Mechanism based modeling of ductile void growth failure in multilayer structures

MECHANISM-BASED MODELING OF DUCTILE VOID
GROWTH FAILURE IN MULTILAYER STRUCTURES

THONG CHEE MENG
(B. Eng. (Hons.), NUS)

A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2003


Acknowledgements

Acknowledgements

This section is specially dedicated to all the kindhearted individuals who granted the
author precious advice, guidance and munificent resources on this Master of
Engineering thesis. The researches would not be made possible without their selfless
devotion of time and effort. Appreciation goes out to:

A/Prof. Cheng Li, supervisor of this research work, who has been a relentless source
of motivation to the author in the exploration of Fracture Mechanics. Through her
enthusiasm and dedication, Prof. Cheng shared much expert knowledge and endowed
valuable insights to the author in his research process. The author would like to
express his deep felt gratitude to Prof. Cheng for her teachings, encouragement and
understanding throughout the project.

Dr. Guo Tian Fu, visiting researcher from Tsinghua University, whom the author is

greatly indebted to, for his invaluable supervision and facilitation in the computational
aspect of the author’s research work. Dr. Guo has also been a humble mentor and more
importantly, a sincere friend in sharing his experience and interpretation in the current
field. His patience and generous support is the basis of the research’s completion.

The author is also grateful to Chong Chee Wei, a postgraduate student, for his advice,
guidance and most importantly the moral support he has given; and to Leo Chin
Khim, a fellow colleague, for his assistance, encouragement and of course, friendship.

Sincere gratitude also extends to the technical officers and peers in the Strength of
Materials Laboratory 2, and everyone else who has contributed to the completion of
this thesis.

i


List of Symbols

List of Symbols

SSY

Small Scale Yielding

Φ

Gurson-Tvergaard continuum flow potential

σe

Mises stress

σm

Mean stress or hydrostatic stress

σ0

Tensile Yield stress

G

Energy release rate

K

Mode I stress intensity factor

Γ

Crack growth resistance

T

T-stress, non-singular elastic stress which acts parallel to the crack plane

∆a

Crack propagation length, distance between initial and current crack tip

X

Distance ahead of the current crack tip location

D

Size of Gurson cell element

l1

Length of fracture process zone

E

Young’s modulus

N

Strain hardening exponent

ν

Poisson’s ratio

f

Void volume fraction in a Gurson cell

fE

Critical void volume fraction in a Gurson cell to trigger element
extinction algorithm for the cell

q1,q2

Micromechanics factors introduced by Tvergaard in the GursonTvergaard model

p

Void pressure in a Gurson cell

x1,x2

Horizontal and vertical datum in the SSY models

x1, x2, x3

Cartesian axis directions for the periodic composite models

C0

Initial reinforcement volume fraction in a composite

R0

Initial radius of the fiber or spherical reinforcement

ii

Table of Contents

Table of Contents

ACKNOWLDEGEMENT

i

LIST OF SYMBOLS

ii

TABLE OF CONTENTS

iii

SUMMARY

vi

LIST OF FIGURES

x

CHAPTER 1
Introduction

1

CHAPTER 2

Literature Review

6

2.1
2.2
2.3
2.4
2.5

Ductile Crack Growth under Small Scale Yielding
Fracture Toughness of Constrained Ductile Layer
Modeling Interfacial Decohesion in a Composite
Extended Gurson Model Incorporating Vapour Pressure
Numerical Implementation

6
9
13
17
20

CHAPTER 3
Ductile Crack Growth under Small Scale Yielding

24

3.1 Introduction
3.2 Problem Formulation
3.2.1 Computational Model and Boundary Conditions

3.2.2 Material Properties
3.2.3 Numerical Details
3.3 Results and Discussion
3.3.1 Effects of T-stress
3.3.2 Effects of Internal Void Vapour Pressure
3.4 Conclusion

24
25
25
26
29
31
31
39
44
iii


Table of Contents

CHAPTER 4
Ductile Failure of Centerline Crack in a Constrained Ductile Layer

46

4.1 Introduction
4.2 Problem Formulation
4.2.1 Computational Model and Boundary Conditions
4.2.2 Material Properties

4.2.3 Numerical Details
4.3 Results And Discussion
4.3.1 Effects of Internal Void Vapour Pressure
4.3.2 Effects of Elastic Modulus Mismatch
4.3.3 Effects of T-stress
4.4 Conclusion

