Implementation and Application of Gurtin

Strain Gradient Viscoplasticity

By

Prateek Nath

B.E., University of Pune, 1999

M.Tech., Indian Institute of Technology Bombay, 2001

A dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in the School of Engineering at Brown University

Providence, Rhode Island

May 2011

© Copyright 2011 by Prateek Nath

This dissertation by Prateek Nath is accepted in its present form

by the School of Engineering as satisfying the

dissertation requirement for the degree of Doctor of Philosophy.

Date________________ ________________________________

W. A. Curtin, Advisor

Recommended to the Graduate Council

Date________________ ________________________________

H. Gao, Reader

Date________________ ________________________________

S. Kumar, Reader

Approved by the Graduate Council

Date________________ ________________________________

Peter Weber, Dean of the Graduate School

iii

Curriculum Vitae

Prateek Nath was born on 14th of April, 1977 in the town of Bhagalpur, India. Prateek earned degree

of his bachelors of engineering from University of Pune with specialization in mechanical engineering in

1999 with distinction throughout the bachelors level. Prateek continued further studies to earn a masters

in technology at the Indian Institute of Technology Bombay in 2001. There he served as teaching assistant

in courses of FEM, optimization and as a research assistant. Prateek worked for GE Global Research at

their Bangalore facility as a mechanical engineer utilizing commercial FEM codes for various GE businesses

including GE Plastics, GE Medical Systems, GE Speciality Materials and earned green belt certification

in design for six sigma. Thereafter Prateek started his doctoral studies at Brown University in fall 2004

with major in solid mechanics and minors in material science and applied mathematics. Prateeks research

focussed on viscoplastic implementation of Gurtins strain gradient theory for crack tip problems, he had

exposure to crystal plasticity, discrete dislocation plasticity and cohesive zone modeling during this time.

Prateek did an internship with SIMULIA in summer 2010 towards implementing a crystal plasticity material

model and further extend it to polycrystalline models based on Taylor’s approach. Prateek looks forward to

join Oak Ridge National Lab at Oak Ridge, TN as a Postdoc to further investigate fretting fatigue problems

in engineering.

iv

Preface and acknowledgements

With training in mechanical engineering and design, I was working for GE Global Research at Bangalore,

India, using FEM for industrial research and development. I was fortunate to have exposure to various GE

businesses and projects in the area of polymers, electronic packaging, and medical devices. I experienced

the importance of constitutive modeling of materials first hand while doing FEM simulations and interacting

with experimentalists. This led to the desire to find a PhD program and many senior colleagues recommended

that I apply to Brown. Metallic material and plastic deformation are being used to new frontiers of strength

and deformation, and plastic deformation became fascinating for me. With interest in understanding plastic

deformation and damage it was natural to start working under the guidance of Prof. Curtin and Needleman.

I had a chance to learn about the Discrete Dislocation Plasticity (DDP) and cohesive zones, use commercial

FEM code ABAQUS and develop my own viscoplastic code for isotropic and single crystal plasticity during

initial phases of my research. The coursework focussing on mechanical behavior of material from prof.

Kumar, right when I started at Brown was extremely helpful to understand more about the dislocations which

are the underlying mechanisms of plastic deformation and DDP simulations. Through further courses in

solid mechanics, seminars, literature and conference, the importance of multiscale modeling and realistic

simulation of fracture and damage became my focus. As an engineer I also had a personal preference to

bring the understanding I gained at Brown university, to practicing engineers by making available simulation

method for plasticity which capture underlying dislocation mechanisms in a FEM framework for wider use.

This lead to my investigation of Gurtin’s theory and its implementation in a FEM framework.

For all the learning and experience I had at Brown University, I would like to express my deep thanks to

Prof. Curtin for his unceasing encouragement and support to help me progress in my research and further

v

to find suitable opportunity after my PhD program. I have learnt a lot from prof. Curtin and every time

amazed by his capabilities and overall approach. I also express my thanks to Prof. Needleman for being

in the advisors role initially and when Prof. Curtin was on sabbatical and to enrich my understanding of

important aspects of crystal plasticity and capturing mechanics via simple implementation of rate dependent

approach. I had a chance to learn through a many courses in solid mechanics from all the faculty, and also in

material science and applied mathematics. I am deeply thankful to the faculties for the framework they have

provide me for future. Over all these years at Brown, I have understood and now appreciate a key aspect of

solid mechanics group which is the collegial atmosphere and the group being a close knit community. The

collegial atmosphere is unparallel and has its signature on the research output and distinction widely known.

I am thankful that I have understood this.

I express thanks to friends I have made during my stay at Providence for the support they provided right

from arriving to providence. My wife Sangita has been extremely helpful and loving over the years while

also maintaining her productivity in biology research along with taking care of our infant daughter Sarojini

and keeping her cheerful and curious. My deep thanks goes to my parents for their desire to see me learn and

contribute.

I acknowledge financial support fromMRSEC for all these years and initial support from General Motors

and thankful for the research opportunity they provided.

vi

Contents

Preface and acknowledgements v

1 Introduction 1

2 Delamination of an Elastic Coating on plastically Deforming Substrate 15

2.1 Thin film delamination problem description . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Comparison of prediction using DDP and continuum plasticity . . . . . . . . . . . . . . . . 18

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Formulation of Gurtin Strain Gradient Viscoplasticity 22

3.1 Numerical implementations for strain gradient crystal plasticity . . . . . . . . . . . . . . . . 27

3.2 General solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Special solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Constrained Shear of Uniform Layer 35

5 Modeling Mode I Crack Tip Fields 52

5.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Results for crack problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Implication for fracture and limitations of Gurtin’s theory . . . . . . . . . . . . . . . . . . . 60

5.4 Comparison with previous strain gradient studies for crack tip . . . . . . . . . . . . . . . . 61

vii

6 Conclusion 71

viii

List of Figures

1.1 Single slip system α with two regions undergoing plastic deformation. Region b = 0 has no

net Burgers vector and standard crystal / continuum plasticity is applicable, region b != 0 has

net Burgers vector and gradients of plastic slip, this region is a candidate for gradient methods. 5

1.2 Simulation methods and strain gradient theories arranged according to length scale of use. For

strain gradient theories solid red line indicates strain gradient enhanced hardening, dotted line

indicates energetic hardening. Double arrows between simulation method indicate current

scale bridging methods available. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Elastic thin film on plastically deforming surface subjected to indentation. Substrate is mod-

eled by DDP and J2 isotropic theory. Interface is modeled by a cohesive zone. Process zone

is defined for DDP simulation. Figure from O’Day et al. (2006) . . . . . . . . . . . . . . . . 17

2.2 Comparison of continuum and DD predictions for normalized critical force for the onset of in-

terface delamination with: (a) Ef/Es =2 and (b) Ef/Es =6; work of separation φ =0.1875

J/m2 in all cases. Figure from O’Day et al. (2006) . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Normalized opening stress σ22/σ and dislocation snapshots in 6-8 mm region below indenter:

(a) prenucleation, and (b) post-nucleation. Dislocation structure captured in DDP cause local

stress enhancement and delamination.Thin film with Ef/Es =2, σ22/σ =2, tf =1 mm and

hmax/w 0.125. Figure from O’Day et al. (2006) . . . . . . . . . . . . . . . . . . . . . . . . 21

ix

4.1 Constrained shear problem. Two slip systems are oriented along π3 and

2π3 from x2, no plastic

slip is permitted at the boundary along x1 = L (top), and x1 =0 (bottom). Displacement rate

u2 is applied along the top and bottom boundary or traction rate τ is prescribed . . . . . . . 36

4.2 Normalized plastic slip distribution vs. position in a constrained shear problem as computed

using various implementations discussed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Plastic slip distribution vs. position in a constrained shear problem as computed using general

implementation, comparison between 1D stress driven and 2D implementation at RSS=0.992 46

4.4 Slip distribution evolution in constrained shear with increase in loading calculated by general

and special implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Stress vs. average shear strain curve illustrating hardening due to energetic length scale l.

Significant hardening is observed for length scale of l/L =1 but not for l/L =0.1 . . . . . . 48

4.6 Comparison of slip distribution for constrained shear between DDP and strain gradient plas-

ticity simulation with energetic hardening with plastic dissipation based on plastic slip. Plot

from Bittencourt et al. (2003). The DDP results are from Shu et al. (2001) . . . . . . . . . . 49

4.7 Schematic load vs. displacement curves for constrained shear with dissipative gradient hard-

ening (a) and energetic gradient hardening (b). Elastic plastic material is shown for reference.

Area under the load displacement curves corresponding to purely plastic dissipation in (a),

and plastic dissipation along with strain energy of plastic strain gradients in (b) as shown. . . 50

4.8 Schematic load vs. displacement curves for constrained shear with further release of constrain

of plastic slip with maintaining traction for the case of gradient enhanced dissipative hard-

ening. Elastic plastic material is shown for reference. For material with gradient enhanced

dissipative hardening (a), on release of constraint for plastic deformation no change in plastic

strain distribution happens due to lack of free energy of strain gradients. For material with

energetic hardening (b), on release of constraint for plastic deformation free energy of strain

gradients causes additional plastic deformation. . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Crack tip in single crystal subjected to Mode I loading by K1, the single crystal has two slip

systems are oriented along π/3 and 2π/3 with the crack plane . . . . . . . . . . . . . . . . 53

x

5.2 Coincident three node and ten node triangles are used for solution of crystal plasticity and

balance of plastic strain (3.19) respectively. Triangles are arranged in crossed triangles to

avoid numerical issues due to incompressibility of plastic strains . . . . . . . . . . . . . . . 54

5.3 Effect of length scale l on total slip along π3 , different contour scale, (a) l = 0, (b) l = 0.057,

and (c) l = 0.115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Effect of length scale l on slip rate along π3 , different contour scale, (a) l = 0, (b) l = 0.057,

and (c) l = 0.115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5 Effect of remote load KI on location of maximum plastic slip for l = 0.057 at plastic zone

size of 0.435 for the same contour scale. The location of maximum slip rate is closer to the

crack tip when the plastic zone size is 0.23 in (a), and further away from the crack tip when

the plastic zone size is 0.435 in (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.6 Effect of length scale l on opening stress along π3 , (a) l = 0, (b) l = 0.057, and (c) l = 0.115 59

5.7 Opening stress σ22 and opening traction vs. scaled distance from the crack tip calculated

using theory of Fleck and Hutchinson (1997) calculated for different gradient length scales.

Trends in opening stress σ22 and opening traction t2 are opposite with gradient length scale.

Figure from Wei and Hutchinson (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.8 effect of length scale parameters on hydrostatic stress using theory of Fleck and Hutchin-

son (2001) on Hydrostatic stress vs. scaled distance from crack tip showing. Figure from

Komaragiri et al. (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.9 Figure from Tang et al. (2004) showing lattice incompatibility (GNDs) existing in a small

region of 0.02 microns near the crack tip which arises due to a non-saturating hardening law. 67

5.10 variation of cohesive traction scaled by g0 ahead of the crack tip for different ko parameter to

scale hardness due to GND, NI isKo = 0. Figure from Tang et al. (2004) . . . . . . . . . . 68

5.11 Cumulative plastic slip on all slip systems, (a) with GND hardening, (b) with no GND hard-

ening. Slip is suppressed with GND hardening (a) but maximum slip increments are at the

crack tip. Figure from Tang et al. (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.12 variation of stresses ahead of the crack tip, using MSG theory. Figure from Jiang et al. (2001) 70

xi

Chapter 1

Introduction

Plastic deformation and fracture phenomena of polycrystalline metals is of great importance to engineering

application for reliability in design and service. Technological innovation towards the reduction in size of

components and better engineering, from the perspective of strength and deformation mechanisms, presents

us with a need to better understand deformation phenomena at sub-micron scales. Deformation at sub-micron

scales is very unlike macro-scale deformation of polycrystalline materials, i.e. aggregates of individual crys-

tals, which can be modeled using isotropic plasticity. At sub-micron scales, the anisotropy of deformation due

to individual grain crystal structure is critical, and is addressed using crystal plasticity. Crystal plasticity anal-

ysis was initiated by Taylor (1938) and the numerical implementation developed by Asaro and Needleman

(1985) is widely used in current engineering, for instance to understand the role of polycrystalline texture on

material deformation.

Moreover, recent experimental observations of deformation at micron and sub-micron scales reveal inter-

esting trends which are not found at the macro-scale. Experiments of indentation on crystals reported by Qing

and Clarke (1995), on plastic deformation of free standing thin films reported by Xiang and Vlassak (2006),

on tension and bending of films by Haque and Saif (2003), and in other results, e.g. Evans and Hutchin-

son (2009), reveal size-dependent phenomena at sub-micron scales. Size-dependent plasticity is outside the

scope of traditional models of both continuum isotropic and crystal plasticity. These size-dependent phenom-

ena happen when the size of the plastic deformation and dislocation patterns due to loading at sub-micron

depths approach material or geometric size scales arising from constraints on plastic deformation, geometry

1

such as film thickness, grain size, etc. The size effect has been attributed to the close spacing and spatial

variation in dislocation phenomena underlying the plastic deformation. With the increasing development in

MEMS technologies and nanoscale materials, it is becoming even more important to understand and predict

the deformation, damage, and fracture behavior at these small length scales by using plasticity models that

can demonstrate size effects.

Turning to theoretical developments, length scale effects in plastic deformation at sub-micron size scales

in single crystals have been successfully demonstrated by the Discrete Dislocation Plasticity (DDP) approach

as in Shu et al. (2001), H. Cleveringa et al. (1999), and Nicola et al. (2006). In contrast to conventional

plasticity and crystal plasticity approaches, the DDP approach models plastic flow via the motion of, inter-

actions among, numerous individual dislocations. DDP involves modeling includes a wide range of physical

features, including elastic interactions, dislocation-dislocation junctions, pile-ups and other dislocation struc-

turing, dislocation starvation, and dislocation limited sources, none of which are captured by conventionally

crystal plasticity. DDP approaches canmodel regions where no dislocation motion is permissible, and thereby

apply constraints on plastic deformation, a feature also outside the scope of conventional crystal plasticity.

The discrete dislocation approach includes length scale effects in plastic deformation in a natural way due to

the elastic interaction of dislocations and lower length scale constitutive relations. But, it has so far proven

difficult to associate the observed length scale effects with any one particular aspect of the comprehensive

dislocation-based model. In addition, DDP simulations are computationally expensive due to the need to ac-

count for the mutual interactions between all the dislocations in the system. Multipole methods as discussed

in Arsenlis et al. (2007) have been used to reduce the computational cost associated with the calculation of

interactions between dislocations, but attaining large plastic strains or studying problems under non-uniform

loading remains a challenge. Computational cost has also motivated the development of alternate modeling

multiscale approaches as in Wallin et al. (2008), but these methods are only now emerging.

