Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies

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Leonid M. Zubov

Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies

~

J

Springer

Author Leonid M. Zubov Department of Mechanics and Mathematics Rostov State University, ul. Zorge 5 344104 Rostov-on-Don, Russia

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Zubov, Leonid M.: Nonlinear theory of dislocations and disclinations in elastic bodies / Leonid M. Zubov. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo :Springer, 1997 (Lecture notes in physics : N.s. M, Monographs ; 47) ISBN 3-540-62684-0

ISSN 0940-7677 (Lecture Notes in Physics. New Series m: Monographs) ISBN 3-540-62684-0 Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author Cover design: design ek production GmbH, Heidelberg SPIN: 10550861 55/3144-543210 - Printed on acid-free paper

Contents

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Volterra's Dislocations in Elastic Body. Large Deformation ...... I.I The Relations of Nonlinear Elasticity ............... 1.2 Displacements in Elastic Body with Given Finite Strain Tensor. 1.3 Extension of Weingarten's Theorem to Large Deformation . . . 1.4 Multi-valued Displacements and Volterra Dislocations in Nonlinear Plane Elasticity ................... 1.5 Total Burgers and Frank Vectors for a System of Isolated Defects. Density of Continuously Distributed Dislocations and Disclinations .......................... 2 Stressed State of Nonlinearly Elastic Bodies with Dislocations and Disclinations .............................. 2.1 Finding the Stressed State of Elastic Body with Given Characteristics of Isolated Defect; Set up of the Problem .... 2.2 Variational Formulation of the Problem of Volterra Dislocations in Nonlinear Elasticity ....................... 2.3 The Semi-inverse Method for Solving Quasi-static Problems of Nonlinear Mechanics of Solids .................. 2.4 Combination of Screw Dislocation and Wedge Disclination in Nonlinear Elastic Cylinder ................... 3 Exact Solutions Stressed to the Problems on Volterra Dislocations in Nonlinearly Elastic Bodies ........................ 3.1 The Wedge Disclination in Nonlinearly Elastic Body ....... 3.2 The Stress Field due to Screw Dislocation in the Nonlinear Elastic Body ................... 3.3 Solving the Wedge Disclination Problem with Use of the Compatibility Equations .................. 3.4 Conjugate Solutions in Nonlinear Elasticity and Their Application to the Disclination Problem ....... ' ...... 3.5 The Edge Dislocation in a Nonlinearly Elastic Medium ..... 3.6 A Cavitation near the Line of Dislocation or Disclination ....

1

9 9 14 17 28

35

41 41 44 50 56

65 65 73 79 83 88 98

VI

Contents Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses . 4.1 General Relations in Nonlinear Theory of Asymmetric Elasticity 4.2 Weingarten's Theorem and Volterra Dislocations in Couple Stress Medium Undergoing Large Strains . . . . . . . 4.3 Nonlinear Problems of Screw Dislocation and Wedge Disclination with Regard for Couple Stresses . . . . . . . . . . . . . . . . . . 4.4 Volterra Dislocations in Nonlinearly Elastic Bodies with Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Energy Integrals Characterizing the Force of Action on a Defect in Micropolar Media . . . . . . . . . . . . . . . . . . . . . . . .

103 103

Nonlinear Theory of Dislocations and Disclinations in Elastic Shells . 5.1 General Statements of Nonlinear Theory of Elastic Shells .... 5.2 Volterra Dislocations in Classical Nonlinear Theory of Elastic Shells ........................... 5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells ....................... 5.4 Invariant Contour Integrals in the Nonlinear Theory of Plates and Shells .............................. 5.5 The two Problems: a Wedge Disclination in a Shell of Revolution and a Screw Dislocation in a Cylindrical Shell .......... 5.6 Equilibrium of a Membrane with Disclination ..........

131 131

6 Stability of Equilibrium and Wave Propagation in Bodies with Inherent Stresses ............................ 6.1 The Linearized Equilibrium and Motion Equations for Elastic Bodies with Inherent Stresses ............. 6.2 Stability of Elastic Cylinder with Disclination .......... 6.3 The Effect of Screw Dislocation on the Stability and Wave Propagation in Elastic Cylinder ............ 6.4 Buckling of Thin Elastic Plate with Disclination .........

References Notations

Index

.................................. ...................................

......................................

