.
For each material DP590 and DP780, the data from the monotonic experiments: NT6,
NT20, CH, SH, PU is used. The obtained parameters are summarized in Table 4. The
performance of the calibrated fracture criterion is shown in Fig. 4-10. The loading paths
identified by the hybrid experimental and numerical method are shown in terms of
equivalent plastic strain versus triaxiality and are interrupted at the experimentally
identified point of fracture initiation. The dots represent the predicted initiation of
localization. The point lies on the numerically determined loading path and its distance to
the tip of the loading path (solid line) reveals the error in the prediction. For all materials,
the maximum error across all tests stays within 3% of error.
H
a
c
DP590
146.3
1.89
0.005
DP780
143.1
1.77
0.022
Table 4-5 FractureParameters.
- 150 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
1.6
1.4
1.2
CO
CH
1
0.8
.
PU
SH
NT20
0.6
w
NT6
0.4
0.2
0
0
06
0.66
0.33
Triaxiality
0.99
(a)
0.8
0.6
0
4-0
C
w
0.4
0.2
0
0
0.66
0.33
Triaxiality
0.99
(b)
Figure 4-11 Calibrationofthe criticalHardeningrate H-C model using the monotonic datafor (a)
DP590 (b) DP780.
4.5.4
Validationand discussion
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 151
0 CTR_20
CTR 15
0
CTR_10
.0
CTR 05
CTR_0
0.8
C
-..
4:.'
O.65
0.
*0.4
c0.2
C 'F
0.66
0.33
Triaxiality
(a)
1.6
1
RS_50
A
RS_25
~1.2
PU
CH
(U
_
SH
0.8
NT20
0.6
NT6
B- 0.4
w
0.2
0
0
0.66
0.33
Triaxiality
0.99
(b)
Figure 4-12 Predictionof the onset offracture (dots) using the criticalhardeningrate model and
loading paths to fracture (solid lines) for (a) compression followed by tension experiments on
DP780 (b) Shear reversal experiments on DP590.
- 152 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
The performance of the model is evaluated in the case of reverse loading conditions.
Fracture initiation is predicted for the compression-tension tests on DP780 and the reverse
shear tests on DP590 using the parameters calibrated in section 4.3. The performance of
the model is illustrated in Fig 11.
It is quite remarkable that the points lie almost on the tip of the loading path for two
levels of pre strain in the case of compression followed by tension of DP780. Remember
that no additional parameter was introduced to add flexibility of the fracture model for
complex loading. Only variations in the identification of parameters for complex loading
can affect the prediction of the model, as well as the data from the hybrid method. Small
inaccuracies in the data for the two other tests may also be attributed to experimental
uncertainties. The equivalent plastic strain at fracture is only slightly overestimated after
shear reversal of DP590. Uncertainties in the data could be related to inaccuracies in the
identification of the plastic behavior and the formation of a relatively narrow shear band in
the simulation. Overall, the critical hardening rate fracture criterion predicts the strain at
fracture within 8% accuracy. Keeping in mind all sources of errors, it seems that a critical
hardening rate identified phenomenologically for proportional loading shows some
relevance to predict ductile fracture after loading reversal.
4.6
Summary
An experimental method to identify ductile fracture after shear reversal is presented.
In addition, data of ductile fracture after compression followed by tension on notched
specimens is considered. A plasticity model is introduced to model DP steels under reverse
loading. Parameters related to reverse loading can conveniently be identified independently
of the response in proportional loading. Reverse shear experiments are used for calibration
at large strains after loading reversal (also made at large strains). Starting from the
assumption that ductile fracture is a consequence of localization of deformations within a
narrow band, and the lecture of the conditions for localization of plastic deformations by
Rice (1976), it is proposed that localization occurs at a critical hardening rate, which is a
function of the stress state. It is shown that it exists a critical hardening rate that is
nathematiCally equivalent to the I osrd-Coulomb criterion for proportional loading of a
Levy-Von Mises material with isotropic hardening. A phenomenological formulation is
-
Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture - 153
proposed and calibrated using experimental data for proportional loading. The performance
of the critical hardening rate model is evaluated in the case of loading reversal. Overall, the
critical hardening rate fracture criterion predicts the strain at fracture within 8% accuracy.