Since DDP will not be used to solve engineering-scale problems, an alternate approach is to modify

conventional or crystal plasticity so as to capture size-dependent phenomena. This has been tackled mainly

through so-called strain gradient plasticity (SGP) approaches, which are amenable to FEM simulation at the

continuum scale and can incorporate prescribed boundary conditions on plastic deformation while capturing

2

length scale effects. We will discuss SGP further below.

An macroscale problem of huge engineering importance is crack growth or fracture. Fracture is inherently

a multiscale phenomenon, and crack tip fields can have high plastic strain gradients, which suggests inappli-

cability of conventional plasticity. Crack propagation has been simulated using a cohesive surface approach

due to Xu and Needleman (1993) and Tvergaard and Hutchinson (1992) in various engineering situations,

including separation of interface cracks for microelectronic applications by Tvergaard and W. (2009). But,

simulating crack growth using the cohesive surface approach in a materials with low work hardening has a

fundamental limitation first pointed out by Tvergaard and Hutchinson (1992). Specifically, the maximum

separation traction required to propagate a crack may be unattainable ahead of the crack tip if the material

surrounding the crack flows too easily and does not undergo significant strain hardening. Quantitatively, if

the cohesive strength σ is more than five times the material flow or yield strength σy , the traction ahead of

the crack tip cannot reach the level of the cohesive strength and cohesion is thus practically ruled out Wei and

Hutchinson (1997). Predictions for toughness are then very high and misleading for materials with a high

ratio of cohesive strength to yield strength, σ/σy . Such predictions can lead to designs which will not be

conservative, which is certainly not desirable.

The unrealistic prediction of crack propagation also arises for crack tip fields in single crystal for a non-

hardening material. For single crystals under asymptotic Mode I loading, the crack tip fields for stationary

and growing cracks in elastic-plastic crystals are given by Rice (1987). Such an asymptotic solution predicts

a uniformmaximum opening stress ahead of the crack tip to be constant, and thus rules out crack propagation

for any cohesive strength higher than this uniform opening stress. For single crystals with low strain harden-

ing, the flow strength with saturates at some flow strength σys leading to saturation of the maximum opening

traction and an inability to drive fracture.

DDP simulations resolve some of the issues with crack growth. For single crystals, the crack tip field

solutions obtained from DDP simulations in H. Cleveringa et al. (2000) and Wallin et al. (2008) agree with

the continuum solutions at some modest distance away from the crack tip. However, the stresses just ahead

of the crack tip differ from the asymptotic solutions, are highly influenced by local dislocation structure, and

can exceed the nominal flow stress. Fracture or crack growth is thus observed in the DDP using the cohesive

3

surface framework, and has been applied for realistic fracture prediction in materials with a high ratio of

cohesive strength to yield strength σ/σy in bi-materials by O’Day and Curtin (2005) and for coating delam-

ination by O’Day et al. (2006). The difference highlighted by O’Day et al. (2006) between the predictions

from continuum plasticity and DDP arises from the long range stress fields of individual dislocations forming

dislocation structures in DDP.

Continuum strain gradient plasticity approaches have also been used to predict higher opening stress

ahead of the crack tip by Wei and Hutchinson (1997) using the theory of Fleck and Hutchinson (1997),

by Komaragiri et al. (2008) for continuum plasticity using a gradient enhanced hardening theory of Fleck

and Hutchinson (2001), and by Tang et al. (2004) using gradient crystal plasticity involving non-saturating

gradient-enhanced hardening at the crack tip. In SGP models, the predictions of the elevation of stress ahead

of the crack tip are due to the gradient-enhanced non-saturating hardening mechanism, and not due to long

range stress fields of individual dislocations, an aspect that will be discussed in detail later. Crack propagation

with appropriate gradient enhanced crystal plasticity can be also extended to be used in a polycrystalline

structures for realistic crack propagation simulations.

In light of the successes of SGP theories for capturing some length scale effects, as noted above, we now

address in general terms how the strain gradient theories capture various aspects of plastic deformation due

to motion of dislocations. The kinematic aspects essential to a strain gradient theory are discussed first, then

the factors affecting motion of dislocations are discussed, and common ideas to approximate these factors

in various strain gradient theories are presented. Prominent SGP theories are discussed in some detail with

more discussion related to Gurtin’s strain gradient theory, which will be the basis for the work reported here.

Strain gradient plasticity accounts for spatial gradients of strain or plastic strain in addition to the usual

variables in a mixed displacement traction boundary value problem. Spatial gradients of plastic strain are

physically relevant because they are related to the net Burgers vector or, equivalently, the density of geo-

metrically necessary dislocations (GNDs). This is best understood using figure ( 1.1) which illustrates two

different regions with different net Burgers vector b, for a single slip system α, wheremα is the normal and

sα is the direction of slip for slip system α. The region characterized by b = 0 is a region where there is

no net Burgers vector, i.e. there are equal numbers of dislocations of opposite Burgers vector. The disloca-

4

Figure 1.1: Single slip system α with two regions undergoing plastic deformation. Region b = 0 has no netBurgers vector and standard crystal / continuum plasticity is applicable, region b != 0 has net Burgers vectorand gradients of plastic slip, this region is a candidate for gradient methods.

tion density in this region is also called the statistical stored density (SSD), and the evolution of the SSD is

responsible for normal plastic flow and strain hardening. For such a region, there is no gradient in plastic slip

along the slip direction α and the plastic behavior is well described by conventional crystal plasticity for a

single crystals and continuum plasticity theory for isotropic plasticity. The second region, characterized by

b != 0 has a predominance of dislocation of one sign of Burgers vector and so has a net Burgers vector. The

presence of a net Burgers vector implies the existence of geometrically-necessary dislocations (GNDs) that

accommodate a gradient in plastic slip along a slip direction α in this region. To characterize the net Burgers

vector in these regions (b != 0), strain gradient approaches use geometric relations due to Nye (1953) as in

5

(1.2). The Burgers’ is defined by

G = curl("up) (1.1)

G =∑

("γα) × mα ⊗ s

α. (1.2)

The burgers vector for a loop with a normal n is found by relation (1.3)

b = GTn. (1.3)

Dislocations in elastic medium have long range stress fields. Thus, the presence of net Burgers vector, b != 0

implying a dislocation structure, has long range stress fields. However absence of net Burgers vector, b = 0

at a location, implies no long range stress field of the dislocation structure. The motion of a dislocation

is influenced by stress field due to boundary conditions, body forces, and presence of other dislocations

structures with net Burgers vector. These stress fields cause a force on the dislocations for their motion.

Another important aspect of dislocation motion is resistance to its motion by existing dislocation structures,

this resistance is higher if a region has net Burgers vector for a level of plastic deformation. Thus dislocation

motion is influenced by presence of net Burgers vector in two ways; first in terms of force on a dislocation,

and second, enhanced resistance to dislocation motion. Neither of these effects are modeled in conventional

crystal plasticity. A typical viscoplastic formulation for conventional crystal plasticity models the plastic

strain rate as (1.4)

γα = a|τα

gα|msign(τα), (1.4)

where a is a reference slip rate, m, a large number, is the rate sensitivity exponent, and gα > 0 is the slip

resistance on the slip plane. The slip resistances gα collectively give rise to the macroscopic flow or yield

stress σy . The slip rate γα at any location depends on the local resolved shear stress τα causing plastic

flow and the local flow resistance gα through a power law relationship. Hardening, both self- and latent-, is

captured through evolution laws forgα. Nothing in the above framework considers or accounts for any effects

of dislocation structures with a net Burgers vector on the resolved shear stress τα, the slip resistances gα, or

the hardening.

6

Using appropriate strain gradient measures which can be related to the GNDs, various strain gradient

theories introduce the effect of strain gradients to approximate the underlying dislocation structures. The

important aspects of dislocation motion which strain gradient theories seek to capture are the enhanced forest

hardening and the long-range stress fields due to net Burgers vector, but in a local manner. The approaches

common in various strain gradient theories to incorporating effect of net Burgers vector are in enhancing the

slip resistance and its hardening, gα, and in replacing the resolved shear stress τα by a new stress πα that is

the stress causing plastic flow. The basis of incorporating strain gradient effect in the flow resistance gα is

motivated by the Taylor hardening model. In the Taylor hardening model (1.5), the flow resistance g depends

on the dislocation density ρs which corresponds to the statistical dislocation density as in the b = 0 region

in figure ( 1.1).

g ∼ Gb√ρs (1.5)

With net Burgers vector b != 0 characterized by kinematic considerations, the flow strength gα in the Taylor

hardeningmodel can be modified to include the GND dependence on hardening through the GND density ρg.

During plastic deformation, the increase in flow resistance due to the presence and increase of GNDs causes

the material to offer increased resistance to plastic deformation, which is manifest as a hardening effect. As

more power is dissipated with increasing plastic deformation and hardening, the strain gradient influence on

hardening is a dissipative effect. This is discussed in more detail below. Other theories attribute additional

free energy to the elastic and plastic strain gradients. Associating a free energy to plastic strain gradients

is a means to approximate the long range elastic interaction of dislocations in an approximate local manner.

Gradients of plastic strain effect can either suppress or enhance the driving stress πα relative to the resolved

shear stress τα. Attribution of an additive free energy to plastic strain gradients is an important feature of

the Gurtin theory. Although the resulting equations are similar to a theory of Kuroda and Tvergaard (2008),

they differ in the thermodynamic motivation as discussed later. Gradients of elastic strain can have a similar

effect of increasing the free energy, as in the strain gradient theories of Fleck and Hutchinson (1997), and

MSG theory due to Gao et al. (1999), and Huang et al. (2000). Elastic strain gradients appear because these

theories involve gradients of total strain in a coupled stress framework. However, the effects of elastic strain

gradients in these theories are deemed to be minimal as elastic strain gradients are expected to be negligible

7

compared to the plastic strain gradients.

Incorporating strain gradient effects to enhance hardening gα or to change the stress causing plastic flow

πα may not be a straightforward extension to the crystal/continuum plasticity description. Incorporating

these strain gradient ideas may require that the rate of plastic slip be calculated from a solution of a partial

differential equation (PDE), with appropriate boundary conditions, instead of a simple point-wise relation as

in (1.4). This is discussed in more detail later. The consideration of boundary conditions for the flow rate γα

from the solution of a PDE also allows prescription of boundary conditions on plastic strain, which can arise

physically and in DDP.

For isotropic plasticity, a prominent class of theories incorporating the idea of enhanced Taylor hardening

due to GND are due to Nix and Gao (1998), Gao et al. (1999), Huang et al. (2000) and Huang et al. (2004).

The theory of Nix and Gao (1998) is based on indentation experiments where the GNDs are related to an

elevation in hardness using a length scale based on material parameters. This inspired later theories by Gao

et al. (1999) and Huang et al. (2000) which is a multiscale theory. For this theory of Gao et al. (1999) and

Huang et al. (2000), the microscale Taylor model of hardening based on a summation of SSD and GND is

used, while in the mesoscale model linear variation in the strain fields is permitted. The micro and macro

scale are linked by a plastic work inequality involving the gradients of plastic strain and higher order traction.

The plastic strain gradient is decomposed into geometrical dislocation density configurations by four com-

binations of plastic deformation modes instead of using invariants of the plastic strain gradient tensor. This

theory thus allows for higher-order traction boundary conditions. Huang et al. (2004) present a simplified

theory derived from Gao et al. (1999) and Huang et al. (2000) in which the spatial plastic strain gradients

enter the constitutive relations. The quadratic invariant of the spatial plastic strain gradient ηp affects the

current flow stress, reducing the effective plastic strain rate via a GND-modified Taylor hardening relation.

εp = εo[σε

σy

√

f2(εp) + lηp]m (1.6)

where the parameter l is introduced as a material length scale. No higher order boundary traction boundary

conditions and boundary conditions of plastic strain allowed in this simplified theory of Huang et al. (2004).

The phenomenological idea of accounting for power dissipation due to the motion of SSDs and GNDs is

8

the focus of another class of theories proposed by Fleck and Hutchinson (1997) and Fleck and Hutchinson

(2001) that aim to generalize conventional isotropic J2 plasticity theory using Fleck and Hutchinson (1997)

invariants of strain gradient tensor η(I) or Fleck and Hutchinson (2001) invariants of the plastic strain gradient

tensor η(I)p, and allowing for more than one length scale l(I) that can be derived by fitting to different

experimental observations. For Fleck and Hutchinson (1997), the plastic work rate Up is modified by the

presence of strain gradients as (1.7)-(1.9)

Up = Σεp, (1.7)

where Σ is the effective stress as in (1.8), and εp is the effective plastic strain rate as in (1.9). The effective

stress is

Σ2 = σ2e +

3∑

I=1

[(τ (I)e

l(I))2], (1.8)

where τ (I)e is effective higher-order stress associated with the elastic strain gradient tensor ηe(I), and the

effective strain is

εp =

√

√

√

√

2

3εpij ε

pij +

3∑

I=1

[l2(I)η(I)pijk η(I)p

ijk ]. (1.9)

Since the gradients of elastic strain are expected to be smaller than the gradients of plastic strain, the energetic

effect due to the elastic strain gradients is again deemed to be small. The primary effect of strain gradients

thus is to increase the plastic work rate Up as in (1.7). These theories of Fleck and Hutchinson (1997)

and Fleck and Hutchinson (2001) allow for prescribing higher-order traction boundary conditions, and Fleck

and Hutchinson (2001) allows for boundary conditions on plastic strain. For a detailed comparison of these

prominent class of strain gradient theories for isotropic material the reader is referred to Evans and Hutchinson

(2009).

Strain gradient plasticity theories for single crystals can be divided into two categories: non-work-

conjugate theories and work-conjugate theories. The simplest approaches of non-work-conjugate theories

assume enhanced hardening due to GND as in Acharya and Bassani (2000), J. Bassani (2001), and Han et al.

(2005). In such theories the statement of virtual work involves displacement and surface traction, but no

plastic slip or higher order traction. Apart from calculation of GNDs and the associated enhanced hardening

in flow strength g, the implementation of such theories is similar to conventional crystal plasticity. For exam-

9

ple, Tang et al. (2004) study a crack tip using the theory of Acharya and Bassani (2000) which involves an

additional non-saturating dissipative hardening contribution due to GNDs. Specifically, the GND parameter

λα, which is proportional to GNDs, introduces an additional rate of hardening gg as

gg ∼∑

α

λα|γα|. (1.10)

However, the energetic interactions between dislocations are not modeled in this theory. Practical situations

can arise in which the plastic flow on some slip system is restricted at a material boundary. Imposing a

constraint on plastic slip, in such situation is not obvious in the framework of non-work-conjugate theories

since plastic slip is not a field quantity directly involved in the statement of virtual work.