108 112 118 125

140 144 154 158 161

167 167 170 178 188

193 197

203

Introduction

We will investigate Volterra's dislocations in the framework of nonlinear elasticity. Volterra's dislocation is a specific state of a multi-connected elastic body for which the strain tensor and the density of free energy are continuous and single-valued functions but the field of displacements cannot be uniquely defined. The non-single-valuedness is of a certain nature. Volterra's dislocation is a mathematical model of an one-dimensional defect of the crystal structure of solids. The defects mainly determine the properties of strength and plasticity of materials. In general, a Volterra's dislocation consists of a dislocation of translation type and a disclination. For a translation-type dislocation, the field of rotations of the particles in the solid is single-valued and the nonuniqueness of the field of rotations causes appearance of a disclination. To illuminate the above notions, consider a ball with some drilled tubes; the tubes are either closed, i.e. their axes are closed curves in the ball, or have their initial and final points on the ball surface. Let the diameters of the tubes tend to zero. The limit domain is simply connected. Note that on a curve being the limit state of the tubes, continuity of the strain tensor and its derivatives is broken in general. Thus the fields of displacements and rotations are not single-valued and the character of nonuniqueness has the same nature as that in multi-connected domains. In the physics of solids it is of particular interest to consider Volterra's dislocations as singular curves (i.e., linear defects). Within a small neighbourhood of the union of these curves, the strain tensor and its derivatives fail to be continuous and single-valued. The physics of strength and plasticity mainly treats the theory of translational-type defects but it is well known that the rotational defects (disclinations) also appear in solids, liquid crystals, biology structures, polymers, and other materials. Along with the physics of solids, the dislocation and disclination theory is useful in the design of elastic constructions such as plates or shells with holes. Stresses due to a dislocation essentially affect the stability of an elastic body and its dynamic characteristics under vibrations. The mathematical tools of dislocation theory are also needed to solve boundary value problems of equilibrium for multi-connected elastic bodies, even if there are no dislocations. In the framework of linear elasticity, the mathematical theory of dislocations and disclinations dates back to V. Volterra (1907), G. Weingarten (1901) and C. Somigliana (1914). For the state of the art of the linear theory of iso-

2

Introduction

lated (discrete) and continuously distributed dislocations and disclinations, see J.D. Eshelby (1956), E. KrDner (1958a), R. de Wit (1960, 1973a-c), C. Teodociu (1982), V.I. and Vladimirov and A.E. Romanov (1986). The only part of nonlinear dislocation theory that is well developed presents the theory of continuously distributed defects; it is based on the tools of differential geometry. E. Kr5ner (1958b, 1960), K. Kondo (1952), B.A. Bilby (1960), I.A. Kunin (1965), V.L. Berdichevski and L.I. Sedov (1967), and A.A. Vakulenko (1991) have made significant contributions to it. The caliber theory of dislocations and disclinations, which belongs to the same circle of ideas, was recently developed by A. Kadic and D.G.B. Edelen (1983). The nonlinear theory of isolated dislocations and disclinations, which is of practical importance for many reasons, is less developed. According to the linear elasticity, values of stresses and strains are singular at the axis of an isolated defect, i.e. they are unbounded near the axis. This contradicts the main assumption of linear theory that the strains are small. If the strains are not small it is not reasonable to apply the tools of linear elasticity; the nonlinear approach is more appropriate. One expects of the nonlinear theory that it can produce a more accurate and realistic picture of the distribution of the stresses and strains near the defect axis than the picture given by the linear theory. With nonlinear tools, the distortion of a crystal lattice can be estimated in a more realistic way. In addition, the length of Frank's vector, which determines the power of disclination, is not small in general, this makes the relations of the linear theory invalid. If we wish to use the equations in stresses and the equations of strain compatibility in solving problems for multi-connected domains, we need integral relations that express Volterra's dislocations in terms of the strain tensor. If there are no dislocation, these relations imply that the displacement field is single valued. These integral relations are well known in the linear elasticity, but they are not applicable to large deformations and rotations, which are typical for thin-walled flexible plates and shells. In what follows, I will present some of the developments made by me and my students and collaborators since 1984. The monograph consists of six chapters. The extension of Weingarten's theorem to large deformations is given in Chap. I. A body is supposed to be multi-connected. Let the field of the finite strain tensor be single-valued and twice continuously differentiable. It is shown that a cut, which makes the domain simply connected and the body free of internal stresses, brings the body to a state such that the locations of the cut faces differ by a finite motion of a rigid body. In the nonlinear theory we distinguish two cases: (a) the body in the reference frame is multi-connected, and (b) the domain, which the body occupies in the deformed state, is multi-connected. In case of (a), the statement of Weingarten's theorem is different from one of (b). The extension of Weingarten's theorem to finite strains enables us to consider Volterra's dislocation (or the isolated defect) in the nonlinear theory. As in the linear theory, an isolated defect of a nonlinearly elastic medium is characterised by two vector parameters, namely Burgers' vector and Frank's vector. These vectors are expressed in terms of the strain tensor; for this, mul-

