Defect and material mechanics

ISDMM 2007

Defect and Material Mechanics

Proceedings of the International Symposium on

Defect and Material Mechanics (ISDMM),

held in Aussois, France, March 25–29, 2007

Edited by

CRISTIAN DASCALU

University J. Fourier, Grenoble, France

GE´ RARD A. MAUGIN

University Pierre et Marie Curie, Paris, France

and

CLAUDE STOLZ

Ecole Polytechnique, Palaiseau, France

Reprinted from International Journal of Fracture

Volume 147, Nos. 1–4 (2007)

Published by Springer,

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ISBN-13: 978-1-4020-6928-4 e-ISBN-13: 978-1-4020-6929-1

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2008 Springer

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Table of Contents

Preface

C. Dascalu and G.A. Maugin

Reciprocity in fracture and defect mechanics

R. Kienzler

Configurational forces and gauge conditions in electromagnetic bodies

C. Trimarco

The anti-symmetry principle for quasi-static crack propagation in Mode III

G.E. Oleaga

Configurational balance and entropy sinks

M. Epstein

Application of invariant integrals to the problems of defect identification

R.V. Goldstein, E.I. Shifrin and P.S. Shushpannikov

On application of classical Eshelby approach to calculating effective elastic moduli of

dispersed composites

K.B. Ustinov and R.V. Goldstein

Material forces in finite elasto-plasticity with continuously distributed dislocations

S. Cleja-T¸ igoiu

Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

P.A. Gourgiotis and H.G. Georgiadis

Bifurcation of equilibrium solutions and defects nucleation

C. Stolz

Theoretical and numerical aspects of the material and spatial settings in nonlinear

electro-elastostatics

D.K. Vu and P. Steinmann

Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics

M. Scherer, R. Denzer and P. Steinmann

Variational design sensitivity analysis in the context of structural optimization and

configurational mechanics

D. Materna and F.-J. Barthold

An anisotropic elastic formulation for configurational forces in stress space

A. Gupta and X. Markenscoff

Conservation laws, duality and symmetry loss in solid mechanics

H.D. Bui

Phase field simulation of domain structures in ferroelectric materials within the context of

inhomogeneity evolution

R. Müller, D. Gross, D. Schrade and B.X. Xu

An adaptive singular finite element in nonlinear fracture mechanics

R. Denzer, M. Scherer and P. Steinmann

Moving singularities in thermoelastic solids

A. Berezovski and G.A. Maugin

1

3–11

13–19

21–33

35–43

45–54

55–66

67–81

83–102

103–107

109–116

117–132

133–155

157–161

163–172

173–180

181–190

191–198

Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial

half-space

T. Profant, O. Ševec9ek, M. Kotoul and T. Vysloužil

Study of the simple extension tear test sample for rubber with Configurational Mechanics

E. Verron

Stress-driven diffusion in a deforming and evolving elastic circular tube of single

component solid with vacancies

C.H. Wu

Mode II intersonic crack propagation in poroelastic media

E. Radi and B. Loret

Material forces for crack analysis of functionally graded materials in adaptively refined

FE-meshes

R. Mahnken

A multiscale approach to damage configurational forces

C. Dascalu and G. Bilbie

199–217

219–225

227–234

235–267

269–283

285–294

The volume presents recent developments in the

theory of defects and the mechanics of material forces.

Most of the contributions were presented at the Inter￾national Symposium on Defect and Material Forces

(ISDMM2007), held in Aussois, France, March 25–

29, 2007.

Originated in the works of Eshelby, the Material or

Configurational Mechanics experienced a remarkable

revival over the last two decades. When the mechanics

of continua is fully expressed on the material manifold,

it captures the material inhomogeneities. The driving

(material) forces on inhomogeneities appear naturally

in this framework and are requesting for constitutive

modeling of the evolution of inhomogeneities through

kinetic laws.

In this way, a general scheme for describing struc￾tural changes in continua is obtained. The Eshelbian

mechanics formulation comes up with a unifying treat￾ment of different phenomena like fracture and damage

evolution, phase transitions, plasticity and dislocation

motion, etc.