Keeping in mind all sources of errors, it seems that a critical hardening rate identified
phenomenologically for proportional loading shows some relevance to predict ductile
fracture after loading reversal.
- 154 - Chapter 4: Critical Hardening Rate Model for Predicting Path Dependent Ductile Fracture
-
Chapter 5: Conclusion - 155
Chapter 5: Conclusion
5.1
Summary of findings
5.1.1
Hosford-Coulomb
In chapter 2, a fracture initiation model was proposed to predict the onset of fracture
in advanced high strength steels at low stress triaxialities. Assuming that the onset of
ductile fracture is imminent with the formation of a band of localization at the mesoscale
or microscale, a stress triaxiality and Lode angle parameter dependent fracture initiation
model was formulated based on the results from a fully three-dimensional localization
analysis. The resulting model is an extended Mohr-Coulomb model, which makes use of
the Hosford equivalent stress and the normal stress acting on the plane of maximum shear
to predict the onset of fracture. A consistent transformation from the principal stress space
to the space of equivalent plastic strain, stress triaxiality and Lode angle parameter was
performed to obtain a fracture initiation model for non-proportional loading. The particular
feature of this transformation is that it makes use of the material's isotropic hardening law
and therefore preserves the physical meaning of the underlying stress-based localization
model.
Experimental results were reported from ductile fracture experiments on three different
advanced high strength steel sheets (DP590, DP780 and TRIP780). The experimental
program included a shear experiment, tension with a central hole, notched tension and a
punch experiment, thereby covering a wide range of stress states. The comparison of
experiments and simulations showed good agreement of the Hosford-Coulomb model
predictions with all experiments for all materials. It is worth noting that without any shear
localization analysis results at their disposal, Stoughton and Yoon (2011), and Bai and
Wierzbicki (2010), had already correctly hypothesized on the existence of a MohrCoulomb type of criterion for predicting the onset of ductile fracture. However, the analysis
of chapter 2 showed that the direct use of a Mohr-Coulomb criterion systematically
- 156 - Chapter 5: Conclusion
underestimated the strain to fracture for biaxial loading, while it could be predicted with
great accuracy when using the Hosford-Coulomb model.
5.1.2
Ductile fracture of sheets after in-plane compression-tension
Chapter 3 deals with large strain compression-tension fracture experiments performed
on uniaxial and notched flat specimens extracted from dual phase steel sheets (DP780).
High compressive in-plane strains (of up to 13%) were achieved using a floating antibuckling device. The relative displacement of the specimen boundaries, as well as local
strains on the specimen surface, were measured using digital image correlation. A
Combined Chaboche-Yoshida (CCY) model was proposed to account for the observed
Bauschinger effect, transient softening and work hardening stagnation. The material
parameter identification based on notched compression-tension experiments for very large
strains was shown in detail. Subsequently, the model was applied to predict the material
response in different monotonic experiments (notched tension, tension with a central hole
and pure shear), and for different levels of reverse loading. The plasticity model predictions
agreed well with the results from all experiments, including the evolution of the surface
strains at the specimen center.
The extracted loading paths to fracture showed a significant increase of the strain to
fracture as a monotonic function of the applied pre-strain. For example, applying a prestrain of 0.13 increases the strain to fracture from 0.57 (for monotonic loading) to 0.77.
The transient hardening of the material and the local thickening of the sheet during
compression delay the formation of a neck and the consequent increase in triaxiality (as
well as the consequent decrease in the Lode parameter). A Hosford-Coulomb damage
indicator model with a non-linear damage accumulation rule was calibrated and validated
based on the experimental data. The model predictions agreed well with all experimental
results for the DP780 steel with a maximum relative error of 5% in the strains to fracture.
It is worth noting that the same phenomenological damage accumulation rule provided an
accurate description of proportional and non-proportional experiments on aluminum 2024-
T351 (Papasidero et al., 2014).
-
Chapter 5: Conclusion - 157
5.1.3
CriticalHardeningRate
Chapter 4 presented an experimental method to identify ductile fracture after shear
reversal. In addition, it referred to considerations in chapter 3: The data of ductile fracture
after compression followed by tension on notched specimens. An improved plasticity
model was introduced to model DP steels under reverse loading. The main added feature
compared to chapter 3 was that the parameters related to reverse loading could
conveniently be identified independently of the response in proportional loading. Reverse
shear experiments were used for calibration at large strains after loading reversal (also
made at large strains). Starting from the assumption that ductile fracture is a consequence
of localization of deformations within a narrow band, and the lecture of the conditions for
localization of plastic deformations by Rice (1976), it was proposed that localization occurs
at a critical hardening rate, which is a function of the stress state. It was shown that there
exists a critical hardening rate that is mathematically equivalent to the Hosford-Coulomb
criterion for proportional loading of a Levy-Von Mises material with isotropic hardening.
