Computational Aspects of Ductile Crack Growth
in Three Dimensions
by
Vaibhaw Vishal
Bachelor of Technology in Mechanical Engineering,
Indian Institute of Technology, Bombay, India (2000)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
MASSACHUSETTS INST11TUTE
OFTECHNOLOGY
at the
JUL 16,2003
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
June 2003
© Massachusetts Institute of Technology 2003. All rights reserved.
A uthor ................................
......
Department of Mechanical Engineering
Vy 23, 2003
Certified by.....................
David M. Parks
Professor of Mechanical Engineering
esis Supervisor
Accepted by ......................
..
Ami A. Sonin
Chairman, Department Committee on Graduate Students
r
BARKER
Computational Aspects of Ductile Crack Growth in Three
Dimensions
by
Vaibhaw Vishal
Submitted to the Department of Mechanical Engineering
on May 23, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Ductile crack growth can occur in many engineering applications, such as penetrating
cracks in pressure vessels. Simulations of ductile crack growth are performed on a
3-D open-bend specimen. Two different finite element schemes namely, implicit finite
element scheme and explicit finite element scheme are used to perform the simulations.
Various type of meshes consisting of first-order and second-order tetrahedral elements
(along with/without a brick or prism elements layer next to the crack-containingplane) are used. Explicit finite element scheme takes about nine times higher cputime (on a Compaq AlphaStation DS20E, two processors at 667MHz each). Loaddisplacement curves show a steeper drop off after peak value when simulations are
performed with explicit finite element scheme compared to when simulations are
performed with implicit finite element scheme.
Thesis Supervisor: David M. Parks
Title: Professor of Mechanical Engineering
2
Acknowledgments
I want to express my deep sense of gratitude to my advisor, Prof. D.M.Parks, for his
invaluable guidance and constant encouragement throughout this work. I have learnt
a lot while working with him and have enjoyed interacting with him. I would also like
to thank Prof. F.A.McClintock for his insightful guidance into the field of fracture
mechanics.
A want to thank my fellow graduate students, Dora, Rajdeep, Nuo, Nici, Jin, Yin,
Hang, Ekrem, Ted, for their support. A special thanks to Ray Hardin for handling
all the administrative works.
I would also like to thank my friends Sachin and Shweta for making a homely atmosphere in Boston.
Support for this research was provided by the D.O.E. under grant number DE-FG0285ER13331 to MIT.
3
Contents
1
11
Introduction
1.1
Motivation and Application . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
Organization of Present Work . . . . . . . . . . . . . . . . . . . . . .
12
2 Finite Element Modeling
16
2.1
GTN Failure M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Elements Used in Simulations . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Advantages and Disadvantages of Various Tetrahedral Elements
23
2.3
Modified Second-Order Tetrahedral Element . . . . . . . . . . . . . .
24
2.4
Issues in Using C3D10M With the GTN Failure Model . . . . . . . .
26
2.5
Procedure for Adding Brick-Element Layer to a surface C3D10M Element Layer
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Finite Element Results with Abaqus/Standard
27
30
3.1
M esh Details
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
M aterial M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.3
Comparison of Load-Displacement Curves
. . . . . . . . . . . . . . .
32
3.4
Void Volume Fraction Evolution for Different Meshes . . . . . . . . .
33
3.5
CPU Time Comparison . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4 Finite Element Results with Abaqus/Explicit
51
4.1
M ass Density Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.2
Comparison of Material Models . . . . . . . . . . . . . . . . . . . . .
53
4
4.3
5
Comparison of Various Meshes . . . . . . . . . . . . . . . . . . . . . .
70
Summary and Conclusions
5.1
55
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Fortran code for attaching brick layer
5
73
78
List of Figures
1-1
A typical specimen. (a) Full 3-D geometry. (b) 2-D idealization (plane
strain ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2
2-D idealization of pressure vessel showing a longitudinal penetrating
crack. ........
2-1
14
...................................
15
True stress-strain curve from ASTM standard compression test on
A572 Gr. 50 steel (taken from [1]).
. . . . . . . . . . . . . . . . . . .
17
2-2
Effect of void volume fraction on yield surface (schematic). . . . . . .
19
2-3
Uniaxial behavior of a porous material against fully dense material
(schem atic). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2-4
Modified second-order tetrahedral element (C3D1OM).
. . . . . . . .
24
2-5
Tetrahedral element (C3D1OM) divided into four hexahedral elements.
25
2-6
Brick elements attached to tetrahedral elements. . . . . . . . . . . . .
28
2-7
Typical mesh detail showing a layer of brick elements attached to
C3D1OM elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
Specimen geometry (a) Full geometry. (b) Quarter geometry which is
modeled in finite element method. All dimensions are in mm. ....
3-2
29
35
Schematic of void nucleation, growth, and coalescence in ductile metals
(taken from [2]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3-3
Details of mesh-1. (a) Isometric view. (b) Zoomed isometric view.
.
37
3-4
Details of mesh-2. (a) Isometric view. (b) Zoomed isometric view.
.
39
3-5
Details of mesh-3. (a) Isometric view. (b) Zoomed isometric view.
.
41
3-6
Details of mesh-4. (a) Isometric view. (b) Zoomed isometric view.
.
43
6
3-7
Load-displacement curves. . . . . . . . . . . . . . . . . . . . . . . . .
3-8
Contours of void volume fraction plotted on the crack-containing-plane
45
(at a load-point displacement of 10.4 mm), obtained with (a) Mesh-1.
(b) M esh-2.
3-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
O-33 Contours of stress (0o
33 ) plotted on the crack-containing-plane
46
(at
a load-point displacement of 10.4 mm), obtained with (a) Mesh-1. (b)
M esh-2.
. . . ....
....
.... ......
. . . . . . . ...
. . ....
.
3-10 Brick elements attached to a tetrahedral element. . . . . . . . . . . .
4-1
50
Kinetic energy and total deformation energy curves for (a) mesh-1, (b)
mesh-3, and (c) mesh-4.
4-2
48
. . . . . . . . . . . . . . . . . . . . . . . . .
Material model distribution in finite element mesh (schematic).
57
(a)
Case-1: GTN model with element failure restricted to only one layer
next to the crack-containing-plane, classical plasticity in the rest of the
mesh. (b) Case-2: GTN model throughout the mesh, element failure
restricted to only one layer next to the crack-containing-plane.
(c)
Case-3: GTN model with element failure throughout the mesh. . . . .
4-3
Load-displacement curves obtained with mesh-3 for the three different
cases of material model distribution.
4-4
58
. . . . . . . . . . . . . . . . . .
59
Contours of normal stress (O3 3 ) obtained with mesh-3 (view of the
crack-containing-plane at a load-point displacement of 10.5 mm ). (a)
Case-1: GTN model with element failure in only one layer next to the
crack-containing-plane, classical plasticity in the rest of the mesh. (b)
Case-2: GTN model throughout the mesh, element failure in only one
layer next to the crack-containing-plane. (c) Case-3: GTN model with
element failure throughout the mesh. . . . . . . . . . . . . . . . . . .
7
60
4-5
Contours of void volume fraction obtained with mesh-3 (isometric view
at a load-point displacement of 10.5 mm). (a) Case-1: GTN model
with element failure in only one layer next to the crack-containingplane, classical plasticity in the rest of the mesh. (b) Case-2: GTN
model throughout the mesh, element failure in only one layer next
to the crack-containing-plane. (c) Case-3: GTN model with element
failure throughout the mesh. . . . . . . . . . . . . . . . . . . . . . . .
4-6
61
Contours of equivalent plastic strain obtained with mesh-3 (isometric
view at a load-point displacement of 10.5 mm). (a) Case-1: GTN model
with element failure in only one layer next to the crack-containingplane, classical plasticity in the rest of the mesh. (b) Case-2: GTN
model throughout the mesh, element failure in only one layer next
to the crack-containing-plane. (c) Case-3: GTN model with element
failure throughout the mesh. . . . . . . . . . . . . . . . . . . . . . . .
4-7
62
Contours of void volume fraction obtained with mesh-3 (view of the
crack-containing-plane at a load-point displacement of 10.5 mm). (a)
Case-1: GTN model with element failure in only one layer next to the
crack-containing-plane, classical plasticity in the rest of the mesh. (b)
Case-2: GTN model throughout the mesh, element failure in only one
layer next to the crack-containing-plane. (c) Case-3: GTN model with
element failure throughout the mesh. . . . . . . . . . . . . . . . . . .
4-8
63
Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-1.
(a) at a load-point displacement of
5.5 mm. (b) at a load-point displacement of 8.8 mm. (c) at a load-point
displacement of 10.5 mm . . . . . . . . . . . . . . . . . . . . . . . . .
64
4-9 Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-2.
(a) at a load-point displacement of
5.25 mm. (b) at a load-point displacement of 8.8 mm. (c) at a loadpoint displacement of 10.5 mm. . . . . . . . . . . . . . . . . . . . . .
8
65
4-10 Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-3.
(a) at a load-point displacement of
5.25 mm. (b) at a load-point displacement of 8.8 mm. (c) at a loadpoint displacement of 10.5 mm. . . . . . . . . . . . . . . . . . . . . .
66
4-11 Contours of normal stress (o33) at a load-point displacement of 10.5 mm
obtained with (a) mesh-3. (b) mesh-4.
. . . . . . . . . . . . . . . . .
67
4-12 Contours of void volume fraction, at various load-point-displacements,
obtained with mesh-4; GTN model with element failure throughout the
mesh. (a) at a load-point displacement of 5.0 mm. (b) at a load-point
displacement of 8.0 mm. (c) at a load-point displacement of 10.0 mm.
68
4-13 Load-displacement curves obtained with mesh-3 and mesh-4. . . . . .
69
5-1
Load-displacement curve obtained with mesh-3 using Abaqus/Standard
and Abaqus/Explicit. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2
Load-displacement curve obtained with mesh-4 using Abaqus/Standard
and Abaqus/Explicit. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
74
75
Contours of void volume fraction plotted on the crack-containing-plane
for mesh-3 (at a load-point displacement of 10.5 mm). (a) Simulation performed with Abaqus/Standard. (b) Simulation performed with
Abaqus/Explicit.
5-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Contours of -33 plotted on the crack-containing-plane for mesh-3, showing positive values of c3 3 behind crack front. . . . . . . . . . . . . . .
9
77
List of Tables
2.1
Flow stress values at different plastic strain levels for A572 Gr. 50 steel
(taken from [1]) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Parameters used in the Gurson-Tvergaard-Needleman model. . . . . .
22
3.1
CPU run-time comparison for various meshes. . . . . . . . . . . . . .
34
4.1
CPU run-time comparison for various cases and meshes.
. . . . . . .
56
5.1
CPU run-time comparison for implicit and explicit finite element scheme. 73
10
Chapter 1
Introduction
1.1
Motivation and Application
The resistance of ductile metals to stable crack extension strongly affects the safety
and reliability of critical engineering structures. Fully-plastic crack growth or ductile
crack growth can occur in many engineering applications such as long part-through
cracks in tension in wide plates, penetrating cracks in pressure vessels, ship hulls etc.
