BEM Solutions for Linear Elastic and Fracture Mechanics Problems with Microstructural Effects

BEM Solutions for Linear Elastic

and Fracture Mechanics

Problems with Microstructural

Effects

Gerasimos F. Karlis

Department of Mechanical Engineering and Aeronautics

University of Patras

Doctoral Thesis

ThesisFigs/UnivShield.eps

2

I would like to dedicate this thesis to my loving parents and brother.

ii

Acknowledgements

First and foremost, I would like to express my deep and sincere grat-

itude to my mentor, Professor Demosthenes Polyzos, for taking me

under his scientific supervision. His faith in me was of utmost impor-

tance. Without his guidance, support and personal work none of this

would have been possible.

It was a great honor for me to have the privilege of collaborating

with Professor Dimitri E. Beskos, whose valuable contribution in the

Boundary Elements field has set a precious example.

I shall always be grateful to Assistant Professor Stefanos V. Tsinopou-

los, for being one of the most important sources of motivation and

guidance. In fact he was there for me at every step, from the begin-

ning of my graduate studies to the end of it and without his guidance

I would have never been able to accomplish the work of this thesis.

I would like to acknowledge the contribution of the Mechanical Engi-

neering and Aeronautics department, as well as the Civil Engineering

department of the University of Patras for providing the necessary

resources and a fertile environment for research.

Finally, I would like to express my thanks to the European Social Fund

(ESF), Operational Program for Educational and Vocational Training

II (EPEAEK II), and particularly the Greek Program PYTHAGO-

RAS II, for funding part of this work.

iv

Abstract

During this thesis, a Boundary Element Method (BEM) has been

developed for the solution of static linear elastic problems with mi-

crostructural effects in two (2D) and three dimensions (3D). The

second simplified form of Mindlin’s Generalized Gradient Elasticity

Theory (Mindlin’s Form II) has been employed. The fundamental so-

lution of the 4th order partial differential equation, that describes the

aforementioned theory, has been derived and the integral equations

that govern Mindlin’s Form II Gradient Elasticity Theory have been

obtained. Furthermore, a BEM formulation has been developed and

specific Boundary Value Problems (BVPs) were solved numerically

and compared with the corresponding analytical solutions to verify

the correctness of the formulation and demonstrate its accuracy.

Moreover, two new partially discontinuous boundary elements with

variable order of singularity, a line and a quadrilateral element, have

been developed for the solution of fracture mechanics problems. The

calculation of the unknown fields near the crack tip (or front) de-

manded the use of elements that could interpolate abruptly varying

fields. The new elements were created in a way that their interpolation

functions were no longer quadratic but their behavior depended on the

order of singularity of each field. Finally, the Stress Intensity Factor

(SIF) of the crack has been calculated with high accuracy, based on

the element’s nodal traction values. Static fracture mechanics prob-

lems for Mode I and Mixed Mode (I & II) cracks, have been solved

in 2D and 3D and the corresponding SIFs have been obtained, in the

context of both classical and Form II Gradient Elasticity theories.

vi

Per�lhyhKat� th di�rkeia th paroÔsa didaktorik diatrib , anaptÔqjhkeMèjodo Sunoriak¸n Stoiqe�wn (MSS) gia thn ep�lush statik¸n pro-blhm�twn elastikìthta me epidr�sei mikrodom se dÔo kai trei di-ast�sei . H jewr�a sthn opo�a efarmìsthke h MSS e�nai h deÔte-rh aplopoihmènh morf th genikeumènh jewr�a elastikìthta touMindlin. Gia th sugkekrimènh jewr�a eurèjh h jemeli¸dh lÔsh th merik diaforik ex�swsh 4h t�xh pou perigr�fei th sumperifo-r� twn sugkekrimènwn ulik¸n kai kataskeu¸n. Ep�sh diatup¸jhke holoklhrwtik ex�swsh twn ant�stoiqwn problhm�twn kai ègine h arij-mhtik efarmog mèsw th MSS. EpilÔjhkan arijmhtik� sugkekrimè-na probl mata sunoriak¸n tim¸n kai ègine sÔgkrish twn apotelesm�-twn me ta ant�stoiqa jewrhtik�.Sth sunèqeia, anaptÔqjhkan dÔo nea asuneq stoiqe�a metablht t�-xh idiomorf�a me skopì thn ep�lush problhm�twn jraustomhqani-k , èna gia disdi�stata kai èna gia trisdi�stata probl mata. Sugkekri-mèna, epeid ta ped�a twn t�sewn apeir�zontai sthn koruf mia rwg-m kai perièqoun sugkekrimènwn tÔpwn idiomorf�e den htan dunatì oakrib upologismì twn ped�wn aut¸n kont� sth rwgm me ta sun jhtetragwnik� sunoriak� stoiqe�a. W ek toÔtou ta nèa stoiqe�a kata-skeu�sthkan me tètoio trìpo ¸ste oi sunart sei parembol tou namhn einai tetragwnikè , all� na exart¸ntai apì ton tÔpo idiomorf�a tou k�je ped�ou. 'Epeita, ègine akrib upologismì tou suntelest èntash t�sh th rwgm me b�sh ti timè tou ped�ou twn t�sewn ko-nt� se aut . Tèlo epilÔjhkan statik� probl mata jraustomhqanik se dÔo kai trei diast�sei kai upolog�sthkan oi suntelestè èntash t�sh gia rwgmè se ulik� me ep�drash mikrodom .

viii

Contents

Nomenclature xviii

1 Introduction 1

1.1 Linear elastic theories with microstructural effects . . . . . . . . . 1

1.2 Numerical solutions in gradient elastic theories . . . . . . . . . . . 5

1.3 Gradient elastic fracture mechanics . . . . . . . . . . . . . . . . . 7

1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Mindlin’s Theory of Elasticity with Microstructure 11

2.1 General Strain Gradient Theory of Elasticity . . . . . . . . . . . . 13

2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Equations of Equilibrium and Boundary Conditions . . . . 15

2.1.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . 18

2.2 Form I, II and III Gradient Elasticity Theories . . . . . . . . . . . 21

2.3 Form II Gradient Elasticity Theory . . . . . . . . . . . . . . . . . 24

2.4 Integral Representation of the Form II Gradient Elastic Problem . 29

2.4.1 Reciprocal Integral Identity . . . . . . . . . . . . . . . . . 29

2.4.2 2D and 3D Fundamental Solutions . . . . . . . . . . . . . 31

2.4.3 Boundary Integral Representations . . . . . . . . . . . . . 34

3 Boundary Element Formulation 37

3.1 BEM Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Symmetry and antisymmetry . . . . . . . . . . . . . . . . . . . . 45

3.3 Subregioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Numerical Integrations . . . . . . . . . . . . . . . . . . . . . . . . 47

ix

CONTENTS

3.4.1 Normal and nearly singular integration . . . . . . . . . . . 48

3.4.2 Singular Integration . . . . . . . . . . . . . . . . . . . . . 50

3.4.2.1 Treating weak singularities . . . . . . . . . . . . . 50

3.4.2.2 Treating strong and hyper singularities . . . . . . 52

3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.1 Hollow Cylinder under pressure . . . . . . . . . . . . . . . 56

3.5.2 Radial deformation of a Sphere . . . . . . . . . . . . . . . 58

3.5.3 Tension of a bar . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Fracture in Elasticity with Microstructure 63

4.1 Displacement and Stress Fields near the Crack . . . . . . . . . . . 64

4.2 Crack Elements for Linear and Gradient Elastic Fracture . . . . . 67

4.2.1 Two dimensional crack element . . . . . . . . . . . . . . . 67

4.2.2 Integrations over a three noded quadratic line special element 71

4.2.2.1 Integrals involving the field R . . . . . . . . . . . 71

4.2.2.2 Integrals involving the field P . . . . . . . . . . . 73

4.2.2.3 Integrals involving the field q . . . . . . . . . . . 75

4.2.3 Three dimensional crack element . . . . . . . . . . . . . . 75

4.2.4 Integrations over an eight-noded quadrilateral special element 78

4.2.4.1 Integrals involving the field R . . . . . . . . . . . 79

4.2.4.2 Integrals involving the field P . . . . . . . . . . . 80

4.2.4.3 Integrals involving the field q . . . . . . . . . . . 82

4.3 BEM Stress Intensity Factor Calculation . . . . . . . . . . . . . . 83

4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 Square plate with horizontal line crack under tension . . . 85

4.4.2 Square plate with diagonal line crack under tension . . . . 90

4.4.3 Cube with central horizontal rectangular crack . . . . . . . 92

5 Conclusions and Future Work 97

A Form I, II & III Constants 101

A.1 Form I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 Form II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.3 Form III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

x

CONTENTS

B Form II: Total Potential Energy Calculation 105

C Mindlin’s Form II: Kernels 107

D Boundary Elements 117

D.1 Surface Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

D.1.1 Eight Noded Quadratic Quadrilateral Element . . . . . . . 117

D.1.2 Six Noded Quadratic Triangular Element . . . . . . . . . . 120

D.2 Line Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D.2.1 Three Noded Quadratic Line Element . . . . . . . . . . . . 122

E Diving elements into triangles 125

E.1 Quadrilateral Elements . . . . . . . . . . . . . . . . . . . . . . . . 125

E.1.1 Triangle 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

E.1.2 Triangle 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

E.1.3 Triangle 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

E.1.4 Triangle 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

E.1.5 Triangle 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

E.1.6 Triangle 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

E.1.7 Triangle 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

E.1.8 Triangle 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

E.2 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . 129

E.2.1 Triangle 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

E.2.2 Triangle 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

E.2.3 Triangle 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

E.2.4 Triangle 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

E.2.5 Triangle 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

E.2.6 Triangle 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

F Taylor expansion of the position vector 133

G Hollow Cylinder Under Pressure: Analytical solution constants135

H Eight Noded Special Element: Interpolation Functions 137

References 153

xi

CONTENTS

xii

List of Figures

2.1 The one dimensional continuum . . . . . . . . . . . . . . . . . . . 11

2.2 1D continuum with quadratically varying displacements and smaller

element size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Kinematic parameters of Mindlin’s theory of elasticity with mi-

crostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Typical components of the double stress tensor and gradient micro-

deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Transformation of elements from the global to their local coordi-

nate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Elements broken down to triangles . . . . . . . . . . . . . . . . . 52

3.3 The hollow cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Radial displacement of the internal points . . . . . . . . . . . . . 57

3.5 Radial displacement of the internal points . . . . . . . . . . . . . 59

3.6 The gradient elastic bar . . . . . . . . . . . . . . . . . . . . . . . 60

3.7 Axial displacement of the internal points . . . . . . . . . . . . . . 61

4.1 Rectangular components of the crack tip stresses . . . . . . . . . . 64

4.2 Mode I: Opening or tensile mode; Mode II: Sliding or in-plane

shear mode; Mode III: Tearing or anti-plane shear mode . . . . . 66

4.3 Variable order of singularity discontinuous boundary element and

its transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 A 2D discontinuous variable order of singularity element . . . . . 70

4.5 Transition from the real 3D space to the parametric representa-

tion of the element and nodal renumbering, for the case of a fully

discontinuous element . . . . . . . . . . . . . . . . . . . . . . . . . 77

xiii

LIST OF FIGURES

4.6 Projection of point x to the crack front . . . . . . . . . . . . . . . 77

4.7 Position of the variable singularity order elements w.r.t. the crack

and domain division . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.8 Gradient elastic plate with a horizontal line crack . . . . . . . . . 85

4.9 Upper right quarter of the COD profile . . . . . . . . . . . . . . . 86

4.10 Position of the variable singularity order elements w.r.t. the crack

and domain division . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.11 Traction values near the crack tip . . . . . . . . . . . . . . . . . . 91

4.12 Gradient elastic plate with a central diagonal line crack . . . . . . 91

4.13 Mixed Mode SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.14 Mixed Mode SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.15 Shape of mode I crack for different values of the gradient coefficient

g compared to the 2D case . . . . . . . . . . . . . . . . . . . . . . 94

4.16 3D Mode I SIFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 Crack opening displacements (CODs) and tractions near the crack

tip for gradient and classical elasticity . . . . . . . . . . . . . . . . 98

D.1 The geometrical and functional nodes of an eight noded quadratic

quadrilateral element . . . . . . . . . . . . . . . . . . . . . . . . . 117

D.2 The geometrical and functional nodes of a six noded quadratic

triangular element . . . . . . . . . . . . . . . . . . . . . . . . . . 120

D.3 The geometrical and functional nodes of a three noded quadratic

line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

E.1 Elements broken down to triangles . . . . . . . . . . . . . . . . . 126

E.2 A random triangle of a quadrilateral element with θ ∈ [θ1, θ2] and

R ∈ [0, Rmax (θ)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

E.3 A random triangle of a triangular element with θ ∈ [θ1, θ2] and

R ∈ [0, Rmax (θ)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xiv

List of Tables

1.1 Works on size effects on specific materials . . . . . . . . . . . . . 5

2.1 The renumbering of the element nodes, so that the crack front

always resides on the first side. . . . . . . . . . . . . . . . . . . . 25

3.1 Material constants for the hollow cylinder . . . . . . . . . . . . . 56

3.2 Average percentage error w.r.t. the analytical solution of Zervos

et al. (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Average percentage error w.r.t. the analytical solution of Tsepoura

et al. (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 The material characteristics used in the hollow cylinder . . . . . . 60

3.5 Average percentage error w.r.t. the analytical solution of Tsepoura

et al. (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1 Orders of magnitude of the asymptotic fields . . . . . . . . . . . . 70

4.2 The renumbering of the element nodes, so that the crack front

always resides on the first side. . . . . . . . . . . . . . . . . . . . 76

4.3 SIF convergence for the classical elastic case . . . . . . . . . . . . 88

4.4 SIFs convergence for the gradient elastic case (g = 0.01, 0.05 and

0.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D.1 The geometrical node coordinates of an eight noded quadratic

quadrilateral element . . . . . . . . . . . . . . . . . . . . . . . . . 118

D.2 The functional node coordinates of a discontinuous eight noded

quadratic quadrilateral element . . . . . . . . . . . . . . . . . . . 119

D.3 The geometrical node coordinates of a six noded quadratic trian-

gular element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xv

LIST OF TABLES

D.4 The functional node coordinates of a discontinuous six noded quadratic

triangular element . . . . . . . . . . . . . . . . . . . . . . . . . . 121

D.5 The geometrical node coordinates of a three noded quadratic line

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D.6 The functional node coordinates of a discontinuous three noded

quadratic line element . . . . . . . . . . . . . . . . . . . . . . . . 123

xvi

Nomenclature

Roman Symbols

fj Coefficient of the body forces per unit volume vector f , see equation (2.22),

page 17

ni Coefficient of the outward pointing unit normal vector n, see equation (2.24),

page 18

N i (ξ1, ξ2) The interpolation function that corresponds to the i-th node of an

element, see equation (3.9), page 41

sij Coefficient of the relative stresses tensor s, see equation (2.19), page 16

Tjk Coefficient of the double forces per unit area tensor T, see equation (2.22),

page 17

tj Coefficient of the traction vector t, see equation (2.22), page 17

ui Coefficient of the macro-displacements u, see equation (2.6), page 13

u′i Coefficient of the micro-displacements u′, see equation (2.6), page 13

W The potential energy density, see equation (2.15), page 16

Xi Coefficient of the material position vector X of a material particle, see

equation (2.6), page 13

xi Coefficient of the spatial position vector x of a material particle, see equa-

tion (2.6), page 13

xvii

NOMENCLATURE

X ′i Coefficient of the material position vector X′ of a material particle with

respect to a rectangular coordinate system that has its origin fixed in the

particle, see equation (2.6), page 13

x′i Coefficient of the spatial position vector x′ of a material particle with

respect to a rectangular coordinate system that has its origin fixed in the

particle, see equation (2.6), page 13

Greek Symbols

ǫij Coefficient of the macro-strain tensor ǫ, see equation (2.12), page 14

γij Coefficient of the relative deformation tensor γ, see equation (2.13), page 15

κijk Coefficient of the micro-deformation gradient ˜κ, see equation (2.14), page 15

λ, µ Lame constants

µijk Coefficient of the double stresses tensor ˜µ, see equation (2.19), page 16

ωij Coefficient of the macro-rotation tensor ω, see equation (2.12), page 14

Φjk Coefficient of the double forces per unit volume tensor Φ, see equation (2.22),

page 17

Φi (ξ1, ξ2) The shape function that corresponds to the i-th node of an element,

see equation (3.4), page 40

ψjk Coefficient of the micro-deformation ψ, see equation (2.9), page 14

σij Coefficient of the total stresses tensor σ, see equation (2.19), page 16

τij Coefficient of the Cauchy stresses tensor τ , see equation (2.19), page 16

Subscripts

L[i,j] The anti-symmetric part of second order tensor L

L(i,j) The symmetric part of second order tensor L

xviii

Chapter 1

Introduction

1.1 Linear elastic theories with microstructural

effects

Experimental observations have shown that macroscopically many materials are

significantly affected by their microstructure and exhibit a mechanical behavior

which is different than that expected classically. Polycrystals, polymers, metal-

lic and polymeric foams, granular materials, graphite, concrete, asphalt, porous

media, cellular materials, bones and particle or fiber reinforced composites are

some examples of such materials. These microstructural effects become more

pronounced especially when the size or a dimension of the considered structure

becomes small and comparable to the microstructure of the constituent materials.

Representative examples are those of membranes, very thin plates and shells, mi-

croelectronic devices, micromechanical systems, layered plates, bones and smart

structures. However, even in large structures there are mechanical responses lo-

calized to small areas of the structural material (cracks, shear bands, dislocations)

the size of which is comparable to the dimensions of the microstructure. Some

representative works dealing with size effects in the aforementioned materials are

provided in Table 1.1.

Due to the lack of internal parameters, which would correlate the microstruc-

ture with the macrostructure, classical theory of linear elasticity fails to describe

such behavior. Thus, resort should be made to other enhanced elastic theories

1

1. INTRODUCTION

where internal length scale constants correlating the microscopic representative

volume elements with the macrostructure are involved in the constitutive equa-

tions of the considered elastic continuum. Such theories and the most general are

the Cosserat elasticity theory (Cosserat & Cosserat (1909)), the Cosserat the-

ory with constrained rotations or couple stresses theory (Grioli (1960), Toupin

(1962), Mindlin & Tiersten (1962), Koiter (1964)), the strain gradient theory

(Toupin (1964)), the multipolar theory of continuum mechanics (Green & Rivlin

(1964)), the elastic theory with microstructure (Mindlin (1964), Mindlin (1965)),

the micromorphic, microstretch and micropolar elastic theories (Eringen (1999))

and the non-local elasticity (Eringen (1992)). Most of the aforementioned the-

ories have been developed in the decade of 60’s and excellent historical reviews

and comments on the subject can be found in the review articles of Mindlin

& Tiersten (1962) and Tiersten & Bleustein (1974), in the paper of Exadakty-

los & Vardoulakis (2001), in the thesis of Tekoglou (2007) and in the books of

Vardoulakis & Sulem (1995) and Eringen (1999).

Cosserat brothers were the first to develop a general mechanics framework of

continuous media where each point possesses six degrees of freedom like a rigid

body and not three (position of a point in Euclidean space) as the classical elas-

ticity does. These three extra degrees of freedom correspond to three directors,

which represent rotations created by the so called “couple stresses”. As mentioned

by Toupin (1964), the most novel feature of Cosserat theory is the appearance

of couple stresses in the equilibrium equations and equations of motion. How-

ever, although this theory is a landmark in the development of enhanced elastic

theories, it did not receive much attention due to the lack of specific constitutive

relations and the non-symmetry of the considered stresses. By the end of the 50’s

and during the beginning of 60’s the subject of the theory of elasticity with couple

stresses reopened and a plethora of new couple stresses elastic theories were pro-

posed in the literature (Grioli (1960), Toupin (1964), Mindlin & Tiersten (1962),

Koiter (1964)). Most of them explored the special case where the three rotations

coincide with the local rotations of classical elasticity leading thus to a couple

stresses theory with three independent variables, i.e. the three components of dis-

placement vector. Using this consideration, beyond the six components of strains,

other eight of the eighteen components of the first gradient of strain were inserted

2

1.1 Linear elastic theories with microstructural effects

in the expression of the strain energy density function. All the components of the

first gradient of the strain were introduced into the strain energy density function,

in a non-linear fashion, by Toupin (1964) proposing the strain gradient theory.

Considering higher order gradients of strains, Green & Rivlin (1964) developed a

very complicated, but the most general enhanced theory of elasticity called mul-

tipolar theory of continuum mechanics. In 1964 and 1965 Mindlin developed a

general and comprehensive elastic theory with microstructure which is actually

the linear version of Toupin’s strain gradient theory and equivalent to the dipolar

gradient theory of Green and Rivlin. In order to balance the dimensions of strains

and higher order gradients of strains as well as to correlate the micro-strains with

macro-strains, Mindlin (1964) utilized eighteen new constants rendering thus his

general theory very complicated from physical and mathematical point of view.

In the sequel, considering long wave-lengths and the same deformation for macro

and micro structure Mindlin proposed three new simplified versions of his theory,

known as Form I, II and III, utilizing in the constitutive equations seven material

and internal length scale parameters instead of eighteen employed in his initial

model. In Form-I, the strain energy density function is assumed to be a quadratic

form of the classical strains and the second gradient of displacement; in Form II

the second displacement gradient is replaced by the gradient of strains and in

Form III the strain energy function is written in terms of the strain, the gradient

of rotation, and the fully symmetric part of the gradient of strain. Although the

three forms are equivalent to each other and conclude to the same equation of

motion, the Form-II leads to a symmetric total stress tensor, as in the case of clas-

sical elasticity, avoiding thus the problems associated with non-symmetric stress

tensors introduced by Cosserat and couple stress theories. Almost simultane-

ously with Mindlin, Eringen (see Eringen (1999)) proposed three general elastic

theories with microstructural considerations called micromorphic, microstretch

and micropolar theories. The micromorphic continuum is none other than the

classical continuum endowed with extra degrees of freedom represented by three

deformable directors, which represent the degrees of freedom arising from mi-

crodeformations of the physical particle. The linear form of the micromorphic

theory (see Eringen (1999)) coincides with the micro-structure theory of Mindlin

3

1. INTRODUCTION

(1964). In the microstretch version of the above theory, the deformable direc-

tors contain only stretches and not microshears, while in the case where the

three directors become rigid and represent three independent rotations of the

microparticle the micromorphic becomes micropolar theory. Finally, a different

theory, which takes into account microstructural effects in a complete non-local

manner, is the non-local theory of elasticity proposed by Eringen (1992). As it is

mentioned in the corresponding works of Eringen, nonlocal continuum mechan-

ics differs from classical and other enhanced continuum mechanics in two basic

ways: (a) balance laws are postulated to be nonlocal (global). This is achieved

by introducing some nonlocal residuals into localized balance equations. Global

(integral) values of these residuals are assumed to vanish; (b) constitutive equa-

tions are nonlocal, i.e., they are functionals of the independent variables over all

points of the body. Although elegant, its treatment is a very difficult task, due

to the integral form of the constituent relations.

After the aforementioned pioneering works, the last two decades a plethora

of papers dealing with new versions of these enhanced elastic theories as well as

with solutions of couple stresses and gradient elastic boundary value problems

have appeared in the literature. This published work is so large that it is not

possible to be mentioned in this chapter. Since the present thesis is referred to

Mindlin’s Form II gradient elasticity theory, from now and further the literature

review will be confined to this kind of theories. One can mention here the simple

gradient elasticity theory of Aifantis (1992), the gradient elasticity theory with

surface energy of Vardoulakis & Sulem (1995) and the gradient theory of Fleck

& Hutchinson (1997) and Fleck & Hutchinson (2001). Aifantis (1992) and Ru &

Aifantis (1993) proposed a very simple gradient elastic model requiring only one

new gradient elastic constant plus the standard Lame ones. This gradient elastic

model can be considered as the simplest possible special case of Form-II version

of Mindlin’s theory. The main problem with Aifantis’ model is that due to the

complete lack of a variational formulation, the considered boundary conditions

are not compatible with the corresponding correct ones provided by Mindlin. The

correction on the boundary conditions is made later in the paper of Vardoulakis

et al. (1996). The gradient elastic with surface energy theory of Vardoulakis &

Sulem (1995) is slightly more complicated than that proposed by Aifantis and

4

1.2 Numerical solutions in gradient elastic theories

co-workers but it is a direct consequence of the continuum model proposed by

Casal (1972) and not a special case of Mindlin’s general theory. Finally, Fleck

& Hutchinson (1997) and Fleck & Hutchinson (2001) decomposed the second

gradient of displacement into the stretch gradient and the rotation gradient ten-

sors proposing thus an alternative version of Mindlin’s Form I gradient elasticity

theory.

Composite Materials Lloyd (1994), Nan & Clarke (1996), Groh et al.

(2005)

Foams Lakes (1983), Lakes (1986), Tekoglou (2007)

Polycrystals Smyshlyaev & Fleck (1996), Dillard et al. (2006)

Metals Fleck et al. (1994), Nix & Gao (1998)

Bones Yang & Lakes (1982), Lakes (1995)

Concrete Vliet & Mier (1999), Dessouky et al. (2006)

Polymers Lam et al. (2003), McFarland & Colton (2005),

Chen & Lakes (1989), Lakes (1983)

Granular Materials Vardoulakis & Sulem (1995)

Porous Materials Lakes (1983), Lakes (1986)

Graphite Tang (1983)

Table 1.1: Works on size effects on specific materials

1.2 Numerical solutions in gradient elastic the-

ories

As in the case of classical elasticity, the solution of gradient elastic problems

with complicated geometry and boundary conditions requires the use of numer-

ical methods such as the finite element method (FEM), the boundary element

method (BEM), the finite differences method (FDM) or the meshless local Petrov-

Galerkin (MLPG) method.

The FEM is the most widely used numerical method for solving applied me-

chanics problems. Shu et al. (1999) were the first to use the FEM for solving elas-

tostatic problems in the framework of the gradient elasticity theories of Mindlin.

5

1. INTRODUCTION

Since then, many papers dealing with FEM solutions of gradient elastic problems

have appeared in the literature. Here one can mention the FEM formulations

of Amanatidou & Aravas (2002), Engel et al. (2002), Tenek & Aifantis (2002),

Matsushima et al. (2002), Peerlings & Fleck (2004), Soh & Wanji (2004), Imatani

et al. (2005), Askes & Gutierrez (2006), Dessouky et al. (2003), Dessouky et al.

(2006), Akarapu & Zbib (2006), Giannakopoulos et al. (2006), Markolefas et al.

(2007), Markolefas et al. (2009), Askes et al. (2007), Askes et al. (2008), Pa-

panicolopulos (2008), Papanicolopulos et al. (2009), Zervos et al. (2001), Zervos

(2008), Zervos et al. (2009), Bennett & Askes (2009) and Zybell et al. (2009). It

should be mentioned that from the above papers only the works of Papanicolopu-

los (2008), Papanicolopulos et al. (2009) and Zervos et al. (2009) deal with three

dimensional problems. The main problem with a conventional FEM formulation

is the requirement of using elements with C1 continuity, since the presence of

higher order gradients in the expression of potential energy leads to an equilib-

rium equation represented by a forth order partial differential operator. Although

a displacement formulation is conceptually simpler and the most convenient for

implementation in existing finite element codes, only the works of Akarapu &

Zbib (2006) and Papanicolopulos et al. (2009) implement C1 elements with the

later being the most comprehensive and complete, since it derives both two and

three dimensional C1 finite elements. The other works bypass the problem via

mixed formulations, Lagrange multipliers and penalty methods.

On the other hand, the BEM is a well-known and powerful numerical tool,

successfully used in recent years to solve various types of engineering problems

(Beskos (1987); Beskos (1997)). A remarkable advantage it offers as compared

to other numerical methods, such as the FDM and the FEM, is the reduction of

the dimensionality of the problem by one. Thus, three dimensional problems are

accurately solved by discretizing only two-dimensional surfaces surrounding the

domain of interest. In the case where the problem is characterized by an axisym-

metric geometry, the BEM reduces further the dimensionality of the problem,

requiring just a discretization along a meridional line of the body. These advan-

tages in conjunction with the absent of C1 continuity requirements, render the

BEM ideal for analyzing gradient elastic problems. Tsepoura et al. (2002) were

the first to use BEM for solving elastostatic problems in the framework of the

6

1.3 Gradient elastic fracture mechanics

gradient elasticity theories of Mindlin. This work was followed by the publica-

tions of Tsepoura & Polyzos (2003), Polyzos et al. (2003), Tsepoura et al. (2003),

Polyzos et al. (2005), Polyzos (2005), Karlis et al. (2007), Karlis et al. (2008),

which are the only papers dealing with two and three dimensional BEM solutions

of static and dynamic gradient elastic and fracture mechanics problems. The

present thesis is the continuation of this research to Mindlin’s Form II gradient

elastic theory.