Introduction

3

tiplicative contour integrals are used. The representations differ from ones of the linear theory, where the ordinary curvilinear integrals appear. It is shown in the nonlinear theory that for two or more defects, the nonuniqueness of the values of the displacement field is of a complex nature. To be more specific, the displacement at a point depends not only on the number of full turns of the integration contour about the lines of defects, but also on the order of the passage. If both Burgers' vector and Frank's vector for every defect equal zero, we obtain a condition of the single-valuedness of the displacements in terms of finite strains. We can use this relation for solving static problems of nonlinear elasticity in a multi-connected domain. Volterra's dislocations in plane nonlinear elasticity are also studied in this chapter. Now we don't use the multiplicative integrals so the expressions for Burgers' and Frank's vectors contain the ordinary contour integrals. We present a proof of Weingarten's theorem for plane finite deformation, which is independent of its three-dimensional modification. A problem of the limit passage from the discrete distribution of dislocations and disclinations to a continuous one is also studied in Chap. I. The noncommutativity of finite rotations implies a complex nature of the multiplicative integrals. We could not find a well-defined expression for the resulting Burgers' and Frank's vectors when disclinations occur. This makes it difficult to formulate the notion of the dehsity of disclinations. This notion is given for the nonlinear theory of plane deformation. If there are no disclinations, i.e. Frank's vector for each defects is equal to zero, then all the dislocations are of the translation type. In this case, Burgers' vector is now expressed in terms of the ordinary curvilinear integral, and the dislocation density is easily defined as a function of the rotor of plastic distortion. In Chap. 2 we formulate boundary value problems for stresses and strains in nonlinearly elastic bodies with prescribed characteristics of isolated or continuously distributed defects. We also present the variational formulation of equilibrium problems for nonlinear bodies with Volterra's dislocation. We propose a half-inverse method to solve static and quasi-static problems of the mechanics of solids. We construct some classes of continuum deformations for which the equilibrium equations are reduced to a system of equations with respect to one spatial variable. These classes are further used to solve nonlinear problems for screw dislocations and wedge disclinations. We also formulate a two-dimensional boundary value problem to which a nonlinear problem for a screw dislocation in a prismatic elastic rod reduces. In Chap. 3 we present a number of exact solutions of some problems for rectilinear dislocations and disclinations; the setting is strictly nonlinear. Considering some solutions of screw dislocation and wedge disclination problems, we show that in the nonlinear theory, the behaviour of the fields of stresses and strains near the defect axis essentially differs from that by the linear theory. In many cases, the nonlinear approach removes the singularities of stresses on the axis of a dislocation or disclination, which are common in the linear theory. At the same time~ it is shown that the behaviour of stresses near the defect axis es-

4

Introduction

sentially depends on a choice of a model for the elastic material. We prove that the energy of a screw dislocation is finite for a wide class of constitutional laws of nonlinearly elastic bodies, whereas the linear theory states that this energy is unbounded. Thus, the nonlinear approach enables us to evaluate the energy of the dislocation kernel and avoid the infinite-energy paradox. The analysis of exact solutions shows that Signorini's perturbation method cannot be applied to evaluate the stresses near the defect axis (Recall that this method is based on the expansion of stresses and displacements in power series with respect to a small parameter).

In Chap. 3, we also prove a theorem on conjugate solutions in the theory of nonlinear elasticity. It asserts roughly the following. Let a homogeneous isotropic body be free of mass load. Then for each solution of equilibrium equations, there corresponds another solution, expressed via the former but related, in general, to another material. The theorem on conjugate solutions can be used to obtain new exact solutions of nonlinear elasticity, in particular, for bodies having dislocations and disclinations. As an example, we find a new exact solution to the problem of the wedge disclination in elastic body. A specific feature of this solution is that the fields of stresses and strains are given as functions of the Euler coordinates, i.e. of coordinates of the deformed state. The problems for screw dislocations and wedge disclinations are relatively simple because they are one-dimensional and thus, can be reduced to ordinary differential equations (in general, nonlinear ones). The problem of rectilinear edge dislocation is more complex since the fields of stresses and strains depend on two coordinates in the plane orthogonal to the dislocation axis. Thus, the problem of an edge dislocation, in the nonlinear setting, is a two-dimensional boundary value problem for a nonlinear system of partial differential equations. Since no approximate approach can give a correct picture of the stresses near the dislocation axis, we tried to find an exact solution. For the model of harmonic material, using the complex potentials of nonlinear plane elasticity, we have found such a solution. The solution crucially defers from the one that the linear theory presents near the dislocation axis. Chap. 3 also deals with singular (discontinuous) solutions to the systems of nonlinear elasticity for bodies with dislocations and disclinations. These solutions describe how a cylindrical cavity appears around the axis of a rectangular screw dislocation or wedge disclination; the radius of the cavity depends on Burgers' or Frank's vectors. It is shown that in certain cases the potential energy of the equilibrium of the body with the cavity is less then the energy of the body without cavity. This demonstrates that in an elastic body the formation of a cavity may be energetically preferable. In Chap. 4 we develop the nonlinear theory of isolated dislocations and disclinations under large deformations of an elastic medium with couple stresses. Given some fields of strain and bending strain tensors, we find the fields of displacements and rotations in a multi-connected domain. We establish the existence of defects of Volterra's type in the nonlinear elastic Cosserat medium. Using the multiplicative contour integral, we express characteristics of Volterra's