For many steels and aluminum alloys, a large increase in plastic deformation ahead
of the crack front during the first few millimeters of (stable) crack extension permits
increase in the crack driving force (e.g., the J-integral) to levels several times the
values present at the initiation of growth.
At the micro-scale, crack growth commonly involves a mechanism of void nucleation, growth, and coalescence driven by the plastic strain field and (tensile) mean
stress ahead of crack front. However, the macroscopic geometry and loading mode,
combined with material flow properties, largely determines the level of mean stress
and plastic strain attainable along the crack front. To better understand this complex
interaction and to provide reliable prediction of crack growth resistance, robust and
efficient computational tools which couple macro-scale features with realistic models
of the micro-scale growth mechanisms are required.
Many models exist in the literature to capture nucleation, growth, and coalescence
of voids. Some of the most-cited models are the Gurson-Tvergaard-Needleman model
11
[3, 4, 5, 6], the Rousselier model [7], and the Rice & Tracey model [8].
Simulations of ductile crack growth can be performed using the finite element
method. In many specimens (such as shown in Fig. 1-1), it is often attractive to
use the 2-D idealization (Fig. 1-1) of the problem to perform the simulation, as
the latter requires much less computational effort, compared to modeling the 3-D
specimen.
Simulating ductile crack growth using a 2-D idealization has a major
disadvantage; namely, it can not capture crack tunneling behavior which occurs in
reality. This disadvantage (of 2-D idealizations) may result in predicting unstable
crack growth when the crack may actually be stable. An experiment performed by
INEEL shows a similar result. In the experiment, hydro test on a cylindrical pressure
vessel containing a long, nearly constant depth longitudinal part-through crack was
performed, a simulation based on a 2-D model of the vessel (Fig. 1-2) predicted
a violently unstable crack growth whereas in experiment the pressure vessel leaked
stably. Thus, it is important to simulate ductile crack growth in 3-D rather than only
in its 2-D idealization.
Simulating ductile crack growth in 3-D specimens requires very high computational effort. To solve the problem in a reasonable time, with desired accuracy, it is
important to wisely choose the finite element scheme along with the type of elements
to be used in the simulations. It is important to understand the relative advantages
and disadvantages of each finite element scheme (e.g., implicit static scheme versus
explicit dynamic scheme).
It is also important to understand the relative merits
of available elements such as, tetrahedral versus brick elements, second-order versus
first-order elements, and how these elements perform within each of the finite element
schemes.
1.2
Organization of Present Work
Ductile crack growth is simulated in a 3-dimensional, 4-point loading open-bend specimen using an implicit finite element scheme (Abaqus/Standard) and an explicit finite
element scheme (Abaqus/Explicit). Various types of elements are used. Chapter two
12
describes the Gurson-Tvergaard-Needleman (GTN) material model, which is used to
simulate the growth of voids and resultant material softening. The material properties and the parameters of the GTN model used in the simulations are also given.
This chapter further discusses the various types of elements available for simulations
and discusses advantages and disadvantages of each. This chapter also shows how a
brick-element-layer can be attached to the second-order modified tetrahedral elements
(C3D1OM elements of Abaqus).
Chapter three gives the specimen geometry and dimensions; it also provides all
the meshes which are used in the simulations. The open-bend simulations performed
on these meshes using Abaqus/Standard are presented in this chapter. The loaddisplacement curves obtained from all the meshes are compared and discussed. The
void volume fraction values, equivalent plastic strain values, and stress values for
all the meshes are compared using contour plots. The cpu-time required for all the
meshes are also compared.
Chapter four presents the results obtained with Abaqus/Explicit. Various types of
material modeling are used in this chapter, such as using isotropic hardening Misesbased plasticity away from the crack-containing-plane and the GTN model next to the
crack-containing-plane. The load-displacement curves obtained with these differing
material models are compared. The void volume fraction values and crack advance
patterns are compared and discussed through contour plots. The cpu-time required
for each of them is also tabulated.
Finally, chapter five summarizes the results and provides interpretations of the
differences between the results obtained from the two finite element schemes. It also
gives recommendations for future work in this area.
13
Crack
(a)
Crack
(b)
Figure 1-1: A typical specimen. (a) Full 3-D geometry. (b) 2-D idealization (plane
strain).
14
Internal Pressure
Applied
Figure 1-2: 2-D idealization of pressure vessel showing a longitudinal penetrating
crack.
15
Chapter 2
Finite Element Modeling
Finite element simulations of ductile crack growth are performed on 4-point loading
open-bend specimen with different meshes. The details of the specimen are given in
the next chapter. The material properties used in all the simulations are of A572
Gr. 50 steel. The Youngs's modulus of this steel is taken as E = 207GPa and the
Poisson's ratio is taken as v
=
0.30. The strain hardening (Table 2.1) of this material
is calculated from the true stress-strain curve shown in Fig. 2-1, obtained from an
ASTM standard compression test [1] on A572 Gr. 50 steel.
The failure model used in the simulations is the Gurson-Tvergaard-Needleman
(GTN) porous plasticity model for void-containing materials. This model was first
proposed by Gurson [3] and later modified by Tvergaard and Needleman. The model
is described below.
2.1
GTN Failure Model
The Gurson model analyzes plastic flow in a void-containing material by averaging
the effect of voids throughout the material. The amount of voids is quantified by
using porosity (or void volume fraction),
f, which
is defined as the ratio of void-
volume to the total volume of the material. In the Gurson model, unlike classical
plasticity, hydrostatic stress also has an effect on the yield surface. Typical yield
surfaces for different levels of porosity are shown in Fig. 2-2. In this figure, the ratio
16
900
I
I
I
I
I
800
700
600
0O
500
Cf)
400
300
200
100
0
0
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
True Strain
Figure 2-1: True stress-strain curve from ASTM standard compression test on A572
Gr. 50 steel (taken from [1]).
17
Flow Stress, S (MPa)
3.53E+2
4.76E+2
5.24E+2
5.55E+2
5.79E+2
5.99E+2
6.16E+2
6.27E+2
6.36E+2
6.44E+2
6.51E+2
6.57E+2
6.63E+2
6.69E+2
6.74E+2
6.79E+2
6.83E+2
Plastic Strain, egm
0
0.009406
0.019175
0.029022
0.038906
0.048811
0.058730
0.068675
0.078633
0.088595
0.098561
0.108530
0.118501
0.128474
0.138449
0.148425
0.158403
Table 2.1: Flow stress values at different plastic strain levels for A572 Gr. 50 steel
(taken from [1]).
18
of mean stress,
Urn,
to tensile flow strength, S, is plotted along the X-axis, and the
ratio of Mises stress,
req,
to tensile flow strength, S, is plotted along the Y-axis (only
positive portions of or/S and ueq/S axes are shown because in the GTN model, yield
surface, <D, is an even function of Ur/S and aUq/S). The figure shows that, for
f
= 0
yield surface is a straight line; i.e., mean-stress has no effect on yield. But as the
porosity increases from
f
=
0, mean stress also has an effect on yield surface. Also as
the porosity increases, the yield surface shrinks in size; i.e., yielding occurs at lower
values of stress. Figure 2-3 compares the uniaxial stressing behavior of a porous metal
with initial void volume fraction fo against a fully dense (void volume fraction = 0)
perfectly plastic material. In this figure, true strain is plotted along the X-axis and
true stress is plotted along the Y-axis. The figure shows that porous metal yields at
a lower value of stress (both in tension and compression), compared to fully-dense
metal. Also in compression, the porous material hardens due to closing of the voids,
while in tension, it softens due to nucleation and growth of voids.
ceq
S
f
0 (Mises)
f
0.4
0
0
OM
S
Figure 2-2: Effect of void volume fraction on yield surface (schematic).
19
f0 =0 (Mises)
S
tension (fO)
C,
compression (fO)-
-s
Figure 2-3: Uniaxial behavior of a porous material against fully dense material
(schematic).
20
In the Gurson model, the yield surface is given by,
2
=(e)
where
ueq
3
+ 2f cosh
[1 + f2=
(2.1)
0,
is Mises stress, S is tensile flow strength of fully dense matrix at the
current level of matrix work hardening, urn is mean hydrostatic stress, and f is void
volume fraction. Tvergaard and Needleman [4], [5], [6] modified the above model
by introducing two dimensionless parameters, qi and q2 , and by replacing
effective void volume fraction,
f*
f*,
if f < fc,7
iff-ff,(2.2)
+
where
fc
with an
defined by
f
=
f
-_'7
f
if fc < f
< fF,
and fF are fitting parameters. The new yield surface is given by
2
S= ()
This addition of qi and
the replacement of
±+ 2qif* cosh
q2
3q2m
-
[1 + (qi f*)2
(2.3)
= 0.
amplifies the effect of hydrostatic stress on yielding, and
f with f* weakens
with complete failure (no load-supporting capacity) at f =
To calculate the evolution of
f,
f
the material rapidly as
ranges from
fc
to fF,
fF.
an initial void volume fraction, fo, is assumed.
The rate of change of f consists of two parts, namely, nucleation and growth:
f
(2.4)
f nucleation + fgrowth7
where fnucieation is the rate of change in void volume fraction due to the creation of
new voids, and jgrowth is due to the rate of growth of existing voids. The rate of
nucleation of new voids is assumed to be plastic-strain-controlled and is given by
{2
SfN
fnucleation
where
P
=2r
exp-
2
J
pN
q,m,
is equivalent plastic strain rate in the fully dense matrix, EN and
21
(2.5)
SN
are
the mean and standard deviation of normal distribution of void nucleation strain, and
fN
is the volume fraction of nucleating voids. The growth of existing voids is based
on conservation of mass, and is given by
fgrowth=
(2.6)
(1 - f) it,
where P is rate of volumetric plastic strain. The plastic strain increment is given by
the normality flow rule
(2.7)
,4
P =A
where cxj are the components of the Cauchy stress tensor. During plastic flow the
parameter A is obtained from the consistency condition <> = 0. The hardening of
fully-dense matrix material is given as a function of equivalent plastic strain in fully
dense matrix by (Table 2.1),
S
= S (&eqm).
(2.8)
The evolution of the equivalent plastic strain in the matrix is given by the equivalence
of plastic work
(1 - f) S
qm
(2.9)
= Auij a.
=iX'
The numerical values of void nucleation parameters EN, SN and fN, as well as for the
initial void volume fraction, fo, and the material failure parameters,
f,
and fF are
taken from [9] and are summarized in Table 2.2.
fo
eN
SN
IfN
f
fF
q,
q2
0.002
0.32
0.10
0.03
0.15
0.25
1.5
1.0
Table 2.2: Parameters used in the Gurson-Tvergaard-Needleman model.
