Mechanism based modeling of ductile void growth failure in multilayer structures

2.1 Ductile Crack Growth under Small Scale Yielding 6 2.2 Fracture Toughness of Constrained Ductile Layer 9 2.3 Modeling Interfacial Decohesion in a Composite 13 2.4 Extended Gurson Mo

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

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

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

l 1 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

f E Critical void volume fraction in a Gurson cell to trigger element

extinction algorithm for the cell

q 1 ,q 2 Micromechanics factors introduced by Tvergaard in the

Gurson-Tvergaard model

p Void pressure in a Gurson cell

x 1 ,x 2 Horizontal and vertical datum in the SSY models

x 1 , x 2 , x 3 Cartesian axis directions for the periodic composite models

C 0 Initial reinforcement volume fraction in a composite

R 0 Initial radius of the fiber or spherical reinforcement

2.1 Ductile Crack Growth under Small Scale Yielding 6

2.2 Fracture Toughness of Constrained Ductile Layer 9

2.3 Modeling Interfacial Decohesion in a Composite 13

2.4 Extended Gurson Model Incorporating Vapour Pressure 17

4.3.1 Effects of Internal Void Vapour Pressure 54

4.3.2 Effects of Elastic Modulus Mismatch 58

CHAPTER 5

Two-Dimensional Modeling of Void-Induced Interfacial Decohesion in a

Fiber-Reinforced Polymeric Matrix Composite 72

5.3.1 Effect of Interface Damage Zone Size, D/R 0 78

5.3.2 Effect of Interfacial Cell Element Porosity, f 0 91

5.3.3 Effects of Void Vapour Pressure, p 0 /σ0 98

5.4 Discussion and Conclusion 104

CHAPTER 6

Three-Dimensional Modeling of Void-Induced Interfacial Decohesion in a

Spherical Particle-Reinforced Polymeric Matrix Composite 107

6.3.1 Perfect Particle-Matrix Interface 113

6.3.2 Imperfect Particle-Matrix Interface 118

6.3.3 Mean stress-Mean strain Response 119

6.3.4 Effect of Interfacial Cell Element Porosity, f 0 121

6.3.5 Effects of Void Vapour Pressure, p 0 /σ0 127

6.3.6 Effects of Particle Volume Fraction, C 0 133

6.4 Discussion and Conclusion 137

CHAPTER 7

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

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

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

T-stress value is applied, results show that void cavitation is initiated at distances in the

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

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

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 f 0 = 0.01, σ0 /E

= 0.002, p 0 /σ0 = 0.0

Figure 3.4: Crack growth resistance curve for four values of T/σ0 for (a) f 0 = 0.03; (b)

f 0 = 0.05; with σ0 /E = 0.002, p 0 /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 f 0 = 0.01, σ0 /E = 0.002, p 0 /σ0

= 0.0

Figure 3.6: Porosity distribution ahead of crack for three different T-stress levels at

∆a/D = 2 and 15; with f 0 = 0.01, σ0 /E = 0.002, p 0 /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 f 0 =

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 f 0 = 0.01, σ0 /E = 0.002, p 0 /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 f 0 =

0.01, σ0 /E = 0.002, p 0 /σ0 = 1.0

Figure 4.1: Schematic of constrained ductile layer model

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 p 0 /σ0 for (a) σ0 /E =

0.01, (b) σ0 /E = 0.004

Figure 4.4: Distribution of mean stress for (a) p 0 /σ0 = 0, (b) p 0 /σ0 = 1; and porosity

ahead of crack for (c) p 0 /σ0 = 0, (d) p 0 /σ0 = 1, for the case of σ0 /E = 0.01

Figure 4.5: Crack growth resistance curve for different values of E s /Ε; with σ0 /E =

Figure 4.11: Plastic zone profile of accumulated plastic strainε pin the ductile layer at

different Τ-stress values at ∆a/D = 20 for (a) − (c) and X1/D = 8 for (d);

Figure 5.3: Plot of normalized mean stress vs mean strain for various D/R 0 values

under (a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, f 0 = 0.05,

p 0 /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/R 0 values for uniaxial loading

corresponding to Figure 5.3(a)