Recently, Atluri and co-workers proposed the Local Boundary Integral Equa-

tion (LBIE) method (Zhu et al. (1998)) and the Meshless Local Petrov-Galerkin

(MLPG) method (Atluri & Zhu (1998)) as alternatives to the BEM and FEM,

respectively. Both methods are characterized as “truly meshless” since no back-

ground cells are required for the numerical evaluation of the involved integrals.

At the same time the so-called element-free Galerkin methods appear also in

the literature Belytschko et al. (1996).In all these methods properly distributed

nodal points, without any connectivity requirement, cover the domain of inter-

est as well as the surrounding global boundary instead of any boundary or fi-

nite element discretization. All nodal points belong to regular sub-domains (e.g.

circles for two-dimensional problems) centered at the corresponding collocation

points. The fields at the local and global boundaries as well as in the interior of

the subdomains are usually approximated by the Moving Least Squares (MLS)

approximation scheme or Radial Basis Functions (RBF). Since mesh-free approx-

imations possess nonlocal properties, they automatically satisfy the higher order

continuity requirement. Representative works on the subject are those of Tang

& Atluri (2003), Pamin et al. (1998) and Sun & Liew (2008).

1.3 Gradient elastic fracture mechanics

As it is explained in the excellent paper of Exadaktylos & Vardoulakis (2001), in

linear elastic fracture analysis, where large strain and stress gradients occur, the

gradient elastic theories seem to be ideal for studying the strain and stress fields

near the crack tip at the microscale. For this reason many analytical works dealing

mainly with two dimensional, gradient elastic, fracture mechanics problems un-

der conditions of plane strain or anti-plane strain have appeared in the literature.

7

1. INTRODUCTION

One can mention the analytical works of Vardoulakis et al. (1996), Exadakty-

los et al. (1996), Vardoulakis & Exadaktylos (1997), Exadaktylos (1998), Huang

et al. (1997), Shi et al. (2000), Fannjiang et al. (2002), Georgiadis (2003), Geor-

giadis & Grentzelou (2006), Tong et al. (2005), Chan et al. (2008), Radi (2008),

Giannakopoulos & Gavardinas (2008) and Gourgiotis & Georgiadis (2009). The

main conclusion they reach, is that near the crack tip displacements and strains

behave as r3/2 and r1/2 functions, respectively, with r being the distance from the

crack tip, while double stresses and total stresses exhibit a singular behaviour of

order r−1/2 and r−3/2, respectively. The important part of these results is that

gradient elastic theories predict the same cusp-like crack shape with Barenblatt’s

cohesive zone theory (Barenblatt (1962)) without demanding extra interatomic

forces beyond those imposed by the non-classical boundary conditions. On the

other hand, stress fields near to the tip of the crack remain singular.

In all the above works no computation of stress intensity factors (SIF) has

been reported, because of the complexity of the problem. It is obvious that for

the solution of complex gradient elastic fracture mechanics problems, the use

of numerical methods is imperative. Amanatidou & Aravas (2002) proposing a

two dimensional mixed FEM formulation for Mindlin’s Form I, II and III theory,

solve the mode III crack problem providing results for the antiplane stress and

displacement fields around the tip of the crack. Although their findings are in

agreement with the theoretical ones of Georgiadis (2003), there are no results

defining explicitly the mode III SIF or correlating the SIF with the constants

inserted by the considered gradient elastic model. Imatani et al. (2005) exploiting

a mixed FEM formulation for the Mindlin’s Form II gradient elastic theory solved

a plane mode I crack problem providing mainly results concerning the variation

of the energy release rate with respect to the length of the crack and for specific

values of the gradient elastic constants. Akarapu & Zbib (2006) and Markolefas

et al. (2009) forming a mixed FEM formulation for the simplified Form II gradient

elastic theory, they calculate stresses and displacements near the tip of a mode I

crack without giving any information about the SIF and its dependence on the

considered gradient elastic constant. Wei (2006) based on triangular C1 elements

solved Mode I, II and III gradient elastic fracture problems in two dimensions. As

previous investigators, he calculated stresses and displacements near to the tip

8

1.4 Structure of the thesis

of a mode I crack without giving any information about the corresponding SIFs.

Finally Askes et al. (2008) based on Ru-Aifantis theorem solved through a direct

FEM formulation a Helmholtz type partial differential equation instead of the

forth order equation of gradient elasticity. However, their results are questionable

since they satisfy boundary conditions which are different from those established

in Mindlin theory.

Very recently, Karlis et al. (2007) addressed a numerical methodology, which

combines the BEM proposed by Polyzos et al. (2003) and Tsepoura et al. (2003)

with special crack tip boundary elements for the numerical determination of the

Stress Intensity Factor (SIF) in plane mode I and mixed mode (I & II) fracture

mechanics gradient elastic problems. Adopting the idea of variable-order singu-

larity boundary elements around the tip of the crack for the evaluation of the

corresponding stress intensity factor (SIF) (Lim et al. (2002), Zhou et al. (2005)),

a new special variable-order singularity discontinuous element was proposed for

the treatment of singular fields around the tip of the crack. The SIFs determi-

nation was accomplished by a displacement type of formulation in connection

with the multiregion approach. As it is mentioned in the review papers of Beskos

(1997), Aliabadi (1997) and Dominguez & Ariza (2003), the displacement based

BEM has the disadvantage of subregioning but is associated with lower order sin-

gularity kernels, than those of either the traction-based or the dual BEM. Later

the same authors extended their work to three dimensional fracture mechanics

problems and their results for Mode I cracks are presented in Karlis et al. (2008).

1.4 Structure of the thesis

This thesis is organized as follows:

Chapter 2 introduces the generalized gradient elasticity theory of Mindlin,

which is used throughout this thesis. Furthermore, the simplified versions of his

theory, known as Form I, II and III, are mentioned and the second simplified form

is derived from the generalized theory. Finally, the integral representation of a

Form II gradient elastic boundary value problem is presented.

In chapter 3 the Form II boundary element formulation is described, as well

as the techniques used therein, i.e. symmetry and subregioning. In addition, the

9

1. INTRODUCTION

method used for the calculation of the boundary integrals is presented in detail

and the chapter closes with numerical examples that are solved and compared to

the corresponding analytical solutions.

Chapter 4 starts with a brief introduction to fracture mechanics paying special

attention to the abrupt changes of the fields that occur near the crack tip. Two

new boundary elements are presented, a line and a quadrilateral one, that address

the occurring singularities and calculate efficiently the unknown fields, as well as

the stress intensity factor of the crack. Finally, some numerical examples are

presented, regarding mode I and mixed mode I & II cracks in elastic and gradient

elastic materials.

Chapter 5 concludes this thesis, summarising the presented work and drawing

concluding remarks. A discussion on possible future research follows.

1.5 Novelty

This thesis comes as a continuation of the work done in the field of higher or-

der strain gradient elasticity theories. The first implementation of such theories

in BEM was made by Tsepoura et al. (2002), for the simplest possible case of

Mindlin’s gradient elasticity theory. In the formulation described therein, only

one gradient elastic constant has been utilized for the correlation of the micro-

structure to the characteristic lenght of the structure. After that, the works of

Tsepoura & Polyzos (2003), Polyzos et al. (2003), Tsepoura et al. (2003), Polyzos

et al. (2005), Polyzos (2005), Karlis et al. (2007) and Karlis et al. (2008) followed,

dealing with 2D and 3D gradient elastic and fracture mechanics problems.

Throughout the preparation of this thesis a series of new results have been

obtained. Since they are not always strongly pointed out in the text, a brief list

containing the new results is provided here.

1. The 2D and 3D, static fundamental solutions of Mindlin’s Form II gradient

elasticity theory have been derived.

2. The 2D and 3D BEM integral formulation of a static Form II gradient

elastic boundary value problem has been obtained.

10

1.5 Novelty

3. A new three-noded line special element, with variable order of singularity

has been developed, for dealing with the unknown fields of classical and

gradient elasticity near the tip of the crack in two dimensional fracture

mechanics problems.

4. A new eight-noded quadrilateral special element, with variable order of

singularity has also been created for treating classical and gradient elasticity

near the tip of the crack in 3D.

5. The numerical results presented in Chapter 4, that indicate the accurate

calculation of the displacement and traction fields near the crack tip, as well

as the calculation of the stress intensity factors in classical and gradient

elasticity.

11

1. INTRODUCTION

12

Chapter 2

Mindlin’s Theory of Elasticity

with Microstructure

It is well known that in classical elasticity all the fundamental quantities – ma-

terial constants, displacements, strains and stresses – at any point x of the ana-

lyzed domain are taken as mean values over very small volume elements around

x, the size of which must be sufficiently large in comparison with the material’s

microstructure. Exadaktylos & Vardoulakis (2001), based on this assumption,

presented a very enlightening and simple example, which reveals the necessity of

enhanced elastic theories. They considered a one-dimensional continuum, a point

x in it, centered in a small volume element l (Figure 2.1) and the mean value of

displacement throughout the element l, i.e.

〈u〉|l;x =1

l

l/2∫

−l/2

u (x+ ξ) dξ (2.1)

Taking Taylor expansion of u (x+ ξ) near the point x and keeping only the

Figure 2.1: The one dimensional continuum

13

ch2-Gradient_Elasticity_Theories/Figs/EPS/1Dcontinuum.eps

2. MINDLIN’S THEORY OF ELASTICITY WITH

MICROSTRUCTURE

constant term u (x+ ξ) one easily obtains from (2.1) that

〈u〉|l;x = u (x) (2.2)

which means that for constantly varying displacements in l, the aforementioned

assumption of classical elasticity is fulfilled. The same happens when the linear

term of Taylor’s expansion is also kept, i.e.

(x+ ξ) = u (x) + u′ (x) ξ ⇒ 〈u〉|l;x = u (x) (2.3)

However, things change when displacements vary quadratically (Figure 2.2(a)) in

l and the third term of Taylor’s expansion, u (x+ ξ) = u (x)+u′ (x) ξ+ 12u′′ (x) ξ2,

should be taken into account. In this case, one can find that

〈u〉|l;x = u (x) − l2

24

d2u

dx2

∣

∣

∣

∣

x

= u (x) − l2

24

de

dx

∣

∣

∣

∣

x

(2.4)

Relation (2.4) leads to the following very interesting remarks:

(a) (b)

Figure 2.2: 1D continuum (a) with quadratically varying displacements and (b)

smaller element size

i. The main value of displacements is not equal to the displacement at point

x. Of course as shown in Figure 2.2(b), one can consider smaller elements

l, where displacements vary constantly or linearly. However, in this case l

becomes comparable to the microstructure and the assumption “the size of

l must be sufficiently large in comparison with the material microstructure”

is violated.

ii. The extra term in relation (2.4) is l2

24dedx

∣

∣

x. This reveals that the locality of

classical elasticity is not able to satisfy the non-local requirements of dedx

∣

∣

x.

14

ch2-Gradient_Elasticity_Theories/Figs/EPS/1Dcontinuum_Step1.eps

ch2-Gradient_Elasticity_Theories/Figs/EPS/1Dcontinuum_Step2.eps

2.1 General Strain Gradient Theory of Elasticity

iii. The term l2

24dedx

∣

∣

xindicates that the problem can be solved if one formulates

a new theory of elasticity where higher order gradients of strains are taken

into account in the expression of the elastic potential energy density.

iv. Finally, the most interesting remark is the appearance of l in (2.4). Actually,

l is an internal length scale parameter, which gives a comparison between

microstructure and macrostructure.

The main conclusion of this example is that in elastic problems where abrupt

changes of displacements, strains and stresses occur, a new elastic theory, en-

hanced by higher order gradient terms and internal length scale parameters, is

required. As it is mentioned in the introduction of the present thesis, such a

theory, namely the generalized elastic theory with microstructure of Mindlin, is

adopted and presented in what follows.

2.1 General Strain Gradient Theory of Elastic-

ity

2.1.1 Kinematics

In 1964, R.D. Mindlin (Mindlin (1964)) formulated an elastic continuum theory

which contained some of the properties of a crystal lattice. This resulted from the

theoretical assumption of a unit cell that was incorporated in his theory. The unit

cell can be interpreted as a molecule of a polymer, a crystallite of a polycrystal

or a grain of a granular material.

In short, Mindlin considered a macro-volume V bounded by a surface S and

a micro-volume V ′, included in V , defining that way the macro- and micro-

displacements as following

ui = xi −Xi (2.5)

u′i = x′i −X ′i (2.6)

with i = 1, 2, 3 and Xi and xi being the components of the material and spatial

position vectors of a material particle with respect to a fixed origin. Accord-

15


MICROSTRUCTURE

ingly, X ′i and x′i are the material and spatial position vectors with respect to a

rectangular coordinate system that has its origin fixed in the particle.

Then, after the macro- and micro-displacements have been defined, he re-

quired that their gradients be small (|∂ui/∂Xi| ≈ |∂ui/∂xi| ≪ 1, |∂u′i/∂X ′i| ≈

|∂u′i/∂x′i| ≪ 1), which resulted to

∂uj

∂Xi≈ ∂uj

∂xi= ∂iuj (2.7)

∂u′j∂X ′

i

≈∂u′j∂x′i

= ∂′iu′j (2.8)

Furthermore, assuming that the micro-displacements can be expressed as a

sum of products of functions of x′i and other functions of xi and t (time), he wrote

the micro-displacement as an approximation, retaining only a single, linear term

of the series

u′j = x′kψkj (2.9)

The function ψkj is the micro-deformation. Differentiating the above equation

to obtain the displacement gradient results to

∂′iu′j = ψij (2.10)

which can be interpreted as the micro-deformation ψij being homogeneous in the

micro-volume V ′ and non-homogeneous in the macro-volume V .

The tensor ψij can be split into symmetric and antisymmetric parts, defining

that way the micro-strain ψ(ij) and the micro-rotation ψ[ij] respectively.

In addition, the macro-strain and macro-rotation tensors can be defined as in

classical elasticity,

ǫij =1

2(∂iuj + ∂jui) (2.11)

ωij =1

2(∂iuj − ∂jui) (2.12)

and the relative deformation (the difference of the macro-displacement gradient

and the micro-deformation) can be defined.

γij = ∂iuj − ψij (2.13)

16


Finally, the micro-deformation gradient is defined as the macro-gradient of

the micro-deformation.

κijk = ∂iψjk (2.14)

Note that all three tensors ǫ, γ and κ are independent of the micro-coordinates

x′i.

(a)

(b)

Figure 2.3: Kinematic parameters of Mindlin’s theory of elasticity with mi-

crostructure

2.1.2 Equations of Equilibrium and Boundary Conditions

The potential energy density is assumed to be a function of ǫij , γij and κijk.

W = W (ǫij , γij, κijk) (2.15)

17

ch2-Gradient_Elasticity_Theories/Figs/EPS/Stencil1.eps

ch2-Gradient_Elasticity_Theories/Figs/EPS/Stencil2.eps


MICROSTRUCTURE

Then the following quantities are defined

τij =∂W

∂ǫij= τji (2.16)

sij =∂W

∂γij(2.17)

µijk =∂W

∂κijk(2.18)

σij = τij + sij (2.19)

which correspond to the Cauchy stresses, relative stresses, double stresses and

total stresses respectively. As Mindlin (1964) explains, the nature of double

stresses can be explained via the Figures 2.4.

Figure 2.4: Typical components of the double stress tensor and gradient micro-

deformation

18

ch2-Gradient_Elasticity_Theories/Figs/EPS/Mindlin.eps


The variation of the potential energy density function W is written as

δW = τijδǫij + µijkδκijk

= τij∂iδuj + µijk∂i∂jδuk

= ∂j [(τjk − ∂iµijk) δuk] − ∂j (τjk − ∂iµijk) δuk + ∂i (µijk∂jδuk)

(2.20)

Utilizing the divergence theorem, the total potential energy becomes∫

V

δW dV =

∫

S

nj (τjk − ∂iµijk) δuk dS

−∫

V

∂j (τjk − ∂iµijk) δuk dV +

∫

S

niµijk∂jδuk dS

(2.21)

with n being the normal unit vector of the surface S. The form of the above

equation, implies the following form for the variation of work done by external

forces.

δW1 =

∫

V

Fjδuj dV +

∫

S

Φjkδψjk dV +

∫

S

tjδuj dS +

∫

S

Tjkδψjk dS (2.22)

The definitions of ui and ψjk, and the fact that the integrands of the volume and

surface integrals represent variations of work per unit volume and area, yield the

physical significances of the coefficients of δui and δψjk. Φjk = ∂iµijk +σjk can be

interpreted as double force per unit volume, fi is the body force per unit volume,

tj the traction vector and Tjk the double forces per unit area.

Substituting (2.21) and (2.22) into the equation of equilibrium

δ

t1∫

t0

W =

t1∫

t0

W1 (2.23)

and dropping the integration with respect to time, we obtain the variational

equation of motion∫

V

(∂iτij + ∂iσij + fj) δujdV

+

∫

V

(∂iµijk + σjk + Φjk) δψjkdV (2.24)

+

∫

S

[tj − ni (τij + σij)] δujdS +

∫

S

(Tjk − niµijk) δψjkdS = 0

19


MICROSTRUCTURE

where ni are the components of the outward pointing unit normal vector of the

surface S.

From the above equation it is easy to extract the stress equations of equilib-

rium

∂iσij + fj = 0 (2.25a)

∂iµijk + sjk + Φjk = 0 (2.25b)

and the boundary conditions

tj = ni (τij + sij) = niσij (2.26)

Tjk = niµijk (2.27)

2.1.3 Constitutive Equations

In order to extract the constitutive equations, Mindlin considered the following

quadratic form for the potential energy density function

W =1

2cijklǫijǫkl +

1

2bijklγijγkl +

1

2αijklmnκijkκlmn

+ dijklmγijκklm + fijklmκijkǫlm + gijklγijǫkl (2.28)

Taking into account relations (2.16-2.24), the Cauchy stresses, relative stresses,

double stresses and total stresses are given by

τpq = cpqijǫij + gijpqγij + fijkpqκijk (2.29)

spq = gpqijǫij + bijpqγij + dpqijkκijk (2.30)

µpqr = fpqrijǫij + dijpqrγij + fpqrijkκijk (2.31)

σpq = τpq + spq (2.32)

In the case of isotropic material, the coefficients dijklm and fijklm vanish, be-

cause there are no isotropic tensors of odd rank. Since the most general form of

20


fourth and sixth order isotropic tensors is a linear function of tensor products of

Kronecker deltas

Lijkl = aδijδkl + bδikδjl + cδilδjk

Fijklmn = C1δijδklδmn + C2δijδkmδln + C3δijδknδlm

+ C4δikδjlδmn + C5δikδjmδln + C6δikδjnδlm

+ C7δilδjkδmn + C8δilδjmδkn + C9δilδjnδkm

+ C10δimδjkδln + C11δimδjlδkn + C12δimδjnδkl

+ C13δinδjkδlm + C14δinδjlδkm + C15δinδjmδkl

the remaining coefficients are written as

cijkl = λδijδkl + µ1δikδjl + µ2δilδjk

bijkl = b1δijδkl + b2δikδjl + b3δilδjk

gijkl = g1δijδkl + g2δikδjl + g3δilδjk

aijklmn = a1δijδklδmn + a2δijδkmδnl + a3δijδknδlm (2.33)

+ a4δjkδilδmn + a5δjkδimδnl + a6δjkδinδlm

+ a7δkiδjlδmn + a8δkiδjmδnl + a9δkiδjnδlm

+ a10δilδjmδkn + a11δjlδkmδin + a12δklδimδjn

+ a13δilδjnδkm + a14δjlδknδim + a15δklδinδjm

Finally taking into account the symmetry of the macro-strain and the com-

mutative property of multiplication, one can see that from the 1458 coefficients

of eq (2.28) only the 903 are independent. In addition, considering an isotropic

material the constants of eqs (2.33) become

µ1 = µ2 = µ, g2 = g3

a1 = a6, a2 = a9 (2.34)

a5 = a7, a11 = a12

21


MICROSTRUCTURE

and the potential energy density function is simplified to

W =1

2λǫiiǫjj + µǫijǫij +

1

2b1γiiγjj +

1

2b2γijγij +

+1

2b3γijγji + g1γiiǫjj + g2 (γij + γji) ǫij +

+ a1κiikκkjj + a2κiikκjkj +1

2a3κiikκjjk +

+1

2a4κijjκikk + a5κijjκkik +

1

2a8κijiκkjk +

+1

2a10κijkκijk + a11κijkκjki +

1

2a13κijkκikj +

+1

2a14κijkκjik +

1

2a15κijkκkji (2.35)

Accordingly, the constitutive equations become

τpq = λδpqǫii + 2µǫpq + g1δpqγii + g2 (γpq + γqp) (2.36)

σpq = g1δpqǫii + 2g2ǫpq + b1δpqγii + b2γpq + b3γqp (2.37)

µpqr = a1 (κiipδqr + κriiδpq) + a2 (κiiqδpr + κiriδpq) + a3κiirδpq

+ a4κpiiδqr + a5 (κqiiδpr + κipiδqr) + a8κiqiδpr + a10κpqr

+ a11 (κrpq + κqrp) + a13κprq + a14κqpr + a15κrqp (2.38)

Mindlin’s theory is not confined to spatially homogeneous material proper-

ties. Taking for instance the elastic coefficients and the densities to be periodic

functions with period 2d, equal to the edge length of the unit cell, would de-

scribe the structure of a crystal lattice. However, this would increase the model’s

complexity and would have made it highly intractable. In order to avoid exactly

that, Mindlin considered the macro-material to be homogeneous, having in mind

that for wavelengths greater than the dimensions of the unit cells it would be a

sufficiently good approximation for demonstrating the main features of his theory.

Considering an isotropic macro-material, replacing eqs (2.11), (2.13) and (2.14)

into the constitutive equations and inserting them into the equations of equilib-

rium (2.25), one obtains the equilibrium equations in terms of the displacements.

(µ+ 2g2 + b2) ∂j∂jui + (λ+ µ+ 2g1 + 2g2 + b1 + b3) ∂i∂juj

− (g1 + b1) ∂iψjj − (g2 + b2) ∂jψji − (g2 + b3) ∂jψij + fi = 0(2.39a)

22

2.2 Form I, II and III Gradient Elasticity Theories

(a1 + a5) (∂k∂lψklδij + ∂i∂jψkk) + (a2 + a11) (∂j∂kψki + ∂i∂kψjk)

+ (a13 + a14) ∂i∂kψkj + a4∂k∂kψllδij + (a8 + a15) ∂j∂kψik

+ a10∂k∂kψij + a13∂k∂kψji + g1∂kukδij + g2 (∂iuj + ∂jui)

+ b1 (∂kuk − ψkk) δij + b2 (∂iuj − ψij) + b3 (∂jui − ψji) + Φij = 0

(2.39b)

2.2 Simplified Versions of Mindlin’s General The-

ory: Form I, II and III Gradient Elasticity

Theories

Mindlin found that the modes that appear in a gradient elastic material due to

a micro-vibration, ψij = Aijeiωt, (dilatational, shear, equivoluminal extensional

and rotational), are analogous to the thickness modes of vibration that appear

in homogeneous plates. Having in mind the derivation of the low frequency

approximation in plate theory, he used the same process to simplify his gradient

elasticity theory.

In homogeneous plates, when the excited frequencies are low compared to the

thickness modes of the plate and the wavelengths long compared to the thickness

of the plate, the coupling of the flexural and extensional modes with the thickness

modes is negligible. As the frequencies of the flexural and extensional modes

approach zero, the thickness-shear deformation approaches zero, but the thickness

stretch deformation does not. However, the stress associated to the thickness

stretch tends to zero. This means that the symmetric and anti-symmetric parts

of deformation and stress have to be treated differently when deriving the low

frequency approximation.

To obtain the low frequency approximation for flexure in plate theory, the

thickness shear deformation is sent to zero and the associated modulus of elasticity

is sent to infinity. Furthermore, in the case of extension, the thickness stress is

set equal to zero and the resulting constitutive equation is used to eliminate

the thickness strain from the remaining equations. In both cases, flexure and

extension, the thickness velocities are set equal to zero in the kinetic energy,

because their contribution is negligible.

23


MICROSTRUCTURE

As mentioned earlier, the modes appearing in gradient elasticity can be as-

sociated with the ones appearing in homogeneous plate theory. Specifically, the

micro-modes are analogous to the thickness modes of vibration and the trans-

verse and longitudinal acoustic modes are analogous to the flexural and exten-

sional modes of the plate. Furthermore, the micro-velocities ψij correspond to the

thickness velocities of the plate and the dimensions of the unit cell 2d are similar

to the plate thickness. Finally, the antisymmetric part of the relative deformation

γ[ij] is analogous to the thickness shear deformation, the coefficients b2 and b3 of

eq. (2.35) correspond to the thickness shear moduli and the symmetric part of

the relative stress s(ij) is associated with the stress resulting from the thickness

stretch of the plate.

Using the above analogies, the assumptions for the derivation of the low fre-

quency approximation in plate theory can be translated in terms of gradient

elasticity.

s(ij) = 0 (2.40)

γ[ij] → 0 (2.41)

b2 − b3 → 0 (2.42)

These assumptions form the basic hypothesis for the low frequency approximation

of Mindlin’s theory. Obviously all the above are also valid for static problems.

The constitutive equations for the Cauchy and relative stresses (2.36-2.38)

become

τpq = λδpqǫii + 2µǫpq + g1δpqγii + 2g2γpq (2.43)

s(pq) = g1δpqǫii + 2g2ǫpq + b1δpqγii + (b2 + b3) γ(pq) (2.44)

s[pq] = (b2 − b3) γ[pq] (2.45)

Equation (2.44) can now be solved for γ(pq)

γ(pq) = −αδpqǫii + (1 − β) ǫpq (2.46)

with α and β depending on the potential energy density function coefficients.

α =1

b2 + b3

(

g1 −b1 (3g1 + 2g2)

3b1 + b2 + b3

)

β = 1 +2g2

b2 + b3

24

2.2 Form I, II and III Gradient Elasticity Theories

Since γ[pq] has been sent to zero, the symmetric and antisymmetric parts of

the micro-deformation are functions of the macro-strains and the macro-rotations

respectively.

ψ(pq) = αδpqǫii + βǫpq (2.47a)

ψ[pq] = ωpq (2.47b)

Accordingly the micro-deformations κijk become

κijk → ακillδjk +1

2(1 + β) κijk −

1

2(1 − β) κikj (2.48)

with κijk = ∂i∂juk = κjik. This means that in static problems, κijk becomes κijk,

a function of the second gradient of displacements. Then the potential energy

density function is written

W → W =1

2λǫiiǫjj + µǫijǫij + α1κiikκkjj

+ α2κijjκikk + α3κiikκjjk + α4κijkκijk + α5κijkκkji

(2.49)

with the constants λ, µ and α1-α5 depending on the potential energy density

function coefficients as shown in Appendix A. This is the first form for the

potential energy density function, also referred to as Form I.

The components κijk may be arranged in tensors, whose components are inde-

pendent linear combinations of ∂i∂juk in more than one ways, resulting in different

forms of he potential energy density function. Specifically, arranging the terms

in such a way that the gradient of strain is formed, leads to the second form of

Mindlin’s gradient elasticity theory.

κijk ≡ ∂iǫjk =1

2(∂i∂juk + ∂i∂kuj) = κikj (2.50)

Then, the potential energy density becomes

W → W =1



(2.51)

with λ = λ, µ = µ and α1-α5 presented in Appendix A. This form is refereed to

as Form II.

25


MICROSTRUCTURE

Finally, separating the curl of the strain κij from the second gradient of dis-

placements results to the third form, which is refereed to as Form III.

κij ≡ ejlm∂lǫmi =1

2ejlm∂i∂lum (2.52)

¯κijk = κijk +1

3eiljκkl +

1

3eilkκjl =

1

3(∂i∂juk + ∂k∂iuj + ∂j∂kui) (2.53)

κijk = ¯κijk −1

3eilj κkl −

1

3eilkκjl (2.54)

The potential energy density expression for Form III is now expressed as a

function of κij and ¯κijk.