A subclass of non-work-conjugate gradient theories has been introduced by Levkovitch and Svendsen

(2006), Yefimov and Giessen (2005), and Kuroda and Tvergaard (2006). Here, the flow stress πα is modified

by including terms involving the slip gradients, and in some cases these flow rule for such theories can be

equivalent to flow rules from work-conjugate theories as mentioned in Kuroda and Tvergaard (2006). The

flow rule in Kuroda and Tvergaard (2006) is

γα = a0sgn(τα − ταb )(|τα − ταb |

gα)m, (1.11)

where ταb is the back stress, and is a function of the gradient of net Burgers vector density. It modifies

the stress which causes the plastic flow and is the resolved shear stress τα only in absence of local net

Burger’s vector. The motivation to introduce the back stress ταb is to approximate the elastic interactions of

dislocations in a local manner which is energetic in nature. The flow rule (1.11) agrees with flow rule in

Gurtin’s work conjugate theory (see below) which is grounded in thermodynamics, but such agreement is

not expected in general. Not all plastic deformations involve boundary conditions of nonzero higher order

traction. For situations involving boundary conditions of plastic slip along with no higher order traction,

work-conjugate and non-work-conjugate theories can be equivalent as discussed by Kuroda and Tvergaard

(2008). For numerical implementation of these subclass of non-work-conjugate theories, additional boundary

conditions on plastic flow are imposed through local GNDs to restrict plastic flow. The solution procedure

10

in Kuroda and Tvergaard (2008) remains similar to conventional crystal plasticity but includes calculations

of slip gradients from GNDs due to imposed plastic BC’s and modification to the flow stress from resulting

back stress slip gradients. A similar approach for applying boundary conditions to strain gradient isotropic

plasticity was discussed by Acharya et al. (2004) for a non-work-conjugate theory.

Work-conjugate theories can introduce a free energy contribution associated with the plastic strain gra-

dients in addition to the elastic free energy, as typified by Gurtin (2002), M. Gurtin and Needleman (2005)

and M. Gurtin et al. (2007). Gurtin proposed that the classical free energy ψ be augmented by an additional

defect energyΨ(G) due to the GND densities as

ψ =1

2εe : C : εe + Ψ(G). (1.12)

The work-conjugate theory of Borg (2007) also involves a higher-order traction that is work conjugate to the

plastic slip gradients and influences hardening. In Gurtin’s model, the work conjugate to the Burgers tensor

G is a higher-order stress T defined as (1.13)

T =∂Ψ(G)

∂G. (1.13)

The work conjugate to the gradient of plastic slip ξα is the micro-stress, which is related to the higher-order

stress T by (1.14)

ξα = mα × Ts

α. (1.14)

The flow rule in work-conjugate theories is a partial differential equation that requires boundary conditions

of plastic strain and of a higher-order traction Ξα(n) (1.15), given by

πα − τα − divξα = 0, n · ξα = Ξα(n). (1.15)

Augmenting the classical free energy ψ by the defect energy Ψ(G) in (1.12) is a unique feature of Gurtin’s

model. The defect energyΨ(G) influences the flow rule (1.15) and tries to capture the energetic interaction of

adjacent regions of net Burger’s vector in a local manner. The additional effect of gradient-enhanced Taylor

11

hardening is also possible in this type of theory.

Comparing the work conjugate and non-work conjugate theories, the attractiveness of the non-work-

conjugate theories lies in the simplicity of problem definition, numerical implementation, and absence of

higher-order tractions. Thermodynamic consistency and the ability to prescribe boundary conditions on plas-

tic slip make the work conjugate theories more relevant to a general situation, with higher-order traction

boundary conditions also admissible, but the numerical implementation is more involved.

Before closing this section, we note several applications of strain gradient plasticity models for single

crystals. These include the investigation of length scale effects in thermal stresses by thin films by Yefimov

and Van Der Giessen (2005) and Kuroda and Tvergaard (2008), fracture of single crystals by Tang et al.

(2004) and Tang et al. (2005), and indentation by Lele and Anand (2009). Composite materials of elastic

particles in single crystal matrix have been analyzed by Bittencourt et al. (2003), and void and inclusions

problems in single crystals have been analyzed by Borg et al. (2006) and Hussein et al. (2008).

Figure 1.2: Simulation methods and strain gradient theories arranged according to length scale of use. Forstrain gradient theories solid red line indicates strain gradient enhanced hardening, dotted line indicates ener-getic hardening. Double arrows between simulation method indicate current scale bridgingmethods available.

Figure ( 1.2) shows the various numerical simulation methods arranged in the order of the length scale

where they are primarily used. Adjacent numerical methods in this figure will also agree on the predic-

tion of deformation and fracture mechanisms, such agreements provide the basis of design and evaluation

12

of numerical schemes which allow to bridge or co-simulate numerical schemes of adjacent length scales.

Strain gradient plasticity approaches which falls in-between DDP and crystal plasticity or continuum plas-

ticity should be probed for deformation mechanisms which can be simulated in the DDP simulations and in

crystal / continuum plasticity solution. Using suitable canonical problems which can cover a broad range of

deformation patterns, the agreement between different numerical methods of adjoining length scales can be

investigated.

The focus of this research is on Gurtin strain gradient plasticity considered with energetic hardening only,

and with no dissipative hardening due to plastic deformation since the energetic hardening is an important

and differentiating feature of thermodynamically consistent Gurtin’s theory. Two situations are investigated

to understand the predictions of Gurtin’s theory with energetic hardening and compare with other results for

similar situation available in literature. Shear of a layer with constraints of no plastic slip on the boundary is

investigated to see if length scale effects due to energetic hardening is in agreement with lower scale DDP.

In light of the importance of polycrystalline effects in deformation and fracture, and because crack tips will

usually reside within individual grains of material, strain-gradient models based on crystal plasticity seem

most appropriate for examining fracture. The mode I fracture problem is also investigated to see if the Gurtin

strain gradient theory predictions are in agreement to crack tip predictions form DDP simulations. For the

Gurtin strain gradient theory with only energetic hardening, the influence of strain gradient effect is scaled

by the energetic length scale l. The situation with no length scale, l = 0, is same as crystal plasticity. Since

l = 0 involves only conventional crystal plasticity calculations, additional numerical procedure due for strain

gradients is not involved and the results are same as a crystal plasticity simulation.

Two viscoplastic implementations of the Gurtin strain-gradient plasticity theory Gurtin (2002), M. Gurtin

and Needleman (2005) are developed in the FEM framework. One of the implementation is a general one

that can accommodate higher order boundary conditions and is distinctly different from conventional crystal

plasticity implementations. The second implementation is one that is suitable for problems when no higher-

order traction are present, and becomes similar to the approaches discussed by Kuroda and Tvergaard (2008).

The remainder of this thesis is organized as follows. In the next chapter, we discuss a fracture situation

- delamination of an elastic coating on a plastically deforming substrate - to bring out in more detail the

13

practical situation of coating fracture and delamination where realistic fracture trends can be predicted from

DDP approach, but not from conventional plasticity approach. Discussion in the chapter also highlights

the deficiencies of conventional continuum plasticity models which, if used for design, will lead to non-

conservative designs. The third chapter discusses the theoretical aspects of Gurtin’s theory and the numerical

implementation methods for a viscoplastic version of Gurtin’s strain gradient theory using FEM. We discuss

two methods. The ‘general method’ applicable for boundary conditions of higher order traction as well as

plastic slip. The ‘special method’ is limited to the prescription of boundary conditions of plastic slip only,

but even then remains very practical for various plastic deformation situations. The fourth chapter discusses

the problem of constrained shear, shows predictions of plastic slip distribution from both numerical solution

methods, and compares to existing DDP and other strain gradient results. Further this chapter also highlights

the difference between energetic hardening and dissipative hardening effect attributed to strain gradients,

which are motivated by different underlying dislocation interactions but are not discriminated in a situation

of constrained shear. The fifth chapter discusses the mode I fracture problem for single crystals and the

predictions from Gurtin’s theory using the special implementation. This chapter also discusses the numerical

difficulty in implementing the general approach for a fracture problem. The discussion in this chapter also

compares the results from the present study to existing crack tip results from different strain gradient theories,

and the implications of using Gurtin’s theory for fracture problem. The conclusions from this research and

recommendations for future work to address the open challenges of engineering simulations for constrained

plastic deformation, length scale effects, and fracture, are discussed in the last concluding chapter.

14

Chapter 2

Delamination of an Elastic Coating onplastically Deforming Substrate

Thin film coatings are of great importance in engineering encompassing automotive, manufacturing, elec-

tronics, aerospace and other applications. Elastic coatings of e.g. ceramic on a metal substrate that can deform

plastically upon mechanical loading is a complex problem which is affected by mismatch of material param-

eters, substrate plasticity and interface properties. Failure of thin ceramic coatings can start from localized

delaminaion zones at the interface. For numerical study of crack initiation and propagation, when the loca-

tion of crack or debonding failure is known, cohesive zones framework due to Xu and Needleman (1993)

and Tvergaard and Hutchinson (1992) provides a unified approach to crack initiation and propagation along

a predefined cohesive zone. Using cohesive zone approach along with conventional plasticity models is not

straightforward. For crack propagation in a simplified configuration of a single low hardening and plastically

deforming materials around a crack tip, modeling challenges for high strength ratio of cohesive strength to

yield strength σ/σy using conventional plasticity is discussed by Tvergaard and Hutchinson (1992) as well

as Wei and Hutchinson (1997). Similar modeling challenges exist for elastic coatings on plastically deform-

ing substrate when conventional plasticity is used to model the substrate with cohesive zone at the interface.

Modeling plastic deformation by continuum approach rules out initiation of tensile delamination if interface

strength σ is more than twice the yield strength σy of the substrate for a non hardening substrate as discussed

by Abdul-Baqi and Giessen (2001) and Gao and Bower (2004). Since modeling of crack initiation and prop-

agation of high strength interfaces in challenging and not realistic, any design based on continuum plasticity

15

is likely to be non-conservative for service conditions. Initial delamination zones for thin films are likely to

be of micron size scale and size dependent plasticity mechanisms due to underlying dislocation activity may

be operative and become very pertinent for initiation of delamination and interface crack propagation. Since

propagation of Mode I crack with high cohesive strength ahead of crack tip in plastically deforming medium

is possible in DDP simulations as shown by H. Cleveringa et al. (2000), and for bimaterials by O’Day and

Curtin (2005), the DDP method was used to investigate delamination of thin films by O’Day et al. (2006).

A direct and quantitative comparison between predictions for the different plasticity models, DDP and con-

tinuum plasticity, was made by O’Day et al. (2006) in this thin film delamination, which is discussed in this

chapter. We discuss the physical situation modeled and show the predictions from DDP and J2 isotropic con-

tinuum plasticity model for the substrate to point out the differences and highlight the need for size-dependent

plasticity based on dislocation mechanisms.

2.1 Thin film delamination problem description

The geometry of the thin film indentation problem studied here is shown in Fig. 2.1. An elastic film on an

elastic-plastic substrate is intended to represent a typical hard ceramic coating on a ductile metal substrate.

The origin of an x1 − x2 coordinate frame is located on the film/substrate interface, directly below the

centerline of the rigid punch. The left and right edges of the specimen are located at x1 = −75 and 75 µm,

respectively. The substrate thickness is 100 µm and the film thickness varies between 1 and 4 µm. All

displacements are constrained (u1 = u2 = 0) along the left, right and bottom edges. Simple displacement

boundary conditions are used to simulate the rigid punch: the indentation depth u2 = −h is specified on the

film surface under the indenter, simulating a frictionless, rigid flat punch. The half-indenter width w is taken

to be 1 µm.

The existence of singularities at the indenter corners in such problems is well-known, but since we are

not concerned with the stress distribution in the film (and no plasticity occurs there) the singularities are

not expected to influence the present results. We use a small-strain formulation and limit the maximum

indentation depth to 25% of the film thickness.

To make the predictions as quantitative as possible, we have performed continuum plasticity calculations

16

Figure 2.1: Elastic thin film on plastically deforming surface subjected to indentation. Substrate is modeledby DDP and J2 isotropic theory. Interface is modeled by a cohesive zone. Process zone is defined for DDPsimulation. Figure from O’Day et al. (2006)

using exactly the same geometry, loading and boundary conditions, finite element mesh, for both the DDP

and continuum plasticity simulation the material properties of substrate are consistent with Aluminium. For

the J2 plasticity the material is assumed to be elastic-perfectly plastic with yield strength of σy =60 MPa and

relevant DDP material parameters are given in O’Day et al. (2006). In the DDP model the substrate is a single

crystal with three slip systems oriented along 0, π/3, and 2π/3, and a tensile yield strength of 60 MPa and

essentially elastic perfectly plastic. Thus, the only difference between the two models is in the description of

the plasticity. Although the continuum calculations could use crystal plasticity to match the DD model slip

systems more closely, the models are as similar as possible in all other respects. For continuum plasticity

calculations commercial FE code ABAQUS was used. For the cohesive surface model of Tvergaard and

Hutchinson (1992) is used with properties confirming to a metal/ceramic interface. Different values of the

ratio of cohesive strength to yield strength σ/σy between 0.5 to 5 are tested for onset of delamination. The

same FE mesh is used for both models and consideration of resolution of cohesive length scale is described

17

in O’Day et al. (2006). The film of thickness 1 µm is assumed to be elastic. Calculations are performed for

two ratios of elastic modulus of film to substrate of Ef/Es =2 and Ef/Es =6.

2.2 Comparison of prediction using DDP and continuum plasticity

The most important parameter with respect to tensile delamination is the ratio of the interface strength to the

metal yield stress, σ/σy . Figures 2.2(a) and (b) show the normalized critical indentation force for delami-

nation as a function of the ratio σ/σy for a system with film thickness tf = 1µm and for Ef/Es = 2 and

Ef/Es = 6, respectively. At low values of σ/σy ≤ 0.75, the DD and continuummodels predict very similar

results and the magnitudes are within a factor of 1.5. At higher ratios of σ/σy , delamination is essentially

prohibited within the continuum plasticity model. For the plane-strain model here, delamination becomes

very difficult for σ/σy >∼ 2, with the critical force rising rapidly for larger ratios and for both ratios of

elastic modulus. The increase in critical force is accompanied by a much larger increase in the corresponding

critical displacement, many times larger than the DD results, due to the perfectly-plastic behavior.

2.3 Summary

A rapid increase in Fc with interface strength is analogous to the rapidly increasing toughness as the limiting

value of σ/σy = 5 is approached in continuum plasticity simulations of interface fracture Tvergaard and

Hutchinson (1992). In contrast, the DD model predicts no such threshold in behavior, but rather a slow,

nearly-linear increase in the critical force or displacement with increasing ratio of σ/σy . As shown Fig. 2.3.

