Micromechanics of Toughening in Polymeric
Composites
by
Rajdeep Sharma
B.E. (Hons.) Mechanical Engineering
Birla Institute of Technology and Science, Pilani, India, 1998
M.S. Mechanical Engineering
University of Maryland at College Park, 2000
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2006
Massachusetts Institute of Technology 2006. All rights reserved.
Author .......................
Department of Mechanical Engineering
Augjus19, 2006
C ertified by ...........................
Mary &,'Boyce
Gail E. Kendall Professor of Mechanical Engineering
Thesis Sqpervisor
7,)
Certified by ...........................
Simona Socrate
d'Arbeloff Assistant ProfesA of Mechanical Engineering
Thess Supervisor
Accepted by ...
Lallit Anand
Chairman, Department Committee on Graduate Students
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2
Micromechanics of Toughening in Polymeric Composites
by
Rajdeep Sharma
Submitted to the Department of Mechanical Engineering
on August 19, 2006, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The macroscopic tensile toughness of glassy homo-polymers is often limited due to
either localized crazing followed by failure [e.g., polymethyl-methacrylate (PMMA),
polystyrene (PS)], or cavitation-induced brittle failure under high triaxiality [e.g.,
polycarbonate (PC)]. Judicious choice of polymer composites alleviates the above
concerns by spreading the inelastic deformation and damage throughout the material,
thereby increasing the macroscopic toughness. This thesis focuses on the microand macro-mechanics of two polymer composite systems - ductile/brittle PC/PMMA
microlaminates and rubber-toughened PS.
Ductile/brittle microlaminates are comprised of alternating layers of ductile polymer (e.g., PC) that inelastically deform by shear yielding, and brittle polymer (e.g.,
SAN, PMMA) that undergo crazing in tension. The layer thicknesses are typically in
the sub-micron to tens of micron range. We have modeled the deformation and axial tensile toughness of PC/PMMA micro-laminates. Experiments indicate that the
macroscopic ductility of ductile/brittle polymeric laminates depends on several factors such as the thickness of the brittle and ductile layers, the volume fraction of the
ductile component, as well as strain rate. The nominally brittle layer can undergo inelastic deformation by both crazing and shear-yielding, with the relative contribution
of these mechanisms being dependent on the laminate morphology and strain rate. In
particular, with decreasing brittle layer thickness, the inelastic behavior of the brittle
layer is dominated by shear-yielding. We present a micromechanical model for twophase ductile/brittle laminates that enables us to capture the macroscopic behavior,
as well as the underlying micro-mechanisms of deformation and failure, in particular
the synergy between crazing and shear yielding. The finite element implementation
of our model considers a two-dimensional and three-dimensional representative volume element (RVE), and incorporates continuum-based physics-inspired descriptions
of shear yielding and crazing, along with failure criteria for the ductile and brittle
layers. The interface, between the ductile and brittle layers, is assumed to be perfectly bonded. The model is used to probe the effect of laminate parameters, such as
the absolute and relative layer thicknesses, and material properties on the behavior
during tensile loading. In addition, our modeling approach can be generalized to
3
other laminate systems, such as two-phase brittle-1/brittle-2 and three-phase ductile1/brittle/ductile-2 laminates, as well as to more complex loading conditions. Results
from our studies reveal that the 2D RVE does not adequately capture the effect of
volume fraction of the constituents on the laminate toughness. However, the 3D RVE
captures the effect of volume fraction based on the extent of craze tunneling through
the width of the specimen; at high volume fractions of PC, crazes emanating from the
surface do not tunnel through the specimen width significantly, while at low volume
fractions of PC, the crazes tunnel through the entire specimen width. The 3D RVE
captures the strain-rate effect on toughness based on the greater rate-sensitivity of
shear yielding compared to craze initiation, thereby increasing the craze density in the
laminate at higher rates. The length-scale effect is captured by the 3D RVE, based
on decrease in the craze opening rate and damage confinement by the PC layers.
It is well-known that the incorporation of a small volume fraction (10-25 %)
of micron-order size, compliant and well-dispersed rubbery particles in (brittle and
crazeable) polystyrene (PS) yields considerable dividends in tensile toughness at the
expense of reduction in stiffness and yield strength. In commercial rubber-toughened
PS, the rubbery particles often have a composite "salami" morphology, consisting
of 70-80 % volume fraction of sub-micron PS occlusions dispersed in a topologically,
continuous polybutadiene (PB) phase. While it is recognized that these composite
particles play the dual role of providing multiple sites for craze initiation in the PS matrix and allow the stabilization of the crazing process through cavitation/fibrillation
in the PB-phase within the particle, the precise role of particle morphology, as well
as the particle-matrix interface are not well understood or quantified. This work
probes the micromechanics and macromechanics of uni-axial tensile deformation and
failure in rubber-toughened PS through axi-symmetric finite element representative
volume element (RVE) models that can guide the development of blends of optimal
toughness. The RVE models reveal the effect on craze morphology and toughness by
various factors such as particle compliance, particle morphology, particle fibrillation
and particle volume fraction. The principal result of our study is that particle compliance and particle heterogeneity alone cannot account for the macroscopic behavior
of HIPS, as well as the experimentally observed craze profile. Fibrillation/cavitation
of PB domains within the heterogeneous particle provides the basic key ingredient to
account for the micro- and macro-mechanics of HIPS.
Thesis Supervisor: Mary C. Boyce
Title: Gail E. Kendall Professor of Mechanical Engineering
Thesis Supervisor: Simona Socrate
Title: d'Arbeloff Assistant Professor of Mechanical Engineering
4
Acknowledgments
I feel greatly privileged and honored to have worked on my doctoral dissertation
with Professor Mary Boyce and Professor Simona Socrate; their guidance throughout
my research was quite valuable.
I have great professional respect for their work
and their high academic standards, but perhaps as importantly, I have found them
to be friendly and approachable.
They possess the admirable trait of encouraging
independent thinking in their students. I would like to thank Professor Robert Cohen
and Professor Gareth McKinley for serving on my Doctoral Committee, and providing
valuable input on my research.
I gratefully acknowledge Professor Ali Argon for indulging me in numerous discussions, primarily on the dilatational plasticity in amorphous materials; he was instrumental in directing me to the key scientific papers in this field. On a personal
note, I have quite enjoyed my long conversations with him.
I am grateful to Sharon Soong and Ben Wang for our brief collaboration on polymeric laminates, and for several stimulating conversations. I would also like to thank
the entire Mechanics and Materials group, especially Michael Demkowicz, Nici Ames,
Sauri and Prakash, for the good and enjoyable times at MIT. Many thanks to Una
and Ray for providing administrative support, and for creating a "sanity" buffer in
a (crazy) research environment. I would like to take this opportunity to thank Leslie
Regan for her commendable work, and for her going out of her way to help students
in their time of need.
It is perhaps impossible to convey in words the critical role that my parents have
played in almost every aspect of my life. I am eternally indebted to them for their selfeffacing love. I feel very fortunate for having a loving and caring brother, Pradeep, and
sister-in-law, Haleh. My wife, Smitha, is perhaps the best thing that has happened
in my life, and has given me great joy. Her love, support and understanding make
me feel truly blessed.
5
6
Contents
1
Introduction
2
Constitutive models for large strain inelastic deformation of poly-
35
2.1
Shear yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.2
Material properties for shear yielding model
. . . . . . . . . . . . .
43
2.3
C razing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.3.1
Craze initiation . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.3.2
Craze growth
. . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.3.3
Craze breakdown
. . . . . . . . . . . . . . . . . . . . . . . .
55
2.3.4
A continuum framework of crazing
.
.
.
.
.
.
meric materials: shear yielding and crazing
57
Material properties for crazing model . . . . . . . . . . . . . . . . .
60
.
. . . . . . . . . . . . . .
.
2.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.2
M odel
. .. . . . . . . . . . . . . . . . .
70
. . . . . . . . . . . . . .
71
72
.
. . . . . . . . . . .....
Description of 2D and 3D RVE
3.2.2
Constitutive description for the ductile and brittle layers
3.2.3
Crazing parameters for PC/PMMA microlaminates . . .
80
3.2.4
PC/PMMA interface description
. . . . . . . . . . . . .
83
Resu lts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
.
.
.
3.2.1
3.3.1
Prelim inaries
. . . . . . . . . . . . . . . . . . . . . . . .
87
3.3.2
Results from 2D simulations . . . . . . . . . . . . . . . .
88
.
3.3
.
3.1
.
63
.
Defc rmation and failure of ductile/brittle polymeric laminates
.
3
25
7
3.4
4
Results from 3D simulations . . . . . . . . . . . . . . . . . . .
96
3.3.4
R ate effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
3.3.5
Length-scale effect
. . . . . . . . . . . . . . . . . . . . . . . .
126
3.3.6
Sensitivity and design studies . . . . . . . . . . . . . . . . . .
126
3.3.7
Role of craze flow stress and craze initiation stress on ductility
139
. . . . . . . . . . . . . . . . . . . . . . . .
143
Summary and conclusions
Deformation and failure of toughened polystyrene
149
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
4.2
M odel
152
4.3
5
3.3.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1
Geometry of the RVE
. . . . . . . . . . . . . . . . . . . . . .
152
4.2.2
Constitutive model . . . . . . . . . . . . . . . . . . . . . . . .
156
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
4.3.1
Parametric study on compliant, homogeneous particles
. . . .
159
4.3.2
Response of RVE with homogenized particle . . . . . . . . . .
162
4.3.3
Response of RVE with heterogeneous particle (no fibrillation)
170
4.3.4
Response of RVE with heterogeneous particle (with fibrillation) 171
4.3.5
Effect of particle volume fraction
. . . . . . . . . . . . . . . .
174
4.3.6
Effect of critical fibrillation stress -fib . . . . . . . . . . . . . .
177
4.3.7
Summary and Conclusions . . . . . . . . . . . . . . . . . . . .
183
Summary and future work
185
5.1
Ductile/brittle polymeric laminates . . . . . . . . . . . . . . . . . . .
186
5.2
Toughened polystyrene . . . . . . . . . . . . . . . . . . . . . . . . . .
188
5.3
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
5.3.1
Ductile/brittle polymeric laminates . . . . . . . . . . . . . . .
190
5.3.2
Toughened polystyrene . . . . . . . . . . . . . . . . . . . . . .
192
References
195
A Evaluation of the Inverse Langevin Function
207
8
B Effective elastic properties of the craze "element"
9
211
10
List of Tables
2.1
Shear yielding material parameters for PC and PMMA
2.2
Craze parameters for PS in HIPS (Piorkowska et al., 1990) and PMMA
hom opolym er
3.1
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Comparison of craze parameters for PMMA microlayers and PMMA
hom opolym er. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
44
80
Weibull parameters for C within the bulk of the specimen and at the
surface.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Values of 7% and
vo0 for the baseline study and for the new study
.
4.1
Craze parameters for PS matrix in HIPS (Piorkowska et al., 1990)
.
4.2
Particle elastic properties used to investigate the role of particle com-
83
139
.
159
pliance on the deformation and toughness of rubber-toughened PS . .
162
11
12
List of Figures
1-1
Davidenkov schematic showing the qualitative effect of strain-rate and
temperature on the ductile-to-brittle transition in a glassy polymer.
The effect of superposed triaxiality would be to raise the ay curve (relative to cf), thereby shifting the ductile-to-brittle transition to lower
strain rates and higher temperatures.
1-2
. . . . . . . . . . . . . . . . . .
Uni-axial tension and compression response of PC (from Boyce and
A rruda, 1990) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-3
26
27
Shearbands in (a) PS and (b) PMMA, obtained from plane-strain compression experiments at room temperature (from Bowden and Raha,
19 70 ).
. . . . . . . . . . .
29
Tensile stress-strain response of PS and HIPS (from Bucknall, 1977).
31
simple schematic blowup of the craze structure.
1-5
28
Crazing in a tensile PMMA specimen (from Bucknall, 1977), and a
13
.
1-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-6
(a) Shear yielding and crazing envelopes for PMMA at 70'C (adapted
from Sternstein and Ongehin (1969)). Solid lines for the shear yielding
and crazing envelopes in the first quadrant are based on experimental
data; the dashed lines for the envelopes in the remaining quadrants are
extrapolations based on the pressure sensitive von-Mises shear yielding
model and the Sternstein and Ongchin crazing model. (b) Biaxial experiments of Kawagoe and Kitagawa (1981) on PMMA in air at 65"C.
Curves (a), (b), (c) and (d) are based, respectively, on the Sternstein
and Ongchin (1969) model, Oxborough and Bowden (1973) model,
Argon and Hannoosh (1977a) model and the Kawagoe and Kitagawa
(1981) model. (c) Temperature dependence of uni-axial shear yielding
and crazing envelopes for PS (Haward, 1972).
1-7
. . . . . . . . . . . . .
33
(a) Optical micrograph of the ductile/brittle PC/SAN laminate (Gregory et al., 1987) (b) TEM micrograph of a thin HIPS section, prepared
from a tensile specimen beyond yield (Bucknall, 2001).
2-1
. . . . . . . .
34
(a) True stress-strain experimental data (dashed lines) for uni-axial
compression of PC at strain rates -1.0 s-- , -0.1 s--
and -0.01 s-,
showing the increased yield and flow strength with increasing strain
rate.
Model predictions (solid lines) adequately capture the experi-
mental data. From Boyce and Arruda (1990).
(b) True stress-strain
experimental data for uniaxial compression and plane strain compression of PMMA at a strain rate of -0.001 s-1, showing the pressure
dependence of yield, shear flow and strain-hardening.
Model predic-
tions, based on the eight chain model of Arruda and Boyce (1993a),
adequately capture the experimental data. From Arruda and Boyce
(1993b ).
2-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
One-dimensional spring-dashpot arrangements to model shear yielding;
(a) proposed by Haward and Thackray (1968); (b) proposed by Boyce
et al. (2001).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
39
2-3
Isothermal uni-axial stress-strain response of PC and PMMA at true
strain-rates of 0.0017 s 1
2-4
and 0.017 s
1
.
. . . . . . . . . . . . . . . .
44
(a) A simplified craze schematic, showing the internal structure of a
craze with the fibrils perfectly aligned in the direction of the maximum principal stress.
(b) A refined representation of the craze in-
ternal structure showing cross-tie fibrils.
Primary fibrils are slightly
misaligned from the direction of the maximum principal stress. From
D onald (1997).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2-5
Biaxial craze initiation data of Sternstein and Ongchin (1969) for PMMA. 49
2-6
(a) Sternstein-Ongchin (solid lines) and Oxborough-Bowden (dashed
lines) do not capture the biaxial craze initiation data for PMMA in
the presence of kerosene.
a and b curves correspond to two different
sets of material parameters for the Sternstein-Ongchin model, while
c and d correspond to two different sets of material parameters for
the Oxborough-Bowden model.
(b) Argon-Hannoosh and Kawagoe-
Kitagawa models capture the biaxial craze initiation data for PMMA
in the presence of kerosene.
2-7
U1 -
U2
space, (b)
U, -
9h
space.
. . . . . . . . . . . . . . . . . .
53
TEM micrograph showing that craze breakdown initiates from the edge
of the craze, and not from the midrib. From Donald (1997).
2-9
52
Comparison of the Sternstein-Ongchin and Argon-Hannoosh locus in
(a)
2-8
. . . . . . . . . . . . . . . . . . . . . . .
. . . . .
56
Schematic of the craze "finite element". Cross-tie fibrils, which bridge
the primary craze fibrils, are not shown.
. . . . . . . . . . . . . . . .
61
2-10 Example of continuum behavior of a (a) PMMA craze element, and
(b) PS craze element. The craze element is of dimension 1 pum, with
im posed velocity vi ... . . . . . . . . . . . . . . . . . . . . . . . . . .
15
62
3-1
(a) The effect of PC volume fraction on the nominal stress-strain response of 32-layered PC/PMMA. The loading direction is coincident
with the co-extrusion direction.
rate= 1.7 x 10-
3
Laminate thickness=1 mm; strain
s- 1 . Redrawn from the experimental data of Kerns et
al. (2000). (b) The specimen geometry (drawn to scale) used by Kerns
et al. (2000). The exact profile of the variation of the width along
the 1-direction, from 30 mm at the ends to 5 mm at the center of the
specimen, is not known. The loading (and coextrusion) direction is the
1-direction .
3-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
Micrograph of necked region of 32-layered PC/PMMA: fpc = 0.7 deformed at a nominal strain rate of 1.7 x 10-
s- 1 . Laminate thickness=1
mm. The loading (and co-extrusion) direction is the 1-direction. From
Kerns et al. (2000). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
66
Stress-strain response of PC/PMMA microlaminates with fpc = 0.7 as
a function of number of layers, and strain rate. Laminate thickness=1
mm. From (Kerns et al., 2000).
3-4
. . . . . . . . . . . . . . . . . . . . .
67
Optical micrographs, in the 1-2 plane, of PC/SAN with fpc = 0.65.
(a) and (b) respectively show the craze morphology at a depth of 20Pm
and 60pm from the free surface, indicating that crazes emanating from
the surface do not necessarily propagate completely in the 3-direction.
From Sung et al. (1994c).
3-5
. . . . . . . . . . . . . . . . . . . . . . . .
68
Optical micrographs, in the 1-3 plane, of PC/SAN with layer thicknesses (a) tpc = 13 pm,
Pm, tSAN
=
16 ,um. (fpc
tSAN =
=
33 i-m (fpc
0.28), (b) tpc = 29
0.65). In both (a) and (b), micrographs
corresponding to point (1) show the presence of surface crazes.
At
point (2), the surface crazes in (a) tunnel through the width, while in
(b), they do not. From Sung et al. (1994c) . . . . . . . . . . . . . . .
3-6
69
The geometry of the 2D RVE along with relevant dimensions and
boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
73
3-7
The geometry of the 3D RVE. The in-plane dimensions and boundary
conditions are same as that for the 2D RVE. . . . . . . . . . . . . . .
74
. . . . . . . . . . . . . . . .
75
3-8
Mesh for (a) 2D RVE, and (b) 3D RVE.
3-9
(a) The dependence of the craze initiation stress a, and the fracture
stress
UF
on 0, which is the angle between the molecular orientation
direction and the tensile loading direction. From Beardmore and Rabinowitz (1975). (b) The dependence of the median strain to craze nucleation, breakdown and fracture, on the pre-strain cor in PS films. The
pre-strain is in the direction of loading. From Maestrini and Kramer
(1991)..........
....................................
77
3-10 (a) Stress-strain behavior of unannealed PC/SAN laminates for fpc =
0.65 as a function of number of layers.
extrusion direction.
(b) Comparison of stress-strain behavior of 776-
layered laminate with fpc
annealed", "11
Tensile loading is in the co-
=
0.65 for four cases: "I
unannealed" and "II annealed".
unannealed", "I
"I" and "11" indicate
that the direction of loading is, respectively, perpendicular and parallel
to the coextrusion direction. . . . . . . . . . . . . . . . . . . . . . . .
79
3-11 (a) Weibull probability density function (pdf), (b) Weibull cumulative density function (cdf), for craze initiation parameter C and craze
initiation stress at the free surface of the RVE. The values of C corresponding to /2 - a',p' p' + a' (where p
and u' are, respectively,
the mean and standard deviation of C at the surface of the RVE) are
indicated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3-12 (a) Weibull probability density function (pdf), (b) Weibull cumulative
density function (cdf), for craze initiation parameter C and craze initiation stress in the bulk of the RVE. The values of C corresponding
to t
-b a b
11 b
+ a
(where pb and ac
are, respectively, the mean
and standard deviation of C in the bulk of the RVE) are indicated.
17
.
85
3-13 Shear loci of PC and PMMA, and craze loci of PMMA based on the
following four values of craze initiation parameter C: minimum (-Y,),
1p1w,
/pw
- o-, and pw
+ o-w (a) for free surface of the RVE, (b) for bulk
of the RV E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3-14 Macroscopic response of 2D laminate with fpc = 0.3 at nominal strain
rate
e
= 1.7 x 10-s-1.
Based on craze initiation properties corre-
sponding to 7.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-15 Contour plots of craze opening
2 Oc
90
(left column) and accumulated
plastic shear strain -yP (right column), corresponding to Figure 3-14
(2D plane strain, fpc = 0.3).
. . . . . . . . . . . . . . . . . . . . . .
3-16 Contour plots of craze opening
2
91
Oc and accumulated plastic shear
strain 7P, corresponding to Figure 3-14 (2D plane strain, fpc = 0.3).
92
3-17 Macroscopic response of 2D laminate with fpc = 0.7 at nominal strain
rate e = 1.7 x 103s1.
Based on craze initiation properties corre-
sponding to -.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-18 Contour plots of craze opening
2
93
Oc (left column) and accumulated
plastic shear strain -y1 (right column), corresponding to Figure 3-17
(2D plane strain, fpc = 0.7) . . . . . . . . . . . . . . . . . . . . . . .
3-19 Contour plots of craze opening
2
94
Oc (left column) and accumulated
plastic shear strain -yP (right column), corresponding to Figure 3-17
(2D plane strain, fpc = 0.7) . . . . . . . . . . . . . . . . . . . . . . .
95
3-20 Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 2D plane strain RVEs at nominal strain
rate
e=
1.7 x 10-3 S-1, as a function of fpc. Based on craze initiation
properties corresponding to -y (Figure 3-11) . . . . . . . . . . . . . .
3-21 Evolution of (a)
#PMMA
97
and (b) qpc,with nominal axial strain for 2D
plane strain RVEs at nominal strain rate e = 1.7 x 10-
s -,
as a
function of fpc. Based on craze initiation properties corresponding to
-y (Figure 3-11).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
98
3-22 Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 2D plane strain RVEs at nominal strain
rate
e
=
1.7 x 10- 3 s-1 , as a function of fpc. Based on craze initiation
properties corresponding to -% (Figure 3-12). . . . . . . . . . . . . . .
#PMMA
3-23 Evolution of (a)
99
and (b) qpc,with nominal axial strain for 2D
plane strain RVEs at nominal strain rate
e
=
1.7 x 10-
3
s- 1 , as a
function of fpc. Based on craze initiation properties corresponding to
b%(Figure
3-12).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-24 Macroscopic response of 3D laminate with fpc
rate
e=
100
0.3 at nominal strain
=
1.7 x 10- 3 s-1. . . . . . . . . . . . . . . . . . . . . . . . . . .
102
3-25 Contour plots of craze opening 2 o (left column) and accumulated plastic shear strain -yP (right column) corresponding to Figure 3-24 (3D,
fpc = 0.3).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
3-26 Contour plots of craze opening 2 0 and accumulated plastic shear strain
-yP corresponding to Figure 3-24 (3D, fpc = 0.3).
. . . . . . . . . . .
104
3-27 Contour plots of failed PC and PMMA elements corresponding to Fig-
ure 3-24 (3D , fpc = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . .
105
3-28 Accumulated plastic shear strain at e = 0.202, corresponding to Fig-
ure 3-24 (3D, fpc = 0.3) . . . . . . . . . . . . . . . . . . . . . . . . .
3-29 Macroscopic response of laminate with fpc
rate
e
=
106
0.7 at nominal strain
= 1.7 x 10- 3 s-1. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3-30 Contour plots of craze opening 2 0 and accumulated plastic shear strain
-yP corresponding to Figure 3-29 (3D, fpc = 0.7).
. . . . . . . . . . .
110
3-31 Contour plots of craze opening 2 o and accumulated plastic shear strain
-yP corresponding to Figure 3-29 (3D, fpc = 0.7).
. . . . . . . . . . .111
3-32 Contour plots of craze opening 2 o for 3D RVE with fpc = 0.7, viewed
in the 1-2 plane, at different depths: (a) x 3
pm; (c) x 3
=
40 pm.
=
0 (surface); (b) x 3 = 20
Contour plots correspond to Figure 3-29 at
macroscopic strain e = 0.07.
. . . . . . . . . . . . . . . . . . . . . .
19
112
3-33 Contour plots of failed PMMA and PC elements corresponding to Fig-
ure 3-29 (3D , fpc = 0.7). . . . . . . . . . . . . . . . . . . . . . . . . .
113
3-34 Accumulated plastic shear strain at e = 0.34, corresponding to Fig-
ure 3-24 (3D , fpc = 0.7). . . . . . . . . . . . . . . . . . . . . . . . . .
114
3-35 Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 3D RVEs at nominal strain rate of 1.7 x
10-
s
1
, as a function of fpc. . . . . . . . . . . . . . . . . . . . . . .
3-36 Evolution of (a)
#PMMA
and (b)
#pc,with
115
nominal axial strain for 3D
RVEs at nominal strain rate of 1.7 x 10-3 S-1, as a function of fpc.
.
116
3-37 Statistical scatter in the macroscopic stress-strain and the normalized
craze volume response, as a function of fpc. .
. . .. . . . . . . . . . 117
3-38 Statistical scatter in volume fraction of failed PC and PMMA, as a
. . .. . .......... .. 118
function of fpc. . . . . . . . . . .
.
3-39 Effect of rate on craze loci (at laminate surface) and shear loci. Loci
in solid lines correspond to
lines correspond to
e=
e
= 1.7 x 102 s- 1 , while loci in dashed
1.7 x 10-3 s-1 . . . . . . . . . . . . . . . . . .
119
3-40 The effect of strain rate on (a) nominal stress-strain and (b) normalized
craze volume, for fpc = 0.6. . . . . . . . . . . . . . . . . . . . . . . .
3-41 The effect of strain rate on (a)
and (b)
OPMMA
3-42 Contour plots for craze opening at (a)
e
#pc
, for fpc = 0.6.
= 1.7 x 10-
3
and (b)
e
120
121
=
1.7 x 10-2, at e = 0.34 for fpc = 0.6. . . . . . . . . . . . . . . . . . .
122
3-43 Contour plots for accumulated plastic shear strain in PC layers at (a)
e
= 1.7 x 10' and (b)
e
= 1.7 x 10-2, at e = 0.34 for
fpc = 0.6. . . . 123
3-44 Contour plots for accumulated plastic shear strain in PMMA layers at
(a)
e=
1.7 x 10-3 and (b)
e=
1.7 x 102, at e = 0.34 for fpc = 0.6.
3-45 Failed PMMA and PC elements at (a) 6 = 1.7 x 10-3 and (b)
1.7 x 10-2, at e = 0.34 for fpc = 0.6
e
124
=
. . . . . . . . . . . . . . . . . .
125
3-46 The effect of doubling the number of layers (with fixed laminate thickness), .i.e., decreasing absolute thicknesses on (a) nominal stress-strain
and (b) normalized craze volume, for fpc = 0.6 at
20
e=
1.7 x 10-2 s
1
127
3-47 The effect of doubling the number of layers (with fixed laminate thickness), i.e. decreasing laminate thicknesses on (a)
for fpc = 0.6 at
e=
1.7 x 10-
s-1
OPMMA
and (b) /pc,
. . . . . . . . . . . . . . . . . . .
128
3-48 The effect of doubling the number of layers (with fixed laminate thickness), i.e. decreasing absolute layer thicknesses on (a) nominal stressstrain and (b) normalized craze volume, for fpc = 0.3,0.5 at e =
1.7 x 10-
3
s- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
3-49 The effect of doubling the number of layers (with fixed macroscopic
thickness) on (a) #PMMA and (b) #pc, for fpc = 0.3,0.5 at
10-3 s-1 .
. . . . . . . . .
e=
1.7 x
. . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . .
130
3-50 Contour plots for craze opening for (a) thick-layered and (b) thinlayered laminate with fpc = 0.6 at e = 0.34 and
e
= 1.7 x 10-2 s
1
. .
131
3-51 Contour plots for accumulated plastic shear strain in PC layers for (a)
thick-layered and (b) thin-layered laminate with fpc = 0.6 at e = 0.34
and e = 1.7 x 10-2 s 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
3-52 Contour plots for accumulated plastic shear strain in PMMA layers
for (a) thick-layered and (b) thin-layered laminate with fpc = 0.6 at
e = 0.34 and
e=
1.7 x 10-2 s l. . . . . . . . . . . . . . . . . . . . . .
133
3-53 Failed PMMA and PC elements for (a) thick-layered and (b) thinlayered laminate with fpc = 0.6 at e = 0.34 and 6 = 1.7 x 10-2 s 1 .