2.2
Elements Used in Simulations
The quality of 3-D meshes generated with tetrahedral elements has been consistently
improving over the past decade. Generating a 3-D mesh is easy and fast with tetra22
hedral elements. Unfortunately, mesh generation with brick elements is not so well
developed. For the present work tetrahedral-element-based meshes are used. Various
types of tetrahedral elements are available in Abaqus, and each has some advantages
and disadvantages, which are described below.
Advantages and Disadvantages of Various Tetrahedral
2.2.1
Elements
* 4-noded first-order tetrahedral element (C3D4); these elements are computationally efficient and fast as they require only one integration point per
element, but these elements have problems of volumetric and shear locking [10].
Very fine meshes are often needed for good accuracy if these elements are used
in a simulation.
" 10-noded second-order tetrahedral element (C3D1O); being second-order
elements, they have four integration point per element, and so require more computational effort per element. They give good accuracy with finite deformation.
However these elements are not appropriate for contact problems because in
uniform pressure situations the contact forces are non-uniform at the corner
and mid-side node (corner nodes carry zero nodal forces) [10]. Further, these
elements may exhibit significant volumetric locking when incompressibility is
approached.
" 10-noded modified second-order tetrahedral element (C3D10M); these
elements exhibit minimal volumetric and shear locking and provide robustness
and convergence for contact simulations, but are computationally expensive,
especially in explicit dynamics simulations, because they require four integration
points per element and also have linear multi-point constraints applied to some
of its nodes. With these elements, relatively coarse meshes can be used for
reasonable accuracy. This element is described below.
23
2.3
Modified Second-Order Tetrahedral Element
The modified second-order tetrahedral element (C3D1OM) is shown in Fig. 2-4.
111
Figure 2-4: Modified second-order tetrahedral element (C3D1OM).
This element has ten user-defined nodes (node numbers 1-10). Node numbers
1-4 are vertex nodes, and node numbers 5-10 are mid-edge nodes. In these elements
Abaqus adds five internal nodes (node numbers 11-15, which are invisible to user), and
divides each tetrahedral element into four hexahedral elements, as shown in Fig. 2-5.
These hexahedral sub-elements use a uniform strain formulation as developed
in [11]; hourglass control is applied to the assembly of hexahedral elements and is
described in [12].
The coordinates of node 11, which is a mid-body node, are given by
Xii-
Atet(Xi + X 2 + X 3 + X 4 ) + Btet(X
24
5
± X 6 + X 7 ± X 8 ± X 9 + X 1 0 ), (2.10)
4
fO
01
5
3
13
9
B
2
Figure 2-5: Tetrahedral element (C3D10M) divided into four hexahedral elements.
25
where Atet = -0.006803281,
ith
Btet = 0.162131146, and Xi are the coordinates of the
node. This mid-body node is free to move; i.e., three degrees of freedom, one in
each Cartesian direction, are added to the system by the addition of this node. The
initial coordinates of nodes 12-15, which are mid-face nodes, are given by,
X 12 = Atri(X1 + X 2 + X 3 ) + Btri(X 5 + X 6 + X 7 ),
X 13 = Atri(XI + X 2 + X4 ) + Btri(X 5 + X8 + X 9 ),
X14-
Atri(X
2
+ X 3 + X 4 ) + Btri(X 6 + X 9 + X 10 ),
X15= Atri(Xi + X 3 + X 4 ) + Btri(X
where Atri = -1/15,
(2.11)
7
+ X 8 + X 1 0 ),
Btri = 2/5, and Xi are the coordinates of the
ith
node. These
nodes are constrained to move in a specific way; e.g., node 12, which is on the face
formed by nodes 1, 2, 3, 5, 6, and 7, is constrained to move according to
U12 =Atri(Ui + u 2 + u3 ) + Btri(U5 + U 6 + u 7 )
where Ati = -1/15
(2.12)
and Btri = 2/5, and ui is the displacement of the ith node. Node
numbers 13-15 are also constrained to move in a similar fashion. Further details about
this element are given in [12].
2.4
Issues in Using C3D10M With the GTN Failure Model
Many metals which fail by a void growth mechanism often display a macroscopically
planar fracture surface. Very little or no void growth is observed away from the
crack-containing plane [13]. Since many calculations are involved in finding the yield
surface and void growth rate in the GTN failure model, it may be computationally
inefficient to use the GTN model throughout the mesh. Therefore, in some of the
simulations, to reduce computational effort, the GTN failure model is used in only
one element layer (attached to the crack plane), and classical plasticity is used away
from this layer. Using this type of modeling will have the disadvantage that we will
26
never know if the crack "wanted" to turn off-plane in a shear-band.
To limit fracture to the crack-containing plane, a layer of brick elements is attached
to the tetrahedral element layer. The procedure for adding the brick element layer
is described in the next section. Similar approaches of using two different types of
elements in a mesh, one attached to the crack plane, and a different one away from
it, have been used by many researchers; e.g., Xia & Shih [14] used computational
cells near the crack plane for modeling ductile crack growth and Ortiz & Pandolfi [15]
used irreversible cohesive elements near the crack plane for 3-D crack propagation
analysis.
2.5
Procedure for Adding Brick-Element Layer to
a surface C3D10M Element Layer
To add brick elements on a face of a C3D1OM element, first, an additional node (node
16) is defined (Fig. 2-6) with the coordinates given by
X 16 - Atri(Xi + X 2 + X 3 ) + Btri(X 5 + X6 + X7 )
where Atrj = -1/15,
(2.13)
Btri = 2/5, and Xi is the position vector of the ith node. This
node (node 16) has the same coordinate as the Abaqus added internal node (node 12)
of Fig. 2-4. This newly-added node is constrained to move according to the equation
U 16
where Atri = -1/15
Atri (U + u 2 + u 3 ) + Btri(u5 + u 6 + u7 )
(2.14)
and Btri = 2/5, and ui is the displacement of the ith node.
Constraining this newly-added node in the way described above makes sure that this
node coincides all the time with the moving, but hidden node (node 12 of Fig 2-4) of
Abaqus.
To complete the formulation of the brick element layer, new nodes are defined at
a height h below each node of the face of the tetrahedra (Fig 2-6), and the newly-
27
Face of
tetrahedral element
h
5
16
6
3 added
brick elements
Figure 2-6: Brick elements attached to tetrahedral elements.
28
--
-
added node (node 16). This way, three cylindrical brick elements can be attached to
one face of each tetrahedral element. A Fortran code is written to attach a layer of
brick elements to a given C3D1OM element mesh. The code is given in appendix A.
Figure 2-7 shows a mesh detail in which a layer of brick elements is attached to
C3D1OM elements.
Brick elements
C3D1OM elements
Figure 2-7: Typical mesh detail showing a layer of brick elements attached to C3D1OM
elements.
29
Chapter 3
Finite Element Results with
Abaqus/Standard
Finite element simulations are performed using the commercially-available finite element software Abaqus/Standard, Versions 5.8, 6.1, and 6.3 [16, 17, 18]. The specimen
geometry is shown in Fig. 3-1. A quasi-static analysis is performed in which a displacement, u, is applied on the center of top rollers while the bottom rollers are held
fixed. Because of symmetry only one-quarter of the specimen is meshed with finite
elements. In all the simulations of this chapter, the rollers are assumed to be frictionless and rigid. Four different meshes, mesh-1, mesh-2, mesh-3, and mesh-4, are
considered. The details of these meshes are given below.
3.1
Mesh Details
Element size and mesh design play an important role in simulating ductile crack
growth ([19],[20]). On the physical level, ductile fracture occurs because of void nucleation, growth and coalescence. Once voids nucleate (around second phase particles
or inclusions), further plastic strain and hydrostatic stress causes them to grow and
eventually coalesce. Figure 3-2 schematically illustrates the growth and coalescence
mechanism of micro-voids. If the initial void volume fraction of voids is low (less
than 10%) [2], each void can be assumed to grow independently; upon further growth,
30
neighboring voids interact. Plastic strain gets concentrated along a sheet of voids,
and local necking instability develop. To capture this local instability, element size
to be used in the finite element calculations should be of the order of inter-particle
spacing (spacing between large inclusions). This way the element size provides the
otherwise missing physical length-scale over which continuum damage occurs.
Mesh-1 is shown in Fig. 3-3; part (a), and part (b) of this figure show the isometric
view of the specimen mesh, whereas parts (c), (d), and (e) show front view, top view,
and a view of the crack-containing-plane, respectively. In this mesh C3D1OM elements
are used away from the crack-containing-plane, and a 0.3 mm-thick layer of 8-noded
brick elements (C3D8) is used on the crack-containing-plane. These brick elements are
attached to the C3D1OM elements in the manner described in the previous chapter.
The brick-layer can be seen in zoomed isometric view, as shown in Fig. 3-3(b). The
number of (user) nodes and the number of elements in this mesh are 41,783 and 27,881,
respectively. If we define the effective element size on the crack-containing-plane by
the equation,
4 x Area of an element-face on the crack-containing-plane
Perimeter of the element-face on crack-containing-plane '
the effective element size for this mesh is 0.23 mm.
Mesh-2 is shown in Fig. 3-4. In this mesh, again C3D1OM elements are used (away
from the crack-containing-plane), along with a 0.3 mm-thick brick-element layer (on
the crack-containing-plane). The number of (user) nodes and the number of elements
in this mesh are 53,346 and 35,930, respectively. The only difference between mesh-1
and mesh-2 is in their number of nodes and elements. Mesh-2 is about 25% denser
than mesh-1. The effective element size for this mesh is 0.20 mm.
In mesh-3 (Fig. 3-5), instead of using C3D1OM elements away from the crackcontaining-plane, first-order 4-noded tetrahedral elements (C3D4) are utilized. A
layer of 6-noded triangular prism elements (C3D6) is attached to these tetrahedral
elements; the thickness of this prism layer is 0.3 mm. Displacement compatibility
between C3D4 elements and C3D6 elements is ensured by extruding C3D6 elements
31
from the base of C3D4 elements. The number of nodes and the number of elements
in this mesh are 16,174 and 78,173, respectively. The effective element size for this
mesh is 0.23 mm.
In mesh-4 (Fig. 3-6), first-order 4-noded tetrahedral elements (C3D4) are used
throughout the mesh. The number of nodes and the number of elements in this
mesh are 15,030 and 75,996, respectively. The effective element size for this mesh is
0.23 mm.
3.2
Material Model
In all the simulations, the matrix hardening of Fig. 2-1, along with the Gurson parameters of Table 2.2 are used. Since Abaqus/Standard does not easily accommodate
element deletion (failure), the GTN material model without element failure is used.
In this chapter, the GTN material model is used throughout the mesh. Later, for
comparison purposes, some simulations are performed with the GTN material model
in one layer next to the crack-containing-plane, and classical plasticity in the rest of
the mesh. It will be noted that using the GTN material model throughout the mesh
does not significantly increases the CPU time.