The DD model shows local hardening and local high stresses at the micron scale underneath the indenter,

and the stress fields of individual dislocations and pile-ups are able to induce delamination in a way that the

averaged continuum plasticity fields can not. The “threshold” for delamination found in the continuummodel

is thus considered to be an artifact of the application of continuum plasticity at small scales. The mechanics of

crack nucleation at a coating/substrate interface during the indentation of a coated material has been studied

within the DDP and continuumplasticity frameworks. There is qualitative agreement for critical conditions of

delamination but quantitative agreement exists only for the scenario of low ratio of cohesive strength to yield

18

strength σ/σy . The divergence between DDP and continuum plasticity in the predictions of delamination

at high ratios of σ/σy can be attributed to details of underlying dislocation structure which are modeled in

DDP but missing in the continuum plasticity approach. High local stresses due to long range dislocation

stress fields and forest type hardening due to buildup of excessive dislocation underneath the indenter act as

local stress enhancers and local enhancers of plastic flow resistance respectively to cause delamination trends

simulated by the DDP approach.

For the practical issue of design of robust interface coatings, the smooth trends exhibited in the predictions

of the DDP model do suggest for reasonable extrapolation of continuum results of low cohesive strength

to yield strength ratio σ/σy to estimate delamination behavior at higher ratios of σ/σy for more-realistic

and conservative coating design, until the time that plasticity models accounting for the above mentioned

dislocation mechanisms become an accepted practice in engineering.

Towards realistic simulation of plastic deformation along with cohesive zone framework in engineering

practice, the thin film delamination simulations comparing DDP and continuum plasticity brings out the need

for a plasticity approach which can capture the stress enhancement due to individual numerous dislocations

or equivalently net Burgers vector corresponding to a GND density. The availability of a continuum plasticity

FEM modeling approach which can accounts for the underlying mechanisms of the dislocation network for

stress fields and enhanced hardening will be a significant step to address reliable coating design and to better

simulate general scenarios involving damage and plasticity. Since strain gradient approaches aim to capture

the underlying dislocation mechanisms, they providemotivation to develop numerical methods to incorporate

strain gradient effects in FEM framework for wide acceptance in engineering practice. Towards addressing

the need for strain gradient approach in engineering, in the next chapters we focus on the Gurtin strain gradient

theory. The Gurtin strain gradient theory attributes a free energy to GNDs, or gradients of plastic strains, to

approximate the stress enhancements due to dislocation densities in an indirect and local manner. We discuss

numerical implementation and applications to the situations of constrained shear and mode I fracture, and

compare with predictions using other strain gradient theories that are motivated by enhancing hardening due

to GNDs.

19

Figure 2.2: Comparison of continuum and DD predictions for normalized critical force for the onset ofinterface delamination with: (a) Ef/Es =2 and (b) Ef/Es =6; work of separation φ =0.1875 J/m2 in allcases. Figure from O’Day et al. (2006)

20

Figure 2.3: Normalized opening stress σ22/σ and dislocation snapshots in 6-8 mm region below indenter: (a)prenucleation, and (b) post-nucleation. Dislocation structure captured in DDP cause local stress enhancementand delamination.Thin film with Ef/Es =2, σ22/σ =2, tf =1 mm and hmax/w 0.125. Figure from O’Dayet al. (2006)

21

Chapter 3

Formulation of Gurtin Strain GradientViscoplasticity

Confining attention to the regime of small strain, the gradient of the rate of displacement vector u can be

decomposed into elastic and plastic parts as

"u = "ue + "u

p. (3.1)

The symmetric part of the rate of displacement gradient u is the total strain rate

ε = ("u)sym. (3.2)

The total strain rate is related to the elastic strain rate and the plastic strain rate as

ε = εe + εp. (3.3)

The plastic strain rate is given in terms of the slip rate on each slip system as

εp =∑

β γβPβ (3.4)

Pβ = 12 (sβ ⊗ mβ + mβ ⊗ sβ) (3.5)

22

where for a given crystal slip system β,mβ is the unit vector normal to the slip system, sβ is the unit vector

along the slip direction, and γβ is the slip rate on that system.

The geometrically necessary dislocation density tensorG is defined following Nye (1953) and M. Gurtin

and Needleman (2005) as

G = curl("up) (3.6)

G =∑

β

("γβ) × mβ ⊗ s

β . (3.7)

Following M. Gurtin and Needleman (2005), the statement of virtual power is modified to include strain

gradient effects. The internal virtual power is written as

Wint(R) =

∫

R

(σ : "ue + T : G)dV +

∑

α

∫

R

παγαdV. (3.8)

where the stress σ expends power over the rate of elastic displacement gradient "ue, a defect stress T

expends power over temporal changes of the net Burgers vector G, and the internal microforce or flow stress

πα expends power over the slip rate γα. Noting the kinematic relation (3.5) and the definition of the GND

density tensor (3.6), a microstress ξα can be defined as

ξα = mα × Ts

α (3.9)

such that

T : G =∑

α

ξα ·"γα. (3.10)

The internal power (3.8) is balanced by the external power given by

Wext(R) =

∫

∂Rt(n) · udA +

∫

R

f · udV +

∫

∂RX(n) : "u

pdA. (3.11)

Here, t(n) is the traction on surface with unit normal n and is power conjugate to the displacement rate u,

f is the body force, and the external defect tractionX(n) is power conjugate to the distortion rate"up. A

23

microtractionΞα(n) can be defined and related to the defect tractionX(n) as

Ξα(n) = s

α ⊗ mα : X(n) (3.12)

which leads to

X(n) : "uP =

∑

α

Ξα(n)γα. (3.13)

Thus, the internal and external virtual powers (3.8) and (3.11) can be written as

Wint(R) =

∫

R

σ : "uedV +

∑

α

∫

R

(παγα + ξα ·"γα)dV. (3.14)

And

Wext(R) =

∫

∂Rt(n) · udA +

∫

R

f · udV +∑

α

∫

∂RΞ(n)αγαdA. (3.15)

Now considering a situation when γα ≡ 0, so there is no plastic deformation, and"ue ≡ "u, then the

equivalence of (3.14) and (3.15) reduces to the conventional virtual power relation

∫

R

σ : "udV =

∫

∂Rt(n) · udA +

∫

R

f · udV. (3.16)

Using the divergence theorem on (3.16) yields the conventional equilibrium equation and traction-stress re-

lations as

div(σ) + f = 0, t = σ · n. (3.17)

Considering a situation when u ≡ 0, which implies ε ≡ 0 and εe = −∑

α γαPα, then the equivalence of

(3.14) and (3.15) reduces to a microscopic virtual power relation

∑

α

∫

R

(παγα − ταγα + ξα ·"γα)dV =∑

α

∫

∂RΞ(n)αγαdA (3.18)

where τα = σ : P is the resolved shear stress. Since R is arbitrary, from (3.18) we have microforce balance

24

and microtraction relations given by

πα − τα − divξα = 0, n · ξα = Ξα(n). (3.19)

For ξα = 0 everywhere in the domain, the microforce balance law (3.19) reduces to the conventional vis-

coplastic relation wherein the resolved shear stress τα equals the flow stress πα. The power contribution

from the higher order tractions acting on plastic slip at the boundary is accounted for by the terms∫

R Ξα · γα

in (3.15). Powerless microscopic boundary conditions (n · ξα)γα = 0, can be satisfied by either

γα = 0, (3.20)

the so-called microhard boundary condition (3.20) and/or microtraction free boundary conditions (3.21)

n · ξα = 0. (3.21)

The material constitutive relations are specified by defining the stress σ, microstress ξα and the flow stress

πα in a thermodynamically consistent manner. Gurtin proposed that the classical free energyψ be augmented

by an additional defect energyΨ(G) due to the GND densities as

ψ =1

2εe : C : εe + Ψ(G). (3.22)

Stress and higher order stress then follow from (3.22) as

σ = C : εe (3.23)

T =∂Ψ(G)

∂G. (3.24)

Gurtin assumed a quadratic form ofΨ(G) in (3.22) given by

Ψ(G) =1

2l2π0G : G. (3.25)

25

Using (3.25) in (3.24) we have the higher order stress as

T = l2π0G, (3.26)

where l is the plastic strain gradient length scale, which is a parameter to be specified. The macroscopic

balance of linear momentum is

divσ + f = 0. (3.27)

The flow stress on the slip plane is πα is related to the resolved shear stress τα and the back stress due to

GND densities ξα by a flow rule which is the microscopic balance

divξα + τα = πα. (3.28)

The microstress ξα is calculated from (3.9) in the balance of microforce (3.18) and is given by

ξα = mα × Ts

α. (3.29)

The evolution of the plastic flow is taken to conform with conventional crystal plasticity so that the current

slip rate on the slip system α is related to the flow stress πα via the viscoplastic constitutive relation

γα = a|πα

gα|msign(πα), (3.30)

where a is a reference slip rate, m is the rate sensitivity exponent, and gα > 0 is the slip resistance on the

slip plane. The evolution of gα is associated with self and latent-hardening in the standard manner given by

gβ =∑

κ

h(βκ)|γκ|, gβ|t=0 = τ0, (3.31)

where h(βκ) are the slip hardening moduli and τ0 is the initial value of gβ at time t = 0.

26

3.1 Numerical implementations for strain gradient crystal plasticity

We now discuss numerical implementations of the M. Gurtin and Needleman (2005) theory outlined in the

previous section as generalizations of the widely used numerical crystal plasticity modeling approach of

Asaro and Needleman (1985). First we discuss the essential steps of crystal plasticity and then introduce

additional steps for two different implementations of strain gradient crystal plasticity.

For crystal plasticity we seek an incremental solution procedure that is applied at a time t for time incre-

ment ∆t. Here we discuss the forward Euler approach, the backward Euler approach which is more stable

and computationally efficient and is outlined by Asaro and Needleman (1985). At any given time t, and for

the time increment ∆t, we know the current stress σ, the resolved shear stress τα, the viscoplastic slip rate

on each slip system γα from the viscoplastic relation (3.30), the viscoplastic strain rate εvpkl , the total slip for

each slip system γα, and the hardening parameter for each slip system gα. The solution procedure can be

viewed as a sequence of two steps. The first step is solution of the rate form of balance of momentum (3.17)

at current time t by solving∫

vσ : δεdV =

∫

vfδudV. (3.32)

The next step is calculation of the stress σ, the resolved shear stress τ , the slip rate γα, the viscoplastic

strain rate εvpkl , the total slip γα, and the hardening parameter gα, at time t + ∆t. For calculation of the stress

σ at t + ∆t we first calculate the increment of stress σ during the time increment∆t as

σij = Cijkl : ( ˙εkl − εvpkl ) (3.33)

where the viscoplastic strain rate εvpkl is known for the increment at time t. The total stress σ at end of time

t + ∆t is calculated from the stress rate σ during increment increment ∆t and the stress σ at time t. From

the total stress σ we can calculate the resolved shear stress as

τα = sα.σmα. (3.34)

The slip rate is then calculated from (3.30), the viscoplastic strain rate is calculated from (3.5), and the

27

hardening on the slip planes computed from (3.31). Thus the two step procedure, consisting of a solution

for the rate form of the balance of momentum and then updating the solution-dependent quantities, can be

repeated at each time increment.

The plastic strain γα so obtained by crystal plasticity does not have to satisfy any microscopic force

balance and corresponding boundary conditions (3.19), (3.20). For strain gradient viscoplastic calculations

the plastic strain rates γα calculated at each time step must be corrected at each time increment so that

microscopic force balance (3.19) and corresponding boundary conditions (3.20) are satisfied. Next we discuss

modifications to the crystal plasticity solution method to accommodate strain gradient crystal plasticity. Both

the methods seek to change the plastic strain rate γα calculated from crystal plasticity so that microforce

balance and boundary conditions (3.19), (3.20) are satisfied.

3.2 General solution approach

Whereas in conventional crystal plasticity the microforce balance (3.19) reduces to πα = τα and the vis-

coplastic strain rate γα evolves based on resolved shear stress τα from the viscoplastic relation (3.30), in

strain gradient plasticity, the gradients of plastic slip γα,i enter into the for slip rates from (3.19). To solve

(3.19), we adopt an FE solution approach for the weak form of (3.19) which is

∫

V[(πα − τα) ˙γα + ξαi γ

α,i ]dV = 0 (3.35)

In an FEM solution of (3.35), we have the plastic slips γα as the nodal variables. The resolved shear stress τα

at the element integration points is known from the crystal plasticity solution above at the start of the current

time increment.

For a plane strain deformations with respect to coordinates (x1, x2), under the assumption that s(α)3 = 0

andm(α)3 = 0 for all slip systems and using the orthogonality relations of the slip direction and normal of a

slip plane, the microstress from (3.7), (3.26), (3.9) can be expressed as

ξαi = l2π0sαi

∑

β

sαj sβj sβpγβ,p. (3.36)

28

γα(x) and its derivatives are interpolated to the integration points from the nodal values as

γα(x) =N

∑

I=1

φI(x)γαI

(3.37)

and

γα,i(x) =N

∑

I=1

φI(x),iγαI

(3.38)

where φI(x) are the interpolation functions, N is the number of nodes of an element, and γαI are the nodal

slip rates.

Thus the microstress can be calculated at integration points using the derivatives of nodal slip γβI by

using (3.38) which can written as

ξαi = l2π0s(α)i

∑

β

s(α)j s(β)

j s(β)p

N∑

I=1

φI(x),pγ(β)I (3.39)

To simplify, we can rewrite this as

ξαi = l2π0sαi

∑

β

S(αβ)sβpN(x)I,pγ

βI (3.40)

where

S(αβ) = sαj sβj (3.41)

and

N(x)I,p =

N∑

I=1

φI(x),p (3.42)

Consider the term ξαi γ,i in the integrand in (3.35). For a Finite Element implementation using the nodal

interpolation (3.38) and (3.40) we calculate for each element

∫

V eleξαi γ,idV = l2π0s

(α)i

∑

β

S(αβ)s(β)p γ(β)I

∫

V eleN(x)I

,pN(x)J,idV γJ = kIJ γJ (3.43)

29

where we define the element stiffness matrix kIJ as

kIJ = l2π0sαi

∑

β

S(αβ)sβp

∫

V eleN(x)I

,pN(x)J,idV (3.44)

This can be assembled into the global stiffness matrix K. Now consider the term (πα− τα)γ in the integrand

in (3.35) for the element load vector,

∫

V ele(πα − τα)γdV =

∫

V ele(πα − τα)N(x)I γIdV = −rαI γI (3.45)

where rαI is the element force vector. Using the viscoplastic relation (3.30) for πα we have

rαI =

∫

V eleN(x)I [τα − g(γ)(

γα

a)1/m]dV (3.46)

The element force vector rαI can be assembled into global force vectorR for FE implementation. Combining

(3.43) and (3.45) the solution to (3.35) for the total slip γα must satisfy

K{γ} = R (3.47)

where {γ} is the vector of all the current nodal slips, R captures the difference between the local resolved

shear stress τα and the stress causing plastic flow πα, which determines the current plastic slip rate γα at

each gauss point. Since, at any time t, slip γα(t) is related to the slip at end of previous increment γαold as

γα(t) = γαold + γα∆t (3.48)

and, via (3.46), R also involves the current slip rate γα, the solution to (3.35), or its FEM equivalent (3.47),

is actually a solution for {γα}.