46
47
48
49
51
53
54
58
63
70

CHAPTER 5
Two-Dimensional Modeling of Void-Induced Interfacial Decohesion in a FiberReinforced Polymeric Matrix Composite
72
5.1 Introduction
5.2 Problem Formulation
5.2.1 Plane Strain Periodic Array
5.2.2 Material Models
5.3 Numerical Results
5.3.1 Effect of Interface Damage Zone Size, D/R0
5.3.2 Effect of Interfacial Cell Element Porosity, f0
5.3.3 Effects of Void Vapour Pressure, p0/σ0
5.4 Discussion and Conclusion

72
74
74
76
78
78
91
98
104

CHAPTER 6
Three-Dimensional Modeling of Void-Induced Interfacial Decohesion in a
Spherical Particle-Reinforced Polymeric Matrix Composite

107

6.1 Introduction
6.2 Problem Formulation
6.2.1 Three-dimensional Periodic Array
6.2.2 Material Models
6.3 Numerical Results
6.3.1 Perfect Particle-Matrix Interface
6.3.2 Imperfect Particle-Matrix Interface
6.3.3 Mean stress-Mean strain Response
6.3.4 Effect of Interfacial Cell Element Porosity, f0
6.3.5 Effects of Void Vapour Pressure, p0/σ0
6.3.6 Effects of Particle Volume Fraction, C0
6.4 Discussion and Conclusion

107
108
108
111
112
113
118
119
121
127
133
137

CHAPTER 7
Summary Of Conclusions

142

iv


Table of Contents

7.1 Ductile Crack Growth under Small Scale Yielding
142
7.2 Ductile Failure of Centerline Crack in a Constrained Ductile Layer
144
7.3 2-D Modeling of Void-Induced Interfacial Decohesion in a Fiber-Reinforced
Polymeric Matrix Composite
146

7.4 3-D Modeling of Void-Induced Interfacial Decohesion in a Spherical
Particle-Reinforced Polymeric Matrix Composite
148

REFERENCES

146

APPENDIX A

A-1

APPENDIX B

B-1

APPENDIX C

C-1

v


Summary

Summary

Inspiration for the present research work comes from industrial developments in the
field of electronic packaging. Motivated by the moisture-induced failure phenomenon
in the integrated circuit (IC) packages, commonly known as “popcorn cracking”, this

literature serves to gain a deeper understanding on the micro-mechanics of void
pressure-assisted ductile fracture. IC packages assembly usually consists of an intricate
multilayer structure. A simple example is that of the thin layer of ductile adhesive (die
attach), sandwiched between the much stiffer silicon die chip and die pad base. In
addition, many constituents in the IC packages are made from polymeric matrix
composites, e.g. Ag-filled epoxy being used as moulding compound or die attach.
These ductile and porous materials are typically susceptible to moisture absorption at
the reinforcement-matrix interface and are therefore prone to fail by void vapour
pressure assisted decohesion and ductile crack growth.

Investigation begins with a preliminary study on the mechanisms of ductile failure by
void growth and coalescence. Effects of internal void pressure as well as crack tip
constraints, implemented via the application of T-stress, on the SSY mode I crack
growth fracture toughness are studied. It is found that a low constraint crack under
negative T-stresses greatly elevates the fracture toughness of the material. This is due
to greater degree of plastic dissipation at the crack front, which effectively raises the
total work of fracture for crack advancement. Conversely, a highly constrained crack
under positive T-stress shows no significant effect on the fracture toughness. Upon
vi


Summary

introduction of internal void pressure, high pressure levels significantly reduce the
fracture resistance of the model. The effect of void pressure is seen to promote void
growth and pre-softening to the cell, thereby resulting in a lower work of separation in
rupturing a cell during crack advancement. Combined effect from high internal void
pressure and restricted plastic dissipation from highly constrained crack is shown to
greatly escalate cell damage and is extremely detrimental to the stability of the system.

Investigation next proceeds to discuss on the different fracture modes in a constraint
layer system. The model consists of a centerline crack in a thin ductile layer, which is
sandwiched between two rigid substrates. Three competing void interaction
mechanisms are demonstrated in this study, namely: (i) near-tip void growth
interactions, (ii) large scale cavitation spanning to a distance of several layer thickness
from the crack tip, and (iii) voids cavitation at site of highest triaxialities ahead of the
current crack tip. Findings show that presence of void pressure significantly lowers the
overall fracture toughness by diminishing the material’s work of separation, while
having negligible effect on the plastic dissipation around the crack tip. Therefore the
size of the fracture process zone remained relatively unaffected under the different
pressure levels. Hence void pressure does not have much influence on the void
interaction mechanism of the growing crack.