Figure 5.5: Plot of normalized effective stress vs effective strain for various D/R 0

values under (a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, f 0 =

0.05, p 0 /σ0 = 0

Figure 5.6: Contour plots of normalized effective stress σe /σ0 at εe = 0.00766 at

different D/R 0 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/R 0 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/R 0 values for biaxial loading corresponding to

Figure 5.5(b)

Figure 5.9: Plot of normalized mean stress vs mean strain for various f 0 values under

(a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, D/R 0 = 0.0785,

p 0 /σ0 = 0

Figure 5.10: Contour plots of normalized mean stress σm /σ0 at (a) εm = 0.00251; (b) εm

= 0.00575 at different f 0 values for uniaxial loading corresponding to

Figure 5.9(a)

Figure 5.11: Plot of normalized effective stress vs effective strain for various f 0 values

under (a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, D/R 0 =

0.0785, p 0 /σ0 = 0

Figure 5.12: Contour plots of normalized effective stress σe /σ0 at (a) εe = 0.00802 at

different f 0 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 f 0 values for biaxial loading corresponding to

Figure 5.9(b)

Figure 5.14: Plot of normalized mean stress vs mean strain for various p 0 /σ0 values

under (a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, f 0 = 0.05,

D/R 0 = 0.0785

Figure 5.15: Plot of normalized effective stress vs effective strain for various p 0 /σ0

values under (a) uniaxial loading; (b) biaxial loading; with C 0 = 0.1, D/R 0

= 0.0785, p 0 /σ0 = 0

Figure 5.16: Contour plots of normalized mean stress σm /σ0 at (a) εm = 0.00201; (b) εm

= 0.00501 at different f 0 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 f 0 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

of spherical particles; (c) Symmetry configuration; (d) Schematics of unit

model

Figure 6.2: (a) Finite element mesh for C 0 = 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 f 0 values under uniaxial

loading, with C 0 = 0.1, D/R 0 = 0.0785, p 0 /σ0 = 0

Figure 6.7: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various f 0 values under equi-biaxial

loading, with C 0 = 0.1, D/R 0 = 0.0785, p 0 /σ0 = 0

Figure 6.8: Plot of normalized mean stress vs mean strain, for various f 0 values under

equi-triaxial loading, with C 0 = 0.1, D/R 0 = 0.0785, p 0 /σ0 = 0

Figure 6.9: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various p 0 /σ0 values for uniaxial

loading, with C 0 = 0.1, D/R 0 = 0.0785, f 0 = 0.05

Figure 6.10: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various p 0 /σ0 values for

equi-biaxial loading, with C 0 = 0.1, D/R 0 = 0.0785, f 0 = 0.05

Figure 6.11: Plot of normalized mean stress vs mean strain, for various p 0 /σ0 values

for equi-triaxial loading, with C 0 = 0.1, D/R 0 = 0.0785, f 0 = 0.05

Figure 6.12: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various p 0 /σ0 values for uniaxial

loading, with C 0 = 0.1, D/R 0 = 0.0785, f 0 = 0.15

Figure 6.13: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various C 0 values for uniaxial

loading, with D/R 0 = 0.0785, f 0 = 0.05, p 0 /σ0 = 0

Figure 6.14: Plot of (a) normalized mean stress vs mean strain; (b) normalized

effective stress vs effective strain, for various C 0 values for equi-biaxial

loading, with D/R 0 = 0.0785, f 0 = 0.05, p 0 /σ0 = 0

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

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

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

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

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

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

Multi-axial 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

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

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

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

T-stress 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

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

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

The above-mentioned experimental researches also documented an interesting

phenomenon of debonded patches occurring well ahead of the crack tip This suggests

that voids have been found to nucleate and grow at distances comparable to the order

of foil thickness ahead of the crack tip Interlayer constraint can thus cause the point of

maximum mean stress to occur at some distance ahead of an existing crack tip [38]