To solve (3.47) for the slip increment γα we proceed as follows. We aim to obtain a slip rate correction

of δγ to the current slip rate estimate γ that will satisfy the microhard boundary conditions in (3.35). The

initial guess of nodal plastic slip rate γ is calculated from the element integration point values of the crystal

30

plasticity solution γxp. Then at the current time we have

K{γold + ∆t(γ + δγ)} = R (3.49)

where

r(α) =

∫

VN(x)[τ − g(γ)(

γ + δγ

a)1/m]dV (3.50)

Assuming a small correction in strain rate δγ * γ, we can expand as

g(γ)(γ + δγ

a)1/m = [g(γold) +

∂g

∂γ(∆t(γ + δγ))](

γ

a)1/m[1 +

1/mδγ

γ] (3.51)

For the iterative procedure, we must solve for the nodal {δγ} satisfying

[K∆t + D]{δγ} = R − K{γold + ∆tγ} (3.52)

whereD is a vector assembled from each element d defined by

d =

∫

V eleN(x)[({g(γold) +

∂g

∂γγ∆t}

1/m

γ) +

∂g

∂γ∆t][

γ

a]1/mdV (3.53)

and R is a vector assembled from element r defined by

r =

∫

V eleN(x)[τ − {g(γold) +

∂g

∂γγ∆t}[

γ

a]1/m]dV (3.54)

Solving for the nodal {δγ} and updating γ at each iteration for the correct γ at the current time increment

satisfies (3.35).

Now we discuss a robust numerical implementation approach. Considering (3.53), if we ignore hardening

then only the first term is significant. This term can be approximated as

∫

V eleN(x)[(g(γold)

1/m

γ)][γ

a]1/mdV :=

∫

V eleN(x)

1/m

γτdV. (3.55)

31

If (3.55) for any node is exceptionally large then this implies an infinitesimally small correction to the current

slip increment estimate γ. However, numerically (3.52) cannot be inverted accurately due to ill-conditioning

of the matrix. To avoid numerical issues that have no bearing on the solution (δγ = 0), we select a minimum

strain rate γtol, below which δγ = 0 is enforced exactly. In other words, any node where the crystal plasticity

strain rate increment is smaller than γtol will have no correction, and the slip rate for that node is implemented

as a constraint in the solution of (3.52). This situation typically arises when the initial crystal plasticity strain

rate γtol is extremely small, so the constraint acts primarily on the first iteration. This also ensures that, in the

limit of zero plastic length scale, the crystal plasticity solution is obtained. Using γtol defines a region where

γ < γtol for which the gradient correction is not necessary.

The converged solution satisfies the balance of microforce in weak form (3.35) as well as any prescribed

boundary conditions of plastic strain and higher order traction.

3.3 Special solution approach

Situations in which the higher order traction Ξα(n) is zero in some regions of the boundary, and where

boundary conditions on plastic slip γα are prescribed in other regions, an alternate solution method is possi-

ble. Most physical problems fall into this category of boundary conditions, so the special solution approach

is quite useful. We first discuss the special implementation where no boundary conditions on plastic slip γα

are prescribed and then discuss incorporating plastic slip boundary conditions.

In this approach the balance of microforce is rewritten as (3.56),

πα = τα + ξαi,i, (3.56)

and used as a constitutive relation to calculate the viscoplastic strain increment. In other words, the stress

causing plastic flow, πα, results from the additive effect of the resolved shear stress τα, calculated in the

crystal plasticity step, and the divergence of microstress ξαi,i due to the plastic slip distribution. The latter

term physically aims to account for the “back stress” generated by accumulated GNDs in the system, but in

a local manner. The basic idea here is similar to that proposed by Kuroda and Tvergaard (2008) and also by

32

Bayley et al. (2006). Kuroda and Tvergaard (2008) proposed (3.56) on physical grounds based on dislocation

mechanics, where as (3.56) arises naturally from Gurtin’s Gurtin (2002) thermodynamic formulation.

In (3.56) using the Gurtin constitutive model from (3.36) we have the divergence of microstress as

ξαi,i(x) = l2π0sαi

∑

β

sαj sβj sβpγβ,pi. (3.57)

For numerical solution by the finite element method, elements which allow quadratic variation of the nodal

quantities, which are total plastic strains of each slip system in this case γαI , are necessary to enable cal-

culation of ξαi,i(x) using the second derivatives of element shape functions. From (3.37) and (3.57), the

divergence of microstress ξαi,i(x) can be calculated at integration points by using the second derivatives of the

shape functions as

ξαi,i(x) = l2π0s(α)i

∑

β

s(α)j s(β)

j s(β)p

N∑

I=1

φI(x),piγ(β)I

(3.58)

In the solution procedure, at any given time t the nodal values of the strain rate increment γα are calculated

from crystal plasticity using their values at the integration points. Then, from the total slip as in (3.48), the

divergence of the microstress ξαi,i is calculated using the element shape functions. The stress causing plastic

flow πα is then calculated from the microforce balance (3.56) and the plastic strain rate is calculated from the

viscoplastic relation (3.30) using πα.

Evidently, implementing the balance of plastic flow (3.19) as a constitutive relation as in (3.56) is a

simpler implementation but requires elements that are amenable to computation of second derivatives. Thus,

for situation when there is no boundary condition prescribed for plastic slip, we have the solution method for

the microforce balance in plastic strain gradient sensitive material. Such situations are discussed by Lele and

Anand (2009)

However a modification to above approach is required to implement prescribed boundary conditions on

plastic slip. So, we now discuss implementation of plastic slip boundary conditions. If plastic slip boundary

conditions are specified on a boundary then, in the numerical solution procedure, the value of plastic slip

increment is prescribed on the nodes of the boundary. At other nodes, the plastic slip and slip rates is

computed as above, i.e. by averaging the plastic slip and rates at the integration points calculated from πα.

33

Prescribing plastic slip at the boundary influences the divergence of microstress ξαi,i, resulting in a change in

πα according to the flow rule (3.56), consequently changing the plastic flow increment at the integration point

for elements adjacent to the boundary where plastic slip is prescribed. Thus microforce balance (3.19) and

the essential boundary conditions of plastic slip can be satisfied by the special approach using appropriate

higher order elements to support quadratic variation of plastic slip as nodal quantities.

In the next two chapters we illustrate the effects of plastic strain gradients in two different situations using

Gurtin theory in the viscoplastic framework discussed above. The two situations, constrained simple shear

and mode I fracture in a single crystal, have a uniform and highly non-uniform stress field, respectively. A

special feature of the constrained shear problem is the restriction of zero plastic slip on the boundary, which

is an essential boundary condition. The important feature of the Mode I fracture problem in a single crystals

is the strong slip localization along sector boundaries that arises in conventional crystal plasticity.

34

Chapter 4

Constrained Shear of Uniform Layer

Consider a thin strip of material of infinite extent in x2 and x3 and of thickness L in x1 direction, where

it is bounded by impenetrable walls preventing plastic slip, as shown in Figure ( 4.1). A displacement rate

u2 is applied on the upper boundary, while the lower boundary is held fixed. The material is a single crystal

with two slip systems oriented at π3 and2π3 with respect to the x1 axis. The plastic strain gradient length

scale in the problem l is a material parameter for the single crystal. Dissipative hardening that depends on

local plastic slip can influence the plastic distribution and can be an additional material parameter for this

problem, but we have not considered any dissipative hardening as it is not the focus when illustrating strain

gradient effects. Dissipative hardening due to net Burgers vector accumulation resulting from the total slip

distribution as discussed in Gurtin (2002) can also be studied, but is not considered here.

This problem of constrained shear is a widely discussed test case for gradient plasticity theories. Bitten-

court et al. (2003) presented a rate-independent FEM implementation of Gurtin (2002) gradient theory and

compared to DD simulations by Shu et al. (2001). Kuroda and Tvergaard (2008) solved this problem using

a non-work-conjugate strain gradient plasticity theory. Anand et al. (2005) implemented a rate-dependent

version of Gurtin (2002) for the one-dimensional case. The analogous problem of constrained shear of a thin

layer for an isotropic material is discussed by Niordson and Hutchinson (2003) using an FEM implementa-

tion of Fleck and Hutchinson (2001), and by Acharya et al. (2004) using a lower order strain gradient theory

implemented with a finite difference scheme.

Here, we present and discuss the results obtained by the two different numerical implementations of

35

Figure 4.1: Constrained shear problem. Two slip systems are oriented along π3 and

2π3 from x2, no plastic

slip is permitted at the boundary along x1 = L (top), and x1 =0 (bottom). Displacement rate u2 is appliedalong the top and bottom boundary or traction rate τ is prescribed

the Gurtin theory, the general and special methods, discussed before in section 3.1, and compare to the

rate independent solution of Bittencourt et al. (2003). The following material properties are used for the

viscoplastic material parameters: rate exponent m =100 or 200; a = 0.01 or 0.001 per unit time; initial

yield strength gαo = 1.0 per unit strength; length scale l =1; and no dissipative hardening hαβ = 0. Elastic

properties are a shear modulusG = 424gαo and Poisson ratio of 0.33.

The analytical solution for this problem in the rate independent limit is discussed by Bittencourt et al.

(2003). The important features of the solution are the state of stress and plastic slip distribution. The state of

stress is uniform, as in simple shear, which is a consequence of stress equilibrium. The plastic slip distribution

for this one dimensional situation has a quadratic variation of the form (x/L − (x/L)2) for 0 ≤ x/L ≤ 1.

The plastic slip is zero at the boundaries x/L = 0 and x/L = 1. The slip distribution for different levels

of displacement, corresponding to uniform resolved shear stress on the slip planes, and for different length

scales have the same functional form and differ only by a numerical factor. Thus the slip distribution for

different levels of displacement level u or for the corresponding resolved shear stress τα, and for different

length scales l, can be normalized to a single curve.

The solution procedure discussed in section 3.1 is a sequence of two steps at each time increment, so-

lutions of (i) balance of linear momentum (3.27) and (ii) balance of plastic strain (3.19). Noting that the

balance of linear momentum for this problem is always a uniform shear stress σ12, with a uniform resolved

36

shear stress τα on the slip planes, a simplification of the solution procedure is possible. For this simplifica-

tion, a spatially uniform resolved shear stress rate τα, is imposed and the balance of plastic strain (3.19) is

then solved for the plastic slip distribution. The displacement rate can be integrated from the elastic strain

rate due to the imposed uniform resolved shear stress rate τα, and from the solution of the plastic slip rate

distribution on the slip planes. We call this approach the ”stress loading” approach, which can be used in both

the general and special solution methods. This stress loading approach is used within the general approach to

calculate the slip distribution by imposing a resolved shear stress rate in a one dimensional implementation

for a rate exponent of m = 200 and a = 0.01. The one dimensional finite element mesh consists of 100

one dimensional elements of equal size with three nodes, extending over one-half of the domain. A starting

resolved shear stress of 0.974 gαo is imposed along with a rate of resolved shear stress of 1 per unit time.

Solutions are obtained up to a final resolved shear stress of 1.4 gαo . Fig. (4.2) shows the predicted slip dis-

tributions at the end of loading, normalized by the maximum plastic slip calculated in the rate-independent

solution at the same final resolved shear stress. The rate-independent result is shown for comparison.

The same problem and material, with the stress loading approach and the general solution method, can be

applied to a two dimensional domain by using periodic boundary conditions to simulate the infinite extent of

the sample in the x1 direction. Periodic boundary conditions are implemented using a penalty method. The

two dimensional mesh uses three-node elements in a combination of four elements to form a crossed triangle

mesh. One-half of the domain is resolved using 8000 crossed triangle elements, with 20 crossed triangle

elements along the x1 the direction where periodic conditions are applied and 100 elements along the x2

direction. The two-dimensional rate-dependent results are also shown in Fig. (4.2), normalized again by the

maximum plastic strain in the rate-independent theory.

We next compare the solution from the special method to the general method for this problem, where the

essential boundary conditions of no slip at the boundary are prescribed in the special solution approach. The

viscoplastic parameters are m =200 and a = 0.001 per unit time, with displacement rate loading as used

just above. The one dimensional finite element mesh is of 100 one dimensional elements of equal size with

three nodes over half of the domain for both the general and special approach. The comparison is made at the

resolved shear stress level of τα=1.219 gαo , which is an outcome of the solution for both the methods. The

37

results are shown in Fig. (4.2), normalized by the maximum plastic slip of rate-independent solution at the

resolved shear stress of 1.219.

A complete finite element implementation involves displacement loading and the solution of stress equi-

librium and balance of plastic strain for each time increment. So we now compare a two dimensional imple-

mentation with displacement loading to the one dimensional general implementation with stress loading at

the same level of final resolved shear stress. For the one dimensional stress loading using the general solution

method, the resolved shear loading rate is one per unit time, and the viscoplastic parameters are m =100

and a = 0.01, which differ from the earlier cases. The displacement rate for the two dimensional general

implementation is 8/3000 per unit time on the top surface, and the resolved shear stress is an outcome of the

solution procedure. The meshes for the one and two dimensional implementations were discussed above. The

comparison is done for the slip distribution when the resolved shear stress level reaches 0.992 gαo . This low

stress level is used because use of the penalty method for periodic boundary conditions along x1 introduces

small non-uniformities in the plastic slip along x1, which then affects the uniformity of resolved shear stress

in subsequent increments, so that the true solution is not obtained at larger stress levels. We monitor the

deviation by the departure from a uniform state of stress; comparison at τα of 0.992 gαo is acceptable. The

slip distribution is shown in (4.3) and indicated by the level of final resolved shear stress of 0.992. Since at

the resolved shear stress of 0.992 rate independent analytical solution does not exist and the comparison is

between rate dependent solutions.