134
3-54 Effect of the gradient of craze initiation properties on (a) nominal
stress-strain and (b) normalized craze volume-strain response, for fpc =
0.3 and fpc = 0.7. "HG" stands for the higher gradient studies.
3-55 Effect of the gradient of craze initiation properties on (a)
#PMMA
. . .
136
and
(b) #pc, for fpc = 0.3 and fpc = 0.7. "HG" stands for the higher
gradient studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
3-56 Contour plots for fpc = 0.3 and fpc = 0.7 with higher gradient of
craze initiation properties.
. . . . . . . . . . . . . . . . . . . . . . . .
138
3-57 Effect of critical stretch and craze strength on (a) nominal stress-strain
and (b) normalized craze volume-strain response, for fpc = 0.7 . . . .
21
140
3-58 Effect of critical stretch and craze strength on (a)
#PAMA
and (b)
#pc
, for f c = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
3-59 Contour plots for effect of lower craze strength and lower craze stretch
142
3-60 An alternative route to accounting for laminate ductility: evolution of
macroscopic nominal stress and normalized craze volume with nominal
m acroscopic strain.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
144
3-61 An alternative route to accounting for laminate ductility: evolution of
Opc and
4-1
#PMMA
with nominal macroscopic strain. . . . . . . . . . . .
145
(a) The stress-strain response of HIPS (right) with particle volume fraction
fp
= 0.2, average particle size=2.5 pim at a strain-rate=1 x 104
s_. Micrograph (left) reveals the composite nature of the particle and
shows multiple crazing in the PS matrix. From Dagli et al.
(1995).
(b) Micrograph showing multiple crazing in the PS matrix and fibril-
lation of PB in the particle in thin films of HIPS (tested within TEM).
From Cieslinski (1991). (c) TEM micrograph of a thin HIPS section,
prepared from a tensile specimen beyond yield (Bucknall, 2001). . . .
4-2
Axisymmetric V-BCC cell in the deformed configuration, along with a
periodic neighbor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
.
157
. . . . . . . . . . . . . . .
158
4-4
The craze locus of the PS matrix in HIPS.
4-5
The macroscopic RVE response for particle volume fraction of
0.28, as a function of particle shear modulus G.
fp
=
............
164
Contour plots of the craze profile for Gp = 0.62 MPa (corresponding
to Figure 4-6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-8
163
Macroscopic response for RVE with PB particle G = 0.62 MPa (fp =
0 .28 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7
155
Axisymmetric RVE geometry for the homogeneous particle and the
heterogeneous particle. PS subinclusions in (b) are shown in white.
4-6
150
165
Contour plots of the craze profile for Gp = 0.62 MPa (corresponding
to Figure 4-6) (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
166
4-9
Comparison of the contour plots of the craze profile for Gp = 6.2, 62, 620
MPa at an axial strain Ez = 0.067
167
. . . . . . . . . . . . . . . . . . .
4-10 The macroscopic RVE response for particle volume fraction of f, =
0.28. The 2-phase particle was represented by its effective elastic properties (KHOM = 1940 MPa and GHOM
=
168
111 MPa) . . . . . . . . . .
4-11 Contour plots of the craze profile, corresponding to Figure 4-10. The 2phase particle was represented by its effective elastic properties (KHOM
1940 MPa and GHOM =111 M a) . . . . . . . . .
-. . . .
=
. . .
4-12 The macroscopic RVE response for particle volume fraction of
169
f, =
0.28. The particle is modeled as a 2-phase system, but with no fibrillation in the PB domains.
. . . . . . . . . . . . . . . . . . . . . . . .
172
4-13 Contour plots of the craze profile, corresponding to Figure 4-12. The
particle is modeled as a 2-phase system, but with no fibrillation in the
PB domains. Particle volume fraction f, = 0.28. . . . . . . . . . . . .
173
4-14 The macroscopic RVE response for particle volume fraction of fp =
0.28. The particle is modeled as a 2-phase system, with fibrillation in
the PB dom ains.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
4-15 Contour plots of fibrillation in PB (left), and craze profile (right), corresponding to Figure 4-14. The particle is modeled as a 2-phase system,
with fibrillation in the PB domains. Particle volume fraction f, = 0.28. 176
4-16 (a) The predicted effect of volume fraction on the deformation and
toughness of HIPS, (b) The effect of particle volume fraction. From
experiments of Correa and de Sousa (1997).
. . . . . . . . . . . . . .
178
4-17 Craze damage contour plots, at E, = 0.144, for (a) fp = 0.08, (b)
fp = 0.18, and (c) f, = 0.28. . . . . . . . . . . . . . . . . . . . . . . . 179
4-18 Enhanced representation of craze damage contour plots, at E, = 0.144,
for (a) f, = 0.08, (b) f, = 0.18, and (c) fp = 0.28 (see Figure 4-17).
4-19 Effect of
c-fib
180
on the macroscopic response of HIPS at various levels of
f1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1
4-20 Craze damage contour plots for f, = 0.28 at E, = 0.04. . . . . . . . .
23
182
A-1
(a) Inverse Langevin function vs. normalized chain, (b) % absolute
error for the Cohen and Bergstrom-Boyce approximation. . . . . . . .
24
209
Chapter 1
Introduction
Glassy polymers, such as polycarbonate (PC), poly(methyl-methacrylate) (PMMA)
and polystyrene (PS), are finding widespread use in innumerable consumer products,
automotive parts, impact resistance windows, transparent eye shields and armor, and
even in aerospace applications.
Some of the key properties that make amorphous
polymers attractive include low density, transparency, ease of processing and fabrication, low cost, and a remarkable flexibility in tailoring properties - by physical
or chemical means - that facilitates an optimum product design.
A key challenge,
however, posed by the use of glassy polymers is related to the strong dependence of
toughness on loading conditions, such as strain rate, temperature, and tri-axiality.
The Davidenkov schematics (e.g., Orowan, 1949), shown in Figure 1-1, illustrate the
effect of strain-rate and temperature on the ductile-to-brittle transition observed in
many glassy polymers. They embody two fundamental ideas
" The operative inelastic mechanism - brittle fracture or yield - is the one that
occurs at the lower stress.
" Relative to the axial stress at yield ay, the axial stress at fracture
Uf
is typically
less sensitive to strain-rate and temperature. For example, Vincent (1960) has
shown that o 1 increases by a factor of 1.6 for PMMA when the temperature is
decreased from 20"C to -196"C, while in an almost similar temperature regime
of -180"C to 20 0 C, Ward and Sweeney (2004) indicate a typical factor of 10 for
25
Axial stress at
yield cy
Axial stress at
fracture ar,
i
00
CO
U,
Ductile
Brittle
Strain rate or (Temperature)"
Figure 1-1: Davidenkov schematic showing the qualitative effect of strain-rate and
temperature on the ductile-to-brittle transition in a glassy polymer. The effect of
superposed triaxiality would be to raise the o-, curve (relative to uf), thereby shifting
the ductile-to-brittle transition to lower strain rates and higher temperatures.
the yield strength o of glassy polymers.
An important implication of the latter idea for designers is the danger inherent in
an unfavorable shift of the ductile-to-brittle transition (defined by the intersection of
f/ay curves), rendering a "tough" polymer "brittle". The detrimental effect of high
triaxiality y =
Uh/e
(where Uh is the hydrostatic stress and o-e is the von-Mises stress)
is most clearly observed in polymers such as PC, which display considerable tensile
toughness at room temperature and quasi-static strain-rates (see Figure 1-2), but fail
in a brittle manner in the presence of cracks or sharp notches that locally increase the
triaxiality ahead of the crack-tip or notch (e.g., Nimmer and Woods, 1992). There
are two underlying inelastic mechanisms that affect the macroscopic toughness of
glassy homo-polymers and the brittle-to-ductile transition - "shear yielding" 1 and
crazing. Shear yielding, a nearly isochoric deformation process, initiates via localized
shear banding that can stabilize depending upon the strain softening and hardening
behavior of the polymer, leading to plastic deformation that spreads through large
volumes of material.
Figure 1-3 shows discrete shearbands in PS, and relatively
1
"distortional plasticity" is more accurate, as this mechanism involves both yield and plastic flow
due to shear processes. In the literature, "shear yielding" is more commonly used.
26
300
200
V1
VI)
4)
S100
-
g'tensio
compression
0.0
1,5
1.0
0.5
True Strain
Figure 1-2: Uni-axial tension and compression response of PC (from Boyce and Ar-
ruda, 1990).
diffuse shearbands in PMMA, which occur during plane-strain compression. Figure 12 shows the uni-axial tension and compression behavior of PC. The behavior shown
in Figure 1-2 is typical of the response of glassy polymers deforming inelastically
by shear yielding, and is characterized by a stiff (visco)-elastic response, followed
by yield and post-yield strain-softening, and finally substantial strain-hardening; the
mechanisms underlying the phenomenology of shear yielding will be discussed in more
detail in Chapter 2. Here, we refer to a "ductile" polymer as one which exhibits shear
yielding plasticity at room temperature in uni-axial tension, and a "brittle" polymer
as one which exhibits brittle failure at room temperature in uni-axial tension.
In
contrast to shear yielding, crazing is a manifestation of dilatationalplasticity unique
to polymers, and is characterized by the formation and growth of crazes (essentially
crack-like structures bridged by highly drawn and load-bearing polymeric fibrils),
followed by their breakdown and crack initiation.
Figure 1-4 shows several crazes
in a tensile PMMA specimen. The crazes are oriented perpendicular to the tensile
loading direction, a typical feature of unoriented polymers.
Figure 1-4 also shows
a schematic blow-up of a craze and its surrounding, and in particular the highly
drawn polymeric fibrils.
Typical craze thickness and length dimensions are found
27
Figure 1-3: Shearbands in (a) PS and (b) PMMA, obtained from plane-strain compression experiments at room temperature (from Bowden and Raha, 1970).
to be in the range 0.1-2 pm and 50-1000 pm, respectively (e.g., Kramer, 1983).
The typical stress-strain response of a craze-able homo-polymer such as PS, shown
in Figure 1-5, comprises an elastic response followed by brittle failure; the stressstrain response of high impact polystyrene (HIPS) shown in the same figure will be
discussed shortly. The brittle failure of PS in tension is associated with the premature
localization of deformation and failure along a dominant craze that traverses the entire
cross-section of the specimen. Other brittle polymers such as PMMA and styreneacrylonitrile (SAN) show similar behavior in tension. An important consequence of
the dilatational nature of crazing is that the crazing mechanism is inhibited under
compressive loadings so that the normally brittle polymers such as PS and PMMA
undergo extensive shear yielding in compression, as has already been seen in Figure 13.
The operative inelastic mechanism - crazing or shear yielding - depends on polymer chemistry, morphology, loading history and temperature. Figure 1-6(a) shows
the shear yielding and craze envelopes, from biaxial experiments of Sternstein and
Ongchin (1969), for PMMA at 700C, indicating that crazing is the dominant mech28
Loading
id Craze fibril
Figure 1-4: Crazing in a tensile PMMA specimen (from Bucknall, 1977), and a simple
schematic blowup of the craze structure.
anism in the first quadrant. Interestingly, the figure also shows that the crazing and
shear yielding envelopes intersect in the second and fourth quadrant, highlighting
that crazing and shear yielding can coexist under certain loading conditions; this has
been experimentally verified by Sternstein and Myers (1973). Figure 1-6(b) shows the
biaxial experimental data of Kawagoe and Kitagawa on PMMA in air at 650C, and
also indicates the craze loci predicted from the Sternstein and Ongchin (1969) model,
Oxborough and Bowden (1973) model, Argon and Hannoosh (1977a) model and the
Kawagoe and Kitagawa (1981) model; these models will be discussed in Chapter 3.
Figure 1-6(c) shows the temperature dependence of the craze and shear yielding envelopes for PS under uni-axial tension, and indicates that the shear yielding envelope
is significantly more sensitive to temperature than the crazing envelope. Hence, while
crazing is the operative inelastic mechanism at room temperature, both crazing and
shear yielding will be operative at a (higher) temperature of 750C.
29
It is clear from the preceding discussion that design of components using glassy
homo-polymers, particularly where toughness is desirable, may be untenable in view
of either the notch sensitivity of otherwise tough polymers such as PC, or premature localization along a dominant craze in polymers such as PMMA, PS, or SAN.
Indeed, glassy homo-polymers are known to have low plane strain fracture toughness
values: e.g., (Gc)pAIMA-
0.5 kJ/m2 (Berry, 1961) and (G1 c)pc - 1.9 kJ/m2 (es-
timated from K1 , = 2.2 MPam1 /2, Parvin and Williams, 1975). The achievement
of optimal macroscopic toughness relies both on the local dissipative nature of the
inelastic mechanisms, and on the participation of a significant volume of the material
in the inelastic deformation. Since both shear yielding and crazing are locally ductile
mechanisms of deformation, a key strategy to improve the macroscopic toughness of
glassy polymers is to distribute the deformation and damage processes throughout
the volume of the deforming material. In view of such considerations, polymeric composites have been successfully used over the recent decades. An example of a tough
polymeric composite is rubber-toughened PC (e.g., Yee, 1977; Cheng et al., 1994);
the notch sensitivity of PC has been found to be relieved by the incorporation of a
small volume fraction of rubber particles that help in reducing the high triaxial stress
states ahead of crack-tips and/or notches (e.g., Socrate and Boyce, 2000; Johnson,
2001; Danielsson et al., 2002; Parsons et al., 2004).
The purpose of this thesis is to explore the micro- and macro-mechanics of deformation and toughening in two different polymeric composites - ductile/brittle microlaminates and rubber-toughened PS - subjected to uni-axial and plane strain tension
loading. Ductile/brittle microlaminates are comprised of alternating layers of a ductile polymer (e.g., PC) that inelastically deforms by shear yielding, and a brittle
polymer (e.g., SAN, PMMA) that undergoes crazing in tension. The layer thicknesses are typically in the sub-micron to tens of micron range. This system has been
extensively studied by Baer and co-workers (e.g., Gregory et al, 1987; Ma et al.,
1990; Sung et al. 1994c) for PC/SAN microlaminates, along with a limited study on
PC/PMMA microlaminates (Kerns et al., 2000). A principal finding of their work
is the synergistic interactions between shear yielding in PC and crazing in SAN (or
30
50j
PS
40
E
7 30
U) Z
H1IPS
10
STRAIN
%)
Figure 1-5: Tensile stress-strain response of PS and HIPS (from Bucknall, 1977).
PMMA) (see Figure 1-7(a)) and the stabilization of the crazing process in the brittle
layers, resulting in multiple crazing. The second system to be studied in this thesis,
rubber-toughened PS, is comprised typically of 10-30 % volume fraction of micronorder sized compliant, rubbery particles that are incorporated within a PS matrix.
Such an arrangement allows for multiple crazing in the PS matrix (e.g., Bucknall
and Smith, 1965; Seward, 1970; Kambour and Russell, 1971; Dagli et al., 1995),
thereby increasing the volume of material participating in inelastic deformation. The
rubbery particles are typically heterogeneous, consisting of about 70-80 % PS in the
form of sub-micron size PS occlusions, incorporated within about 20-30 % polybutadiene (PB). The overall composite so formed is referred to as high impact polystyrene
(HIPS). Multiple crazing in the PS matrix, and the fibrillation of thin, bridging PB
layers that are confined between larger occluded PB particles, can be seen in Figure 1-7(b). The typical stress-strain response for HIPS is shown in Figure 1-5. While
both ductile/brittle laminates and rubber-toughened PS have been extensively tested
experimentally, detailed modeling efforts that clarify the underlying micromechanics,
and provide a springboard for design of these material systems to optimize toughness,
are lacking; the modeling efforts in this thesis are directed to bridge this gap.
31
The outline of this thesis follows. Chapter 2 discusses the physics and modeling
of shear yielding and crazing. The constitutive models for shear yielding and crazing
have been implemented into the finite element software ABAQUS, and are extensively
used in the micromechanical modeling of ductile/brittle microlaminates and rubbertoughened PS.
Chapter 3 begins with summarizing the pertinent experimental work on ductile/brittle PC/SAN and PC/PMMA microlaminates, and is followed by the analysis
of laminate micromechanical models to give insight into the effect of volume fraction
of the ductile polymer, absolute layer thickness, and strain-rate. Several parametric
studies are included to understand the effect of material parameters on the deformation and tensile toughness of the microlaminates.
Chapter 4 is devoted to the micromechanical modeling of rubber-toughened PS,
and probes the effect of several factors including particle compliance, particle heterogeneity, particle volume fraction, and the role of particle fibrillation on the deformation and failure in this material system. The results are compared with experiments.
Chapter 5 summarizes the contributions and conclusions of the thesis, and gives directions for future work.
32
Crazlng,*..
envelope
Equblaidal
3000 psi
Shear
yeldlng
envelope
Pure
shear
(a)
(a)
q (14n)
a0
zr'q arn
42
W
. p/ K*
-30
-20
-10
0
10
20
(b)
YIELD STRESS
CANG
goo
STRESS
PROASLE MJO.
S
--IC
-so
///US
0
so
TUAPO*URE (*c)
100
(C)
Figure 1-6: (a) Shear yielding and crazing envelopes for PMMA at 70 0C (adapted
from Sternstein and Ongchin (1969)). Solid lines for the shear yielding and crazing envelopes in the first quadrant are based on experimental data; the dashed lines for the
envelopes in the remaining quadrants are extrapolations based on the pressure sensitive von-Mises shear yielding model and the Sternstein and Ongchin crazing model.
(b) Biaxial experiments of Kawagoe and Kitagawa (1981) on PMMA in air at 65"C.
Curves (a), (b), (c) and (d) are based, respectively, on the Sternstein and Ongchin
(1969) model, Oxborough and Bowden (1973) model, Argon and Hannoosh (1977a)
model and the Kawagoe and Kitagawa (1981) model. (c) Temperature dependence
of uni-axial shear yielding and crazing envelopes for PS (Haward, 1972).
33
k
Loading
(a)
r
(b)
Figure 1-7: (a) Optical micrograph of the ductile/brittle PC/SAN laminate (Gregory
et al., 1987) (b) TEM micrograph of a thin HIPS section, prepared from a tensile
specimen beyond yield (Bucknall, 2001).
34
Chapter 2
Constitutive models for large
strain inelastic deformation of
polymeric materials: shear yielding
and crazing
This chapter is comprised of two parts. The first part begins with the description of
the physics and finite strain constitutive modeling of the shear yielding mechanism,
and briefly explains the underlying mechanisms governing the yield, post-yield softening and orientation hardening at large strains. A simple model, based on critical
chain stretch, is used to describe ductile failure. The second part describes the physics
and modeling of the three stages of crazing - initiation, growth and breakdown, and
presents a continuum formulation for crazing. Both shear yielding and crazing mechanisms have been implemented in the commercial finite element software ABAQUS.
2.1
Shear yielding
The typical response of glassy polymers undergoing inelastic deformation via shear
yielding is shown in Figure 2-1. Figure 2-1(a) shows the true stress-strain behavior
of PC, subjected to uni-axial compression, at three strain rates. The initial response
35
consists of a relatively stiff (visco)-elastic slope before yield, followed by post-yield true
strain softening and, finally, significant strain-hardening at large strains. Figure 21(a) also shows that, with increasing applied strain-rate, the yield and flow strength
increase.
Figure 2-1(b) shows the true stress-strain behavior of PMMA, subjected
to uni-axial and plane strain compression, and highlights the pressure-dependence
of yield, flow and strain-hardening.
Model predictions, shown in Figure 2-1, are in
excellent agreement with the experimental data, and will be discussed shortly.
Over the last few decades, a substantial amount of literature on glassy polymers
has aimed to clarify the mechanisms underlying the phenomenology of shear yielding.
The purpose of this section is to provide a brief review of the physics and modeling
of shear yielding.
Motivated by the features of deformation in glassy polymers, Haward and Thackray (1968) introduced a one-dimensional spring-dashpot arrangement shown in Figure 2-2(a). Their model includes a linear elastic spring to model the initial deformation, a viscous dashpot described by Eyring's (Eyring, 1936) rate and temperature
dependent viscosity equation to capture the viscoplastic deformation, and a nonlinear rubber elastic spring based on Langevin statistics to capture the substantial
strain-hardening in several glassy polymers. It is noted that the initial deformation
of glassy polymers at room temperature is actually visco-elastic and time-dependent.
However, since the predominant contribution to the total deformations of interest in
this thesis are viscoplastic, the errors incurred in assuming a linear elastic spring are
usually negligible, and will not be discussed any further.
While Eyring's viscosity
equation, mentioned above, was the first thermally activated model that successfully
described the viscoplastic nature of glassy polymers, his model suffered from lack of a
clear connection between the model parameters and the local mechanisms that govern
shear yielding. Argon (1973a) and Argon and Bessenov (1977) developed a mechanistic theory of shear yielding, and conceived the viscoplasticity in glassy polymers
to be due to thermally activated rotation of molecular segments under stress.
An important advance in the modeling of the three dimensional elastic-plastic
behavior was due to Parks et al. (1985), who extended the one-dimensional model of
36
300
.01)
200
J
0
(nI
100
0
1.0
0 .0
1.5
True _Strain
(a)
200
//
L.
100
. niaxial data
- - -- - plane strain data
uniaxial eight chain
_
-.-- plane strain eight ch &in
...
n
0.0
1.0
0.5
TRUE STRAIN
1,5
(b)
Figure 2-1: (a) True stress-strain experimental data (dashed lines) for uni-axial compression of PC at strain rates -1.0 s- 1, -0.1 s- and -0.01 s- 1 , showing the increased
yield and flow strength with increasing strain rate. Model predictions (solid lines)
adequately capture the experimental data. From Boyce and Arruda (1990). (b) True
stress-strain experimental data for uniaxial compression and plane strain compression
of PMMA at a strain rate of -0.001 s- 1 , showing the pressure dependence of yield,
shear flow and strain-hardening. Model predictions, based on the eight chain model of
Arruda and Boyce (1993a), adequately capture the experimental data. From Arruda
and Boyce (1993b).
37
Haward and Thackray into a rate-independent three-dimensional framework. Their
model considered a constant inter-molecular resistance, resulting in plastic flow being
independent of rate, temperature, and pressure.
Boyce et al. (1988) incorporated
the effect of rate, temperature, pressure, and captured the experimentally observed
strain-softening in glassy polymers. Both Parks et al. (1985) and Boyce et al. (1988)
modeled the entropic resistance due to developing molecular orientation with strain
through Langevin statistics using Wang and Guth's (1952) 3-chain network idealization. Wang and Guth's 3-chain network model has the drawback in its inability
to adequately distinguish the effect of imposed deformation states, such as uni-axial
vs. biaxial deformation. Arruda and Boyce's (1993a) 8-chain network model constitutes a major improvement over Wang and Guth's model, and other network models
(e.g., Treloar, 1975) in that it successfully captures the deformation dependence of
the network response, and hence the strain-hardening behavior of glassy polymers
and elastomers.
Boyce et al.
(2001) used a spring-dashpot arrangement different
from the one used in their previous work, as shown in Figure 2-2(b). This kinematic
framework was developed from the time-dependent elastomer deformation model of
Bergstrom and Boyce (1998). Both spring-dashpot representations yield almost identical stress-strain results for glassy polymers, except for the slight difference in the
initial effective elastic stiffness and the stiffness upon unloading. In this thesis, the
spring-dashpot arrangement in Figure 2-2(b) is used, with the associated finite deformation constitutive framework of Boyce et al. (2001).
Compatibility considerations imply that the deformation gradient FN acting on
the rubber elastic spring N is equal to the deformation gradient FM acting on the
elasto-viscoplastic element M:
FN = FM = F
(2.1)
Using the Kroner-Lee decomposition (e.g., Lee, 1969), the deformation gradient of the
elasto-viscoplastic element can be decomposed into an elastic and viscoplastic part:
38
Linear
Elastic
Rubber
Shear
Elastic
Flow
(a)
Linear
Elastic
Rubber
Elastic
Shear
N
M
Flow
(b)
Figure 2-2: One-dimensional spring-dashpot arrangements to model shear yielding;
(a) proposed by Haward and Thackray (1968); (b) proposed by Boyce et al. (2001).
F = FeFP; detFP = 1
(2.2)
where the slight dilatancy of the viscoplastic shear flow is neglected. The reference
configuration' is mapped into the current configuration via F.
The intermediate
configuration, obtained by "pulling back" the current configuration via F- 1 , is stressfree and known as the "relaxed" configuration.
Invoking the polar decomposition theorem, the deformation gradient can be written as
F=VR=RU
(2.3)
where R is the rotation tensor, and U and V are respectively the right and left stretch
tensors. The polar decomposition theorem is also taken to apply separately to the
elastic and plastic parts: Fe = V*Re
=
ReUe and FP = VPRP = RPUP.
The velocity gradient L can be expressed as
'The reference configuration is frequently taken to be the initial, unstressed configuration
39
L=
F- 1 =
PeFe
+ FeNPFP-lFe-1
(2.4)
The velocity gradient in the relaxed configuration is LP = FPFP-1 = DP + WP,
where DP and WP are, respectively, the stretching and spin tensors associated with
the relaxed configuration.
Re and RP can be made unique in a variety of ways.
Boyce et al. (1989) examined the consequences of using three different constraints:
Re = R, RP
=
R, or WP = 0, and found that the material response is independent
of the choice of constraint when one properly handles the pull-back and push-forward
aspects of the calculations. Here, we take with no loss of generality: WP
=
0. The
flow rule for shear yielding is then given by:
LP=
DP =
PN
(2.5)
where N is the direction of inelastic flow for shear yielding and is given by
-M'
N =_
IITM'11
(2.6)
Here, IM' represents the stress acting on the elasto-viscoplastic element expressed in
the relaxed configuration, with the prime and bar respectively indicating the deviator
and the unloaded configuration.
The plastic shear strain rate
'P
is constitutively
prescribed as:
iP =
where
AO
Oexp
[As()f
-
1
_
-/6)
(5/6)3i
(2.7)
is the pre-exponential factor proportional to the attempt frequency, As(p)
is the zero stress level activation energy, k is the Boltzmann's constant, and
e
is the
absolute temperature. The athermal shear strength is taken to be pressure-dependent:
s(p) =
s
.
+ ap, where p is the pressure and a is the pressure sensitivity. The resistance
evolves with deformation from an initial value AO = 0.077p/(1 - v) (p and v are
respectively the shear modulus and Poisson's ratio for isotropic elastic materials) to
a steady-state value
s,,
according to the equation
40
s
=-h
I -
(2.8)
)
where h is the softening modulus, taken to be a positive constant.
equivalent shear stress is given by -r=
The applied
I TM' I/,'2.
Equation 2.8, which is used to describe the strain-softening phenomenon observed
in most glassy polymers, has as its underlying basis the very local molecular-level dilatation induced by shear flow that results in a decrease in resistance to deformation.
It is noted that this local molecular-level dilatation has little effect on macroscopic volume change such that the assumption of plastic incompressibility, detFP = 1, holds.
The shear yielding kinetics described by Equation (2.7), based on the model originally
proposed by Argon 2 (1973a), is formulated to capture the inter-molecular interactions
that govern the barriers to chain segment rotation. This model is based on a single
activation process for yield. We note that yield in many homo-polymers, including
PC and PMMA, is more accurately modeled, especially in the low-temperature and
moderate-to-high strain rate regime, by also including a barrier to a secondary process, associated with the /-transition of the polymer (see Mulliken, 2004; Mulliken
and Boyce, 2006). Here, for simplicity we ignore the
-transition contribution.
The Cauchy stress TM is obtained as:
TA"=
1
detFe
(2.9)
Le[lnV]
where Le is the fourth-order linear, isotropic elasticity tensor given by
(2.10)
Ice = 2pI +
r - 2
1
where I and 1 are respectively the fourth and second order identity tensors, and
K
is
the elastic bulk modulus. The stress TM associated with the relaxed configuration is
connected to the Cauchy stress TM by TM = ReTTMR
.
-t
The stress acting on the network orientation element, TN, is given by the 8-chain
model of Arruda and Boyce (1993a):
2
Argon's original equation did not incorporate the pressure dependence, or the strain softening.