However, varying the elastic modulus mismatch between the substrates and ductile
layer shows that a smaller modulus mismatch promotes the mechanism of near-tip void
growth. But when higher mismatch values are imposed, large constraint on the
deformation within the layer caused the failure mechanism to shift into the second
mechanism of large-scale multi-void cavitation. Likewise, when a large negative Tstress value is applied, results show that void cavitation is initiated at distances in the

vii


Summary

order of the layer’s thickness ahead of the current crack tip, thereby forming a new
crack front. This corresponds to the third void interaction mechanism described above.

The next two case studies presented in the report focus on the stress-strain behaviour
of polymeric matrix composites, particularly pertaining to those used in IC packages.
Analysis is first conducted on a 2-D plane strain model of fiber-reinforced composite

where the sole failure mechanism is reinforcement-matrix decohesion. Stress contour
plots under uniaxial loading indicate that stress carrying capacity of a composite
relates more or less proportionately to the extent of void growth damage at the
interface. Peak stress carrying capacity is attained when approximately half of the
interfacial surface area becomes severely softened by void growth. At the same time, a
45° shear band develops fully across the cell diagonal when peak tensile strength is
reached. Furthermore, higher values of both interfacial porosity and internal void
pressure are observed to reduce the composite’s stress carrying capacity and tensile
strength. They also cause macroscopic yielding to initiate earlier, especially so under
the influence of internal void pressure.

The framework is then extended to a full 3-D study on multi-axial loading states on a
spherical particle-reinforced polymeric matrix composite, with a Gurson damage
constitutive model lining the reinforcement-matrix interface. Under an imperfect
interface susceptible to void growth and decohesion, the stress strain behaviour of the
composite is seen to exhibit macroscopic yielding followed by strain hardening phase
before attaining a maximum stress peak. Beyond peak stress, macroscopic softening
sets in, similar to the response found in the previous plane strain fiber-matrix model.
Effects of internal void pressure combined with an interface of high porosity again
prove to greatly erode the stress carrying capacity and tensile strength of the

viii


Summary

composite. High triaxial loading like the equi-triaxiality, results in sudden “brittle”-like
load release due to occurrence of massive interfacial voids cavitation upon reaching a
critical strain. Tensile behaviour also displays a distinctive dual-peak profile due to the
subsequent strain hardening effects from the matrix after stress redistribution from the

interfacial decohesion.

ix


List of Figures

List of Figures

Figure 3.1: Schematic of plane strain SSY crack growth model.
Figure 3.2: Finite element mesh for small scale analysis: (a) Entire mesh showing
remote boundary, (b) Refined mesh of inner region, (c) Gurson cells
ahead of crack tip with size D/2 under half-plane symmetry.
Figure 3.3: Crack growth resistance curve for four values of T/σ0; with f0 = 0.01, σ0/E
= 0.002, p0/σ0 = 0.0.
Figure 3.4: Crack growth resistance curve for four values of T/σ0 for (a) f0 = 0.03; (b)
f0 = 0.05; with σ0/E = 0.002, p0/σ0 = 0.0.
Figure 3.5: Distribution of mean stress ahead of crack for three different T-stress
levels for (a) ∆a/D = 2, (b) ∆a/D = 15; with f0 = 0.01, σ0/E = 0.002, p0/σ0
= 0.0.
Figure 3.6: Porosity distribution ahead of crack for three different T-stress levels at
∆a/D = 2 and 15; with f0 = 0.01, σ0/E = 0.002, p0/σ0 = 0.0.
Figure 3.7: Distribution of mean stress ahead of crack for four different stages of
crack growth for (a) T/σ0 = -0.5, (b) T/σ0 = 0.0, (c) T/σ0 = 0.5; with f0 =
0.01, σ0/E = 0.002, p0/σ0 = 0.0.
Figure 3.8: Crack growth resistance curve for four values of T/σ0 for (a) p0/σ0 = 1.0,
(b) p0/σ0 = 0.0; with f0 = 0.01, σ0/E = 0.002.
Figure 3.9: Crack growth resistance curve for four values of p0/σ0 for (a) T/σ0 = -0.5,
(b) T/σ0 = 0.0, (c) T/σ0 = 0.5; with f0 = 0.01, σ0/E = 0.002.
Figure 3.10: Distribution of mean stress ahead of crack for three different T-stress

levels for (a) ∆a/D = 2, (b) ∆a/D = 15; with f0 = 0.01, σ0/E = 0.002, p0/σ0
= 1.0.
Figure 3.11: Distribution of mean stress ahead of crack for four different stages of
crack growth for (a) T/σ0 = -0.5, (b) T/σ0 = 0.0, (c) T/σ0 = 0.5; with f0 =
0.01, σ0/E = 0.002, p0/σ0 = 1.0.
Figure 4.1: Schematic of constrained ductile layer model.