From the different plastic slip distributions computed by different numerical methods in Fig. (4.2),

(4.3) and (4.4), we have demonstrated the presence of a plastic strain gradient length scale l leads to a

non-uniform plastic strain distribution starting right from initial loading and evolving to a non-uniform and

nearly-quadratic profile that is a manifestation of energetic hardening due to strain gradients. From the agree-

ment between the general and the special implementation shown in Fig. (4.4) for the slip distribution and the

resolved shear stress for different levels of plastic slip, we have demonstrated that the special implementa-

tion serves as a candidate method when boundary conditions of higher order traction are zero, and that the

method is quite practical. The special implementation is also attractive because it does not involve an iterative

solution for balance of plastic flow mieq. However, the special implementation requires quadratic elements

38

to calculate the corrections to the stress causing plastic flow πα due to the accumulated plastic strain. Our

calculations using a rate-dependent viscoplastic model, but with large m value, yield predictions that are

very similar to the rate-independent model, as expected in this limit and confirming the robustness of the

rate-dependent models and implementation.

We now examine the evolution of the slip distribution with displacement loading as obtained by the two

different methods for the material with m =100 and a = 0.01, with the loading and mesh as described just

above. Fig. (4.4) shows the results, scaled so that the maximum plastic slip is unity and labeled by the level

of resolved shear stress τα and maximum plastic slip. Agreement in the slip distribution is obtained from

the two numerical methods at all different levels of resolved shear stress τα and plastic slip. Fig. (4.4) also

illustrates the transition of the slip distribution to a near quadratic distribution starting from a near-uniform

distribution with a boundary region.

As shown in Fig. (4.4), at lower levels of plastic strain and resolved shear stress the slip distribution

is almost uniform with a high gradient near the boundary region. At higher levels of plastic strain the slip

distribution approaches a near-quadratic distribution and does not change the quadratic form with further

increases in level of plastic strain and resolved shear stress. At all times, the resolved shear stress τα is

uniform. At low levels of plastic slip, the zero plastic slip boundary condition causes a backstress ξαi,i and a

plastic slip gradients γαx in elements adjacent to the boundary. This causes πα to be less than the resolved

shear stress τα near the boundary. Lower πα adjacent to the boundary in turn causes less plastic flow near the

boundary. But, the absence of backstress or plastic slip gradients away from the boundary allows for uniform

plastic strain in these regions. At any point, the spatial difference in plastic slip in the neighboring regions

via the back stress ξαi,i influences the πα, and in-turn the plastic slip rates. This effect has two consequences.

First, if a location at the boundary between region of uniform slip and region with gradients, there is a back

stress ξαi,i at that location as a result after the increment in time due to lower current slip rate the location

becomes a part of region where there is a gradient. This is how the effect of constraint at boundary moves

towards the center. When the slip distribution becomes quadratic and the back stress is uniform, further strain

rates are uniform and the quadratic slip distribution is maintained. But this does not exactly happen due to

constraint of no slip at the boundary. The second consequence is to cause the slip rate at a location between

39

the slip rate value of the neighborhood points, and consequently cause the value of slip and back stress to

be between the values of the neighborhood region. The first effect increases the region with backstress with

further plastic deformation and the second effect tries to maintain a uniform backstress via the neighborhood

bounds. The first effect becomes weaker with increase in plastic slip and the slip distribution thus keeps

changing towards a near quadratic distribution with a near uniform backstress.

The stress-strain curve for length scales of l =0.1 and l =1 are shown in Figure (4.5). The calculation

is done using stress loading in the general solution approach by imposing resolved shear stress rate of unity

starting from a resolved shear stress level of 0.974 in a one dimensional implementation for a rate exponent

of m = 200 and a = 0.01. The stress corresponds to the boundary traction along x2 and the strain is the

shear strain ε12. Because there is no dissipative hardening due to plastic slip, the length scale of l/L =0.1

shows very little hardening and is close to the case of an elastic-perfectly-plastic material. For the length

scale l/L = 1, corresponding to all the results for plastic slip shown above, significant hardening is observed

due to the strain gradient effect.

The constrained shear problem has been investigated using DDP simulations by Shu et al. (2001) and a

comparison of the DDP study to the rate-independent Gurtin theory was done by Bittencourt et al. (2003).

This comparison is shown in Figure ( 4.6) from Bittencourt et al. (2003). The zero plastic slip boundaries

are at x2/h=0 and x2/h=1. The slip distribution between the two methods matches well, given the discrete

nature of DDP. The slip distribution profile is not quadratic; rather, there is a boundary region near the

slip-constrained surfaces and a near-uniform slip distribution in the middle. The deviation from quadratic

slip distribution is due to the presence of dissipative hardening in both the DDP simulation and the rate-

independent strain gradient calculations. An analytical expression for the slip distribution was derived by

Bittencourt et al. (2003). The characteristic length scale λ−1 for the boundary layer in the slip distribution is

given by (4.1)

λ−1 = Sin2θ

√

go

Hol, (4.1)

where l is the energetic length scale, Ho is the dissipative hardening coefficient associated with the plastic

slip on a slip system, go is the initial flow strength, and θ is the orientation of the slip system in Figure ( 4.1).

From (4.1) we see that presence of both dissipative hardening due to plastic slip Ho and an energetic length

40

scale l are required for the formation of boundary layer between the region of near uniform slip. Higher

dissipative hardening will result in thinner boundary layer. In the absence of hardening Ho both boundary

layers interact, resulting in a quadratic profile.

The qualitative agreement for slip distribution in a constrained shear problem between DDP study of Shu

et al. (2001) and an earlier study using strain gradient theory by Fleck and Hutchinson (1997) is discussed

by Shu et al. (2001). Also, agreement between DDP results in Shu et al. (2001) and rate-independent Gurtin

gradient plasticity is discussed in Figure 4.6 from Bittencourt et al. (2003). However the underlying disloca-

tion mechanisms for the strain gradient theories of Fleck and Hutchinson (1997) and Gurtin are different; the

primary hardening mechanisms are based on enhanced dissipation due to GND’s and increase in free energy

due to GNDs, respectively. To probe the agreement observed between different gradient theories with DDP

simulations further, we next discuss a situation where the differences based on the underlying primary dislo-

cation mechanism can be seen. Since the distinctive underlying mechanism for Gurtin’s theory the existence

of a free energy associated with strain gradients, a situation that allows for a reduction of stored free energy

due to strain gradients is a good candidate for comparison.

To bring out the difference between strain gradient theories motivated by main aspects of free energy due

gradients of fields vs. flow strength attributed to field gradients, one can use the situation of constrained shear

with a modification to the boundary conditions after some level deformation. Imagine a constrained shear

situation with a certain level of displacement at the boundarywhich corresponds to a state of uniform resolved

shear stress in the domain and a quadratic slip profile with energetic hardening only. At this stage, we switch

the loading so that the state of deformation is maintained by an appropriate traction at the boundary instead of

by displacement boundary conditions and, most importantly, the constraint on plastic deformation is relaxed.

After this change in boundary conditions, the response of the material, and in particular the changes in the

plastic slip distribution and displacement near the boundary can be considered. A DDP simulation will predict

different dislocation configurations and boundary displacements depending on the dislocation structure. In

particular, in a DDP simulation with low hardening, there may be outflow of those dislocations that were

previously piled up near the boundary and responsible for the hardening. Flow of these dislocations out of

the sample would cause additional plastic deformation and increase of displacement at the boundary. If the

41

dislocation structure at the boundary has associated hardening (e.g. forest hardening), which impedes the

outflow of the piled-up dislocations, then there would be little change in the plastic strain distribution and

displacement at the boundary. The two different outcomes expected from the DDP simulations represent the

effects of gradient-based energetic and dissipative hardening, respectively.

The above scenarios can also be understood from the schematic load displacement curves in Figure ( 4.7

(a), and (b)) for the case of constrained shear and Figure ( 4.8 (a), and (b)) following relaxation of the

boundary condition on plastic strain, for dissipative and energetic gradient hardening. Assuming elastic

deformation is negligible, for the case of no dissipative gradient hardening and no energetic length scale l,

the area under the load displacement curve represents plastic dissipation for an elastic plastic material. For

the case of dissipative hardening attributed to the strain gradients, the area under the load displacement curve

represents the total energy dissipated in plastic deformation. The excess energy dissipated, compared to the

elastic-perfectly-plastic material, is due to the gradient-enhanced dissipative hardening. In the presence of an

energetic length scale l with no dissipative hardening, the area under the load displacement curve represents

the cumulative free energy due to the plastic strain gradients and the plastic dissipation. Comparing to the

elastic-perfectly-plastic material, the excess energy under the load-displacement curve is the free energy due

to the plastic strain gradients. The load displacement curves in Figure ( 4.7 (a), and (b)) look alike but the

underlying dislocation mechanisms are different and also the free energy is different.

When the boundary constraints of plastic deformation are released and equivalent boundary tractions

are applied to maintain the same state of stress, the expected load-displacement behavior for the different

mechanisms of gradient hardening are illustrated in Figures 4.8 (a), and 4.8 (b). For the case of gradient-

enhanced dissipative hardening, Figure 4.8 (a), no change in the load-deformation curve is expected since

there is no free energy available. Also, because there is no free energy available to be used for evolution of

the underlying dislocation structure, the plastic strain distribution and the displacement at the boundary do

not change. For the case with energetic gradient hardening, Figure 4.8 (b), the plastic slip distribution is

expected to change to allow the gradients of plastic strain, or equivalently the free energy due to plastic strain

gradients, to reduce. The reduction in free energy results in an increase in the overall deformation. This is

reflected in the load-deformation curve as an increase in average deformation at constant load.

42

Numerical simulations that relax the zero plastic slip boundary condition while maintaining boundary

tractions are not straight forward because of the change of boundary condition on plastic slip. The change of

boundary condition can cause an abrupt change of free energy, causing additional plastic dissipation. Detailed

consideration of the balance of the reduction in free energy, due to plastic strain gradients, and the increase in

energy dissipated in concomitant plastic strain evolution, associated with reduction in strain gradients, also

have to be incorporated in the numerical scheme along with balance laws for evolution of GND’s, which are

necessary to simulate abrupt GND evolution. No restriction of plastic strain at the boundary also implies

no higher order traction Ξα(n). Thus, relaxing the constraint of plastic strain at the boundary should lead

to a slip profile consistent with zero higher order traction at the boundary via some intermediate transition,

modeled by the evolution of GND’s consistent with the reduction on free energy due to strain gradients and

concomitant plastic deformation.

From the examples of constrained shear and the discussion of relaxing the constraint of no plastic slip at

the boundary,Gurtin’s theory of strain gradient plasticity via the additive free energy of plastic strain gradients

is closer to DDP simulations when the energetic interaction of dislocations dominate over forest hardening

mechanisms of dislocations. For practical situations, when the level of plastic deformation is low to cause

forest hardening but a length scale effect is observed due to constraints on plastic deformation, Gurtin’s

theory with above outlined numerical procedures will be applicable. Such situations can arise in deformation

of thin structures with passivation to block dislocation motion and also in polycrystalline structure with

constraints on plastic deformation at grain boundaries due to grain orientation or grain boundary interface

structure. Predictions of theories motivated by enhanced dissipative effects due to GNDs will agree with

DDP predictions if hardening prevents dislocation motion, resulting in no changes of plastic slip profile

on release of the constraints on plastic slip. Consideration of the underlying dislocation structure in terms

of the mobility of dislocation will be the governing criteria for applicability of the energetic and dissipative

strain gradient mechanism of hardening. Further insight fromDDP simulation of relaxing the constraint of no

plastic strain at the boundarywill be helpful to enhance currently developed numerical simulation capabilities

for Gurtins strain gradient plasticity theories to simulate change of boundary conditions of plastic strain and

including consideration of evolution of GND structures. Such capabilities for strain gradient theories will

43

advance the applicability of strain gradient theories based on energetic hardening to simulate a wider range of

plastic deformation situation for e.g. opening of interface at inclusions surrounded by plastically deforming

medium.

44

X/L

NormalizedPlasticSlip

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

general, stress rate, 1D, m=200, RSS=1.4general, stress rate, 2D periodic, m=200, RSS=1.4rate independent, analytical, RSS > 1special, disp rate, 1D, m=200, RSS=1.219general, disp rate, 1D, m=200, RSS=1.219

Figure 4.2: Normalized plastic slip distribution vs. position in a constrained shear problem as computedusing various implementations discussed

45

x/L

PlasticSlip

0 0.1 0.2 0.3 0.4 0.50

0.0002

0.0004

0.0006

0.0008

general, stress rate, 1D, m=100, RSS=0.992general, disp rate, 2D Periodic XP, m=100, RSS=0.992

Figure 4.3: Plastic slip distribution vs. position in a constrained shear problem as computed using generalimplementation, comparison between 1D stress driven and 2D implementation at RSS=0.992

46

x/L

Plasticslipnormalizedbymaximum

slip

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

General, RSS=0.9608, Max Slip=1.91e-6General, RSS=0.9947, Max Slip=0.128e-3General, RSS=0.9991, Max Slip=0.475e-3General, RSS=1.0702, Max Slip=8.37e-3Special, RSS=0.9608, Max Slip=1.91e-6Special, RSS=0.9947, Max Slip=0.128e-3Special, RSS=0.9991, Max Slip=0.476e-3Special, RSS=1.0702, Max slip=8.37e-3

Figure 4.4: Slip distribution evolution in constrained shear with increase in loading calculated by generaland special implementation

47

Figure 4.5: Stress vs. average shear strain curve illustrating hardening due to energetic length scale l.Significant hardening is observed for length scale of l/L =1 but not for l/L =0.1

48

Figure 4.6: Comparison of slip distribution for constrained shear between DDP and strain gradient plasticitysimulation with energetic hardening with plastic dissipation based on plastic slip. Plot from Bittencourt et al.(2003). The DDP results are from Shu et al. (2001)

49

Figure 4.7: Schematic load vs. displacement curves for constrained shear with dissipative gradient hardening(a) and energetic gradient hardening (b). Elastic plastic material is shown for reference. Area under the loaddisplacement curves corresponding to purely plastic dissipation in (a), and plastic dissipation along withstrain energy of plastic strain gradients in (b) as shown.

50

Figure 4.8: Schematic load vs. displacement curves for constrained shear with further release of constrain ofplastic slip with maintaining traction for the case of gradient enhanced dissipative hardening. Elastic plasticmaterial is shown for reference. For material with gradient enhanced dissipative hardening (a), on release ofconstraint for plastic deformation no change in plastic strain distribution happens due to lack of free energyof strain gradients. For material with energetic hardening (b), on release of constraint for plastic deformationfree energy of strain gradients causes additional plastic deformation.

51

Chapter 5

Modeling Mode I Crack Tip Fields

5.1 Problem description

We analyze the deformation and stress fields around a crack tip in a single crystal subject to Mode I loading

as a function of the strain gradient length scale l within the context of M. Gurtin and Needleman (2005)

theory. As shown in the Figure ( 5.1) The single crystal has a crack along the x1 axis with two slip systems

oriented at π3 and2π3 relative to the crack line, and with a symmetric geometry around the x1 axis. The crack

surface x1 < 0 is traction free. Since there is no restriction on the slip along this free surface, there are no

higher-order traction or slip conditions imposed. By symmetry, ahead of the crack tip there are no higher

order traction boundary conditions. A displacement-rate boundary condition is applied at the outer boundary

of the domain corresponding to a Mode I loading K1. For the domain size chosen and the range of applied

load levels reported, there is no significant plastic slip at the outer boundary, which is approximately 30 times

larger than the size of the maximum plastic zone.