The implication is that voids in this region may nucleate and undergo cavitation

instability, leading to additional cracks being formed, whenever mechanism for crack

growth of current tip is suppressed He et al [38] also provided a theoretical analysis

to explain such interface cracking phenomena in constrained metal layers In another

important study on stationary crack in a constrained metal foil, Varias et al [86]

derived the predictive criteria for the three competing crack growth mechanisms: (i)

near-tip void growth and coalescence, (ii) large scale cavitation and (iii) interfacial

debonding at site of highest triaxialities ahead of the current crack tip The latter two

failure modes are induced high triaxiality, which take place over a distance of several

foil thickness ahead of the crack tip

In the area of mechanism-based failure modeling of crack growth in constrained metal

layers, Tvergaard and Hutchinson [78;80;82] successfully applied their cohesive zone

model for the fracture process A traction separation law was embedded along the

crack plane and the numerically computed crack growth resistance is characterized by

the work of separation per unit area and the peak traction Their work [80;82]

discussed extensively on the contribution of plastic deformation to the effective work

of fracture for a crack lying along one of the interfaces of a thin ductile layer joining

two elastic solids However, the model is not adequate for ductile crack growth lying

in the void by void crack advance regime [81] This is because the intense deformation

immediate to the tip amplifies the growth of the void at the tip above what it would

experience in the separation plane further away from the tip This same deformation

may bring about the nucleation of new voids and hence effectively lowering the peak

separation stress, which is held constant in the cohesive zone model

In accurately modeling transient crack growth behaviour by void growth and

coalescence, Xia and Shih’s porous ductile cell model [87] is able to more accurately

characterize the crack growth resistance for any void interaction regime This is

because the size of the fracture process zone is incorporated into the Gurson cell model

and in the order of the void spacing The advantage is that crack growth is computed

with a microstructural basis through the process of discrete incremental advances from

void to void The cell model is therefore able to capture the effect of intense near-tip

plastic deformation as well as any other void interaction mechanisms depicted by

Varias et al [86] Xia and Shih’s methodology is adopted in Chapter 4 to further

investigate the different failure mechanisms of a constrained ductile because the

voided cells present a good prospect for the incorporation of internal pressure

Presence of vapour pressure integrated with the constrained layer model provides a

good framework to study moisture-induced failure in multilayer structure of IC

packages, as discussed in the previous section

Effects of crack tip constraint on the near-tip stress and strain, under different

specimen geometries, on fracture toughness has attracted many research works over

the years Crack tip constraint can be implemented by means of a two-parameter

approach which characterizes the triaxial state of stress ahead of the crack in several

mode I geometries Betegon and Hancock [4] utilize a J-T approach where the

non-singular elastic T-stress, acting parallel to the crack plane, is correlated with finite size

crack geometries under large scale yielding Xia and Shih [87] and Tvergaard and

Hutchinsin [79] have similarly observed that crack tip constraint can significantly

affect the fracture toughness of ductile materials Fleck et al [20] and several other

researches [1;6;7], on the other hand, targeted another interesting phenomenon: the

directional stability of the propagation of the centerline crack in the constraint system

under the influence of T-stress Conventional fracture models with crack growth

mechanism embedded only in the horizontal crack plane are unable to accurately

simulate such situation Due to the complexity involved, directional crack propagation

will not be covered in Chapter 4

2.3 Modeling Interfacial Decohesion in a Composite

Reinforcements such as fibers, whiskers and particulates are commonly added to

metals or metallic alloys to enhance their mechanical properties Although

improvements to the elastic modulus, yield strength and ultimate tensile strength can

be achieved, the reinforcement on the other hand, usually leads to poor ductility and

low fracture toughness [87;34] Under ideal conditions, remote tensile loading results

in monotonically increasing tensile hydrostatic stresses within the composite Apparent

flow strength also rises with macroscopic strain and hence continuous strain hardening

is expected In practice, the interplay of different micromechanical failure processes

limits the composite to a maximum stress carrying capacity, beyond which softening

and eventual failure occurs Needleman et al [52] categorized the failure mechanisms

in metal matrix composites into three main groups: (i) debonding along the

matrix-reinforcement interface, (ii) ductile failure of the matrix material, and (iii) brittle

fracture of the hard reinforcements

Due to the drastic difference in mechanical properties between the matrix and

reinforcement, large hydrostatic stresses are induced by the constrained deformation of

the matrix in close vicinity to the reinforcement Llorca et al [46] argued that if the

cohesive bonds between the matrix and the reinforcement are weak, interfacial void

formation may be triggered On the other hand, when good fabrication conditions

promote a strong interface, the high stress triaxialities can in turn result in failure

within the ductile matrix Voids are usually nucleated at various potential matrix sites

such as intermetallic inclusions and dispersoids [18;87] and grow to coalescence,

ultimately forming a macro-crack This has been observed for various aluminum alloy

matrices reinforced with Al2O3 or SiC particles [10;46]