Introduction

5

dislocation in terms of the field of strain tensors. As a special case, we consider a plane deformation, for which the characteristics of dislocations and disclinations can be expressed by means of ordinary contour integrals. In the framework of the nonlinear couple theory of elasticity, exact solutions of problems of screw dislocation and wedge disclination are found, and the influence of couple stresses and nonlinearity on the behaviour of the solutions near defect axis is analysed. In the model of Cosserat's continuum, a particle of a medium enjoys all the degrees of freedom which a rigid body has. In a more general model with a micro-structure, a particle enjoys properties of a deformable medium, that is it can undergo a homogeneous deformation. This model can describe the behaviour of micro-nonhomogeneous bodies such as polycristal and granular materials, composites, polymers, etc. A state of a medium with micro-structure is characterised not only by stresses and couple stresses, but also by so-called double stresses which represent the surface density of distribution of force dipoles. To develop the nonlinear theory of dislocations and disclinations in a medium with micro-structure, we introduce tensor measures of deformation which are invariant under rigid motions of the medium and then, for given fields of strain tensors in the multi-connected domain, we find the field of displacements. As for Cosserat's continuum, the field of displacements is expressed in terms of the multiplicative integral. This enables us to establish existence of Volterra's dislocations in nonlinearly elastic bodies with microstructure and to formulate the problem how to find the fields of stresses and strains in microononhomogeneous bodies with given characteristics of Volterra's dislocations. In the theory of defects in solids, the integrals over a closed curve or closed surface, which are independent of the choice of the curve (or the surface), called the invariant integrals, play an important part. In linear elasticity, the first invariant integral was constructed by J.D. Eshelby (1956). In Chap. 4, invariant integrals are constructed for the nonlinear elastic Cosserat's continuum and the nonlinear elastic medium with microstructure. These integrals determine the resistance due to a change in the position of the defect, they can be used to study the motion of defects, dislocation climbs, etc. If the defect is a crack, the invariant integral defines an energy criterion of the development of the crack. In addition, the invariant contour integrals can be used to estimate the order of singularity of stresses and strains on the dislocation axis or at the tips of the crack. Dislocations and disclinations in thin elastic shells are considered in Chap. 5. The main attention is paid to a model of a two-dimensional material continuum described by a nonlinear shell theory of Love's type, no restrictions on the smallness of displacements, rotations or stretching are imposed. Given the components of the tangential and bending deformation, we solve the problem of displacements of the multi-connected middle surface of the shell. This problem serves as the base for the notion of Volterra's dislocation in a two-dimensional nonlinear elastic continuum. We also use this problem when we derive the formulas defininig the parameters of an isolated defect. We formulate a boundary value problem for the strained state of a multi-connected elastic shell, the char-

6

Introduction

acteristics of Volterra's dislocations being given. The theory of dislocations and disclinations in shells can be employed as a theory of defects of two-dimensional systems (surface crystals); it can also be used in the design of multi-connected thin-walled constructions. In some special but important cases, the problem of Volterra's dislocations in shells can be essentially simplified, for instance, it can be reduced to ordinary differential equations. Examples are the problem of screw dislocation in a cylindric shell of arbitrary cross-section and the problem of wedge disclination in a shell of revolution. For a circular cylindric shell, these problems have exact solutions. An exact solution is also found for an equilibrium problem for a plane membrane with disclination. It is shown that after a positive wedge disclination appears, the plane membrane transforms into a conic surface. An equilibrium form of a membrane with a negative disclination is more complex, it is described by the elliptic functions. For nonlinear elastic plates and cylindric shells under heavy bending and stretching, we present contour invariant energy integrals, similar to those of Eshelby and Cherepanov and Rice. For this, we use a new simple form of the equilibrium equations for the nonlinearly elastic Love-type shells. We also pay some attention to a model of an elastic shell of the Cosserat type. We formulate the main statements of the nonlinear theory of isolated dislocations and disclinations in the Cosserat-type shells. In the framework of the geometrically linear theory of Cosserat shell, that is under the assumption that the strains are small, we manage to develop the study of dislocations and disclinations much further as compared to the general nonlinear theory. In particular, in the geometrically linear theory of shells, it is easy to pass from a discrete set of defects to a continuous one and to introduce well-defined notions of the density of dislocations and the density of disclinations. We establish that theses densities are vector-valued, whereas in the three-dimensional problems, they are tensor-valued. The theory of shells with continuously distributed defects can be used for establishing the relations between micro- and macro-properties of two-dimensional physical or biological systems such as surface crystals, films and membranes. The inner (proper) stresses in solids, due to dislocations or other defects, can have a great impact on the critical buckling load and the wave propagation in solids. We discuss these topics in the last chapter. The study is based on equations describing small static and dynamic strains in a preliminary strained medium. The equations are derived by linearization of the general nonlinear equations of statics and dynamics in a neighbourhood of an equilibrium state. The stability of an equilibrium is investigated by static tools, which are valid for conservative loads. Here we solve a problem for plane forms of non-stability of a hollow elastic cylinder with disclination under an uniform hydrostatic load on the external surface. We also investigate the influence of a screw dislocation on the stability of a circular cylinder under an axial compression and on the buckling of a thin annular plate with disclination. Finally, we study the impact

Introduction

7

of defects on the characteristics of elastic waves, an example of a rectilinear screw dislocation in a circular cylinder being considered. I am most grateful to Professor A.A. Vakulenko for his help and support in my work on nonlinear disclinations and dislocations. Our long-time discussions have illuminated many topics of this book. I wish to express gratitude to Professor E. KrSner for his interest in my work. My especial gratitude is given to Dr. S.I. Moiseenko who translated the book into English, and Dr. L.P. Lebedev and Prof. Levendorsky who transformed the translation into somewhat readable. A great assistance in writing the book was rendered by my colleagues, Dr. M.I. Karyakin, Dr. V.A. Eremeyev, and E.S. Nikitin. Thanks to all!