The material parameters used are a rate exponentm = 50 and a = 0.00001 per unit time, and as before

we reference all stresses to the initial yield strength of material gαo =. A Young’s Modulus of E = 1000

times the yield strength and Poisson ratio of ν = 0.33 are used. There is no dissipative hardening considered.

The size of plastic deformation zone at the crack tip depends on the current load K1. This is an evolving

length scale in the problem. The other length scale in the numerical computation is the due to the fixed FE

mesh. We start loading from a finite value ofK1 at which the maximum resolved shear stress τα is less than

52

Figure 5.1: Crack tip in single crystal subjected to Mode I loading by K1, the single crystal has two slipsystems are oriented along π/3 and 2π/3 with the crack plane

0.95 times the yield strength for any given mesh so as to capture the strain gradient effects from the initial

stages of plastic deformation in the numerical simulation. For strain gradient effects, three length scales are

considered, zero and two non-zero values. The non-zero length scales are scaled to current plastic zone size.

The current plastic zone size can be approximately calculated by assuming the material is linear elastic and

then calculating the location where the resolved stress on the dominant slip plane with slip direction π3 equals

the flow strength. The actual plastic zone will be larger than predicted by the approximate calculation. Plastic

zone sizes calculated in this way will depend on the remote Mode I load on the crack. The plastic length

scales l used in the calculation of the crack tip fields are reported in terms of the current plastic zone size.

For the crack problem, triangular elements arranged in crossed triangle form are used for both balance

of linear momentum (3.27) and balance of plastic strain (3.19). As shown in Figure (5.2), constant strain

triangles are used for interpolation of the displacement for the solution of balance of linear momentum (3.27).

The slip rates γxp are calculated at the element integration points after solution of balance of linear momentum

(3.27). For the balance of plastic strain (3.19) involving calculation of plastic strain gradients or divergence of

53

Figure 5.2: Coincident three node and ten node triangles are used for solution of crystal plasticity and balanceof plastic strain (3.19) respectively. Triangles are arranged in crossed triangles to avoid numerical issues dueto incompressibility of plastic strains

the microstress ξαi,i, ten-node elements with quadratic shape functions are used. The integration point values

of slip and slip rate from the crystal plasticity solution of (3.27) are used to calculate the nodal values of

slip and slip rate for the corresponding ten-node elements. For the three nodes at the vertices, the nodal slip

rate and plastic slip values are calculated by averaging from the integration point values of the three node

elements. The slip and slip rate values at the inner node are the values at the element integration point in the

three-node triangle. The values at the remaining nodes at element boundary are calculated from the values at

the vertices of triangle.

Higher order elements like the ten node triangle elements are necessary for the general solution for the

following reason. In the incremental solution procedure for the crack problem, a single gauss point close to

the crack tip will yield first. All the neighboring elements do not yield and do not have significant levels of

plastic strain. Nodal averaging of the plastic strain in such a situation is likely to yield a near-uniform plastic

strain distribution for the yielded element and zero plastic strain gradient but for the adjoining elements nodal

averaging will lead to large gradients of plastic slip without significant plastic slip. In the general approach

this leads to large a gradient correction at the adjoining elements which do not have sufficient levels of plastic

strain. This issue cannot be avoided by prescribing a cutoff γtol below which a correction to the crystal

plasticity solution is not calculated (δγ = 0). This is an issue with discretization using three-noded elements.

Thus the constant strain triangles are not suitable for crack tip problems in the general approach of numerical

solution.

54

For the crack problem using the general approach, essential boundary conditions of plastic slip are re-

quired on the outer boundary. Prescribing the plastic slip boundary condition is not straight forward, because

in a viscoplastic implementation plastic slip develops for all load levels and there is no elastic-plastic bound-

ary. Consistency with the rate of Mode I loading K1, i.e. small scale yielding, requires zero plastic slip

γα = 0, and slip rate γα = 0, at the outer boundary. Furthermore, from (3.55) one can prescribe a mini-

mum plastic slip rate γtol for crystal plasticity analysis, which may be corrected in the subsequent gradient

correction, and the boundary condition at the outer boundaries is then satisfied approximately, because the

plastic slip rate at the boundary is consistent with crystal plasticity solution, which is negligible at the outer

boundaries γ = γxp ≈ 0. However, we have found prescribing a fixed minimum plastic slip rate, γtol is

not satisfactory in the numerical solution procedure for crack tip situation. A possible reason is that a fixed

γtol causes abrupt change in the degree of freedom for plastic slip which become a candidate for gradient

correction when plastic slip rates in the adjoining elements become higher than γtol. This abrupt change in

degrees of freedom for which a gradient correction is calculated causes an abrupt change in the plastic strain

gradient profile between increments which is not handled satisfactorily by the numerical procedure in crack

problems. Because of the issues discussed here associated with using the general method for a crack tip

problem, further analysis and results are obtained using only the special implementation approach.

5.2 Results for crack problem

We present results for the dominant slip system, along π3 , which is representative of similar phenomena

occurring along the non-dominant plane along 2π3 . First we discuss the effect of the length scale on total

plastic slip. Figure 5.3(a) - 5.3(c) shows the effect of length scale on the total slip distribution along π3 ,

for length scales of l = 0, l = 0.057, and l = 0.115. The lengh scales are in terms of current plastic zone

size. The results are reported load the plastic zone size is 0.435 unless stated otherwise. The contours of

plastic slip along π3 are scaled differently for each length scale. The presence of a gradient length scale l

has a drastic effect on the slip distribution. When the length scale parameter l is zero, the slip distribution is

non-uniform with a maximum slip at the crack tip. However, when the material is sensitive to plastic strain

gradients i.e. l > 0, then the magnitude of the maximum slip decreases with increasing length scale l, and the

55

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8s10.10.090.080.070.060.050.040.030.020.010-0.01

(a) Length scale l = 0

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8s1

0.0060.00550.0050.00450.0040.00350.0030.00250.0020.00150.0010.00050-0.0005-0.001

(b) Length scale l = 0.057

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8s1

0.00180.00160.00140.00120.0010.00080.00060.00040.00020-0.0002-0.0004

(c) Length scale l = 0.115

Figure 5.3: Effect of length scale l on total slip along π3 , different contour scale, (a) l = 0, (b) l = 0.057,

and (c) l = 0.115

slip distribution is almost uniform along the slip direction away from the crack tip. The effect of remote load

and gradient length scale l leads to more uniform distribution of total plastic strain along the slip direction

when plastic strain gradient length scale is non-zero. The plastic slip distribution is not only more uniform in

the presence of a length scale l, but the magnitude of the maximum plastic strain is also lower with a larger

gradient length scale l. The near-uniform plastic slip distribution along the slip direction is very unlike the

highly localized plastic slip distribution near the crack tip when the length scale is zero (corresponding to

crystal plasticity).

The origin of the behavior of total slip in Figure ( 5.3(a) - 5.3(c)) is best understood by examining the

56

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8sd10.0240.0220.020.0180.0160.0140.0120.010.0080.0060.0040.0020-0.002

(a) Length scale l = 0

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8sd1

0.00340.00320.0030.00280.00260.00240.00220.0020.00180.00160.00140.00120.0010.00080.00060.00040.00020-0.0002

(b) Length scale l = 0.057

Xf

Yf

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8sd1

0.00280.00260.00240.00220.0020.00180.00160.00140.00120.0010.00080.00060.00040.00020

(c) Length scale l = 0.115

Figure 5.4: Effect of length scale l on slip rate along π3 , different contour scale, (a) l = 0, (b) l = 0.057, and

(c) l = 0.115

slip rates. Figure ( 5.4(a) - 5.4(c)) shows the effect of length scale l on the slip rate distribution. For a given

load, the location of the maximum slip rate depends on the strain gradient length scale l. The maximum slip

rate for l = 0 is at the crack tip, where the resolved shear stress is maximum. However, the location of the

maximum plastic strain rate moves away from the crack tip with increasing strain gradient sensitivity, as seen

from Fig. 5.4(b) - 5.4(c) for different length scales. The location of the maximum plastic strain is further

from the crack tip for a larger length scale l at any given level of remote Mode I loadK1 on the crack tip.

The stress rates are plotted on the same contour scale for the same length scale of l =0.057 for a plastic

zone size of 0.435, but for different remote load corresponding to plastic zone sizes of 0.23 and 0.435 in

57

-0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.80.0030.00280.00260.00240.00220.0020.00180.00160.00140.00120.0010.00080.00060.00040.00020-0.0002

γ1.

(a) Length scale l = 0

-0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.80.0030.00280.00260.00240.00220.0020.00180.00160.00140.00120.0010.00080.00060.00040.00020-0.0002

γ1.

(b) Length scale l = 0.057

Figure 5.5: Effect of remote load KI on location of maximum plastic slip for l = 0.057 at plastic zone sizeof 0.435 for the same contour scale. The location of maximum slip rate is closer to the crack tip when theplastic zone size is 0.23 in (a), and further away from the crack tip when the plastic zone size is 0.435 in (b)

Figures ( 5.5(a)) and ( 5.5(b)) respectively. Figures ( 5.5(a) and 5.5(b)) indicate that the location of the

maximum plastic strain rate, for a given length scale l, is further away from the crack tip with increasing

remote Mode I load K1. For non-zero strain gradient length scale l, the strain rate near the crack tip is very

low compared to the maximum strain rate and also compared to the strain rate at the crack tip for crystal

plasticity. The change in location of maximum plastic slip rate, being further away from crack tip with

increase in load KI for a material with length scale l, causes a near-uniform plastic slip distribution as seen

in the total slip distribution shown in Figure ( 5.3(a) - 5.3(c))

Figure 5.6(a) - 5.6(c) shows the contours of opening stress for different length scales. We see that the

Rice (1987) sector solutions begin to emerge when the length scale is zero. The opening stress profile for

the Rice sector solutions are 4√

(3) for the sector between angles 0 to π3 ahead of the crack, 2

√

(3) for the

sector between angles π3 to

2π3 and 0 for the sector between angles 2π

3 to π. When a gradient length scale

l is involved we see deviations from the Rice sector solution with increasing length scale. The maximum

opening stress ahead of the crack tip is lower with a larger length scale l. The lower opening stress ahead of

the crack tip with increase in gradient length scale l, and as a deviation from Rice (1987) sector solutions,

can be explained based on the location of the maximum slip rates as shown in Fig. 5.4(a) - 5.4(c). The

sector solution for the stress distribution is achieved at the crack-tip after sufficient slip accumulation along

58

x1

x2

-0.1 0 0.10

0.05

0.1

0.15

0.2syy

6.9285.938294.948573.958862.969141.979430.9897140

(a) Length scale l = 0

x1

x2

-0.1 0 0.10

0.05

0.1

0.15

0.2syy

6.9285.938294.948573.958862.969141.979430.9897140

(b) Length scale l = 0.057

x1

x2

-0.1 0 0.10

0.05

0.1

0.15

0.2syy

6.9285.938294.948573.958862.969141.979430.9897140

(c) Length scale l = 0.115

Figure 5.6: Effect of length scale l on opening stress along π3 , (a) l = 0, (b) l = 0.057, and (c) l = 0.115

the slip planes. Slip localization along the slip directions at the crack tip is a requirement for the formation

of discontinuous stress sectors. Localization of plastic slip at the crack tip can happen only if the maximum

plastic slip rate is always at the crack tip. For material with a length scale l, the location of maximum slip rate

is not near the crack tip and so does not lead to the slip accumulation and consequent slip localization near

the crack-tip that is necessary for stress sector formation. Further, for a given load the location of maximum

slip is more distant for a larger length scale and the distance increases with increase in load. Thus, for larger

length scales the slip localization near the crack tip ceases earlier and at a corresponding lower opening stress

level ahead of the crack tip. Thus, the maximum opening stress ahead of the crack tip is lower with a larger

length scale l.

59

5.3 Implication for fracture and limitations of Gurtin’s theory

Gurtin strain gradient theory, with no dissipative hardening, predicts a lower magnitude of total plastic slip

and more uniformity in the plastic slip along the slip plane directions as compared to a crystal plasticity

solution with no dissipative hardening. It also leads to a prediction of lower opening stress ahead of the

crack with increase in gradient length scale l at any given remote load. Since the opening stress levels are

predicted to be lower, the Gurtin theory with only energetic hardening will not predict crack propagation if

a traction separation law is used ahead of the crack tip where the peak cohesive strength is more than three

times the flow strength gαo in the current configuration of two slip systems oriented at π/3 and 2π/3. Thus

enhancements to crystal plasticity within the Gurtin theory with only energetic hardening are not sufficient

to drive fracture and remedy the unrealistic prediction of high load levels required for propagation crack in

single crystals if the cohesive zone strength ahead of the crack is greater than three times the flow strength

gαo .

Predictions from Gurtins theory indicate that GNDs near the crack tip try to reduce the available free

energy and reduce the plastic slip and slip localization, in-turn causing plastic flow away from the crack tip.

Since the free energy due to GNDs does not cause any stress σ and only causes higher-order stress T or a

correspondingmicrostress ξα to influence the plastic flow as in ( 3.19), the effect of a stress field due to GNDs

near the crack tip cannot be simulated to cause any additional stress ahead of the crack tip with Gurtin theory.

This situation can be remedied by following alternative approaches such as introducing a non-saturating

hardening based on GND’s, as done for a crack tip problem by Tang et al. (2004) using the theory of Acharya

and Bassani (2000). Along the lines of DDP approach, crystal plasticity can be enriched to account for long

range stress fields due to GND’s and their evolution. Accounting for stress long range fields in general will

require solution to a boundary value problem to satisfy traction and displacement boundary conditions. In

this approach the long range nature of stress fields of GNDs is not amenable to local approximation of strain

gradient approach.

60

5.4 Comparison with previous strain gradient studies for crack tip

Fields around crack tips have been analyzed by others researchers, using different strain gradient models for

single crystal as well as isotropic materials. We now discuss some of the results available in the literature and

compare with the results from the current implementation of Gurtin’s strain gradient plasticity. We discuss

results from results from numerical studies studies based on following theories. For isotropic material the

existing results are based on the strain gradient theory of Fleck and Hutchinson (1997), Fleck and Hutchinson

(2001), and MSG theory of Gao et al. (1999), and Huang et al. (2000). For single crystals, the theory of

Acharya and Bassani (2000) is used. All the theories mentioned above introduce measures for plastic strain

gradients with higher dissipative hardening. The Fleck and Hutchinson (1997) theory based on gradients of

total strain has additional elastic strain gradients which add to the free energy density. The present results from

Gurtin’s theory for crystal plasticity implemented with GNDs influencing the free energy has no dissipative

hardening, which is unlike the theories used in the existing literature.