A phenomenological formulation was proposed and calibrated using experimental data for
proportional loading. The performance of the critical hardening rate model was evaluated
in the case of loading reversal. Overall, the critical hardening rate fracture criterion
predicted the strain at fracture within 8% accuracy. Keeping in mind all sources of errors,
it seems that a critical hardening rate identified phenomenologically for proportional
loading shows some relevance to predict ductile fracture after loading reversal.
5.2
Ongoing and future work
5.2.1
OrthogonalLoading
This study focused on a specific type of complex loading: reverse loading. The
plasticity of the material is affected by other well-known effects such as the crosshardening for different types of non-linear hardening. This suggests that the response of
the material under orthogonal loading conditions undergoes different physical mechanisms
of deformation. Two types of tests have already been developed. Large dogbone specimens
were pre-strained in tension. Uniaxial tensile specimens, as well as tensile specimens with
- 158 - Chapter 5: Conclusion
notches, were extracted and tested. Alternatively, the butterfly geometry may be used to
perform shear followed by tension. Early results suggest that the DP980 material features
a very rapid transient behavior after re-yielding.
IQ
35
30
25
20
-J
0
15
10
0a 0
04
0
008
Displacement (mm)
01
012
0
0.1
0.2
0.3
Displacement (mm)
(a)
0.4
(b)
Figure 5-1 Comparison of the experimental load-displacement (dashedlines) with the numerical
prediction (red line) using the plasticity model calibratedin chapter 4 for DP590 on orthogonal
tests (a) notched tension at 90 degrees with respect to a 5% tensile pre-strain (b) shear to 50%
equivalentplastic strainfollowed by plane strain tension to fracture.
Some constitutive models describing such effects already exist. In addition, the response
of the material at large strain after orthogonal loading tests must be studied. Once the strain
hardening response of the material is well predicted, it would be of great interest to
investigate the relevance of the critical hardening rate model for such types of tests.
5.2.2
Validations studies
The model proposed in chapter 3, including the plasticity and the fracture models, has
been applied in order to predict fracture during three point bending of a hot formed
martensitic hat-shaped profile. At a certain displacement of the punch, fracture is imminent
at two competing locations. One is mostly under plane strain tension A, while the other, B,
-
Chapter 5: Conclusion - 159
undergoes plane strain compression followed by plane strain tension. Using the CCY
plasticity model with a non-linear damage accumulation rule, the fracture initiation was
predicted at point A, in agreement with the experiment; while at point B, a linear damage
accumulation with isotropic hardening predicts fracture, since the increase in ductility is
not predicted for reverse loading conditions.
(a)
(b)
(c)
Figure 5-2 Comparison of (b) the crack during experimental three point bending of a martensitic
hat-shapedprofile with (a) the crack predictedby FEA using a lineardamage indicatorand (c) the
crackpredictedby FEA using the non-lineardamage indicator.
In another attempt to understand the effect of loading histories on ductile fracture, three
point bending tests of a cold formed hat-shaped profile made in DP980 were performed.
Numerical simulations of the process using several modelling approaches were considered.
It has been found that fracture cannot be predicted if a virgin state of the material is
assumed after the cold forming process. On the other hand, simulating the multi-stage
process leads to an early prediction of the crack. Early results suggest that taking into
account the effect of complex loading on ductile fracture allows for an accurate prediction
of crack initiation.
- 160 - Chapter 5: Conclusion
5.2.3
Strain rate and temperature effects
It is well known in the literature that strain rate and temperature also have a strong
influence on the plastic behavior and ductility of materials. In particular, it is often read
that strain rate increases both the hardening of the material and its ductility. Interestingly,
the increase in hardening capacity would induce an increase in ductility according to the
critical hardening rate model. It would be of great interest to investigate experimentally the
relevance of the current model at different strain rates.
-
References - 161
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