Brittle fracture of hard reinforcement phase such as whiskers and particulates is also

possible when a critical average tensile stress builds up within the reinforcement

[19;46;71;72] For certain specific situations, there can also be other controlling failure

mechanisms besides the above-mentioned ones An example is the contribution of

reinforcement distribution (e.g staggered arrangement) and/or for certain loading

conditions (e.g imposed hydrostatic pressures), which lead to intense strain

localization in the matrix at regions separating the brittle reinforcements [10;46;64]

Under such circumstances, shear failure of the matrix becomes the dominant

mechanism

Over the past decade, numerical analyses have proven that significant levels of tensile

hydrostatic stresses develop in the composite matrix as a consequence of constrained

deformation [9;10] Such stresses develop primarily at the interface between the matrix

and reinforcement where there exist a steep gradient in mechanical properties Perfect

periodic arrangements of continuous fibers or whiskers with aligned ends produce the

highest levels of hydrostatic stresses because of high constraints on the matrix plastic

flow If the reinforcements arrangement are staggered either periodically or randomly,

a significant reduction in the level of triaxiality is predicted by the finite element

analyses [10] High levels of triaxiality along the matrix-reinforcement interface make

it susceptible to debonding Interfacial decohesion together with fiber breakage result

in cavity formation and hence ductile fracture occurs at relatively small strains [72]

Thus ductility of metal matrix composites is often dictated by the bond strength

between the reinforcement and matrix

Tvergaard [69] studied the onset of failure by interfacial decohesion at the fiber ends,

and the reference therein have found good qualitative agreement between experimental

observations of initial void shapes and theoretical predictions based on a cohesive zone

model of the interface behaviour Further analyses have been carried out by Christman

et al [10], to study the effects of fiber volume fraction and fiber spacing on the initial

void formation at fiber ends In their work, influence of different reinforcement types

on the evolution of stress and strain field quantities in the matrix of the composite is

also investigated together with effects of particle clustering on the tensile properties

Most numerical studies employed the regular arrays of end-to-end fibers, whiskers or

particles In all the numerical models for fiber analysis, it has been assumed that fiber

orientations are parallel, which is most effective towards improving tensile properties

in the direction of fiber axes From the fabrication point of view, this is justified since

extrusion processing can lead to almost perfectly aligned fibers [28] In addition,

observations of matching pairs of fracture surfaces for Al-SiC composites in

microscopy also showed the occurrence of failure by debonding as well as by brittle

fracture of elongated SiC particles aligned with the tensile direction [50], hence further

supporting the parallel arrangement approximation

As described above, interfacial debonding is widely researched upon for metal matrix

composites In recent years, there has been growing interest on polymeric materials

and foams, fuelled by advancement in both composite and semiconductor industry

Thin polymeric composite films bonded between stiff elastic materials are becoming

increasingly common in applications such as integrated circuits, electronic packaging

and multilayer passivation Ag-filled epoxy, an adhesive to join the die to its die-pad in

IC packaging is a common example Fleck and co-workers [21] in their recent research

suggested constitutive relations for polymeric foams that are derived on the same

context as ductile metallic materials Guo and Cheng [33;34] developed an approach

along this line on an extended Gurson model [36;70] incorporating constitutive

relations for porous ductile polymeric material which absorbs moisture The advantage

of their model is closely related to the “popcorn” cracking problem [23] involving

rupture in plastic electronic packages due to pressure built up in the polymeric

adhesives and moulding composites Popcorning is a common phenomenon that occurs

during reflow soldering, when the package temperature is rapidly raised to about

220°C to 260°C Such values are near to the glass-transition temperature Tg of the

polymeric material, which can behave like elastic-plastic solids exhibiting extensive

ductility [29]