3.3
Comparison of Load-Displacement Curves
Figure 3-7 compares load displacement curves for all the meshes. Mesh-1 and mesh2 give almost identical load-displacement curves, but mesh-3 shows a lower loaddisplacement curve than that of the other meshes. This is because mesh-3 has a higher
value of void volume fraction on the crack-containing-plane than other meshes (Fig. 38). Since elements soften and carry less load at high void volume fraction values,
therefore the load-displacement curve for mesh-3 is lower than the load-displacement
curves of other meshes.
Since the simulations in this chapter are performed using Abaqus/Standard, elements on the crack-containing-plane never fail and continue to carry load in direction-
32
3. This can be seen in Fig. 3-9, which show contours of -3 3 on the crack-containingplane for all the meshes. The figure shows positive values of O-33 in the crack-growth
region.
Some elements on the crack-containing-plane, which should have actually
failed (because of high value of void volume fraction) and carry no load, are carrying
load in direction-3, leading to a higher load-displacement curve. This condition is
further worsened because of the long "moment arm" of these elements. To resolve
this problem of overestimating the load-displacement curve and thereby falsely indicating more stability, in the next chapter, another finite element software program,
Abaqus/Explicit is used, which accommodates element failure easily.
3.4
Void Volume Fraction Evolution for Different
Meshes
Figures 3-8 (a), (b), (c), and (d) show contours of void volume fraction for mesh-1,
mesh-2, mesh-3, and mesh-4, respectively. The contours are shown on the crackcontaining-plane for a load-point displacement of u =10.4 mm. Lower values of void
volume fraction are observed in mesh-1 and mesh-2, compared to mesh-3. This is
because the evolution of void volume fraction depends upon plastic strain increments
(eq. 2.4,
2.5, and 2.6), and plastic strain increments would be smaller in a brick
element of mesh-1 and mesh-2, compared to a prism element of mesh-3, as explained
below.
Consider brick-element 1 shown in Fig. 3-10. The plastic strain increment of this
element will depend on the displacement increment of node 3 (because node 16 is
constrained to move according to eq. 2.14), which is far away from crack front, and
therefore has a small value of displacement increment. This in turn gives a low value
of plastic strain increment in element 1. Whereas no such constraint exists in prism
elements of mesh-3, and hence lower void volume fraction values are observed in
mesh-1 and mesh-2, compared to that observed in mesh-3.
In other words effective element size for mesh-1 and mesh-2 is not 0.23 mm and
33
0.20 mm (as mentioned in section 3.1, which is calculated based on the area and
perimeter of a brick element); rather, it is 0.36 mm and 0.33 mm, respectively (which
is calculated based on the area and perimeter of a C3D1OM element-face just above
the brick layer).
Because of this higher effective element size, lower void volume
fraction values are observed in mesh-1 and mesh-2, compared to that observed in
mesh-3.
Further, poor contours of void volume fraction are observed in mesh-4. This is
because mesh-4 contains only C3D4 elements on the crack-containing-plane. These
elements have three nodes on the crack-containing-plane and only one node away
from it. When these element have softened because of void growth, they become
very compliant and essentially carry uniaxial load (low-triaxiality) in direction three.
Because of low-triaxiality, the volumetric plastic strain increment is smaller, which
results in slower void growth (eq. 2.6).
3.5
CPU Time Comparison
The CPU-time (on a Compaq Alpha Station DS20E, two processors at 667MHz each)
for all the simulations are tabulated in Table 3.1. Mesh-1 and Mesh-2, which utilize
second-order modified tetrahedral elements with a brick layer, result in significantly
higher CPU-time, compared to that required for mesh-3 and mesh-4, which utilize
first-order tetrahedral elements.
Mesh
Mesh-1
Mesh-2
Mesh-3
Mesh-4
No. of nodes
41783
53346
16174
15030
No. of elements
27881
35930
78173
75996
CPU Time
50 hours 44 minutes
84 hours 28 minutes
10 hours 40 minutes
9 hours 56 minutes
Table 3.1: CPU run-time comparison for various meshes.
34
-~~Jnu-T-~
-
-
--
-=
u
P/2
-----.
ii
--
~-----
-~--------
(applied)
P/2 (calculated)
d=38.1
D = 25.4
B = 25.
................
.
t=25.4
16.5
D
s = 63.5
2.4-
L = 203.2
(a)
2
3
U
Uncracked Surface
(fixed in direction-3
Crack Front
Cracked Surface
Front symmetry plane
(fixed in direction-2)
A77
(b)
Figure 3-1: Specimen geometry (a) Full geometry. (b) Quarter geometry which is
modeled in finite element method. All dimensions are in mm.
35
-.-
.4
e
-
~--
--
-
4PI4
.4-.
.4..
(b) Void nucleation.
(a) Inclusions in a ductile matrix.
o
00
0
0
(d) Strain localiation between voids
(c) Void growth.
(f) Void coalescence and fracture.
(e) Necking between voids.
Figure 3-2: Schematic of void nucleation, growlth, and coalescence in ductile metals
(taken from [2]).
36
.
- -
Top view
Crack-containing-plane
2
3
(a)
V
C3D1OM elements
Brick elements
(b)
Figure 3-3: Details of mesh-1. (a) Isometric view. (b) Zoomed isometric view.
37
* y 1 r~
*
--
-~
2
3
(c)
3
(d)
2
1
uncracked
ligament
cracked ligament
(e)
Figure 3-3 (Continued): Details of mesh-1. (c) Front view. (d) Top view. (e) View of the
crack-containing-plane.
38
-~------
-U
Crack-containing-plane
3
(a)
I
Brick elements
C3D1OM elements
(b)
Figure 3-4: Details of mesh-2. (a) Isometric view. (b) Zoomed isometric view.
39
2
3
(c)
3
(d)
2
I
uncracked
ligament
cracked ligament
(e)
Figure 3-4 (Continued): Details of mesh-2. (c) Front view. (d) Top view. (e) View of the
crack-containing-plane.
40
Top view
Crack-containing-plane
2
3
Front View
(a)
1
C3D4 elements
Prism elements
(b)
Figure 3-5: Details of mesh-3. (a) Isometric view. (b) Zoomed isometric view.
41
32
(c)
1
(d)
2
cracked ligament
uncracked
ligament
(e)
Figure 3-5 (Continued): Details of mesh-3. (c) Front view. (d) Top view. (e) View of the
crack-containing-plane.
42
Top view
Crack-containing-plane
2
3
1I
Only C3D4 elements
(b)
Figure 3-6: Details of mesh-4. (a) Isometric view. (b) Zoomed isometric view.
43
2
3
(c)
2
(d)
2
I
cracked ligament
uncracked
ligament
(e)
Fig. 3-6 (Continued): Details of mesh-4. (c) Front view. (d) Top view. (e) View of the
crack-containing-plane.
44
8000
7000 . ..
.... ......
. ......
...... ...................... .............. ......
.
......
Mesh-2
.. .......................
6000
Mesh-1
.... .........................
INX
.
............
Mesh-4
5000
..............
.............................. M O S4_3
....... ..
\\
z
-0
CO 4 0 0 0 . .............................
....................... ..... ..
.....................
............... .... ........
3 0 0 0 . .............................. ............ ....................
...........
....... .....
2000
............................
1 000
.. ........................... .............
0
0
..... ...........
................... .................. . ..
10
5
Displacement of rollers (mm)
Figure 3-7: Load-displacement curves.
45
..............
.....
15
VVF
(Ave. Crit.: 75%)
+6.682e-01
+6.125e-01
+5.568e-01
+5.011e-01
+4.454e-01
+3.898e-01
+3.341e-01
+2.784e-01
+2.227e-01
+1.670e-01
+1.114e-01
+5. 568e-02
+0.000e+00
(a)
VVF
(Ave.
Crit.: 75%1
+6.6 94e-01
+6. 137e-01
+5.-579e -01
+5.021e-01
+4.463e-01
+3.905e-01
+3. 347e- 01
+2. 789e-0
+2.231e-01
+1 .674e -OI1
+1.116e-.O
+5.
579e-02
(b)
Figure 3-8: Contours of void volume fraction plotted on the crack-containing-plane
(at a load-point displacement of 10.4 mm), obtained with (a) Mesh-1. (b) Mesh-2.
46
VVF
(Ave. Crit.: 75%)
+6.844e-01
+6 .274e-01
+5. 703e-01
+5.133e-01
+4.563e-01
+3.992e-01
+3.422e-01
+2.852e-01
+2 .28le-01
+1. 71le-01
+1.141e-01
+5.703e-02
+0.000e+00
(c)
VVF
(Ave. Crit.: 75%)
+6.672e-01
+6.116e-01
+5.560e-01
+5.004e-01
+4.448e-01
+3. 892e-01
+3. 336e-01
+2. 780e-01
+2. 224e-01
+1. 668e-01
+1.112e-01
+5.560e-02
+0.000e+00
(d)
Fig. 3-8 (Continued): Contours of void volume fraction plotted on the crack-containingplane (at a load-point displacement of 10.4 mm), obtained with (c) Mesh-3. (d) Mesh-4.
47
S, S33
(Ave. Crit.: 75%)
+3.109e+03
+1.655e+03
+1.449e+03
+1.242e+03
+1.036e+03
+8.300e+02
+6.238e+02
+4.175e+02
+2.112e+02
+5.000e+00
-2.012e+02
-4.075e+02
-6.138e+02
-8.200e+02
-2.528e+03
(a)
S, S33
(Ave. Crit.: 75%)
+3.742e+03
+1.655e+03
+1.449e+03
+1. 242e+03
+1.036e+03
+8.300e+02
+6.238e+02
+4 . 175e+02
+2.112e+02
+5.000e+00
-2.012e+02
-4 . 075e+02
-6. 138e+02
-8.200e+02
-2.999e+03
(b)
Figure 3-9:
-3 3 Contours of stress (Or 33 ) plotted on the crack-containing-plane (at a
load-point displacement of 10.4 mm), obtained with (a) Mesh-1. (b) Mesh-2.
48
S,
S33
(Ave. Crit.: 75%)
+2.492e+03
+1. 830e+03
+1.592e+03
+1.354e+03
+1.116e+03
+8.783e+02
+6.404e+02
+4.025e+02
+1.646e+02
-7.333e+01
-3.112e+02
-5 .492e+02
-7. 871e+02
-1.025e+03
-4 .632e+03
(c)
S,
S33
(Ave. Crit.: 75%)
+2.174e+03
+1.696e+03
+1.460e+03
+1.223e+03
+9.870e+02
+7.507e+02
+5.143e+02
+2.780e+02
+4. 167e+O1
-1.947e+02
-4.310e+02
-6 .673e+02
-9. 037e+02
-2.140e+03
-4. 174e+03
(d)
Fig. 3-9 (Continued): Contours of stress (- 33 ) plotted on the crack-containing-plane
(at a load-point displacement of 10.4 mm), obtained with (c) Mesh-3. (d) Mesh-4.