41
TN
-- 1hr
(-1 chain
Achain
vN
B'
(2.11)
where J =detF, F=J- 1/ 3 F, B=PFT, and Achain=(tr[B/3)1/2. The material parameters PR and N are, respectively, the initial hardening modulus and the number of
rigid molecular units between entanglements. L-
is the inverse Langevin function,
where L(O) = coth(3) - 1/0 and 13 = L- 1 (Achain/ N).
The inverse Langevin function becomes unbounded as the chain stretch Achain approaches the locking stretch AL = vN, and hence accounts for finite chain extensibility. The numerical evaluation of the inverse Langevin function is discussed in
Appendix A.
The total stress is given by:
T=TN+TM
(2.12)
While failure in glassy polymers has been considered to be a kinetic process
(Zhurkov, 1965), simplified time-independent approaches have focussed on two limiting cases (e.g., Danielsson, 2003; Gearing and Anand, 2004a):
" Brittle failure, which is associated with cavitation in the unoriented material
subjected to (local) hydrostatic stress, is taken to occur when the local elastic
volumetric strain E,,, reaches a critical value E,,
EVl = Ec,
(2.13)
This criterion is equivalent to a critical hydrostatic stress criterion at negligible
plastic strain, i.e. prior to any substantial molecular motion.
" Ductile failure, which is associated with chain scission and disentanglement in
highly oriented, plastically deforming material, is conceived to occur when the
chain stretch Achain reaches a critical value A,:
42
(2.14)
\ckhain =Acr
We note that PC layers of the PC/PMMA microlaminates (see Chapter 3) undergo
ductile failure. Conditions for brittle failure in PC are reached at large hydrostatic
stresses ~ 85 MPa and negligible plastic strain, in the presence of notches or cracks
(e.g, Nimmer and Woods, 1992; Narisawa and Yee, 1993; Johnson, 2001; Gearing and
Anand, 2004a); this condition is not met in the uni-axial deformation of PC/PMMA
laminates. The failure in PMMA layers occurs due to craze breakdown.
2.2
Material properties for shear yielding model
The shear yielding parameters for PC and PMMA are given in Table 2.1. Except for
the values of a and A, all values have been taken from Arruda and Boyce (1993b).
The values of a are due to Mulliken and Boyce (2006).
For PC, Ac, has been esti-
mated 3 to be 0.9AL.Based on these material properties, the isothermal shear yielding
response of PC and PMMA subjected to uni-axial tension at true strain rates of
1.7 x 10-3 s-
and 1.7 x 10-2 s-
are shown in Figure 2-3', which represents a homo-
geneous response with no shear localization, and no (suppressed) crazing. Figure 2-1
shows that the model results of Boyce and coworkers are in excellent agreement with
experimental results, and adequately capture the rate and pressure dependence of
yield, post-yield softening and strain-hardening.
2.3
Crazing
The tensile toughness of several glassy polymers such as PS, PMMA and SAN is negligible due to the phenomenon of crazing. Figure 1-4 shows several crazes in a tensile
3
compare this value to that used by Johnson(2001): 0.85AL, and by Gearing and Anand (2004a):
0.82AL. In Chapter 3, we discuss the results of a sensitivity study conducted to investigate the
effects of varying the value of Ac on the micro- and macro- behavior of ductile/brittle laminates.
4
These strain-rates correspond to the nominal strain-rates used in tensile simulations of
PC/PMMA ductile/brittle micro-laminates in Chapter 3.
43
Table 2.1: Shear yielding material parameters for PC and PMMA
E (MPa)
PC
2300
v
0.33
PMMA
3250
0.33
5
1)
2 x 101
Yo (s
3.31 x 10-27
A (in 3 )
0.16
a
99
so (MPa)
77
seS (MPa)
500
h (MPa)
2.78
N
18
pr (MPa)
1.5
Acr
150
8
1.3
A: 0.0017/s
-PM
-
2.8 x 10 7
1.39 x 10-27
0.26
137
113
315
2.1
PMMA: 0.017/s
PC: 0.0017/s
PC: 0.017/s
1001
(I,
L.
a')
a)
1...
50
'o
'
0c
0.2
0.4
True strain
0.6
0.8
Figure 2-3: Isothermal uni-axial stress-strain response of PC and PMMA at true
strain-rates of 0.0017 s- and 0.017 s-.
44
PMMA specimen. The deformation typically localizes, along one of the crazes, leading to premature cracking and failure. However, crazes themselves are manifestations
of dilatational plasticity, and result in substantial toughening in glassy polymers,
provided a large density of crazes develop throughout the material. The stabilization
of multiple crazes in HIPS is perhaps one of the best examples of toughened glassy
polymers. Crazes are comprised of elongated voids and fibrils of highly oriented, load
bearing polymeric material. Figure 2-4(a) shows a simple craze schematic, indicating
the porous nature of a craze. The fibrils are shown to be aligned in the direction of
the maximum principal stress. Typical values for the fibril diameter and the fibril
spacing for PS are respectively, D
1983).
~ 4 - 10 nm and Do ~ 20 - 30 nm (Kramer,
Figure 2-4(b) provides a more realistic representation of the craze structure
and shows the existence of cross-tie fibrils that bridge the main fibrils. The cross-tie
fibrils provide a shear stiffness to the craze structure; the shear stiffness is about 2
orders of magnitude lower than the axial stiffness. The main fibrils, instead of being
perfectly aligned in the direction of the maximum principal stress, are misaligned by
about t5'
(e.g., Donald, 1997). The void volume fraction in the craze structure for
PS is about 75% (Kramer, 1983).
Since the 1960s, crazing has been the subject of numerous studies that have shed
light on the craze microstructure; the different stages of crazing (initiation, growth
and breakdown); and the dependence of crazing on various factors including environment (presence of solvents), molecular weight, orientation, defects, temperature and
strain-rate. These studies have been summarized and consolidated in several excellent reviews (e.g., Rabinowitz and Beardmore, 1972; Kambour, 1973; Kramer, 1983;
Narisawa and Yee, 1993; Argon, 1993; Donald, 1997); some of the relevant details
will be briefly discussed in the following sub-sections.
Detailed, numerical studies that incorporate the physics of crazing are a recent
phenomenon. Van der Giessen and co-workers (Estevez et al., 2000; Tijssens et al.,
2000a,b) incorporated the three stages of crazing in a cohesive zone finite element
formulation. Estevez et al. (2000) performed a finite element study of Mode I cracktip plasticity that accounted for both crazing and shear yielding.
45
Tijssens et al.
D.
fibril
prindpal
growth direcdon
(a)
undeformed polymer
crosstie
fibril
(b)
Figure 2-4: (a) A simplified craze schematic, showing the internal structure of a craze
with the fibrils perfectly aligned in the direction of the maximum principal stress.
(b) A refined representation of the craze internal structure showing cross-tie fibrils.
Primary fibrils are slightly misaligned from the direction of the maximum principal
stress. From Donald (1997).
46
(2000a,b) studied crazing in a plate with a circular hole subjected to far-field uniaxial tension, and Mode I crack growth due to crazing. Socrate et al. (2001) proposed
a mechanism-inspired cohesive zone micro-mechanical crazing model that accounts for
craze initiation and growth, and applied it to the study of multiple crazing in HIPS.
Gearing and Anand (2004b) developed a macroscopic continuum model that accounts
for both crazing and shear yielding, and applied it to numerical studies on a thin plate
with a hole subjected to far-field tension, as well as a notched-beam under four-point
bending. 5
The following sub-sections discuss the three stages of crazing and the associated
models, along with a continuum framework for crazing.
2.3.1
Craze initiation
An unambiguous understanding of craze initiation has defied researchers to this date,
partly on account of the significant sensitivity of craze initiation to geometric flaws
and material heterogeneities. Presence of defects can clearly ensure large discrepancy
between the globally applied stress state and the local stress state in the vicinity
of the defect where craze initiation occurs preferentially. Therefore, craze initiation
experiments performed by various researchers, and their proposed models show some
degree of incongruity.
The prominent models for craze initiation include the Sternstein-Ongchin model,
the Oxborough and Bowden model, and the Argon-Hannoosh model. The SternsteinOngchin model (Sternstein and Ongchin, 1969; Sternstein and Myers, 1973), based on
biaxial plane-stress experiments on PMMA, postulates craze initiation to occur when
the "stress-bias"
Ub
reaches a critical value that depends on the hydrostatic stress:
'The above mentioned researchers have used different models for crazing. For craze initiation:
van der Giessen and co-workers used the Sternstein-Ongchin model (1969,1973); Socrate et al. (2001)
used the Argon-Hannoosh criterion (1977a); Gearing and Anand (2004b) used Oxborough and Bowden's criterion (1973). For craze thickening: van der Giessen and co-workers and Gearing and Anand
devised phenomenological expressions; Socrate et al. relied on the mechanistic expressions derived
by Argon and co-workers. For shear yielding studies, van der Giessen and co-workers and, Gearing
and Anand, each followed similar but somewhat modified versions of the constitutive model of Boyce
and co-workers (1988).
47
b
(2.15)
I0 ~- U 2 1 > A(O) + B(6)
where A(O) and B(O) are temperature-dependent material properties, a, and -2 are
the non-zero in-plane principal stresses, I,
the stress tensor, and
ch
= 3og
= o1
+
92
is the first invariant of
is the hydrostatic stress. With this model, Sternstein and
Ongchin (1969) were able to successfully consolidate their biaxial test data on PMMA
at different temperatures.
The biaxial experiments were performed on thin-walled
cylindrical specimens, which were subjected to a tangential stress o
= o-o (by inter-
nally pressurizing the thin-walled cylinder) and an axial stress U 2 = a.
To initiate
crazes preferentially at the center of the specimen, the samples were slightly tapered.
The imposed stress states (o, o), ranging from uniaxial tension to equi-biaxial tension, were held constant for 10 minutes. Figure 2-5 shows their experimental data
and the model fit obtained using Equation 2.15 at three different temperatures.
While the Sternstein-Ongchin model successfully described their biaxial craze iniFirst, the stress-bias
term of their model is ambiguous at best; for a principal stress state al
Sternstein and co-workers considered ub = a,
-
> U 2 > U-3
,
tiation data, the model suffers from a number of drawbacks.
U 2 when og, o 2 > 0 and ob = a, -
73
when a, > 0, -2 < 0 (Oxborough and Bowden, 1973). And second, Wang et al.(1971),
through their uniaxial tension experiments on PS containing a well-bonded spherical
steel inclusion 6 , found crazes to initiate at the boundary of the steel particle at an
angle of 370 to the tensile axis, which is not in agreement with the Sternstein-Ongchin
model. 7
Oxborough and Bowden (1973), in an effort to eliminate the inconsistencies of the
Sternstein-Ongchin model, proposed a hydrostatic stress dependent critical strain
criterion for craze nucleation:
+ Y'(e)
c=
6
(2.16)
The diameter of the steel inclusion was 0.125 in.
'Wang et al. estimated the material parameters A and B of the Sternstein-Ongchin model from
their uniaxial tension experiments on PS samples embedded with a steel inclusion, and PS samples
containing a rubber inclusion.
48
4000-
50'C
70* C
2000-
0//
0
2000
1000
C~2
3000
4000
PS)
Figure 2-5: Biaxial craze initiation data of Sternstein and Ongchin (1969) for PMMA.
where c, is the critical strain for craze nucleation, and X' and Y' are temperature and
time-dependent material parameters. This equation successfully captures the biaxial
experimental data of Sternstein and Ongchin on PMMA, as well as their own biaxial
data on PS generated by pulling a PS plate with a hole in uni-axial tension, and by
imposing a compressive stress state at the hole.
Apart from recognizing the dilatational nature of crazing (through I,), the SternsteinOngchin and the Oxborough-Bowden models do not incorporate the physical mechanisms of crazing in their model, and in particular do not consider the well-known
time-dependence 8 of the craze initiation process. Among the craze initiation models,
the model by Argon and co-workers (1973b, 1975, 1977a, 1977b) is perhaps the most
detailed and well-tested.
Recognizing the time-dependence of craze nucleation, Ar-
gon and co-workers developed the theory of crazing within a kinetic framework, which
finally led to the Argon-Hannoosh model (1977a). The Argon-Hannoosh (1977a) criterion is based on detailed experiments on PS where craze initiation times were studied
on surfaces with controlled topography subjected to combinations of deviatoric and
hydrostatic stresses. The Argon-Hannoosh model postulates that the critical event in
8
There is a time-lag between the application of a tensile stress and the visual appearance of crazes
49
craze nucleation is the formation of sub-micron size cavities by the arrest of thermally
activated micro-shear processes due to the local von-Mises stress o, (Argon, 1975).
The cavities undergo plastic growth in the presence of a critical hydrostatic stress,
which is dependent on the level of local porosity. These considerations lead to the
following expression for the craze initiation time tI (Piorkowska et al., 1990):
ti = rexp
[-
1(e
where
T
(2.17)
-
2QYI
is a characteristic time constant, C is a temperature dependent constant pro-
portional to the activation energy of the micro-shear processes,
Q is
a multiplicative
factor that reduces the critical hydrostatic stress needed for plastic growth of cavities,
and Y is the tensile yield strength.
The Argon-Hannoosh model, not only describes the experimental data on homogeneous PS samples, but is also consistent with the uni-axial tension results of Wang
et al. (1971) on PS with an embedded steel ball (described above), as detailed in
Argon (1975).
Further, through their biaxial experiments on PMMA in air and in
kerosene, Kawagoe and Kitagawa (1981) found that the craze-locus obtained from
the Argon-Hannoosh model describes well the experimental data obtained for both
air and environmental crazing'. In particular, Kawagoe and Kitagawa found that the
Sternstein-Ongchin and Oxborough-Bowden models could not capture craze initiation data for PMMA in the presence of kerosene, while their own model and that of
Argon and Hannoosh successfully described the same (see Figure 2-6). In this thesis,
we will use the Argon-Hannoosh model to predict craze initiation.
As pointed out by Socrate et al. (2001), Equation (2.17) gives the craze initiation
time for fixed values of Oh and oe. For the monotonic (and proportional) loading
conditions of interest in this thesis, Equation (2.17) is interpreted to give the instantaneous value of the craze initiation time ti(t*) due to the applied
ch(t*)
and ocr(t*)
at some time t*. For a time-varying stress history, Equation (2.17) is generalized to:
9crazing in the presence of solvents and plasticizers is known as "environmental crazing", while
crazing in the absence of such agents is referred to as "air crazing"
50
ftdt*
0
ti(t*)
(2.18)
= 1
It is noted in passing that for stress histories, in which o-e and
ch
do not change
proportionally, Equation 2.18 may not hold. Socrate et al. (2001) have cited the aging
experiments of Argon and Hannoosh (1977b), in which PS samples were subjected
to a pure deviatoric stress for increasing time intervals followed by the application of
a hydrostatic stress. A non-intuitive result of their aging experiments was that the
craze initiation times increased with increasing aging times under deviatoric stress
(followed by the application of hydrostatic stress). This highlights the complexity of
craze nucleation kinetics under non-proportional loads.
Figure 2-7 compares the Sternstein-Ongchin and the Argon-Hannoosh locus in
the oi -
02
and
U, -
Oh
space for biaxial states of stress at room temperature. The
Sternstein-Ongchin parameters at room temperature were taken to be A
=
-70 MPa
and B = 7500 (MPa)2 (Tijssens et al., 2000a). To compare the time-independent
Sternstein-Ongchin model with the time-dependent Argon-Hannoosh criterion, the
latter was rendered time-independent:
C
Je
where K -- ln(ti/T).
3
ch
K
(2.19)
2QY
The Argon-Hannoosh material parameters for PMMA ho-
mopolymer are not known.
Here, in order to bring the Argon-Hannoosh craze lo-
cus close to the Sternstein-Ongchin craze locus in the ci
-
02
space, we estimate
Y = 90 MPa, Q = 0.0133, K = 18.5, and C = 2600 MPa. (Figure 2-7(a)). In
Figure 2-7(b), both loci are replotted in the
Ue -
Uh
space. Figure 2-7(b) shows the
relatively higher sensitivity of the Argon-Hannoosh locus to hydrostatic stress, than
the Sternstein-Ongchin locus1 0 . Detailed experiments performed on several crazeable
glassy polymers under identical conditions should help resolve the observed discrepancy between the two (and other) craze initiation models, particularly at biaxial and
10Under biaxial states of stress, the Oxborough-Bowden model is similar to the Sternstein-Ongchin
model
51
(a)
10
* CGNg
NO
*
0
-50
-60
Shear y e4dng /
-30
-20
Rure SJeOr
(a
10
0
-10
20
3.
o
30
)
-
OMzing
te-ng hi
2:()tr
Figure
.i30
(si
tbo g
/
/
/
-80
-70
00
10
0Qcfl9
*
-60
-50
No ami~rng
-40
-30
-20
-10
0
10
20
30
(b)
Figure 2-6: (a) Sternstein-Ongchin (solid lines) and Oxborough-Bowden (dashed
lines) do not capture the biaxial craze initiation data for PMMA in the presence
of kerosene. a and b curves correspond to two different sets of material parameters
for the Sternstein-Ongchin model, while c and d correspond to two different sets of
material parameters for the Oxborough-Bowden model. (b) Argon-Hannoosh and
Kawagoe-Kitagawa models capture the biaxial craze initiation data for PMMA in the
presence of kerosene.
triaxial states of stress.
2.3.2
Craze growth
Craze growth" consists of two main aspects (e.g., Kramer, 1983) (see Figure 2-4(a))
* craze lengthening by the advance of the craze-tip, in the plane of the craze,
which results in the generation of more fibrils.
* craze thickening, which occurs perpendicular to the craze plane and results in
the separation of the craze surfaces as well as lengthening of the craze fibrils,
"In the literature, "growth" often refers only to craze lengthening, while "thickening" or "widening" are associated with the separation of the craze surfaces.
52
60
58
56
54
52
2 50
48
Sternstein-Ongchin
Argon-Hannoosh
---- Equi-biaxial stress
46
44
42
40
0
10
20
30
a 2 (MPa)
40
50
60
(a)
60
58
56
Cu
~5452
.T 50CD
E
4846-
0
4 -42 --
40
15
Stemstein-Ongchin
Argon-Hannoosh
35
30
25
20
Hydrostatic stress ah (MPa)
40
(b)
Figure 2-7: Comparison of the Sternstein-Ongchin and Argon-Hannoosh locus in (a)
Ol - U2 space, (b) -e - clh space.
53
either by drawing in material from the bulk, or by fibril stretching (creep).
Initially, the advance of the craze-tip was believed to occur by the repeated nucleation of new craze matter (Argon, 1973b), which has been discussed in Section 2.3.1.
This mechanism suffers from two main drawbacks. First, a process of repeated nucleation of new craze matter at the craze-tip predicts the formation of isolated microvoids within the bulk polymer. However, experimental observations on crazes have
shown the existence of the opposite topology, i.e., voids are found to be the topologically continuous phase. Second, this mechanism requires very large stresses to
account for the experimentally observed craze growth rates.
These considerations
prompted Argon and Salama (1977) to propose a new mechanism - craze-tip advance
due to meniscus instability, according to which the polymer at the craze-tip breaks up
and leads to craze-tip advance by a process of repeated convolution. This meniscus
instability mechanism predicts the correct topology of the craze matter, and predicts
According to the Argon-Salama
craze growth kinetics at reasonable stress levels.
model (1977), the velocity of the advancing craze-tip is given by:
v
where
Y
= vo -exp
a1
{-
k6
1 -
A/ -
1
(2.20)
0.133p/(1 - v) is the athermal plastic resistance to plastic flow, B/ks is
the effective activation energy for plastic flow, A' is an orientation-hardening modified
extension ratio, and a, is the maximum principal stress. It is noteworthy that the
craze-tip advance depends only the local maximum principal stress, and can occur
even if ah < 0, unlike craze initiation (Kramer, 1983).
As the craze-tip advances, the craze also thickens in the direction of the maximum
principal stress. Two mechanisms have been proposed to account for craze thickening
- fibril creep and the surface-drawing mechanism.
As the name implies, the fibril
creep mechanism postulates that craze thickening occurs by creep due to the local
maximum principal stress at the craze-bulk interface.
This mechanism naturally
predicts the thinning of the fibrils (thereby increasing the magnitude of true stresses
in the fibrils), and a decrease in the volume fraction of the fibrils in the craze with
54
craze thickening. While fibril creep may play an important role for craze thickening
in the presence of solvents (Kramer, 1983), air crazing occurs primarily due to the
surface drawing mechanism (Lauterwasser and Kramer, 1979), according to which
craze matter from the bulk is drawn into the fibrils at the craze-bulk interface in a
manner similar to the drawing of the undeformed polymer into the neck of a tensile
specimen undergoing shear yielding. According to Henkee and Kramer (1986), the
formation of the fibril surface area during craze thickening requires a 25-50 % loss of
entanglement strand density. This loss of entanglement density is achieved by chain
scission and/or disentanglement at the craze-bulk interface.
The drawing of bulk
polymer into the craze ensures that the fibril extension ratio remains constant for a
fixed level of stress at the craze surfaces. The work in this thesis is confined to the
study of air crazing.
Argon and co-workers (Piorkowska et al., 1990; Dagli et al.,
1995; Argon, 1999) have considered the velocity of the craze surfaces o to be:
O = &v=
YB
exp
&voO71
k
L (A Y1-
5-/6-
(2.21)
where & is considered to be a constant here. The relative velocity of the craze surfaces
is 240.
2.3.3
Craze breakdown
The breakdown of crazes due to the failure of craze fibrils was long thought to be
governed by creep of the fibrils (e.g., Trassaert and Schirrer, 1983; Verhuelpenheymans, 1984; Kramer and Hart, 1984). Fibril failure due to creep would be expected
to be at the midrib, which is the oldest and the most heavily drawn part of the craze.
However, it was shown by Kramer and Berger (for e.g., Berger (1990) and Kramer
and Berger (1990)) through careful microscopy on thin PS films that craze breakdown
invariably initiates at the craze-bulk interface or the active zone. TEM observations
indicate that pear-shaped voids, initiated at the active zone, grow and result in the
failure of the entire craze structure. Figure 2-8 shows damage initiation at the edge
of the craze, and not at the midrib. While the presence of dust particles in the active
55
Figure 2-8: TEM micrograph showing that craze breakdown initiates from the edge
of the craze, and not from the midrib. From Donald (1997).
zone aids in the void initiation process, their role is by no means critical; TEM micrographs indicate nucleation of voids in the absence of dust particles as well (Kramer
and Berger, 1990). This mode of craze breakdown is also seen in other brittle glassy
polymers, such as PMMA and PSAN (Berger, 1990). Kramer and Berger (1990)
further note that fibril breakdown cannot be completely accounted by chain scission
required to form craze fibrils in the active zone, but that there must be a further loss
in entanglement density.
The failure of the entire craze structure following crack initiation in the active
zone can be understood from the work of Brown (1991), who showed that cross-tie
fibrils, which bridge the primary craze fibrils parallel to the maximum principal stress
direction, transmit stress from the broken fibrils to the unbroken ones resulting in the
failure of the entire craze structure. Subsequently, Kramer and coworkers (e.g., Hui
et al. (1992) and Sha et al. (1995, 1997, 1999)) have extended and further quantified
56
Brown's work. In particular, Sha et al. (1997) developed a micromechanical model
that links the critical craze thickness and fracture toughness of the craze to the fibril
network structure. Sha et al. (1999) have shown that a rate-dependent craze failure
criterion is needed to obtain the correct dependency of the critical craze thickness
and fracture toughness on crack growth rate.
In our preliminary modeling effort, we use a rate-independent critical craze opening criterion for craze breakdown, relying on the experiments of Doll (1983), who
showed that for crack speeds in the range 10-8 to 10 mms'
imum craze width is weakly rate-dependent.
for PMMA, the max-
In reality, the critical craze opening
depends on several factors, such as molecular weight, craze fibril network, manufacturing/processing conditions; the use of a single material parameter
2
Oc is a phe-
nomenological simplification, and actual values of this parameter would need to be
obtained experimentally.
2.3.4
A continuum framework of crazing
In our finite element studies, the deformation, prior to craze initiation, is based on
the shear yielding model described in Section 2.1. During the initial (small strain)
deformation, the contribution of the viscoplastic dashpot to the overall deformation
is negligible, and the deformation can be effectively considered elastic. The stress in
network M, TM, given by Equation (2.9), is used to calculate the local hydrostatic
stress ch
=
TM/3 (k=1 to 3) and the local von-Mises stress a-
=
3TYT7/2 (i,j=1
to 3), the summation convention being implied. A craze is taken to nucleate when
Equation (2.18) is satisfied12 , and the process of craze thickening in the direction
of the maximum principal stress begins. Due to the very nature of finite element
discretization, each craze will be associated with a layer of finite elements, whose
surface will be perpendicular to the maximum principal stress direction. Finite elements that deform by crazing will be conveniently referred to as "craze" elements.
12
since TM ~ T unless the plastic strains are large, using TM instead of T to calculate Ch and
0e, is justified. The use of TM is that it precludes the possibility of un-physical craze nucleation in
highly drawn polymer when the network stress TN substantially raises the total stress T.
57
Figure 2-9 shows a schematic of a craze element (Socrate et al., 2001). The element
includes craze fibrils of current length 2
0 (t),
as well as bulk matter of current length
2iqo(t) that acts as a source of material for craze fibril drawing by the surface drawing mechanism. The zero subscript indicates the elastically unloaded configuration.
For convenience, cross-tie fibrils are not shown in Figure 2-9. Initially, the element
contains only uncrazed (or bulk) matter, i.e., o(0) = 0 and 'qo(0) = ho/2, where ho
is the initial element thickness. The craze element thickens in the direction of the
maximum principal stress (in Figure 2-9, the maximum principal stress direction is
ei) with thickening rate 2( o +
). The rate of bulk matter consumption, Q
1 (t), and
the normal rate of propagation of the craze/bulk interface,
o, are related through
the conservation of mass:
(- )(2.22)
M =
Af (t)
where the instantaneous fibril extension ratio is Af(t).
Consistent with the requirements for a continuum framework of crazing, we define an
effective strain rate across the craze element as
c =
(2.23)
where 2 and 2,q are the craze and bulk thicknesses in the loaded (current) configuration, and are respectively estimated by
= G [1 + U1 /Ecraze] and y = qo[1
+ Oc /EbuIk1.
Here, Ecraze = Cii (see Appendix B) and EbuIk are, respectively, the effective elastic
stiffness of the craze matter, and the elastic stiffness of the bulk matter in the direction of the maximum principal stress al. Elastic properties for the craze structure
are discussed in more detail in Appendix A. The craze physics is now used within
the following three-dimensional, finite deformation framework, which is based on the
spring-dashpot schematic shown in Figure 2-2(b), but with TN = 0.
The finite deformation kinematics begin with
F = FeFP; detFP > 1
58
(2.24)
where detFP > 1 due to the dilatant nature of the crazing process. The Cauchy stress
is obtained by:
T
1
detFe
TI [nVej
(2.25)
where LTI is the effective fourth-order linear, transversely isotropic elasticity tensor
for the craze element. The elastic properties of the craze element are obtained through
the homogenization of the elastic properties of the bulk matter, which is assumed to
be isotropic, and the elastic properties of the craze, which is taken to be transversely
isotropic with the circular symmetry plane being perpendicular to the maximum
principal stress direction (see Appendix B for more details). We note here that LTI
depends on o and qo. In particular, LTI(qo = ho/2) is the isotropic elasticity tensor
for bulk PMMA, and L TI(,o
=
0) is the transversely isotropic elasticity tensor for
the craze matter. Spectral decomposition of T yields the maximum principal stress
ai (in direction el), which is then used to obtain the kinematic quantities o, i o, Af
and finally the "craze strain rate" given in Equation (2.23).
The craze flow rule is
written as:
DP = 'ej 0 el
(2.26)
The critical craze opening for breakdown is taken to be equivalent to the craze
opening when 71o = 0, i.e., when the bulk matter in the element is consumed. This
can be shown to imply, by Equation 2.22 and the initial conditions on g7o and o, that
2
Oc = Af ho provided Af is a constant. This craze breakdown condition is clearly ad-
hoc, and artificially ties the maximum craze opening with the initial element thickness
ho in the direction of the principal tensile loading, and the fibril extension ratio Af.