C. Dascalu (B)

Laboratoire Sols Solides Structures, Université Joseph

Fourier, Grenoble, Domaine Universitaire, B.P. 53,

38041 Grenoble cedex 9, France

e-mail: [email protected]

G. A. Maugin

Université Pierre et Marie Curie, Institut Jean Le Rond

d’Alembert, Case 152, 4 place Jussieu,

75252 Paris cedex 05, France

e-mail: [email protected]

This special issue aims at bringing together recent

developments in Material Mechanics and the more clas￾sical Defect Mechanics approaches. The contributions

are highlighting recent research on topics like: fracture

and damage, electromagnetoelasticity, plasticity, dis￾tributed dislocations, thermodynamics, poroelasticity,

generalized continua, structural optimization, conser￾vation laws and symmetries, multiscale approaches and

numerical solution strategies.

We expect the present volume to be a valuable

resource for researchers in the field of Mechanics of

Defects in Solids.

We dedicate this special issue to the memory of

the late Professor George Herrmann (1921–2007).

G. Herrmann was a prestigious scientist, well-known

in the international mechanics community. In the last

years, he was an active researcher in the field of Mate￾rial Mechanics. George Herrmann supported the orga￾nization and registered for attending the ISDMM2007

Symposium in Aussois, before his sudden dead on

January 7, 2007. None of us will forget his passion for

mechanical sciences, his enthusiasm and generosity.

Preface

C. Dascalu · G. A. Maugin

Defect and Material Mechanics. C. Dascalu, G.A. Maugin & C. Stolz (eds.),

doi: 10.1007/978-1-4020-6929-1_ , © Springer Science+Business Media B.V. 200

1

1 8

Abstract For defects in solids, when displaced within

the material, reciprocity relations have been established

recently similar to the theorems attributed to Betti and

Maxwell. These theorems are applied to crack- and

defect-interaction problems.

Keywords Reciprocity · Fracture · Defect

interaction · Material forces

1 Introduction

When treating problems of linear elastic systems, such

as beams, frames or two- and three-dimensional contin￾uous elastic solids, the reciprocity theorems associated

with the names of Betti and Maxwell have proven to

be quite valuable. In its simplest form, Betti’s theorem

states that if a linear elastic body is supported properly

such that rigid body displacements are precluded and

if an external force F1 at point 1 which produces a dis￾placement u21 at some other point 2, then a force F2

at 2 would produce a displacement u12 at 1 where (cf.

e.g., Marguerre 1962)

F1 · u12 = F2 · u21. (1)

The contents of the present paper has been developed together

with Prof. Dr. Dr. h. c. George Herrmann, Stanford University,

California, who passed away on January 7, 2007.

R. Kienzler (B)

Department of Production Engineering,

University of Bremen, Bremen, Germany

e-mail: [email protected]

The dot marks the scalar product between the two

vectors. In double-indexed terms, the first index indi￾cates the position at which the quantity is measured

(effect), and the second index indicates the cause due

to which this quantity occurs.

To reach a scalar version of Betti’s theorem the dis￾placement component of u12 in the direction of F1 is

introduced as u P

12 and the component of u21 in the direc￾tion of F2 as u P

21 (cf. Fig. 1). With the magnitude F1

and F2 of F1 and F2, respectively, it is

F1u P

12 = F2u P

21. (2)

Since u P

12 is proportional to F2 and u P

21 is proportional

to F1 influence coefficient may be defined as

u P

12 = δ12F2, (3a)

u P

21 = δ21F1, (3b)

and according to Marguerre (1962), Maxwell’s theo￾rem states

δ12 = δ21. (4)

The reciprocity relations are based on the result that the

energy stored in an elastic body after application of two

forces is independent of their sequence of application,

and equals the external work done on the body. Various

applications of these theorems are to be found in , e.g.,

Timoshenko and Goodier (1970), Barber (2002).

During the recent decades a new topic has emerged

in mechanics of elastically deformable media which

is variously described as Defect Mechanics, Fracture

Mechanics, Configurational Mechanics, Mechanics in

Reciprocity in fracture and defect mechanics

R. Kienzler

Defect and Material Mechanics. C. Dascalu, G.A. Maugin & C. Stolz (eds.), 3

doi: 10.1007/978-1-4020-6929-1_2, © Springer Science+Business Media B.V. 2007

4 R. Kienzler

u12

u21

F1

F2

12

P u

21

P u

1 2

u12

u21

F1

F2

12

P u

21

P u

1 2

u12

u21

F1

F2

12

P u

21

P u

1 2

Fig. 1 Elastic body subjected to two forces

Material Space or Eshelbian Mechanics (cf. Maugin

1993; Gurtin 2000; Kienzler and Herrmann 2000). The

importance of this topic is based on the necessity of

improved material modeling of defects of various types

and scales in deformable solids. And this necessity in

turn is the result of developing new technologies in a

variety of applied fields such as devices in IT, aerospace

and energy sectors.