49
-
-
-
~44
I
I
I
44
I
I
I
I
I
-
7
4,
h
Face of
tetrahedral element
N
44
44
44
44
4443
'I
16
5
2
----
-~
Brick-element 1
Direction of motion of crack-front
3 added
brick elements
Figure 3-10: Brick elements attached to a tetrahedral element.
50
Chapter 4
Finite Element Results with
Abaqus/Explicit
In the last chapter, finite element results obtained using Abaqus/Standard were
summarized.
Since Abaqus/Standard does not easily accommodate element dele-
tion/failure (which is required to simulate the evolution of the crack-front), another
software package, Abaqus/Explicit [21, 22, 23], is used in this chapter. Abaqus/Explicit
easily accommodates element failure. When an element fails in Abaqus/Explicit, its
internal nodal forces are set to zero while its mass remains allocated to its nodes.
The specimen geometry is shown in Fig. 3-1.
An explicit-dynamic analysis is
performed in which a displacement, u, is applied at the center of the top roller while
the bottom rollers are held fixed. This displacement, u, is applied through a smoothly
ramped-up displacement boundary condition. Again, because of symmetry, only one
quarter of the specimen is meshed with finite elements. The meshes considered in this
chapter are mesh-1 (Fig. 3-3), mesh-3 (Fig. 3-5), and mesh-4(Fig. 3-6). The details
of these meshes have already been given in the last chapter (section 3.1).
4.1
Mass Density Scaling
Abaqus/Explicit uses an explicit time integration scheme to march forward in time
with very small time increments. These time increments have to be smaller than a
51
critical value for a stable solution. For a given mesh (element size), the critical value
of stable time increment is proportional to square-root of the material mass density
[10],
Ate,. c VP.
(4.1)
Since the problem under consideration is quasi-static (material mass density does
not affect the stresses in a quasi-static problem), but is being solved by an explicit
dynamic analysis, therefore to reduce the total computation time, material mass
density is scaled-up from its actual value of p
=
7.83 x 103kg/m 3 . Density is scaled-
up by a factor of 1.3 x 108 for mesh-1 and by a factor of 6400 for mesh-3 and mesh-4.
Scaled-up mass density results in larger stable time increments, thereby reducing total
number of increments, and so the total computation time decreases. Higher mass
density also results in larger values of system kinetic energy, all the other factors
being equal. Since the problem under consideration is quasi-static, the ratio of total
deformation energy to kinetic energy is monitored for all the simulations, so as to
keep the kinetic energy of the system at a suitably small value. These kinetic energy
and total deformation energy curves are plotted in Fig. 4-1. For mesh-3 and mesh-4,
kinetic energy is indeed very small compared to total deformation energy. But for
mesh-1, kinetic energy is considerably large, which will give poor results. A lesser
mass density scaling is required for mesh-1 to provide acceptable results. With the
current mass scaling (1.3 x 108), the cpu-time for mesh-i is 12-days, 18-hours, so
further reducing mass density will reduce the stable time increment, but the total
computation time will increase by a factor of 2-3, resulting in a cpu-time of about
30-days, which is prohibitively high. The reason for a very high total computation
time for mesh-1 is explained below.
Mesh-1 utilizes C3D1OM elements along with an attached brick layer. This attached brick layer has many nodes which are constrained to move along with the
attached mid-face nodes of C3D1OM elements (as explained in section 2.5). These degrees of freedom, corresponding to the constrained nodes, are statically condensed at
each increment, which results in very high total computation time as Abaqus/Explicit
52
takes a large number of increments to solve the problem.
Because of the above,
meshes which have large number of constrained nodes are not suitable for analysis
with Abaqus/Explicit.
4.2
Comparison of Material Models
Plasticity modeling is done in three different ways in this chapter; these ways are
described schematically in Fig. 4-2. In the first case (case-1), the GTN model with
element failure is restricted to only one layer near the crack-containing-plane, while
all other elements have been assigned a material model of classical (non-porous, nondilating), strain hardening (Fig. 2-1), metal plasticity, as shown in Fig. 4-2(a). This
type of model has been previously used by [13] for crack growth in compact tensile
(bend) specimens. In the second case (case-2), the GTN porous plasticity model is
assigned throughout the mesh. Again, the GTN model is augmented with element
failure and deletion in only one layer near the crack-containing-plane; in the remainder
of the mesh no element failure/deletion is implemented, as shown in Fig. 4-2(b). In
this case matrix hardening of Fig. 2-1 along with the Gurson parameters of Table 2.2
are used. In the last case (case-3), the GTN plasticity with element deletion is used
throughout the mesh, as shown in Fig. 4-2(c); and again matrix hardening of Fig. 2-1
along with the Gurson parameters of Table 2.2 are used.
The load-displacement curves (for mesh-3) for the above three idealizations of
material models are plotted in Fig. 4-3. These load-displacement curves show some
oscillations since the quasi-static problem is analyzed using explicit dynamics.
Here, case-2 and case-3 give identical load-displacement curves. In case-3, elements
were free to fail anywhere in the mesh, but they failed only in the first layer next to
the crack-containing-plane, as can be seen in Fig. 4-4 (which plots contours of normal
stress
(U-33)
on the crack-containing-plane).
Since elements failed only in one layer
next to the crack-containing-plane in case-3, and they were allowed to fail in the same
layer in case-2 as well, therefore the load-displacement curves obtained for these two
cases are identical.
53
Also, the load-displacement curve of case-1 is below the load-displacement curve
of case-2 or case-3. This is explained as follows: Void growth causes the elements
to soften, i.e., they deform more at the same load level. In case-2 and case-3, elements soften away from the crack-containing-plane (elements attached to the crackcontaining-plane also soften), as can be seen in the void volume fraction contours
shown in Fig. 4-5. While in case-1 softening of elements was allowed only in one layer
next to the crack-containing-plane, the rest of the mesh was not allowed to soften
(non-porous, non-dilating classical plasticity).
This causes the case-1 specimen to
deform less (since there is no void growth away from the crack-containing-plane) at
the same load level, compared to case-2 or case-3, and so the load-displacement curve
for case-1 is below the load-displacement curve of case-2 and case-3. Thus, case-1
falsely indicates that the system is less stable.
The contours of equivalent plastic strain and void volume fraction for all three
cases, at load point displacement of 10.5 mm, are shown in Fig. 4-6 and Fig. 47, respectively. These figures show identical contours for case-2 and case-3, while
slightly different contours for case-1 are observed. Figure 4-7 shows the crack profile
(fracture surface) for all three cases at load point displacement of 10.5 mm. Crack
tunneling is observed in these simulations, because the rate of void growth is higher
in high stress triaxiality than in a low stress triaxiality environment. In our specimen
the back-free-surface has a low stress triaxiality (as it is traction free) and the frontsymmetry-plane has a high stress triaxiality (behaves like plane strain near frontsymmetry-plane). This results in faster crack growth near the front-symmetry plane
and a much slower growth near the back-free-surface, resulting in significant crack
tunneling. Void volume fraction contours showing faster crack growth near the frontsymmetry-plane and slower growth near back-free-surface are shown in Fig. 4-8, 4-9,
and 4-10. These figures are showing the position of crack front at increasing load-point
displacement for all three cases.
The cpu-time for the three cases are given in Table 4.1. The table shows almost
identical cpu-times for case-2 and case-3, and a little less cpu-time for case-1. The advantage of lesser cpu-time in case-1 is only marginal; hence the case of using the GTN
54
material model throughout the mesh, with failure, is a better choice for simulating
ductile crack growth.
4.3
Comparison of Various Meshes
Results obtained on mesh-3 and mesh-4 are compared in this section. The GTN
model with element failure is used throughout the mesh for both the meshes.
Fig. 4-11 shows the fracture surface for mesh-3 and mesh-4 at load-point displacement of 10.5 mm, contours of normal stress (-3 3 ) are plotted in this figure. The
fracture surface obtained with mesh-3 is smooth, whereas the fracture surface obtained with mesh-4 is very rough, as some of the tetrahedral elements have not failed,
though the crack front has advanced beyond these elements. This is happening in
mesh-4 because it has only C3D4 elements on the crack-containing-plane, and when
elements around a tetrahedral element fail, the stress state of the center element becomes uniaxial (because surrounding elements have failed, so no constraint in plane
1-2). Since void growth is very slow in a low-triaxiality environment (as explained
in section 3.4), these elements never reach the critical value of void volume fraction
(which is 0.25 in our case), resulting in a very rough fracture surface. The evolution
of crack front for mesh-3 and mesh-4 is shown in Fig. 4-10 and Fig. 4-12, respectively.
The load-displacement curves for these two meshes are compared in Fig. 4-13.
Mesh-3, which comprises C3D4 elements with an attached layer of C3D6 elements, has
a lower load-displacement curve than that of mesh-4, which has only C3D4 elements
throughout the mesh. The reason for a higher load-displacement curve for mesh-4
was already given in section 3.3. Again, these load-displacement curves show some
oscillations since the quasi-static problem is analyzed using explicit dynamics. The
cpu-time comparison for these two meshes are given in Table 4.1. As can be seen from
the table, mesh-4, which comprises only C3D4 elements, takes a little less cpu-time
than mesh-3, but this difference is only marginal, and given the other problems with
mesh-4 (bad fracture surface and higher load-displacement curve), mesh-3, which
comprises C3D4 elements with an attached prism (C3D6) layer, is a better choice for
55
simulating ductile crack growth.
Mesh
Mesh-3
Mesh-3
Mesh-3
Mesh-4
Material model
Gurson with failure
in one layer,
classical plasticity
in rest (case-1)
Gurson throughout
fail one layer
(case-2)
Gurson with failure
throughout (case-3)
Gurson with failure
No. of nodes
No. of elements
CPU Time
41783
27881
95 hours 17 minutes
53346
35930
95 hours 38 minutes
16174
78173
95 hours 45 minutes
15030
75996
93 hours 23 minutes
throughout (case-3)
Table 4.1: CPU run-time comparison for various cases and meshes.
56
I
20.
I
i
i
15.
.~.
4
0 0
10.P
Total Def. Energy
Kinetic Energy
04
0.2
0.0
0.4
0.8
0.6
Time
1.0
(a)
[x10 3]
[xlO 3]
r-4
:
30.00
30.00
25.00
25.00 :
20.00 -
0
15.00 -
15.00
0%
10.00
20.00 -
>1
>1
/
0)
5.00
1z1Energy
Total Def.
0
Energy
0)
Kinetic
Total Def.
Energy
Kinetic
Energy
.00
5u~
0.00
0.00
10.00
0.00
0.20
0.40
0.60
0.80
1.00
Time
0.00
-
0.20 0.40
0.60
0.80
Time
(c)
(b)
Figure 4-1: Kinetic energy and total deformation energy curves for (a) mesh-1, (b)
mesh-3, and (c) mesh-4.