W → W =1

2λǫiiǫjj + µǫijǫij + 2d1κijκij + 2d2κijκji

+3

2α1 ¯κiij ¯κkkj + α2 ¯κijk ¯κijk + f eijkκij ¯κkll

(2.55)

with the constants d1, d2, α1, α2 and f presented in Appendix A.

2.3 Form II Gradient Elasticity Theory

In this section the equilibrium equations corresponding to Mindlin’s Form II

gradient elasticity theory are derived.

As mentioned earlier, the potential energy density function W of the gen-

eral gradient elasticity theory becomes W in the case of the Form II, when the

components ∂i∂juk are arranged in such a way that the gradient of strains is

formed.

W =1



(2.56)

The stresses are now defined with respect to W as

τij =∂W

∂ǫij= τji (2.57a)

µijk =∂W

∂κijk= µikj (2.57b)

26


Theory Potential Energy Density function

General Strain Gradient

Elasticity Theory

W = 12λǫiiǫjj + µǫijǫij + 1

2b1γiiγjj + 12b2γijγij +

12b3γijγji + g1γiiǫjj + g2 (γij + γji) ǫij + a1κiikκkjj +

a2κiikκjkj + 12a3κiikκjjk + 1

2a4κijjκikk + a5κijjκkik +12a8κijiκkjk+

12a10κijkκijk+a11κijkκjki+

12a13κijkκikj+

12a14κijkκjik + 1

2a15κijkκkji

Mindlin’s Form I W = 12 λǫiiǫjj + µǫijǫij + α1κiikκkjj + α2κijj κikk +

α3κiikκjjk + α4κijkκijk + α5κijkκkji

Mindlin’s Form II W = 12 λǫiiǫjj + µǫijǫij + α1κiikκkjj + α2κijj κikk +

α3κiikκjjk + α4κijkκijk + α5κijkκkji

Mindlin’s Form III W = 12 λǫiiǫjj + µǫijǫij + 2d1κij κij + 2d2κij κji +

32 α1 ¯κiij ¯κkkj + α2 ¯κijk ¯κijk + f eijkκij ¯κkll

Simple Gradient Elasticity

theoryW = 1

2 λǫiiǫjj + µǫijǫij + 12 λg2κijjκikk + µg2κijkκijk

Simple Gradient Elasticity

theory with Surface En-

ergy

W = 12λǫiiǫjj + µǫijǫji + 1

2λℓ2∂kǫii∂kǫjj +

µℓ2∂kǫij∂kǫji + 12λℓk∂k (ǫiiǫjj) + µℓk∂k (ǫijǫji), ℓ:

characteristic length of the material, ℓk: surface

energy characteristic directors

Table 2.1: The renumbering of the element nodes, so that the crack front always

resides on the first side.

27


MICROSTRUCTURE

and after some calculations they finally become

τpq = λδpqǫii + 2µǫpq (2.58a)

µpqr =1

2α1 (δpqκrii + 2δqrκiip + δrpκqii) + 2α2δqrκpii

+ α3 (δpqκiir + δprκiiq) + 2α4κpqr + α5 (κrpq + κqrp)(2.58b)

or in vector notation

τ = λ (∇ · u) I + µ (∇u + u∇) (2.59a)

µ =1

2α1

[

∇2u ⊗ I + I ⊗∇∇ · u + ∇∇ · u⊗ I +(

∇∇ · u ⊗ I

)213]

+ 2α2∇∇ · u⊗ I

+1

2α3

[

I ⊗∇2u + I ⊗∇∇ · u +(

∇2u ⊗ I

)213

+(

∇∇ · u ⊗ I

)213]

+ α4 (∇∇u + ∇u∇) +1

2α5 (2u∇∇ + ∇∇u + ∇u∇)

(2.59b)

where I = δijxi ⊗ xj, a ⊗ b = aibjxi ⊗ xj and (a⊗ b ⊗ c)213 = b ⊗ a ⊗ c. The

variation of the potential energy density function (2.56) is written as

δW = τijδǫij + µijkδκijk

= τij∂iδuj + µijk∂i∂jδuk

= ∂j [(τjk − ∂iµijk) δuk] − ∂j (τjk − ∂iµijk) δuk + ∂i (µijk∂jδuk)

(2.60)

Utilizing the divergence theorem, the total potential energy becomes

∫

V

δW dV =

∫

S


−∫

V


∫

S

niµijk∂jδuk dS

(2.61)

with n being the normal unit vector of the surface S.

Mindlin assumed that the surface S is composed of two portions, S1 and S2,

that intersect forming an edge C. In that context, he used the Stokes theorem

and tensor manipulations that resulted to the following expression for the total

28


potential energy. These calculations are presented in detail in Appendix B.

∫

V

W dV = −∫

V

∂j (τjk − ∂iµijk) δuk dV

+

∫

S

[nj τjk − ninjDµijk − 2niDiµijk + (ninjDlnl −Djni) muijk] dS

+

∫

S

ninjµijkDδuk dS +

∮

C

JnimjµijkKδuk dS

(2.62)

with D ≡ nl∂l, Dj ≡ (δjl − njnl) ∂l, mj = emljsmnl and sm being the components

of the unit vector that is tangent to the edge C, whereas the double brackets J·Kindicate the difference between the values of the enclosed quantities on S1 and S2.

This suggests the following form for the variation of the work done by external

forces.

δW1 =

∫

V

Fkδuk dV +

∫

S

Pkδuk dS +

∫

S

RkDδuk dS +

∮

C

Ekδuk ds (2.63)

where Fk, Pk dS, Rk dS, Ek dS are external body forces, surface forces, double

surface forces and jump line forces, respectively.

Equilibrating (2.62) with (2.63) one obtains the following fundamental rela-

tions.

i. The relative and total stresses tensors are

sjk = −∂iµijk and σjk = τjk − ∂iµijk (2.64)


s = −∇ · µ and σ = τ −∇ · µ (2.65)

ii. The equation of equilibrium is

∂i (τjk − ∂iµijk) + Fk = 0 or

∇ · (τ −∇ · µ) + F = 0 (2.66)

29


MICROSTRUCTURE

iii. The essential boundary conditions, concerning the determination of the

kinematic variables and the external boundary are{

uk = u0k on S, and/or

∂uk

∂n= qk, on S

(2.67)

iv. The natural boundary conditions dealing with stress type variables are

nj τjk − ninjDµijk − (njDi + niDj) µijk

+ (ninjDlnl −Djni) µijk = Pk

(2.68)

with Pk representing the traction vector. The above boundary conditions

is written in vector form as

n · τ − (n ⊗ n) :∂µ

∂n− n · (∇S · µ) − n ·

(

∇S · µ213)

+ (∇S · n) (n⊗ n) : µ− (∇Sn) : µ = P

where the inner product (:) is defined as a ⊗ b : c ⊗ d = (b · c) (a · d) and

∇S =(

I − n⊗ n)

· ∇. Also

ninjµijk = Rk or (n⊗ n) : µ = R on S (2.69)

andJnimjµijkK = Ek or J(m⊗ n) : µK = E on S (2.70)

with Rk, Ek representing the double traction and jump traction vectors,

respectively.

It is worth noting that since µijk = µikj, the term (τjk − ∂iµijk) in the stress

equation (2.66) is symmetric, which simplifies the introduction of an Airy stress

function.

The equilibrium equation in terms of displacements is derived by substituting

eqs (2.50) and (2.11) into (2.58) and the latter in the stress equations of motion

(2.66).(

λ+ 2µ)(

1 − l21∂i∂i

)

∂j∂kuk − µ(

1 − l22∂i∂i

)

ejmnenpq∂m∂puq + Fj = 0 (2.71)

or vector notation(

λ+ 2µ)(

1 − l21∇2)

∇∇ · u − µ(

1 − l22∇2)

∇×∇× u + F = 0 (2.72)

with l21 = 2 (α1 + α2 + α3 + α4 + α5) /(

λ+ 2µ)

, l22 = (α3 + 2α4 + α5) /2µ.

30

2.4 Integral Representation of the Form II Gradient Elastic Problem

2.4 Integral Representation of the Form II Gra-

dient Elastic Problem

2.4.1 Reciprocal Integral Identity

In the previous section, the tilde (·) has been used to describe the fields related

to the Form I gradient elasticity theory, in accordance to Mindlin (1964). From

now on, the tilde (·) will be used over a symbol to specify that it is a tensor of

second or higher order.

In order to proceed with the integral representation of a static Form II gra-

dient elastic problem, a reciprocal integral identity must be derived, analogous

to Betti’s reciprocal identity for the classical elasticity (Brebbia & Dominguez

(1992)).

Consider a gradient elastic material with volume V and surrounding surface

S. Also consider two different deformation states for this material, denoted as

(u, σ) and (u∗, σ∗) with u, u∗ being the displacement vectors and σ, σ∗ being

the total stress tensors of the first and second deformation state respectively.

Betti’s theorem for classical elasticity states that the work done by the external

forces of the first state in the displacements of the second state is equal to the

work of the external forces of the second state in the displacements of the first

state. In order to derive an identity analogous to Betti’s the same procedure as

followed by Polyzos et al. (2003) is applied.

First, a vector w involving the two deformation states is defined.

w = σ · u∗ − σ∗ · u (2.73)

Replacing the total stresses from eq (2.65), calculating the divergence of w and

exploiting the identities ∇· (τ · u) = (∇ · τ ) ·u+ τ : ∇u and τ : ∇u− τ : ∇u = 0

yields

∇ ·w = ∇ · (τ −∇µ) · u∗ − [∇ · (τ ∗ −∇ · µ∗)] · u− (∇ · µ) : ∇u∗ + (∇ · µ∗) : ∇u

(2.74)

31


MICROSTRUCTURE

Now, applying the Gauss divergence theorem for w over the volume V produces

the following:∫

V

[∇ · (τ −∇ · µ)] · u∗ − [∇ · (τ ∗ −∇ · µ∗)] · u dV

−∫

V

(∇ · µ) : ∇u∗ − (∇ · µ∗) : ∇u dV

=

∫

S

[n · (τ −∇ · µ)] · u∗ − [n · (τ ∗ −∇ · µ∗)] · u dS

(2.75)

Using the equilibrium equation (eq (2.66)) for both deformation states with forces

f and f∗ respectively, the above integral equation becomes:∫

V

f∗ · u − f · u∗ dV

+

∫

V

(∇ · µ∗) : ∇u− (∇ · µ) : ∇u∗ dV =

∫

S

t · u∗ − t∗ · u dS(2.76)

with t = n · (τ −∇ · µ) and t∗ = n · (τ ∗ −∇ · µ∗) being the traction vectors for

the two states, acting on the boundary surface S.

Furthermore, using eq (2.65) and Green’s integral identity, the second volume

integral of the above equation can be written as a surface integral over S (Polyzos

et al. (2003)).

∫

V

(∇ · µ∗) : ∇u−(∇ · µ) : ∇u∗ dV =

∫

S

(n · µ∗) : ∇u−(n · µ) : ∇u∗ dS (2.77)

In view of the above, eq (2.76) becomes

∫

V

f∗ · u − f · u∗ dV

+

∫

S

(n · µ∗) : ∇u− (n · µ) : ∇u∗ dS =

∫

S

t · u∗ − t∗ · u dS(2.78)

Finally, using the identities

n · µ : ∇u∗ = (n · µ · n) (n · ∇u∗) + (n · µ) : ∇Su∗ (2.79)

32


(n · µ) : ∇Su = ∇S · [(n · µ) · u∗] −[

∇Sn : µ+ n ·(

∇S · µ213)]

· u∗ (2.80)

∇S · [(n · µ) · u∗] = n · ∇S × [n× (n · µ · u∗)] + [(∇S · n) (n⊗ n) : µ]u∗ (2.81)

the reciprocal integral identity becomes

∫

V

f∗ · u − f · u∗ dV +

∫

S

P∗ · u− P · u∗ dS

=

∫

S

R · ∂u∗

∂n− R∗ · ∂u

∂ndS +

∑

a

∮

Ca

E · u∗ − E∗ · u dC(2.82)

with

P = n · τ − (n⊗ n) :∂µ

∂n− n · (∇S · µ) − n ·

(

∇S · µ213)

+ (∇S · n) (n⊗ n) : µ− (∇Sn) : µ (2.83)

R = (n⊗ n) : µ (2.84)

E = J(m⊗ n) : µK (2.85)

and a being the total number of edges Ca of the surface S. It must be noted

here, that in the two dimensional case, the line integral of eq (2.82) reduces to a

sum of distinct values, over the corners of S. It should also be mentioned here

that Giannakopoulos et al. (2006) have also derived the same reciprocity relation

in their paper dealing with the Saint-Venant principle for linear strain-gradient

elastic bodies.

2.4.2 2D and 3D Fundamental Solutions

The fundamental displacement is defined as the displacement of point x due to

a unit excitation δ (x,y) I at point y. This means that the fundamental solution

u∗ (x,y) of the Form II gradient elasticity is a tensor satisfying the following

partial differential equation.

(

λ+ 2µ)(

1 − l21∇2)

∇∇·u∗ (r)−µ(

1 − l22∇2)

∇×∇×u∗ (r) = −δ (r) I (2.86)

with r = |x − y| and δ (r) being the Dirac delta function.

33


MICROSTRUCTURE

It is worth noting here, that using the identity ∇2 = ∇∇·+∇×∇× and after

some calculations we end up with a more familiar form of eq (2.86).

(

1 − l22∇2)

∆∗u∗ (r) +(

λ+ 2µ)(

l22 − l21

)

∇2∇ (∇ · u∗ (r)) = −δ (r) I (2.87)

with ∆∗ = µ∇2 +(

λ+ µ)

∇∇· being the differential operator of classical elas-

ticity.

The Dirac delta function in eq (2.86) may be replaced by the Laplacian of a

scalar function γ (r), which has the following form:

γ (r) =

{

12π

ln(

1r

)

, for 2D1

4πr, for 3D

(2.88)

since γ (r) is the fundamental solution of the Laplacian operator, i.e. ∇2γ (r) =

−δ (r).

Furthermore, Dassios & Lindell (2001) have proven that any second order

tensor can be decomposed in three parts. The first is the gradient of the gradient

of a scalar function φ (r), the second is the gradient of the curl of a vector function

A (r) and the last part is the curl of the curl of a dyadic function G (r).

u∗ (r) = ∇∇φ (r) + ∇∇× A (r) + ∇×∇× G (r) (2.89)

Utilizing the identity ∇2[

γ (r) I

]

= ∇∇ ·[

γ (r) I

]

− ∇ × ∇ ×[

γ (r) I

]

, and

replacing all the above in eq (2.86), the latter becomes

∇∇[(

λ+ 2µ)(

∇2φ (r) − l21∇4φ (r))]

+∇∇×[(

λ+ 2µ)(

∇2A (r) − l21∇4A (r))]

+∇×∇×[

µ(

∇2G (r) − l22∇4G (r))]

= ∇∇γ (r) I −∇×∇× γ (r) I

(2.90)

For the above equation to hold true each part of the left hand side must be

equated to the corresponding part on the right hand side.

(

λ+ 2µ)(

∇2φ (r) − l21∇4φ (r))

= γ (r) (2.91)(

λ+ 2µ)(

∇2A (r) − l21∇4A (r))

= 0 (2.92)

µ(

∇2G (r) − l22∇4G (r))

= −γ (r) I (2.93)

34


The radial nature of the fundamental solution exists only when A = 0. The

order of eqs (2.91) and (2.93) can be reduced by replacing f (r) = ∇2φ (r) and

g (r) = ∇2G respectively.(

λ+ 2µ)(

f (r) − l21∇2f (r))

= γ (r) (2.94)

µ(

g (r) − l22∇2g (r))

= −γ (r) I (2.95)

For the 3D case, the last two equations admit solutions of the form

f (r) = ∇2φ (r) =1

4π(

λ + 2µ)

(

1

r− e−r/l1

r

)

(2.96)

g (r) = ∇2G (r) = − 1

4πµ

(

1

r− e−r/l2

r

)

I (2.97)

which yield

φ (r) =1

4π(

λ+ 2µ)

(

r

2− l21r− l21

e−r/l1

r

)

(2.98)

G (r) = − 1

4πµ

(

r

2+l22r− l22

e−r/l2

r

)

I (2.99)

Accordingly for the 2D case, the corresponding functions φ (r) and G (r) are

found to be

φ (r) = − 1 − 2ν

4πµ (1 − ν)

[

r2

4(ln r − 1) + l21 ln r + l21K0

(

r

l1

)]

(2.100)

G (r) =1

2πµ

[

r2

4(ln r − 1) + l22 ln r + l22K0

(

r

l2

)]

I (2.101)

with 2ν = λ/(

λ+ µ)

and K0 (·) the modified Bessel function of the second kind

and zeroth order.

Substituting the above results for φ (r), A (r) and G (r), into the decomposed

expression for the displacement (eq (2.89)) we end up with the final form of the

displacement fundamental solution.

u∗ (r) =1

(a− 1) 8πµ (1 − ν)

[

Ψ (r) I −X (r) r ⊗ r]

(2.102)

35


MICROSTRUCTURE

with a being equal to the problem spatial dimensions, i.e. two for 2D and three

for 3D, and X (r), Ψ (r) being scalar functions. For the three dimensional case,

X (r) and Ψ (r) are

X (r) = −1

r+ 2 (1 − 2ν)

1

r

[(

3l21r2

+3l1r

+ 1

)

e−r/l1 − 3l21r2

]

−4 (1 − ν)1

r

[(

3l22r2

+3l2r

+ 1

)

e−r/l2 − 3l22r2

]

(2.103)

Ψ (r) = (3 − 4ν)1

r+ 2 (1 − 2ν)

1

r

[(

l21r2

+l1r

)

e−r/l1 − l21r2

]

−4 (1 − ν)1

r

[(

l22r2

+l2r

+ 1

)

e−r/l2 − l22r2

]

(2.104)

whereas for the two dimensional case the corresponding functions are given below.

X (r) = −1 + 2 (1 − 2ν)

[

−2l21r2

+K2

(

r

l1

)

]

+4 (ν − 1)

[

−2l22r2

+K2

(

r

l2

)

]

(2.105)

Ψ (r) = − (3 − 4ν) ln r + (2ν − 1)

[

2l21r2

+K0

(

r

l1

)

−K2

(

r

l1

)

]

+2 (ν − 1)

[

−2l22r2

+K0

(

r

l2

)

+K2

(

r

l2

)

]

(2.106)

2.4.3 Boundary Integral Representations

The integral representation of a Form II gradient elastic problem is obtained by

means of the reciprocal integral identity derived in the previous section.

Assuming that the second deformation state (u∗, σ∗) has an excitation of the

form

f∗ (y) = δ (x − y) e (2.107)

36


with e being a unit constant vector at point y. Then the displacement vector u∗

is given by

u∗ (y) = u∗ (x,y) · e (2.108)

Replacing the above equation into the reciprocal integral identity and assuming

zero body forces f = 0, we get

∫

V

δ (x − y)u (y) dVy

· e

+

∫

S

(

P∗T (x,y) · u (y) − u∗ (x,y) ·P (y))

dSy

· e

=

∫

S

(

q∗T (x,y) · R (y) − R∗T (x,y) · q (y))

dSy

· e

+

∑

a

∮

Ca

(

u∗ (x,y) · E (y) − E∗T (x,y) · u (y))

dCy

· e

(2.109)

with q∗T (x,y) = ∂u∗ (x,y) /∂ny and q (y) = ∂u (y) /∂ny.

Since eq (2.109) must hold for every possible e and u (x,y) is symmetric, the

following integral equation is obtained.

c (x) · u (x) +

∫

S

[

P∗T (x,y) · u (y) − u∗ (x,y) · P (y)]

dSy

=

∫

S

[

q∗T (x,y) · R (y) − R∗T (x,y) · q (y)]

dSy

+∑

a

∮

Ca

[

u∗ (x,y) ·E (y) − E∗T (x,y) · u (y)]

dCy

(2.110)

with c (x) being the jump tensor used in the classical elasticity case (Brebbia &

Dominguez (1992)) taking the values

c (x) =

0, x ∈ (R3 − V )I, x ∈ V12I, x ∈ S

and the kernels u∗, q∗, P∗, R∗ and E∗ as presented in Appendix C.

37


MICROSTRUCTURE

The above equation has five unknown fields, i.e. u (y), P (y), q (y), R (y) and

E (y). Since there are three boundary conditions available, one more equation is

required to evaluate the unknown fields. This equation is obtained by applying

the differential operator ∂/∂nx on the first boundary integral equation.

c (x) · ∂u∂nx

(x) +

∫

S

[

∂P∗T

∂nx

(x,y) · u (y) − ∂u∗

∂nx

(x,y) · P (y)

]

dSy

=

∫

S

[

∂q∗T

∂nx(x,y) · R (y) − ∂R∗T

∂nx(x,y) · q (y)

]

dSy

+∑

a

∮

Ca

[

∂u∗

∂nx(x,y) · E (y) − ∂E∗T

∂nx(x,y) · u (y)

]

dCy

(2.111)

The kernels ∂u∗/∂nx, ∂q∗/∂nx, ∂P

∗/∂nx, ∂R

∗/∂nx and ∂E

∗/∂nx are given in

Appendix C.

The integral equations (2.110) and (2.111), along with the classical boundary

conditions ( u or P prescribed) and the non-classical boundary conditions (q or

R prescribed and E prescribed) form the integral representation of a Form II

gradient elastic problem.

38

Chapter 3

Boundary Element Formulation

3.1 BEM Formulation

Consider a volume V composed of a gradient elastic material and surrounded by

a surface S. The boundary integral equations that describe the problem are

c (x) · u (x) +

∫

S

[

P∗T (x,y) · u (y) − u∗ (x,y) · P (y)]

dSy

=

∫

S

[

q∗T (x,y) · R (y) − R∗T (x,y) · q (y)]

dSy

+∑

a

∮

Ca

[

u∗ (x,y) · E (y) − E∗T (x,y) · u (y)]

dCy

(3.1a)

c (x) · ∂u∂nx

(x) +

∫

S

[

∂P∗T

∂nx(x,y) · u (y) − ∂u∗

∂nx(x,y) · P (y)

]

dSy

=

∫

S

[

∂q∗T

∂nx(x,y) · R (y) − ∂R∗T

∂nx(x,y) · q (y)

]

dSy

+∑

a

∮

Ca

[

∂u∗

∂nx

(x,y) · E (y) − ∂E∗T

∂nx

(x,y) · u (y)

]

dCy

(3.1b)

If we discretize the domain into E boundary elements then equations (3.1) can

39

3. BOUNDARY ELEMENT FORMULATION

be written in discretized form

c (x) · u (x) +E∑

e=1

∫

Se

[

P∗T (x,ye) · u (ye) − u∗ (x,ye) · P (ye)]

dSye

=

E∑

e=1

∫

Se

[

q∗T (x,ye) · R (ye) − R∗T (x,ye) · q (ye)]

dSye

+E∑

e=1

∮

Ce

[

u∗ (x,ye) · E (ye) − E∗T (x,ye) · u (ye)]

dCye

(3.2a)

c (x) · ∂u (x)

∂nx+

E∑

e=1

∫

Se

[

∂P∗T

∂nx(x,ye) · u (ye) − ∂u∗

∂nx(x,ye) · P (ye)

]

dSye

=E∑

e=1

∫

Se

[

∂q∗T

∂nx

(x,ye) · R (ye) − ∂R∗T

∂nx

(x,ye) · q (ye)

]

dSye

+

E∑

e=1

∮

Ce

[

∂u∗

∂nx(x,ye) · E (ye) − ∂E∗T

∂nx(x,ye) · u (ye)

]

dCye

(3.2b)

with Se and Ce being the surface and the boundary of element e respectively.

Supposing that element e has S sides, then its boundary Ce can be expressed as

a sum of the boundaries of each side Ce =∑

S

i=1Cie. Then, the last integrals of

eqs (3.2) become respectively

∮

Ce

[


dCye

=

S∑

i=1

∮

Cie

[


dCiye

(3.3a)

∮

Ce

[

∂u∗



]

dCye

=

S∑

i=1

∮

Cie

[

∂u∗



]

dCiye

(3.3b)

40

3.1 BEM Formulation

Since each integral of the above involves only one side of each element and E

is expressed as a difference between two sides of two adjacent elements, if the

elements are coplanar, E becomes equal to zero on their common side. For a

smooth domain boundary and a dense discretization this is a good approximation.

However, in the case of a non-smooth domain boundary or a coarse mesh, the

above line integrals must be taken into account.

Before proceeding with the discretization of the integrals, some aspects of

the elements must be discussed. In order to treat all elements in a unified man-

ner, they are represented parametrically with the aid of a transformation. This

transformation involves the shape functions of the element. If the element is

quadrilateral, it is transformed to a square on the plane ξ1, ξ2 with −1 ≤ ξ1 ≤ 1

and −1 ≤ ξ2 ≤ 1 (Figure 3.1(a)). If it is triangular, it is transformed to an

equilateral triangle represented on a skew coordinate system defined by its two

sides as shown in Figure 3.1(b).

(a)

(b)

(c)

Figure 3.1: Transformation of (a) a quadrilateral, (b) a triangular and (c) a line

element from the global to their local coordinate systems

41

ch3-The_Boundary_Element_Method/Figs/EPS/quadrilateral_element.eps

ch3-The_Boundary_Element_Method/Figs/EPS/triangular_element.eps

ch3-The_Boundary_Element_Method/Figs/EPS/line_element.eps


The location of a point on the element with local coordinates (ξ1, ξ2) can be

used to calculate its coordinates in the global coordinate system, by computing

the sum of the shape functions of the element multiplied by the coordinates of

the corresponding geometrical node, over all element nodes.

x (ξ1, ξ2) =

Ag∑

i=1

Φi (ξ1, ξ2)xi (3.4)

Ag is the number of geometrical nodes of the element, Φi (ξ1, ξ2) the shape func-

tion of the element that corresponds to the i-th geometrical node and xi the

location vector of the i-th geometrical node expressed in global coordinates. The

shape functions of the element are usually polynomial functions of the variables

ξ1, ξ2 that are linearly independent with one another, equal to one on the corre-

sponding node and zero on all other geometrical nodes. This is called the delta

property. Furthermore, their sum is always equal to one and the sum of their

derivatives with respect to any of the varialbes ξ1, ξ2 over the geometrical nodes

is equal to zero.

In order to calculate the fields at any point of the element, the element’s

interpolation functions are used. Specifically, the value of a field at any point on

the element is approximated by the sum of the interpolation functions multiplied

by the corresponding functional node of the element.

u (x) =

Af∑

i=1

N i (ξ1, ξ2)ui (3.5)

q (x) =

Af∑

i=1

N i (ξ1, ξ2)qi (3.6)

P (x) =

Af∑

i=1

N i (ξ1, ξ2)Pi (3.7)

R (x) =

Af∑

i=1

N i (ξ1, ξ2)Ri (3.8)

E (x) =

Af∑

i=1

N i (ξ1, ξ2)Ei (3.9)

42

3.1 BEM Formulation

with Af being the number of functional nodes of the element, N i (ξ1, ξ2) the

interpolation function of the element that corresponds to the i-th functional node

and ui, qi, Pi, Ri and Ei the values the field u, q, P, R and E on the i-th

functional node respectively. The interpolation functions of the element are also

linearly independent with one another, have the delta property and their sum as

well as the sum of their derivatives are equal to one and zero respectively.

Since an element is used to interpolate the geometry as well as the fields, it

can be considered as being a geometrical and a functional element at the same

time. However, its geometrical nodes are independent of its functional ones. In

the present work, the geometrical nodes reside on the element’s boundaries mak-

ing it geometrically continuous. On the other hand, its functional nodes may or

may not reside on the elements boundaries. In cases where the prescribed fields

are different for two adjacent elements or the boundary forms an edge or a corner,

causing the normal vector to be discontinuous on the nodes residing there, the

functional nodes of that side are slightly retracted towards the element’s center.

Then the element is said to be functionally discontinuous. More than one sides

of the element are allowed to be discontinuous at the same time. In the present

work, since no geometrically discontinuous elements are used, the functionally

discontinuous elements will be referred to simply as discontinuous. Placing dis-

continuous elements over an edge of the boundary has also the advantage that

the tensor c (x) of eqs (3.2) is easier to calculate since no edges ared contained

in the boundary and it is considered smooth. A thorough description of all the

elements used can be found in Appendix D.