However, in the case of PC/PMMA microlaminates (see Chapter 3), meshing with
ho= 1 um and noting that Af ~ 2 for PMMA (Kramer, 1983), the condition 77
=
0 is
equivalent to a maximum craze opening of 2 pm, which is close to the maximum craze
thickness values for PMMA reported by Doll (1983). For HIPS (see Chapter 4), with
ho ~~25 nm and Af ~ 4, the maximum craze opening achieved within this framework
59
is about 100 nm.
We note that, due to computational expense considerations, in our implementation
of the craze model, the propagation of the craze is implemented by repeated nucleation
of new crazes and not by meniscus instability, which is the operative mechanism. A
challenge in the numerical implementation of the latter is that it requires an a-priori
knowledge of which finite elements have initiated crazes, so that adjacent elements
(oriented perpendicular to the local maximum principal stress) can obey the craze
propagation model given by Equation 2.20. The ramifications of this approximation
in our implementation are discussed in Chapter 5.
2.4
Material properties for crazing model
Table 2.2 reports the material properties for the crazing model for the PS matrix
of HIPS and PMMA homopolymer; the craze properties of PMMA in PC/PMMA
microlaminates will be discussed in Chapter 3. All of the material properties, except
2&, for the PS matrix in HIPS have been taken from Piorkowska et al.
(1990).
For HIPS, 2oc = 0.1 pm is imposed by our model implementation, as discussed in
Section 2.3.4.
For PMMA homopolymer, the craze initiation parameters i and
Q are not
known
for PMMA, and were taken to be the same as that for PS. Based on the shear
yielding parameters for PMMA given in Table 2.1, the tensile yield strength for
PMMA is Y = 90 MPa at an applied strain rate of 1.7 x 10-3 s-. The parameter
C = 2600 Ma
was obtained by bringing the Argon-Hannoosh craze locus close to
the Sternstein-Ongchin craze locus (see Figure 2-7). The parameters B/KE,
#
and
A' are taken from Argon and Salama (1977), while Af is given by Kramer (1983).
The maximum critical craze opening, 2& , for PMMA is based on Doll's data (Doll,
1983). The parameter & for PMMA was taken to the same as for the PS matrix in
HIPS: & = 0.282 (Piorkowska et al., 1990). vo for PMMA is given by Argon and
Salama (1977).
Figure 2-10(a) shows the stress-strain response of a single PMMA craze element
60
Maximum
principal stress o
7(t)
+Craze fibril
2F.,(t)
-+,Void
O(t)
e
e2
Figure 2-9: Schematic of the craze "finite element". Cross-tie fibrils, which bridge
the primary craze fibrils, are not shown.
subjected to two velocities v, = 1.7 x 10- 3 , 1.7 x 10-2 yms-
1
at the boundary of
the square element of dimension 1 um. The response prior to initiation is elastic,
after which the stress drops to a steady value associated with the craze flow stress.
Note that both the craze initiation and flow stress are rate dependent.
The rate
dependence of the craze initiation stress was modeled by rendering the tensile yield
strength Y in Equation 2.17 strain-rate dependent. Finally, craze breakdown occurs,
resulting in the inability of the craze element to sustain stresses. Figure 2-10(a) also
shows the corresponding craze initiation loci 13 for v, = 1.7 x 10- 3 , 1.7 x 10-2 Pms-1.
Figure 2-10(b) shows the stress-strain response of a PS element of dimension 1
Pm at vi = 1 x 10-3 pms
1,
as well as the corresponding craze initiation locus".
We emphasize that unlike Y, which is strain-rate dependent, the craze flow stress is
dependent on the velocity vi (and not on the strain-rate across the element).
= ln(ti/-r) was found to be 18.5 for v, = 1.7 x
o1
Kms"For HIPS, K was found to be K = 27.
13K
61
10-3
pms- 1 and 15.5 for v, = 1.7 x 10-2
10
70
=65 MPa
=60 MPa
60
0
u.1.7 -10-
80
~70
540
r
Go0
~30
20
.1
.
~30
*
...
10
Or
1 0.1
0.2
0.3
0.4
05
True strain
0.6
08
07
1UnicDxal
loading
20
60
80
40
Hdrosta00 stress ah (Mpa)
1s0
(a)
100
40,
-
-
-
07
ru.s103
7
2
40
go
,.*
80
Unlaxial
loading
70
0
20
215
40
10
20
30
5
0
0.2
0.4
0.6
Toue
0.8
sasl
1
1
1.4
0
60
80
40
Hydrostaic stress a. (MPa)
20
100
(b)
Figure 2-10: Example of continuum behavior of a (a) PMMA craze element, and (b)
PS craze element. The craze element is of dimension 1 pm, with imposed velocity vi.
Table 2.2: Craze parameters for PS in HIPS (Piorkowska et al., 1990) and PMMA
homopolymer
PS (of HIPS)
6.0 x 10-8
PMMA homopolymer
1650
0.0133
70
3.4686 x 103
6.0 x 10-8
2600
0.0133
90
2.314 x 102
B/ks
218.5
44.72
237.5
49
A'
1.853
2.313
Af
4
2
2&oc(pm)
0.1
2
f (s)
C (MPa)
Q
Y (MPa)
&vo (ms- 1)
Y (MPa)
62
Chapter 3
Deformation and failure of
ductile/brittle polymeric laminates
3.1
Introduction
The strategy of toughening polymer systems by forming alternating layers of two different polymers goes back to the early 1970s. Some of the earliest work on polymeric
laminates can be traced to Schrenk and Alfrey and co-workers (e.g., Schrenk and
Alfrey 1969, 1978; and Radford et al., 1973). These authors made important contributions to the technology of coextruding microlayers, and also studied the physical
and mechanical properties of polymeric laminates.
However, Baer and co-workers
(e.g., Gregory et al., 1987; Ma et al., 1990; Shin et al., 1993a,b; Sung et al, 1994a,b,c;
Kerns et al., 2000) were the first to perform systematic experiments to probe the tensile toughness1 of ductile/brittle laminates, and make connections with the operative
micromechanisms of deformation. Most of their detailed work has been focussed on
the PC/SAN system, with limited work on the PC/PMMA system. By performing
tensile tests on un-notched and semi-circular-notched specimens, they showed that
high tensile ductility in ductile/brittle laminates is favored with (a) increasing volume fraction of the ductile component in the laminate, (b) increasing number of layers
for fixed macroscopic thickness (i.e., decreasing absolute layer thicknesses), and (c)
1Here, we define tensile toughness as the area under the stress-strain curve in uniaxial tension.
63
decreasing strain rate.
Figure 3-1(a) (Kerns et al., 2000) depicts the nominal stress-strain curves for a set
of 32-layered PC/PMMA laminates (with laminate thickness 1 mm), and indicates
the beneficial effect of increased PC volume fraction, fpc, on the tensile toughness.
Increase of fpc in the range 0.3 to 0.5 leads to minimal increases in the macroscopic
ductility, after which at fpc = 0.6, there is a jump in ductility.
by a smaller increase in ductility at fpc
=
0.7.
This is followed
Figure 3-1(b) shows the specimen
geometry, drawn to scale, used by Kerns et al. (2000) in their uni-axial experiments on
PC/PMMA laminates. Note that the cross-sectional area 2 of the specimen varies in
the (axial) 1-direction, which is the loading direction, resulting in an inhomogeneous
axial stress state in the specimen. The authors used the minimum cross-sectional area
to define their measure of nominal stress. The axial nominal strain and axial nominal
strain-rate are based on the total specimen length of 60 mm; quantitative details of the
local strain and local strain-rate are not provided in their paper. Figure 3-2 shows the
surface craze microstructure at the 1-2 surface in a necked PC/PMMA specimen with
fpc = 0.7, subjected to a nominal tensile strain rate of 1.7 x 10-3 s
1
; crazes originate
at the surface and do not significantly penetrate or tunnel in the 3-direction at this
composition and strain rate. Corresponding micrographs for other fpc, particularly
below the transition, were not provided by Kerns et al. A principal finding of Baer
and co-workers, most clearly seen in optical micrographs of PC/SAN (see for e.g.,
Figure 3-4) are the interactions of crazes and shearbands:
shearbands in PC are
associated with the crazes in SAN; the stress concentration provided by crazes in a
given SAN (or PMMA) layer nucleate shearbands in PC, which propagate to traverse
the entire PC layer.
Figure 3-3 shows the stress-strain response of the laminate with
fpc = 0.7 as a function of number of layers 3 at a nominal strain rate of 8.5 x 10-3
s- 1 , five times higher than for results in Figure 3-1. The increase of number of layers
results in a considerable improvement in the ductility at a given strain rate. Based on
2
The
3
exact manner in which the cross-sectional area changes is not known.
For fixed laminate thickness, increasing the number of layers leads to decreasing layer thicknesses.
In this context, notions such as "increasing number of layers" and "decreasing layer thicknesses" are
equivalent and will be used interchangeably
64
~-
--
-
1001
80
(n
(0
60-- fPc=0. 3
(0
z
40-
-f P=0.4
-fPc=0.5
..-fPc=0.6
20[
.- -fPc=0.7
0'0
0.05
0.1
0.15
Nominal axial strain
0.2
0.25
(a)
Gage length
=60 mm
3
2
(b)
Figure 3-1: (a) The effect of PC volume fraction on the nominal stress-strain response
of 32-layered PC/PMMA. The loading direction is coincident with the co-extrusion
direction. Laminate thickness=1 mm; strain rate=1.7 x 10-3 s- 1 . Redrawn from
the experimental data of Kerns et al. (2000). (b) The specimen geometry (drawn to
scale) used by Kerns et al. (2000). The exact profile of the variation of the width
along the 1-direction, from 30 mm at the ends to 5 mm at the center of the specimen,
is not known. The loading (and coextrusion) direction is the 1-direction.
65
-
-.---
-
-.
'I.'..'.
Figure 3-2: Micrograph of necked region of 32-layered PC/PMMA: fpc = 0.7 deformed at a nominal strain rate of 1.7 x 10-3 s-1. Laminate thickness=1 mm. The
loading (and co-extrusion) direction is the 1-direction. From Kerns et al. (2000).
their work on PC/SAN (e.g., Ma et al., 1990), the authors attributed the increased
ductility with increasing number of layers to cooperative shearbanding in the both
the (thinner) ductile and brittle layers. While data on volume changes in PC/PMMA
laminates is not given by Baer and coworkers, a study on PC/SAN laminates (Ma et
al., 1990) revealed that with decreasing layer thicknesses, the volume change becomes
negligible. This is associated with reduced crazing and a concomitant increase of
shear yielding in the brittle layers. The macroscopic yield in PC/PMMA laminates
does not change with the number of layers. The low ductility (and lower yield) of
the 32-layered laminate was attributed by the authors to premature breakdown of
crazes that initiated on the surface leading to crack propagation and early failure
of the laminate; this did not happen for 256 layered and 2048 layered laminate. By
comparing data for the 32-layered laminate subjected to axial strain rate of 1.7 x 10-3
s-1, and data for the laminates subjected to axial strain rate of 8.5 x 10' s-1, it
is noted that the macroscopic yield increases with strain rate. Another noteworthy
feature of Figure 3-3 is the decrease in ductility as the strain-rate is increased, for a
given number of layers (32) and composition (fpc = 0.7). The "strain-hardening"
66
100
80-
-
S60
U)
-
E
-
Z 20
--
Z 0
0
32 layers: 0.0017 s32 layers: 0.0085 s
256 layers: 0.0085 s2048 layers: 0.0085 s~
'
.05
-
40
0.4
0.2
Nominal axial strain
0.6
Figure 3-3: Stress-strain response of PC/PMMA microlaminates with fpc = 0.7 as
a function of number of layers, and strain rate. Laminate thickness=1 mm. From
(Kerns et al., 2000).
seen in Figure 3-3 is due to the tapered specimen geometry, shown in Figure 3-1, and
the consequent increase in load to axially propagate the inelastic deformation events.
While the understanding of Baer and coworkers of the micromechanisms is based
mostly on optical micrographs that reveal shearband and craze interactions at the
surface and immediate sub-surface of the laminate, Sung et al. (1994c) have explored
the shearband and craze interactions through the width of the specimen (in the 3direction).
Their essential conclusion is that the extent to which crazes emanating
from the surface penetrate through the width of the specimen depends on the PC
volume fraction of the laminate - the lower the fpc, the more is the tendency of the
crazes to propagate or "tunnel" through the width of the specimen. The "tunneling"
of the crazes through the width of the specimen is associated with decreased ductility.
The fact that crazes may not tunnel all the way through the width of the specimen
is shown in Figure 3-4; craze density in Figure 3-4(b) is lower than in Figure 3-4(a),
which is closer to the free surface (Sung et al., 1994c). Figure 3-5 compares the craze
67
(a)
e,
e2
Figure 3-4: Optical micrographs, in the 1-2 plane, of PC/SAN with fpc = 0.65. (a)
and (b) respectively show the craze morphology at a depth of 20pm and 60pm from
the free surface, indicating that crazes emanating from the surface do not necessarily
propagate completely in the 3-direction. From Sung et al. (1994c).
propagation in the 3-direction for laminates with fpc = 0.28 and fpc = 0.65. The
micrographs, associated with point (1) of the stress-strain curve, for both (a) and (b)
show the presence of surface crazes. At point (2), while the crazes penetrate in the
3-direction for fpc = 0.28, the crazes do not penetrate significantly in the 3-direction
for fpc = 0.65.
PC/PMMA and PC/SAN microlaminates have been quantified to have peel energies of GI, ~ 950J/m2 and GI, ~ 70-90J/m2 , respectively, guaranteeing stress transfer from layer to layer. Work by van der Sanden and co-workers (e.g., van der Sanden
et al., 1993, 1994) on PS/PE and PS-PPE/PE brittle/ductile microlayers highlights
the weak adhesion between PS and PE layers (peel energy of GI, ~ 5 - 25J/m2 ),
with consequent reduction in stress-transfer between the layers, and in shear yielding and crazing interactions; laminate systems with weak adhesion are not the focus
of our current work. The brittle/brittle laminate system PS/PMMA, investigated
recently by Ivan'kova et al. (2004), deforms due to cooperative crazing in both PS
68
0
22
-
20-
sai%
8
6
(1)
(a)
604020
0
4
2
a
Strainj%!
1
(2)
(1)
3
(b)
Figure 3-5: Optical micrographs, in the 1-3 plane, of PC/SAN with layer thicknesses
(a) tpc = 13 pm, tSAN= 33 ym (fpc = 0.28), (b) tPC = 29 pm, tSAN = 16 um
(fpc = 0.65). In both (a) and (b), micrographs corresponding to point (1) show the
presence of surface crazes. At point (2), the surface crazes in (a) tunnel through the
width, while in (b), they do not. From Sung et al. (1994c).
69
and PMMA, and shows superior tensile toughness compared to the individual constituents, with decreasing layer thickness, but the maximum observed tensile failure
strain is quite low at 0.1, compared with failure strains as high as 0.5 for PC/PMMA
microlaminates. These results indicate that the synergy between shear yielding and
crazing in ductile/brittle laminates provides a significantly higher gain in macroscopic
toughening, as compared to cooperative crazing in brittle/brittle laminates.
As can be inferred from the above discussion, a key feature of ductile/brittle laminates is the complex synergy between crazing and shear yielding, which can be tailored by optimizing the absolute and relative layer thicknesses, and the ease/difficulty
of craze initiation compared to shear yielding in the brittle layer.
Although duc-
tile/brittle laminates have been experimentally studied in detail, and some understanding of the shear yielding and crazing interactions and competition has been
achieved, microstructural models of ductile/brittle laminates that can give detailed
insight into the laminate behavior, and can guide the design of optimal laminates are,
to this date, not available. In particular, there is a need to understand the effect of
fpc, the number of layers (or layer thicknesses) and strain-rate on the macroscopic
toughness of the laminate. Here, we apply our fundamental understanding of shear
yielding and crazing behavior of the constituent materials to construct finite element
based simulations of the tensile behavior of PC/PMMA microlayers. This new simulation capability is then used to investigate the effect of volume fraction of the ductile
component, number of layers for fixed laminate thickness, and strain rate on the
synergy and competition between shear yielding and crazing in the layers, and the
resulting consequence on macroscopic tensile behavior. The modeling approach can
be easily generalized to other laminate systems, such as two-phase brittle-1/brittle-2
and three-phase ductile-i /brittle/ductile-2 laminates.
3.2
Model
In order to explore and understand the underlying physics that account for the
macroscopic tensile behavior of ductile/brittle laminates, 2D and 3D micromechanical
70
models that are representative of the laminate morphology are used. The proposed
micromechanical model for microlayered polymeric laminates includes three components: the geometric description of the laminate morphology of the representative
volume element (RVE); constitutive descriptions of the material constituents - PC
and PMMA; description of the material interface and its idealization.
3.2.1
Description of 2D and 3D RVE
Figure 3-6 shows the morphology of a 2D RVE, with PC volume fraction fpc = 0.7,
used in finite element simulations to investigate the micro- and macro-mechanics
of deformation and failure in micro-layered laminates.
boundary conditions ui = 0 and
U2
The indicated displacement
= 0 reduce the model size to 1/4 by symmetry.
Similar 2D RVEs were constructed for fpc = 0.1, 0.3,0.4,0.5, and 0.6, such that the
RVE aspect ratio was equal to 1 in each case, and that each bi-layer, consisting of
one PC and one PMMA layer, is spanned by 20 finite elements in the 2-direction.
To adequately capture craze breakdown (2 oc = 2 pm) and noting that )
~ 2
(Kramer, 1983), each finite element has a dimension of 1 pm in the 1-direction. The
aspect ratio of the elements is unity for the 2D RVEs.
The 2D-RVE is displaced
at the top boundary in the 1-direction such that a nominal macroscopic strain rate
e
= 1.7 x 10-
s-1 for the RVE is kept constant during the process of deformation;
this nominal strain-rate is consistent with that used by Kerns et al.
(2000).
The
right edge of the 2D RVE is a free edge. The 2D RVE was subjected to plane strain
constraint in the 3-direction.
Figure 3-8(a) shows the mesh for the 2D RVE with
fpc = 0.7.
Figure 3-7 shows the 3D RVE with fpc = 0.7. The details described above for the
2D RVE also hold for the 3D RVE; the 2D and the 3D RVE are identical in the 1-2
plane. In the 3-direction, the 3D RVE is 175 pm wide and is spanned by 70 elements.
The mesh in the 3-direction is biased such that the mesh is finer closer to the free
4Baer and co-workers have investigated microlayered laminates in which the outermost layers
were constituted of PC. We have kept this feature in our geometric models; Figure 3-6 actually
depicts a laminate with fpc = 0.724, but we will nominally refer to it as fpc = 0.7
71
surface X3 = 0 pm. In particular, starting from X3 = 0 pDm, the first 10 elements
are 1pm wide, the second set of 10 elements are 1.5pm wide, and so on, so that
the seventh set of 10 elements are 4pm wide.
U3 = 0 at
X3=
The symmetry boundary condition,
175 pm along with u 1 = 0 and U 2 = 0 reduces the model size to 1/8.
Figure 3-8(b) shows the mesh for the 3D RVE with fpc = 0.7.
The actual specimen geometry used by Kerns et al.
(2000) in their uni-axial
experiments on PC/PMMA micro-laminates has already been described in Figure 31(b). In our simulations, the use of a "small" finite element size of 1 pm, in the 1direction, stems from the need to adequately capture craze breakdown, which typically
occurs in PMMA at a critical craze opening of
2
oc = 2 pm. Due to computational
constraints, the 3D RVE is much smaller than the actual specimen geometry employed
by Kerns et al. (2000). The total number of finite elements used in our 3D simulations
is over 500,000.
3.2.2
Constitutive description for the ductile and brittle layers
In the micromechanical models, the behavior of PC is governed by shear yielding,
discussed in detail in Section 2.1.
For the PMMA layers, there are select regions,
shown in Figure 3-6 and Figure 3-7 in which PMMA can undergo either crazing or
shear yielding; the remaining PMMA is allowed to undergo only shear yielding 5 . The
plane of the crazeable PMMA regions are oriented perpendicular to the 1-direction.
Their arrangement, from one layer to another, is staggered.
The basic theory and
modeling of crazing has already been discussed in Section 2.3. Here, we will develop
some remaining crucial aspects pertaining to the subject of craze initiation.
5
This restriction reduces the computational requirements, and stems from experimental observations in which a minimum inter-craze distance is established due to the local unloading of the
PMMA regions surrounding each craze.
72
PMMA
PMMA
(non-crazeable)
PMMA
(crazeable)
t
PC
t
A
(D
Symmetry
3
C,,
*N-H
7 pm
(7 elements)
Symmetry
u,=0
20 pm
(20 elements)
14
87 pm
(87 elements)
t2
Figure 3-6: The geometry of the 2D RVE along with relevant dimensions and boundary conditions.
73
Symmetry
u 3 =0 on x3 =1 75
Symmetry
ob1
"
k\
Symmetr y
3
kz2
u,=0 on x,=0
Figure 3-7: The geometry of the 3D RVE. The in-plane dimensions and boundary
conditions are same as that for the 2D RVE.
74
.
......................
....
...................
..........
44444-141-1-1-1-1-11-1-1 ............ ........
...
...
....
......
.......
....
.............
.......
.....
..........
.........
. . ....
... ...
....
...............
.
....... ......
-- ...
.
.....
.........
....
...
...
.........
..... ...
.......
....
.......
....
.....
.....
....
.......
....
.....
.....
.... ---.......
...
II
... .. .....
.... .. .....
...
....... ..
.......
.....
....... .....
.....
.......
......
.
...............
...........
..............
..............
...............
......
........
............
..... ......
......
.....
...
......
----.. ...... ...
.....
..... .. .....
.. ........
. .. . .
.....
..............
.......
.....
....
..... ......
.....
......
..... ......
---ifift
....
HI'll'
HT,
t 1
T:
Figure 3-8: Mesh for (a) 2D RVE, and (b) 3D RVE.
75
2
Factors controlling craze initiation
Below, we discuss some relevant factors that affect craze initiation, and thereby result
in a finely balanced competition between shear yielding and crazing in ductile/brittle
laminates.
* Molecular network orientation: It is well known that craze initiation is
quite sensitive to molecular network orientation. Figure 3-9(a), taken from an
early study on PMMA due to Beardmore and Rabinowitz (1975), shows the
craze initiation stress oc (and the fracture stress JF) plotted against 0, which is
the angle between the loading direction and the molecular orientation direction.
The authors report c, = 16 ksi (103.1 MPa) for isotropic PMMA. When the
direction of molecular orientation is aligned with the tensile loading direction,
i.e. 0 = 00,
c is about twice the isotropic value.
With 0 = 900, a, attains
its minimum value of about 12 ksi (82.7 MPa). Figure 3-9(b), taken from
Maestrini and Kramer (1991), shows a similar trend for PS. In Figure 3-9(b),
the median strain to craze nucleation is reported for pre-orientated PS film
samples. The orientation is quantified in terms of the pre-strain level c.
The
median strain is seen to increase with increasing Eor. These results highlight
that when the molecular orientation of specimens coincides with the tensile
loading direction, the craze initiation stress increases, compared to unoriented
specimens. Figure 3-9(b) also shows an increase in the median strain to craze
breakdown with increasing levels of cor. Maestrini and Kramer have noted that
in view of the increase in backstress with increasing network orientation, the
craze flow stress increases with network orientation.
* Defects:
Commercially available glassy polymers invariably contain defects,
which manifest themselves in several ways including surface irregularities, internal impurities, and molecular-level "defects". Craze initiation preferentially
occurs at defect sites, particularly at surface scratches, because of the higher
local stress levels in the vicinity of defects.
76
40
PMMA 77 K
wt.
.0 c
3632
24
-
28-
20
i
16-
84-
0
0
15
30
I
45
60
75
90
8'
(a)
(b)
Figure 3-9: (a) The dependence of the craze initiation stress a-, and the fracture stress
~F on 0, which is the angle between the molecular orientation direction and the tensile
loading direction. From Beardmore and Rabinowitz (1975). (b) The dependence of
the median strain to craze nucleation, breakdown and fracture, on the pre-strain Eo,.
in PS films. The pre-strain is in the direction of loading. From Maestrini and Kramer
(1991).
77
Crazing in PMMA in PC/PMMA microlaminates
Ductile/brittle microlaminates are manufactured by co-extruding the ductile and brittle glassy polymeric constituents to form tens to thousands of layers, resulting in some
level of molecular orientation in the direction of co-extrusion. Figure 3-10(a), taken
from Ma et al. (1990) shows the stress-strain behavior of unannealed PC/SAN microlaminates with PC volume fraction fpc = 0.65. The direction of tensile loading
%
coincides with the coextrusion direction. A large increase in ductility, from 7-8
strain to 130 % strain, is observed as the number of layers is increased from 49 to
776. Ma et al. found that the SAN layers show profuse crazing in the 49-layered laminate, but negligible crazing in the 776-layered laminate. The inelastic deformation
in the SAN layers for the latter was accommodated mostly by shear yielding. The
increase in ductility is associated with reduced crazing and increased shear yielding
in the laminate. Along expected lines, when Ma et al. annealed the 776 layered laminate, the ductility reduced from 130% (for the unannealed) to 27% (annealed), as can
be seen in Figure 3-10(b). More crazing was observed in the annealed sample, compared to the unannealed one; annealing resulted in the loss of molecular orientation
.
and a consequent decrease in the craze initiation stress 6
Interestingly, comparing the 49 layered unannealed laminate with the 776 layered
annealed laminate, it can be seen that the ductility increases from 7-8 % to 27 %.
Micrographs indicate reduced crazing in the annealed 776 layered laminate, compared
to the unannealed 49 layered laminate. This shows that orientation is not the only
factor that can result in increased craze initiation stresses, and that arguably the
comparatively reduced availability of defects where crazing can occur preferentially
in thin-layered laminates (compared to thick-layered laminates) reduces the craze
density within the brittle layers.
Reduced craze density within the brittle layers
results in more inelastic deformation via shear yielding.
6
9.1
For the 776-layered laminate, annealing resulted in a decrease of laminate length and width by
2.0 %.
1.0 %. respectively. The laminate thickness increased by 10.1
1.2 % and 1.7
78
80 49 LAYER
60
40
20-
0
LU
194 LAYER
776 LAYER
7 60
4020
0
10
10
STRAIN
130
(%)
(a)
7X 1UNANNEALED
/WANNEALED
27%
I ANNEALED
.28%
W
I-
UNANNEALED 130%
4
2
0
10
STRAIN (0/)
(b)
Figure 3-10: (a) Stress-strain behavior of unannealed PC/SAN laminates for fpc =
0.65 as a function of number of layers. Tensile loading is in the co-extrusion direction.
(b) Comparison of stress-strain behavior of 776-layered laminate with fpc = 0.65 for
four cases: "I unannealed", "I annealed", "II unannealed" and "II annealed". "_L"
and "II" indicate that the direction of loading is, respectively, perpendicular and
parallel to the coextrusion direction.
79
Table 3.1: Comparison of craze parameters for PMMA microlayers and PMMA homopolymer.
PMMA microlayers
PMMA homopolymer
Y (MPa)
voO (ms- 1 )
Y (MPa)
B/ks
A'
6.0 x 10-8
7000 (mean)
0.0133
90
1.45 x 10-1
237.5
49
2.313
6.0 x 10-8
2600
0.0133
90
2.314 x 102
237.5
49
2.313
Af
2
2
2&(/tpm)
2
2
f (s)
C (MPa)
Q
3.2.3
Crazing parameters for PC/PMMA microlaminates
Based on the above discussion, we can infer that
* Due to effects of network orientation, the craze initiation resistance is higher in
the PMMA microlayers, compared to the PMMA homopolymer. We model this
effect by raising the value of C in Equation 2.17 beyond that given in Table 2.2.
For convenience, the material parameters
Q,
Y and i for PMMA microlayers
are kept the same as that for PMMA homopolymer.
The expected increased craze flow stress (due to network orientation) in PMMA
microlayers is modeled by decreasing the value of dvo in Equation 2.21 compared
to PMMA homopolymer. For simplicity, the maximum craze opening in PMMA
microlayers is kept the same as that for PMMA homopolymer. Table 3.1 summarizes the craze properties of PMMA microlayers and PMMA homopolymer
(taken from Table 2.2). Note that Table 3.1 shows the mean value for parameter
C for PMMA microlayers in PC/PMMA laminates. The mean value for C is
associated with its Weibull probability density function, which will be discussed
shortly.