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List of Figures

Figure 4.2: Finite element mesh for small scale analysis: (a) Entire mesh showing
remote boundary, (b) Refined mesh of inner region, (c) Gurson cells
ahead of crack tip with length D/2 under symmetry
Figure 4.3: Crack growth resistance curve for different values of p0/σ0 for (a) σ0/E =
0.01, (b) σ0/E = 0.004.
Figure 4.4: Distribution of mean stress for (a) p0/σ0 = 0, (b) p0/σ0 = 1; and porosity
ahead of crack for (c) p0/σ0 = 0, (d) p0/σ0 = 1, for the case of σ0/E = 0.01
Figure 4.5: Crack growth resistance curve for different values of Es/Ε; with σ0/E =
0.01.
Figure 4.6: Distribution of porosity ahead of crack for the case of σ0/E = 0.01 at Es/E
= 1, 7, 10 and 20.
Figure 4.7: Distribution of mean stress ahead of crack for the case of σ0/E = 0.01 at
Es/E = 1, 7, 10 and 20.
Figure 4.8: Crack growth resistance curve for four values of T/σ0 with σ0/E = 0.01
and Es/E = 10.
Figure 4.9: Distribution of porosity ahead of crack for the case of σ0/E = 0.01 and
Es/E = 10 at T/σ0 = 0, 5, -2 and -5.
Figure 4.10: Distribution of mean stress ahead of crack for the case of σ0/E = 0.01 and

Es/E = 10 at T/σ0 = 0, 5, -2 and -5.
Figure 4.11: Plastic zone profile of accumulated plastic strain ε p in the ductile layer at
different Τ-stress values at ∆a/D = 20 for (a) − (c) and X1/D = 8 for (d);
with σ0/E = 0.01 and Es/E = 10.
Figure 5.1: (a) Periodic array of transversely aligned fibers; (b) Schematics of unit
model.
Figure 5.2: (a) Finite element mesh for C0 = 5%; (b) Closed up view showing
interface elements.
Figure 5.3: Plot of normalized mean stress vs. mean strain for various D/R0 values
under (a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, f0 = 0.05,
p0/σ0 = 0.
Figure 5.4: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00151; (b) εm
= 0.00351; (c) εm = 0.0055 at different D/R0 values for uniaxial loading
corresponding to Figure 5.3(a).
Figure 5.5: Plot of normalized effective stress vs. effective strain for various D/R0
values under (a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, f0 =
0.05, p0/σ0 = 0.
xi


List of Figures

Figure 5.6: Contour plots of normalized effective stress σe/σ0 at εe = 0.00766 at
different D/R0 values for uniaxial loading corresponding to Figure 5.5(a).
Figure 5.7: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00346; (b) εm
= 0.00637 at different D/R0 values for biaxial loading corresponding to
Figure 5.3(b).
Figure 5.8: Contour plots of normalized effective stress σe/σ0 at (a) εe = 0.00336; (b)
εe = 0.00508 at different D/R0 values for biaxial loading corresponding to
Figure 5.5(b).

Figure 5.9: Plot of normalized mean stress vs. mean strain for various f0 values under
(a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, D/R0 = 0.0785,
p0/σ0 = 0.
Figure 5.10: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00251; (b) εm
= 0.00575 at different f0 values for uniaxial loading corresponding to
Figure 5.9(a).
Figure 5.11: Plot of normalized effective stress vs. effective strain for various f0 values
under (a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, D/R0 =
0.0785, p0/σ0 = 0.
Figure 5.12: Contour plots of normalized effective stress σe/σ0 at (a) εe = 0.00802 at
different f0 values for uniaxial loading corresponding to Figure 5.11(a).
Figure 5.13: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00386; (b) εm
= 0.00551 at different f0 values for biaxial loading corresponding to
Figure 5.9(b).
Figure 5.14: Plot of normalized mean stress vs. mean strain for various p0/σ0 values
under (a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, f0 = 0.05,
D/R0 = 0.0785.
Figure 5.15: Plot of normalized effective stress vs. effective strain for various p0/σ0
values under (a) uniaxial loading; (b) biaxial loading; with C0 = 0.1, D/R0
= 0.0785, p0/σ0 = 0.
Figure 5.16: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00201; (b) εm
= 0.00501 at different f0 values for uniaxial loading corresponding to
Figure 14(a).
Figure 5.17: Contour plots of normalized mean stress σm/σ0 at (a) εm = 0.00253; (b) εm
= 0.00501 at different f0 values for biaxial loading corresponding to
Figure 14(b).
Figure 6.1: (a) SEM pictures showing evidence of particle-matrix decohesion in
polyamide 6 reinforced with 25 vol % glass beads [48]; (b) Periodic array