57
1.00
Classical plasticity
(a)
GTN with failure
Crack- ContainingPlane
GTN with failure
Crack- ContainingPlane
GTN with failure
Crack- ContainingPlane
GTN without failure
(b)
GTN with failure
(c)
Figure 4-2: Material model distribution in finite element mesh (schematic). (a) Case1: GTN model with element failure restricted to only one layer next to the crackcontaining-plane, classical plasticity in the rest of the mesh. (b) Case-2: GTN model
throughout the mesh, element failure restricted to only one layer next to the crackcontaining-plane. (c) Case-3: GTN model with element failure throughout the mesh.
58
8000
7 0 0 0 -.
. . .. . . .. . . . . - - - - - - - ---
. . . . ..
Case-2 & Case-3
6000 -\-
Case--
5000
4000 -
3000-
2000
1000-
0
0
2
6
4
8
10
Displacement of rollers (mm)
Figure 4-3: Load-displacement curves obtained with mesh-3 for the three different
cases of material model distribution.
59
12
S, S33
(Ave. Crit.:
7S)
+7.631e+03
+1 .826e-+03
+1.567e+03
+1.308e+03
+1.,049e+,03
+7,902e+02
+5.313e+02
+2 .725e+02
+1.3S9e+03
-2.453e+02
-5.041e+02
-7.630e+02
-1.022e+03
-1.281e+03
-5 529e+03
(a)
S, S33
(Ave.
CriL.: 7SU
+2.219e+03
+1.742e+03
+1.294e+03
+1,046e+03
+7.983e+02
+S .P04e+02
-1.933e+02
-4.412e+02
1+1.702e+03
-6.892e+02
-9.37!e+02
-4 .87e+03
(b)
S,533
;,Ave. Crit.: 75%)
+2.219e+03
+1. 790e+03
+1.542e-03
+1.294e+03
+1.046e+03
+7.983e+02
+5.504e+02
+3.025e+02
+5.458e+02
-1.933e+02
-4.412e+02
-6.892e+02
-9.371e+02
-c.)5te+03
Li -4. 567e-03
( C)
Figure 4-4: Contours of normal stress (o-33) obtained with mesh-3 (view of the crackcontaining-plane at a load-point displacement of 10.5 mm ). (a) Case-1: GTN model
with element failure in only one layer next to the crack-containing-plane, classical
plasticity in the rest of the mesh. (b) Case-2: GTN model throughout the mesh,
element failure in only one layer next to the crack-containing-plane. (c) Case-3:
GTN model with element failure throughout the mesh.
60
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.000e-02
+1.850e-02
+1.700e-02
+1.550e-02
+2.400e-02
+1.250e-02
(a)
+1.100e-02
+9.500e-03
+8.000e-03
+6.500e-03
+5.000e-03
+3.500e-03
+2.000e-03
+1.041e-04
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.000e-02
+1.850e-02
+1.700e-02
+1.550e-02
+1.400e-02
+1.250e-02
+1.100e-02
+9.500e-03
+8.000e-03
+6.500e-03
+5.000e-03
+3.500e-03
+2.000e-03
+0.000e+00
(b)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.000e-02
+1.850e-02
+1.700e-02
+1.550e-02
+1. 400e-02
+1.250e-02
+1.100e-02
(C
+6.500e-03
+5.000e-03
+3.500e-03
+2.000e-03
+0.000e+00
Figure 4-5: Contours of void volume fraction obtained with mesh-3 (isometric view at
a load-point displacement of 10.5 mm). (a) Case-1: GTN model with element failure
in only one layer next to the crack-containing-plane, classical plasticity in the rest
of the mesh. (b) Case-2: GTN model throughout the mesh, element failure in only
one layer next to the crack-containing-plane. (c) Case-3: GTN model with element
failure throughout the mesh.
61
PEEQ
(Ave. Crit.:
75%)
+1.128e+00
+5.000e-01
+4.583e-01
+4.167e-01
+3.750e-01 )
+3.333e-01(a
+2.917e-01
+2 .500e-01
+2 .083e-01
+1. 667e-01
+1.250e-01
+8 .333e-02
+4 .167e-02
+0.000e+00
PEEQ
(Ave. Crit.:
75%)
+1.090e+00
+4.583e-01
+4 .167e-01
+3.750e-01
+3.333e-01
+2.917e-01
(b)
+2.500e-01
+2.083e-01
+1.667e-01
+1.250e-01
+8.333e-02
+4 .167e-02
+0.000e+00
+5.000e-01.
PEEQ
(Ave. Crit.:
75%)
+1.090e+00
+4.583e-01
+4.167e-01
+3.750e-01
+3 .333e-01
+2.917e-01
+2 .500e-01
+2.083e-01
+1.667e-01
+1.250e-01
+8.333e-02
+4.167e-02
(C)
+0.000e+00
Figure 4-6: Contours of equivalent plastic strain obtained with mesh-3 (isometric
view at a load-point displacement of 10.5 mm). (a) Case-1: GTN model with element
failure in only one layer next to the crack-containing-plane, classical plasticity in the
rest of the mesh. (b) Case-2: GTN model throughout the mesh, element failure
in only one layer next to the crack-containing-plane. (c) Case-3: GTN model with
element failure throughout the mesh.
62
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2. 292e-01
+2. 084e-01
+1.875e-01
+1. 667e-01
+1.459e-01
+1.251e-01
+1. 042e-01
+8.340e-02
+6.258e-02
+4.-175e-02
+2.093e-02
+1.041e-04
(a)
VVF
(Ave. Crit.: 75%)
+2. 500e-01
+2. 292e-01
+2.083e-01
+1.875e-01
+1. 667e-01
+1. 458e-01
+1. 042e-01
+8.333e-02
+1. 250e-01
+6.250e-02
167e-02
+4.
+2.083e-02
+0.000e+00
(b)
VVF
(Ave. Crit.: 75%)
+2.500~e-01
+2. 292e-01.
+2.083e-01
+1. 875e-01.
+1. 667e-01.
+1.458e-01
+1.-250e-01
+1. 042e-01
+8.333e-02
+6 .250e-02
+4. 167e-02
+2. 083e-02
+0. 000e+00
(c)
Figure 4-7: Contours of void volume fraction obtained with mesh-3 (view of the crackcontaining-plane at a load-point displacement of 10.5 mm). (a) Case-1: GTN model
with element failure in only one layer next to the crack-containing-plane, classical
plasticity in the rest of the mesh. (b) Case-2: GTN model throughout the mesh,
element failure in only one layer next to the crack-containing-plane. (c) Case-3:
GTN model with element failure throughout the mesh.
63
......
.......
VVF
(Ave. Crit.: 75%)
+2.500~e-01
5+2
.292e-01.
+2.084e-01.
+1.875e-01.
+1.667e-01.
+1..459e-01.
+.25.e-3
+1.042e-01.
+8.343.e-02
+6 .258e- 02
+4.1376e-02
+2.094e-02
S1.23e-04
(a)
VVF
(Ave. Crit.: 75%)
+2 .500e-01.
5+2 .292e-01.
+2. 084e-01.
+1.875e-01.
+1.667e-01.
+1 .459e-01.
~+.253.e-0.
+1.042e-01.
+8. 340e-02
+6 .258e-02
+4. 175e-02
+2093e-02
+1041.e-04
(b)
VVF
(Ave. Crit.: 75%)
+2 .500e-01.
5+2
.292e-01.
+2. 084e-01.
+1.875e-01.
+1.667e-01.
+1.459e-01.
~+.253.e-03.
+1.042e-01.
+8.340e-02
+6 .258e-02
+4.1375e-02
+2093e-02
+1041.e-04
(C)
Figure 4-8: Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-i. (a) at a load-point displacement of 5.5 mm. (b) at a
load-point displacement of 8.8 mm. (c) at a load-point displacement of 10.5 mm.
64
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2 .292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.334e-02
+6.251e-02
+4.167e-02
+2.084e-02
+8.717e-06
(a)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8. 333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+7.242e-32
(b)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(c)
Figure 4-9: Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-2. (a) at a load-point displacement of 5.25 mm. (b) at a
load-point displacement of 8.8 mm. (c) at a load-point displacement of 10.5 mm.
65
VVF
(Ave. Crit.: 75%)
+2. 500e-01
+2.292e-01
+2. 083e-01
+1. 875e-01
+1. 667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.334e-02
+6.251e-02
+4.167e-02
+2.084e-02
+8.717e-06
(a)
VVF
(Ave. Crit.: 75%)
+2. 500e-01
+2 .292e-01
+2. 083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+7.242e-32
(b)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(c)
Figure 4-10: Contours of void volume fraction, at various load-point-displacement,
obtained with mesh-3, case-3. (a) at a load-point displacement of 5.25 mm. (b) at a
load-point displacement of 8.8 mm. (c) at a load-point displacement of 10.5 mm.
66
S,
S33
(Ave. Crit.: 75%)
+2.148e+03
+1.760e+03
+1.517e+03
+1.273e+03
+1.030e+03
+7.867e+02
+5.433e+02
+3.000e+02
+5.667e+O2
-1.867e+02
-4.300e+02
-6.733e+02
-9.167e+02
-1.160e+03
-4.186e+03
(a)
S,
S33
(Ave. Crit.: 75%)
+2.346e+03
+1.745e+03
+1.502e+03
+1. 260e+03
+1.017e+03
+7.747e+02
+5.321e+02
+2.895e+02
+4.692e+01
-1.957e+02
-4.382e+02
-6.808e+02
-9.234e+02
-1.166e+03
-4 . 363e+03
(b)
Figure 4-11: Contours of normal stress (C-33) at a load-point displacement of 10.0 mm
obtained with (a) mesh-3. (b) mesh-4.
67
NEI effni IE RNi ii M
VVF
(Ave. Crit.: 75%)
+2. 500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(a)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(b)
VVF
(Ave. Crit.: 75%)
+2 .500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6 .250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(c)
Figure 4-12: Contours of void volume fraction, at various load-point-displacements,
obtained with mesh-4; GTN model with element failure throughout the mesh. (a) at
a load-point displacement of 5.0 mm. (b) at a load-point displacement of 8.0 mm.
(c) at a load-point displacement of 10.0 mm.
68
8000
1
1
7000-
Mesh-4
. ...
-- -
6000 - -
.
5000 - - ,
-... .
... ....
..
Mesh+-3
. ....
---
--
. ..-
. ..
.
-..
- --- -
4000
.-
3000
2000
-.-.-
-
01
0
-
-
1000-
2
6
4
8
10
Displacement of rollers (mm)
Figure 4-13: Load-displacement curves obtained with mesh-3 and mesh-4.
69
12
Chapter 5
Summary and Conclusions
In this study ductile crack growth in an open-bend specimen is simulated using the
finite element method. The constitutive material model used is the GTN model for
void-containing materials. Finite element analysis is performed using an implicit
scheme as well as an explicit scheme; the results obtained with these schemes are
presented in chapter 3 and chapter 4, respectively.