* The presence of defects in the PC/PMMA microlaminates has two important
80
effects.
First, since the magnitude and type of local stress concentration in-
duced by the defects varies from one defect to another, the propensity of craze
initiation in the vicinity of defect sites also varies.
We model defects in the
crazeable portion of the PMMA layers through a 3-parameter Weibull distribution in the craze initiation material parameter C. In this setting, "low" values
of C correspond to defects. Second, crazing is known to preferentially initiate
from the surface and the immediate sub-surface of specimens. We model this by
reducing the mean value of C at the surface, compared to the mean value of C
within the bulk of the material. The statistics of craze initiation will be developed next. We note in passing that craze breakdown is also defect sensitive, and
hence naturally leads to statistical variations (e.g., Kramer and Berger, 1990).
In this thesis, we have kept the critical craze opening constant for simplicity.
Since craze initiation is the first inelastic deformation process in ductile/brittle
laminates, it is crucial to model the craze initiation statistics.
Statistics of craze initiation
Here, we discuss the operational aspects pertinent to our modeling of craze initiation
in ductile/brittle laminates.
The Weibull distribution, which is well known to be
particularly suited to fit data on strength and failure (Weibull, 1951), will be used to
model defects. The probability density function (pdf) for the Weibull distribution is
given by:
f(x)
f(x)
0 if x <y
=
w (- X
exp
[-
(x
W
if x >y
(3.1)
where - , Ow, and q, are respectively the location, shape and scale parameters of the
distribution. The cumulative density function (cdf) is given by:
81
F(x)
=
0 if x <
F(x)
=
1 - exp
; )1
-
if x > y
(3.2)
The mean p, and standard deviation oaw are given as:
pm=-
(3.3)
I + 1
W + y.
+1
Ow2
(7w=
(>-2
W\/
('I>3
2
1)(3.4)
Table 3.2 gives the Weibull parameters for C within the bulk and at the surface
of the laminate; the other craze initiation parameters are considered constants. Note
that the Weibull parameters corresponding to the bulk of the laminate differ from their
"surface" counterparts only in the -yy term. Figure 3-11(a) and (b) show respectively
the pdf and cdf for C applicable to PMMA on the surface of the laminate. While these
values are not founded directly on experimental data, they were chosen to capture
the experimentally observed competition between shear yielding and crazing in the
brittle layers. Figure 3-11(a) and (b) also show the resulting pdf and cdf of the craze
initiation stress.
Figure 3-13(a) shows the shear yielding loci for PC and PMMA,
and the craze loci for PMMA. The shear yielding loci are based on the pressure and
rate-sensitive von-Mises yield function o, = V/5[so( ) - ach], where so is the strainrate dependent initial shear strength of the material, and a is the pressure sensitivity.
The craze loci are based on the time-independent form of Equation 2.17:
C
Oe
3
h
2QY
K
(3.5)
where the value of K = 20 was chosen to closely match the craze initiation stress give
by the time-dependent Equation 2.17.
shows four craze loci in oe
-
Uh
Based on these parameters, Figure 3-13(a)
space corresponding to the values of C equal to -ys,
82
Table 3.2: Weibull parameters for C within the bulk of the specimen and at the
surface.
Bulk
7 (MPa) 3400
mq(MPa) 3950
4
ow
Surface
1700
3950
4
11- -as, ps , and ps + os. Also shown are the shear yielding loci for PC and PMMA.
Similarly, Figure 3-12 shows the pdf and cdf for C, and the craze initiation stress
for PMMA in the bulk of the laminate. In our model, the Weibull parameters
#w and
77 for C are independent of x 3 , but -, varies according to:
-Y(X 3 )
=
'Y(X 3 )
=
+ ('Y% - -Y)
7 if 25
<
x3
(3)
if X3 < 25
175
(3.6)
(3.7)
where the dimensions of X3 are in yum. It is evident from Equations 3.6 that the
craze loci at the surface and the sub-surface are lower than that of the bulk. This
is in accord with experimental observations that crazing almost invariably initiates
at the surface or the sub-surface, because of surface imperfections.
According to
Equation 3.6, 7, increases linearly from the surface until X3 = 25 pim, after which
it remains constant at its "bulk" level. In Section 3.3.6, a sensitivity study will be
conducted in which the gradient at which 'yw changes from the surface to the bulk
region was increased.
3.2.4
PC/PMMA interface description
It is well recognized that laminate interfaces play an important role because they
govern the degree of stress transfer between layers, and in the case of ductile/brittle
laminates will influence the synergy between crazing and shear yielding. From the
work of Kerns et al. (2000), it is clear that the PC/PMMA system has a very high
83
.10
0.03,
-
3
0.025
p
2.5
0.02
2
0.015
1.5
0.01
0.5
0.005
4000
000
C (MPa)
2
6000
0
7000
60
80
100
Craze initiation stress (MPa)
12-
120
(a)
09
0.8
-,
0.'8
+o
p -
te
+
.
0.9
0.7
0.7
0.6
0.5
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
2000
3000
4000 5000
C (MPa)
8000
7000
60
80
100
Craze initiation stress (MPa)
120
(b)
Figure 3-11: (a) Weibull probability density function (pdf), (b) Weibull cumulative
density function (cdf), for craze initiation parameter C and craze initiation stress at
+ u
the free surface of the RVE. The values of C corresponding to p' - o,P,4P
(where p4 and a' are, respectively, the mean and standard deviation of C at the
surface of the RVE) are indicated.
84
x 10
-
35
0.03
b
3
b
!Ib Cb
,b
0.025
+b
-
2.5
0.02
21
0.015
1.5
0.01
0,005
05
4000 5000
6000 7000 800
0
9000
10000
810
C (MPa)
120
140
100
Craze initation stress (MPa)
160
(a)
0.9
0.8
0.9
pb -
b
p +a$
0.7
0.7
0,6
0.5
0.54
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
4000 5000
7000 8000 9000 10000
C (MPa)
140
ton120
Craze indaton stress (MPa)
C0
160
(b)
Figure 3-12: (a) Weibull probability density function (pdf), (b) Weibull cumulative
density function (cdf), for craze initiation parameter C and craze initiation stress in
+ ab (where
the bulk of the RVE. The values of C corresponding to pIL - ab P bp
p3 and a b are, respectively, the mean and standard deviation of C in the bulk of the
RVE) are indicated.
85
300
250
(U
0~
C)
200
Uni-axial
loading
U,
(I)
a,
C,,
C,,
150
a,
s
s
C,,
100
-------
C
-------
p
50
MMA shear
locus
l~cus
0
0
20
80
60
40
100
Hydrostatic stress a h (MPa)
(a)
300
250
200
Uni-axial
loading
b
t)
150
b
b
b
Ib +(b
W
W 1W
W
100
--------W
-----50
--
PC sfiear
lcQus
20
U0
20
PMMA shear
locus
60
40
40
60
8
80
100
Hydrostatic stress ah (MPa)
(b)
Figure 3-13: Shear loci of PC and PMMA, and craze loci of PMMA based on the
following four values of craze initiation parameter C: minimum (-yw), p1 , p
and p1, + O-
(a) for free surface of the RVE, (b) for bulk of the RVE.
86
w- 0
interfacial strength (Gic ~~950J/m 2 , as measured by the T-peel test), compared to
G1, ~~70 - 90J/m2 for PC/SAN, and G1 , ~ 5 - 25J/m 2 for PS-PPE/PE (van der
Sanden et al., 1994b). Unlike PC/SAN, which is known to undergo local delamination
at sites of craze opening, there seems to be no observable delamination in the case
of PC/PMMA. The lack of delamination, however, comes at the expense of notching
of the PC layers because craze opening in the adjacent PMMA layers leads to large
local stretching of PC; notching of PC in PC/SAN is less severe because of the stress
relief provided by local delamination. In our simulations on PC/PMMA in this work,
we assume that the interface continues to be perfectly bonded to layers throughout
the progression of deformation.
3.3
3.3.1
Results
Preliminaries
The detailed connections amongst the microstructure, deformation and failure mechanisms, and the macroscopic response for both 2D and 3D RVE simulations are
explored via macroscopic metrics, as well as contour plots of craze opening 2 o and
accumulated plastic shear strain -yP =
f
P(r)dT. The macroscopic metrics include
the macroscopic nominal stress, the normalized craze volume and the normalized volume of failed PC,
#pc,
and PMMA,
#PMMA.
plotted against the macroscopic nominal strain.
The evolution of these metrics are
The normalized craze volume is
defined as
Volume of crazed region in laminate
Undeformed volume of PMMA in laminate
Failure volume fractions are defined as
Undeformed volume of failed PC elements
Undeformed volume of PC in laminate
87
(39)
Undeformed volume of failed PMMA elements
9PMMA
Note that
#PIMA
=
(3.10)
Undeformed volume of PMMA in laminate
is associated with both craze breakdown and failure due to
excessive plastic stretch. In all cases discussed below, the predominant contribution
of
#PM AA
3.3.2
turns out to be from craze breakdown.
Results from 2D simulations
Detailed 2D analysis for case fpc = 0.3
Here, we first investigate in detail the connections between the macroscopic behavior
and the behavior of the underlying microstructure, as revealed from the 2D plane
strain RVE model for the case fpc = 0.3. The craze initiation properties corresponding to free surface conditions 7{ (Figure 3-11) have been used. As the 2D RVE is
displaced in the 1-direction, the initial deformation of the laminate is linear elastic
until point (a) (e = 0.02) (see Figure 3-14). The 2 o contour plot in Figure 3-15
shows the nucleation of a few isolated crazes, at locations of low resistance to craze
initiation. No shear plasticity is evident from the -yP contour plot. From point (a) to
(b) (e = 0.024), the nominal stress increases with nominal strain; there is very little
deviation from the linear elastic slope, consistent with the observations of Haderski
et al. (1994) on PC/SAN ductile/brittle laminates. Due to crazing in the PMMA
layers, the normalized craze volume increases at a steady rate from (a) to (b). The
2 o contour plot indicates the opening of crazes throughout the laminate, and the
.YP
contour plot shows the development of plastic zones in PC at locations where crazes
in PMMA impinge on the adjacent layers. It is evident that the macroscopic yield is
associated with both inelastic deformation mechanisms - crazing and shear yielding.
Between points (b) and (c) (e = 0.07), the nominal stress drops, because of strainsoftening in the shear-yielding response, as well as due to the drop in the stress after
craze initiation (see Figures 2-3 and 2-10). The normalized craze volume continues to
increase steadily. The -yP contour plot shows shearbands in PC connecting the crazed
regions in PMMA layers. Between (c) and (d) (e = 0.1), the nominal stress decreases
88
further, due to strain-softening, as well as due to craze breakdown and crack formation, which can be inferred from the increase in
#PMMA,
as well from contour plots
associated with point (d) shown in Figure 3-16. The reduced load-bearing capacity
of the PMMA layers due to craze breakdown manifests itself locally through highly
drawn PC regions adjacent to PMMA cracks, as can be seen in the -yP contour plot.
The rate of increase of normalized craze volume decreases in this regime. From point
(d) to (e) (e = 0.25), the nominal stress further decreases, with the stress at point
(e) dropping to about 10 MPa. The wavy nature of the stress drop from (d) to (e)
is due to the competing factors of failure in PMMA and PC, and the reinforcement
provided by un-failed PC due to orientation hardening. The contour plots show the
cracking in the PC layers, which initiate due to the large local stretching of PC in
the vicinity of the failed PMMA crazes.
Detailed 2D analysis for case fpc = 0.7
The detailed analysis for case fpc = 0.7, shown in Figures 3-17 through 3-19 is almost
identical to the case of fpc = 0.3, discussed above. A noteworthy point is that due
to the higher PC content of fpc = 0.7, the macroscopic stress drops more gradually
compared to the case fpc = 0.3, and Opc becomes non-zero at a higher nominal
strain.
2D results for different fpc
Figures 3-20 and 3-21 show the macroscopic results for varying levels of fpc obtained
with the surface craze initiation properties, characterized by y, (Figure 3-11). A similar study using 2D plane strain RVEs was conducted with craze initiation properties
corresponding to the "bulk" condition characterized by the Weibull parameter
,.
Figures 3-22 and 3-23 summarize the macroscopic results of this study as a function
of fpc. The nominal stress-strain curves in both sets of figures indicate the correct
trend in predicting the increased ductility with increasing fpc. However, comparing
these estimates with the experimental results (see Figure 3-1), the effect of increase
in fpc is not adequately captured with the 2D plane-strain models. Particularly, the
89
100
80
b)
CO
C)
2 60
C
40
Z
d)
20
(e)
0 0.05
0.1
0.15
0.2
0. 25
Nominal axial strain
(a)
0.
0.08-
a)
E
:)
( 01
(d)
0.06
N
(C)
*0
0.04
a)
0.02
(rb)(c
(a
0.05
0.1
0.15
0.2
0.25
Nominal axial strain
(b)
0.04
0.035
4:
(
PMMA
0.03
0
0.025
a-
-a)-
-oC
0.02
(d)
(V
0~
0.015
10
-
C
4
Pc
0.01
0
_0
(C
0.05
0.05
0.1
-
0.005
0.15
0.15
0.1
0.2
0.25
Nominal axial strain
(C)
Figure 3-14: Macroscopic response of 2D laminate with fpc = 0.3 at nominal strain
rate e = 1.7 x 10- 3 s 1 . Based on craze initiation properties corresponding to -y,.
90
Plastic shear strain
Craze opening
.Y5
04
05
0
I
(a) I
e=0.02
2eo
12
.2
io
0
(b) U
e=0.024
I
2eO
2o
1
(c)
e=0.06
Figure 3-15: Contour plots of craze opening 2 Oc (left column) and accumulated
plastic shear strain -yP (right column), corresponding to Figure 3-14 (2D plane strain,
fpc = 0.3).
91
2
Plastic shear strain
Craze opening
2
0
(d)
(e)
e=0.25
I
-
e=O.1
2
Figure 3-16: Contour plots of craze opening
2 &0c
and accumulated plastic shear strain
yP, corresponding to Figure 3-14 (2D plane strain, fpc = 0.3).
92
100
80
0-
(b)
(C)
60
(d)
(a)
a
X
40
z
20
0
(e'
0
0
0.05
0.1
0.15
Nominal axial strain
0.2
0.25
(a)
0.1
(e
0 0.08E
if
0
(d)
0.06.
C.)
-0
)
.L
(c
0.04
E
0
0.02
C a)
0
(b)
0.05
0.1
0.15
Nominal axial strain
0.2
0.25
(b)
0.04
0.035
0.03
0.025
(e
PMI
0.02
a0.015
(d)
0.01
Pc
f
0.005
0
0.05
0.1
0.15
Nominal axial strain
0.2
0. 25
(C)
Figure 3-17: Macroscopic response of 2D laminate with fpc = 0.7 at nominal strain
rate e = 1.7 x 10- 3 s 1 . Based on craze initiation properties corresponding to y",.
93
Plastic shear strain
Craze opening
2eo
yp
.06
0. 04
io
0
(a)
e=0.02
2o
06
.25
0
0
(b)
e=0.025
2eO
0
0
(c)
e=0.07
Figure 3-18: Contour plots of craze opening 2 Oc (left column) and accumulated
plastic shear strain f (right column), corresponding to Figure 3-17 (2D plane strain,
fpc = 0.7).
94
2
Plastic shear strain
Craze opening
2
yp
0
:0
(d)
e=O.1
(e)
e=0.25
1
2
Figure 3-19: Contour plots of craze opening 2 Oc (left column) and accumulated
plastic shear strain -yP (right column), corresponding to Figure 3-17 (2D plane strain,
fpc = 0.7).
95
2D plane strain models fail in capturing the experimentally observed transition in
ductility with fpc increasing from 0.5 to 0.6. The plots of normalized craze volume,
and
#PMMA
do not show any significant trend. With increasing fpc,
#pc
generally
becomes non-zero at a higher magnitude of nominal strain, and hence explains the
somewhat tougher response at high fpc. It is noteworthy that the normalized craze
volume plot for a given fpc is lower in Figure 3-23 as compared to Figure 3-21. This
is because the former is associated with yt; the greater craze resistance results in less
crazing in the RVE. Compared to the simulations based on
, the simulations based
on -y show a higher macroscopic yield strength for the same fpc, indicating that the
macroscopic yield is a function of the craze initiation stress.
3.3.3
Results from 3D simulations
Detailed analysis for case fpc = 0.3
Figures 3-24 through 3-26 show the macroscopic response and the associated contour
plots at different stages of the deformation, corresponding to points (a)-(e) in Figure 324 for the case of fpc = 0.3. For clarity, (unlike the 2D RVE contour plots) the 2 o
contour plots show only those PMMA elements that have a non-zero craze opening.
The -yP contour plots for the PC layers have been rendered translucent to enable
visualization of shear plasticity within the 3D RVE. Figure 3-27 shows only the failed
PMMA and PC elements at points (c)-(e), indicating the crack morphology of the
failed laminate.
In Figure 3-24, prior to point (a), the specimen deforms elastically subjected to
uni-axial loading. The initial (elastic) slope is the volume-averaged uni-axial elastic
modulus of PC and PMMA. At point (a), the contour plot for 2 o in Figure 3-25
indicates that 5 elements in PMMA have initiated crazes. These elements correspond
to the surface and immediate sub-surface of the specimen; this is consistent with the
relatively weaker resistance to craze initiation at and near the free surface, compared
to that found in the bulk (interior) of the RVE. The -yP plot shows the absence of
shear plasticity.
At point (b), which corresponds to the macroscopic yield stress,
96
10(
I
fPC =0.3
0
PC
80-
f PC=0.5
fPC=0.6
4U
C,
60-
=0.7
---
40
z 20
00G
0.05
0.2
0.15
0.1
Nominal axial strain
0.25
(a)
-
-
0.1
N
-
a) 0.08
E
-..----------------
0.06
--- fPC=0. 1
CU
N
0.04
-PC=0.4
fPC=0.5
0
Z
fPC=0.3
... f P C= 0 . 6
0.02
0
'J
0.05
0.05
0.1
0.15
0.15
0.1
Nominal axial strain
0.2
0.25
(b)
Figure 3-20: Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 2D plane strain RVEs at nominal strain rate e = 1.7 x
10-3 s-, as a function of fpc. Based on craze initiation properties corresponding to
-y (Figure 3-11).
97
4T
fpc
fpc
fpc
fpc
fpc
fpc
0.035.
0. 0310.025-
=0. 1
=0.3
=0.4
=0.5
=0.6
=0.7
- ----- -----
0.02-
a-
-
-
0.0
0.0150. 010.005-
O0
0.05
0.15
0.1
Nominal axial strain
0.2
0.25
0.0 4
50.03
5-
0.0 3
fPC=0.5
PC
... ... f c=0.6
_ fPc=0.7
0.02
0
0~
4PC .
-- fPc=0.1
fpc=0.3
.
(a)
0.0 20.01 5
0.0
0.005
0C
0
0.05
0.05
0.15
0.1
0.1
0.15
Nominal axial strain
0.2
0.25
(b)
Figure 3-21: Evolution of (a) #PMMA and (b) #pc,with nominal axial strain for 2D
plane strain RVEs at nominal strain rate e = 1.7 x 10-3 s--1, as a function of fpc.
Based on craze initiation properties corresponding to -y,, (Figure 3-11).
98
=01
f
100
..... fPC=0.1
-f PC =0.3
(U
80
fPC=0.5
..5 0fPC=0.6
----
60
0.7
CU
x
C
40
E
0
z 20
SC
0.05
0.15
0.*1
Nominal axial strain
0.2
0.25
(a)
0.1
fPC=0.1
-
WU
--
0.08-
E
-
70
-
0.06-
fPC=0.3
PC=0.4
fPC=0.5
fPC=0.6
fPC=0.7
0)
L4
(U
E
0.04
/.7.
0
z
0.02
uT
0.05
0.15
0.1
Nominal axial strain
0.2
0.2E
(b)
Figure 3-22: Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 2D plane strain RVEs at nominal strain rate e = 1.7 x
10- s--1, as a function of fpc. Based on craze initiation properties corresponding to
-'y, (Figure 3-12).
99
0.0 4 I
f
PC
=0.1
- PC=0.3
- -fPC=0.4
f =0.5
VU
- fPC=0.6
_ f P=0.7
0.030.025-
-------
0.035-
PC,J
0.02-6-
0.015
0.01
0.005
CL
0
0.05
0.1
0.15
Nominal axial strain
0.2
0.25
(a)
0.041
0.035
fPC=0.1
-- fPC=0.3
- fPC=0.4
0.03- fPC=0.5
. .. f C=0.6
0.0251 ___ fPC=0.7
PC
0.02
0.015-
I'
0.01.
0.005[
CI0
-I
0.05
0.15
0.1
Nominal axial strain
0.2
0.25
(b)
Figure 3-23: Evolution of (a) #PMMA and (b) #pc,with nominal axial strain for 2D
1
plane strain RVEs at nominal strain rate e = 1.7 x 10-' s- , as a function of fpcBased on craze initiation properties corresponding to 7b (Figure 3-12).
100
2& 0-contour plot shows that crazes initiating from the surface and sub-surface have
started to propagate towards the center of the specimen (which is the X 3 = 175 pm
face by symmetry). Several "intrinsic" crazes, not emanating from the surface, have
also developed in the specimen. These intrinsic crazes initiate at locations of weak
craze initiation resistance, and are physically associated with internal defects in the
specimen, such as voids, heterogeneities, or molecular-level defects. The 'YP plot shows
shear-banding in the PC layers, especially at the free-surface X3 = 0 where the craze
density in the PMMA layers is high.
is negligible.
At point (b), the normalized craze volume
It is worth emphasizing that the macroscopic yield is associated with
both shear yielding and crazing. Between points (b) and (c), the nominal axial stress
decreases, while the normalized craze volume increases. Also, the region between (b)
and (c) marks the beginning of craze breakdown at the surface, which can be inferred
both from a non-zero
#PMMA
at point (c) and from the 2 o contour plots at (c),
where it is clear that several crazes have reached the maximum craze opening. The
failed elements can also be seen in Figure 3-27. The decrease in the nominal stress
is due to the shear softening in PC and uncrazed-PMMA, and softening associated
with the reduction of stress in a craze element from the initiation to the flow stress,
and also due to a small amount of craze breakdown. In the region between (c) and
(d), the nominal stress decrease is sharp, and is associated with the sharp increase in
#PMMA.
There is also a non-zero Opc. The normalized craze volume also increases
sharply. The 2 o and the failure contour plot at (d) indicate a crack that grew from
the free-surface of the specimen towards the center of the specimen, and relatively
smaller cracks that grew from the center towards the surface. The -}P plot for point
(d) shows extensive plastic strain in the vicinity of the crack. The failure from point
(d) to (e) is marked predominantly by failure in PC. The fact that the two dominant
cracks are not in the same plane accounts for the non-zero stress at (e). Figure 3-28
qualitatively indicates an overall higher plastic shear strain in the PC layers, compared
to the PMMA layers, at a nominal strain e = 0.202.
101
100
(b
80
0-
C)
Ii)
60
a)
(d)
40
C
E
0
z 20
e)
0.1
0.2
Nominal axial strain
0.3
I
.
0
0.07
a),
E 0.06
e)
0.05
N
_0
ci)
N
E
(d)
0.04
0.03
0
z 0.02
c)
0.01
(a)
0
br)
0.1
0.2
Nominal axial strain
0.3
0.03
0.025
(e)
0.02
0.015
(d
0,0
0.005[
~0
(d)/
,
(C
(allb)
0.2
0.1
Nominal axial strain
0.3
Figure 3-24: Macroscopic response of 3D laminate with fpc
rate e = 1.7 x 10-3-1.
102
=
0.3 at nominal strain
Plastic shear strain in PC
Craze opening in PMMA
2
0
0
(a)
e=0.02
0 ''.0
-)
---
(b)
Ole
r)
0
-
Yp
-.-
0
(C)
e=0.068
3
2
Figure 3-25: Contour plots of craze opening 2 o (left column) and accumulated plastic
shear strain yP (right column) corresponding to Figure 3-24 (3D, fpc = 0.3).
103
Craze opening in PMMA
Plastic shear strain in PC
-
yp 1
2&
10
10
e d .0 9 8
e=
.202
3
2
Figure 3-26: Contour plots of craze opening 2 o and accumulated plastic shear strain
-yP corresponding to Figure 3-24 (3D, fpc = 0.3).
104
Failed PMMA elements
Failed PC elements
(C)
e=0.068
w
'V
(d)
e=0.098
3
(e)
e=0.202
2
Figure 3-27: Contour plots of failed PC and PMMA elements corresponding to Figure 3-24 (3D, fpc = 0.3).
105
yp
Plastic shear strain in PMMA
Plastic shear strain in PC
Figure 3-28: Accumulated plastic shear strain at e = 0.202, corresponding to Figure 324 (3D, fpc = 0.3)
106
Detailed analysis for case fpc = 0.7
For the 3D RVE with fpc = 0.7, until point (b) (see Figure 3-29) the deformation is
similar to that of the specimen fpc = 0.3. The deformation is locally and globally
linear elastic until point (a).
Point (a) is associated with initiation of crazes in
elements at or close to the free surface x 3 = 0.
The macroscopic stress from (a)
to (b) increases in a linear elastic manner because the volume fraction of region
deforming inelastically, by either shear yielding and crazing, is very small. At point
(b), associated with the macroscopic yield stress, the normalized craze volume is
negligible and there has been no failure.
Between (b) and (c),
#PMMA
slightly, and the normalized craze volume increases. The contour plot of
increases
yP
at (c)
in Figure 3-30 shows a significant increase in the diffusion and percolation of shear
plasticity, when compared with point (c) in Figure 3-25. Between points (c) and (d)
(see Figure 3-31), the nominal stress decreases, and is associated with a sharp increase
in
#PMMA.
The normalized craze volume increases gradually. Several surface crazes
have undergone breakdown and formed cracks. But these cracks have not propagated
in the x 3 direction.
This is in contrast to the case fpc = 0.3.
much smaller compared to
#PMMA.
Note that
#pc
is
A feature that distinguishes the case fpc = 0.7
from fpc = 0.3 is the relative dominance of shear yielding (inferred from the lower
normalized craze volume), and the considerable decrease in
#pc.
The latter feature,
in particular, accounts for the higher tensile ductility for fpc = 0.7.
A point of
central importance is that unlike the fpc = 0.3 laminate, the tunneling of the surface
crazes into the bulk of the RVE is significantly less in the case fpc = 0.7; this is in
accord with the observations of Sung et al. (1994c) and provides an explanation for
the higher toughness for fpc = 0.7 laminate. Figure 3-32 shows the contour plots
of craze opening, as viewed in the 1-2 plane; it highlights that the craze density in
the 1-2 plane decreases as one traverses from the surface x 3
=
0 pum into the bulk of
the laminate, and is qualitatively in accord with Figure 3-4. Figure 3-33 shows the
cracked morphology of the laminate.
Comparing these contours with Figure 3-27,
we note that damage and failure is more diffused and not localized to critical crack-
107
zones. Unlike the case fpc
shear plasticity in the fpc
0.3, Figure 3-34 shows the well distributed nature of
=
=
0.7 laminate. Further, and again in contrast with the
fpc = 0.3 case, the PMMA layers undergo substantial shear yielding for fpc = 0.7
laminate.
3D results for different fpc
Figures 3-35 and 3-36 summarize the macroscopic results for uni-axially loaded 3D
RVEs, as a function of fpc. Unlike the 2D results shown earlier, results with 3D
RVEs are able to adequately capture the transition in ductility at fpc = 0.6, consistent with the experimental results of Kerns et al. (2000) shown in Figure 3-17. The
normalized craze volume plot shows that at a fixed strain level, the normalized craze
volume for fpc = 0.6, 0.7 is much lower than the other volume fractions, indicating that at high fpc, shear yielding in both PC and PMMA layers is the dominant
mechanism.
The failure plots indicate a staggering difference in
creased, and a somewhat less difference for
#PMMA.