In approaching configurational mechanics one can

say that a material force is associated with a defect in

a stressed elastic body, because if this defect were dis￾placed within the body, the total energy of the system

would be changed and the negative ratio of this change

and the displacement, in the limit as the displacement

is made to approach zero, is the material force on the

defect.

Let us consider a linearly elastic body of arbitrary

geometry properly supported and subjected to surface

tractions and/or body forces. And let us assume that the

body contains an arbitrary number and type of distrib￾uted or localized (point) defects, such as dislocations,

cracks, inclusions, cavities, etc. Let us focus attention

on two localized defects placed at positions 1 and 2

and let the associated material forces be the vectors

B10 and B20 respectively. Now, defect 1 will be dis￾placed by an amount λ1 relatively to the material in

which the defect is positioned (material displacement).

In turn the material force at 1 will be changed by an

amount B11 and the material force 2 will be changed by

B21. The material displacement is assumed to be small

in the sense that linearity is implied between material

displacements and material forces, e.g., B21 ∝ λ1. If

defect 2 would be materially displaced by λ2 the mate￾rial force at 1 and 2 would be changed by B12 and B22,

respectively.

B12

B21

1

2

B12

P

21

P B

1

2

F

q

B

B

B

B

1

2 B

B

B

B

1

2

F

q

Fig. 2 Stressed elastic body containing two point defects

Based on the similar argument (as applied in Physi￾cal Space) that the change of energy stored in the body

after application of two material displacements is inde￾pendent of their sequence of application, and equals the

work done of the material forces in the material dis￾placements, a material, Betti-like reciprocity theorem

is established (Herrmann and Kienzler 2007a) as

λ2 · B21 = λ1 · B12, (5)

stating that the work in the material translation at 2 of

the change of the material force due to a material trans￾lation at 1 equals the work in the material translation at

1 of the change of the material force due to a material

translation at 2.

Note the difference between Physical and Material

Space: in Physical Space, the applied physical forces

are the causes of (the change of) physical displacements

(effects) whereas in Material Space the material dis￾placement cause (changes of) material forces (effects).

In analogy, a scalar version of Eq. 5 is reached by intro￾ducing the component of B21 in the direction of λ2 as

BP

21, the component of B12 in the direction of λ1 as BP

12

(cf. Fig. 2) and the magnitudes of λ1 and λ2 as λ1 and

λ2, respectively, and if follows

λ2BP

21 = λ1BP

12. (6)

Since linearity is implied, material influence coeffi￾cients may be defined as

BP

21 = β21λ1, (7a)

BP

12 = β12λ2, (7b)

and a material, Maxwell-like reciprocity theorem is

obtained (Herrmann and Kienzler 2007a)

β21 = β12. (8)

It states that the change of the material force at 2 in

the direction of the material displacement λ2 due to a

Reciprocity in fracture and defect mechanics 5

Fig. 3 Crack configuration

within a bar in tension

N N

b

b

1 a

1 a 2 a

2 a

N N

b

b

1 a

1 a 2 a

2 a

unit-material translation at 1 equals the change of the

material force at 1 in the direction of λ1 due to a unit￾material translation at 2.

The implications of a non-linear formulation of the

reciprocity relation (5) have been dealt with in Kienzler

and Herrmann (2007). By considering a stressed elastic

body subjected sequentially to a material displacement

of a defect and the application of a physical force a

novel type of coupling of Physical and Material

Mechanics by means of a reciprocity theorem has been

established in Herrmann and Kienzler (2007b).

It is the intention of the present contribution to

explore some applications of the two theorems (5) and

(8) in fracture and defect mechanics.

2 Interacting cracks

Interacting cracks have been the object of research over

several years (cf., e.g., Erdogan 1962; Panasyuk et al.

1977; Gross 1982). The material forces at crack tips are

usually calculated from the path-independent J inte￾gral (Rice 1968). Reciprocity relations are, therefore,

concerned with the change of the J integral due to the

translation of some other defect, e.g., the change of

length of a crack 2 in the neighbourhood of the origi￾nal crack 1. Usually, the solution of crack-interaction

problems involve either some advances analytical tools

or an extended numerical investigation. Based on the

strength-of-materials theories, a simple first estimate

for interacting edge cracks in an elastic bar under ten￾sion have been given recently by Rohde and Kienzler

(2005) and Rohde et al. (2005), and the reciprocity rela￾tions are applied within this simplified problem setting.