The integrals of eqs (3.2) are calculated using numerical integration. If the

problem is 3D, the integrals over Se and Ce are surface and line integrals respec-

tively. In the two dimensional case however, the integrals over Se reduce to line

integrals and the ones over Ce reduce to a sum of distinct values, one for each

side of the line element e.

It is possible to write the integral equations (3.2) for a random node k of

the boundary, with 1 ≤ k ≤ L and L being the total number of nodes of the

discretized boundary. To this end, we replace eqs (3.5-3.8) into (3.2) and the

43


nodal field values are moved outside of the integrals.

c(

xk)

· uk +

E∑

e=1

Aef

∑

a=1

(

Hkea · uk − Gk

ea · Pk)

=E∑

e=1

Aef

∑

a=1

(

Ikea · Rk − Jk

ea · qk)

+E∑

e=1

Aef

∑

a=1

(

Kkea ·Ek − Lk

ea · uk)

(3.10a)

c(

xk)

· qk +E∑

e=1

Aef

∑

a=1

(

Nkea · uk − Mk

ea · Pk)

=E∑

e=1

Aef

∑

a=1

(

Okea · Rk − Sk

ea · qk)

+E∑

e=1

Aef

∑

a=1

(

Tkea ·Ek − Vk

ea · uk)

(3.10b)

Here, uk, qk, Pk, Rk and Ek are the values of the fields on node k, Aef is the

number of functional nodes of element e and Gkea, Hk

ea, Ikea, Jk

ea, Kkea, Lk

ea, Mkea,

Nkea, Ok

ea, Skea, Tk

ea and Vkea are the integrals

Gkea =

1∫

−1

1∫

−1

u∗ (xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.11)

Hkea =

1∫

−1

1∫

−1

P∗T (xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.12)

Ikea =

1∫

−1

1∫

−1

q∗T (xk,ye (ξ1, ξ2))

Na (ξ1, ξ2) JL (ξ1, ξ2) dξ1dξ2 (3.13)

Jkea =

1∫

−1

1∫

−1

R∗T (xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.14)

Kkea =

S∑

i=1

bi∫

ai

u∗ (xk,ye (ξ1 (ti) , ξ2 (ti)))

Na (ξ1 (ti) , ξ2 (ti))

JL (ξ1 (ti) , ξ2 (ti)) |γ′i (ti)| dti (3.15)

Lkea =

S∑

i=1

bi∫

ai

E∗T (xk,ye (ξ1 (ti) , ξ2 (ti)))

Na (ξ1 (ti) , ξ2 (ti))


44

3.1 BEM Formulation

Mkea =

1∫

−1

1∫

−1

∂u∗

∂nx

(

xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.17)

Nkea =

1∫

−1

1∫

−1

∂P∗T

∂nx

(

xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.18)

Okea =

1∫

−1

1∫

−1

∂q∗T

∂nx

(

xk,ye (ξ1, ξ2))

Na (ξ1, ξ2) JL (ξ1, ξ2) dξ1dξ2 (3.19)

Skea =

1∫

−1

1∫

−1

∂R∗T

∂nx

(

xk,ye (ξ1, ξ2))

Na (ξ1, ξ2)JL (ξ1, ξ2) dξ1dξ2 (3.20)

Tkea =

S∑

i=1

bi∫

ai

∂u∗

∂nx

(

xk,ye (ξ1 (ti) , ξ2 (ti)))

Na (ξ1 (ti) , ξ2 (ti))


Vkea =

S∑

i=1

bi∫

ai

∂E∗T

∂nx

(

xk,ye (ξ1 (ti) , ξ2 (ti)))

Na (ξ1 (ti) , ξ2 (ti))


Note that the Jacobian of the transformation from the global coordinate system

to the element local coordinate system JL is present in the above integrals.

In the case of triangular elements, the limits of the above integrals, instead

of −1 and 1 become 0 and 1 respectively. Furthermore the integrals Kkea, Lk

ea,

Tkea and Vk

ea of eqs (3.15-3.16) and (3.21-3.22) correspond to the line integrals

that appear in the boundary integral equations. For each element, these integrals

are written as a sum over the element’s sides. Since they are line integrals, the

local variables ξ1 and ξ2 are expressed as functions of the parameter ti, ξ1 (ti) =

ξ1 (γi (ti)) and ξ2 (ti) = ξ2 (γi (ti)), with γi being the parametric representation of

the i-th element side ti ∈ [ai, bi] → γi.

For the two dimensional case, the above surface integrals become line integrals

and the line integrals Kkea, Lk

ea, Tkea and Vk

ea reduce to a sum of distinct values.

Gkea =

1∫

−1

u∗ (xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.23)

45


Hkea =

1∫

−1

P∗T (xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.24)

Ikea =

1∫

−1

q∗T (xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.25)

Jkea =

1∫

−1

R∗T (xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.26)

Kkea =

S∑

i=1

u∗ (xk,ye (ξ))

Na (ξ) (3.27)

Lkea =

S∑

i=1

E∗T (xk,ye (ξ))

Na (ξ) (3.28)

Mkea =

1∫

−1

∂u∗

∂nx

(

xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.29)

Nkea =

1∫

−1

∂P∗T

∂nx

(

xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.30)

Okea =

1∫

−1

∂q∗T

∂nx

(

xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.31)

Skea =

1∫

−1

∂R∗T

∂nx

(

xk,ye (ξ))

Na (ξ)JL (ξ) dξ (3.32)

Tkea =

S∑

i=1

∂u∗

∂nx

(

xk,ye (ξ))

Na (ξ) (3.33)

Vkea =

S∑

i=1

∂E∗T

∂nx

(

xk,ye (ξ))

Na (ξ) (3.34)

Assuming a smooth boundary the boundary integral equations (3.10) are sim-

46

3.2 Symmetry and antisymmetry

plified to

c(

xk)

· uk +

E∑

e=1

Aef

∑

a=1

(

Hkea · uk − Gk

ea ·Pk)

=

E∑

e=1

Aef

∑

a=1

(

Ikea ·Rk − Jk

ea · qk)

(3.35a)

c(

xk)

· qk +E∑

e=1

Aef

∑

a=1

(

Nkea · uk − Mk

ea · Pk)

=E∑

e=1

Aef

∑

a=1

(

Okea · Rk − Sk

ea · qk)

(3.35b)

Collocating the above equations for all nodes k of the boundary to a linear

system of 2 · 3 ·L scalar equations that have 3 · 4 ·L unknowns in total. Utilizing

the boundary conditions that are prescribed on every node (one classical and one

non-classical) the number of unknowns reduces to 3 · 2 · L, leading to a linear

system with that many unknowns as the number of equations. The matrix of

the linear system is fully populated and the system can be solved using the LU

decomposition.

The solution of the system contains the values of the unknown fields on the

nodes of the discretized boundary.

3.2 Symmetry and antisymmetry

It is well known that a symmetric geometry combined with symmetric bound-

ary conditions yields a symmetric solution. The analogous statement is true for

antisymmetry. This means that symmetry and antisymmetry can be exploited

to reduce the utilized memory and the computational time required to solve a

symmetric or antisymmetric problem.

In order to take advantage of a symmetry relative to a plane, only the one

half of the domain needs to be discretized. The other half of the domain is

constructed on demand using projections with respect to the symmetry plane.

The boundary integral equations are calculated over the whole domain, but the

47


degrees of freedom of the symmetric part are expressed with respect to the degrees

of freedom of the discretized part.

Consider a plane of symmetry/antisymmetry located at the center of the

global Cartesian coordinate system and parallel to the plane formed by any two of

the unit normal vectors ei of the the global coordinate system. Then, the values

of any field f l on a node l can be expressed with respect to the field values fk of

its symmetric/antisymmetric counterpart, i.e. node k, as

{

Symmetric case: f li = cif

ki

Antisymmetric case: f li = −cifk

i

(3.36)

with i = 1, · · · , 3 and ci =

{

−1, if ei ⊥ plane of symm./antisymm.;

1, otherwise.

3.3 Subregioning

Subregioning is a simple technique based on the assumption that a domain with

one region can be solved equivalently by dividing it into subregions and applying

appropriate boundary conditions on their interfaces.

This technique is used in a variety of scenarios. For instance when the material

properties of the domain change from one subregion to another or when the

domain contains cracks. In the latter case, there are nodes that belong to different

crack faces, but reside on the same location, affecting the condition of the final

matrix. This problem is addressed by dividing the domain into two subregions is

such a way that the two crack faces belong to different subregions.

The idea behind the subregioning technique is that all regions are discretized

retaining common discretization on their interfaces with other subregions. Then

each subregion is collocated separately and its equations are added to the final

linear system after continuity conditions have been applied on its interfaces. The

final matrix is sparse instead of fully populated and can be solved faster.

The boundary conditions that have to be satisfied on the interface, for the

48

3.4 Numerical Integrations

coupling of the two subregions are

uRegion1 = uRegion2 (3.37a)

qRegion1 = −qRegion2 (3.37b)

RRegion1 = RRegion2 (3.37c)

PRegion1 = −PRegion2. (3.37d)


The fundamental solutions of the employed gradient elasticity theory depend on

the distance r of the node k, for whom the integral representation is written, from

the node a of the element we are integrating over.

The integrals involved can be divided into three categories depending on the

distance of the current node k from the current element node a (see eqs (3.35)).

The first category contains all the integrals that their node k is far from their

element node a, with far meaning a distance greater than the element’s sides.

The integrals of this category are called normal, they possess no singularities and

can be integrated with a simple Gauss-Legendre quadrature. The second category

includes the integrals that their node k coincides with their element node a. These

integrals are called singular, because the fundamental solutions involved contain

several negative powers of the distance r (O (1/r), O (1/r2), O (1/r3)), that tend

to infinity when r tends to zero. The last category contains the integrals that have

a distance r not equal to zero, but equal or smaller than their element’s sides.

In these cases, the calculations of negative powers of r lead to abrupt changes of

the fundamental solutions over the element causing instabilities. These integrals

are called nearly singular.

The singular and nearly singular integrals have been treated semi-analytically

following the methodology proposed by Guiggiani (1992, 1998); Guiggiani &

Gigante (1990). The numerical integrations have been performed with Gauss-

Legendre quadrature, using points optimally distributed according to Bu (1997).

49


3.4.1 Normal and nearly singular integration

As mentioned above the Gauss-Legendre quadrature has been utilized to calcu-

late the values of the integrals. In order to calculate the value of a typical one

dimensional integral I the following procedure is followed.

I =

b∫

a

f (u) du (3.38)

The fist step is to compute the Gauss points and weights. This is done using the

method of Bu (1997) which dictates that in order to compute the above integrals

with an error less than 0.01% the required number of Gauss points is given by

Gi = 1 +

⌊

2.8

(

R

Li

)−0.8⌋

(3.39)

with R being the distance of the current node from the current element and Li

the characteristic length of the element, measured in the i-th direction.

The second step is to transform the integral from [a, b] to the interval [−1, 1],

taking into account the Jacobian of the transformation Jξu

b∫

a

f (u) du =

1∫

−1

f (ξ)Jξu dξ (3.40)

Then, the value of the integral is given by the sum of the integrands, evaluated

at the Gauss points, multiplied by the corresponding Gauss weights

1∫

−1

f (ξ)Jξu dξ =

G∑

i=1

f (ξi) Jξu (ξi)w

i (3.41)

with G being the number of gauss points and ξi, wi, i = 1, · · · , G being the Gauss

points and the corresponding weights.

In the case of surface integrals the procedure is basically the same, but the

integration is conducted in two dimensions.

b∫

a

d∫

c

f (u1, u2) du1du2 =

1∫

−1

1∫

−1

f (ξ1, ξ2) Jξu dξ1dξ2

=

G1∑

i=1

G2∑

j=1

f (ξi, ξj)Jξu (ξi, ξj)w

i1w

j2

(3.42)

50


where G1, G2 are the number of Gauss points in the ξ1 and ξ2 directions, respec-

tively, Jξu is the Jacobian of the transformation from (u1, u2) to (ξ1, ξ2) and wi,

wj are the Gauss weights in the two directions.

In the present work, for a typical normal integral over an element, the function

f is equal to the product of a fundamental solution and an interpolation function

and the Jacobian Jξu is the Jacobian of the transformation from the global coor-

dinate system to the element local coordinate system. Since the local variables of

the element (ξ1, ξ2 for quadrilateral and ξ for line) already belong in the interval

[−1, 1], no extra transformation in required and Jξu = 1.

However, in the case of a triangular element, an additional transformation has

to be performed. This is due to the fact that the local coordinates ξ1, ξ2 of the

element belong in the interval [0, 1]. Thus, in order to apply the Gauss-Legendre

quadrature a transformation γ : [0, 1] × [0, 1] → [−1, 1] × [−1, 1] is introduced.

Then, the local coordinates of the element are related to the transformed coordi-

nates γ1, γ2 according to the following relations

ξ1 = 14(γ1 + 1) (1 − γ2) (3.43a)

ξ2 = 14(γ1 + 1) (1 + γ2) (3.43b)

dξ1dξ2 = 18(γ1 + 1) dγ1dγ2 (3.43c)

For example, the integral Gkea (eq (3.11)) over a triangular element becomes

Gkea =

G1∑

i=1

G2∑

j=1

wi1w

j2u

∗ (xk,ye (ξ1 (gi) , ξ2 (gj)))

Na (ξ1 (gi) , ξ2 (gj)) JL (ξ1 (gi) , ξ2 (gj))1

8(gi + 1)

(3.44)

with gi and wi1 being the gauss points and the gauss weights in the ξ1 direction

and gj and wj2 the gauss points and weights in the ξ2 direction.

It should be noted that the method of Bu (1997) works well also for the nearly

singular integrals, allowing a unified treatment for both cases ( normal and nearly

singular).

More details about the Gauss-Legendre quadrature and the determination of

Gauss points can be found in Bu (1997) and Press et al. (2007).

51


3.4.2 Singular Integration

Singular integrals are categorized according to their order of singularity. Namely,

the integrals with singularity order O (1/r) are called weakly singular, the inte-

grals with order O (1/r2) are said to possess a strong singularity and the ones

with singularity O (1/r3) are called hypersingular integrals.

3.4.2.1 Treating weak singularities

In order to treat a weak singularity, it is a common practice to introduce a

transformation from the local element coordinate system to a local polar coor-

dinate system, centered at the point of singularity, i.e. the current node k (see

eqs (3.35)). Then, as explained below, the Jacobian of the transformation cancels

out the singularity.

For a quadrilateral element the transformation from the local coordinate sys-

tem (ξ1, ξ2) to the new, local polar coordinate system (R, θ) is straightforward

ξ1 = ξk1 +R cos θ (3.45a)

ξ2 = ξk2 +R sin θ (3.45b)

dξ1dξ2 = RdRdθ (3.45c)

with(

ξk1 , ξ

k2

)

being the coordinates of the singular point.

For a triangular element however, an intermediate transformation is required

in order to make the transaction from the skew coordinate system of the element

(ξ1, ξ2) to an orthonormal one (η1, η2)

ξ1 = η1 − η2 tan (π/6) (3.46a)

ξ2 =η2

cos (π/6)(3.46b)

dξ1dξ2 =1

cos (π/6)dη1dη2 (3.46c)

After that, the transformation to the polar coordinate system, centered at the

singular point can be performed to the new coordinates (η1, η2)

η1 = ηk1 +R cos θ (3.47a)

52


η2 = ηk2 +R sin θ (3.47b)

dη1dη2 = RdRdθ (3.47c)

with(

ηk1 , η

k2

)

being the coordinates of the singular point expressed in orthonormal

coordinates.

At this point, it is easy to see why the transformation to polar coordinates can

treat weak singularities. After the transformation, since the distance r has be-

come a function of the radius R, the singularity has also become a function of R,

retaining however its order (O (1/R)).It is due to the Jacobian of the transforma-

tion, which multiplies the integrand with R, that the singular term is eliminated.

In order to demonstrate the details of this technique, the integral Gkea (eq (3.11))

is presented as an example, for a triangular element.

Gkek =

2π∫

0

Rmax(θ)∫

0

u∗ (xk,ye (R, θ))

Na (R, θ)JL (R, θ)JηξRdRdθ (3.48)

In the above integral, Jηξ is the Jacobian of the transformation (3.46) from the

skew coordinate system (ξ1, ξ2) to the orthonormal one (η1, η2). This Jacobian is

absent in the case of a quadrilateral element, since no intermediate transformation

is required.

In order to proceed with the numerical integration, the range of the polar

radius R must be determined. Since the elements are not symmetric around the

singular point, the radius is a function of the polar angle R (θ). In addition, the

element is broken down into triangles, so that each triangle has one of its vertices

on the singular point, according to figure 3.2.

Now the singular integral over the element can be written as a sum of integrals

over the triangles.

Gkek =

Te∑

t=1

θt2∫

θt1

Rtmax(θ)∫

0

u∗ (xk,ye (R, θ))

Na (R, θ)JL (R, θ)JηξRdRdθ (3.49)

with Te being the total number of triangles in the element.

The advantage of this technique, is that there are no abrupt changes of R over

a single triangle, making thus the integrals easy to calculate even with few Gauss

53


(a) (b)

Figure 3.2: (a) A quadrilateral and (b) a triangular element broken down to

triangles

points. Extensive details about the calculation of Rmax (θ) for quadrilateral and

triangular elements are given in Appendix E.

3.4.2.2 Treating strong and hyper singularities

An integral is characterized as strongly singular when it possesses a singularity

of the order of O (1/R2), with respect to the variable of integration R, within the

interval of integration. Similarly, hypersingular is the integral whose dominant

term is of the order of O (1/R3).

In order to treat these integrals, the methodology of Guiggiani (1998) is used

for the unified treatment of strong and hypersingular integrals.

The basic idea behind Guiggiani’s technique is that the singular parts of the

kernels are first subtracted from the integrals; then the integrals are computed

as normal ones, using the Gauss-Legendre quadrature and the integrals of the

singular parts are added back after having been analytically computed.

Consider an integral of the form

Nkea =

1∫

−1

1∫

−1

∂P∗T

∂nx

(

xk,ye (ξ1, ξ2))

Na (ξ1, ξ2) JL (ξ1, ξ2) dξ1dξ2 (3.50)

with ∂P∗T /∂nx being the fundamental solution for the normal derivative of trac-

tions. Assume that the kernel of integral (3.50) possesses two kinds of singularity,

54

ch3-The_Boundary_Element_Method/Figs/EPS/quadrilateral_to_triangles.eps

ch3-The_Boundary_Element_Method/Figs/EPS/triangular_to_triangles.eps


O (1/R3) and O (1/R2). In that case, the integral is transformed from the lo-

cal element coordinate system (ξ1, ξ2) to a local polar coordinate system (R, θ)

and broken down to a sum of integrals over a set of triangles, as in the previous

section. Each of these integrals is of the form

I =

θ2∫

θ1

Rmax(θ)∫

0

∂P∗T

∂nx

(

xk,ye (R, θ))

Na (R, θ) JL (R, θ) JηξRdRdθ (3.51)

with Na (R, θ) being the a-th interpolation function of the element e, written in

terms of the new polar coordinates and k being the current node, which coincides

with the node a of the current element. Furthermore, Jηξ is the Jacobian of the

transformation (3.46) if the element e is triangular and it is equal to 1 if the

element is quadrilateral. After the transformation to the local polar coordinates,

the kernel of the integral (3.51),

K (R, θ) = ∂P∗T/∂nx

(

xk,ye (R, θ))

Na (R, θ)JL (R, θ) JηξR,

still possesses two kinds of singularities, but of lower order than before, O (1/R2)

and O (1/R). In order to proceed the kernel K (R, θ) must be expanded with

respect to the polar radius R.

To this end, the position vector r can be written as a Taylor expansion around

the singular point xk.

r = ye (η1 (R, θ) , η2 (R, θ)) − xk(

ηk1 , η

k2

)

= R

(

∂ye (η1, η2)

∂η1

∣

∣

∣

∣

η=xk

cos θ +∂ye (η1, η2)

∂η2

∣

∣

∣

∣

η=xk

sin θ

)

+R2

(

∂2ye (η1, η2)

∂η21

∣

∣

∣

∣

η=xk

cos2 θ

2+∂2ye (η1, η2)

∂η1∂η2

∣

∣

∣

∣

η=xk

cos θ sin θ

+∂2ye (η1, η2)

∂η22

∣

∣

∣

∣

η=xk

sin2 θ

2

)

+ O(

R3)

= RA (θ) +R2B (θ) + O(

R3)

(3.52)

Since ye can be written as a sum of the element’s shape functions Φi multi-

plied by the i-th geometrical node, the functions A (θ) and B (θ) can easily be

calculated. This calculation can be found in Appendix F.

55


Furthermore, all quantities included in the integral (3.50) can be expressed in

terms of their expansions around the singular point ηk.

∂P∗T

∂nx=

A (θ)

R3+

B (θ)

R2+O(1) (3.53)

Na (η1, η2) = Na0 +RNa

1 + O(

R2)

(3.54)

JL = JL0 +RJL1 + O(

R2)

(3.55)

It is easy to see from the above, that the whole kernel, due to the transformation

to the local polar coordinate system, has a singularity of the order of O (1/R2).

K (R, θ) =A (θ)

R2Na

0 JL0Jηξ

+1

R

[

B (θ)Na0 JL0J

ηξ + A (θ)

(

Na1 JL0J

ηξ +Na

0 JL1Jηξ

)]

+ O (1)

=K2 (θ)

R2+K1 (θ)

R+ O (1)

(3.56)

The next step is to subtract the singular part of the kernel and calculate the

integral over the remaining terms as normal. The integrals of the singular parts

are then added back.

I =

θ2∫

θ1

Rmax∫

0

K (R, θ) − K2 (θ)

R2− K1 (θ)

RdR+

Rmax∫

0

K2 (θ)

R2+K1 (θ)

RdR

dθ

(3.57)

The first integral of the above equation has a singularity of the order of O (1). The

second integral can be calculated analytically. Namely, according to Guiggiani

(1998) the integral becomes

Rmax∫

0

K2 (θ)

R2+K1 (θ)

RdR

= K2 (θ) limǫ(θ)→0

Rmax∫

ǫ(θ)

1

R2dR+K1 (θ) lim

ǫ(θ)→0

Rmax∫

ǫ(θ)

1

RdR

= K1 (θ) ln

∣

∣

∣

∣

Rmax (θ)

β (θ)

∣

∣

∣

∣

−K2 (θ)

(

γ (θ)

β2 (θ)+

1

Rmax (θ)

)

(3.58)

56


with β (θ) = 1/ |A (θ)| and γ (θ) = (A (θ) · B (θ)) / |A (θ)|4. The integral I of

eq (3.57) is now equal to

I =

θ2∫

θ1

Rmax(θ)∫

0

K (R, θ) − K2 (θ)

R2− K1 (θ)

RdR

+ K1 (θ) ln

∣

∣

∣

∣

Rmax (θ)

β (θ)

∣

∣

∣

∣

−K2 (θ)

(

γ (θ)

β2 (θ)+

1

Rmax (θ)

)]

dθ (3.59)

For the 2D case, the process is basically the same. A similar technique is used

for treating weakly and strongly singular integrals. However, there is a significant

difference. Two cases have to be considered, depending on the location of the

singular point with respect to the current element. If the singular point lies inside

the element, then a line integral like (3.30) would be treated for singularities

according to the following formula.

I =

1∫

−1

[

K(

ξk, ξ)

− K2

(

ξk)

(ξ − ξk)2 − K1

(

ξk)

ξ − ξkdξ

]

+ K1

(

ξk)

ln

∣

∣

∣

∣

1 − ξk

−1 + ξk

∣

∣

∣

∣

+K2

(

ξk)

(

− 1

1 − ξk+

1

−1 − ξk

)

(3.60)

with K1

(

ξk)

and K2

(

ξk)

corresponding to the coefficients of the 1/(

ξk − ξ)2

and

1/(

ξk − ξ)

singularities of the integrand, respectively and ξ ∈ (−1, 1) being the

coordinate of the singular point expressed in the local element coordinate system.

If the singular point resides between two adjacent elements, then both ele-

ments must be taken into account for the treatment of the singularity.

I =

2∑

m=1

1∫

−1

[

Km(

ξk, ξ)

− Km2

(

ξk)

(ξ − ξk)2 − Km1

(

ξk)

ξ − ξk

]

dξ

+ Km1

(

ξk)

ln

∣

∣

∣

∣

2

βm (ξk)

∣

∣

∣

∣

sgn(

ξ − ξk)

− Km2

(

ξk)

(

sgn(

ξ − ξk) γm

(

ξk)

βm (ξk)+

1

2

)}

(3.61)

with ξk = 1 if m = 1, ξk = −1 if m = 2, βm

(

ξk)

= 1/ |A|, γm

(

ξk)

= A · B/ |A|4

and A, B being the coefficients of the Taylor expansion of the distance between

57


the current point x from the current singular point y.

xi − yi =dxi

dξ

∣

∣

∣

∣

ξ=ξk

(

ξ − ξk)

+d2xi

dξ2

∣

∣

∣

∣

ξ=ξk

(

ξ − ξk)2

2+ . . .

= Ai

(

ξ − ξk)

+Bi

(

ξ − ξk)2

+ . . . , i=1,2 (3.62)

3.5 Numerical Examples

This section contains some numerical examples, in 2D and 3D, that demonstrate

the accuracy of the above methodology.

3.5.1 Hollow Cylinder under pressure

Consider a hollow cylinder with internal radius ri = 1.05m and external radius

ro = 2.1m. The cylinder has the material characteristics shown in Table 3.1.

Young’s modulus 4.0 GPa 4.0 GPa 4.0 GPaPoisson’s ratio 0.4 0.4 0.4Mindlin’s α1 13.86 MNt 1.264 MNt 2.0 KNtMindlin’s α2 11.240 MNt 1.424 MNt 1.553 KNtMindlin’s α3 7.226 MNt 1.36 MNt 0.1 KNtMindlin’s α4 8.252 MNt 1.376 MNt 1.63 KNtMindlin’s α5 4.31 MNt 5.504 MNt 0.1 KNt

Table 3.1: Material constants for the hollow cylinder

In order to model the cylinder in 2D, quarter symmetry has been used and the

problem was solved as plane strain, for which an analytical solution is available

( Papanicolopulos (2008) and Zervos et al. (2009)).

Specifically, the analytical solution for the displacements has radial symmetry

and is given by

ur (r) = C1r +C2

r+ C3l1I1

(

r

l1

)

+ C4 l1K1

(

r

l1

)

(3.63)

with I1 (·) and K1 (·) being the modified Bessel functions of the first and sec-

ond kind respectively and of order one. The constants C1–C4 are provided in

Appendix G.

58


Radial tractions Ti = −100KPa and To = −200KPa have been applied to

the internal and external of the cylinder respectively. The double tractions R are

considered to be equal to zero on the boundary. Internal points have been placed

in the center of the cylinder, along the radial direction. Figure 3.4 shows the

Figure 3.3: The hollow cylinder

radial displacements on the internal points of the hollow cylinder with respect

to the distance r from the center of the cylinder as compared to the analytical

solution.

1.0 1.2 1.4 1.6 1.8 2.0 2.20.055

0.060

0.065

0.070

0.075

0.080

Rad

ial d

ispl

acem

ent o

f int

erna

l poi

nts

Radial distance r

2D Form II

Analytical solution l

1 l

2 0.1

l1 l

2 0.05

l1 l

2 0.001

Figure 3.4: Radial displacement of the internal points

59

ch3-The_Boundary_Element_Method/Figs/EPS/2D_hollow_cylinder.eps

ch3-The_Boundary_Element_Method/Figs/EPS/2d_hollow_cylinder_under_pressure_ips.eps


The percentage error of the applied BEM methodology as compared to the

mesh density is presented in Table 3.2 for the first material case (l1 ≃ l2 ≃ 0.1).

Elements % Error

12 0.17317 0.12026 0.06950 0.028

Table 3.2: Average percentage error w.r.t. the analytical solution of Zervos et al.

(2009)

3.5.2 Radial deformation of a Sphere

A gradient elastic sphere of radius α = 0.5m is subjected to a uniform displace-

ment on its boundary ur = 0.01m. The material characteristics of the sphere are

the same used in the previous example and are presented in Table 3.1.

In order to model the problem, octant symmetry has been used and the same

classical and non-classical boundary conditions have been applied to all elements.