Various meshes containing tetrahedral elements (in most of the mesh) are considered for this study, as meshing with tetrahedral elements is well-developed compared
to with other types of elements (such as brick elements). Since ductile crack growth
is largely planar, to restrict the crack-growth only near the crack-containing-plane,
a layer of elements is attached to the tetrahedral elements. A 0.3 mm thick layer
of 8-noded brick elements is attached to C3D1OM elements, while a similar layer of
6-noded prism elements is attached to C3D4 elements. A fortran code is written to
make and attach a layer of 8-noded brick elements to C3D1OM elements.
In the implicit finite element scheme (Abaqus/Standard), various meshes are
used. Those meshes which utilize C3D1OM elements along with an attached brickelement-layer tend to overestimate the load value (post-peak) at a given load-pointdisplacement, as the characteristic element size for these meshes are larger (0.6 mm)
than other meshes which utilize first-order tetrahedral elements (0.3 mm). Also the
meshes which had only C3D4 (first-order tetrahedra) elements overestimated the
(post-peak) load values, compared to the mesh in which C3D4 elements are attached
70
with a prism-element-layer (C3D6) next to the crack-containing-plane. Further, it
is seen that meshes with C3D10M elements take significantly. higher cpu-time (50
hours, 44 minutes, for mesh-1 and 84 hours, 28 minutes, for mesh-2), compared to
the meshes which utilize first-order tetrahedral elements (10 hours, 40 minutes, for
mesh-3 and 9 hours, 56 minutes, for mesh-4). This lead to the conclusion that among
the meshes considered, mesh-3 gives the best result. Mesh-3 utilizes C3D4 elements
throughout the mesh except in one layer next to the crack-containing-plane, where it
uses 6-noded prism elements.
Since the implicit finite element scheme of Abaqus/Standard does not accommodate element failure easily, Abaqus/Explicit, which uses an explicit finite element
scheme, is used to model growth of crack. In this scheme, elements whose void volume
fraction reaches a critical value are deleted from the mesh. It is noted that C3D10M
elements with an attached brick layer can not be used efficiently in Abaqus/Explicit
as the cpu-time required for analysis is prohibitively high (12 days, 18 hours) because
of the multi-point-constraints present on many of the nodes of the brick-layer.
In the explicit finite element scheme, modeling of plastic deformation is done in
three different ways (section 4.2). It is shown that the case in which classical plasticity
is used in all the elements except in one layer next to the crack-containing-plane
(where the GTN model with element failure is used) tends to underestimate the load
values calculated on the center of the rollers. Other cases in which the GTN model is
used throughout the mesh, with failure in only one layer next to the crack-containingplane, and with failure throughout the mesh, give identical results. The cpu-time for
these three cases (Table 4.1) shows that using classical plasticity away from the crackcontaining-plane does not give any significant advantage in cpu-time, and is therefore
not desirable. Load-displacement curves obtained from Abaqus/Standard are higher
than those obtained with Abaqus/Explicit, as can be seen in Fig. 5-1 and Fig. 5-2.
Figure 5-1 shows the load-displacement curves obtained with mesh-3, and Fig. 5-2
shows the load-displacement curves obtained with mesh-4. In both the figures, we
can see that Abaqus/Standard overestimates the load values (post-peak).
Since Abaqus/Standard does not advance the crack front, for the purpose of com71
paring the crack front profiles obtained with Abaqus/Explicit and Abaqus/Standard,
it is assumed in the case of Abaqus/Standard that the elements whose void volume fraction values are higher than 0.25 have "failed". With this assumption, the
location of the crack front at a load-point-displacement of 10.5 mm is shown in
Fig. 5-3(a); the contours of void volume fraction are plotted in this figure. It can be
seen that crack growth on the back-symmetry-plane is 3.64 mm in Abaqus/Explicit,
but 3.74 mm with Abaqus/Standard.
The shape of the crack front is very simi-
lar with both finite element schemes.
But the cpu-time required to perform the
simulation with Abaqus/Explicit is very high compared to the cpu-time required
with Abaqus/Standard, as given in Table 5.1. Though the cpu-time required with
Abaqus/Explicit is higher, Abaqus/Explicit is a better choice for performing simulations involving large (ductile) crack growth, as Abaqus/Standard highly overestimates the (post-peak) load values at a given load-point-displacement. If studies are
to be performed with small amount of cracking, Abaqus/Standard would be a better
choice. One can also write a user material model which will fail the elements when
their void volume fraction value reaches 0.25; this way cpu-time can be reduced (as
an implicit finite element scheme will be used) and a better load-displacement curve
can be obtained.
In Abaqus/Standard, elements on the crack-containing-plane never fail (even at
high void volume fraction values), and continue to carry load in direction-3, this
contributes to higher load values at the same load-point-displacement, compared to
the load obtained with Abaqus/Explicit. The contours, showing positive values of
33 ,
obtained with Abaqus/Stadard for mesh-3 on the crack-containing-plane are shown
in Fig. 5-4. These small stresses in unfailed elements (whose void volume fraction
value exceeds 0.25) have big effect on the load values because of the long "moment
arm" of these elements.
Slope of the load-displacement curve in the crack advance region is of great importance in understanding the stability of ductile crack growth. Less steep load drop (in
crack advance region) predicted by Abaqus/Standard indicates false stability, which
is very undesirable, and therefore Abaqus/Explicit is a better choice for simulating
72
ductile cracking.
Mesh
Mesh-3
Mesh-3
Mesh-4
Mesh-4
No. of nodes
16174
16174
15030
15030
No. of elem ents
78173
78173
75996
75996
Finite element scheme
Implicit
Explicit
Implicit
Explicit
CPU Time
10 hours, 40 minutes
95 hours, 45 minutes
9 hours, 56 minutes
93 hours, 23 minutes
Table 5.1: CPU run-time comparison for implicit and explicit finit e element scheme.
5.1
Future Work
The GTN model used in the current work assumes that the shape of the voids remain
spherical throughout the growth regime, though that is not the case as voids change
their shape and become ellipsoidal. Void shape change effect can be incorporated into
the GTN model, as published in [24] and [25].
73
8000
70001
-
with Abaqus/Standard
--
60001
-
- -
50001
Ca;
0
-j
40001
with Abaqus/Explicit
-
3000
-
2000
1000
0
0
10
5
Displacement of rollers (mm)
Figure 5-1: Load-displacement curve obtained with mesh-3 using Abaqus/Standard
and Abaqus/Explicit.
74
15
7000
with A baqus/Standard
6000 -
-F
5000
with Abaqus/Explicit
4000
I-O
0
3000
............ .... ... .... . . ... .... ... .... ... ..-
2000
-.
. . .. . . . .-.. .
1000
0
0
10
5
Displacement of rollers (mm)
Figure 5-2: Load-displacement curve obtained with mesh-4 using Abaqus/Standard
and Abaqus/Explicit.
75
15
VVF
(Ave. Crit.: 75%)
+6.844e-01
+2 .500e-01
+2.292e-01
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(a)
VVF
(Ave. Crit.: 75%)
+2.500e-01
+2.292e-01.
+2.083e-01
+1.875e-01
+1.667e-01
+1.458e-01
+1.250e-01
+1.042e-01
+8.333e-02
+6.250e-02
+4.167e-02
+2.083e-02
+0.000e+00
(b)
Figure 5-3: Contours of void volume fraction plotted on the crack-containing-plane
for mesh-3 (at a load-point displacement of 10.5 mm). (a) Simulation performed with
Abaqus/Standard. (b) Simulation performed with Abaqus/Explicit.
76
S, S33
(Ave. Crit.: 75%)
+2.650e+03
+1.745e+03
+1.500e+03
+1.254e+03
+1.009e+03
+7.633e+02
+5.179e+02
+2 725e+02
+2:708e+O1
42.13e+02
- 46e+02
-7092e+02
-200e+03
-097e+03
2-
Figure 5-4: Contours of -33 plotted on the crack-containing-plane for mesh-3, showing
positive values of -33 behind crack front.
77
Appendix A
Fortran code for attaching brick
layer
Program makebricks
Implicit none
Integer size
Parameter (size = 80000)
Integer i,j,k,n-nodes,n-el-surf,elem(size/20,6),n-nodesurf,
* surf-nodes(size/10),brick(size/7,8),njlayers,
*
nofbricks, nof-elem,elements(10),bc-nodes(size/10),
*
sizeof_bc-nodes,top-tet(100),no-of
*
noofnodebacktet,below(size/10),back-tet(size/10)
node_top-tet,
start,
Real*8 coord(size,3) ,templ,temp2,thickness,ratio,crack tip_ coord
Real*8 backsurfcoord
Common coordnoofnodeback-tet,crack-tip-coord,back-surfcoord
Character comment*50
Open (unit=2,
c 'n_nodes'
file="2_elements")
is total number of nodes
Read(*,*) comment
Write(*,*)
"Should be *Node,
",comment
Read(*,*) nnodes
78
(corresponding to tetrahedras).
Read(*,*)
(j, coord(i,1),coord(i,2),coord(i,3),
i=1,n_nodes)
c The following reads ALL tetrahederal elements and their connectivity.
Read(*,*) comment
Write(*,*)
"Should be *Element,
",comment
Read(*,*) nofelem ! This is total number of tet elements.
Write(2,3)
3 Format("*Element,
type=C3D10M,
elset=tets")
Do i=1,n-ofelem
Read(*,*)
k, (elements(j), j=1,10)
Write(2,4) i, (elements(j), j=1,10)
4
Format(16,',',9(I6,','),6)
End Do
c 'nelsurf' is number of surface tetrahedral elements.
c "elem" is an array which stores nodes of a surface element in each row.
Read(*,*) comment
Write(*,*) "Should be *Reading tet ...
",
comment
Read(*,*) nelsurf
Read(*,*)
((elem(i,j), j=1,6),
i=1,n_elsurf)
Read(*,*) comment
Write(*,*)
"Should be *Reading layer ...
",
comment
Read(*,*) njlayers
Read(*,*) thickness
Read(*,*) crack-tip-coord
Read(*,*) backsurfcoord
ratio = thickness/n-layers
c "nnodesurf" is number of nodes on crack surface.
c "surfnodes" is an array which stores NODE NUMBERS of crack surface nodes.
Read(*,*) comment
Write(*,*)
"Should be *Reading Interface
Read(*,*) nnodesurf
79
...
",
comment
Read(*,*) (surfnodes(i), i=1,n_node-surf)
c start
=
1
c i=o
c Do while (start .ne. 0)
C i=i+1
c Read(*,*) surfnodes(i)
c If (surf-nodes(i) .eq. 0) Then
c start = 0
c i=i-1
c End If
c End Do
c nnodesurf = i
Write (*,*) "surfnode(last)
= ",
surfnodes(n-node-surf)
c MAKE MID TRIANGLE NODES, ie STORE NODAL COORDINATES.
Open(unit=3,file="3_equation")
Do i=1,n-elsurf
!1,2,3 for X,Y,Z coordinates.