1. Volterra's Dislocations in Elastic Body. Large D e f o r m a t i o n

1.1 T h e Relations of Nonlinear Elasticity For elastic materials the specific free energy function I/Y (per unit volume of reference configuration) is given by I/Y = W(R, grad R, 0),

grad R = r ~Rs, (1.1.1)

R~ = OR/Oq ~,

rk = Or/Oq k

r~ . rk = 5~

Here R is the radius-vector of a particle of the body in the deformed configuration, r is its radius-vector in reference configuration, curvilinear coordinates qS (s = 1,2,3) refer to the reference configuration (Lagrange's coordinates ), r s is a vector basis of the reference configuration, R~ is a vector basis of the deformed configuration, 0 is the temperature and (f~ is Kronecker's symbol. By a . b we denote the inner product of a and b, whereas by ab the tensor (or dyad) product of a by b. The transformation of the reference configuration into the current one is assumed to be sufficiently smooth and one-to-one so that the second order tensor C, the deformation gradient, is nonsingular and thus has a unique inverse. There is another representation for C,

C = (OQM/Oq ~) r~RM,

RM = OR/OQ M,

(1.1.2)

where QM are curvilinear space coordinates (Euler's coordinates) and RM is a vector basis in the space. Let us consider the polar decomposition of the deformation gradient C=U.A=A.V, (1.1.3) where U and V are positive-definite symmetric tensors and A is an orthogonal tensor. It is possible to choose a coordinate system in which det C is positive so that we can consider A to be proper orthogonal. The tensors U and V characterize pure deformation, i.e. the change of length of material lines; they are called the left and right stretch tensors, respectively, and A is called the rotation tensor. The representation (1.1.3) splits the deformation of a neighborhood of the particle into two parts, the pure deformation and the rigid rotation. To

10

1. Volterra's Dislocations in Elastic Body. Large Deformation

express U and V in terms of function R ( r ) which represents deformation of medium it is necessary to extract the square root from positively definite tensors C . C T and c T . c , where C T is the transposition of C. It is difficult, in general, to perform this explicitly so more convenient symmetric tensors A and )~, called the Cauchy's and Almansi's measures of deformation, respectively, are frequently used, A = U 2 = C . C T, A = V -2 = C -1 • C -T, (1.1.4) c - T _ ( c - l ) T, as well as their inverse tensors A -1 = C - T . C -1

A-I --- C T . C .

and

(1.1.5)

A-1 is called Finger's measure of de/ormation. In the bases rs, Rs the following representations hold

A = G,~,~r'~r n,

A -1 = G'~'~rmrn,

)k = gmn R m R n ,

.~- 1 _ gmn a m R n ,

e m n ~-- a m " R n ,

gmn ~- ?Pro " ~'n,

GmkG kn = ~5~,

(1.1.6)

gmkg kn = an,

R m = Gm~Rn. The free energy of the elastic body is invariant under rigid motions of the medium. The invariance of 142 with respect to translations implies its independence from R in (1.1.1). The requirement of independence with respect to rotations of reference system leads to the condition

w ( c . o, e) = w ( c , e) with any orthogonal tensor 0 ( i.e. for O satisfying O T --- o - l ) . A T in (1.1.7), with regard for (1.1.3) we obtain 1/Y =

~/Y(U, e).

(1.1.7) Putting O =

(1.1.8)

The relation (1.1.8) is a consequence of (1.1.7). It is easy to verify that (1.1.8) is sufficient for the free energy to be invariant with respect to rigid motions. As U 2 = A, we can consider ~V to be a function of the Cauchy's strain measure and temperature. In what follows, we will consider isothermal processes of deformation, that is when the temperature is constant, ~ = 00. In these processes, the free energy is usually called the potential energy of deformation and the constant parameter 00 is not shown as its argument. We note the function ]/V is determined by a set of parameters which are constant in deformation processes but depend on a choice of reference configuration; say, tensors characterizing the material anisotropy are of this kind. To

1.1 The Relations of Nonlinear Elasticity

11

derive equilibrium equations and boundary conditions of nonlinear elasticity we will use the Lagrange's variational equation

(5~l,'Ydv: f pok.SRdv+~J'.e51:lda,

(1.1.9)

where v is the body volume in the reference configuration, a is the boundary surface, P0 is the material density in non-deformed state, k is the volume force, J' is the intensity of surface load per unit square in reference configuration, 5 is the symbol of variation. With regard for (1.1.4) we have 5142 = tr ( P . C . (~CT)

=

tr (C T. P . grad 51:t), (1.1.10)

p = p T = 201/Y/0A.

Integrating by parts in (1.1.9) and then applying the divergence theorem we obtain f f (n . D - f ) . 5 R d a - ]v(div D + pok) . ~ R d v = O, d i v D - r s. OD/Oq ~,

(1.1.11)

D = P . C.

n being the unit normal to a. The variational equation (1.1.11) implies the following differential equilibrium equations and stress boundary conditions n . DI~ = f.

div D + pok = 0 in v,

(1.1.12)

Along with the operation of gradient in reference configuration, one uses the space gradient of tensor field • of arbitrary order, Grad • = R MOo/OQ M -- R sOO/Oq s

(1.1.13)

The first equality in (1.1.13) refers to a case when • is a function of Euler coordinates, the second to a case of • to be a function of Lagrange coordinates. The space gradient is given by Grad • = C-1

.

grad ~.