Namely, (ur, uθ, uφ) = (1.0e−02, 0, 0) and (qr, qθ, qφ) = (0, 0, 0), with the sub-

scripts r, θ and φ indicating the coefficients of a spherical coordinate system

located at the center of the sphere. A set of internal points have been deployed

inside the sphere, along its radius. The analytical solution was given in Tsepoura

et al. (2003).

ur (r) =

[

C1r + C2

(

−l21sinh(r/l1)

r2+ l1

cosh(r/l1)

r

)]

r (3.64)

with

C1 = −2l1u0α cosh(α/l1) − 2l21u0 sinh(α/l1) − u0α2 sinh(α/l1)

α[

−3l1α cosh(α/l1) + 3l21 sinh(α/l1) + α2 sinh(α/l1)] (3.65)

C2 = − u0α2

−3l1α cosh(α/l1) + 3l21 sinh(α/l1) + α2 sinh(α/l1)(3.66)

Figure 3.5 shows the radial displacement on the internal points with respect to

their distance from its center as compared to the analytical solution. In Table 3.3

60


0.0 0.1 0.2 0.3 0.4 0.50.000

0.002

0.004

0.006

0.008

0.010R

adia

l dis

plac

emen

t of i

nter

nal p

oint

s

Radial distance r

2D Form II Analytical solution l

1~=l

2~=0.1

l1~=l

2~=0.05

l1~=l

2~=0.001

Figure 3.5: Radial displacement of the internal points

the average relative error of the internal radial displacement is presented with

respect to the analytical solution for various mesh sizes is examined for the first

material case (l1 ≃ l2 ≃ 0.1).

Elements % Error

7 0.31212 0.20548 0.05675 0.038

Table 3.3: Average percentage error w.r.t. the analytical solution of Tsepoura

et al. (2003)

3.5.3 Tension of a bar

Consider a 2D gradient elastic bar of length h = 1.2m, width d = 4.2m and

rounded edges with radius re = 0.05m (Figure 3.6). The material characteristics

of the bar are those presented in Table 3.4. Note that Mindlin’s Form II theory

is equivalent to the simplified Form of Mindlin’s theory with gradient elastic

61

ch3-The_Boundary_Element_Method/Figs/EPS/3d_radial_deformation_of_a_sphere_ips.eps


Figure 3.6: The gradient elastic bar

E 2.0e+05Paν 0.0a1 0.0a2 0.0a3 0.0a4 4.0KNta5 0.0

Table 3.4: The material characteristics used in the hollow cylinder

62

ch3-The_Boundary_Element_Method/Figs/EPS/bar_under_tension.eps


coefficient g = l1 = l2 when a1, a3 and a5 are set equal to zero and a2 = g2λ/2

and a4 = g2µ.

The analytical solution of the gradient elastic bar, with Poisson ratio ν = 0,

in the context of the simplified version of Mindlin’s Form II theory are given in

Tsepoura et al. (2003) and Tsepoura et al. (2002).

u (x) =T0

E|x| + T0g

2E cosh(h/2g)

(

e−|x|/g − e|x|/g)

, |x| ≤ h

2(3.67)

The bar is subjected to tension T0 = 200KPa on its top and bottom sides. The

non classical boundary condition applied to the top and bottom faces of the

bar is (qx, qy) = (0, 0). The sides of the bar are left traction free by imposing

(Tx, Ty) = (0, 0) and (Rx, Ry) = (0, 0).

In order to solve the problem quarter symmetry has been used, requiring thus

only the one fourth of the domain to be discretized. A set of internal points has

been placed along the central vertical axis of the bar.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Axi

al d

ispl

acem

ent u

y

y

2D Form II Analytical Solution BEM

Figure 3.7: Axial displacement of the internal points

In Figure 3.7 the axial displacements of the internal points are presented. For

all the displayed results, the relative error with respect to the analytical solution

(provided by Tsepoura et al. (2003)) is provided in Table 3.5.

63

ch3-The_Boundary_Element_Method/Figs/EPS/2d_tension_of_a_bar_ips.eps


Elements % Error

6 0.10819 0.10454 0.043

Table 3.5: Average percentage error w.r.t. the analytical solution of Tsepoura

et al. (2003)

64

Chapter 4

Fracture Mechanics in Elasticity

with Microstructure

It is well known that in classical elastic fracture mechanics exist mainly two

approaches: the energy approach and the stress intensity factor (SIF) approach.

The first concerns energy theorems dealing mainly with the concept of energy

release rate and it is association with the J-integral, while the second approach

concerns the evaluation of critical parameters, like Stress Intensity Factors (SIFs),

near to the tip of the crack via classical stress and strain analysis. First Griffith

(1924) utilized the idea of energy release rate in order to explain crack growth. In

many situations the energy and stress intensity approaches are equivalent and give

the same predictions. Especially in linear elastic fracture mechanics the energy

release rate is explicitly associated to SIFs corresponding to three fundamental

crack modes, i.e. Mode I, II and III. However, as it is pointed out in O’Dowd

(2002), it is important to be familiar with both approaches. The energy approach

is appropriate mainly for elastic materials while the SIF approach can be applied

to a wider range of materials.

As it is expected, in linear gradient elastic fracture mechanics both approaches

have appeared so far in the literature. Most of these results can be found in

the theoretical papers of Vardoulakis et al. (1996), Exadaktylos et al. (1996),

Vardoulakis & Exadaktylos (1997), Exadaktylos (1998), Huang et al. (1997), Shi

et al. (2000), Fannjiang et al. (2002), Georgiadis (2003), Georgiadis & Grentzelou

(2006), Tong et al. (2005), Chan et al. (2008), Radi (2008), Giannakopoulos

65

4. FRACTURE IN ELASTICITY WITH MICROSTRUCTURE

& Gavardinas (2008)and Gourgiotis & Georgiadis (2009) and the numerical of

Amanatidou & Aravas (2002), Imatani et al. (2005), Akarapu & Zbib (2006),

Markolefas et al. (2009), Wei (2006) and Askes et al. (2008). As mentioned in the

introduction, the main conclusion they reach is that near the crack tip displace-

ments and strains behave as r3/2 and r1/2 functions, respectively, with r being

the distance from the crack tip, while double stresses and total stresses exhibit

a singular behaviour of the order r−1/2 and r−3/2, respectively. The important

fact about these results is that gradient elastic theories predict the same cusp-like

crack shape with Barenblatt’s cohesive zone theory (Barenblatt (1962)) without

demanding extra interatomic forces, beyond those imposed by the non-classical

boundary conditions. On the other hand, stress fields near to the tip of the crack

remain singular. However, although there are results concerning the J-integral

defined in a closed line around the crack tip, there are no results dealing with the

determination of SIFs and its association with the energy release rate. The goal

of the present section is to give a numerical estimation of SIFs for two and three

dimensional mode I and mode II cracks in gradient elastic medium, via the BEM

formulation described in Chapter 3.

4.1 Displacement and Stress Fields near the Crack

Consider a crack in a linear elastic material. It is convenient to define a polar

coordinate system centered on the crack tip, as show in Figure 4.1. The crack

faces are considered to be stress free, having

σθθ (r,±π) = 0 (4.1)

τrθ (r,±π) = 0 (4.2)

For a linear elastic material, this problem can be solved using an Airy stress

function. In short, the stress field is represented as an infinite series

σ (r, θ) =

∞∑

i=1

Airλifi (θ) . (4.3)

If sufficient terms are taken, the exact solution of any linear elastic problem can be

obtained. However, the possible values of λi reduce down to only one, λi = −1/2.

66

4.1 Displacement and Stress Fields near the Crack

Figure 4.1: Rectangular components of the crack tip stresses

This happens due to the fact that all positive values are excluded, because as r

tends to zero, rλi tends to zero, if λi > 0. Additionally, if λi = 1, limr→0 rλi = 1,

excluding thus all non-negative values for λi. On the other hand, values less −1

are also discarded because they lead to an unbounded form of the strain energy

on the crack tip. Consequently, the only possible choices for λi lie in the interval

(−1, 0). However, the only acceptable value is that of −1/2, because it is the

only one in (−1, 0) that satisfies the equilibrium conditions. The stress field can

now be approximated as

σ ∼ A√rf (θ) + . . . . (4.4)

Conventionally, the constant A is called Stress Intensity Factor and is denoted by

KI , KII or KIII , depending on the type of loading; tension, shear or anti-plane

shear respectively, which also characterizes the crack as Mode I, Mode II or Mode

III. The displacements and stresses in classical elasticity are of the order of r1/2

and r−1/2 respectively, for all three crack modes. Specifically, setting

κ = (3 − ν) / (1 + ν) , ν ′ = 0 and ν ′′ = ν, for plane stress (4.5)

κ = (3 − 4ν) , ν ′ = ν and ν ′′ = 0, for plane strain (4.6)

the Cartesian coefficients of the stresses and the displacements are given by

Mode I:

{

σxxσyyσxy

}

=KI

(2πr)1/2

{

cos (θ/2) [1 − sin (θ/2) sin (3θ/2)]cos (θ/2) [1 + sin (θ/2) sin (3θ/2)]

sin (θ/2) cos (θ/2) cos (3θ/2)

}

67

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/crackRVE.eps


Figure 4.2: Mode I: Opening or tensile mode; Mode II: Sliding or in-plane shear

mode; Mode III: Tearing or anti-plane shear mode

σzz = ν ′ (σxx + σyy)

σxz = σyz = 0{

uxuy

}

=KI

2E

r1/2

√2π

{

(1 + ν) [(2κ− 1) cos (θ/2) − cos (3θ/2)](1 + ν) [(2κ+ 1) sin (θ/2) − sin (3θ/2)]

}

uz = −ν′′z

E(σxx + σyy)

Mode II:

{

σxxσyyσxy

}

=KII

(2πr)1/2

{ − sin (θ/2) [2 + cos (θ/2) cos (3θ/2)]sin (θ/2) cos (θ/2) cos (3θ/2)

cos (θ/2) [1 − sin (θ/2) sin (3θ/2)]

}

σzz = ν ′ (σxx + σyy)

σxz = σyz = 0{

uxuy

}

=KII

2E

r1/2

√2π

{

(1 + ν) [(2κ+ 3) sin (θ/2) + sin (3θ/2)]− (1 + ν) [(2κ− 3) cos (θ/2) + cos (3θ/2)]

}

uz = −ν′′z

E(σxx + σyy)

Mode III:{

σxzσyz

}

=KIII

(2πr)1/2

{

− sin (θ/2)cos (θ/2)

}

σxx = σyy = σzz = 0

68

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/ModeI,II,III.eps

4.2 Crack Elements for Linear and Gradient Elastic Fracture

{

uxuy

}

={

00

}

uz =4KIII

E

r1/2

√2π

[(1 + ν) sin (θ/2)]

where E is the Young’s modulus, ν the Poisson’s ratio, KI , KII and KIII the

SIFs for the mode I, II and III respectively.

On the other hand, this is not the case for gradient elasticity theories. Ac-

cording to Vardoulakis et al. (1996), Exadaktylos et al. (1996), Vardoulakis &

Exadaktylos (1997), Exadaktylos (1998), Shi et al. (2000), Fannjiang et al. (2002)

and Georgiadis (2003) the fields u, q, R and P vary near the crack as r3/2, r1/2,

r−1/2 and r−3/2 respectively,with r being the distance from the crack tip or front.

Since the elements that are typically used in BEM interpolate the unknown fields

either linearly or quadratically, the behavior of the fields near the crack can

never be represented correctly. To this end, two new boundary elements have

been designed, a line and a quadrilateral element, that take the above field sin-

gularities into account by incorporating them into their interpolation functions,

as explained in the next section. Note that these elements have variable order of

singularities, which means that they can be used in classical as well as in gradient

elasticity.

4.2 Crack Elements for Linear and Gradient Elas-

tic Fracture

In this section, adopting the idea of using boundary elements of variable-order of

singularity around the tip of the crack, for the description of the near tip behavior

and the evaluation of the corresponding SIFs (Lim et al. (2002) and Zhou et al.

(2005)), a new special, discontinuous element with variable order of singularity

has been designed. The advantage of this approach is that the fields around the

tip of the crack are treated in a unified manner. Furthermore, the side of the

element that resides on the crack is always discontinuous avoiding that way the

calculation of the fields on the crack tip or front, where they become infinite.

69


4.2.1 Two dimensional crack element

In this special element, the functional nodes are identical to those of a classi-

cal discontinuous three-noded quadratic line element, with one geometrical node

residing always at the crack tip. The main advantage of using discontinuous el-

ements is that no functional nodes are located at the tip of the crack and thus,

despite the singularity of R and P at the tip, their nodal values are finite and

can be computed.

Figure 4.3: Variable order of singularity discontinuous boundary element and its

transformation

As shown in Figure 4.3, the tip of the crack can be located either at ξ = −1

or at ξ = 1 for the special element being to the left or right of the tip. In order

to unify these two possible cases, a new variable p is introduced via the linear

transformation

p =1 + cξ

2(4.7)

with c = ±1 for the tip located at ξ = ∓1, respectively. Thus, the tip of the

crack is always located at p = 0 and the interval ξ ∈ [−1, 1] is transformed into

the interval p ∈ [0, 1]. Consider a point x (p) on the element and a point y (0)

70

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2D_crack_element.eps


located at the crack tip. The fields of interest F at the point x, can be expressed

in terms of the asymptotic solution as

F = Krλ1 + Lrλ2 + C (4.8)

where K, L and C are constant vectors to be determined. Vector F could rep-

resent u, q, R or P and thus λ1 and λ2 take the values displayed in Table 4.1.

Considering that suitable interpolation functions N i exist, fields F can be ap-

proximated as

F = N iFi, i = 1, 2, 3 (4.9)

with Fi being the three nodal values of F. In view of eq (4.8) N i should have the

form

N i (r, p, λ1, λ2) = airλ1 + bir

λ2 + di (4.10)

where r = |xj (p) − y (0)| is the distance of the functional node j from the crack

tip, as illustrated in Figure 4.4, while the vectors K, L and C of eq (4.8) are

given by

K = aiFi

L = biFi (4.11)

C = diFi

The constants ai, bi, di can be easily obtained by solving a set of three linear

systems, consisting of three equations each, which arise from the requirement

that each interpolation function must satisfy the relations

N i (p corresponding to node j) = δij , i, j = 1, 2, 3 (4.12)

where δij is the Kronecker delta.

The coefficients ai, bi and di have been found to be

a1 =rλ1

2 rλ2

3 − rλ2

2 rλ1

3

rλ1

2 rλ2

3 − rλ2

2 rλ1

3 + rλ1

1

(

rλ2

2 − rλ2

3

)

+ rλ2

1

(

−rλ1

2 + rλ1

3

) (4.13)

a2 =rλ1

1 rλ2

3 − rλ2

1 rλ1

3

−(

rλ1

2 rλ2

3

)

+ rλ2

2 rλ1

3 + rλ1

1

(

−rλ2

2 + rλ2

3

)

+ rλ2

1

(

rλ1

2 − rλ1

3

) (4.14)

71


a3 =rλ1

1 rλ2

2 − rλ2

1 rλ1

2

rλ1

2 rλ2

3 − rλ2

2 rλ1

3 + rλ1

1

(

rλ2

2 − rλ2

3

)

+ rλ2

1

(

−rλ1

2 + rλ1

3

) (4.15)

b1 =1

rλ1

1 − rλ1

2 +[(

rλ2

1 − rλ2

2

) (

rλ1

2 − rλ1

3

)]

/(

−rλ2

2 + rλ2

3

) (4.16)

b2 =1

−rλ1

1 + rλ1

2 +[(

rλ2

1 − rλ2

2

) (

rλ1

1 − rλ1

3

)]

/(

rλ2

1 − rλ2

3

) (4.17)

b3 =1

−rλ1

1 + rλ1

3 +[(

rλ1

1 − rλ1

2

) (

rλ2

1 − rλ2

3

)]

/(

rλ2

1 − rλ2

2

) (4.18)

d1 =1

rλ2

1 − rλ2

2 +[(

rλ1

1 − rλ1

2

) (

rλ2

2 − rλ2

3

)]

/(

−rλ1

2 + rλ1

3

) (4.19)

d2 =1

−rλ2

1 + rλ2

2 +[(

rλ1

1 − rλ1

2

) (

rλ2

1 − rλ2

3

)]

/(

rλ1

1 − rλ1

3

) (4.20)

d3 =1

−rλ2

1 + rλ2

3 +[(

rλ2

1 − rλ2

2

) (

rλ1

1 − rλ1

3

)]

/(

rλ1

1 − rλ1

2

) (4.21)

It can be verified that∑

N i = 1 for all the combinations of λ1, λ2 provided

in Table 4.1. Finally it should be mentioned that in the present 2D boundary

element formulation the distance r between points x and y (Figure 4.4) is taken

as a straight line between these points as it should be and not as a curved one

along the coordinate p.

F λ1 λ2

u 3/2 1q 1/2 1R −1/2 1P −3/2 −1/2

Table 4.1: Orders of magnitude of the asymptotic fields

4.2.2 Integrations over a three noded quadratic line spe-

cial element

In the boundary integral equations (3.35), the integrals involving the fields P and

R and defined over the special boundary elements, in addition to the usual fun-

damental solution type of singularities (Tsepoura et al. (2003)), exhibit an extra

72


Figure 4.4: A 2D discontinuous variable order of singularity element

singularity due to the singular behavior of the interpolation functions (4.10) near

the tip of the crack. Thus, even in cases where the source point does not reside in

the element, i.e. in cases where a so-called regular integration is performed, there

is always a singularity present near the tip of the crack. The methodology for

the treatment of these integrals deals first with the handling of the singularities

coming from the interpolation functions of the special element and then addresses

any possible singularities that are introduced by the fundamental solutions (in

case the source point resides in the element).

4.2.2.1 Integrals involving the field R

The integrals involving the field R and defined over a special boundary element,

appear in the boundary integral equations as

1∫

−1

∂u∗

∂ny(x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.22a)

1∫

−1

∂2u∗

∂nx∂ny(x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.22b)

where N i is the i-th interpolation function, with the parameters λ1 and λ2

being equal to 1/2 and 1, respectively (Table 4.1) and Je is the Jacobian of the

transformation from the global coordinates to the intrinsic local coordinate ξ.

73

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_r_distance.eps


Applying the transformation (4.7) to the integrals (4.22), one obtains them

in the form

1∫

−1

∂u∗

∂ny

(x (p) ,y)N i (p, λ1, λ2) JeJp dp (4.23a)

1∫

−1

∂2u∗

∂nx∂ny

(x (p) ,y)N i (p, λ1, λ2) JeJp dp (4.23b)

with Jp = 2c being the Jacobian of the transformation.

In order to overcome the singularities introduced by the interpolation func-

tions near the crack tip, the non-linear transformation

1 + s

2= pa, (4.24)

due to Zhou et al. (2005) is applied to the integrals (4.23), where the parameter

a > 0 remains to be determined. Thus, integrals (4.23) become

1∫

−1

∂u∗

∂ny(x (s) ,y)N i (p (s) , λ1, λ2) JeJpJnl ds (4.25a)

1∫

−1

∂2u∗

∂nx∂ny(x (s) ,y)N i (p (s) , λ1, λ2) JeJpJnl ds (4.25b)

with Jnl being the Jacobian of the non-linear transformation (4.24) reading

Jnl =1

2a

(

1 + s

2

)1

a−1

=1

2ap1−a. (4.26)

According to eq (4.10), the interpolation functions N i, for the field R can be

written in the general form

N i

(

p,1

2, 1

)

= aip−1/2 + O (1) (4.27)

Substituting eqs (4.26) and (4.27) into integrals (4.25), it is easily observed that

the singularity of the interpolation function is cancelled out when a takes values

in the interval

0 < a ≤ 1

2. (4.28)

74


The value of a = 1/2 is adopted. As long as the singular behavior of the inter-

polation functions has been overcome, integrals (4.25) are treated in the same

manner as the ones corresponding to non-special elements.

4.2.2.2 Integrals involving the field P

The integrals involving the field P and defined over a special boundary element,


1∫

−1

u∗ (x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.29a)

1∫

−1

∂u∗

∂nx(x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.29b)

where the parameters λ1 and λ2 are equal to −3/2 and −1/2, respectively (Ta-

ble 4.1). To overcome the singularities of the interpolation functions appearing

in the integrals 4.29, one would expect that a methodology, similar to the one

followed for the field R, in the previous section, would be adequate. However,

taking into account that the interpolation functions for P can be written as

N i

(

p,1

2, 1

)

= aip−3/2 + bip

−1/2 + O (1) . (4.30)

Substituting eqs (4.26) and (4.30) into eq (4.29), one can observe that there

is no any value of the parameter a that completely removes the interpolation

function singularities. Nevertheless, the non-linear transformation (4.24) is used

again with a = 1/2 and a reduction of the order of the interpolation functions

singularity from O(

p−3/2)

to O (p−1) is achieved. Thus, the integrals (4.29) take

the form

1∫

−1

u∗ (x (s) ,y)N i (p (s) , λ1, λ2) JeJpJnl ds (4.31a)

1∫

−1

∂u∗

∂nx

(x (s) ,y)N i (p (s) , λ1, λ2) JeJpJnl ds (4.31b)

75


At the tip of the crack, located at s = −1, the integrals (4.31) still exhibit

a singular behavior of O (p−1) and are evaluated by applying the methodology

for direct treatment of singular integrals due to Guiggiani & Gigante (1990).

According to this methodology, the kernels of the integrals are expanded asymp-

totically in power series with respect to the local coordinate s around the singular

point. Then the divergent part of the integrals becomes regular by subtracting the

corresponding singular terms that were produced during the expansion. These

subtracted terms are finally added back after having been calculated by analytical

integration.

Applying the above briefly described procedure, the integrals (4.31) take the

form

1∫

−1

u∗ (x (s) ,y)N i (p (s) , λ1, λ2)

(

1 + s

2

)

Je (s) Jp

− u∗|s=−1 aiJpJe|s=−1

(

1 + s

2

)−2

ds (4.32a)

+ u∗|s=−1 aiJpJe|s=−1 limǫ→0

1∫

−1+ǫ

(

1 + s

2

)−2

ds

1∫

−1

∂u∗

∂nx

(x (s) ,y)N i (p (s) , λ1, λ2)

(

1 + s

2

)

Je (s) Jp

− ∂u∗

∂nx

∣

∣

∣

∣

s=−1

aiJpJe|s=−1

(

1 + s

2

)−2

ds (4.32b)

+∂u∗

∂nx

∣

∣

∣

∣

s=−1

aiJpJe|s=−1 limǫ→0

1∫

−1+ǫ

(

1 + s

2

)−2

ds

where ǫ is the radius of a circle including the tip of the crack. The analytical

calculation of the last integrals appearing in eq (4.32) yields

limǫ→0

1∫

−1+ǫ

(

1 + s

2

)−2

ds = limǫ→0

(

− 4

1 + s

∣

∣

∣

∣

1

−1+ǫ

)

= −2 + limǫ→0

4

ǫ.

(4.33)

76


Considering the contribution of all elements around the tip of the crack with

the same size of ǫ, the last term in eq (4.33) must be zero (Guiggiani & Gi-

gante (1990)). As long as the singular behavior of the interpolation functions

has been overcome, the integrals (4.32) are treated in the same way as the ones

corresponding to non-special elements.

4.2.2.3 Integrals involving the field q

The integrals involving the field q and defined over a special boundary element,


1∫

−1

R∗T (x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.34a)

1∫

−1

∂R∗T

∂nx

(x (ξ) ,y)N i (p (ξ) , λ1, λ2) Je dξ (4.34b)

where the parameters λ1 and λ2 are equal to 1/2 and 1, respectively (Table 4.1).

The interpolation functions involved in integrals (4.34) do not exhibit any singu-

larity as one approaches the crack tip. Thus, one would expect that a standard

Gauss quadrature would be adequate for an accurate integration. However, a

slow convergence was observed due to the order O(

p1/2)

of the interpolation

functions. In order to achieve a better convergence, the non-linear transforma-

tion (4.24) is used again with a = 1/2, this time in an effort to increase the

order of the integrand from O(

p1/2)

to O(

p3/2)

. Thus the integrals (4.34) finally

become

1∫

−1

R∗T (x (s) ,y)N i (p (s) , λ1, λ2)JeJpJnl ds (4.35a)

1∫

−1

∂R∗T

∂nx(x (s) ,y)N i (p (s) , λ1, λ2) JeJpJnl ds, (4.35b)

where Jnl is the Jacobian of the non-linear transformation given by (4.26).

77


4.2.3 Three dimensional crack element

As in the 2D case, case, near the crack front, the fields u, q, R and P vary as r3/2,

r1/2, r−1/2 and r−3/2 respectively, with r being the distance from the crack front.

Once more, adopting the idea of using variable-order continuous elements (Lim

et al. (2002), Zhou et al. (2005)), a new discontinuous, quadrilateral, eight-nodded

element with variable-order singularity has been constructed for the treatment of

the fields around the crack front.

In this special element, the crack side is always discontinuous, while disconti-

nuity on the other sides is optional. The main advantage of using discontinuity

on the crack side is that no functional nodes are located on the crack front and

thus, despite the singularity of R and P there, the field nodal values are finite

and can be computed. The local coordinates of the functional nodes are identical

to those of a classical, partially or fully discontinuous, eight-noded, quadratic,

quadrilateral element. Practically, the crack front can be located at any of the

element’s sides. In order to be able to deal with all the possible cases of the crack

front location, the local numbering of the element nodes is changed, so that the

crack front always resides on the first side of the element. The result of the local

renumbering is described in Table 4.2 for all the possible cases. An example of an

element having the crack front located at its third side is illustrated in Figure 4.5.

Nodes Crack on:Side 1 Side 2 Side 3 Side 4

Node 1′ 1 2 3 4Node 2′ 2 3 4 1Node 3′ 3 4 1 2Node 4′ 4 1 2 3Node 5′ 5 6 7 8Node 6′ 6 7 8 5Node 7′ 7 8 5 6Node 8′ 8 5 6 7

Local coord. ξ′1 ξ1 ξ2 −ξ1 −ξ2

Local coord. ξ′2 ξ2 −ξ1 −ξ2 ξ1

Table 4.2: The renumbering of the element nodes, so that the crack front always

resides on the first side.

78


Figure 4.5: Transition from the real 3D space to the parametric representation of

the element and nodal renumbering, for the case of a fully discontinuous element

Consider a point x (ξ′1, ξ′2) on the element and a point y (ξ′1,−1) located at

the crack front having the same ξ1-coordinate as x, as shown in Figure 4.6.

Figure 4.6: Projection of point x to the crack front

The field of interest F at the point x, can be expressed in terms of the asymp-

totic solutions as

F (ξ′1, r) = K (ξ′1) rλ1 + L (ξ′1) r

λ2 + C (ξ′1) (4.36)

where r is the distance r = |x − y|, the symbol F represents u, q, R and P and

λ1, λ2 take the values of Table 4.1. In addition, the fields F can be approximated

using the interpolation functions N i and their corresponding nodal values Fi as

follows

F (ξ′1, r) = N i (ξ′1, r)Fi, i = 1, . . . , 8 (4.37)

79

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3D_crack_element_transformation.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_r_distance.eps


Combining eqs (4.36) and (4.37) and assuming a quadratic behaviour for the

functions K (ξ′1), L (ξ′1) and C (ξ′1), the interpolation functions N i should be of

the form

N i (ξ′1, r) =(

ei1 + ei

2ξ′1 + ei

3ξ′21

)

rλ1

+(

ei4 + ei

5ξ′1 + ei

6ξ′21

)

rλ2

+ ei7 + ei

8ξ′1 + ei

9ξ′21

(4.38)

where eij , for j = 1, . . . , 9 are constants to be determined. Due to the use of

eight-noded elements, one of the nine terms of the above expression must be

omitted. Here, having in mind that the coefficients of rλ1 and rλ2 will be used for

the SIF calculation, this term is taken to be ei9. The remaining eight constants

eij can be easily obtained by solving a set of eight linear systems, consisting of

eight equations each, which arise from the requirement that each interpolation

function N i must satisfy the relations

N i(

ξ′j1 , rj)

= δij , i, j = 1, . . . , 8 (4.39)

where δij is the Kronecker delta, rj =∣

∣x(

ξ′j1 , ξ′j2

)

− y(

ξ′j1 ,−1)∣

∣ and(

ξ′j1 , ξ′j2

)

are

the local coordinates of the j functional node. The interpolation functions N i

have been calculated and are presented in Appendix H. It can be verified that∑

N i = 1 for all the combinations of λ1, λ2 provided in Table 4.1.