xii

List of Figures

of spherical particles; (c) Symmetry configuration; (d) Schematics of unit
model.
Figure 6.2: (a) Finite element mesh for C0 = 10%; (b) Closed up view showing
interface elements.
Figure 6.3: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain for various σ0/E values under uniaxial
loading.
Figure 6.4: Plot of normalized effective stress vs. effective strain for various N values
under uniaxial loading.
Figure 6.5: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain for various C0 values under uniaxial
loading.
Figure 6.6: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various f0 values under uniaxial
loading, with C0 = 0.1, D/R0 = 0.0785, p0/σ0 = 0.
Figure 6.7: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various f0 values under equi-biaxial
loading, with C0 = 0.1, D/R0 = 0.0785, p0/σ0 = 0.
Figure 6.8: Plot of normalized mean stress vs. mean strain, for various f0 values under
equi-triaxial loading, with C0 = 0.1, D/R0 = 0.0785, p0/σ0 = 0.
Figure 6.9: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various p0/σ0 values for uniaxial
loading, with C0 = 0.1, D/R0 = 0.0785, f0 = 0.05.
Figure 6.10: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various p0/σ0 values for equibiaxial loading, with C0 = 0.1, D/R0 = 0.0785, f0 = 0.05.
Figure 6.11: Plot of normalized mean stress vs. mean strain, for various p0/σ0 values

for equi-triaxial loading, with C0 = 0.1, D/R0 = 0.0785, f0 = 0.05.
Figure 6.12: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various p0/σ0 values for uniaxial
loading, with C0 = 0.1, D/R0 = 0.0785, f0 = 0.15.
Figure 6.13: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various C0 values for uniaxial
loading, with D/R0 = 0.0785, f0 = 0.05, p0/σ0 = 0.
Figure 6.14: Plot of (a) normalized mean stress vs. mean strain; (b) normalized
effective stress vs. effective strain, for various C0 values for equi-biaxial
loading, with D/R0 = 0.0785, f0 = 0.05, p0/σ0 = 0.
xiii


List of Figures

Figure 6.15: (a) SEM picture of polyamide 6 (PA6) reinforced with 25 vol % of
untreated glass beads (arrows indicate (d) debonded bead and (b) bonded
bead) [48]; (b) Numerical predictions vs. experimental results [48] for
unreinforced PA6 matrix, 25 vol % treated (T) glass beads and 25% vol
% untreated (NT) glass beads reinforcement.

xiv


Introduction

Chapter 1
Introduction

The drive towards high performance composites and recent advancement in the

semiconductor industry has propelled the development toward miniaturization of
multilayer structures. In order to ensure that the desired mechanical properties are
attained, such as layers adhesion, sheet resistance, surface uniformity, tensile strength
etc., multilayer systems require stringent quality control of the ceramics, intermetallics
and ductile metallic alloys or polymeric materials constituents. Due to the complex
structures within the multilayers, together with the varied mechanical behaviour of the
material contents, failure modes in such system are highly complicated and diverse. In
fact, several competing mechanisms may occur within the multi-layer structures at the
same time depending on the loading conditions.

Ductile failure in metallic alloys and polymeric materials constituents of the multilayer
structures is commonly driven by void growth and coalescence. This forms the basis of
the current research work on which all investigations into the interplay of failure
mechanisms in different case models are built upon. Ductile fracture begins with the
nucleation of cavities from brittle cracking, or decohesion of inclusion, dispersoids or
any second-phase sites in the matrix. These cavities grow in size under the high triaxial
tension which causes plastic flow in the surrounding material. The intense local fields
generated by the void growth in turn nucleates other neighbouring potential voids

Page 1


Introduction

leading to the ductile crack growth process of nucleation, growth and coalescence.
Studies on the micro-mechanics of ductile fracture began as early as the 1980s [2;3]
and extend for a decade before the milestone work of Xia and Shih [87] introduces a
realistic mechanism-based failure approach for modeling ductile failure numerically.