In Rohde and Kienzler (2005) and Rohde et al. (2005)

bars are investigated with a set of 2 × 2 edge cracks

symmetrically positioned with respect to its length axis

(a)

a a

a

(b)

1 a b

1 a

2 a

2 a

1

2

1 a b

1 a

2 a

2 a

1

2

Fig. 4 Single crack (a), two neighboured cracks (b)

and loaded by an axial force in tension as depicted in

Fig. 3.

If a single crack is considered first, the key idea is

that the energy-release rate is proportional to the length

of the shaded strip in Fig. 4a, i.e.,

G = J ∝ 2a

√

2; J = C2a

√

2. (9)

The constantC, say, would be proportional to the square

of the applied load, inverse proportional to Young’s

modulus, and would combine some information about

the geometry of the bar.

In the presence of a second crack, cf. Fig. 4b, both

strips can not develop completely due to shielding such

6 R. Kienzler

that the energy-release rate of crack 1 is proportional

to the length of the darkly shaded area on the left and

the energy release rate of crack 2 is proportional to the

lightly shaded area on the right, i.e.,

G1 = J1 = B10 = 1

2

√

2C(3a1 − a2 + b), (10a)

G2 = J2 = B20 = 1

2

√

2C(3a2 − a1 + b), (10b)

where a1 and a2 are the crack lengths of crack 1 and

crack 2, respectively, and b is the distance of both

cracks. Eq. 10 is valid as long as

|a1 − a2| < b < a1 + a2. (11)

It is observed, of course, that the crack configuration

represents a mixed-mode problem and the vector J

does not point in the direction of a potential self-similar

crack extension. It is sufficient, however, to consider

only the component of J in the direction of a1 or

a2, corresponding to BP

12 or BP

21. Details of the anal￾ysis and the validation of the results are given in Rohde

and Kienzler (2005) and Rohde et al. (2005).

With the simple relations (10) at hand it is easy to

check the reciprocity theorems. A crack extension of

crack 1, a1, corresponds to a material translation λ1.

Due to λ1, the material forces B10 and B20 change to

B11 = B1λ1 − B10

= 1

2

√

2C [3(a1 + λ1) − a2 + b − 3a1 + a2 − b]

= +

3

2

√

2Cλ1, (12a)

B21 = B2λ1 − B10

= 1

2

√

2C [3a2 − (a1 + λ1) + b − 3a2 + a1 − b]

= −1

2

√

2Cλ1,

= β21λ1. (12b)

If crack 2 is extended by an amount λ2 the change of the

material forces at crack 1 and 2 is calculated similarly

as

B12 = −1

2

√

2Cλ2 = β12λ2, (12c)

B22 = +

3

2

√

2Cλ2. (12d)

The changes of the various energy-release rates have

been made graphic in Fig. 5

Substitution of (12b) and (12c) into relations (5)

or (6) and (8) confirms and illustrates the validity of

1 2

1

2

1 2

(b)

(a)

1 2

Fig. 5 Change of energy-release rates due to crack extension λ1

of crack 1 (a) and λ2 of crack 2 (b)

reciprocity relations in Material Space. In words: the

change of the J integral at crack 1 due to a change of

the crack length of crack 2 equals the change of the J

integral at crack 2 due to the change of the crack length

of crack 1. This relation may also be verified within a

more rigorous approach either analytically or numeri￾cally and may be used to establish influence surfaces

for defects within the stress state of a crack tip. This

problem will be further treated in the next section.

3 Interaction of an edge dislocation with a circular

hole

As a further example for the application of the material

reciprocity relations let us consider the following defect

configuration. Within an infinitely extended plane elas￾tic sheet an edge dislocation is situated in the origin

of a plane Cartesian coordinate system (x1, x2) with

Burgers vector pointing (without loss of generality) in

x2-direction. At xi = ξi(i = 1, 2) a circular hole with

radiusr0 is centred, or, in polar coordinates, at distance

d > r0(ε = d/r0 > 1) under the angle ϕ measured

from the x1-axis as depicted in Fig. 6.