4.2.4 Integrations over an eight-noded quadrilateral spe-

cial element

In this section, the treatment of the integrals appearing in the boundary inte-

gral equations for the 3D case is explained. The integrals involving the fields P

and R and defined over the special boundary elements, in addition to the usual

fundamental solution type of singularities (section 3.4.1, Tsepoura et al. (2003)),

exhibit an extra singularity due to the singular behavior of the interpolation func-

tions near the crack front. Thus, even in cases where the source point does not

reside in the element, i.e., in cases where a so-called regular integration is per-

formed, there is always a singularity present near the front of the crack. The

methodology for the treatment of these integrals deals first with the handling of

80


the singularities coming from the interpolation functions of the special element

and then addresses any possible singularities that are introduced by the funda-

mental solutions (in case the source point resides in the element). Without loss

of generality, one can assume that the crack front resides at the first side of the

element. If it does not, one can renumber the element nodes so that it does. This

assumption is useful for the simplification of the following paragraphs.

4.2.4.1 Integrals involving the field R

The integrals involving the field R are defined over a special boundary element

and appear in the discretized form of the boundary integral equations as

1∫

−1

1∫

−1

∂u∗

∂ny(x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2)Je dξ1 dξ2 (4.40a)

1∫

−1

1∫

−1

∂2u∗

∂nx∂ny(x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2)Je dξ1 dξ2 (4.40b)

where N i is the i-th interpolation function given by eqs (4.38), with the param-

eters λ1 and λ2 being equal to −1/2 and 1, respectively (Table 4.1) and Je is the

Jacobian of the transformation from the global coordinates to the intrinsic local

coordinates ξ1, ξ2. It is important to see that there is a singularity of the form

r−1/2, attributed to the new interpolation functions (4.38). If r is expanded in

series with respect to ξ2, around the singular point ξ2 = −1 it is easy to see that

r is of the following form

s1 = ξ1 (4.41a)

s2 = 2

(

1 + ξ22

)a

− 1 (4.41b)

which was introduced by Zhou et al. (2005) and the integrals (4.40) become

1∫

−1

1∫

−1

∂u∗

∂ny(x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2) JeJnl ds1 ds2 (4.42a)

1∫

−1

1∫

−1

∂2u∗

∂nx∂ny

(x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2) JeJnl ds1 ds2 (4.42b)

81


where Jnl is the Jacobian of the non-linear transformation (4.41) of the form

Jnl =1

a

(

1 + s2

2

)(1−a)/a

(4.43)

and the parameter a > 0 is a constant to be determined. For 0 < a ≤ 1/2, the

transformation completely removes the interpolation function singularity. The

value a = 1/2 has been chosen. As long as the singular behavior of the interpo-

lation functions has been overcome, integrals 4.42 are treated in the same way

as the integrals corresponding to non-special elements (section 3.4.1, Tsepoura

et al. (2003)).

4.2.4.2 Integrals involving the field P

The integrals involving the field P and defined over a special boundary element,

appear in the discretized form of the boundary integral equations as

1∫

−1

1∫

−1

u∗ (x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2) Je dξ1 dξ2 (4.44a)

1∫

−1

1∫

−1

∂u∗

∂nx

(x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2) Je dξ1 dξ2 (4.44b)

where the parameters λ1 and λ2 are equal to −3/2 and −1/2, respectively (Ta-

ble 4.1). This time to deal with the interpolation function singularities, the

aforementioned non-linear transformation is not adequate. Since the singulari-

ties introduced by the interpolation functions are more than one (of the orders of

r−3/2 and r−1/2), the non-linear transformation raises their order only partially.

Again, expanding r−3/2 in series around the singular point ξ2 = −1, one can see

that r is given by

r = f (ξ1) (ξ2 + 1)−3/2 + g (ξ1) (ξ2 + 1)−1/2 + O (ξ2 + 1)1/2 (4.45)

82


After applying the nonlinear transformation (4.41) the integrals (4.44) become

1∫

−1

1∫

−1

u∗ (x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2) JeJnl ds1 ds2 (4.46a)

1∫

−1

1∫

−1

∂u∗

∂nx(x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2) JeJnl ds1 ds2 (4.46b)

Observing the transformation, one can notice that there is no value for the param-

eter a that raises the order of both interpolation function singularities. However,

by choosing a = 1/2 one can reduce the singularities to the order of r−1. To

address this type of singularity one also applies the methodology for direct treat-

ment of singular integrals, introduced by Guiggiani & Gigante (1990). In short,

the kernels of the integrals are expanded asymptotically to power series with re-

spect to the local coordinate s2 around the point s2 = −1. Then the singular

terms of the divergent part of the integrals are subtracted and the integral is cal-

culated with the Gauss quadrature, as it is now regular and finally the subtracted

terms are added, after integrating them analytically. Applying the above briefly

described procedure, the integrals (4.46) take the form

1∫

−1

1∫

−1

u∗N i (s1, r (s1, s2) , λ1, λ2)

(

1 + s2

2

)

Je (s1, s2)

− u∗|s2=−1 f (s1)Je|s2=−1

(

1 + s2

2

)−2

ds2 (4.47a)

+ u∗|s2=−1 f (s1)Je|s2=−1 limǫ→0

1∫

−1+ǫ

(

1 + s2

2

)−2

ds2

ds1

1∫

−1

1∫

−1

∂u∗

∂nx

N i (s1, r (s1, s2) , λ1, λ2)

(

1 + s2

2

)

Je (s1, s2)

− ∂u∗

∂nx

∣

∣

∣

∣

s2=−1

f (s1)Je|s2=−1

(

1 + s2

2

)−2

ds2 (4.47b)

+∂u∗

∂nx

∣

∣

∣

∣

s2=−1

f (s1)Je|s2=−1 limǫ→0

1∫

−1+ǫ

(

1 + s2

2

)−2

ds2

ds1

83


where ǫ is the radius of a sphere including the singular point, which resides on the

crack front. The analytical calculation of the last integrals appearing in eqs (4.47)

yields

limǫ→0

1∫

−1+ǫ

(

1 + s2

2

)−2

ds2 = limǫ→0

(

− 4

1 + s2

∣

∣

∣

∣

1

−1+ǫ

)

= −2 + limǫ→0

4

ǫ

(4.48)

Considering the contribution of all elements around the singular point within a

neighbourhood of size ǫ, the last term in eq (4.48) must be zero (Guiggiani &

Gigante (1990)). As long as the singular behavior of the interpolation functions

has been overcome, the integrals (4.47a) and (4.47b) are treated in the same way

as the integrals corresponding to non-special elements (section 3.4.1, Tsepoura

et al. (2003)).

4.2.4.3 Integrals involving the field q

The integrals involving the field q and defined over a special boundary element,

appear in the discretized form of the boundary integral equations as

1∫

−1

1∫

−1

R∗T (x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2) Je dξ1 dξ2 (4.49a)

1∫

−1

1∫

−1

∂R∗T

∂nx

(x (ξ1, ξ2) ,y)N i (ξ1, r (ξ1, ξ2) , λ1, λ2) Je dξ1 dξ2 (4.49b)

where the parameters λ1 and λ2 are equal to 1/2 and 1, respectively (Table 4.1).

The interpolation functions involved in integrals 4.50 do not exhibit any singu-

larity as one approaches the crack front. Thus, one would expect that a standard

Gauss quadrature would be adequate for an accurate integration. However, a slow

convergence was observed due to the O(

r1/2)

term of the interpolation functions.

In order to achieve a better convergence, the non-linear transformation (4.41) is

used again with a = 1/2, this time in an effort to increase the order of the

84

4.3 BEM Stress Intensity Factor Calculation

integrand from O(

r1/2)

to O(

r3/2)

. Thus the integrals (4.50) finally become

1∫

−1

1∫

−1

R∗T (x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2)JeJnl ds1 ds2 (4.50a)

1∫

−1

1∫

−1

∂R∗T

∂nx(x (s1, s2) ,y)N i (s1, r (s1, s2) , λ1, λ2)JeJnl ds1 ds2 (4.50b)

where Jnl is the Jacobian of the non-linear transformation given by (4.43).

4.3 BEM Stress Intensity Factor Calculation

As mentioned in Chapter 2, the goal of the BEM is to solve numerically the

well-posed boundary value problem described by the boundary integral equa-

tions (2.110), (2.111) and the corresponding boundary conditions. To this end

the global boundary S is discretized into quadratic, continuous and discontin-

uous isoparametric boundary elements, while special variable-order singularity,

discontinuous elements are placed on both sides of the crack tip or crack front

as it is illustrated in Figure 4.7(a). Note that in order to calculate the SIFs the

problem domain must be divided into two subregions, created by extending the

crack plane. For the 2D case the domain is divided by extending the crack line

from both crack tips (Figure 4.7(b)). Once the boundary value problem has been

solved, the calculation of SIFs is done via the nodal traction values of the special

elements.

Approaching the crack tip or front (r → 0), the traction P, according to

eq (4.8) for the 2D case and eq (4.36) for the 3D case, admits a representation of

the form

P =K1 (P1, . . . ,PN)√

2πlimr→0

r−3/2 +K2 (P1, . . . ,PN)√

2πlimr→0

r−1/2 + C (P1, . . . ,PN)

(4.51)

where N = 3 for 2D and N = 8 for 3D. Taking into account relations (4.11)

and (4.38) the stress intensity factors corresponding to x, y and to x, y and z

85


(a) (b)

Figure 4.7: (a) Position of the variable singularity order elements w.r.t. the crack

and (b) domain division and element positioning in 2D

directions for 2D and 3D respectively, are obtained by

K1 =√

2πK =√

2πajPj (4.52a)

K2 =√

2πL =√

2πbjPj (4.52b)

and

K1 (ξ1) =√

2π(

D1 + ξ1D2 + ξ21D3

)

(4.53a)

K2 (ξ1) =√

2π(

D4 + ξ1D5 + ξ21D6

)

(4.53b)

with j = 1, 2, 3 for two dimensions and Di = ejiPj, i = 1, . . . , 6 and j = 1, . . . , 8

for three dimensions and aj, bj , eji taking values from eqs (4.13-4.18) and (H.2-

H.65) respectively.


This section contains three numerical examples regarding a 2D mode I, a 2D

mixed-mode (I & II) and a 3D mode I crack, that demonstrate the accuracy of

the above methodology and lead to interesting conclusions regarding the nature of

the stress intensity factors and the stress field around a crack in a microstructured

material.

86

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_crack_element_positioning.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_crack_element_positioning.eps


4.4.1 Square plate with horizontal line crack under ten-

sion

Consider a square gradient elastic plate with rounded corners of very small radius

of curvature (in order to have a smooth boundary) in a state of plane stress. The

plate contains a central horizontal line crack and is subjected to a uniform tensile

traction P0 = 100MPa applied normal to its top and bottom sides, as shown

in Figure 4.8. The crack length is chosen to be equal to 2a = 1m and the side

of the square plate is L = 16a. The Young modulus and the Poisson’s ratio

of the gradient elastic plate are E = 210GPa and ν = 0.2, respectively. The

constants a1, a3 and a5 were set equal to zero and the constants a2 and a4 were

set equal to λg2/2 and µg2, for various values of g (0.01, 0.1, 0.3, 0.5).Then,

the gradient coefficients l1 and l2 become equal to each other and equal to g

and Form II gradient theory is downgraded to its simplified version. Due to the

double symmetry of the problem, only one quarter of the plate is discretized, with

the following boundary conditions along the axes of symmetry: P (0,y) = 0 and

R (0, y) = 0 for 0 ≤ y < a, uy (0,y) = 0 and R (0, y) = 0 for a ≤ y ≤ L/2 and

ux (x, 0) = 0 and R (x, 0) = 0 for 0 ≤ x ≤ L/2. Figure 4.9 displays the upper-

Figure 4.8: Gradient elastic plate with a horizontal line crack

87

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeI_square.eps


right-quarter of the crack opening displacement profile obtained by the present

BEM for the aforementioned values of the coefficient g (0.01, 0.1, 0.3, 0.5). In

the same figure, the crack profile provided by the classical elasticity theory is also

shown. As it is apparent, the crack profile in the gradient elastic case remains

sharp at the crack tip and is not blunted as in the classical case. This cusp type

of profile is identical to the one coming out of Barenblatt’s (Barenblatt (1962))

cohesive zone theory.

Barenblatt explains that the two faces of the crack, right at the tip, are sub-

jected to very strong interatomic forces. Thus, considering these atomic attraction

forces as compressive stresses larger than the tensile ones due to external loading,

he obtained a cusp-like crack opening near the tip of the crack. The important

conclusion here is that the results depicted in Figure 4.9 are fully compatible with

Barenblatt’s findings without, however, to consider other forces than those im-

plied by the Mindlin’s Form II gradient elasticity theory. Also, it should be noted

that as the coefficient g increases, i.e. the gradient coefficients l1, l2 increase, the

crack becomes stiffer.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

classical g = 0.01 g = 0.1 g = 0.3 g = 0.5

Dis

plac

emen

t uy (n

orm

al to

the

crac

k)

Distance from the origin (crack tip at x=0.5)

Figure 4.9: Upper right quarter of the COD profile

In order to assess the accuracy of the computed SIFs by the proposed method,

a convergence analysis of the results with mesh refinement is performed. The

problem is solved using a large number of uniform discretizations with the crack

88

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeI_cods.eps


half-length to special boundary element length ratio a/le varying from 1 to 50.

The computed SIFs for the classical elastic and the gradient elastic case with

three characteristic values of the coefficient g (0.01, 0.05 and 0.3) are listed in

Tables 4.3 and 4.4, respectively. In addition, Table 4.3 also lists the percentage

error of the results for the case of classical elasticity in comparison with the

available analytical solution (Broek (1974)). Regarding classical elasticity, the

mesh convergence analysis reveals that for a wide range of ratios a/le (from 3 to

50) the percentage error is below 1% and the convergence appears to be affected

insignificantly by the element size. A ratio of a/le = 5 leads almost to the

exact result. As far as the gradient elasticity is concerned, the computed SIF

(KI)1 values converge as the ratio a/le increases independently of g, while the

SIF (KI)2 values appear to be sensitive to the element size, especially as the

coefficient g increases. Specifically, for the worst possible case here of g = 0.3,

there is a narrow range of ratios a/le (from 2 to 5) where the SIF (KI)2 indicates

a subtle convergence, while for greater ratios tends to zero, which is obviously not

correct. As described in the previous section, the SIFs are computed as functions

of the nodal traction values (P1,P2,P3) corresponding to one of the two special

elements attached to the crack tip. As it is obvious, the smaller the size of the

special element is, the closer to the tip of the crack its nodes reside. As mentioned

earlier the traction field P on the special element, varies as

P =(KI)1√

2πr−3/2 +

(KI)2√2π

r−1/2 + C. (4.54)

In the neighborhood of the crack tip, i.e. for small values of r, the first term

of eq (4.54) becomes dominant and the contribution of the second term to the

traction P is insignificant. As a result, decreasing the size of the special element

with its nodes residing in the area of domination of the first term, makes (KI)2

difficult to calculate. In the extreme case, where all the nodes of the special

element are located in the area of domination of the first term, the computed

(KI)2 tends to zero, as it is evident from Table 4.4. However this extreme case

of high values of g, like g = 0.3 may not be realistic and one should pay more

attention to results corresponding to smaller values of g, like g = 0.01, for which

the results are very good.

89


a/le KI/ (P0√

πa) % Error1 0.95428 5.49382 0.99058 1.898153 1.00165 0.801924 1.00691 0.281085 1.00996 0.021146 1.01195 0.217787 1.01354 0.375778 1.01457 0.4776810 1.01599 0.6181712 1.01692 0.7102914 1.01745 0.7624818 1.01832 0.8483124 1.01903 0.9185430 1.01945 0.9610950 1.01961 0.97629

Analytical solution: 1.00975Broek (1974)

Table 4.3: SIF convergence for the classical elastic case

One can observe from Tables 4.3 and 4.4 that gradient elastic SIFs are smaller

than those of the classical elastic case, especially for increasing values of g.

This is in agreement with many other studies (Ru & Aifantis (1993), Chang &

Gao (1997), Tsepoura et al. (2002), Papargyri-Beskou et al. (2003b), Papargyri-

Beskou et al. (2003a), Georgiadis et al. (2004), Giannakopoulos & Stamoulis

(2007)) indicating a stiffness increase in gradient elasticity. Tables 4.3 and 4.4

also indicate that gradient elasticity requires a slightly denser mesh around the

tip of the crack, compared to the classical elasticity. This is due to the higher

order of singularity (r−3/2). The results that follow have been obtained with

a/le = 8.

In Figures 4.10(a) and 4.10(b) the two mode-I SIFs for the gradient elastic

case, (KI)1 and (KI)2, are plotted versus the coefficient g. The interesting remark

here is that the SIF (KI)1 tends to zero as g tends to zero. As a result of that,

eq (4.51) becomes Py = (KI)2 /√

2π limr→0 r−1/2 with (KI)2 being the mode-I

SIF as defined in classical elasticity theory. Moreover, Figure 4.10(b) depicts the

behavior of the SIF corresponding to r−1/2 traction term as a function of the

90


a/le g = 0.01 g = 0.05 g = 0.3

(KI)1

(P0

√πa)

(KI)2

(P0

√πa)

(KI)1

(P0

√πa)

(KI)2

(P0

√πa)

(KI)1

(P0

√πa)

(KI)2

(P0

√πa)

1 0.95428 5.4938 5.4938 5.4938 5.4938 5.49382 0.99058 1.89815 1.89815 1.89815 1.89815 1.898153 1.00165 0.80192 0.80192 0.80192 0.80192 0.801924 1.00691 0.28108 0.28108 0.28108 0.28108 0.281085 1.00996 0.02114 0.02114 0.02114 0.02114 0.021146 1.01195 0.21778 0.21778 0.21778 0.21778 0.217787 1.01354 0.37577 0.37577 0.37577 0.37577 0.375778 1.01457 0.47768 0.47768 0.47768 0.47768 0.4776810 1.01599 0.61817 0.61817 0.61817 0.61817 0.6181712 1.01692 0.71029 0.71029 0.71029 0.71029 0.7102914 1.01745 0.76248 0.76248 0.76248 0.76248 0.7624818 1.01832 0.84831 0.84831 0.84831 0.84831 0.8483124 1.01903 0.91854 0.91854 0.91854 0.91854 0.9185430 1.01945 0.96109 0.96109 0.96109 0.96109 0.9610950 1.01961 0.97629 0.97629 0.97629 0.97629 0.97629

Table 4.4: SIFs convergence for the gradient elastic case (g = 0.01, 0.05 and 0.3)

coefficient g. It should be noted that for g greater than 0.1, the contribution of

this term is much smaller than that of the term corresponding to r−3/2. For this

reason the evaluation of the SIF (KI)2 for large g requires further investigation.

However, for small g it is apparent from Figure 4.10(b) that as g approaches zero

(KI)2 becomes dominant and goes to the classical elastic case. However, the most

important observation here is that the SIF (KI)1 takes only negative values. This

means that in gradient elasticity the stresses near the crack tip not only go to

infinity with a different order (r−3/2) than those of classical elasticity (r−1/2), but

are also compressive and not tensile as in classical elasticity. This explains the

different shapes of the crack profile in gradient and classical elasticity theories,

as shown in Figure 4.9.

This behaviour becomes more pronounced in Figure 4.11 where the traction

field near to the crack tip, for various values of the gradient coefficient g is dis-

played.

91


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

(KI)

1 (Order of singularity: r-3/2)

KI/[

P 0(pa)

1/2 ]

Gradient coefficient g

(a)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.100.80

0.85

0.90

0.95

1.00

1.05

1.10

(KI)

2 (Order of singularity: r-1/2)

KI/[

P 0(pa)

1/2 ]


(b)

Figure 4.10: SIFs (a) (KI)1 and (b) (KI)2 as functions of the coefficient g

92

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeI_sif1.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeI_sif2.eps


0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48

-1.40E+009

-1.20E+009

-1.00E+009

-8.00E+008

-6.00E+008

-4.00E+008

-2.00E+008

0.00E+000

2.00E+008

4.00E+008

Trac

tions

Distance from crack tip

classical g=0.01 g=0.1 g=0.3

Figure 4.11: Traction values near the crack tip

4.4.2 Square plate with diagonal line crack under tension

The square plate of the previous example with an inclined at an angle of 45 deg

central slant crack is analyzed here again by the proposed method. This is a

mixed mode (I & II) crack problem. The plate domain is divided here into two

Figure 4.12: Gradient elastic plate with a central diagonal line crack

93

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeI_tractions2.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/2d_modeIandII_square.eps


subregions, as shown in Figure 4.12, which are both treated by the BEM and

combined together through the continuity and equilibrium eqs (3.37) at their

interface, except of course the crack faces, that are left traction free. This is

necessary in view of the displacement based BEM employed here (Beskos (1997),

Aliabadi (1997), Dominguez & Ariza (2003)). A traction based BEM would

probably be a better choice as not requiring subregioning, but this is associated

with higher order kernel singularities (Beskos (1997), Aliabadi (1997), Dominguez

& Ariza (2003)).

In Figures 4.13(a) and 4.13(b) the SIFs, (KI)1, (KII)1 and (KI)2, (KII)2,

respectively, are plotted as functions of the gradient coefficient g and exhibit

decreasing values for increasing values of g. Similarly to the mode I case, the

SIFs (KI)1, (KII)1 tend to zero as the gradient coefficient g tends to zero and

take only negative values.

4.4.3 Cube with central horizontal rectangular crack

Consider a gradient elastic cube with rounded corners of very small radius of

curvature (in order to have a smooth boundary). The cube contains a central

horizontal rectangular crack and is subjected to a uniform tensile traction P0 =

100MPa applied normal to its top and bottom sides. The side of the cube L is

chosen to be equal to 16a = 8 and the crack dimensions are 2a× L, as shown in

Figure 4.14(a). The Young modulus and the Poisson ratio of the gradient elastic

plate are E = 210GPa and ν = 0.2, respectively.

Due to the octant symmetry of the problem, the analysis is performed by

taking into account two Cartesian symmetries with respect to the X-Z and Y-Z

planes, while on the X-Y symmetry plane the following boundary conditions are

considered: P (x, y, 0) = 0 and R (x, y, 0) = 0 for 0 ≤ x < a and 0 ≤ y < L/2

and uz (x, y, 0) = 0 and R (x, y, 0) = 0 for a ≤ x ≤ L/2 and a ≤ y ≤ L/2. The

mesh used is shown in Figure 4.14(b), where 4 × 8 elements are placed at the

crack surface.

Figure 4.15 displays the lower right of the crack opening displacement profile,

at y = 0 , obtained by the present 3D BEM for four different values of the

gradient coefficient g (0.05, 0.1, 0.3, 0.5), as well as the profile corresponding to

94


0.00 0.05 0.10 0.15 0.20 0.25 0.30-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

(KI) 1,(K

II) 1/ [

P 0( a

)1/2 ]


(KI)1

(KII)1

Order of singularity: r-3/2

(a)

0.00 0.05 0.10 0.15 0.20 0.25 0.300.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

0.54

(KI) 2,(K

II) 2/ [

P 0( a

)1/2 ]


(KI)2

(KII)2

Order of singularity: r-1/2

(b)

Figure 4.13: SIFs (a)(KI)1, (KII)1 and (b)(KI)2, (KII)2 as functions of the gra-

dient coefficient g for the mixed mode (I & II) crack

95

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/MixedMode_r32.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/MixedMode_r12.eps


(a) (b)

Figure 4.14: (a)The gradient elastic cube with a central horizontal rectangular

crack and (b) the discretized domain (one eighth of the cube)

0.0 0.1 0.2 0.3 0.4 0.5-5.0x10-4

-4.0x10-4

-3.0x10-4

-2.0x10-4

-1.0x10-4

0.0g=0.5

g=0.3

g=0.05

Linear

Dis

plac

emen

ts u

z at y

=0

Distance from the origin (crack front at x=0.5)

2D BEM 3D BEM

g=0.1

Figure 4.15: Shape of mode I crack for different values of the gradient coefficient

g compared to the 2D case

96

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_modeI_cube.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_modeI_cube_mesh.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_ModeI_cods.eps


the classical elastic case (g = 0). The profiles are compared to the 2D ones of

Figure 4.9 and found to be the same as expected. The same conclusion is valid

for the calculated (KI)1 and (KI)2 SIFs plotted in Figures 4.16(a) and 4.16(b)

respectively for different gradient coefficient g.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

KI/[P 0

()1/

2 ]


Component: r-3/2

2D SIF (KI)

1

3D SIF (KI)

1

(a)

(b)

Figure 4.16: SIFs (a)(KI)1 and (b)(KI)2 SIFs as functions of the gradient coeffi-

cient g for the 3d mode I crack

97

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_ModeI_r32.eps

ch4-Fracture_Mechanics_in_Elasticity_with_Microstructure/Figs/EPS/3d_ModeI_r12.eps


98

Chapter 5

Conclusions and Future Work

In the context of this thesis, Betti’s reciprocal identity has been obtained, as well

as the 2D and 3D fundamental solutions of a Form II gradient elastic problem.

Furthermore, the integral formulation of the problem has been derived and the

corresponding integral kernels have been calculated.

A displacement based BEM has been employed for the solution of 2D and 3D

static Form II gradient elastic problems.

It has been verified that for appropriate values of the gradient constants α1-

α5 of Form II, it is downgraded to its simplified version with only one gradient

coefficient g and if g is sent to zero, the classical elasticity theory is obtained as

well.

Two new elements have been created; a three noded line and an eight noded

quadrilateral element. Both of them have been equipped with interpolation func-

tions that adapt themselves to the field singularities. These elements have been

used to calculate the unknown fields near the crack and to evaluate the crack’s

stress intensity factor in elastic materials with and without microstructural ef-

fects. This approach requires the use of subregions, but is associated with lower

order kernel singularities than other approaches, such as the traction based BEM

or the dual (traction/displacement) BEM.

The employed BEM requires a discretization which is restricted only to the

boundaries and possible interfaces of the domain considered. The new elements

were placed on the crack tip (or front) and the SIF was calculated by means

of the element’s nodal traction values. Very accurate results, with respect to

99

5. CONCLUSIONS AND FUTURE WORK

the analytical solutions, have been obtained in the context of both classical and

Mindlin’s Form II gradient elasticity theories. One can notice that a hardening

effect is introduced due to the microstructure, which makes the crack stiffer than

that of classical elasticity. Furthermore, the gradient Form II results have been

found to be more physically acceptable then those of the classical elastic case.

Namely, in contrast to the classical elasticity, the crack profile remains sharp and

is not blunted, as illustrated in Figure 5.1. This result can also be observed in the

Figure 5.1: Crack opening displacements (CODs) and tractions near the crack

tip for gradient and classical elasticity

traction field, where near the crack tip, the stresses become compressive. These

results are fully compatible with Barenblatt’s findings, without however consid-

ering other forces than those implied my Mindlin’s Form II gradient elasticity

theory.

Finally, the calculated SIFs indicate that when Form II approaches classical

elasticity, the gradient elasticity SIF (coefficient of the term r−3/2) becomes small

compared to the coefficient of the term r−1/2, which becomes dominant and tends

to the classical elasticity SIF.

In the near future, the following two issues will be addressed:

100

ch5-Conclusions/Figs/EPS/composition.eps

– The calculation of the stress tensor on the domain boundary, which involves

the evaluation of hypersingular integrals.

– The enhancement of the current BEM formulation so that it is applicable to

non-smooth domains, by taking into account the extra boundary condition

E = prescribed, as proposed by Mindlin.

101

5. CONCLUSIONS AND FUTURE WORK

102

Appendix A

Form I, II & III Constants

This section contains all the constants used in Mindlin’s Form I, II and III gradient

elasticity theories.