Through years of development, the mechanisms of simple ductile crack growth have

been fairly well understood. However, the high deformation constraint within the
multilayer structure induces a variety of intriguing ductile fracture modes in the
integrated system, which are not normally observed in homogeneous matrix. Stresses
arising from residual, thermal or mechanical factors are common due to the large
mismatch in mechanical and thermal properties between the different material layers,
from stiff ceramic substrates to metallic thin film layers to polymeric matrix
composites. Thus plastic deformation within the ductile layers is constrained by the
surrounding elastic layers and high triaxialities usually develop. Such stresses are
detrimental to the integrity of the multilayer system as interfacial decohesion, film
peeling/buckling, and catastrophic rupture may result.

Inspiration for the present research work comes from industrial developments in the
field of electronic packaging. In integrated circuits (IC), the die chip is securely
bonded to the die pad by an adhesive commonly known as the die attach. After proper
mounting and connection of wire bonds from the chip to the I/O leads, the remaining
volume within the IC plastic encapsulation is filled with a moulding compound. The
structure of the IC package represents a small-scale multilayer structure with the
ductile die attach being a polymeric matrix composite (e.g. Ag particle-filled epoxy),
sandwiched between two stiff substrates of the silicon chip and die pad. Under storage
in humid ambient conditions, moisture is readily absorbed by the porous molding

Page 2


Introduction

compound as well as die attach of the electronic packages. When exposed to high
processing temperature during processing (e.g. reflow soldering), the absorbed
moisture rapidly vaporizes, forming microvoids filled with high internal vapour
pressure. The consequence is thus void pressure-assisted ductile fracture in the IC

packages, commonly known as “popcorn cracking”.

Motivated by this moisture-induced failure phenomenon, investigations presented in
this report seek to gain a deeper understanding into the micro-mechanics of ductile
crack growth induced by void pressure and crack-tip constraint. Investigation next
proceeds to study the different fracture modes in a constraint layer system. Research
also encompasses material studies on the failure of different polymeric matrix
composites commonly used in the electronic packaging industry. Due to the vast work
that has been done in the areas mentioned above, a detailed literature review is first
documented in Chapter 2, to give the reader a clear overview of the background
knowledge and relevant findings to date. Specific references will be made to
fundamental research works that are both closely related to, as well as those providing
the framework upon which some case studies are based in this report.

Damage parameters and material variables, e.g. void volume fraction and elastic
modulus, has been well researched by many in the field of ductile void growth
modeling. However, void pressure-assisted ductile crack growth remains relatively
new in this area. Thus investigations begin with a preliminary study in Chapter 3 to
understand the micro-mechanics behind the effect of void pressure on the fracture
toughness of a crack. A plane strain small-scale yielding model with a semi-infinite
crack under mode I loading sets the case study. The methodology of Xia and Shih [87]
is adopted, which uses computational voided cells for modeling ductile crack growth.

Page 3


Introduction

Such approach not only has a sound microstructural basis, but also provides a
convenient means in void pressure application. Progressive growth and interactions of

the voided cell elements is governed by an extended Gurson constitutive relation
incorporated with internal vapour pressure [34;36;70]. Near-tip stress field is simulated
by a two parameter J-T approach [4] so as to study effects of crack-tip constraints on
the system.

With better knowledge on the mechanisms of ductile crack growth triggered by void
pressure, Chapter 4 proceeds to apply the crack growth model to a constraint
multilayer system. In this study, a thin ductile polymeric layer is sandwiched between
two stiff elastic substrates in order to model the situation of constraint deformation
within the ductile layer, such as the die attach film of IC packages. A semi-infinite
centerline crack is introduced to the ductile layer and a layer of extended Gurson cell
elements again lined the crack front. Parameters such as internal void pressure, elastic
mismatch between the substrate and polymeric layer, and T-stress effects are
investigated. Through varying such parameters, different fracture modes and void
interactions induced by the constrained plastic dissipation within the ductile layer are
discussed.