Due to a material translation ξi → ξi + λi the total

energy  of the system changes and the energy-release

rate is defined as

Reciprocity in fracture and defect mechanics 7

Fig. 6 Edge dislocation

and circular hole in a plane

elastic plate

0r

b 1 x

2 x

2

1

d

0r

b 1 x

2 x

2

1

d

Fig. 7 Free-body diagram

for material quantities

L

1 x

2 x

2

1

1

H J

2

H J

M

1

D J

2

D J

L

1 x

2 x

2

1

1

H J

2

H J

M

1

D J

2

D J

lim

λi→0

(ξi + λi) − (ξi)

λi

= −Ji . (13)

As usual, the energy-release rate is calculated by means

of a path-independent contour integral involving the

energy-momentum tensor (cf., e.g., Maugin 1993;

Gurtin 2000; Kienzler and Herrmann 2000). The con￾tour of integration is arbitrary as long as the same de￾fect, either the hole or the inclusion, is included. In

addition, two further path-independent integrals have

been introduced (Günther 1962; Knowles and Stern￾berg 1972; Budiansky and Rice 1973) designated as

L and M integrals. The L-integral is a material vector

moment (r × J) and calculates the energy-release rate

due to a rotation of the defect ϕ → ϕ + ω. It is path

independent if the material is isotropic. The M integral

is a material scalar moment (r · J) and calculates the

energy-release rate due to a self-similar expansion of

the defect, here r0 → αr0. L and M depend on the

choice of the point of reference. Choosing for L the

origin of the coordinate system and for M the center

of the hole, the path-independent integrals have been

evaluated (Kienzler and Herrmann 2000; Kienzler and

Kordisch 1990) and are given as

J1 = − E∗b2 cos ϕ

4πr0ε3

1

ε2 − 1 + 1 + 2 sin2 ϕ

, (14a)

J2 = − E∗b2 sin ϕ

4πr0ε3

1

ε2 − 1 + 2 sin2 ϕ

, (14b)

L = − E∗b2 sin ϕ cos ϕ

4πε2 , (14c)

M = +

E∗b2

4πε2

1

ε2 − 1 + 1 + sin2 ϕ

, (14d)

with E∗ = E for plane stress, E∗ = E/(1 − ν2) for

plane strain, Young’s modulus E and Poisson’s ratio

ν. As in Physical Space, free body diagrams can be

sketched in Material Space. This is shown in Fig. 7,

where for the virial M, in the sketch a pressure-like

symbol and in the equation the symbol ⊗ is used.

Likewise, material equilibrium conditions apply,

where the expanding-moment condition (“Fliehmo￾ment”, cf. Schweins 1849) is normally not used nei￾ther in physical nor in material space

→ : J H

1 − J D

1 = 0 (15a)

↑ : J H

2 − J D

2 = 0 (15b)

∩ : L + ξ1 J H

2 − ξ2 J H

1 = 0 (15c)

⊗ : M + ξ1 J D

1 + ξ2 J D

2 = 0 (15d)

8 R. Kienzler

1 x

2 x

d

d

1

d

1 x

2 x

d

d

1

d

Fig. 8 Material translation λ1 of the hole

The change of material dynamic quantities Ji, L, M

due to the small material kinematic quantities λi,ω,α

can be calculated by differentiation

λi( ) = ∂( )

∂λi

λi + 0

λ2

i



(16)

and by geometrical considerations.

From Fig. 8, e.g., we read off

d¯ cos(ϕ − dϕ) = d cos ϕ + λ1, (17a)

d¯ sin(ϕ − dϕ) = d sin ϕ (17b)

and after linearization (dϕ  1, λ1  d) we find

ε → ¯ε = ε +

λ1

r0

cos ϕ (18a)

ϕ → ¯ϕ = ϕ − λ1 sin ϕ

r0ε . (18b)

Inserting in (16) for i = 1 and applying the chain role

leads to

λ1 ( ) =

∂( )

∂ε

cos ϕ

r0

− ∂( )

∂ϕ

sin ϕ

r0ε

λ1, (19a)

and in a similar way we obtain

λ2 ( ) =

∂( )

∂ε

sin ϕ

r0

+ ∂( )

∂ϕ

cos ϕ

r0ε

λ2. (19b)

Due to a rotation of the hole around the dislocation,

ϕ → ϕ + ω, and a self-similar expansion of the hole,

r0 → αr0 we find

ω( ) = ∂( )