(1.1.14)

The following relations are evident but useful grad r = Grad R = E,

Grad r = C -1,

(1.1.15)

where E is the identity tensor. The equilibrium conditions (1.1.12) can be rewritten in geometry of deformed configuration of elastic body, then the body occupies the volume V with the boundary surface Z and N , the normal to Z. To derive the new conditions

12

1. Volterra's Dislocations in Elastic Body. Large Deformation

we shall use formulae of transformation of elementary volume and elementary oriented area element under deformation

dV = J dr,

N d Z = J C -1 • n da, (1.1.16)

J=

detC=

detU.

Taking into account the law of mass conservation, p d V = p0dv, p being the density in the deformed state, and the Piola's identity div ( J C -T) = 0,

(1.1.17)

on the ground of (1.1.12) and (1.1.14), we obtain Div T + pk = 0 in V,

N . TIE = F,

(1.1.18)

T = j - 1 c T . D = j - 1 c T . P . C = T T,

(1.1.19)

Div T =_ R M . i)~/OQ M - R s . O~/Oq 8.

F d Z = f da,

The relations (1.1.18) are equivalent to a statement that the principal vector and the principal couple of all forces acting in an arbitrary part V, of the body bounded by a surface ~F, are zero:

/v. pk dV + /E N " T d Z = [ ,IV,

pRx

k d V + [_ R x ( N . T ) d Z = 0 d 2;

where the cross denotes the cross product of vectors. The symmetric tensor T is called Cauchy stress tensor, P is Kirchhoff's stress tensor and non-symmetric tensor D is called the Piola's stress tensor. From (1.1.10) it follows another form of constitutional law of elastic body, J T = 2C T. (01/V/0A). C,

D = 014~/0C.

(1.1.20)

We note Cauchy's stress tensor T has physical meaning, it characterizes contact forces of interaction of body parts in the deformed state. T does not depend on the reference configuration unlike P and D which are dependent. That is why the components of Cauchy's stress tensor in the orthonormal basis are called true stresses. In accordance with (1.1.19), for Cauchy's stress tensor written in the deformed basis and Kirchhoff's tensor in the reference configuration, we have

T = tmnP~P~,

P = Jtmnrmr~.

1.1 The Relations of Nonlinear Elasticity

13

From (1.1.6)and (1.1.10)it follows a component representation of constitutional law of elastic material, Jt mn= ~O]/Y/OGmn, (1.1.21) 77=

2, m = n 1, m ~ n .

An elastic material is called isotropic if there is a reference configuration, called the non-distorted state, such that the specific potential energy of deformation and Cauchy's stress tensor are invariant under the transformation C ~ O . C with any orthogonal tensor O. Since under this transformation, the strain measure A is replaced with O. A. O T we conclude ~/Y to be isotropic, i.e. VO = O -T

W ( O . A. O T) = W(A),

(1.1.22)

Putting O = A T in (1.1.22) we get that in the isotropic material the specific potential energy is an isotropic function of Finger's strain m e a s u r e ~-1 ._ A T. A . A. As is clear W can be also considered as an isotropic function of Almansi's strain measure ~ whereas the Cauchy's stress tensor in the isotropic material is an isotropic function of Almansi's measure. Along with the above measures, the Cauchy's strain tensor,

I=~1 (A -

E)

and Almansi's strain tensor,, 1 (E - ,k) are widely used. Representations of the above tensors expressed in terms of displacements u = R - r follow from (1.1.4), (1.1.5), (1.1.15): I = ~1 [grad u + ( grad u) T + ( grad u ) . ( grad u)T], (1.1.23) i = ~1 [Grad u + ( Grad u) T -- ( Grad u ) . ( G r a d lit)T]. If relative stretching of material lines is small we have IIIII << 1 and Ilill << 1,

(1.1.24)

here I[XI[ denotes the norm of second order tensor X, IlXll - v/tr ( x . xT). Under small rotations of material lines in the deformed body the inequality [ I A - E[[ << 1

(1.1.25)

holds. Simultaneous fulfillment of inequalities (1.1.24) and (1.1.25) is equivalent of smallness of components of distortion tensor grad u:

14

1. Volterra's Dislocations in Elastic Body. Large Deformation II grad ull << 1.

(1.1.26)

In this case, called the small deformation, all products of components of distortion tensor can be neglected so it brings us to the linear strain tensor e 1

e = ~ [grad u + ( grad u)T].

I ~ i ~ e,

(1.1.27)

For small strains with respect to reference configuration, which is the unstressed state of the body, difference between tensors of Cauchy, Kirchhoff and Piola is negligible, that is the difference is of the second or higher order of smallness with respect to the distortion tensor. Under assumption of smallness of strains, a study of elastic body reduces to solving linear boundary value problems for partial differential equations, it is the subject of classical linear elasticity. Nonlinear theory of elasticity in more detail can be found in (Novozhilov 1958), (Green et al. 1960), (Truesdell 1977), (Lurie 1980), (Zubov 1982), (Chernykh 1986), (Ciarlet 1988).

1.2 Displacements in Elastic B o d y with Given Finite Strain Tensor Let a body occupy a volume v in a reference configuration and the vector basis r8 and metric coefficients gsk = r~ .rk be given functions of coordinates qk in the same configuration. We require these functions, as well as gmn = rm. r n, to be continuous and single valued as well as their first and second derivatives in V.