#pc
as fpc is in-
The transition in ductility is due
to the PC constraining the tunneling of the PMMA crazes and thus promote overall
shear yielding, as fpc is increased. These results highlight the superiority of the 3D
RVE over its 2D counterpart in qualitatively predicting the macroscopic response of
ductile/brittle laminates, and also underline the importance of including the effect of
the 3-direction. We note that the statistical scatter in our results is negligible (see
Figure 3-37 and 3-38)
3.3.4
Rate effect
In order to explore the effect of strain-rate on the laminate response, uniaxial simulations were performed on the 3D RVE with fpc = 0.6 at a higher nominal strain-rate
d= 1.7 x 10-2 s-,
compared to
e
=
1.7 x 10-3 s
7
1
used in Section 3.3.3. Figure 3-
As noted earlier, the dimensions of the 3D RVE are much smaller than the size of the test
specimens used by Kerns et al. (2000). In particular the gage length of the 3D RVE and the
specimens are different. Therefore, comparisons of strain-to-failure with the experimental results
are not possible.
108
100
80
(Cb
60
(d)(e
0
a)
w
40
20
0
0.1
0.2
Nominal axial strain
0.3
0.07
d)
E 0.06
0
0.05N
o
0.04-
-
0.03-
(d)
E
z0.021
(C
0.01
alb)
-0(
0.1
0.2
Nominal axial strain
0.3
0.03
0.025|
0.02
(e)
a
<6-
(d)
0.015-
-o
0.01
0.005
0
c)
0.1
(d)
0.2
Nominal axial strain
..
e)
0.3
Figure 3-29: Macroscopic response of laminate with fpc
a = 1.7 x 10-3-1.
109
=
0.7 at nominal strain rate
Craze opening in PMMA
Plastic shear strain in PC
(a)
e=0.02
-
-
.
4
(b)
e=O.03
w<
-.
-..
v
t-
e=0.1I
3
2
Figure 3-30: Contour plots of craze opening 2 o and accumulated plastic shear strain
-yP corresponding to Figure 3-29 (3D, fpc = 0.7).
110
Plastic shear strain in PC
Craze opening in PMMA
yp
2&-
do
A '-;J~
v'
p
0
Or.
(d)
E=0.202
40
-A
e= 3 .34
3
2
Figure 3-31: Contour plots of craze opening 2 o and accumulated plastic shear strain
-yP corresponding to Figure 3-29 (3D,
fpc = 0.7).
111
0
(a) x3=0 pm
0
(b) x3=20 pm
1t
(c) x 3=40
pm
2
Figure 3-32: Contour plots of craze opening 2 0 for 3D RVE with fpc = 0.7, viewed
in the 1-2 plane, at different depths: (a) x 3 = 0 (surface); (b) x 3 = 20 pm; (c) x 3 = 40
pm. Contour plots correspond to Figure 3-29 at macroscopic strain e = 0.07.
112
Failed PMMA elements
Failed PC elements
(c)
e=O.1
40
4
*v
-A'
(d)
e=0.202
0
3
(e)
e=0.34
2
Figure 3-33: Contour plots of failed PMMA and PC elements corresponding to Figure 3-29 (3D, fpc = 0.7).
113
yP
10
Plastic shear strain in PMMA
Plastic shear strain in PC
Figure 3-34: Accumulated plastic shear strain at e = 0.34, corresponding to Figure 324 (3D, fpc = 0.7).
114
100
80
SfPc =0
--- fPC=0. 1
fPC =0.3
60
fPC =0.5
-- f=0.6
f
0
Co
=0.7
40
20
0
0.2
0.1
0.1
0.2
Nominal axial strain
0.3
(a)
-
0.08
fPC =0
fPC=0.1
0.071-
-f
=0.3
PC
E 0.06
PC=0.5
>0 05
N
f c=0.7
0 0.04
-
0.03
E
IP
Z 0.02
0.01
00
0.2
0.1
0.2
0.1
Nominal axial strain
0.3
(b)
Figure 3-35: Evolution of (a) nominal axial stress and (b) normalized craze volume,
with nominal axial strain for 3D RVEs at nominal strain rate of 1.7 x 10-' s- 1, as a
function of fpc.
115
0.031
0.025
0.02[
-
20.015
f =05
PC
f PC =0.3
-
0.01
-
f_ =0.4
-- I
0.005
=0.5
.... f =0.6
.__f=0.7
C'I
0
0.3
0.2
0.1
Nominal axial strain
(a)
0.03
fp
-
0.025- -
=0.1
fPC=0.3
fPc=0.4
fPC=0.5
0.02 -
fPc=0.6
-
fpc=0.7
a 0.015
-9-
0.01[
- P-
- ----
0.005
0'
0
0.2
0.1
Nominal axial strain
0.3
(b)
Figure 3-36: Evolution of (a) #PMMA and (b) #pc,with nominal axial strain for 3D
RVEs at nominal strain rate of 1.7 x 10-3 --1, as a function of fpc.
116
100
--
fpc=0.3
fPc =0.5
PC
80
60
0
Co
40
0.-.
20
0
0
0.1
0.2
Nominal axial strain
0.3
(a)
0.0
0.07-
f =0.3
Pc
PC =0.5
-fpc=0.7
E
0. 06-
---------......--------
>)
0. 05-
0
-
N
0.04[
N
0. 03-
0
z 0. 02-
0. 010.2
0.1
Nominal axial strain
0.3
(b)
Figure 3-37: Statistical scatter in the macroscopic stress-strain and the normalized
craze volume response, as a function of fpc.
117
0.031
- I-
-
-
0.025[
f PC
.1
'C"
=0.5
_ f PC=0.7
0.02
4:
0~
0.01 5
0. 011
0.0051
I0
,
0.2
0.1
Nominal axial strain
0.3
(a)
0.01 -f
0.025
:03
0.3
f = 0.5
f PC 0.7
=
fPC
0. 021
a-0 0.015
0. 01[0.005[
"10
0.2
0.1
Nominal axial strain
0.3
(b)
Figure 3-38: Statistical scatter in volume fraction of failed PC and PMMA, as a
function of fpc.
118
300
250
P
MA craze loci
ni-axial
U) 20020
150-
-
a)
loading
PMMA shear loci
100
50
0
PC shear
20
loci
40
60
Hydrostatic stress a-
80
100
Figure 3-39: Effect of rate on craze loci (at laminate surface) and shear loci. Loci in
solid lines correspond to e
1.7 x 10-2 s-, while loci in dashed lines correspond to
e= 1.7 x 10- 3 s-1.
39 shows that with increasing strain rate, both the shear and craze loci are raised.'
Comparing the PMMA shear and craze loci, we note that shear yielding in PMMA is
more rate dependent than crazing in PMMA. This feature should increase the craze
density in PMMA layers at the higher rate.
Figure 3-40 shows that increasing
e results in reduced ductility,
and increased yield
strength. This is in accord with experimental results of Kerns et al. (2000), shown
in Figure 3-3. The reduced ductility is associated with a relatively high normalized
craze volume and relatively high values of Opc and
#PMMA
(Figure 3-41). Figure 3-42
shows that at the higher rate, crazes penetrate fully through the 3-direction, but not
at the lower rate. This is consistent with the experiments of Sung et al. (1994c), where
the authors found that for PC/SAN laminates with fpc = 0.65, craze penetration in
the 3-direction increased with strain-rate. Figure 3-43 and Figure 3-44 show localized
plastic deformation in the laminate at the higher rate. Figure 3-45 shows pictorially
the failure pattern as a function of strain rate.
8
The rate dependence of the craze loci has been modeled by rendering the tensile yield strength
Y rate dependent in Equation 3.5. Other parameters have been treated as rate-independent.
119
100
-
0.0017s-1
0.017 s-1
80
a)
60
0
Zi
40
20
0
0.1
0.2
Nominal axial strain
0.3
(a)
-
0.08a)
0 .07[
=3
0.06
E
N
(U
N
---
0.0017s
0.017 s
-
0.0 0
0.05
0.04
0.03-
0
IF
Z 0.020.01
0
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-40: The effect of strain rate on (a) nominal stress-strain and (b) normalized
craze volume, for fpc = 0.6.
120
,
0.0
0.025-
0.0017 s'
0.017 s-1
I-
.
0.02
0
-
-
11I
0.015
0.01
0.005
U
-
0
0.1
0.2
Nominal axial strain
0.3
(a)
0.0 I
I ---
0.0017 s
---0.017 s-
0.0250.020 0.015
aI
/
0.01.
,'
0.005I
0
I. -I
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-41: The effect of strain rate on (a)
121
#PMMA
and (b) Opc , for fpc = 0.6.
--
II-.
-.
2&
.,04, "go
10
w,04.
(a) Craze opening (e= 1 .7x1 0- s-')
3
2
Figure 3-42: Contour plots for craze opening at (a)
1.7 x 10-2, at e = 0.34 for fpc = 0.6.
122
e
2
1.7 x 10-
S1
)
(b) Craze opening (e=1 .7x1
and (b)
=
yp
10
(a) Plastic shear strain (e= 1 .7x10 3
s~')
3
(b) Plastic shear strain (e= 1.7x10 2 s-1 )
2
Figure 3-43: Contour plots for accumulated plastic shear strain in PC layers at (a)
e= 1.7 x 10-3 and (b) e = 1.7 x 10-2, at e = 0.34 for fpc = 0.6.
123
yp
:0
(a) Plastic shear strain (e= 1.7x1 0 -S1)
3
2
(b) Plastic shear strain (e= 1.7x10-2 s;*1]
Figure 3-44: Contour plots for accumulated plastic shear strain in PMMA layers at
(a) e = 1.7 x 10- and (b) e = 1.7 x 10-2, at e = 0.34 for fpc = 0.6.
124
Failed PMMA elements
de
.
o
Failed PC elements
I*
~4w
(a) e=1.7x 0-3 S-
3
(b) e=1,7x1
0-2 S1
2
Figure 3-45: Failed PMMA and PC elements at (a)
1.7 x 10-2, at e = 0.34 for fpc = 0.6
125
e
=
1.7 x 10-3 and (b)
e
=
3.3.5
Length-scale effect
To investigate the effect of number of layers on laminate toughness the number of
layers in the RVE were doubled, while keeping the laminate thickness fixed, i.e.,
the absolute layer thicknesses were halved. The (initial) element size (1 ,um in the
direction of tensile loading) was kept the same as in the baseline simulations.
The
number of elements in a bilayer, consisting of one PC and one PMMA layer, were
reduced from 20 (in the baseline simulations) to 10.
Figure 3-46 and 3-47 show the increased ductility for thinner layers of fpc = 0.6 at
e
= 1.7x 10-2 s-1. The normalized craze volume and Opc decrease significantly for the
thin-layered laminate. These results are in qualitative accord with the experimental
results shown in Figure 3-3. The length scale effect is negligible for fpc = 0.3, while
the length scale effect for fpc = 0.5 is less pronounced than for fpc = 0.6 (see
Figures 3-48 and 3-49).
Figure 3-50 shows that with increasing number of layers
(decreasing absolute layer thicknesses), the crazes and cracks in PMMA are better
confined by the adjoining PC layers.
Interestingly, comparing the craze opening
contour plots for "thick" layered and "thin" layered laminate, it seems that the ratio
of the PMMA layer thickness to the depth of craze/crack penetration in the 3-direction
is approximately preserved. Better damage confinement accounts for the improved
ductility with increasing number of layers. Figures 3-51 and 3-52 show shear plasticity
to be better distributed when the number of layers is increased. Consistent with the
other contour plots, Figure 3-53 shows the less localized pattern of failure for the
thin-layered laminate.
3.3.6
Sensitivity and design studies
Here, the results of two sensitivity studies and one design study are presented. The
two sensitivity studies are performed to assess the sensitivity of our baseline results
to the gradient of craze initiation properties, and to the critical failure stretch in PC,
as these parameters are not known with certainty.
A design study is performed in which the brittle layer craze initiation and flow
126
100
-
THICK
THIN
-
80
C')
U)
60
I
--
40
0
z
0.-.
20
0
(
0.1
0.2
Nominal axial strain
0.3
(a)
0.09
0.08
-
THICK
.-- -THIN
a) 0.07
E
0 0.06N
z)
N
0.05
0.04
01..--1
0.03
0
z 0.02
0.01
00
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-46: The effect of doubling the number of layers (with fixed laminate thickness), .i.e., decreasing absolute thicknesses on (a) nominal stress-strain and (b) normalized craze volume, for fpc = 0.6 at a = 1.7 x 10-2 s1
127
0.03
-T HICK
---T HIN
0. 025[
0. 02
CL
0.01500.1
0.01
.2
0.
0.005
0I
0
0.1
0.2
Nominal axial strain
0.3
(a)
(
(0.n
THICK
THIN
0. 0251
0.021-
-e-
0. 015%
0 .0110.0051
0'
0
0.2
0.1
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-47: The effect of doubling the number of layers (with fixed laminate thickness), i.e. decreasing laminate thicknesses on (a) #PMMA and (b) #pc, for fpc = 0.6
at e = 1.7 x 10-2 s-1
128
100
Pc=0.3: THICK
fPc=0.3: THIN
Pc=0.5: THICK
f PC=0.5: THIN
-
80-
60
40
20
U0
0.1
0.2
Nominal axial strain
0.3
(a)
-A.
0
tPC=
0.07
0.06
THIC
f =0.3: THIN
......
fPc=0.5:
THICK
PC
- - pc=0.5: THIN
0.05
0.04
0.03
-
I.
0.020.01
00
0
|
,.
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-48: The effect of doubling the number of layers (with fixed laminate thickness), i.e. decreasing absolute layer thicknesses on (a) nominal stress-strain and (b)
normalized craze volume, for fpc = 0.3,0.5 at e = 1.7 x 10-3 s-1
129
0.03
-
tPC=U.3: I HIGK
fpc=0.3: THIN
fpc=0.5: THICK
fpc=0.5: THIN
0.025
0.02
0.015
0.01
0.005
0.2
0.1
0.1
0.2
Nominal axial strain
0
0.3
(a)
0.03
0.025%
-t
=
_ PC 0:3:
0.:3: THICK
THICK
fPC 0.3: THIN
fPC
f PC
0.5: THICK
0.5: THIN
0.5:.THIN
0.02[
0
o- 0.015
-9-
/-
0.01.
I-
.
0.005
0,---0.0
0
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-49: The effect of doubling the number of layers (with fixed macroscopic
thickness) on (a) #PMMA and (b) #pc, for fpc = 0.3, 0.5 at e = 1.7 x 10- s-
130
2&
2
io
(a) Craze opening (THICK)
0-4
O~tte4P
IZ J
3
2
(b) Craze opening (THIN)
.
Figure 3-50: Contour plots for craze opening for (a) thick-layered and (b) thin-layered
laminate with fpc = 0.6 at e = 0.34 and e = 1.7 x 102 s 1
131
yp
(a) Plastic shear strain (THICK)
1
3
(b) Plastic shear strain (THIN)
2
Figure 3-51: Contour plots for accumulated plastic shear strain in PC layers for
(a) thick-layered and (b) thin-layered laminate with fpc = 0.6 at e = 0.34 and
e = 1.7 x 10-2 S1.
132
yp
II
0
(a) Plastic shear strain (THICK)
3
2
(b) Plastic shear strain (THIN)
Figure 3-52: Contour plots for accumulated plastic shear strain in PMMA layers
for (a) thick-layered and (b) thin-layered laminate with fpc = 0.6 at e = 0.34 and
e = 1.7 x 10-2 s-1.
133
Failed PMMA elements
Failed PC elements
-7
(a) THICK
p
p'
do Ar
000
000-.
-
__lo
3
(b) THIN
2
Figure 3-53: Failed PMMA and PC elements for (a) thick-layered and (b) thin-layered
laminate with fpc = 0.6 at e = 0.34 and e = 1.7 x 10-2 s-1.
134
stress are modified to explore the response of a laminate comprised of the same ductile
layers (PC) but layers of a different, hypothetical brittle polymer.
Effect of gradient in the craze initiation properties
Equation 3.6, which characterizes how sharply the craze initiation stress increases
from the surface to the bulk in the 3-direction, was used in our baseline study in
Section 3.3.3. A sensitivity study was performed to investigate the effect of increasing
the gradient in craze initiation properties, so that the bulk craze locus is reached for
a smaller value of x 3 . For this sensitivity study, Equation 3.6 is replaced by:
Y, +
(3.11)
(, -7) if X3 <)
7Y(X3)
=
Y(X3)
= jt if 5 <x
3
(3.12)
175
The gradient distance according to Equation 3.11 is 5 ym, compared to 25 pm
in the baseline simulation (see Equation 3.6). 3D simulations were performed for the
cases fpc
=
0.3,0.7.
Figure 3-54 and Figure 3-55 show that the macroscopic stress-strain response is
not sensitive to the use of the higher gradient (HG) in 7y, achieved by Equation 3.11.
The normalized craze volume plot indicates that the HG simulations have less crazing
associated with them, but not significantly.
In the failure plots in Figure 3-55, the
only significant difference is that for the HG case,
#PMMA
is smaller for fpc = 0.7.
Contour plots shown in Figure 3-56 are almost identical with their counterparts in
Figure 3-26 and Figure 3-31.
Critical failure stretch
The baseline simulations are based on a critical failure stretch Ac, = 0.9AL.
Since
the critical failure stretch is not known with certainty, simulations were performed
on the case fpc
=
0.7 with a lower critical stretch A,, = 0.8AL to determine the
sensitivity of the macroscopic response to Acr. The stress-strain response (Figure 3-
135
100
.
801
fPC=0.3
--.fPc=0.3:. HG
-- I=0.7
-.f
=0.7: HG
4U)
60
40
E
0
Z 20
0C
0.1
0.2
Nominal axial strain
0.3
(a)
0.08
0.06-
-- fPC=0.7: HG
0.05-
......--------.---
-
0.07-
---fPC=0.3
- =0.3: HG
a)
N
0.04
N
0 .03
-I
0
-
Z 0 .02
0.01
CL
0
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-54: Effect of the gradient of craze initiation properties on (a) nominal stressstrain and (b) normalized craze volume-strain response, for fpc = 0.3 and fpc = 0.7.
"HG" stands for the higher gradient studies.
136
0.031
Pc=0.
0.025
---f c=0.3:. HG
__$4=0.7
... F
=07
....
0.02
:0
0. 015
0. 01F
~
-
0.005-
00
I'
0.1
0.2
Nominal axial strain
0.3
(a)
0.03
-- fPc=0.3
----. fPc=0.3: HG
0.025-.
fy=0.7
fpc=0.7: HG
0 .02F
0. 015F
0 01F
0.0051-
0
0.1
0.2
0.3
Nominal axial strain
(b)
Figure 3-55: Effect of the gradient of craze initiation properties on (a) OPMMA and
(b) /pc, for fpc = 0.3 and fpc = 0.7. "HG" stands for the higher gradient studies.
137
I
,
-
I
Plastic shear strain in PC
Craze opening in PMMA
Yp
=I
2
rv
10
(a) f =0.3 at e=0.202
01
~ 4.4W
(b) f,=0.7 at e=0.34
2
A
Figure 3-56: Contour plots for fpc = 0.3 and fpc = 0.7 with higher gradient of craze
initiation properties.
138
Table 3.3: Values of 'YI and 6vo for the baseline study and for the new study
b
(MPa)
6vo (ms')
Baseline
3400
1.45 x 10-1
New
2000
1.45
57(a)) for Acr = 0.8AL shows slight deviations from the baseline response, which can
be accounted for by the higher
#pc
(Figure 3-58(b)) for Acr = 0.8. The normalized
craze volume curves are almost identical for both cases (Figure 3-57(b)). The contour
plots in Figure 3-59 are not significantly different from the baseline simulations.
Craze properties
A design study was done by lowering the craze initiation stress, as well as the craze
flow stress to simulate a material different from PMMA. This was implemented by
decreasing the Weibull parameter 'y , for C, and by increasing 6vo, respectively. Taand
ble 3.3 provides the new values for _Y
vO0 , and also reports the baseline values
already given in Table ??. The initiation locus was lowered while still using Equation 3.6. A study was performed on fpc = 0.7. Figure 3-57 shows a negligible effect
on the nominal stress. However, the normalized craze volume and
#PMMA
are higher
(Figure 3-58. The 2 o contour plot (Figure 3-59) shows this is because crazes emanating from the surface have tunneled more deeply compared to the baseline results.
3.3.7
Role of craze flow stress and craze initiation stress on
ductility
A study was done on fpc = 0.5 at
e
=
1.7 x 10--
ms
1
, in which the flow stress
was increased by decreasing 6vo from the baseline value of 1.45 x 10-1 ms
1.45 x 10-
ms
1
1
to
. The craze initiation properties were kept the same as that for the
baseline study. This is designated as "HFS" in Figures 3-60 and 3-61. The relatively
high toughness of the "HFS" simulation compared to the baseline simulation is due
to limited craze tunneling in the 3-direction for the former.
139
Next, the high flow
100
-
80
C,)
4.1
U)
Baseline
Low critical stretch
Low craze strength
60
40
0
z 20
of
0
0.1
0.2
Nominal axial strain
0.3
(a)
0.0 8 1
-
0.07-
-
Baseline
Low critical stretch
Low craze strength
a)
E 0. 06
U)
0. 05-
N
0.04N
CU
0.03-
0
Z 0.02
0.011
0
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-57: Effect of critical stretch and craze strength on (a) nominal stress-strain
and (b) normalized craze volume-strain response, for fpc = 0.7
140
--- Baseline
Lo w critical stretch
Lo w craze strength
0.025
0. 02-
-
0.015-
-
0.01
0.005-
0
0.1
0.2
Nominal axial strain
0.3
(a)
0.031
0.025-
-
Baseline
Low critical stretch
Low craze strength
0.02
a. 0.015
0. 010.00510'
0
0.1
0.2
0.3
Nominal axial strain
(b)
Figure 3-58: Effect of critical stretch and craze strength on (a) OPMMA and (b) Opc
, for fpc = 0.7
141
Plastic shear strain in PC
Craze opening in PMMA
2&
0
Y
-0
(a) Lower craze strength (e=0.34)
3
(b) Lower critical stretch (e=0.34)
2r
Figure 3-59: Contour plots for effect of lower craze strength and lower craze stretch
142
stress (obtained by using &vo = 1.45 x 10--
s-1 was used in conjunction with a
lower initiation stress (LIS) of 100 MPa in the bulk of the specimen (compared to
110 MPa in the baseline study).
The 10 MPa drop in the initiation stress was
implemented by reducing -% from 3400 MPa in the baseline simulations to 2000
MPa. This resulted in the macroscopic stress-strain response to be similar to the
original baseline simulation with &vo = 1.45 x 10-1 s-
and yt = 3400 MPa. This
study highlights that toughness in ductile/brittle laminates is controlled by both the
initiation and the craze flow stress, and that the same laminate toughness can be
achieved by "high" initiation stress and "low" flow stress, or by "lower" initiation
stress and "higher" flow stress.
3.4
Summary and conclusions
The objective of this chapter was to investigate the mechanisms underlying the
deformation and tensile toughness in ductile/brittle laminates, with emphasis on
PC/PMMA microlaminates.
The uniaxial experiments of Baer and coworkers on
PC/SAN and PC/PMMA microlaminates highlight three factors that increase the
macroscopic toughness of these composites:
" high volume fraction of PC, fpc
" small absolute layer thicknesses, t, and
" low strain-rate,
E.
High macroscopic toughness in ductile/brittle microlaminates is associated with
the predominance of the shear yielding mechanism (as can be inferred from the work
of Ma et al. (1990)), as well as with negligible tunneling of crazes through the width
of the specimen (Sung et al., 1994c). The well-known dichotomy of crazes, as being
precursors of fracture as well as manifestations of dilatational plasticity that provides
toughness, is clearly seen in these composites. It is important to note that laminate
failure is initiated by the breakdown of crazes in PMMA (or SAN) layers. The cracks
so formed propagate through the PC layers, leading eventually to laminate failure.
143
100
(V
-lBaseline
HFS
HFS &LIS
80
0n
CO
CO
60
0.-.
40
0
Z 20
01
0
0.1
0.2
Nominal axial strain
0.3
(a)
0.12
-
0.1
Baseline
-- HFS
-- HFS & LIS
E
0. 08
N
o.oe
N
EV
0.04
0
Z
0.02
1~
0
0.1
0.2
Nominal axial strain
0.3
(b)
Figure 3-60: An alternative route to accounting for laminate ductility: evolution of
macroscopic nominal stress and normalized craze volume with nominal macroscopic
strain.
144
0.03
-
-
Baseline
HFS
HFS & LIS
0.025
0.02
0.015
0.01
0.005
0'
0.2
0.1
0.3
Nominal axial strain
(a)
0.031
-
0.025- -
Baseline
HFS
HFS &LIS
0. 021
0
Q- 0.015
0.011
0.0051
01
0
0.2
0.1
Nominal axial strain
0.3
(b)
Figure 3-61: An alternative route to accounting for laminate ductility: evolution of
#PC
and
PMMA with nominal macroscopic strain.
145
We developed microstructural models that were able to predict and provide an
explanation for the fine interplay of fpc, I, and
e
on the laminate toughness. In our
modeling approach, we modeled the ductile constituent, PC, with the shear yielding
mechanism. Consistent with experimental findings, the "brittle" constituent PMMA
deforms inelastically by either crazing or shear yielding.
Although representative
crazing parameters for PMMA in PC/PMMA microlaminates are not known from
experiments, our estimates of these parameters allowed us to capture the competition
between shear yielding and crazing in the PMMA layers. Sensitivity studies showed
that our baseline results are robust against uncertainties in our estimates of the
crazing material parameters. In line with the high interfacial strength of the interface
between PC and PMMA layers, the interface was idealized to be perfectly bonded
throughout the deformation. Both 2D and 3D RVE studies were uniaxially deformed.
2D plane strain RVE studies failed to adequately capture the experimentally observed macroscopic behavior of the microlaminates, and in particular failed to predict
a transition in ductility with increasing fpc. On the other hand, 3D RVE studies
adequately captured the effect of fpc on the macroscopic and the microscopic behavior of the laminates. The principal reason for the poor performance of the 2D plane
strain RVEs is that they neglect the distribution of inelastic events in the 3-direction.
In line with the experiments of Sung et al. (1994c), our 3D RVE studies correctly
predict that crazes do not tunnel through significantly from the surface into the 3direction when fpc is large (> 0.6), but tunnel through the entire cross-section at
low fpc (< 0.4); when fpc is high, the PC layers constrain the tunneling of crazes in
the PMMA layers.
3D RVE studies at two different strain-rates (at fixed fpc = 0.6) showed that
ductility decreases with increasing strain-rate. This effect was captured in our model
by the higher rate dependence of shear yielding in PMMA, compared to craze initiation; at the higher rate, the contribution of crazing to the deformation of PMMA
layers increases. Toughening due to small layer thicknesses or large number of layers
(while keeping the laminate thickness constant) was due to better damage confinement and better distribution of plasticity with increasing number of layers.
146
In all
cases, it was observed that low toughness is associated with ease of craze tunneling
in the 3-direction.
147
148
Chapter 4
Deformation and failure of
toughened polystyrene
4.1
Introduction
The incorporation of a rubbery phase into a brittle (craze-able) polymer matrix has
long been recognized as a means to significantly toughen the material. One of such
systems that has been realized, and will be the focus of this chapter, is high impact
polystyrene (HIPS). HIPS was one of the first commercial polymers that was toughened by compliant, micron-order sized second-phase particles. The particles in HIPS
are heterogeneous with a morphology referred to as the "salami" morphology. The
"salami" morphology consists typically of 80 % volume fraction of sub-micron sized
PS sub-inclusions within 20 % PB, which is the topologically continuous phase. PB
in the salami particles occurs in the form of thin layers, typically of thickness 5-10
nm. The actual mechanism that governs the toughening of HIPS was first identified
by Bucknall and Smith (1965).
By stretching thin films1 of the blend, they found
that macroscopic yielding was accompanied by the formation of multiple crazes in
PS around the rubber particles. Figure 4-1(a) shows a typical stress-strain curve for
HIPS, along with a micrograph showing the heterogeneous nature of the particle and
multiple crazing in the PS matrix.
1
with thicknesses in the range 5 - 10
pn
149
24~
0
:1?