∂ϕ ω, (19c)

α( ) = ε

∂( )

∂ε

α. (19d)

Using Eq. 19, the change of the relevant quantities can

easily be calculated with the results

• material translation λ1

λ1 J1 = +λ1

E∗b2

4πr 2

0 ε4

×

2 cos 2ϕ

(ε2 − 1)2 +

6 cos 2ϕ − 1

(ε2 − 1)

−3(4 cos 4ϕ − 6 cos2 ϕ + 1)



(20a)

λ1 J2 = +λ1

E∗b2 sin ϕ cos ϕ

2πr 2

0 ε4

1

(ε2 − 1)2

+

3

(ε2 − 1) + 6 sin 2ϕ

(20b)

λ1 L = −λ1

E∗b2 sin ϕ

4πr0ε3

×

1 − 4 cos 2ϕ



(20c)

λ1 M = −λ1

E∗b2 cos ϕ

2πr0ε3

×

ε4

(ε2 − 1)2 + 2 sin 2ϕ

(20d)

• material translation λ2

λ2 J1 = +λ2

E∗b2 sin ϕ cos ϕ

2πr 2

0 ε4

1

(ε2 − 1)2

+

3

(ε2 − 1) + 6 sin 2ϕ

(21a)

λ2 J2 = +λ2

E∗b2

4πr 2

0 ε4

×

2 sin 2ϕ

(ε2 − 1)2 +

6 sin 2ϕ − 1

ε2 − 1

−6 sin 2ϕ(1 − 2 sin2 ϕ)

(21b)

λ2 L = −λ2

E∗b2 cos ϕ

4πr0ε3

×

1 − 4 sin 2ϕ



(21c)

λ2 M = −λ2

E∗b2 sin ϕ

2πr0ε3

×

ε4

(ε2 − 1)2 + 1 − 2 cos 2ϕ

(21d)

Reciprocity in fracture and defect mechanics 9

• material rotation ω

ω J1 = +ω E∗b2 sin ϕ

4πr0ε3

×

1

ε2 − 1 + 3(1 − 2 cos 2ϕ)

(22a)

ω J2 = +ω E∗b2 cos ϕ

4πr0ε3

×

− 1

ε2 − 1 − 6 sin 2ϕ

(22b)

ωL = −ω E∗b2

4πε2

1 − 2 cos 2ϕ



= +ω E∗b2

4πε2 cos 2ϕ (22c)

ωM = +ω E∗b2

2πε2 sin ϕ cos ϕ

= ω E∗b2

4πε2 sin 2ϕ (22d)

• material self-similar expansion α

α J1 = α

E∗b2 cos ϕ

2πr0ε3

×

− ε4

(ε2 − 1)2 − 2 sin 2ϕ

(23a)

α J2 = α

E∗b2 sin ϕ

2πr0ε3

×

− ε4

(ε2 − 1)2 − 1 + 2 cos2 ϕ

(23b)

α L = −α

E∗b2

2πε2 sin ϕ cos ϕ

= −α

E∗b2

4πε2 sin 2ϕ (23c)

α M = +α

E∗b2

2πε2

×

ε4

(ε2 − 1)2 + sin 2ϕ

. (23d)

Also the modified material quantities have to satisfy

material equilibrium conditions. Especially for the vec￾torial and the scalar moments the change of the lever

arms due to the material displacements have to be ob￾served. As an example consider the material rotation

depicted in Fig. 9

Equilibrium of moments yields

L + ωL + (ξ1 − ξ2ω)(J2 + ω J2)

−(ξ2 + ξ1ω)(J1 + ω J1) = 0. (24)

L L

2

1

1 1 J J

1 x

2 x

d

0r

2 2 J J

L L

2

1

J J

1 x

2 x

d

0r

2 2 J J

Fig. 9 Free-body diagram after material rotation ω

Due to (15c) and neglection of terms of 0(ω2) we arrive

at

ωL + r0ε(cos ϕω J2 − ω sin ϕ J2

− sin ϕω J1 − ω cos ϕ J1) = 0. (25c)

In a similar way we obtain

λ1 L + ξ1λ1 J2 + λ1 J2 − ξ2λ1 J1 = 0, (25a)

λ2 L + ξ1λ2 J2 − λ2 J1 − ξ2λ2 J1 = 0, (25b)

α L + r0ε(cos ϕα J2 − sin ϕα J1 = 0. (25d)

The virial equations are

λ1 M + λ1 J1 + ξ1λ1 J1 + ξ2λ1 J2 = 0, (26a)

λ2 M + λ2 J2 + ξ1λ2 J1 + ξ2λ2 J2 = 0, (26b)

ωM + r0ε(cos ϕω J1 − ω sin ϕ J1

+ sin ϕω J2 + ω cos ϕ J2) = 0, (26c)

α M + r0ε(cos ϕα J1 + sin ϕα J2) = 0. (26d)

On introducing (20)–(23) into (25) and (26) it is

observed that the material equilibrium conditions are

satisfied identically.