Do j=1,3
Do k = 1,3
!1,2,3 just to add up coordinates in a concise way.
tempt
=
tempt + coord(elem(i,k),j)
temp2
=
temp2 + coord(elem(i,k+3),j)
End Do
coord(n-nodes+i,j) = -1.0/15.0*templ + 2.0/5.0*temp2
tempt = 0
temp2 = 0
End Do
c Write "*EQUATION"
******************************************
Do j=1,3
5
Format('',16,',',I1,',',F4.1,',1)
Write(3,1)
1 Format ("*Equation")
80
Write(3,*)
"7"
Write(3,5) nnodes+i,j,15.0
Do k=1,3
Write(3,5) elem(i,k),j,1.0
End Do
Do k=4,5
Write(3,5) elem(i,k),j,-6.0
End Do
Write(3,7) elem(i,6),j,-6.0
7 Format('',16,',',I1,',',F4.1)
End Do
c Finished writing "*EQUATION" *******************************
surf-nodes(nnodesurf+i) = nnodes + i !Updates NODE NUMBERS of surface.
End Do
Close (unit=3)
c MAKE BRICKS, ie STORE NODAL CONNECTIVITY.
Do j=1,3*nelsurf,3
i=(j+2)/3
brick(j,1)
= nnodes + i
brick(j,2)
= elem(i,6)
brick(j,3)
= elem(i,1)
brick(j,4)
= elem(i,4)
brick(j+1,1) = n-nodes + i
brick(j+1,2) = elem(i,4)
brick(j+1,3) = elem(i,2)
brick(j+1,4) = elem(i,5)
brick(j+2,1) = n-nodes + i
brick(j+2,2) = elem(i,5)
brick(j+2,3) = elem(i,3)
brick(j+2,4) = elem(i,6)
81
surf-nodes(nnodesurf + i) = n-nodes + i
End Do
n_nodes = nnodes + nelsurf
!Update total nuber of nodes.
n_nodesurf = nnodesurf + nelsurf
!Updates number of nodes on surface.
c Define node below each node on matching surface for making bricks.
c below(k) = a node no.
which is right below "k" th node.
Do i=1,n-nodesurf
below(surf.nodes(i))
= i + nnodes
coord(i+n-nodes,1)
= coord(surf -nodes(i) ,1)
coord(i+n-nodes,2)
= coord(surf -nodes(i) ,2)
coord(i+nnodes,3)
= coord(surf -nodes(i),3)
- ratio
End Do
c Define the remaining four nodes of a brick.
Do i=1,3*nelsurf
brick(i,5) = below(brick(i,1))
brick(i,6) = below(brick(i,2))
brick(i,7) = below(brick(i,3))
brick(i,8) = below(brick(i,4))
End Do
Write(*,*)
"One Brick Layer Complete"
c ########################## ONE BRICK LAYER COMPLETE #########################
Write(*,*)
"Total number of layers",
n.layers
Do j=1,n-layers-1
Do i=3*n-el-surf*j+1,3*n-el-surf*(j+1)
brick(i,1) = brick(i-3*n-el-surf,5) ! Periodic numbering.
brick(i,2) = brick(i-3*n_elsurf,6) ! Previous brick's 6th node
brick(i,3) = brick(i-3*n-el.surf,7) ! is second node of this brick
brick(i,4) = brick(i-3*n-el-surf,8) ! and so on.
brick(i,5) = brick(i,1)+n-nodesurf
brick(i,6) = brick(i,2)+n-nodesurf
82
! This brick's sixth node is
brick(i,7) = brick(i,3)+n-nodesurf
brick(i,8)
= brick(i,4)+n-nodesurf
! this brick's second node +
! "nnodesurf" and so on.
c Fetching coordinates of newly added nodes.
Do k=1,4
coord(brick(i,k+4),1) = coord(brick(i,k),1)
coord(brick(i,k+4),2) = coord(brick(i,k),2)
coord(brick(i,k+4),3) = coord(brick(i,k),3) - ratio
End Do
End Do
End Do
Write(*,*)
"All Brick layers complete"
c ##########################
ALL BRICK LAYERS COMPLETE ########################
n_nodes = nnodes + njlayers*n-nodesurf
!Update total nuber of nodes.
n_ofbricks = 3*nelsurf*njlayers
Open (unit=1, file='1_node')
Write(1,12)
Write(1,13)
Write(1,14)
12 Format("*Heading")
13 Format("Symmetric Cuboid")
14 Format("**")
Print node numbers and their coordinate ****************
c ***************
Write(1,11)
11 Format("*Node")
Do i=1,n-nodes
Write(1,2)
2
i,
(coord(i,j),
j=1,3)
Format(16,',',2(E13.6,','),E13.6)
End Do
c ***************
Print brick elements and their connectivity ****************
Write(2,8)
83
8 Format("*Element, type=C3D8, elset=bricks")
Do i=1,n-ofbricks
Write(2,6)
6
i+nofelem,
(brick(i,j),j=1,8)
Format(17,',',7(I7,','),I7)
End Do
c "bcnodes" is an array which will (after executing subroutine) store node
c
numbers on which boundary conditions will be applied (ie the uncracked part).
c "sizeofbcnodes" is the size of array "bcnodes".
Call test(size,n-nodes,n-node-surfbcnodes,size-of-bcnodes)
c#############
c The following writes the node nubers of the crack interface surface for BCs.
Write(2,9)
9 Format("*Nset,
Nset=uncracked-interf acesurface")
k = sizeofbcnodes/7 ! I want 7 node numbers in a line for *.inp.
Do i=Ok-1
Write(2,10)
(bc-nodes(7*i+j),
j=1,7)
10 Format('',7(I5,','))
End Do
Write(2,10)
(bc-nodes(i),
i=7*k+1,sizeof_bcnodes)
c#############
c The following writes the node numbers of top tet surface for applying load.
c Write(2,15)
c15 Format("*Nset,
c Read(*,*)
Nset=top-surface-oftets")
comment
c Write(*,*)
"Should be *Reading top tet
...
",
comment
c Read(*,*) noofnode-toptet
c Read(*,*) (top-tet(i), i=1,no-of-nodetop-tet)
c k = noofnodetop-tet/7 ! I want 7 node numbers in a line for *.inp.
c Do i=0,k-1
c Write(2,17)
(top-tet(7*i+j),
j=1,7)
84
c End Do
c Write(2,17) (top-tet(j), j=7*k+1,noofnode-top-tet)
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
c The following writes the element numbers of contact set (closer)
c Write(2,18)
c18 Format("*Elset, elset=closercontact.set)
c Read(*,*) comment
c Write(*,*) "Should be *Reading closer contact set ...
",
comment
c Read(*,*) nnodeclosercontt
c Read(*,*) (closer-contt(i), i=1,nnodeclosercontt)
c k = n-nodecloser-contt/7 ! I want 7 element numbers in a line for *.inp.
c Do i=O,k-1
c Write(2,17) (closercontt(i), i=1,n_nodecloser-contt)
c End Do
c Write(2,17) (closercontt(i), i=7*k+1,nnodeclosercontt)
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
c The following writes the element numbers of contact set (farther)
c Write(2,18)
c18 Format("*Elset, elset=farthercontactset)
c Read(*,*) comment
c Write(*,*) "Should be *Reading farther contact set ...
",
comment
c Read(*,*) nnodefarthercontt
c Read(*,*) (farthercontt(i), i=ln_nodefarthercontt)
c k = nnodefarthercontt/7 ! I want 7 element numbers in a line for *.inp.
c Do i=O,k-1
c Write(2,17) (farthercontt(i), i=1,n-nodefarthercontt)
c End Do
c Write(2,17) (farthercontt(i), i=7*k+1,nnodefarthercontt)
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
85
c The following writes the node numbers of symmetry plane per-r to dir-n 1.
Write(2,16)
16
Format("*Nset,
Nset=backsurfaceof_tets")
Read(*,*) comment
Write(*,*)
"Should be *Reading back surface
",comment
Read(*,*) noofnodebacktet
Write(*,*)
"noofnodebacktet
=",noofnodebacktet
Read(*,*) (backtet(i), i=1,noofnodeback-tet)
Write (* ,*)
"backtet(last) = ", back-tet(noofnodeback-tet)
Call backtest(sizen_nodes,n-node-surf,back-tet,n-layers)
k = noofnodeback-tet/7 ! I want 7 node numbers in a line for *.inp.
Do i=O,k-1
Write(2,17) (back-tet(7*i+j), j=1,7)
17
Format('',7(I5,','))
End Do
Write(2,17) (back-tet(j), j=7*k+1,no-ofnode_backtet)
Read(*,*) comment
Write(*,*)
comment
Close (unit=1)
Close (unit=2)
End
SUBROUTINE test(size,n-nodes,nnode-surf,bc-nodes,size-ofbcnodes)
Implicit none
Integer i,j
,size-of bcnodes,n-node-surf,bcnodes(size/10)
Integer n-nodes,size, no-ofnodebacktet
Real*8 crack-tip-coord, coord(80000,3),back-surfcoord,
* crackfrontcoord
Common coord,no-of-nodeback-tet,crack-tip-coord,back-surf-coord
Write(*,*)
"Executing Subroutine test (subl.f)"
j=0
86
If (size .ne. 80000) Then
Write(*,*)
"*WARNING
coord-size is different in subroutime and main"
End If
Do i=nnodes-n-nodesurf+1,n_nodes
CALL check(coord(i,1), crackfront.coord)
If (coord(i,2).ge.crack-front.coord) Then
j=j+1
bc-nodes(j) = i
End If
End Do
sizeofbcnodes =
j
Write(*,*) "sizeofbc-nodes",j
If (size/10 .le. sizeofbcnodes) Then
Write(*,*)"*WARNING: Size of array bcnodes is not large enough"
End IF
Write(*,*)
"Fininshed Executing Subroutine test (subl.f)"
Return
End
SUBROUTINE backtest(size,nnodesn_nodesurf,backtetn_layers)
Implicit none
Integer i,j ,size,n-nodes,n_nodesurf ,back-tet(size/40),
* njlayers,no-ofnodebacktet
Real*8 backsurfcoord,
coord(80000,3),temp,
crack-tip-coord
Common coordnoofnodeback-tetcrack-tip-coord,back-surfcoord
Write(*,*)
"Executing Subroutine backtest
(sub2.f)"
j=0
If (size .ne. 80000) Then
Write(*,*)
"*WARNING
coord-size is different in subroutine and main"
End If
Do i=n-nodes-n-layers*n-nodesurf+1, nnodes
87
temp=coord(i,1) - backsurf-coord
If (abs(temp) .lt. 1.OE-5) Then
j=j+1
back-tet(no-ofnodebacktet+j)=i
End If
End Do
= noofnodebacktet
noofnodebacktet
Write(*,*)
"noofnodeback-tet",
Write(*,*)
"size =", size
If (size/40 .le.
+
j
noofnodebacktet
noofnode-backtet) Then
Write(*,*)"*WARNING:
Size of array backtet is not large enough"
End If
Write(*,*)
"Fininshed Executing Subroutine backtest
Return
End
88
(sub2.f)"
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