In Chapter 5, the debonding damage mechanics for a fiber-reinforced polymeric matrix

composite is studied A plane strain model of a periodic array of transversely-aligned

continuous fibers is considered while effects of transverse stresses around the

matrix-fiber interface are studied The advantage of plane strain model is that it is able to

provide good qualitatively indication of three-dimensional effects, without the need for

a computationally heavy full three-dimensional cell-model analysis [73] The study

offers insights into the debonding damage mechanism induced by high stress triaxiality

of interfacial constraints, stress relaxation associated with void growth at the

matrix-fiber interface, as well as the influence of void vapour pressure residing in the porous

interface

Most researches conducted on the different failure aspects of composites have

employed axisymmetric models in their analysis for a quick qualitative investigation

on three-dimensional trends However, axisymmetry is limited to uniaxial and the

axisymmetric triaxial loading conditions and is unable to capture plane strain, simple

shear and other multi-axial loading conditions [13] In an attempt to gather reliable

insights into ductile matrix failure of metal matrix composite, Tvergaard [73]

employed a full three-dimensional model for the whisker reinforced model and is able

to follow failure development until regions of final void coalescence have developed

With improved computer processing capability nowadays, 3-D numerical modeling is

gaining popularity Danielsson et al [13] similarly employed the 3-D unit model to

study multi-axial stress-strain behaviour of a polymeric matrix composite reinforced

with particles arranged in a body-centered cubic array In Chapter 6, the framework of

the plane strain model used in Chapter 5 is extended to a full three-dimensional model

to capture debonding damage mechanisms in a spherical particle-reinforced polymeric

matrix composite Effects of interface layer porosity, void vapour pressure and

reinforcement volume fraction on decohesion failure of reinforced polymeric matrix

are investigated

2.4 Extended Gurson Model Incorporating Vapour Pressure

In the approach of Xia and Shih [87] described in the earlier section, a porous ductile

material model is applied in a single row of equally-sized elements ahead of the initial

crack tip along the crack plane Progressive void growth and subsequent macroscopic

material softening in each cell element are governed by the Gurson’s [36] porous

plastic constitutive relations, modified by Tvergaard [70] for improved experimental

predictions The modified Gurson-Tvergaard model is used to represent the nucleation

of voids from the second phase particles and the subsequent growth of voids to

coalescence The analysis account for two populations of particles, (i) large inclusions

with low strength, which result in large voids near the crack tip at an early stage [87],

and (ii) small particles, which require large strains before cavities nucleate [88;90]

Guo and Cheng [33;34] extended the Gurson-Tvergaard continuum flow potential Φ to

take account of vapor pressure p present within the voids It has the form (Gurson [36];

Tvergaard [70]; Guo and Cheng [34]):

(1 ( ) ) 02

)(

3cosh

1 2

1

2

=+

e

σ

σσ

σ

where σe denotes the Mises stress, σm is mean stress and σM is an equivalent tensile

flow stress representing the actual microscopic stress state in the matrix, and f is the

current void volume fraction Factors q 1 and q 2 were introduced by Tvergaard [70] to

improve the model predictions for periodic arrays of cylindrical and spherical voids

The adjustment parameters are set to be q 1 = 1.25, q 2 = 1 Guo and Cheng showed that

the initial yielding imposes a constraint on the magnitude of possible initial vapour

pressure For general q-values, the constraint is p0 /σ0 ≤2ln(q1f0)/3q2

In addition to the inherent variables of σM and f in the Gurson flow potential, extended

form of Equation (2.1) introduces an additional variable p, the internal void vapour

pressure which acts as traction on the interior void surface The void vapour pressure

arises from vaporized moisture residing in the matrix material Being is a function of

temperature T and void volume fraction f, the relationship for fully vaporized moisture

is given by:

T e f

f f

f T

T p

0 0 0

where ∆T = T − T 0 is the temperature rise, f 0 is the initial void volume fraction, or

porosity, of the Gurson material, and α is the thermal expansion coefficient of the

material [34] (refer to Appendix A for detailed derivation) In all the case studies

presented in this report, isothermal conditions are assumed in order to minimize the

number of variables Thus Equation (2.2) reduces to:

0 0

1

f

f f

f p

p

−

−

Validation studies of the extended Gurson model (2.1) as well as its finite element

implementation have been carried out by Guo and Cheng [34] wherein several initial

vapour pressure levels were applied to a range of material parametric combinations

involving σ0 /E = 0.001, 0.01; N = 0, 0.1; f 0 = 0.01, 0.05

Most polymeric compounds used in IC packaging are hydroscopic and tend to absorb

airborne moisture into defects and voids present during storage The absorbed moisture

content vaporized into pockets of microvoids when subjected to high temperatures

manufacturing processing such as reflow soldering The extended Gurson model is

thus useful in studying a potential failure mechanism common to most polymeric

materials used in IC packaging: vapour pressure-induced ductile void growth failure

The additional tractions imposed by the p in Equation (2.1) is representative of the

vaporized moisture induced in microvoids in the polymeric compounds, e.g epoxy

adhesive, moulding compounds, etc Due to the increased hydrostatic stress field at

the void surface, internal void vapour pressure greatly promote the growth of the voids,

thereby resulting in severe damage to the voided matrix even before external load is

applied With higher levels of internal void pressure, void deformation becomes more

severe and thus leads to earlier coalescence with the neighbouring voids Ductile

failure by void growth and coalescence processes is therefore aggravated by the

presence of void pressure As will be seen in later case studies, internal void pressure

strongly influences the stress carrying capacity, tensile strength and fracture toughness

of a system The pressure ratios p 0 /σ0 chosen in this report for parametric study are

consistent with estimates of acceptable stress levels reported in the literature for

electronic package failure by cracking, including popcorn failure [34]

2.5 Numerical Implementation

The nonlinear, implicit finite element research code Warp3D [32] provides the

computational framework for the algorithms and analyses described in this report

Finite element solutions are generated for static nonlinear analysis of fracture models

constructed with three-dimensional (3-D), 8-node hexahedral elements Key features of

the code employed in this work include: (i) the Gurson-Tvergaard dilatant plasticity

constitutive model and Mises constitutive models implemented in a finite strain

formulation, (ii) element extinction using a linear traction-separation model to support

crack growth, (iii) automatic load step sizing based on the rate of damage

accumulation, and (iv) evaluation of the J-integral using a domain integral procedure

The software architecture of Warp3D supports the analysis of very large, 3-D fracture

models on shared and distributed memory parallel computers through explicit message

passing It also uses the so-called B-formulation which precludes mesh lock-ups that

arise as the deformation progresses into fully plastic, incompressible modes

Implementation of the nonlinear finite element method in Warp3D derives from force

equilibrium stated through the principle of virtual work Weak formulation of the

principle of virtual work expressed in the current configuration, denoted n+1, is given

by:

1

0d

1

n

where σn+1 denotes the Cauchy stress, P contains the external nodal forces acting on

the model at n+1 uδ defines virtual displacements at the nodes and δε represents the

symmetric rate of virtual deformation tensor relative to the current configuration [31]

Equation (2.4) involves significant nonlinearities arising from material response, large

geometry changes, and/or crack growth These nonlinearities, coupled with an implicit

time-integration scheme, necessitate the use of full Newton iterations for solution of a

computational load step In advancing the solution from n, each Newton iteration

employs the consistent tangent stiffness computed for the current estimate of the

solution at n+1 Evaluation of final increments of logarithmic strain over n to n+1 use

the linear strain-displacement matrix evaluated on the converged mid-increment

configuration, x n+1/2

For nonlinear analyses, the intensity of deformation along the crack front is generally

characterized by the Crack Tip Opening Displacement or a pointwise value of the

J-integral In two dimensions, the J-integral sets the amplitude of the singular field near

a sharp crack tip, as given by the HRR solutions (Rice and Rosengren [61]; Hutchinson

[42]) under certain limiting conditions involving material constitutive behavior and the

extent of plastic deformation relative to the uncracked ligament size Purely

mechanical arguments concerning the energy flux show that the J-integral provides a

local energy release rate independent of the exact singular form of the near tip fields

Crack growth resistance and fracture toughness is computed using the implicit domain

integral definition in Warp3D To characterize the intensity of far field loading on the

crack front, the code computes the mechanical energy release rate, J Moran and Shih

[49] defined the local value of J at a point s along a stationary crack front is by:

0 1

where Γ0 denotes a planar contour at crack front location s, which is defined on the

undeformed configuration at time t = 0 The contour begins at the bottom crack face

and ends on the top face n j is the outward normal to Γ0 and W denotes the stress-work

density per unit of the undeformed volume P ij and u i are Cartesian components of the

unsymmetric Piola-Kirchoff stress and the displacement in the crack front coordinate

system respectively

The finite element computations in Chapter 3 and 4 employ Warp3D domain integral

procedure for numerical evaluation of Equation (2.5) For crack fronts which

experience ductile growth, J is computed over domains defined well outside material

having the highly non-proportional histories of the near-tip fields and thus retains a

strong domain (path) independence Such J-values provide a convenient parameter to

characterize the average intensity of far field loading on the crack front It also serves

as a check that small-scale yielding conditions are obeyed by tallying the calculated

J-values with the prescribed analytical K-field J-values at the remote matrix boundary

The extended Gurson model (2.1) is also implemented through Warp3D The yield

function and related first-order derivatives in the Gurson material subroutine as well as

other related subroutines has been modified to incorporate the additional tractions

arising from internal void pressure Details of numerical implementation within the

framework of J 2 flow theory of plasticity are given in the Warp3D manual

However the Gurson-Tvergaard constitutive relation does not predict a realistic loss of

macroscopic stress in a cell at large void fractions, e.g f > 0.1 To overcome this

deficiency, an element extinction procedure (proposed by Tvergaard) is employed to

set the cell stiffness to zero when the averaged value of f at the Gauss points in a cell

element reaches a critical value of f E To place force release process of a newly extinct

element on a more physical basis, the remaining stresses are converted to equivalent

nodal forces, which are reduced to zero over subsequent load steps using a linear

traction-separation model The model controls the force release based upon additional

elongation of the cell normal to the crack plane after reaching f E For amounts of crack

growth many times the cell size, element extinction also has the benefit of removing

highly distorted elements from the model This greatly improves convergence of the

global Newton iterations Crack growth analyses for a moderate strength steel

demonstrate that critical porosity values (f E) between 0.1 and 0.2 show almost no effect

on predicted R-curves [31]

Chapter 3

Ductile Crack Growth under Small Scale Yielding

3.1 Introduction

Stresses in thin films and multilayer structures have several origins, namely thermal

mismatch, mechanical constraints and vapour-induced stresses [43] Together, they are

highly detrimental to the reliability and performance of the multilayer system

Eventual failure is a likely consequence, especially in the ductile components such as

multi-levels metallizations of integrated circuits or metal matrix composites Thus the

objective of this chapter is to gain some fundamental understanding of the interaction

between the mechanics of void growth process, the role of crack tip constraint, the

effect of internal void vapour pressure and their relative contribution to plastic

dissipation, as well as the total work of fracture In order to minimize the sheer volume

of generated results, thermal effects are not studied

Xia and Shih’s methodology is adopted here since their voided cell presents a good

prospect for the integration of internal pressure into the current study Sixty cell

elements of dimension D × D are placed ahead of the semi-infinite crack in a

horizontal plane An extended Gurson model by Guo and Cheng [33;34] is used to

describe the void growth of each cell elements which can be internally pressurized For

a preliminary study on the crack growth resistance under the various effects mentioned

in the above paragraph, the background matrix material is simplified to be of a

homogeneous elastic-plastic ductile material The semi-infinite matrix is modeled with

the remote boundary significantly larger than D in order for mode I small-scale

yielding conditions to be applicable Analysis includes: (i) crack growth resistance

under the various parametric studies, (ii) void interaction mechanism ahead of the

propagating crack, and (iii) stress distribution along the crack plane and its

implications on the fracture process zone size

3.2 Problem Formulation

For an elastic material in the absence of plasticity, Griffith criterion [30] for crack

growth states that:

where G is the energy release rate and Γ0 is the work of separation per unit area

required to create the new crack surface However, under small-scale yielding (SSY),

plastic yielding takes place actively around the crack tip region The crack growth

resistance Γ(∆a ) thus exceeds Γ0 and increases with crack propagation ∆a, until an

asymptotic toughness level of a steady-state value Γss is reached Γ(∆a ) depends not

only on the macroscopic plastic dissipation but on the microstructural mechanism of

crack propagation as well

In this chapter, plane strain mode I crack growth analysis for conditions of small-scale

yielding (SSY) is the focus The modified boundary layer (MBL) formulation provides

a convenient means to investigate transient crack growth effects The remote boundary