First we discuss results from Wei and Hutchinson (1997) for model I crack in a power-law hardening

isotropic material. Wei and Hutchinson (1997) analyze crack tip fields for mode I loading using Fleck and

Hutchinson (1997) theory, for an isotropic material with a single plastic gradient length scale l and single

elastic gradient length scale le = l/2 and power-law hardening exponent of N = 0.2. Their predictions,

shown in Fig. 5.7, for opening traction are higher with a larger length scale, but the prediction for opening

stress ahead of the crack tip is lower with a higher length scale. The opening traction ahead of the crack tip

is different from the opening stress because of higher order stresses τ212,1, τ222,2 involved in the definition of

traction (5.1) since the theory Wei and Hutchinson (1997) is in the framework of coupled-stress (CS) solids.

t2 = σ22 − 2τ212,1 − τ222,2. (5.1)

The trend of lower opening stress with larger strain gradient length scale l seems to agree with the opening

stress trend predicted from the present study using Gurtin’s theory. This may be due to the elastic length scale

le, which is one-half of the plastic length scale le = l/2, and contributes to the free energy similarly to GNDs

in Gurtin’s theory. The elastic length scale le enters the calculation of stress increments since the theory is

61

Figure 5.7: Opening stress σ22 and opening traction vs. scaled distance from the crack tip calculated usingtheory of Fleck and Hutchinson (1997) calculated for different gradient length scales. Trends in openingstress σ22 and opening traction t2 are opposite with gradient length scale. Figure from Wei and Hutchinson(1997)

in the framework of CS solids. Wei and Hutchinson (1997) discounted the role of elastic length scale le on

account of the much smaller gradient of elastic strain as compared to gradients of plastic strain. But more

information on stress and traction ahead of the crack tip with different elastic length scales le, the location of

the maximum plastic strain rate, and the strain hardening, will be necessary to fully understand the apparent

agreement in the variation of opening stress with length scale.

Komaragiri et al. (2008) analyzed a similar situation using Fleck and Hutchinson (2001) theory. A detailed

parametric study was performed using a single and also three length scales of different magnitudes for plastic

gradients effect, and for a wider loading range and hardening exponent n, where n = 1/N in the previous

study. They report significant in elevations of hydrostatic stresses ahead of the crack-tip, which is shown

in Fig. 5.8, with opening stresses that follow a similar trend. The effect of zero length scale parameter,

62

plotted with the dashed line, has the smallest increase in stress as one approaches low values of r/rp, where

r is the distance ahead of crack tip and rp the current size of the plastic zone. The increase in stress is due

to the singular nature of the HRR fields and would require appropriate meshes in the FEM calculations to

capture the singularity. The opening stress increases with a single length scale parameter l∗ and the increase

in stress is much more with three length scale when l1 is higher than l2 and l3. Differences in hydrostatic

response with different combinations of length scale parameters are due to the difference in the scaling of

the corresponding plastic strain gradient invariants for gradient-based hardening. For a single length scale

parameter l∗, strain gradient effects are significant only if the HRR stress are greater than four times the yield

strength σy suggesting the role of the mesh to capture the singularity of HRR field. The hydrostatic stress

enhancement ahead of the crack tip is due to a combination of the mesh resolution and larger contributions

from gradient-based hardening due to the l1 parameter. Additionally, the l1 parameter is more relevant, as

its effect is similar for a low-hardening material n = 20 as discussed in Komaragiri et al. (2008) which

emphasizes the important role of the corresponding gradient measure for the gradient hardening law.

Tang et al. (2004) studied crack tip fields of a single crystal with slip plane lattice incompatibility en-

hanced, in other words GND-enhanced, dissipative hardening within crystal plasticity. The GND-enhanced

hardening is in addition to the standard hardening due to cumulative plastic slip in (3.31). λα is the GND

parameter associated with slip plane α, where λα/b is interpreted as net forest dislocations per unit area

threading the slip plane α. The non-saturating dissipative hardening law is as in (5.2),

g =

η2µ2b2(g−go)ko

∑

α λα|γα| + θo(

gs−ggs−go

)∑

α |γα| if g ≤ gs,

η2µ2b2(g−go)ko

∑

α λα|γα| if g > gs.

(5.2)

where θo is the stage II work hardening rate at flow stress go, gs is the saturation strength, µ is the

shear modulus, η ≈ 13 is a non-dimensional material constant, and ko is a non-dimensional material constant

introduced for GND effects.

From the results of Tang et al. (2004) in Fig. 5.9 we see that lattice incompatibility, is concentrated in a

very narrow region near the crack tip. This implies that the hardening due GNDs is operative in only a small

63

region near the crack tip.

From Fig. 5.10, we see the effect of ko, the non-dimensional parameter which scales hardening effects

due to GNDs, on the opening cohesive traction ahead of the crack tip. NI here implies ko = 0, or only

conventional stage II hardening. Fig. 5.10 shows that a significant increase in opening traction can be

simulated depending on the value of ko. The cumulative slip distribution from this study is shown in Fig.

5.11 a, for the case with hardening due GNDs as well as conventional hardening, and in Fig. 5.11 b, for the

same loading with no hardening due to GNDs. The contour levels of accumulated slip are the same, except

the maximum level which is an order of magnitude higher when only conventional hardening is operative.

From these results we can conclude that hardening due to GNDs, operative in a very small region near the

crack tip, causes an order of magnitude lower maximum plastic slip at the crack tip. Hardening due to lattice

incompatibilities also reduces the plastic slip in regions where practically no lattice incompatibility exist,

implying a localization in the spatial extent of plastic slip due to GND-based hardening. This implies that

the GND-based hardening causes the ratio of resolved shear stress to flow strength τα/gα to continue to be

maximum at the crack tip, consequently the slip rate is maximum at the crack tip, leading to a maximum

total slip accumulation at the crack tip. As the hardening due to GNDs is significant only near the crack tip

and not elsewhere, it implies a much higher stress level near crack tip compared to the case with no GND

enhanced hardening, which is also implied with the opening traction plots in Fig. 5.10. These results show

that dissipative hardening due to GNDs or lattice incompatibility causes significant hardening at the crack

tip and high opening stress, which is a different strain gradient based hardening mechanism than the Gurtin

strain gradient effect leading to a difference in prediction of stress at the crack tip.

Results on crack tip stresses from the study by Jiang et al. (2001), using the MSG theory due to Gao

et al. (1999), and Huang et al. (2000) show an elevation in opening stress ahead of the crack tip relative to

predictions of the conventional HRR fields. The stress singularity near the crack tip where strain gradient

effects are operative is the same or higher than the stress singularity for linear elasticity away from the crack

tip, as show in Fig (5.12). The MSG theory is based on Taylor hardening due to the presence GNDs. Such

a hardening mechanism is similar to the GND-induced hardening of single crystals by Tang et al. (2004),

and similar to results from Komaragiri et al. (2008) even if mathematical details of GND quantification and

64

hardening function due to GNDs are different. The singular nature of the stress at crack tip and hardening

enhancement due to GNDs, contribute to higher stress levels at crack tip with furtherK loading and maintain

the spatial distribution of stress so that the maximum plastic strain rate is always at the crack tip, resulting in

enhanced hardening and stresses at the crack tip.

The difference in stresses ahead of the crack tip between the literature discussed above and the present

results is due to the absence of a singularity at the crack tip for zero gradient length scale in a non-dissipative

hardeningmaterial, and non-dissipative hardening due to GNDs when the length scale is present. The location

of the maximum plastic strain rate is also different for energetic hardening due to plastic strain gradients,

which does not lead to extensive slip required for formation of stress sectors, and hence predicts a lower

stress ahead of crack tip as compared to a crystal plasticity solution. Thus, different mechanisms of plastic

deformation due to non-dissipative and dissipative hardening from strain gradients lead to different crack-tip

stresses.

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Figure 5.8: effect of length scale parameters on hydrostatic stress using theory of Fleck and Hutchinson(2001) on Hydrostatic stress vs. scaled distance from crack tip showing. Figure from Komaragiri et al.(2008)

66

Figure 5.9: Figure from Tang et al. (2004) showing lattice incompatibility (GNDs) existing in a small regionof 0.02 microns near the crack tip which arises due to a non-saturating hardening law.

67

Figure 5.10: variation of cohesive traction scaled by g0 ahead of the crack tip for different ko parameter toscale hardness due to GND, NI isKo = 0. Figure from Tang et al. (2004)

68

Figure 5.11: Cumulative plastic slip on all slip systems, (a) with GND hardening, (b) with no GND hard-ening. Slip is suppressed with GND hardening (a) but maximum slip increments are at the crack tip. Figurefrom Tang et al. (2004)

69

Figure 5.12: variation of stresses ahead of the crack tip, using MSG theory. Figure from Jiang et al. (2001)

70

Chapter 6

Conclusion

Understanding mechanisms of deformation and fracture of materials is of paramount importance to de-

sign reliable engineering components and systems which are increasingly utilizing smaller length scales.

Length scale effects in plastic deformation, restrictions on plastic deformation, and better simulation of frac-

ture behavior, as discussed in the introduction and the subsequent chapter on coating failure, are critical for

future engineering design. Plastic deformation is a manifestation of the accumulated motion of the individual

dislocations. Lower-scale simulations using DDP capture the mechanics of individual dislocations, resulting

in natural length scale effects in plastic deformation, and are capable of simulating boundary conditions on

plastic deformation and simulation of realistic fracture. Even though DDP is capable of capturing the under-

lying mechanisms of deformation and fracture in canonical problems, its prohibitive computational costs for

dimensions and details of engineering components are barriers to its wide spread in current engineering simu-

lations. To enable simulation of the above phenomena involving plastic deformation at small length scales in

current engineering practice, a numerical implementation of Gurtin strain gradient plasticity was developed

here. Gurtin theory was a natural choice for the following reasons. It is a theory based on crystal plastic-

ity, which is currently used in engineering simulations. Additionally, Gurtin theory attributes additional free

energy to the plastic slip gradients or net Burgers vector to account the elastic interactions of the underly-

ing dislocations approximately in a local manner. We focussed on the energetic effects of strain gradients,

which are not widely investigated in the current literature, but closer in mechanics and motivation to elastic

interactions among dislocations. Other strain gradient theories motivated by Taylor hardening (1.5) attribute

71

enhanced hardening to net Burgers vector, with a dissipative effect due to strain gradients. Results from these

gradient theories were used to compare the predictions form Gurtins theory.

During the present research, two different numerical solution procedures were developed for implement-

ing Gurtin’s strain gradient plasticity theory. One of the engineering foci for the research was to establish

methods which are in the framework of FEM and amenable to crystal plasticity to analyze and model situ-

ations where length scale effects manifest from restriction on plastic strains at interfaces and boundaries for

e.g. in deformation of passivated metal thin films and grain boundary regions resisting plastic deformation.

Aligned with the interest of bridging length scales using different numerical methods, the present research

probed the the agreement and disagreement between various gradient theories and DDP simulations with

respect to the underlying dislocation structure and proposed a modification to the commonly-used test case

of constrained shear via relaxation of the constraint on plastic deformation as a new test case to establish

domains of applicability of the various strain gradient theories based on underlying dislocation mechanisms.

The limitation of the current numerical implementation, in terms of not accounting for evolution of GNDs

on change of boundary conditions of plastic slip, also points to the potential to increase the applicability of

Gurtin strain gradient approach for situations where GNDs could evolve to minimize the free energy asso-

ciated with them. Damage at an inclusion interface in a plastically deforming medium is one such situation

where abrupt changes in plastic slip distribution can arise if the damage reaches a critical size, simulations of

which may not be possible using theories with gradient enhanced dissipative hardening.

Another critical engineering focus was to understand how the crack tip fields behave for a material with

energetic hardening due to plastic strain gradients and if it can mitigate the limitations of unrealistic prediction

of loading for crack propagation when using crystal plasticity. Such an investigation has wide a interest

encompassing reliable and conservative design of coatings and bi-materials and allowing for simulation of

cleavage fracture in ductule failure without resorting to an expensive computational approach such as DDP.

Analysis of mode I cracks with two slip systems using Gurtins theory predicts, however, lower opening

stresses head of the crack tip with a larger gradient length scale when compared to the Rice sector solution.

This result of lower opening stress rules out prediction of realistic crack propagation using Gurtins theory

with only energetic hardening. While similar studies for crack tips using dissipative gradient theories reveal

72

enhancement of stress ahead of crack tip, in essence they rely on non-saturating gradient-enhanced dissipative

hardening to simulate forest hardening by an appropriate measure of strain gradients. Crack tip situations in

DDP simulations naturally account for forest hardening as well as the long range stress fields of dislocations.

While long range stress fields are absent in all strain gradient theories, Gurtins theory approximates it in a

local manner via attributing free energy to strain gradients, which however is not a good approximation for

crack tip situation. From these differences between the simulation methods for a crack tip situation, some

of the possible approaches for realistic numerical simulation for crack-tip are as follows. One approach

is to develop a crystal plasticity model that is enhanced with GNDs that can evolve based on constitutive

rules of crystal plasticity but including the long range stress fields of dislocations. On the other side of the

scale, for DDP this implies not only simplification of dislocation structure interaction which is already the

basis of multipole method, but also simplification of the motion of the forest dislocation structure which is a

mechanism for dissipative gradient theories.

While strain gradient approaches approximate a particular aspect of dislocation mechanism from a broad

range of mechanisms and enable simulation of a wider range of phenomena in the continuum framework,

strain gradient approaches also have limitations. Gurtins theory in particular has limitations of applicability in

a crack tip situation. However understanding the limitations and differences in predictions of strain gradient

approaches along with understanding agreement with adjoining scale simulations of crystal plasticity and

DDP is helpful towards development of better numerical simulation methods.

The Gurtin strain gradient approach with the implementation developed during the current research is

well suited for situations where length scale effects become important in a crystal plasticity framework in

cases having restrictions on plastic slip at an overall low-level of plastic deformation (to rule out excessive

forest hardening). Engineering applications include applications to MEMS and micro-electronics where un-

derstanding deformation mechanism of passivated multilayered thin structures are of interest. Understanding

deformation mechanisms in engineering of polycrystalline materials and grain boundary structures which

can resist plastic deformation, rigid particle-reinforced composites where particle size, particle separation

distance, and feature size are of similar order of magnitude are also of potential candidates to be analyzed by

the numerical implementations developed here for Gurtins strain gradient theory.

73

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