24
SRAUIN
35
X)
4.8
60
(a)
(b)
(c)
Figure 4-1: (a) The stress-strain response of HIPS (right) with particle volume fraction f, = 0.2, average particle size=2.5 pm at a strain-rate=1 x 10-4 s- 1 . Micrograph
(left) reveals the composite nature of the particle and shows multiple crazing in the
PS matrix. From Dagli et al. (1995). (b) Micrograph showing multiple crazing in the
PS matrix and fibrillation of PB in the particle in thin films of HIPS (tested within
TEM). From Cieslinski (1991). (c) TEM micrograph of a thin HIPS section, prepared
from a tensile specimen beyond yield (Bucknall, 2001).
150
The previously well-accepted notion that the toughening in HIPS is due to multiple crazing in the PS matrix, and that the role of the particle is simply to act as
a stress concentrator and provide a multitude of preferential sites for crazing in the
matrix, has been a subject of serious critical thought over at least the last decade. An
important early study, due to Bubeck et al. (1991), was based on real-time SAXS experiments on commercially thick HIPS tensile impact samples. Importantly, the use
of SAXS technique overcame the inherent limitation of electron microscopy studies
where thin specimens are needed. From the scattering measurements and analysis,
Bubeck et al. were able to estimate the total inelastic strain, as well as the crazing
strain. By subtracting the crazing strain from the total inelastic strain, they determined the magnitude of the inelastic strain due to non-crazing mechanisms, which
in HIPS is attributed mostly to particle cavitation and elastic/inelastic bending of
ligaments between cavitating particles. Bubeck et al. found that the inelastic strain
due to non-crazing mechanisms exceeds the strain due to crazing (in many cases
by about a factor of 2) , and that the non-crazing inelastic strain precedes crazing.
On lines somewhat similar to Bubeck et al., Magalhaes and Borggreve (1995) used
real-time SAXS on HIPS tensile specimens subjected to quasi-static strain rates, and
found that crazing accounts only for 25% of the total volume change in the specimens.
TEM micrographs showing patterns of deformation in thin HIPS films are shown in
Figure 4-1(b) and 4-1(c). These micrographs show that particle deformation is mainly
accomplished through fibrillation of the thin PB layers between PS occlusions. Recently, Bucknall and Soares (2004) subjected HIPS specimens to progressively higher
doses of -y radiation, which has the effect of increasing the crosslink density of PB
(thereby minimizing or eliminating fibrillation), but has a negligible effect on PS. Increasing levels of -y radiation were shown to result in increased yield and flow stress,
and reduced toughness, and thus highlighted the important role of fibrillation on the
toughening in HIPS.
To this date, very limited modeling studies have been done on HIPS that can
shed light on the toughening mechanisms, and in particular on the role of particle
morphology.
Socrate et al. (2001) developed a micromechanical model for crazing
151
using a cohesive zone formulation, and applied their model to a preliminary study of
HIPS, in which the composite "salami" particle was represented by its homogenized,
elastic properties, and hence did not address the role of particle morphology and
particle cavitation on the deformation and toughness of HIPS. Recently, Zairi et al.
(2005, 2006) have modeled the overall constitutive stress-strain behavior of the rubber
toughened polymers- PMMA and PS- using a viscoplastic damage model within the
Gurson-Tvergaard micromechanical framework. While with their approach, they were
able to obtain quantitative agreement with experimental data, it does not address the
role of particle morphology on the deformation and toughening of HIPS.
The objective of our study is to clarify and quantify the precise role of particle morphology on the micromechanics and macromechanics of uni-axial tensile deformation
and failure in rubber-toughened PS. In particular, this study seeks to investigate the
role of particle compliance, particle heterogeneity and particle cavitation/fibrillation.
4.2
Model
In order to investigate the local mechanisms that govern the deformation and failure
of HIPS, a micro-mechanical model that adequately captures the particle morphology
is used. The proposed micro-mechanical model includes two parts: (i) the geometric
description of the RVE and, (ii) the constituent behavior of the three phases - the PS
matrix, the PS within the particle, and the PB within the particle. All inter-phase
boundaries are idealized as sharp and perfectly bonded.
4.2.1
Geometry of the RVE
Similar to perhaps all particle composites, micrographs of HIPS reveal several typical
features such as variations in particle size and shape and spatial variations in interparticle spacing with possibility of particle clustering even in well-dispersed systems.
The case of HIPS has an additional complexity in that the particles are themselves
2-phase composites; the volume fraction and spatial arrangement of the PB and PS
phase in a given particle is expected to differ from other particles. The overwhelming
152
task of modeling the above-mentioned features found in HIPS is greatly simplified by
introducing a one-particle RVE model. By imposing appropriate periodic boundary
conditions on the RVE model, the basic key interactions with neighboring particles
are simulated.
In order to investigate porous plasticity in metals, Tvergaard (1982) used a oneparticle RVE model.
This RVE, which is referred to as Stacked Hexagonal Array
(SHA), is based on a three-dimensional array of space-filling stacked hexagonal cylinders, each containing a spherical particle. Tvergaard further simplified the 3D SHA
model to an axisymmetric version that, despite being only approximately space filling, provides a judicious computational route to modeling particle composites with
applied "axisymmetric" loading conditions at low particle volume fractions
f,
and/or
low triaxiality, where the particle/void interactions are not significant (Socrate and
Boyce, 2000). Since the SHA model is not used in this study, it will not be discussed
further; further details on the SHA model can be found in numerous publications
such as Tvergaard (1982), Tzika (1999) and Socrate and Boyce (2000).
The one-particle RVE used in this thesis was proposed by Socrate and Boyce
(2000), and is an axi-symmetric equivalent to the Voronoi tessellation of a Body
Centered Cubic (V-BCC) array of voids/particles.
In simulations of porous PC,
Socrate and Boyce found that the numerical predictions of the axi-symmetric VBCC model are more in line with experiments than the traditional axisymmetric
SHA model, especially at high void volume fractions and high triaxiality. Figure 4-2
shows the deformed configuration of the axisymmetric V-BCC cell, along with one
of its periodic neighbors, which is identical to the parent cell but rotated by 180'. A
characteristic feature of this model is the staggered arrangement of particles in the
r - z plane, which is closer to the morphology of particle composites, compared to the
SHA model, in which the particles are aligned in the equatorial plane. The boundary
conditions needed to simulate the staggered arrangement of particles were given by
Tvergaard (1996, 1998) in his study of cavity growth and interaction between small
and large voids.
The axial compatibility of the RVE requires that the top (BC) and bottom (OD)
153
of the RVE remain planar throughout the deformation.
Further, points along the
lateral boundary (CD) are constrained by
Uz(r17) + Uz(q7 2 ) = 2uzIF, f or
/i -=
(4.1)
q2
where uZ(r1) and Uz(r/2) are the axial displacements of points located at an axial
displacement ql and
'q2
from point F, which is the midpoint of the initially straight
and axially oriented boundary CD. The axial displacement of the top boundary (BC)
is connected to the motion of F via,
2uzIF
(4.2)
= UzBC = Uz2B
Radial compatibility considerations require that the total cross-sectional area of an
infinite array of cells be independent of the axial coordinate z, leading to the following
constraint on the lateral boundary (CD):
[Ro +
Ur(171)] 2
+ [fR0 + Ur(772)] 2 = 2[Ro + UIF]2,
f or rml = r72
(4-3)
where Ro is the initial RVE radius. Expanding Equation 4.3 gives
Ur(Tl1) + Ur(r7 2 )
-
2UrIF + Ur(71i)
2
+ Ur(22
Ro
_
2
=
2R.
Due to the negligible lateral contraction in HIPS, the bracketed term in the above
equation is negligible, and hence Equation 4.3 simplifies to
Ur (r11)
+
Ur (r7
2
) =
2urIF f or
/1 = 72
(4.4)
In this thesis, we have used Equation 4.4 to implement radial compatibility of the
RVE. Symmetry with respect to the z-axis and z-plane result in
UrIOB = 0 = UzJOD
(4.5)
Operationally, in our simulations the top plane of the RVE is driven by an axial
154
C
B
A
1
............. F ----------772
Z
E
0
D
Figure 4-2: Axisymmetric V-BCC cell in the deformed configuration, along with a
periodic neighbor.
displacement history uz(t)IB that produces a constant applied axial strain rate
E2
on the RVE. The axial reaction force corresponding to node B, PIB, can be used to
determine the axial component of the Cauchy stress Ez by
-z(x)dV =
EZ = - 1
V jev
(4.6)
2
7[Ro+UrIF
where 7r[Ro + UrIF]2 is the average, deformed cross sectional area of the RVE. The
macroscopic axial strain Ez is calculated from the height of the RVE in the deformed
configuration, H, and the initial RVE height, HO:
Ez
=
In
=
ln (1 + UzIB)
(4.7)
HO
The macroscopic volumetric strain
4
is given by the logarithmic ratio of the deformed
RVE volume V to the initial volume V = 7rR'HO:
S= n
(-)
Ur|F)2(1
(o
V
= In ((IO + RIH
uzIBIY
)
(4.8)
The composites investigated in our study can be geometrically classified into two
types - RVEs with homogeneous or homogenized particles, and RVEs with heteroge-
155
neous particles consisting of 70 % PS subinclusions within 30 % topologically continuous PB. The particle size is kept fixed at 1.5 pm for all RVEs. Models are developed
f,
for three different particle volume fractions
= 0.08, 0.18, 0.28 by appropriately in-
creasing the size of the RVE. In all cases, the initial aspect ratio of the RVE is given
.
by Ro/Ho = 1, and the macroscopic applied strain rate is E, = 1 x 10-3 s- 1
The geometry of the axisymmetric RVEs, for particle volume fraction of 0.28,
are shown in Figure 4-3.
The RVE with the homogeneous particle is used in the
study of the effect of particle compliance on toughening of PS. The RVE with the
heterogeneous particle is used to study the effect of particle heterogeneity (in a "nofibrillation" study), and the effect of fibrillation in PB on the macroscopic response.
We note that while the actual thickness of the PB phase in the particle is in the range
5-10 nm, the PB elements are about 25 nm; meshing considerations necessitated this
approximation.
4.2.2
Constitutive model
The constitutive model for crazing in HIPS, along with the associated material parameters have been described in Chapter 2, and are reproduced in Table 4.1 for
convenience. Based on the material parameters given in Table 4.1, the craze locus for
PS is shown in Figure 4-4. PB is modeled as a compressible, rubber-elastic material
with the following constitutive equation:
TN _
-1
1
Achain
B' + KB(J
-
(4.9)
1)
)
Achain
where KB is the bulk modulus; the remaining quantities have been defined in Chap-
ter 2. For PB, KB/p, = 3125 (Boyce et al., 1987) before fibrillation. Fibrillation
results in the formation of voids in PB, which effectively destroys the resistance of
the material to volumetric expansion.
We take KB/p, = 1 after fibrillation.
We
adopt the following simple criterion for fibrillation of PB.
Uh
OUfib
156
(4.10)
Mesh with a homogeneous particle
Mesh with particle showing PS domains
Figure 4-3: Axisymmetric RVE geometry for the homogeneous particle and the heterogeneous particle. PS subinclusions in (b) are shown in white.
157
100
L
90
Uniaxial
loading
''80
7070U
CD
60
50
/
40
/
20
/
30
0
>
10/
0
0
20
40
60
80
100
Hydrostatic stress ch (MPa)
Figure 4-4: The craze locus of the PS matrix in HIPS.
where Ufib is the critical hydrostatic stress that results in fibrillation. The values for
0-fib are discussed in Section 4.3.6.
4.3
Results
The objective of the finite element studies in this section is to understand the role of
particle morphology on the deformation and toughness of HIPS. This goal is accomplished by determining the effect of
" particle compliance through a parametric study on homogeneous particles, whose
compliances are systematically varied.
" particle heterogeneity by comparing the response of the RVE with homogenized
particle vs. heterogeneous particle (with no fibrillation)
" particle fibrillation by comparing the response of the RVE with a non-fibrillating
heterogeneous particle vs. a fibrillating particle.
158
Table 4.1: Craze parameters for PS matrix in HIPS (Piorkowska et al., 1990)
4.3.1
? (s)
C (MPa)
6.0 x 10-8
1650
Q
0.0133
Y (MPa)
&vo (ms 1 )
Y (MPa)
B/kO
A;
Af
2(,um)
70
3.4686 x 10-3
218.5
44.72
1.853
4
0.1
Parametric study on compliant, homogeneous particles
In order to study the effect of particle compliance on the deformation and toughness of
the composite , a parametric study was conducted on RVEs with
f,
=
0.28, wherein
the particle shear modulus was systematically varied, as shown in Table 4.2. The
0.62 MPa in Table 4.2
elastic properties associated with Kp = 1940 MPa and Gp
=
correspond to polybutadiene (PB) (Boyce et al.
The remaining entries in
1987).
Table 4.2 are associated with hypothetical particles with the same bulk modulus
as PB, but with the shear modulus progressively increasing by factors of 10. For
reference, the small strain elastic properties selected for the PS matrix correspond to:
Kps = 2500 MPa and Gps = 1150 MPa (corresponding to Eps
VPs =
=
3000 MPa and
0.3).
Figure 4-5(a) and (b) respectively show the evolution of the macroscopic true
axial stress E, and the macroscopic true volumetric strain
', with the macroscopic
true axial strain E,. The EZ - E, behavior for composites for cases corresponding
to particles with Gp
=
0.62, 6.2, 62 MPa exhibit an initial linear elastic behavior,
followed by departure from linearity and a non-linear rise to a peak stress, and finally
followed by softening; the association of departure from linearity, peak stress, and
softening to the local deformation mechanisms will be discussed in more detail below.
The composite with Gp = 620 MPa indicates a sharp stress drop, followed by a stress
159
rise, and ultimately by softening. The
-
curves show a monotonic increase in
4
for all the particles. The differences in the magnitude of 4 (at a fixed E,) indicate that
the amount of axial deformation that is accommodated by particle shearing increases
slightly with increasing particle compliance.
Detailed analysis for PB particle
Here, we perform a detailed investigation of the macroscopic response of the RVE
with the PB particle (Gp = 0.62 MPa) and the underlying features of the deformation. Figure 4-6 shows the macroscopic response of the RVE with the PB particle.
Figures 4-7 and 4-8 show contour plots of craze nucleation "damage" D, which has
been defined in Equation 2.18, at the macroscopic strain levels marked (a)-(e) on the
E, - E, plot of Figure 4-6. A contour level of D = 1 indicates a nucleated craze;
D = 0 indicates no crazing.
Due to the stress concentration of the compliant PB particle, crazing initiates at
the interface of the particle and matrix, at the equator (as shown in Figure 4-7),
and results in the deviation from linearity of the EZ - Ez response. The softening
of the craze element after nucleation, provides a stress concentration for the lateral
propagation of the craze. The macroscopic stress EZ continues to rise at points (b)
and (c), despite the progression of crazing in the PS matrix, due to increasing levels
of axial stress in the particle At point (d), E, reaches its peak value. Both the local
axial stress and the local hydrostatic stress in the particle reach their peak value of
- 36 MPa, with a slightly higher stress of ~ 38 MPa at the equator, adjacently to
the interface. At point (d), the craze has fully bridged the ligament between nearby
particles and begun to travel around the particle circumference. The subsequent drop
in Ez is associated with crazing around the particle circumference, which relieves the
high axial stress in the particle. Contour plots at point (e) show large distortions in
particle elements close to the interface, at the equator due to the extensive opening
of the adjoining craze element.
In view of the large stresses in the particle at point (d) mentioned above, the particle is likely to undergo degradation/cavitation, particularly at the region adjoining
160
the equatorial craze.
Hence, the E, - E, response beyond (d) is unrealistic.
The
localized nature of crazing in conjunction with the high stresses in the particle would
lead to loss of structural integrity of the RVE at E, < 0.067.
Figure 4-9 shows the craze morphology for RVEs with particle stiffness G
=
6.2, 62, 620 MPa, and can be compared with the craze morphology for the RVE with
PB particle shown in Figure 4-8. A characteristic feature of the craze morphology
for these RVEs is the process of crazing around the circumference of the particle, in
order to relieve the axial stress build-up in the particle. The crazing, particularly in
Figure 4-9(b) and (c), shows uncontrolled and unrealistic crazing, arguably because
the stiffer particles with GP = 62,620 MPa reduce the stress concentration at particle/matrix interface, compared to the more compliant particles2 . If the unrealistic
crazing is suppressed, then localization of crazing at the equator with possible particle
degradation should lead to a low macroscopic toughness for the RVE. The results of
this study provide a micromechanical explanation of the low toughening potency of
homogeneous PB particles. TEM micrographs of homogeneous PB particles (Donald
and Kramer, 1982) show that the elongation of the particle in the direction of the tensile load is accompanied by an equatorial contraction resulting in the debonding of the
particle from the matrix. The large voids formed as a result of the debonding result
in early failure of the composite. In our study, particle-matrix compatibility prevents
debonding, with the consequence that unrealistic crazing occurs in the PS matrix to
accommodate the deformation. Given the low toughening potency of homogeneous
particles, it would seem that incorporating PC particles in PMMA matrix would not
result in significant toughening of the resulting PC/PMMA composite; a laminated
morphology incorporating alternating layers of PC and PMMA (see Chapter 3) would
2
= 0.735
Based on Goodier's solution, Boyce et al. (1987) have given ae/0ox = 1.730 and 0%/,
for PB particle within a PS matrix in an infinite body subjected to a far field uniaxial stress o.
Here o, and Och represent the von-Mises stress and the hydrostatic stress in the matrix just outside
the particle at the equator. For a less compliant particle with K, = 2880 MPa and Gp = 880
MPa,
= 1.130 and Uh/O, = 0.407. While these values are not exactly true for our case of a
=/o
periodic RVE with a given volume fraction of particles, they qualitatively show that with decreasing
particle compliance (relative to the PS matrix), the stress concentration, in the matrix just outside
the particle at the equator, decreases; this delays the initiation of crazes for the less compliant
particles.
161
Table 4.2: Particle elastic properties used to investigate the role of particle compliance
on the deformation and toughness of rubber-toughened PS
Kp (MPa)
1940
1940
1940
1940
Gp (MPa)
0.62
6.2
62
620
Ep (MPa)
vp
0.4998
3.5
35.2
0.4984
351.9
0.4842
3519.1
0.3556
be more effective.
4.3.2
Response of RVE with homogenized particle
The effective (linear) elastic properties of the two phase composite particle of HIPS,
consisting of about 20% topologically continuous PB and abut 80% PS sub-inclusions,
were used in this study with the purpose of comparing the predicted microscopic and
macroscopic features with experiments.
The homogenized elastic properties of the
particle used were taken to be: KHOM = KPB
=
1940 MPa and GHOM = 111 MPa,
where the latter is taken from Argon et al. (1987). Note that these elastic properties
are between the cases Gp = 62 MPa and G, = 620 MPa, discussed in Section 4.3.1.
Figure 4-10 shows the EZ - E, and the b - E, behavior of the RVE. Figure 4-11
shows respectively the contour plots for the craze profile and the local axial stress.
Point (a) (E, = 0.01) corresponds to the initiation of the equatorial craze. At point
(b) (EZ = 0.045), associated with the peak stress, the craze profile contour plots show
that pervasive crazing occurs in the matrix, with a tendency for the crazing to occur
along the particle circumference. Finally at point (c) (Ez = 0.192), crazing can be
seen to occur almost everywhere in the matrix. Neither the Ez - Ez response nor the
craze profiles are in accord with experimental results.
162
40
35
(U
0-
30
25
F20
CU)
-)
15
-
-
G =0.62 MPa
G =6.2 MPa
-..
G =62 MPa
G =620 MPa
p
0
b
2
-
5
0
0.08
0.06
0.04
0.02
Macroscopic axial strain Ez
0.1
(a)
0.1
c 0.08
E
0.06
0.04
0
0
20.02
..
-
0
0
G =0.62 MPa
G =6.2 MPa
G =62 MPa
G P=620 MPa
P
0.08
0.06
0.04
0.02
Macroscopic axial strain Ez
0.1
(b)
Figure 4-5: The macroscopic RVE response for particle volume fraction of
as a function of particle shear modulus G,
163
fp =
0.28,
40
35 I.
0t
N
30
-
(d)
(e)
(C)
CO
cD 25
CO
(b)
~ 20
15
0
0
o 10
2
(a
5
3
0.06
0.08
0.02
0.04
Macroscopic axial strain Ez
0
0 .1
(a)
0.1
(e)
0.08
.a-.
C'
E
0
-0
0.06
(d)
(C)
0.04
0
(b)
U'
0. 021
(a)
C
0
0.02
0.08
0.06
0.04
Macroscopic axial strain E
0 .1
(b)
Figure 4-6: Macroscopic response for RVE with PB particle GP = 0.62 MPa (f,
0.28).
164
=
D
00
(a) E=0.009
I
:1 1t
(b) E =0.03
(C) E =0.05
Figure 4-7: Contour plots of the craze profile for G, = 0.62 MPa (corresponding to
Figure 4-6)
165
Do
(d) EZ=0.067
(e) EZ=0.09
Figure 4-8: Contour plots of the craze profile for G, = 0.62 MPa (corresponding to
Figure 4-6) (cont.)
166
D
10
(a) GP=6.2 Mpa
(b) GP=62 Mpa
(c) GP=620 Mpa
Figure 4-9: Comparison of the contour plots of the craze profile for G, = 6.2,62,620
MPa at an axial strain E, = 0.067
167
30
(b)
25
0
.C
N
(U
C> 20
a)
15
10
(.
0
5
0'-
0.05
0.1
0.15
Macroscopic true axial strain Ez
0
0.2
(a)
0.2
(C)
CO,
0.15
E
0.11
a)
0
C.)
0
0. 051
(b)
(a)
0'
C
0.1
0.05
0.1
0.15
Macroscopic true axial strain E
0.2
(b)
Figure 4-10: The macroscopic RVE response for particle volume fraction of f, = 0.28.
The 2-phase particle was represented by its effective elastic properties (KHOM = 1940
MPa and GROM
=
111 MPa).
168
(a) E
IOIIIi
I
(b) EZ=O.O45
()EZ=O.192
Figure 4-11: Contour plots of the craze profile, corresponding to Figure 4-10. The
2-phase particle was represented by its effective elastic properties (KHOM = 1940
MPa and CHOM = 111 MPa).
169
4.3.3
Response of RVE with heterogeneous particle (no fibrillation)
To investigate the role of particle heterogeneity on the micro- and macro-mechanics
of HIPS, the particle is modeled as a 2-phase composite comprising PS subinclusions
and PB. Fibrillation in the PB domains is suppressed. Figure 4-12 shows the macroscopic response, and Figures 4-13 and ?? shows the contour plots for the craze profile
and the local axial stress. Crazing initiates in the matrix at point (a) (Ez = 0.01)
adjacent to the PB region, a feature that is seen often in thin-film micrographs. The
initial low stiffness of the PB regions provides a local stress concentration in the
matrix and provides preferential sites for crazing in the PS matrix.
At point (b)
(Ez = 0.026), which is associated with the macroscopic yield, a craze has spanned
the net section of the RVE. More realistic craze patterns are predicted, compared to
the studies in previous sections, in that crazes tend to initiate adjacent to PB regions
in the particle. Nevertheless, the (non-fibrillating) particle cannot accommodate a
significant amount of axial stretch since its relatively high bulk modulus compared to
the compliant (crazeable) PS matrix means that the particle would have to undergo
large compressive radial stretch. On the other hand, the PS matrix undergoes inelastic deformation by crazing at negligible radial stretch. The need to enforce radial
compatibility between the matrix and the particle results in a competition between
the axial stretch contribution of the matrix and the particle. It is energetically more
favorable for the axial stretch to be almost entirely accommodated by the PS matrix
through pervasive crazing, as well as crazing along the particle circumference. It is
not surprising, therefore, that
',arctie
~ 0 (see Figure 4-12) for the non-fibrillating
particle (as well as for homogeneous/homogeneized particles studied in previous sections). At point (c) (E, = 0.134), cracks in the matrix have initiated adjacent to the
pole of the particle. Pervasive and unrealistic crazing is seen in the matrix. It is noted
that the post-yield softening behavior is an instability in the Ez - Ez response, and
that in reality should lead to rapid failure after the peak stress is reached. Results of
this study indicate that particle heterogeneity is a key feature to generating multiple
170
crazes from a particle, but it alone cannot explain the macroscopic response of HIPS
and, in particular, the experimentally observed craze profile.
4.3.4
Response of RVE with heterogeneous particle (with fibrillation)
In this study, the additional feature of fibrillation of PB domains is incorporated
for the heterogeneous particle investigated in Section 4.3.3; a fibrillation stress of
0-fib =
10 MPa is used. The effect of the magnitude of afib on the results is explored
in Section 4.3.6. The E, - E, shown in Figure 4-14 is quite typical of HIPS. The
V) - E, result of Figure 4-14 shows a slope of nearly unity due to absence of lateral
contraction during tensile loading, which is consistent with creep experiments on HIPS
(e.g., Bucknall, 1977), as well as the data of G'Sell et al. (2002). In line with the SAXS
results of Bubeck et al. (1991) and Magalhaes and Borggreve (1995), a substantial
amount of the total volumetric strain is accommodated by non-crazing mechanisms,
i.e., by particle cavitation.
However, unlike the experimental results of Bubeck et
al. and Magalhaes and Borggreve, our results predict that the volume change due to
crazing is more than the volume change due to cavitation.
This discrepancy could
perhaps be due to the meshing of the PB domains with element thicknesses of about
25 nm; the actual thickness of PB domains is in the range 5-10 nm.
It is worth
noting that V) - E, behavior is well predicted for all particle morphologies, because
the dilatant nature of crazing results in negligible lateral contraction of the RVE.
At point (a) (E, = 0.004), fibrillation in the particle can be seen in the contour
plots of Figure 4-15.
Crazing in the matrix initiates at point (b) (Ez = 0.006).
It
is noted that in our previous studies, crazing initiated at Ez = 0.01. This highlights
that fibrillation in the PB domains allows crazing to occur earlier by providing local
stress concentrations in the PS matrix.
At point (c) (Ez = 0.01), crazes begin to
extend from one particle to another. At point (c) (E2 = 0.1), the crazes bridge the
particles, after which at points (d) (Ez
=
0.25) and (e) (E, = 0.3), the crazes thicken,
finally leading to craze breakdown.
171
301
(b)
25
(a)
N
-FU
20
(C)
CO
15
CD
10
0
0
5
true axial strain E
Macroscopic
0.1
0.05
Macroscopic true axial strain E
0
0.15
(a)
0.1
~RVE
-
/
'1'particle
C
I
4-a
Co
0.1
I
1
E
I
U)
I
I
I
I
/
051-
I
I
I
I
(b
Ca)
(U
C
'
0
I
I
I
I
I
/
(a
,~
I
I
- --------
-
00
0
Co)
0
I
I
I
I
/
C)
I
0.1
0.05
Macroscopic true axial strain E
0.15
(b)
Figure 4-12: The macroscopic RVE response for particle volume fraction of f, = 0.28.
The particle is modeled as a 2-phase system, but with no fibrillation in the PB
domains.
172
...- Msnm
MIU
= MEMNO
Du~
InMENS Noun
'won
son la
.-..
....
A
m~~nnn~~n
ang alli
-- man
-- 10.
a,.aa
.1M
"1.- ft
..
molami
s EVENum
maim Mn
m ..sm o
.m
.. B..ell
ml
------
--mu ==mo Oum
(b) E =0.026
Mon
0n MO.-NO muum
Paricl voum
arati,
173
ons= M.28
Comparison of the macroscopic response and the contour plots for RVEs with a
heterogeneous particle without and with fibrillation, clarifies the role of fibrillation
in the deformation and toughening of HIPS. First, comparing Figure 4-12(a) with
Figure 4-14(a), we note that in the absence of fibrillation, the macroscopic stress
required to initiate crazes as well as the macroscopic yield are higher compared to
the case where the particle fibrillates. The former exhibits softening beyond the peak
stress, while the latter shows a slightly hardening behavior.
Typical experimental
stress-strain curves for HIPS, such as in Figure 4-1(a), are consistent with particle
fibrillation. Second, the absence of fibrillation renders
7/particle
~ 0, which is not in line
with the SAXS experiments of Bubeck et al. (1991) and Magalhaes and Borggreve
(1995).