Let the field of Cauchy's tensor of finite deformation, I = 1 / 2 ( A - E), be given and twice differentiable, or by (1.1.6) that is the same, the functions G~k (q") be given. Let us consider how to find out the field of medium displacements u if we know a metric G~k of the deformed configuration. Following A.I. Lurie (1980) we will compose a system of equations to determine the deformation gradient C. By the formulae of differentiation of basis vectors,

Or"/Oqm=-7~kr k,

ORk/Oqm= r,~kP~ ,

(1.2.1)

we obtain

OC/Oq k = I I k . C,

YIk - - r k • I-I,

(1.2.2) n = (F~m - 7~m)rkrmrn • Here -y~,~ and Fk~ are the Christoffel's symbols of second kind in the metrics gsk and Gsk, respectively. Components of third order tensor II, called the a]:fine deformation tensor (Lurie 1980), are as follows -

n r

+ v m a k r - VrCkm),

(1.2.3)

1.2 Displacements in Elastic Body with Given Finite Strain Tensor

15

where V k is the symbol of covariant derivative in the non-deformed metric gsk. The integrability conditions of system (1.2.2),

OIIk/Oq ~ - Orl~/Oq k = Hr" IIk - Ilk" I L

(1.2.4)

are equivalent to a condition for Riemann-Christoffel's curvature tensor, corresponding to the metric Gsk, to be zero (we note the curvature tensor for the metric g~k is zero identically). In three-dimensional space the relations (1.2.4) contain six independent equations which constitute the condition of compatibility of strains. We supplement the system (1.2.2) with initial condition for C at a point Ado of the domain v, C (M0) = Co. (1.2.5) Since the tensor field A(r) is given, from the polar decomposition of deformation gradient, C = U . A, U = A 1/2, A being a proper orthogonal tensor, it follows that it is necessary and sufficient to assign only the value of rotation tensor A at Ado. Having found C(r), we can write out the radius-vector R(M) =

d r . C + Ro,

Ro = R (A/~0) •

(1.2.6)

o

Ro being a given vector. In (1.2.6), the integral over a simply connected domain does not depend on the contour of integration if and only if rot C = 0,

rot C = r ~ × OC/Oq ~.

By (1.2.2), this condition is valid since the Christoffel's symbols are symmetric in subscripts. Consider a curve qS = qS(t) which connects a current point M with the starting point A/t0, the latter corresponds to t = to. By (1.2.2), along the curve, we have dq k dC = B ( t ) . C, S(t) = --~-rk. H. (1.2.7) dt Using the multiplicative integral (Gantmakher 1960) we can write down a solution to Cauchy problem (1.2.5), (1.2.7) A

C(.M) =

(E + B d t ) . Co.

(1.2.8)

From the Jacobian identity, det C = a exp

tr B dt

(a = const)

it follows this multiplicative integral is a non-singular tensor, i.e. it has a unique inverse.

16

1. Volterra's Dislocations in Elastic Body. Large Deformation

The multiplicative integral is the limit of integral product /\

i

t

to

[E + B{t) dt] = lim [E + B (Tn) 6tn] ... [E + B (Tl) 6t 1], ~tk-+O

6. tk = tk - tk-l (k = 1,2, ... ,n; t n = t),

(1.2.9)

Tk E (tk-I, tk).

Here the segment (to, t) is divided into n sub-segments (to, t 1 ), (tI, t 2 ),

... ,

(tn-I, t n). We notice some properties of multiplicative integral:

i it i /\

/\

t

to

=

/\

h .

h

(1.2.10)

to '

where the integrands are omitted for brevity. If for any two points t', til E (to, t) the values of B(t) commute mutually then the multiplicative integral can be written in usual terms /\

1:

(E + Bdt) = exp

(1:

(1.2.11)

Bdt) .

The multiplicative integral over qS(t) in (1.2.8) will be called the curvilinear multiplicative integral and be denoted (Zubov 1986a, 1987) as follows /\

i

1M (E + dr . II), dr = rkddt. dqk (E + B dt) = /\

t

to

(1.2.12)

t

Mo

By (1.2.8) and (1.2.12) the solution to (1.2.5), (1.2.7) takes the form 1\

C(M)

=l

M

(E + dr· 11)· Co.

(1.2.13)

Mo

Let C(M o) be given and (1.2.4) holds in the simply connected domain v then, by Cartan's theorem (Cartan 1960), there exists a single-valued solution to (1.2.2) which is continuously differentiable. The single-valuedness means C(M), defined by solving the system (1.2.5), (1.2.7), does not depend on a contour of integration qk(t) connecting M with Mo. From Cartan's theorem it follows

f /\

(E + dr · 11) = E,

(1.2.14)

this is valid for any closed contour in the simply connected domain. Finding the deformation gradient by Cauchy problem (1.2.2), (1.2.5) is an intermediate step to construct the field of displacements of continuous medium.

1.3 Extension of Weingarten's Theorem to Large Deformation

17

Instead of equations (1.2.2), given the strain tensor I, we can form a system to determine the rotation tensor A(r). Substituting C = U - A in (1.2.2), we get OU/Oq k . A + U . OA/Oq k = Hk. U . A .