Chapter 5 and Chapter 6 next emphasize on the fracture mechanics of polymeric
matrix composites, which are commonly used as moulding compounds and die attach
films in electronic packages. Studies from the two chapters aimed at capturing the void
nucleation phase via reinforcement-matrix interfacial decohesion and the subsequent
growth of the debonded cavity. Chapter 5 uses a two-dimensional periodic plane strain
model to simulate the stress-strain response of a fiber-reinforced polymeric matrix
composite, while Chapter 6 extends this framework to a full three-dimensional analysis

Page 4


Introduction

on spherical particle-reinforced composite. In both cases, reinforcements are
approximated as rigid while the matrix are assigned continuum polymeric properties.
In modeling the decohesion damage process zone at the reinforcement-matrix
interface, the extended Gurson model is similarly employed to simulate the growth of
voids and eventual lost of stress carrying capacity at the interface due to ligament
tearing. Studies on void vapour pressure effects are conducted at the interface because
the latter is usually porous and serves as a potential site for moisture to reside. Multiaxial loading is also applied on both models to analyze different deformation
characteristics.

Finally, the report concludes with a summary of all the important findings presented in
the various case studies. As mentioned previously, the research works discussed here
aim to provide a better understanding to the failure mechanisms found in multilayer
systems and composites, and is especially of high relevance to electronic IC packaging
industry. Through better knowledge of fracture toughness under the different
applications, failure predictions can be implemented accurately. In turn, preventive
measures can be aptly applied to design considerations, material selection as well as
processing methods. Under proper planning, the result is, of course, improved product
reliability.

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Chapter 2
Literature Review

2.1

Ductile Crack Growth under Small Scale Yielding

Ductile failure in metals is commonly driven by void growth and coalescence
mechanism. In an attempt to understand the link between microstructural variables and
continuum properties of the material to the measured macroscopic fracture resistance,
mechanism- based ductile fracture approaches has received a great of interest recently.
Early development in this field was contributed by Needleman [51] in his study on
decohesion of interfaces in metal matrix composite. Needleman developed a cohesive
zone interface model that is embedded within an elastic-plastic continuum to model
void nucleation from inclusion debonding and subsequent void growth. A similar
approach was adopted by Tvergaard and Hutchinson [78;80] where the fracture
process is represented in terms of a traction-separation law, specified on the crack
plane to model crack growth initiation and advance. The numerically computed crack
growth resistance depends primarily on two parameters: the work of separation per
unit area and the peak traction, which characterizes the fracture process and continuum
properties of materials.

However, cohesive zone models fall short of capturing the microstructural
phenomenon of void growth to coalescence and hence unable to investigate the

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characteristics of ductile metal porosity arising from inclusions or second-phase
dispersoids. In modeling transient crack growth by void growth and coalescence, the
milestone is probably set by Xia and Shih [87;88]. In their research, the fracture
process zone is represented by void containing elements, known as cell elements.
These cell elements, employed on the crack plane, are represented by the Gurson [36]
− Tvergaard [70] model which governs their hole growth and coalescence process as

the crack propagates. Xia and Shih’s proposed cell model provides a sound mechanism
basis for the void growth process and contains two important microstructurally-linked
parameters: the cell size and its initial void volume fraction. Studies based on the
porous ductile cell model is able to account for the effect of plastic straining on
fracture, due to the mechanism of void growth to coalescence and due to plastic strain
controlled nucleation of voids. By contrast, the traction–separation law of cohesive
zone model is purely stress dependent and does not allow for crack growth if the peak
stress specified by the law is too high compared to the initial yield stress. Xia and
Shih’s cell model methodology is further explored in Chapter 3 to incorporate the
effects of crack tip constraint as well as internal void vapour pressure within the
voided elements.

McClintock [47] first noted that a single parameter of J-value is insufficient to
characterize the near-tip stress and strain states of fully yielded crack geometries.
Reasons being non-hardening plane strain crack tip fields of fully yielded bodies are
not unique but exhibit varying levels of stress triaxiality depending on crack geometry.
In addition, near-tip deformation and hydrostatic stress is weakly coupled especially in
regions undergoing plastic deformation. It follows that crack tip constraint, which
characterizes the triaxial state of stress ahead of the crack, must be scaled by two
parameters that are effectively independent. Betegon and Hancock [4] together with
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Du and Hancock [14] proposed a J-T approach where in addition to the applied
amplitude of the asymptotic stress field J, a non-singular elastic T-stress, which acts
parallel to the crack plane, is correlated with finite size crack geometries under large
scale yielding. Another approach that received considerable attention is the J-Q
expansion of the plastic crack-tip fields by O’Dowd and Shih [55;56]. The first

parameter is characterized by the applied J and with an amplitude Q in the second
hydrostatic stress parameter. Several investigators have noted a one-to-one correlation
between T and Q under small and moderate scale yielding, and both parameters
provides a proper measure of crack-tip constraint imposed by different crack
geometries.