Let us now turn our attention to reciprocity. As a first

application we consider two material displacements λ1

and λ2 of the circular hole. The reciprocity theorem

states that the work of the change of J2 due to the mate￾rial translation λ1 in the material translation λ2 is equal

to the work of the change of the material force J1 due

to a material translation λ2 in the material translation

λ1. Thus

λ2λ1 J2 = λ1λ2 J1. (27a)

10 R. Kienzler

With (20b) and (21a) it is easily seen that (27a) holds.

For the Maxwell-like version we introduce material

influence coefficients as

λ1 J2 = β21λ1, (28a)

λ2 J1 = β12λ2, (28b)

and it is seen, immediately that

β21 = β12. (28c)

Another reciprocity relation may be formulated as: the

work of the change of L due to a material self-similar

expansion α in the material rotation ω is equal to the

work of the change of M due to a rotation ω in the

material self-similar expansion α, i.e.,

− ωα L = αωM. (27b)

The minus sign appears due to the positive definitions

of the involved quantities. The Maxwell-like version is

obvious. Both relations are confirmed with (23c) and

(22d). Likewise, four further reciprocity relations can

be established where we restrict ourselves to the Betti￾like formulation because the Maxwell-like version is

trivially accessible

λ1α J1 = αλ1 M, (27c)

λ2α J2 = αλ2 M, (27d)

− ωλ1 L = λ1(ω J1 + ωJ2), (27e)

− ωλ2 L = λ2(ω J2 + ωJ1). (27f)

The relations (27e) and (27f) are a little more exten￾sive as mostly when rotations are involved. It might be

supporting to give some more ideas on its derivation.

Consider (27e) and apply the rotation ω first. J1 and J2

would change to J1 + ω J1 and J2 + ω J2 the work

W0ω would be

W0ω = J2r0εω cos ϕ +

1

2

r0εω cos ϕω J2

−J1r0εω sin ϕ − 1

2

r0εω sin ϕω J1. (29)

L does not contribute to (29) since the hole is rotated

around the origin whereas the dislocation is not rotated.

An additional translation in x1-direction produces addi￾tional work as

Wωλ1 = (J1 + ω J1)λ1 +

1

2

λ1λ1 J1. (30)

1

i

Fig. 10 Crack in a damaged material

If λ1 is applied first, then the change of work would be

W0λ1 = J1λ1 +

1

2

λ1λ1 J1. (31)

The material forces after this step are λ1 + λ1 J1 and

λ2+λ1 J2. An additional rotation ω causes the change

of work to an amount of

Wλ1ω = (J2+λ1 J2)r0εω cos ϕ+

1

2

r0εω cos ϕω J2

−(J1+λ1 J1)r0εω sin ϕ−1

2

r0εω sin ϕω J1.

(32)

Since finally, the energy stored in the system under con￾sideration is the same independently of the sequence of

the small material displacements, the work W0ω+Wωλ1

and the work W0λ1 + Wλ1ω must be equal

W0ω + Wωλ1 = W0λ1 + Wλ1ω. (33)

After cancelling equal terms we find

ω J1λ1 = r0εω cos ϕλ1 J2 − r0εω sin ϕλ1 J1, (34)

and with (25a), Eq. 27e is verified.

As an example of the usefulness of the reciprocity

relations given above let us consider a crack surrounded

by a damaged material characterized by various holes

of different radii as depicted in Fig. 10. the system may

support the general idea underlying Gurson’s model

(Gurson 1977).

Assume that we would be interested in the change of

the J integral at the crack tip (defect 1) due to self-sim￾ilar growth of each void (defect i). We would thus have

to calculate αi J1. For this purpose we would have to

evaluate the original configuration first and, addition￾ally, construct for each void a new FE mesh with an

extended radius ri → αri , perform the FE calculation