α =1

b2 + b3

[

g1 −b1 (3g1 + 2g2)

3b1 + b2 + b3

]

(A.1)

β = 1 +2g2

b2 + b3(A.2)

A.1 Form I

α1 =1

2

[

(1 + β) (3α + β)α1 +(

1 + 2αβ + β2)

α2

−1

2(1 + β) (1 − 2α− β)α3 − (1 − β) (3α+ β)α5

−1

2(1 − β) (1 + 2α+ β)α8 + 2αβα11 − α (1 − β)α14

+α (1 + β)α15

]

(A.3)

α2 =1

2

{

− (1 − 2α− β) (3α + β)α1 −1

2

[

1 − (2α+ β)2]α2

+1

4(1 − 2α− β)2 α3 + (3α + β)2 α4

103

A. FORM I, II & III CONSTANTS

+ (3α+ β) (1 + 2α + β)α5 +1

4(1 + 2α+ β)2 α8

+α (3α + 2β)α10 + 2α (α + β)α11 + α (3α + 2β)α13

+α (1 + α + β)α14 − α (1 − α− β)α15

}

(A.4)

α3 =1

4

[

−(

1 − β2)

α2 +1

2(1 + β)2 α3 +

1

2(1 − β)2 α8

]

(A.5)

α4 =1

4

[

(

1 + β2)

α10 −(

1 − β2)

(a11 + a13) +1

2(1 + β)2 α14

+1

2(1 − β)2 α15

]

(A.6)

α5 =1

4

[

−(

1 − β2)

α10 +(

1 + 3β2)

α11 +(

1 + β2)

α13

−1

2

(

1 + 2β − 3β2)

α14 −1

2

(

1 − 2β − 3β2)

α15

]

(A.7)

λ+ 2µ = λ+ 2µ− 8g22

3 (b2 + b3)− (3g1 + 2g2)

2

3 (3b1 + b2 + b3)(A.8)

µ = µ− 2g22

b2 + b3(A.9)

A.2 Form II

α1 = 2α1 − 4α3 (A.10)

α2 = −α1 + α2 + α3 (A.11)

α3 = 2α3 (A.12)

α4 = 3α4 − α5 (A.13)

α5 = −2α4 + 2α5 (A.14)

104

A.3 Form III

A.3 Form III

18d1 = −2α1 + 4α2 + α3 + 6α4 − 3α5 (A.15)

18d2 = 2α1 − 4α2 − α3 (A.16)

3α1 = 2 (α1 + α2 + α3) (A.17)

α2 = α4 + α5 (A.18)

3f = α1 + 4α2 − 2α3 (A.19)

105

A. FORM I, II & III CONSTANTS

106

Appendix B

Form II: Total Potential Energy

Calculation

As mentioned in section 2.3 the total potential energy is the integral over the

volume V of the variation of the potential energy density function W .

∫

V

δW dV =

∫

S


−∫

V


∫

S

niµijk∂jδuk dS

(B.1)


∫

V

δW dV =

∫

S

n · (τ −∇µ) · δu dS

−∫

V

∇ · (τ −∇ · µ) · δu dV +

∫

S

n · µ : ∇δu dS(B.2)

The second term of the first surface integral of eq (B.1) can be written as

nj∂iµijk = njDiµijk + ninjDµijk (B.3)

107

B. FORM II: TOTAL POTENTIAL ENERGY CALCULATION

In addition, the integrand of the last integral may be broken down into two

parts; a tangential and a normal to the surface S.

niµijk∂jδuk = niµijkDjδuk + niµijknjDδuk (B.4)

Where Dj ≡ (δjl − ninl) ∂l and D ≡ nl∂l. The first term in the right hand side

of the above equation can be written as a sum of three terms:

niµijkDjδuk = Dj (niµijkδuk) − niDjµijkδuk −Djniµijkδuk (B.5)

On the surface S, the first term of the right hand side of eq (B.5) can be further

decomposed

Dj (niµijkδuk) = (Dlnl)njniµijkδuk + nqeqpm∂p (emljnlniµijkδuk) (B.6)

Utilizing the Stokes theorem, the integral of the last term of eq (B.6) over a

smooth surface vanishes. If however the surface S has an edge C, formed by the

intersection of two portions S1 and S2, then the Stokes theorem for the last term

gives∫

S

nqeqpm∂p (emljnlniµijkδuk) dS =

∮

C

JnimjµijkKδuk ds (B.7)

with mj = emljsmnl, sm the components of the unit vector tangential to C and

the brackets J·K indicating that the enclosed quantity is the difference between

the values on S1 and S2.

Substituting all the above equations into eq (B.1) we end up to the final

expression for the total potential energy.

∫

V

δW dV = −∫

V

∂j (τjk − ∂iµijk) δuk dV

+

∫

S

[nj τjk − ninjDµijk − 2njDiµijk + (ninjDlnl −Djni) µijk] δuk dS

+

∫

S

ninjµijkDδuk dS +

∮

C

JnimjµijkKδuk ds

(B.8)

108

Appendix C

Mindlin’s Form II Gradient

Elasticity Theory: Integral

Representation Kernels

In this section, the expressions for the fundamental quantities q∗, R∗, P∗ and E∗

are given, as well as their normal derivatives ∂u∗/∂nx, ∂q∗/∂nx, ∂R

∗/∂nx and

∂P∗/∂nx.

First some basic quantities are defined, that will simplify the formulas that

follow.

The constant a indicates the dimensionality of the problem.

a =

{

2, for 2D

3, for 3D(C.1)

Furthermore, the constant b is defined to be:

b =1

(a− 1) 8πµ (1 − ν)(C.2)

Then the quantities J1 (r), J2 (r), K1 (r), K2 (r) and L1 (r)-L3 (r) can be defined.

J1 (r) = Ψ′′ (r) − a− 1

rΨ′ (r) − 2X (r)

r2(C.3)

109

C. MINDLIN’S FORM II: KERNELS

J2 (r) = X ′′ (r) +a− 1

rX ′ (r) +

2aX (r)

r2(C.4)

K1 (r) = Ψ′′ (r) − 1

rΨ′ (r) −X ′′ (r) − a− 2

rX ′ (r) + 2 (a− 1)

X (r)

r2(C.5)

K2 (r) =Ψ′ (r)

r− X ′ (r)

r− a− 1

rX (r) (C.6)

L1 (r) = X ′′ (r) − 5X ′ (r)

r+

8X (r)

r2(C.7)

L2 (r) =X ′ (r)

r− 2X (r)

r2(C.8)

L3 (r) = Ψ′′ (r) − Ψ′ (r)

r(C.9)

Then the scalar functions A1 (r)-A3 (r), B1 (r)-B10 (r) and C1 (r)-C9 (r) are de-

fined as:

A1 (r) = −2µb

(

X ′ (r) − 2X (r)

r

)

(C.10)

A2 (r) = µb

(

Ψ′ (r) − X (r)

r

)

(C.11)

A3 (r) = b

[

λ

(

Ψ′ (r) −X ′ (r) − (a− 1)X (r)

r

)

− 2µX (r)

r

]

(C.12)

B1 (r) = −2 (a4 + a5) bL1 (r) (C.13)

B2 (r) = b

[

−1

2a3J2 (r) +

1

2(a1 + a3)K1 (r) − 2 (a4 + a5)L2 (r)

]

(C.14)

B3 (r) = b

[

−1

2a1J2 (r) +

1

2(a1 + 4a2)K1 (r) − 2 (a4 + a5)L2 (r)

]

(C.15)

B4 (r) = b

[

1

2(2a4 + a5)L3 (r) − 1

2(2a4 + 3a5)L2 (r)

]

(C.16)

B5 (r) = B2 (r) (C.17)

B6 (r) = b [− (2a4 + a5)L2 (r) + a5L3 (r)] (C.18)

B7 (r) = b

[

−1

2(2a4 + 3a5)L2 (r) +

1

2(2a4 + a5)L3 (r)

]

(C.19)

B8 (r) = b

[

1

2a3J1 (r) +

1

2(a1 + a3)K2 (r)

+1

2(2a4 + a5)

Ψ′ (r)

r− 1

2(2a4 + 3a5)

X (r)

r2

]

(C.20)

B9 (r) = B8 (r) (C.21)

110

B10 (r) = b

[

1

2a1J1 (r) +

1

2(a1 + 4a2)K2 (r)

− (2a4 + a5)X (r)

r2+ a5

Ψ (r)

r

]

(C.22)

C1 (r) = B′1 (r) − 4B1 (r)

r(C.23)

C2 (r) = A1 (r) − 2B′1 (r) +

2B1 (r)

r− 2 (a− 1)

B1 (r)

r

−B′2 (r) +

2B2 (r)

r−B′

3 (r) +2B3 (r)

r−B′

5 (r) +2B5 (r)

r

−B′6 (r) +

2B6 (r)

r−B′

7 (r) +2B7 (r)

r(C.24)

C3 (r) = B′4 (r) − 2B4 (r)

r(C.25)

C4 (r) = A2 (r) − B3 (r)

r− 2B′

4 (r) +2B4 (r)

r− 2 (a− 1)

B4 (r)

r

−B5 (r)

r− B6 (r)

r− B7 (r)

r−B′

8 (r) (C.26)

C5 (r) =B1 (r)

r+B′

3 (r) − 2B3 (r)

r+B′

5 (r) − 2B5 (r)

r(C.27)

C6 (r) = A3 (r) − B2

r−B′

3 (r) − (a− 1)B3

r

−B′5 (r) − (a− 1)

B5 (r)

r−B′

9 (r) − B′10 (r) (C.28)

C7 (r) =B1 (r)

r+B′

6 (r) − 2B6 (r)

r+B′

7 (r) − 2B7 (r)

r(C.29)

C8 (r) = A2 (r) − B2 (r)

r− B′

6 (r) − (a− 1)B6 (r)

r

−B′7 (r) − (a− 1)

B7 (r)

r−B′

9 (r) − B′10 (r) (C.30)

C9 (r) =B3 (r)

r+B5 (r)

r+B6 (r)

r+B7 (r)

r+B′

9 (r) +B′10 (r) (C.31)

As shown in section 2.4.2, the fundamental displacement for both the 2D and

the 3D cases is

u∗ = b[

Ψ (r) I −X (r) r ⊗ r]

(C.32)

with X (r) and Ψ (r) being scalar functions that are different for the 2D and 3D

cases (see eqs (2.103-2.106)). Based on the fundamental displacement, all other

111


quantities can be calculated.

q∗ (x,y) =∂u∗

∂ny

= −b[

X ′ (r) − 2X (r)

r

]

(n · r) r ⊗ r

+ bΨ′ (r) (n · r) I − bX (r)

r(n⊗ r + r ⊗ n)

(C.33)

µ∗ (x,y) = B1 (r) r ⊗ r ⊗ r ⊗ r

+B2 (r) I ⊗ r ⊗ r

+B3 (r) r ⊗ I ⊗ r

+B4 (r) r ⊗ r ⊗ I

+B5 (r)(

I ⊗ r ⊗ r)1324

+B6 (r)(

I ⊗ r ⊗ r)1342

+B7 (r)(

I ⊗ r ⊗ r)3142

+B8 (r) I ⊗ I

+B9 (r)(

I ⊗ I

)1324

+B10 (r)(

I ⊗ I

)3124

(C.34)

R∗ (x,y) = (n⊗ n) : µ∗

= [B1 (r) (n · r) +B2 (r)] r ⊗ r

+[

B4 (r) (n · r)2 +B8 (r)]

I

+ [B3 (r) +B5 (r)] (n · r) n⊗ r

+ [B6 (r) +B7 (r)] (n · r) r ⊗ n

+ [B9 (r) +B10 (r)] n⊗ n

(C.35)

112

P∗ (x,y) = n · τ + (n⊗ n) :∂µ∗

∂n− n · (∇ · µ∗)

− n ·(

∇ · µ∗2134)+ (∇S · n) (n ⊗ n) : µ∗ − (∇Sn) : µ∗

={ [

C1 (r) (n · r)2 + C2 (r)]

(n · r)+B1 (r) (aS · a′

S + bS · b′S) (n · r)2

− B1 (r) [(aS · r) (a′S · r) + (bS · r) (b′

S · r)]}

r ⊗ r

+{ (

C3 (r) (n · r)2 + C4 (r))

(n · r)+B4 (r) (aS · a′

S + bS · b′S) (n · r)2

− B4 (r) [(aS · r) (a′S · r) + (bS · r) (b′

S · r)]}

I

+{

C5 (r) (n · r)2 + C6 (r)

+ [B3 (r) +B5 (r)] (aS · a′S + bS · b′

S) (n · r)}

n⊗ r

+{

C7 (r) (n · r)2 + C8 (r)

+ [B6 (r) +B7 (r)] (aS · a′S + bS · b′

S) (n · r)}

r ⊗ n

+{

C9 (r) (n · r)+ [B9 (r) +B10 (r)] (aS · a′

S + bS · b′S)}

n⊗ n

− B3 (r) [(a′S · r) aS ⊗ r + (b′

S · r)bS ⊗ r]

− B5 (r) [(aS · r) a′S ⊗ r + (bS · r)b′

S ⊗ r]

− B6 (r) [(aS · r) r ⊗ a′S + (bS · r) r ⊗ b′

S]

− B7 (r) [(a′S · r) r ⊗ aS + (b′

S · r) r ⊗ bS]

− B9 (r) [a′S ⊗ aS + b′

S ⊗ bS]

− B10 (r) [aS ⊗ a′S + bS ⊗ b′

S]

(C.36)

E∗ (x,y) = J(m⊗ n) : µ∗K= J B1 (r) (m · r) (n · r) r ⊗ r +B4 (r) (m · r) (r · r) I

+B5 (r) (m · r) n ⊗ r +B3 (r) (n · r) m ⊗ r

+B6 (r) (m · r) r ⊗ n +B7 (r) (n · r) r ⊗ m

+B9 (r) n⊗ m + (r)B10m ⊗ nK (C.37)

∂u∗

∂nx(x,y) = b

[ (

X ′ (r) − 2X (r)

r

)

(nx · r) r ⊗ r

+X (r)

r(nx ⊗ r + r ⊗ r ⊗ nx) − Ψ′ (r) (nx · r) I

] (C.38)

113


∂q∗

∂nx(x,y) = b

{

[L1 (r) (nx · r) (n · r) + L2 (r) (nx · n)] r ⊗ r

+ L2 (n · r) (nx ⊗ r + r ⊗ nx)

+ L2 (nx · r) (n ⊗ r + r ⊗ n)

+X (r)

r2(nx ⊗ n + n ⊗ nx)

+

[

−(

Ψ′ (r) − Ψ′ (r)

r

)

(nx · r) (n · r)

−(

Ψ′ (r)

r

)

(nx · n)

]

I

}

(C.39)

∂R∗

∂nx(x,y) =

{

−[(

B′1 (r) − 4B1 (r)

r

)

(nx · n) (n · r) +2B1 (r)

r(nx · n)

]

(n · r)

−(

B′2 (r) − 2B2 (r)

r

)

(nx · r)}

r ⊗ r

+

{

−[(

B′4 (r) − 2B4 (r)

r

)

(nx · r) +2B4 (r)

r(nx · n)

]

(n · r)

−B8 (r) (nx · r)}

I

+

[

−B2 (r)

r− B1 (r)

r(n · r)2

]

(nx ⊗ r + r ⊗ nx)

+

{

−[(

B′3 (r) − 2B3 (r)

r

)

(nx · r) (n · r) +B3 (r)

r(nx · n)

]

−[(

B′5 (r) − 2B5 (r)

r

)

(nx · r) (n · r) +B5 (r)

r(nx · n)

]}

(n ⊗ r)

+

{

−[(

B′6 (r) − 2B6 (r)

r

)

(nx · r) (n · r) +B6 (r)

r(nx · n)

]

−[(

B′7 (r) − 2B7 (r)

r

)

(nx · r) (n · r) +B7 (r)

r(nx · n)

]}

(n ⊗ r)

+

[

−B3 (r) +B5 (r)

r(n · r)

]

n⊗ nx

+

[

−B6 (r) +B7 (r)

r(n · r)

]

nx ⊗ n

+ [− (B′9 (r) +B′

10 (r)) (nx · r)] n⊗ n

(C.40)

114

∂P∗

∂nx

(x,y) =

[

−3C1 (r)

r(nx · n) (n · r)2 −

(

C ′1 (r) − 5C1 (r)

r

)

(nx · r) (n · r)3

−(

C ′2 (r) − 3C2 (r)

r

)

(nx · r) (n · r) − C2 (r)

r(nx · n)

− (aS · a′S + bS · b′

S)

(

B′1 (r) − 4B1 (r)

r

)

(nx · r) (n · r)2

− (aS · a′S + bS · b′

S)2B1 (r)

r(nx · n) (n · r)

+ [(aS · r) (a′S · r) + (bS · r) (b′

S · r)] (nx · r)(

B′1 (r) − 4B1 (r)

r

)

+B1 (r)

r(nx · aS) (a′

S · r) +B1 (r)

r(nx · a′

S) (aS · r)

+B1 (r)

r(nx · b′

S) (b′S · r) +

B1 (r)

r(nx · b′

S) (bS · r)]

r ⊗ r

+

[

−C1 (r)

r(n · r)3 − C2 (r)

r(n · r) − (aS · a′

S + b · b′S)B1 (r)

r(n · r)2

+B1 (r)

r(aS · r) (a′

S · r) +B1 (r)

r(bS · r) (b′

S · r)]

(r ⊗ nx + nx ⊗ r)

+

[

−(

C ′5 (r) − 3C5 (r)

r

)

(nx · r) (n · r)2

− 2C5 (r)

r(nx · n) (n · r) −

(

C ′6 (r) − C6 (r)

r

)

(nx · r)

− (aS · a′S + bS · b′

S) (B′3 (r) +B′

5 (r)) (nx · r) (n · r)

+ (aS · a′S + bS · b′

S)

(

2B3 (r) +B5 (r)

r

)

(nx · r) (n · r)

− (aS · a′S + bS · b′

S)B3 (r) +B5 (r)

r(nx · n)

]

(n ⊗ r)

+

[

−(

C ′7 (r) − 3C7 (r)

r

)

(nx · r) (n · r)2

− 2C7 (r)

r(nx · n) (n · r) −

(

C ′8 (r) − C8 (r)

r

)

(nx · r)

− (aS · a′S + bS · b′

S) (B′6 (r) +B′

7 (r)) (nx · r) (n · r)

+ (aS · a′S + bS · b′

S)

(

2B6 (r) +B7 (r)

r

)

(nx · r) (n · r)

− (aS · a′S + bS · b′

S)B6 (r) +B7 (r)

r(nx · n)

]

(n ⊗ r)

(C.41)

115


+

[

−C5 (r) (n · r)2 − C6 (r)

r− (aS · a′

S + bS · b′S)B3 (r) +B5 (r)

r(n · r)

]

(n⊗ nx)

+

[

−C7 (r) (n · r)2 − C8 (r)

r− (aS · a′

S + bS · b′S)B6 (r) +B7 (r)

r(n · r)

]

(nx ⊗ n)

+

[

−(

C ′9 (r) − C9 (r)

r

)

(nx · r) (n · r) − C9 (r)

r(nx · n)

− (aS · a′S + bS · b′

S) (B′9 (r) +B′

10 (r)) (nx · r)]

(n ⊗ n)

+

{

−(

C ′3 (r) − 3C3 (r)

r

)

(nx · r) (n · r)3 − 3C3 (r)

r(nx · n) (n · r)2

−(

C ′4 (r) − C4 (r)

r

)

(nx · r) (n · r) − C4 (r)

r(nx · n)

− (aS · a′S + bS · b′

S)

[(

B′4 (r) − 2B4 (r)

r

)

(nx · r) (n · r)2

+2B4 (r)

r(nx · n) (n · r)

]

+

(

B′4 (r) − 2B4 (r)

r

)

[(aS · r) (a′S · r) + (bS · r) (b′

S · r)] (nx · r)

+B4 (r)

r[ (nx · aS) (a′

S · r) + (nx · a′S) (aS · r)

+ (nx · bS) (b′S · r) + (nx · b′

S) (bS · r)]}

I

+

[(

B′3 (r) − 2B3 (r)

r

)

(a′S · r) (nx · r) +

B3 (r)

r(nx · a′

S)

]

aS ⊗ r

+

[(

B′3 (r) − 2B3 (r)

r

)

(b′S · r) (nx · r) +

B3 (r)

r(nx · b′

S)

]

bS ⊗ r

+B3 (r)

r[(a′

S · r) (aS ⊗ nx) + (b′S · r) (bS ⊗ nx)]

+B7 (r)

r[(a′

S · r) (nx ⊗ aS) + (b′S · r) (nx ⊗ bS)]

+B5 (r)

r[(aS · r) (a′

S ⊗ nx) + (bS · r) (b′S ⊗ nx)]

+B6 (r)

r[(aS · r) (nx ⊗ a′

S) + (bS · r) (nx ⊗ b′S)]

+

[(

B′5 (r) − 2B5 (r)

r

)

(aS · r) (nx · r) +B5 (r)

r(nx · aS)

]

a′S ⊗ r

(C.40)

116

+

[(

B′5 (r) − 2B5 (r)

r

)

(bS · r) (nx · r) +B5 (r)

r(nx · bS)

]

b′S ⊗ r

+

[(

B′7 (r) − 2B7 (r)

r

)

(a′S · r) (nx · r) +

B7 (r)

r(nx · a′

S)

]

r ⊗ aS

+

[(

B′7 (r) − 2B7 (r)

r

)

(b′S · r) (nx · r) +

B7 (r)

r(nx · b′

S)

]

r⊗ bS

+

[(

B′6 (r) − 2B6 (r)

r

)

(aS · r) (nx · r) +B6 (r)

r(nx · aS)

]

r ⊗ a′S

+

[(

B′6 (r) − 2B6 (r)

r

)

(bS · r) (nx · r) +B6 (r)

r(nx · bS)

]

r⊗ b′S

− B′10 (r) (nx · r) (aS ⊗ a′

S + bS ⊗ b′S)

− B′9 (r) (nx · r) (a′

S ⊗ aS + b′S ⊗ bS)

(C.40)

∂E∗

∂nx(x,y) =

{

−(

B′1 (r) − 4B1 (r)

r

)

(nx · r) (n · r) (m · r)

−B1 (r)

r[(nx · m) (n · r) + (nx · n) (m · r)]

}

r ⊗ r

+

{

−(

B′4 (r) − 2B4 (r)

r

)

(nx · r) (n · r) (m · r)

−B4 (r)

r[(nx · m) (n · r) + (nx · n) (m · r)]

}

I

−[(

B′5 (r) − 2B5 (r)

r

)

(nx · r) (m · r) +B5 (r)

r(nx · m)

]

n⊗ r

−[(

B′6 (r) − 2B6 (r)

r

)

(nx · r) (m · r) +B6 (r)

r(nx · m)

]

r ⊗ n

−[(

B′3 (r) − 2B3 (r)

r

)

(nx · r) (n · r) +B3 (r)

r(nx · n)

]

m⊗ r

−[(

B′7 (r) − 2B7 (r)

r

)

(nx · r) (n · r) +B7 (r)

r(nx · n)

]

r ⊗ m

− B1 (r)

r(m · r) (n · r) (nx ⊗ r + r ⊗ nx)

− (B′9 (r) +B′

10 (r)) (nx · r) (n⊗ m + m ⊗ n)

− B5 (r) +B6 (r)

r(m · r) (n⊗ nx + nx ⊗ n)

− B3 (r) +B7 (r)

r(n · r) (m⊗ nx + nx ⊗ m)

(C.42)

117


118

Appendix D

Boundary Elements

D.1 Surface Elements

D.1.1 Eight Noded Quadratic Quadrilateral Element

(a) (b)

Figure D.1: (a) The geometrical and (b) functional nodes of an eight noded

quadratic quadrilateral element

This element has eight geometrical and eight functional nodes. It can be

119

Appendix1/Figs/EPS/8QQ-geometrical.eps

Appendix1/Figs/EPS/8QQ-functional.eps

D. BOUNDARY ELEMENTS

discontinuous at any number of sides. The coordinates of the geometrical nodes

of the element, expressed in its local coordinate system, are given in Table D.1.

Geom. node ξ1 coord. ξ2 coord.

1 −1.0 −1.0

2 1.0 −1.0

3 1.0 1.0

4 −1.0 1.0

5 0.0 −1.0

6 1.0 0.0

7 0.0 1.0

8 −1.0 0.0

Table D.1: The geometrical node coordinates of an eight noded quadratic quadri-

lateral element

The functional node coordinates of the element are given in Table D.2. Note

that if one or more bi are equal to zero, the corresponding element sides are

continuous.

The shape and interpolation functions of an eight noded quadratic quadrilat-

eral element are quadratic functions of the local variables ξ1, ξ2.

Φ1 (ξ1, ξ2) =(−1 + ξ1)(1 + ξ1 − ξ1ξ2 − ξ2

2)

4(D.1)

Φ2 (ξ1, ξ2) =(1 + ξ1)(−1 + ξ1 − ξ1ξ2 + ξ2

2)

4(D.2)

Φ3 (ξ1, ξ2) =(1 + ξ1)(−1 + ξ1 + ξ1ξ2 + ξ2

2)

4(D.3)

Φ4 (ξ1, ξ2) =(−1 + ξ1)(1 + ξ1 + ξ1ξ2 − ξ2

2)

4(D.4)

Φ5 (ξ1, ξ2) =(−1 + ξ1ξ1) ∗ (−1 + ξ2)

2(D.5)

Φ6 (ξ1, ξ2) =(1 + ξ1)(1 − ξ2

2)

2(D.6)

Φ7 (ξ1, ξ2) =(1 − ξ1ξ1)(1 + ξ2)

2(D.7)

120


Func. node ξ1 coord. ξ2 coord.

1 −1+b4 −1+b1

2 1−b2 −1+b1

3 1−b2 1−b3

4 −1+b4 1−b3

5 0 −1+b1

6 1−b2 0

7 0 1−b3

8 −1+b4 0

Table D.2: The functional node coordinates of a discontinuous eight noded

quadratic quadrilateral element

Φ8 (ξ1, ξ2) =(−1 + ξ1)(−1 + ξ2

2)

2(D.8)

N1 (ξ1, ξ2) =(ξ1 − α2) (−ξ2 + α3) (α1α4 + ξ1α1 + ξ2α4)

α1α4 (α1 + α3) (α2 + α4)(D.9)

N2 (ξ1, ξ2) =(ξ1 + α4) (ξ2 − α3) (α1α2 − ξ1α1 + ξ2α2)

α1α2 (α1 + α3) (α2 + α4)(D.10)

N3 (ξ1, ξ2) =(ξ1 + α4) (ξ2 + α1) (−α2α3 + ξ1α3 + ξ2α2)

α2α3 (α1 + α3) (α2 + α4)(D.11)

N4 (ξ1, ξ2) =(−ξ1 + α2) (ξ2 + α1) (−α3α4 − ξ1α3 + ξ2α4)

α3α4 (α1 + α3) (α2 + α4)(D.12)

N5 (ξ1, ξ2) =(ξ1 − α2) (ξ1 + α4) (ξ2 − α3)

α2α4 (α1 + α3)(D.13)

N6 (ξ1, ξ2) =(ξ1 + α4) (−ξ2 + α3) (ξ2 + α1)

α1α3 (α2 + α4)(D.14)

N7 (ξ1, ξ2) =(−ξ1 + α2) (ξ1 + α4) (ξ2 + α1)

α2α4 (α1 + α3)(D.15)

N8 (ξ1, ξ2) =(ξ1 − α2) (ξ2 + α1) (ξ2 − α3)

α1α3 (α2 + α4)(D.16)

with αi = (1 − bi) , i = 1, . . . 4.

121


D.1.2 Six Noded Quadratic Triangular Element

(a) (b)

Figure D.2: (a) The geometrical and (b) functional nodes of a six noded quadratic

triangular element

This element has six geometrical and six functional nodes. It can be discon-

tinuous at any number of sides. The coordinates of the geometrical nodes of the

element, expressed in its local coordinate system, are given in Table D.3.

Geom. node ξ1 coord. ξ2 coord.

1 0.0 0.0

2 1.0 0.0

3 0.0 1.0

4 1/2 0.0

5 1/2 1/2

6 0.0 1/2

Table D.3: The geometrical node coordinates of a six noded quadratic triangular

element



122

Appendix1/Figs/EPS/6QT-geometrical.eps

Appendix1/Figs/EPS/6QT-functional.eps


continuous. with b4 = 1 − b1 − b2 − b3. The shape and interpolation functions

Func. node ξ1 coord. ξ2 coord.

1 b3 b1

2 b3 + b4 b1

3 b3 b1 + b4

4 (b3 + b4) /2 b1

5 (b3 + b4) /2 (b1 + b4) /2

6 b3 (b1 + b4) /2

Table D.4: The functional node coordinates of a discontinuous six noded

quadratic triangular element

of a six noded quadratic triangular element are quadratic functions of the local

variables ξ1, ξ2.