In Chapter 3 and Chapter 4, the former J-T approach is chosen for crack tip constraint
investigation, under small scale yielding (SSY) conditions, due to its ease of
implementation in the incremental displacement loading boundary conditions. Both
Tvergaard and Hutchinson [79], and Xia and Shih [87] investigated the effects of Tstress on the mode I SSY crack growth models in relations to their different
mechanism-based approach models. It is shown that the predicted T-stress dependence
of fracture toughness during crack growth is qualitatively similar to experimental
observations, even though the experiments go beyond small-scale yielding. Generally,
in the context of small scale yielding, it is found that J-dominance of the HRR-field at
crack tip is maintained for zero or a highly constraint state of positive T-stress. On the
contrary, low constraint situation of negative T-stress causes a loss of J-dominance
which also reflects a loss of triaxialities near the crack tip [4;14]. This implies that a
negative T-stress exhibits significant contribution of plasticity to the crack growth
resistance, while positive T-stress shows little contribution to fracture toughness
improvement.
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Another major focus of this report is the role of void vapour pressure in ductile porous
materials, which sparked many interests in the field of electronic packaging
application. Recent advances into “popcorn cracking” as a failure mechanism by
Galloway and Munamarty [23] and, Galloway and Miles [22] in electronic packages
stimulated the current study. Under humid ambient conditions, moisture is readily

absorbed by the porous hydroscopic moulding compounds used in electronic packages.
During a fabrication process called reflow soldering, the absorbed moisture absorbed
suddenly vaporizes under the high processing temperature. The trapped water vapour
subsequently generates numerous voids of high internal pressures in the ductile
moulding compound. This consequently actuates crack propagation to the outer surface
of the packaging in an attempt to release the pressure build-up, causing the familiar
“popping” of IC moulds.

In Chapter 3, an attempt is made to understand the interconnection between the process
of material separation involving hole growth and coalescence, together with the plastic
dissipation that occurs over a larger scale, and their contribution to the total work of
fracture. Void vapour pressure is also incorporated into the model for its applicability
in IC packaging industry, while effects of T-stress are studied for crack tip constraint
imposed by geometric or mechanical considerations.

2.2

Fracture Toughness of Constrained Ductile Layer

Failure analysis in multilayer structures is of considerable importance for its extensive
range of engineering applications, from multilayer protective coatings in the structural
industry to composite laminates used widely in the aircraft and automotive industries.
In recent development, advancement in the semiconductor industry has rapidly

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propelled the drive towards miniaturization of multilayer systems. Thin metallic or

polymeric films bonded between stiff elastic layers of ceramics are becoming
increasing common in applications such as integrated circuits, electronic packaging,
multilayer passivation etc. However, deformation in the ductile layer is constrained by
the stiff elastic adherends when loaded thermally or mechanically. This constraint
leads to stress triaxialities far exceeding the tensile flow strength of the sandwiched
ductile layer material. Regions of high stress concentration produce large component
of hydrostatic stress, causing the ductile layer to be exceptionally susceptible to plastic
cavitation at any inherent defects or crack tips. Hence a detailed analysis of the failure
mechanisms and crack growth resistance are necessary for constrained layer structures,
and this focus is undertaken in Chapter 4.

The hydrostatic stress elevation within the constrained ductile layer in turn has several
repercussions on the failure mechanisms within the ductile layer. Dalgleish et al. [11]
found that large-scaled plastic deformation may be induced during crack propagation,
with the plastic region at failure being much greater or at least comparable to the layer
thickness. In their work, the strength of a sandwiched system of Al2O3 substrates
bonded using thin Platinum layers is investigated. The measured strength levels are
interpreted by conducting elastic-plastic stress analysis in conjunction with weakest
link statistics. Mechanisms of crack propagation for similar joint system have also
been investigated in a sequence of experiments. In the strongly bonded Al/Al2O3
interfaces by Evans and Dalgleish [15], the weaker the Au/Al2O3 sandwich system by
Reimanis et al. [60], and even the Au/sapphire interfaces by Turner and Evans [67],
crack extension by both ductile interface fracture i.e. plastic void growth and brittle
interfacial debonding were captured.

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