When the particle is allowed to fibrillate, a substantial amount of inelastic
volumetric strain is accommodated by the particle. Physically, the fibrillating particle
is more compliant than the non-fibrillating particle, rendering an easy development
of axial stretch in the particle; this eliminates the unrealistic, pervasive crazing seen
for the non-fibrillating particle. The consequent reduced rate of craze opening in the
PS matrix results in delayed craze breakdown (compared to the non-fibrillating case)
and hence a tough composite. A comparison of craze damage contour plots for the
non-fibrillating vs. the fibrillating case indicates a more realistic craze pattern for the
latter.
4.3.5
Effect of particle volume fraction
Figure 4-16(a) shows the EZ-E, behavior of RVEs with fibrillating particles (-fib
MPa), as a function of particle volume fraction
fp,
f,
=
0.18 (or
fp
=
0.28) show more ductility compared to the case
highlighting that ductility increases with increasing
10
with a fixed particle size. The
macroscopic yield and flow stress are predicted to increase, with decreasing
case with
=
fp.
fp
fp.
The
= 0.08,
These trends are consistent
with experimental results such as those of Correa and de Sousa (1997), shown in
4 - E, plot as a function of fp. While
fp, the volumetric strain contributions of
Figure 4-16(b). Figure 4-16(a) also shows the
the overall volume change is independent of
the matrix and the particle change with
fp.
174
In particular, with increasing
fp,
Iparticle
201
(C)
15
(D
0
N
En
0)
10
(b)
a)
5
CI-
0.1
0.2
Macroscopic true axial strain E Z
0
0.3
(a)
0.3
-
RVE
Wmatrix
*
0.25[
~I1matnx(e)
---..Yparticle
0.21
0
0
0.15F
(d)
0.1 [
0.05F
(--
-
0I-
(a g-~
'
r
0
0.1
0.2
Macroscopic true axial strain E
0.3
(b)
Figure 4-14: The macroscopic RVE response for particle volume fraction of fp = 0.28.
The particle is modeled as a 2-phase system, with fibrillation in the PB domains.
175
D
Fib
0
(b) EZ
EZ=O.OO6
(d) EZ=O.OI
Ez=0 1
(d()EZ=0.251
(f) EZ=0.30
Figure 4-15: Contour plots of fibrillation in PB (left), and craze profile (right), corresponding to Figure 4-14. The particle is modeled as a 2-phase system, with fibrillation
in the PB domains. Particle volume fraction f, = 0.28.
176
increases (and bmarix decreases), thereby lessening the accommodation of axial strain
by craze opening. This results in delayed craze breakdown and delayed macroscopic
failure in HIPS. The contour plots for craze damage at E, = 0.144, shown in Figure 417, indicate a more localized pattern of crazing for the case
fp
= 0.08. Figure 4-18 is
a visual enhancement of Figure 4-17 to facilitate the identification of craze patterns,
as a function of particle volume fraction.
4.3.6
Effect of critical fibrillation stress
Our baseline results for the fibrillation study is based on
almost identical with
'fib
0-fib
rfib =
10 MPa. Results are
= 20 MPa, except for the slightly higher yield stress for the
latter case (see Figure 4-19). For a higher critical fibrillation stress of afib = 30 MPa,
crazing in the matrix initiates prior to fibrillation in the particle. The response of the
RVE is the same as that shown in Figure 4-12 (upto 4 % strain). The high yield and
post-yield softening is not consistent with the typical stress-strain curves of HIPS.
Contour plots at E, = 0.04, shown in Figure 4-20, show unrealistic crazing patterns
for the case 9fib = 30 MPa; the contour plots for cfib
=
10 MPa and Ufib = 20 MPa
are similar to each other.
We note that in bulk specimens of rubber (which inherently contains defects),
failure occurs at a critical hydrostatic stress of
-
5E/6, where E is the small strain
Young's modulus (Gent and Lindley, 1958; Gent and Wang, 1991). Since EPB = 1.86
MPa (Boyce et al., 1987), the expected
'fib =
1.5 MPa. However, as noted by
Lazzeri and Bucknall (1993), Gent's model is applicable for bulk rubber specimens
which have defects of size 0.5 - 1000 pm.
Given that the HIPS particle size is of
the order of microns, and that the PB layers within the particles are 5-10 nm thick,
it should not be surprising that the value of
c'fib =
1.5 MPa is significantly lower
than the actual hydrostatic stress estimated here to fibrillate the small length-scale
PB salami HIPS particles.
177
15
0. -I'I
._
Pf=1
-
f =0.18
P
0.25-
6
20 .V0.28
_
l
.
NRE
0.2
)
-2
E
RVE:fp=02
.
RVE p
matrix f
---
~ matrx
particle
-
=0.08
=0.08
f 018
p
02
-
-T
>0.15
.
f =0.08
-
Wparticle fp
a)
------
i'matrix: f =0.28
0
EL0.1.
0
0.
0000
5
0
6 0.050
0..
0.3
0.2
0.1
0
0.2
0.1
Macroscopic true axial strain Ez
0.3
Macroscopic true axial strain Ez
(a)
30
0.08
'-
0.12
.1
0.35
20
10
0
0
10
30
20
40
50
60
Strain (%)
(b)
Figure 4-16: (a) The predicted effect of volume fraction on the deformation and
toughness of HIPS, (b) The effect of particle volume fraction. From experiments of
Correa and de Sousa (1997).
178
Aa
(a)
(b)
(C)
Figure 4-17: Craze damage contour plots, at E, = 0.144, for (a)
fp = 0.18, and (c) f, = 0.28.
179
f,
= 0.08, (b)
(a)
(b)
(C)
Figure 4-18: Enhanced representation of craze damage contour plots, at E, = 0.144,
for (a) f, = 0.08, (b) f, = 0.18, and (c) f, = 0.28 (see Figure 4-17).
180
25
(D
20
U)
x
15
U)
f P=0.08: Cfib=10 MPa
10
a)
f =0.18:rfb=20 MPa
Pf=0.18: a=10 MPa
f =0.28: aW=MPa
Gfib=1
p
fP=0.28: Cri =20 MPa
0
0
U)
5
0
Co)
CU
0.1
0
0.2
0.3
Macroscopic true axial strain E
(a)
0.3
0.3
0.25
0.2
E
PRVE f =0.08
---- r : f =0.08
'RVE'
>
0)
0.15
f =0.08
Wpaticle: f=0.08
RVE f 018
-XWmax: f =0.18
.-p paticle :f 7 =0.18
4 matrx:
00.2
15
E
=02
f
=0.28
>0,15
0.1
pf=
02
S0.1
0
2-.
-
0
0
0 0.05
U'0
particle
WRVEf
l'marx: f =0.28
particle
1particle :7p =0.
z
SkRVE'
aix: f7p~o
=0.08
IVparticle:
WRVE: fp=0.18
--- LP - f =018
,mtix f 018
.
-
0 0.05
01- 02-03-00.1
0.2
Macroscopic true axial strain Ez
0.3
"0
0.1
0.2
Macroscopic true axial strain E_
0.3
-fib-20 MPa
O-,, =10 MPa
(b)
Figure 4-19: Effect of
ifib
on the macroscopic response of HIPS at various levels of
fp-
181
O-fib=10
MPa
Of-b=20
MPa
(a)
(b)
O-,fb=
3 0
MPa
(C)
Figure 4-20: Craze damage contour plots for
182
f,
= 0.28 at E. = 0.04.
4.3.7
Summary and Conclusions
The objective of this chapter was to address the role of particle morphology on the
deformation and toughness of HIPS. To this end, a one-particle axisymmetric RVE
model, proposed by Socrate and Boyce (2000), was used to independently study the
effect of the following factors: (a) particle compliance, (b) particle heterogeneity, and
(c) particle fibrillation.
To study the effect of particle compliance, a parametric study was conducted in
which the bulk modulus was kept fixed (to the value of
was systematically increased, starting from
GPB.
KPB)
and the shear modulus
The study provided a micromechan-
ical interpretation for the experimental finding that homogeneous, compliant particles
do not toughen PS, a fact known from the prior work of Donald and Kramer (1982).
Next, to investigate the role of particle heterogeneity, two studies were conducted.
In the first study, the heterogeneous "salami" particles found in HIPS were represented
by their homogenized elastic properties, while in the second study, both the PB and
the PS phases of the particle were modeled, but the fibrillation of the PB domains
was suppressed. The macroscopic stress-strain behavior predicted from both studies
showed unreasonably large yield stresses, and a softening behavior not consistent with
experimental observations on HIPS. Further, both predicted pervasive and unrealistic
crazing in the PS matrix, although the craze patterns for the heterogeneous particles
(for small strains) were superior, compared to the predicted craze patterns for the
homogenized particle. An important conclusion of these two studies was that particle
heterogeneity is an important ingredient to initiating multiple crazes from a particle,
but it alone cannot adequately explain the deformation and toughness of HIPS, since
it ultimately leads to unrealistic, pervasive crazing.
Finally, the heterogeneous "salami" particles were allowed to fibrillate. Encouragingly, the predicted E, - E, response is well in line with experiments. The substantial
volume change in the particles is consistent with the SAXS experiments of Bubeck
et al. (1991) and Magalhaes and Borggreve (1995), as well as numerous TEM micrographs that show cavitation in the particle. These TEM micrographs show that
183
the initiated crazes in the PS matrix almost invariably adjoin fibrillated regions of
the particle. This feature is predicted by our model predictions with the fibrillating
particle. Increasing particle volume fraction improves the ductility, and is associated
with the an increase in
4
partcle
for a given macroscopic axial strain. We conclude
that both particle heterogeneity and particle fibrillation are necessary to account for
the deformation and toughness of HIPS.
184
Chapter 5
Summary and future work
The toughness of glassy polymers such as PC, PMMA and PS is strongly dependent
on loading conditions, such as strain rate, temperature, and tri-axiality. As in other
material systems, low macroscopic toughness in glassy homopolymers stems either
from brittle' failure mechanisms or from the strong localization of deformation and
failure in a localized region.
The room temperature, uniaxial tension response of
"brittle" polymers such as PMMA and PS is characterized by premature failure due
to the localization of deformation along a dominant craze that traverses the net
section of the specimen, followed by craze breakdown and fracture.
In the case of
"ductile" polymers such as PC, the room temperature uniaxial tension response is
marked by high macroscopic toughness obtained via large scale yielding and plastic
stretching to large strains. However, in the presence of sharp notches or cracks, PC
undergoes cavitation-induced brittle failure due to the high, local stress triaxiality
ahead of the sharp notch/crack. The key problem posed by the propensity of glassy
homopolymers to undergo localized deformation and failure is alleviated by a judicious
design of polymer composites, which enable the development of diffuse patterns of
inelastic deformation and damage that spread throughout the volume of the material.
This thesis focusses on the modeling of the deformation and failure of two polymer composite systems - ductile/brittle microlaminates and toughened polystyrene.
'Recall that "brittle" polymers deform inelastically by crazing, while "ductile" polymers deform
inelastically by shear yielding.
185
While a qualitative understanding for both these composite systems exists, detailed
modeling studies that provide insight into the micromechanics of deformation and
failure, and that can guide the development of optimal polymer blends are lacking.
The modeling efforts in this thesis serve to bridge this gap.
This chapter summarizes the work that was accomplished in this thesis pertaining
to ductile/brittle microlaminates and rubber-toughened polystyrene. Directions for
future work in these areas are also discussed.
5.1
Ductile/brittle polymeric laminates
Ductile/brittle microlaminates are comprised of alternating layers of ductile and brittle layers, with layer thicknesses of the order of microns and the number of layers
between tens to thousands. Two prominent examples of ductile/brittle polymeric laminates that have been experimentally studied, by Baer and co-workers, are PC/SAN
and PC/PMMA. The laminate toughness in PC/SAN and PC/PMMA composites
is controlled by a fine balance of three main factors - volume fraction of PC (fpc),
number of layers (N) for fixed laminate thickness, or equivalently absolute layer thicknesses, and strain-rate (6). Increase in fpc and N improve the laminate toughness,
while increase in
e
reduces the toughness. Numerous optical micrographs, from the
work of Baer and coworkers, of the laminate edge surface indicate crazing in the
PMMA (or SAN) layers, and shearbands in the PC layers. These surface patterns
of inelastic deformation indicate a synergy between crazing in the brittle layers and
shearbanding in the ductile PC layers. With decreasing layer thicknesses, surface micrographs show that the brittle layers undergo both crazing and shear yielding (e.g.,
Ma et al., 1990), revealing that the inelastic deformation in the brittle layers depends
on the absolute layer thickness of the brittle layers.
To provide insight into the underlying micromechanics associated with the macroscopic behavior of ductile/brittle laminates, suitable 2D (plane strain) micromechanical models for the PC/PMMA system were developed. The response of 2D RVEs
was studied as a function of fpc, while keeping N and
186
e
constant.
In line with
experiments, the ductile constituent PC in the micromechanical models inelastically
deformed by shear yielding, while the brittle constituent PMMA deformed by either
crazing or shear yielding, depending on the local stress state. Ductile failure criteria
for PC and PMMA based on critical failure stretch, and a craze breakdown criterion
for PMMA were incorporated in the model. While the crazing parameters for PMMA
in PC/PMMA microlaminates were not known, these parameters were estimated to
capture the experimentally observed competition between crazing and shear yielding
in the PMMA layers. Weibull statistics for craze initiation were introduced to account for the defect sensitivity of craze initiation. While the 2D RVEs captured the
experimentally observed shear yielding and crazing interactions, they were found to
be inadequate in predicting the the macroscopic behavior of PC/PMMA laminates.
In particular, the 2D RVEs do not reveal a transition in ductility with increasing fpc.
The reason for the inadequacy of the 2D RVEs lies principally in the effect of fpc
on the extent of craze penetration through the laminate width. Optical micrographs
of Sung et al. (1994c) showed that with increasing fpc, the extent of craze penetration
(or "tunneling") through the laminate width decreases. In fact, at "high" fpc, the
surface crazes propagate only to the sub-surface of the laminate with the result that
most of the inelastic deformation in the brittle layers is accommodated by shear
yielding. High laminate toughness is associated with reduced craze tunneling.
3D RVE micromechanical models were developed in order to capture the distribution of inelastic deformation in the laminate width (or 3-) direction. Due to the
preponderance of defects on the surface of specimens, the craze initiation resistance
at the surface of the laminate was taken to be lower than in the bulk of the laminate.
Uniaxial tension simulations of 3D RVEs successfully accounted for the toughness
dependence on fpc, and captured the experimentally observed transition in ductility with increasing fpc. A key feature predicted with the 3D RVEs is that at high
fpc, crazes in the PMMA layers do not tunnel significantly through the width of the
laminate, while at low fpc crazes in PMMA layers penetrate through the laminate
width.
At high fpc, the PC layers constrain the tunneling of crazes through the
laminate width which, in turn, results in most of the inelastic deformation in the
187
PMMA layers to be accommodated by shear yielding; the inhibition of craze tunneling combined with shear yielding of both PC and PMMA layers thereby increase the
laminate toughness. The contrasting performance of the 2D and 3D RVEs highlights
the crucial role of the distribution of inelastic events in the 3-direction on the laminate
toughness.
Uniaxial tension simulations of 3D RVEs with two different
0.6 and N fixed, predicted that with increasing
e,
e,
while keeping fpc
=
the laminate yield strength increases
while the laminate ductility decreases. The strain-rate dependence of craze initiation
in PMMA was found to be lower than the strain-rate dependence of shear yielding;
this resulted in increased crazing in the PMMA layers with increasing strain rate.
In particular, the extent of craze tunneling increased with increasing
e,
which is
consistent with experimental findings. The influence of length scale of the layers was
studied using 3D RVEs for two different N (keeping fpc = 0.6 and
e
fixed). The
length scale study predicted a higher level of toughness as N was increased (i.e., with
decreasing layer thicknesses). Results indicated that with increasing N, the PC layers
constrain the crazing in the PMMA layers, thereby increasing the laminate toughness.
Further, deformation and damage were found to be more diffuse at the higher N. In
summary, our micromechanical models successfully capture the effect of PC volume
fraction, absolute layer thicknesses, and strain rate on the macroscopic behavior and
the underlying micromechanics.
5.2
Toughened polystyrene
As discussed above, under uniaxial tension loading and at room temperature, PS
fails in a brittle manner due to uncontrolled crazing, leading to premature craze
breakdown and failure.
The incorporation of 10-30 % volume fraction of micron-
order sized compliant, rubbery particles within a PS matrix results in significant gain
in tensile toughness, at the expense of lower yield strength and lower elastic stiffness.
We have focussed on the prominent example of rubber-toughened polystyrene - HIPS.
In HIPS, the "rubbery" particles are heterogeneous and are comprised of 70-80% PS
188
in the form of sub-micron sized occlusions within a 20-30% topologically continuous
PB phase, where the PB phase takes the form of thin inter-particle ligaments of
thickness 5-10 nm.
The main purpose of our study was to address the role of particle morphology
on the deformation and toughness of HIPS. To this end, we investigated the effect
of particle compliance, particle heterogeneity and particle fibrillation on the macroscopic behavior and the associated craze morphology of toughened PS. One-particle
axisymmetric RVEs were subjected to uniaxial tension deformation.
In the study of the effect of particle compliance, a parametric study was conducted wherein compliant, homogeneous particles (at fixed particle volume fraction
fp) were embedded within the RVE, and the particle shear stiffness was systematically increased. This study predicted unrealistic craze morphology and low levels of
toughness for the composite, highlighting that particle compliance is not the principal factor enabling multiple crazing and toughness of HIPS. The effect of particle
heterogeneity was investigated through two studies. In the first study, the two-phase
particle in HIPS was represented by its homogenized properties.
A significant dis-
crepancy between the predicted yield strength and the typical experimental value, as
well as post-yield softening associated with an unrealistic craze morphology indicated
that the uni-axial tension behavior of HIPS cannot be explained by homogenized
particles. In the second study, the particle was represented as a two phase composite
comprised of occluded PS in a PB matrix.
Similar to the study on homogenized
particle, the RVE response with the heterogeneous particle was characterized by unrealistically high yield followed by post-yield softening. Nevertheless, the predicted
craze morphology at low strains was found to be in good agreement with experimental observations, suggesting that the heterogeneous nature of the particle governs
the initiation of multiple crazes from the particle (a feature widely observed from
micrographs of deformed HIPS).
In the final study, the effect of particle fibrillation was investigated, wherein the
highly confined PB phase within the heterogeneous particle was allowed to fibrillate.
The incorporation of particle fibrillation results in predictions of yield strength and
189
flow strength to be in good agreement with experiments. The predicted craze profiles
agree well with TEM micrographs showing patterns of deformation in HIPS. A study
of the effect of increasing
fp
correctly predicts decreasing yield and flow strength,
and increased toughness. We conclude that the both particle heterogeneity and particle fibrillation are necessary to account for the deformation and toughness of HIPS.
Indeed, particle fibrillation is necessary to explain the SAXS results of various researchers, who have pointed out that a significant volume change in HIPS is due to
non-crazing mechanisms.
5.3
Future work
In this section, we discuss possible avenues for future work on ductile/brittle polymeric
laminates.
Ductile/brittle polymeric laminates
5.3.1
We recommend the following directions for future work on ductile/brittle laminates
* Crazing: Recall that in Chapter 3, crazing material parameters for initiation
and growth for PMMA in PC/PMMA microlaminates were estimated so that
they captured the experimentally observed competition between crazing and
shear yielding in the PMMA layers. While our sensitivity studies indicate that
the baseline results are robust against changes in the crazing parameters, it is
nevertheless important to experimentally obtain the correct values of the parameters, and their dependence on factors such as orientation to enable successful
design.
In view of the crucial role played by craze tunneling on laminate toughness,
another important aspect that needs to be quantified is the extent of craze
tunneling from the surface of the PC/PMMA laminate to within the bulk of
the laminate, as a function of fpc, d and N for a given macroscopic strain.
190
Currently, only a limited set of data for craze tunneling as a function of fpc
and
a
on PC/SAN is available (Sung et al., 1994c).
" Interfacial delamination: Kerns et al.(2000) have noted that the interfacial
strength of the PC/PMMA interface is about an order of magnitude higher
than that for the PC/SAN interface (see Chapter 3 for details). A consequence
of this is that no interfacial delamination is observed in PC/PMMA laminates.
However, PC/SAN interfaces delaminate, with the result that the severity of
the notch in PC layers created by SAN crazes is less compared to that in the
PC/PMMA system. This is arguably one of the reasons for the higher toughness
of PC/SAN compared to PC/PMMA laminates. Incorporating a delamination
criterion will enable investigation of the PC/SAN system.
" Different loading conditions: This thesis has focussed attention exclusively on
the uniaxial tension behavior of ductile/brittle laminates. Other loading conditions such as biaxial tension and ballistic impact will provide further insight
into the micromechanics of ductile/brittle laminate systems, particularly with
respect to crazing.
" Craze propagation: Recall that in Chapter 2, we mention that we have modeled
craze propagation by a process of repeated initiation of new crazes, and not
by the operative mechanism of meniscus instability. This approximation was
made due to computational considerations; within a finite element framework,
propagation of the craze by meniscus instability requires an apriori knowledge
of which elements have crazed, so that adjacent elements (perpendicular to the
local maximum principal stress direction) "propagate" the craze by meniscus
instability. While we do not expect that the conclusions of this thesis will change
with this modification, incorporating the correct mechanism is important to
capture the correct kinetics of craze propagation.
* Design of laminates: The micromechanical models developed in this thesis,
along with the refinements mentioned above, serve as a springboard for the de-
191
sign of ductile/brittle laminates. Based on laminate features such as the volume
fraction of the constituents, and the number of layers for a given laminate thickness, as well as loading conditions, the micromechanical models can be used to
obtain a cost-effective and optimally tough laminate system.
5.3.2
Toughened polystyrene
9 Craze propagation: The discussion on "craze propagation" for ductile/brittle
laminates also applies for toughened PS. There is, however, an additional complexity. Unlike the case of uniaxial tension deformation of ductile/brittle laminates where the direction of the maximum principal stress is known, in the case
of HIPS, the direction of the maximum principal stress evolves with the deformation. Ideally, there should be elements adjacent to a given craze element,
such that they are aligned in the direction perpendicular to the maximum principal stress. This could possibly be achieved by adaptive meshing algorithms.
9 Length-scale of PB: In our RVE studies on HIPS, PB elements were typically
25 nm thick, instead of the actual 5 - 10 nm. The use of thick PB elements
was necessary in order to obtain a good mesh for the RVE. One effect of this
approximation is that it underestimates the effective chain stretch in the PB
elements. This may have two consequences.
First, due to the lower effective
chain stretch in the PB elements, the network stress will be lower as well. Hence,
the hardening behavior of the macroscopic stress-strain response will be underpredicted. Second, a lower network stress in the PB elements means that the
breakdown of the fibrillated PB elements will be delayed; our work does not
address the possibility of initiation of failure in HIPS due to the failure of PB
elements.
9 3D HIPS study: An important limitation of our work, and also that of most
experimental work on HIPS, is that it does not address the craze morphology
in the 3-direction. Finite element modeling studies for three-dimensional study
of HIPS are somewhat challenging owing to meshing difficulties. However, our
192
understanding of crazing in HIPS will be significantly advanced with the 3D
studies.
" Different loading conditions: Our focus on HIPS has been limited to uniaxial
simulations. An extension of our work to complex loading conditions is necessary in order to obtain predictive capability that can be an input in design.
" Study of different particle morphologies: The focus of this thesis was limited
mostly to the HIPS system.
There are several other PS/PB systems (e.g.,
Boyce et al., 1987), in which particles of different morphologies are embedded
within a PS matrix. Two examples of particle morphologies obtained with the
PS/PB system include concentric spherical shell particles with alternating layers
of PB and PS, and particles with PB rods within a topologically continuous PS
phase. The modeling framework developed in this thesis allows us to investigate
the deformation and toughness of composites with different particle morphologies, and can be potentially useful in creating new particle morphologies that
optimize the toughness of the resulting composite.
193
194
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Appendix A
Evaluation of the Inverse Langevin
Function
The Langevin function is given by
(3) = coth(O) -
where 3 =
12(Achain/AL),
1
(A.1)
L1 is the inverse Langevin function, Achain is the effec-
tive chain stretch and AL is the locking stretch. The evaluation of inverse Langevin
function is conceptually possible by solving the transcendental Equation (A.1) for 0.
However, an exact analytical solution to Equation (A.1) is not known to date. Here,
we compare two computationally attractive approximate analytical expressions for
the inverse Langevin function.
Cohen (1991) gave the following Padd approximation to the inverse Langevin
function
,c(x)
where x -A
hain/AL.
=
x3
2
I - X2
+ O(X 6 )
(A.2)
An alternative approximation, due to Bergstrom and Boyce
(2001), used in this thesis is
207
Es33-1(x)
=
1.31446tan(1.58986x) + 0.91209x if x < 0.84136
S
1
if
(A.-3)
x > 0.84136
It is noteworthy that both Equations (A.2) and (A.3) are asymptotically correct:
Lc(x),LB%3I3(X)
-
1
as x
1- x
Figure A-1 compares the approximations due to Cohen and Bergstrom-Boyce.
Both approximations are found to be in close agreement with the exact solution,
although the Bergstrom-Boyce approximation is superior, as can be clearly seen from
the negligible % absolute error (defined as |L-
208
-
B1s~ 11/L- 1 ) in Figure A-1(b).
100
-
90
80
Cohen
Bergstrom-Boyce
Exact
70
0)
60
(V
-j
0)
50-
(j)
U)
4030
20
10
0
0.4
0.6
0.8
0.2
Normalized chain stretch x
(a)
4.5
-
Cohen
Bergstrom-Boyce
4
0
U)
U)
3
2.5
C,)
.0
2
1.5
1
0.5
00
0.8
0.4
0.6
0.2
Normalized chain stretch x
1
(b)
Figure A-1: (a) Inverse Langevin function vs. normalized chain, (b) % absolute error
for the Cohen and Bergstrom-Boyce approximation.
209
210
Appendix B
Effective elastic properties of the
craze "element"
The bulk matter in the craze element is assumed to be isotropic, while the crazed
matter is transversely isotropic with the plane of circular symmetry perpendicular to
the maximum principal stress direction (which is el as shown in Figure 2-9).
For compactness, the stiffness matrix for both the bulk matter and crazed matter
is written as (using Nye's notation):
C
C11
C12 C12
0
0
0
C12
C 22 C23
0
0
0
C 12 C23 C22
0
0
0
0
0
0
(C22 - C23)/2
0
0
0
0
0
0
C66
0
0
0
0
0
0
C66
where for the isotropic bulk matter, the following relations hold: C bk
= Cik,
and
Cbulk = (Cbulk -
Cuk) / 2
.
Cbk
=Clk
The effective behavior of the craze element can now be exactly established by
applying (a) an arbitrary uniform strain field, (b) the local equilibrium equations, and
(c) the constitutive equations for the bulk and crazed matter. Clearly, the effective
stiffness matrix is transversely isotropic with the same symmetry as that of the crazed
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matter. The final result is
Celf =< C- >eff <
C2
=< 1 >-l< C12Cl >
O
>il
1
-
C! =< C12Cj1>2< C- >-1 + <22 - C
Cef =< C 12 C
>2<
C
>
+ <123
-C
C eff =< C 6 >- I
where < ...
>= 2r/o( ... )bu"k
+
2
o(...)"'
represents the volume average of (...) in
the craze element.
Sha et al. (1995) have determined the following elastic parameters for the craze
matter through a 2D idealized spring network under plane strain for PS: Cn = 720
MPa, C22 = 4.8 MPa, C12 = 14 MPa and C66 = 14 MPa. Although C23 is not
known, the positive definiteness of the stiffness matrix ensures that
(C22 - C23) >
0, or
C23 < C22 = 4.8 MPa. Here, we take C23 = 2 MPa. To obtain the corresponding C
for PMMA, we make the following estimate: CPMMA
EPMMA =
=
(EpMMA/EPs)Cs,
where
3250 MPa and Eps = 3000 MPa are the elastic moduli for bulk PMMA
and PS respectively.
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