Φ1 (ξ1, ξ2) = (1 − 2ξ1 − 2ξ2)(1 − ξ1 − ξ2) (D.17)

Φ2 (ξ1, ξ2) = ξ1(−1 + 2ξ1) (D.18)

Φ3 (ξ1, ξ2) = ξ2(−1 + 2ξ2) (D.19)

Φ4 (ξ1, ξ2) = 4ξ1(1 − ξ1 − ξ2) (D.20)

Φ5 (ξ1, ξ2) = 4ξ1ξ2 (D.21)

Φ6 (ξ1, ξ2) = 4(1 − ξ1 − ξ2)ξ2 (D.22)

N1 (ξ1, ξ2) = − (b2−1+ξ1+ξ2)[(1−b1−b3)(2ξ1+2ξ2−3b1b3)−2ξ1b1−2ξ2b3](−1+2b1+b3)(−1+b1+b2+b3)(−1+b1+2b3)

(D.23)

N2 (ξ1, ξ2) = − (b3−ξ1)[(−1+b2)(−1+2ξ1)−2b2ξ2+b1(−3+3b2+4ξ1+2ξ2)](−1+2b1+b2)(−1+b1+2b2)(−1+b1+b2+b3)

(D.24)

N3 (ξ1, ξ2) = − (b1−ξ2)[1−2ξ2+b2(−1+3b3−2ξ1+2ξ2)+b3(−3+2ξ1+4ξ2)](−1+b1+b2+b3)(−1+2b2+b3)(−1+b2+2b3)

(D.25)

N4 (ξ1, ξ2) = 4(b3−ξ1)(−1+b2+ξ1+ξ2)(−1+b1+2b2)(−1+b1+2b3)

(D.26)

N5 (ξ1, ξ2) = 4(b3−ξ1)(b1−ξ2)(−1+2b1+b2)(−1+b2+2b3)

(D.27)

N6 (ξ1, ξ2) = 4(b1−ξ2)(−1+b2+ξ1+ξ2)(−1+2b1+b3)(−1+2b2+b3)

(D.28)

123


D.2 Line Elements

D.2.1 Three Noded Quadratic Line Element

(a) (b)

Figure D.3: (a) The geometrical and (b) functional nodes of a three noded

quadratic line element

This element has three geometrical and three functional nodes. It can be

discontinuous at any number of sides. The coordinates of the geometrical nodes

of the element, expressed in its local coordinate system, are given in Table D.5.

Geom. node ξ coord.

1 −1.0

2 1.0

3 0.0

Table D.5: The geometrical node coordinates of a three noded quadratic line

element



continuous. with αi = (1 − bi) , i = 1, 2. The shape and interpolation functions of

a three noded quadratic line element are quadratic functions of the local variable

ξ.

Φ (ξ) =ξ (ξ − 1)

2(D.29)

124

Appendix1/Figs/EPS/3QL-geometrical.eps

Appendix1/Figs/EPS/3QL-functional.eps

D.2 Line Elements

Func. node ξ coord.

1 −α1

2 α2

3 0.0

Table D.6: The functional node coordinates of a discontinuous three noded

quadratic line element

Φ (ξ) =ξ (ξ + 1)

2(D.30)

Φ (ξ) = 1 − ξ2 (D.31)

N1 (ξ) =ξ (ξ − α2)

α1 (α1 + α2)(D.32)

N2 (ξ) =ξ (ξ + α1)

α2 (α1 + α2)(D.33)

N3 (ξ) =(ξ + α1) (α2 − ξ)

α1α2(D.34)

125


126

Appendix E

Diving elements into triangles

As mentioned in Chapter 3, in order to integrate numerically weakly singular inte-

grals over an element, a transformation from the element local coordinate system

to a local polar coordinate system centered at the point of singularity, is used.

The Jacobian of this transformation cancels out the singularity of the integral

and it can be calculated with high accuracy using Gauss-Legendre quadrature.

The integrals are broken down in triangles, so that each has one of its vertices

on the singular point (Figure E.1). Then, the singular integral is written as a sum

of integrals over these triangles. The polar radius R of the latter transformation

is a function of the polar angle θ. The determination of the maximum radius

Rmax (θ) as a function of the polar angle θ is the subject of this section.

E.1 Quadrilateral Elements

E.1.1 Triangle 1

θ1 = 0 (E.1a)

θ2 = arctan1 − ξk

2

1 − ξk1

(E.1b)

Rmax (θ) =1 − ξk

1

cos θ(E.1c)

127

E. DIVING ELEMENTS INTO TRIANGLES

(a) (b)

Figure E.1: (a) A quadrilateral and (b) a triangular element broken down to

triangles

Figure E.2: A random triangle of a quadrilateral element with θ ∈ [θ1, θ2] and

R ∈ [0, Rmax (θ)]

128

ch3-The_Boundary_Element_Method/Figs/EPS/quadrilateral_to_triangles.eps

ch3-The_Boundary_Element_Method/Figs/EPS/triangular_to_triangles.eps

Appendix2/Figs/EPS/quadrilateral_any_triangle.eps

E.1 Quadrilateral Elements

E.1.2 Triangle 2

θ1 = arctan1 − ξk

2

1 − ξk1

(E.2a)

θ2 =π

2(E.2b)


1

cos θ(E.2c)

E.1.3 Triangle 3

θ1 =π

2(E.3a)

θ2 =pi

2+ arctan

1 + ξk1

1 − ξk2

(E.3b)

Rmax (θ) =1 + ξk

1

cos θ(E.3c)

E.1.4 Triangle 4

θ1 =pi

2+ arctan

1 − ξk2

1 + ξk1

(E.4a)

θ2 = 2π (E.4b)


2

cos θ(E.4c)

129


E.1.5 Triangle 5

θ1 = 2π (E.5a)

θ2 = π + arctan1 + ξk

1

1 + ξk2

(E.5b)

Rmax (θ) =1 + ξk

2

cos θ(E.5c)

E.1.6 Triangle 6

θ1 = π + arctan1 + ξk

2

1 + ξk1

(E.6a)

θ2 =3π

4(E.6b)

Rmax (θ) =1 + ξk

1

cos θ(E.6c)

E.1.7 Triangle 7

θ1 =3π

4(E.7a)

θ2 =3π

4+ arctan

1 − ξk1

1 + ξk2

(E.7b)


1

cos θ(E.7c)

E.1.8 Triangle 8

θ1 =3π

4+ arctan

1 + ξk2

1 − ξk1

(E.8a)

130

E.2 Triangular Elements

θ2 = 2π (E.8b)

Rmax (θ) =1 + ξk

2

cos θ(E.8c)


Figure E.3: A random triangle of a triangular element with θ ∈ [θ1, θ2] and

R ∈ [0, Rmax (θ)]

E.2.1 Triangle 1

θ1 = −φ (E.9a)

θ2 = π/6 (E.9b)

Rmax (θ) =B

cos (θ2 − θ)(E.9c)

with

φ = arcsin(A)

A =ηk

2 cos(π/6)

CB = C sin (π/3 − φ)

131

Appendix2/Figs/EPS/triangular_any_triangle.eps


C =√

1 + ηk1

(

ηk1 − 2

)

+ ηk2 (1 − 2 sin (π/6)) + 2ηk

1ηk2 sin (π/6)

E.2.2 Triangle 2

θ1 =π

6(E.10a)

θ2 =π

6+ arctan (A) (E.10b)

Rmax (θ) =B

cos (θ − θ1)(E.10c)

with

A =1 − C cos (π/3 − φ)

C sin (π/3 − φ)

B = C sin (π/3 − φ)

C =√

1 + ηk1

(

ηk1 − 2

)

+ ηk2 (1 − 2 sin (π/6)) + 2ηk

1ηk2 sin (π/6)

φ = arcsin (D)

D =ηk

2 cos (π/6)

C

E.2.3 Triangle 3

θ1 = π − π

6− arctan (A) (E.11a)

θ2 = π − π

6(E.11b)

Rmax (θ) =B

cos (θ2 − θ)(E.11c)

with

A =1 − C cos (π/3 − φ)

C sin (π/3 − φ)

132


B = C sin (π/3 − φ)

C =√

1 + ηk1

(

ηk1 − 2

)

+ ηk2 (1 − 2 sin (π/6)) + 2ηk

1ηk2 sin (π/6)

φ = arcsin (D)

D =ηk

2 cos (π/6)

C

E.2.4 Triangle 4

θ1 = π − π

6(E.12a)

θ2 = π − π

3+ φ (E.12b)

Rmax (θ) =o′Wffl

cos (θ − θ1)(E.12c)

with

A =ηk

2 cos (π/6)

CB = C sin (π/3 − φ)

C =√

1 + ηk1

(

ηk1 − 2

)

+ ηk2 (1 − 2 sin (π/6)) + 2ηk

1ηk2 sin (π/6)

φ = arcsin (A)

E.2.5 Triangle 5

θ1 = π + φ (E.13a)

θ2 =3π

2(E.13b)

Rmax (θ) =A

cos (θ2 − θ)(E.13c)

133


with

A = ηk2 cos (π/6)

B = ηk2 sin (π/6) + ηk

1

φ = arctan (A/B)

E.2.6 Triangle 6

θ1 =3π

2(E.14a)

θ2 =3π

2+ arctan

(

B

A

)

(E.14b)

Rmax (θ) =A

cos (θ − θ1)(E.14c)

with

A = ηk2 cos (π/6)

B = 1 − ηk2 sin (π/6) − ηk

1

134

Appendix F

Taylor expansion of the position

vector

As discussed in section 3.4.2.2, the vector r is the position vector of the current

integration point ye with respect to the singular point xk. In order to proceed

with the integrations, r is expanded in Taylor series around xk.

r = ye (η1 (R, θ) , η2 (R, θ)) − xk(

ηk1 , η

k2

)

= R

(

∂ye (η1, η2)

∂η1

∣

∣

∣

∣

η=xk

cos θ +∂ye (η1, η2)

∂η2

∣

∣

∣

∣

η=xk

sin θ

)

+R2

(

∂2ye (η1, η2)

∂η21

∣

∣

∣

∣

η=xk

cos2 θ

2+∂2ye (η1, η2)

∂η1∂η2

∣

∣

∣

∣

η=xk

cos θ sin θ

+∂2ye (η1, η2)

∂η22

∣

∣

∣

∣

η=xk

sin2 θ

2

)

+ O(

R3)

= RA (θ) +R2B (θ) + O(

R3)

(F.1)

In the above equation, the point ye can be written as the sum of the elements’

shape functions Φi, since it resides inside the element, multiplied by the corre-

sponding geometrical node of the element yei . Then the coefficients A (θ) and

135

F. TAYLOR EXPANSION OF THE POSITION VECTOR

B (θ) become

A (θ) = yei

[

∂Φi (η1, η2)

∂η1

∣

∣

∣

∣

η=xk

cos θ +∂Φi (η1, η2)

∂η2

∣

∣

∣

∣

η=xk

sin θ

]

(F.2)

B (θ) = yei

[

∂2Φi (η1, η2)

∂η21

∣

∣

∣

∣

η=xk

cos2 θ

2

+∂2Φi (η1, η2)

∂η1∂η2

∣

∣

∣

∣

η=xk

cos θ sin θ +∂2Φi (η1, η2)

∂η22

∣

∣

∣

∣

η=xk

sin2 θ

2

] (F.3)

In the case of a quadrilateral element, the partial derivatives of the shape functions

with respect to η1 and η2 are equal to the corresponding derivatives with respect to

ξ1 and ξ2. In the case of a triangular element however, where the transformation

(3.46) is used, the chain rule must be applied to calculate the partial derivatives.

∂Φi (η1, η2)

∂η1=

∂Φi (ξ1, ξ2)

∂ξ1

ξ1 (η1, η2)

∂η1+∂Φi (ξ1, ξ2)

∂ξ2

ξ2 (η1, η2)

∂η1

=Φi (ξ1, ξ2)

∂ξ1(F.4)

∂Φi (η1, η2)

∂η2

=∂Φi (ξ1, ξ2)

∂ξ1

ξ1 (η1, η2)

∂η2

+∂Φi (ξ1, ξ2)

∂ξ2

ξ2 (η1, η2)

∂η2

= −∂Φi (ξ1, ξ2)

∂ξ1tan

π

6+∂Φi (ξ1, ξ2)

∂ξ2

1

cos (π/6)(F.5)

∂2Φi (η1, η2)

∂η21

=∂2Φi (ξ1, ξ2)

∂ξ21

(F.6)

∂2Φi (η1, η2)

∂η22

=∂2Φi (ξ1, ξ2)

∂ξ21

tan2 π

6+∂2Φi (ξ1, ξ2)

∂ξ22

1

cos2 (π/6)

−(

∂2Φi (ξ1, ξ2)

∂ξ1∂ξ2+∂2Φi (ξ1, ξ2)

∂ξ2∂ξ1

)

tan (π/6)

cos (π/6)(F.7)

∂2Φi (η1, η2)

∂η1∂η2

= −∂2Φi (ξ1, ξ2)

∂ξ21

tanπ

6+∂2Φi (ξ1, ξ2)

∂ξ1∂ξ2

1

cos (π/6)(F.8)

136

Appendix G

Hollow Cylinder Under Pressure:

Analytical solution constants

In this section the constants C1-C4 regarding the analytical solution of a hollow

cylinder under pressure are presented, as provided in Papanicolopulos (2008).

First the following expression are defined.

pa = Ti (G.1)

pb = −To (G.2)

α =ri

l1(G.3)

β =ro

l1(G.4)

ξ =a1 + a2 + a3 + a4 + a5

a4 + a5(G.5)

ζ =1

2

µ

λ+ µ(G.6)

φα =(

α3ξ + 2α)

I1 (α) − α2I0 (α) (G.7)

φβ =(

β3ξ + 2β)

I1 (β) − β2I0 (β) (G.8)

ψα =(

α3ξ + 2α)

K1 (α) + α2K0 (α) (G.9)

ψβ =(

β3ξ + 2β)

K1 (β) + β2K0 (β) (G.10)

137

G. HOLLOW CYLINDER UNDER PRESSURE: ANALYTICAL

SOLUTION CONSTANTS

ξ = ζ

(

1

α2− 1

β2

)

(φαψβ − ψαφβ) −(

I1 (β)

β− I1 (α)

α

)

(ψβ − ψα)

+

(

K1 (β)

β− K1 (α)

α

)

(φβ − φα) (G.11)

Then the constants C1-C4 are give by

C1 = − pb

2(λ+ µ)+

pa − pb

2(λ+ µ)

1

χ

[

ζ

β2(φαψβ − ψαφβ) +

I1 (β)

β(ψβ − ψα)

−K1 (β)

β(φβ − φα)

]

(G.12)

C2 = l21pa − pb

4(λ+ µ)

φαψβ − ψαφβ

χ(G.13)

C3 =pa − pb

2(λ+ µ)

ψα − ψβ

χ(G.14)

C4 =pa − pb

2(λ+ µ)

φβ − φα

χ(G.15)

138

Appendix H

Interpolation functions of the

eight noded quadrilateral element

with variable order of singularity

As mentioned in Chapter 4, the interpolation functions for an eight noded quadri-

lateral element with variable order of singularity are given by

N i (ξ′1, r) =(

ei1 + ei

2ξ′1 + ei

3ξ′21

)

rλ1

+(

ei4 + ei

5ξ′1 + ei

6ξ′21

)

rλ2

+ ei7 + ei

8ξ′1.

(H.1)

The constants eij with i, j = 1, . . . , 8 are given by

e11 =pd2

(

r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8

)

(pd1+ pd2

)W1(H.2)

e12 =pd2

(

r′λ2

5 − r′λ2

7

) (

r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.3)

e13 = −pd2

(

r′λ1

5 − r′λ1

7

) (

r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.4)

e14 =−r′λ1

4 r′λ2

8 + r′λ2

4 r′λ1

8

(pd1+ pd2

)W1(H.5)

139

H. EIGHT NODED SPECIAL ELEMENT: INTERPOLATION

FUNCTIONS

e15 =−pd1

(

r′λ2

5 − r′λ2

7

) (

r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8

)

+ pd2W2

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.6)

e16 =pd1

(

r′λ1

5 − r′λ1

7

) (

r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8

)

+ pd2W3

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.7)

e17 =W2

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W4

(H.8)

e18 =r′λ2

4

(

−r′λ1

5 + r′λ1

7

)

r′λ1

8 +(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

r′λ1

8 + r′λ1

4 W5

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.9)

e21 =pd1

(

r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

)

(pd1+ pd2

)W6

(H.10)

e22 =pd1

(

r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

) (

r′λ2

5 − r′λ2

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.11)

e23 = −pd1

(

r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

) (

r′λ1

5 − r′λ1

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.12)

e24 =r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

(pd1+ pd2

)W6

(H.13)

e25 =pd2

(

r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

) (

r′λ2

5 − r′λ2

7

)

+ pd1W7

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.14)

e26 =−pd2

(

r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6

) (

r′λ1

5 − r′λ1

7

)

+ pd1(W8)

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.15)

e27 =W7

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.16)

e28 =W8

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.17)

e31 =pd1

(

r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

)

(pd1+ pd2

)W9(H.18)

e32 = −pd1

(

r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

) (

r′λ2

5 − r′λ2

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.19)

e33 =pd1

(

r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

) (

r′λ1

5 − r′λ1

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.20)

e34 =r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

(pd1+ pd2

)W9(H.21)

140

e35 =−pd2

(

r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

) (

r′λ2

5 − r′λ2

7

)

+ pd1W10

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.22)

e36 =pd2

(

r′λ1

2 r′λ2

6 − r′λ2

2 r′λ1

6

) (

r′λ1

5 − r′λ1

7

)

+ pd1W11

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.23)

e37 =W10

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.24)

e38 =W11

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.25)

e41 =pd2

(

r′λ1

1 r′λ2

8 − r′λ2

1 r′λ1

8

)

(pd1+ pd2

)W4

(H.26)

e42 = −pd2

(

r′λ2

5 − r′λ2

7

) (

r′λ1

1 r′λ2

8 − r′λ2

1 r′λ1

8

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.27)

e43 =pd2

(

r′λ1

5 − r′λ1

7

) (

r′λ1

1 r′λ2

8 − r′λ2

1 r′λ1

8

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.28)

e44 =−(

r′λ1

1 r′λ2

8

)

+ r′λ2

1 r′λ1

8

(pd1+ pd2

)W4(H.29)

e45 =pd1

(

r′λ2

5 − r′λ2

7

) (

r′λ1

1 r′λ2

8 − r′λ2

1 r′λ1

8

)

+ pd2W12

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.30)

e46 =−pd1

(

r′λ1

5 − r′λ1

7

) (

r′λ1

1 r′λ2

8 − r′λ2

1 r′λ1

8

)

+ pd2W13

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.31)

e47 =r′λ1

1

(

r′λ2

5 − r′λ2

7

)

r′λ2

8 +(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

r′λ2

8 + r′λ2

1 W14

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.32)

e48 =r′λ2

1

(

r′λ1

5 − r′λ1

7

)

r′λ1

8 +[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

r′λ1

8 + r′λ1

1 W15

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.33)

e51 = 0 (H.34)

e52 =1

r′λ1

5 − r′λ2

5 r′−λ2+λ1

7

(H.35)

e53 =1

r′λ2

5 − r′λ1

5 r′λ2−λ1

7

(H.36)

e54 = 0 (H.37)

e55 =(−pd1

+ pd2) r′λ2

7

pd1pd2

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.38)

141


FUNCTIONS

e56 =pd1r′λ1

7 − pd2r′λ1

7

pd1pd2r′λ1

5 r′λ2

7 − pd1pd2r′λ2

5 r′λ1

7

(H.39)

e57 =1

−(

pd1pd2r′λ1

5

)

+ pd1pd2r′λ2

5 r′−λ2+λ1

7

(H.40)

e58 =1

−(

pd1pd2r′λ2

5

)

+ pd1pd2r′λ1

5 r′λ2−λ1

7

(H.41)

e61 =pd1

(

r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

)

(pd1+ pd2

)W6(H.42)

e62 =pd1

(

r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

) (

r′λ2

5 − r′λ2

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.43)

e63 = −pd1

(

r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

) (

r′λ1

5 − r′λ1

7

)

(pd1+ pd2

)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.44)

e64 =r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

(pd1+ pd2

)W6(H.45)

e65 =pd2

(

r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

) (

r′λ2

5 − r′λ2

7

)

+ pd1W16

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.46)

e66 =−pd2

(

r′λ1

2 r′λ2

3 − r′λ2

2 r′λ1

3

) (

r′λ1

5 − r′λ1

7

)

+ pd1W17

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.47)

e67 =W16

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.48)

e68 =W17

pd2(pd1

+ pd2)W6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.49)

e71 = 0 (H.50)

e72 =1

−(

r′−λ2+λ1

5 r′λ2

7

)

+ r′λ1

7

(H.51)

e73 =1

r′λ2

7 − r′λ2−λ1

5 r′λ1

7

(H.52)

e74 = 0 (H.53)

e75 =pd1r′λ2

5 − pd2r′λ2

5

pd1pd2r′λ1

5 r′λ2

7 − pd1pd2r′λ2

5 r′λ1

7

(H.54)

e76 =(−pd1

+ pd2) r′λ1

5

pd1pd2

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

) (H.55)

e77 =1

pd1pd2r′−λ2+λ1

5 r′λ2

7 − pd1pd2r′λ1

7

(H.56)

142

e78 =1

−(

pd1pd2r′λ2

7

)

+ pd1pd2r′λ2−λ1

5 r′λ1

7

(H.57)

e81 =pd2

(

r′λ1

1 r′λ2

4 − r′λ2

1 r′λ1

4

)

(pd1+ pd2

)W1(H.58)

e82 =pd2

(

r′λ1

1 r′λ2

4 − r′λ2

1 r′λ1

4

) (

r′λ2

5 − r′λ2

7

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.59)

e83 = −pd2

(

r′λ1

1 r′λ2

4 − r′λ2

1 r′λ1

4

) (

r′λ1

5 − r′λ1

7

)

(pd1+ pd2

)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.60)

e84 =−(

r′λ1

1 r′λ2

4

)

+ r′λ2

1 r′λ1

4

(pd1+ pd2

)W1

(H.61)

e85 =−pd1

(

r′λ1

1 r′λ2

4 − r′λ2

1 r′λ1

4

) (

r′λ2

5 − r′λ2

7

)

+ pd2W18

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.62)

e86 =pd1

(

r′λ1

1 r′λ2

4 − r′λ2

1 r′λ1

4

) (

r′λ1

5 − r′λ1

7

)

− pd2W19

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.63)

e87 =r′λ1

1 r′λ2

4

(

−r′λ2

5 + r′λ2

7

)

+ r′λ2

4 −(

r′λ1

5 r′λ2

7 + r′λ2

5 r′λ1

7

)

+ r′λ2

1 W20

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.64)

e88 =W19

pd1(pd1

+ pd2)(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

W1

(H.65)

with Wk, k = 1, . . . , 20 being

W1 = r′λ1

4 r′λ2

8 − r′λ2

4 r′λ1

8 + r′λ1

1

(

r′λ2

4 − r′λ2

8

)

+ r′λ2

1

(

−r′λ1

4 + r′λ1

8

)

(H.66)

W2 = r′λ1

4

(

r′λ2

5 − r′λ2

7

)

r′λ2

8 +(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

r′λ2

8

+r′λ2

4

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

7 r′λ1

8 + r′λ2

5

(

r′λ1

7 − r′λ1

8

)]

(H.67)

W3 = r′λ2

4

(

r′λ1

5 − r′λ1

7

)

r′λ1

8 +[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

r′λ1

8

+r′λ1

4

[

−(

r′λ2

5 r′λ1

7

)

+ r′λ1

7 r′λ2

8 + r′λ1

5

(

r′λ2

7 − r′λ2

8

)]

(H.68)

W4 = −(

r′λ1

4 r′λ2

8

)

+ r′λ2

4 r′λ1

8 + r′λ1

1

(

−r′λ2

4 + r′λ2

8

)

+ r′λ2

1

(

r′λ1

4 − r′λ1

8

)

(H.69)

W5 = r′λ2

5 r′λ1

7 − r′λ1

7 r′λ2

8 + r′λ1

5

(

−r′λ2

7 + r′λ2

8

)

(H.70)

W6 = r′λ1

3 r′λ2

6 − r′λ2

3 r′λ1

6 + r′λ1

2

(

r′λ2

3 − r′λ2

6

)

+ r′λ2

2

(

−r′λ1

3 + r′λ1

6

)

(H.71)

W7 = r′λ1

3 r′λ2

6

(

−r′λ2

5 + r′λ2

7

)

+ r′λ2

6

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

+r′λ2

3

[

r′λ1

5 r′λ2

7 − r′λ1

6 r′λ2

7 + r′λ2

5

(

r′λ1

6 − r′λ1

7

)]

(H.72)

W8 = r′λ2

3 r′λ1

6

(

−r′λ1

5 + r′λ1

7

)

+ r′λ1

6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

+r′λ1

3

[

r′λ2

5 r′λ1

7 − r′λ2

6 r′λ1

7 + r′λ1

5

(

r′λ2

6 − r′λ2

7

)]

(H.73)

143


FUNCTIONS

W9 = −(

r′λ1

3 r′λ2

6

)

+ r′λ2

3 r′λ1

6 + r′λ1

2

(

−r′λ2

3 + r′λ2

6

)

+ r′λ2

2

(

r′λ1

3 − r′λ1

6

)

(H.74)

W10 = r′λ1

2 r′λ2

6

(

r′λ2

5 − r′λ2

7

)

+ r′λ2

6

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

+r′λ2

2

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ1

6 r′λ2

7 + r′λ2

5

(

−r′λ1

6 + r′λ1

7

)]

(H.75)

W11 = r′λ2

2 r′λ1

6

(

r′λ1

5 − r′λ1

7

)

+ r′λ1

6

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

+r′λ1

2

[

−(

r′λ2

5 r′λ1

7

)

+ r′λ2

6 r′λ1

7 + r′λ1

5

(

−r′λ2

6 + r′λ2

7

)]

(H.76)

W12 = r′λ1

1

(

−r′λ2

5 + r′λ2

7

)

r′λ2

8 +[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

r′λ2

8

+r′λ2

1

[

r′λ1

5 r′λ2

7 − r′λ2

7 r′λ1

8 + r′λ2

5

(

−r′λ1

7 + r′λ1

8

)]

(H.77)

W13 = r′λ2

1

(

−r′λ1

5 + r′λ1

7

)

r′λ1

8 +(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

r′λ1

8 + r′λ1

1 W5 (H.78)

W14 = −(

r′λ1

5 r′λ2

7

)

+ r′λ2

7 r′λ1

8 + r′λ2

5

(

r′λ1

7 − r′λ1

8

)

(H.79)

W15 = −(

r′λ2

5 r′λ1

7

)

+ r′λ1

7 r′λ2

8 + r′λ1

5

(

r′λ2

7 − r′λ2

8

)

(H.80)

W16 = r′λ1

2 r′λ2

3

(

−r′λ2

5 + r′λ2

7

)

+ r′λ2

3

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7

]

+r′λ2

2

[

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7 + r′λ1

3

(

r′λ2

5 − r′λ2

7

)]

(H.81)

W17 = r′λ2

2 r′λ1

3

(

−r′λ1

5 + r′λ1

7

)

+ r′λ1

3

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

+r′λ1

2

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7 + r′λ2

3

(

r′λ1

5 − r′λ1

7

)]

(H.82)

W18 = r′λ1

1 r′λ2

4

(

r′λ2

5 − r′λ2

7

)

+ r′λ2

4

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

+r′λ2

1

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7 + r′λ1

4

(

−r′λ2

5 + r′λ2

7

)]

(H.83)

W19 = r′λ2

1 r′λ1

4

(

−r′λ1

5 + r′λ1

7

)

+ r′λ1

4

(

r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7

)

+r′λ1

1

[

−(

r′λ1

5 r′λ2

7

)

+ r′λ2

5 r′λ1

7 + r′λ2

4

(

r′λ1

5 − r′λ1

7

)]

(H.84)

W20 = r′λ1

5 r′λ2

7 − r′λ2

5 r′λ1

7 + r′λ1

4

(

r′λ2

5 − r′λ2

7

)

(H.85)

.

144

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BEM Solutions for Linear Elastic and Fracture Mechanics Problems with Microstructural Effects

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