(1.2.15)

From (1.2.15) we obtain OA/i)q k = V -1. (IIk. V - OV/Oqk) • A .

As (OA/Oqk) • A T are anti-skew tensors, we derive equations to determine A(r), OA/Oq k = (~k" A ,

Ok =

1 ( U _ I . Hk. U _v

-1 . 0V/0q

U - Y I k . U -1 + 0V/0q

•v

(1.2.16) =

If the metric Gmn is given then Ok are uniquely defined. In other form and in other way, these equations were derived in (Shamina 1974). The solution to (1.2.16) can be written with use of multiplicative integral: A

A(M) =

A

(E + • dt). A(M0) =

(E + d r . ffJ). A(A40),

(1.2.17)

o

• = _ , ~ T = Okdqk/dt,

~ = rkOk.

In general, use of (1.2.16) instead of (1.2.2) does not bring us advantages. An exclusion is a case of plane deformation when (I)(t) can be represented in the form (I) = E × kw(t), k being a unit vector. For different t, values of (I) are commutative so the multiplicative integral reduces to usual one. In Sect. 4 we will consider the plane variant in detail.

1.3 E x t e n s i o n of W e i n g a r t e n ' s T h e o r e m to Large Deformation Let a volume v occupied by the elastic body in reference configuration be a multi-connected domain. We get a simply connected domain by cutting v; we assume all the sections (or partitions), Tk, (k = 1 , 2 , . . . , m), to be two-sided oriented surfaces which have no points of intersection. Say, a hollow circular cylinder turns into a simply connected volume by cutting with a half-plane through the cylinder axis. On Fig. 1.1 the cylinder cross-section and the trace of cut plane T are shown. Having turned the domain into simply connected, consider a partition T. An one-sided limit at the partition of values of a function is marked with signs "+" or "-" in accordance with the side of partition from which the limit passage is

18

1. Volterra's Dislocations in Elastic Body. Large Deformation

T t

i i

+

Fig. 1.1.

made. Let us integrate twice the system (1.2.2) over a curve from A/t to Af in ~', first the curve lies on one side of T then on the other. By Cartan's theorem, we have A

c_(N) =f~r (E + d r .

I I ) . C_ (A/l), (1.3.1)

A

c+(N) =f~

(E + d r . H ) . C+ (A/l).

Since the tensor YI is continuous through T, the values of the two multiplicative integrals in (1.3.1) coincide. For any two points A/~, Af on T it follows

c:l(M), c+(M) c:~(N), c+(N) q, =

-

(1.3.2)

where Q is a constant second order tensor. From the polar decomposition C = U . A and continuity U through r we have Q to be proper orthogonal, i.e. Q . Q T = E,

det Q = 1.

Taking into account (1.2.1) and (1.3.2) we have on r

1.3 Extension of Weingarten's Theorem to Large Deformation V R + = V R _ . Q,

V R = ( E - n n ) . grad R,

19 (1.3.3)

n being the normal to ~-. Here V is the nabla operator on T (Zubov 1982). Integrating (1.3.3) on T we get

R+ = R _ . q + b,

(1.3.4)

b being a constant vector. On representation of Q in terms of the vector of finite rotation q (Lurie 1961; Zubov 1982) Q = (4 + q. q)-i [(4 - q. q)E + 2qq - 4E × q],

(1.3.5)

the jump of displacement vector is u+ - u_ =

(1)1(1) l + ~q . q

q×

R _ + -~q × R _

+ b.

(1.3.6)

The constant vectors b and q do not depend on a choice of partition, i.e. their values are constant if we change T for ~-' in such a way that T' does not intersect other partitions and these partitions transform the volume v into a simply connected domain. To show this let us consider a partition T" having common parts with both T and ~-' (cf. fig.l.1). For both ~-' and ~-" formulae (1.3.3), with evident replacements q, b for q', b' and q", b", respectively, hold; moreover, q' and b' are constant on T' whereas q" and b" on T". Since ~- and T" as well as T' and T" have common parts, equalities q = q", b = b", q' = q", b' = b" are valid, therefore we have established q = q', b = b'. Relations (1.3.~), (1.3.6) mean the following. If in the reference configuration a multi-connected body is cut by partitions Tk then, in the deformed configuration, the banks of a cut can be superposed by a rigid motion. To realize the motion, it is necessary to remove or add the material, moreover it can be physically impossible. The relation (1.3.6) is a nonlinear variant Weingarten's theorem (Zubov 1986, 1987) of linear elasticity; we have to mention that another variant of nonlinear theorem proposed as a conjecture by de Wit (1977) is not completely correct. In linear elasticity the displacement jump is u+-u_=q×r+b.

De Wit conjecture (1973a) proposes to change the first term by the formula of finite rotation, taken from kinematics of rigid body, this leads to a relation which differs from (1.3.6) by the continuous location vector from the reference configuration in which the body is simply connected which replaces the correct value of R_ of discontinuous vector R. So de Wit's conjecture does not agree with the correct statement of Weingarten's theorem (1.3.6). Continuity and single-valuedness of Cauchy' strain measure A in the multiconnected volume v do not imply continuity of Almansi's, A, and Finger's, A-1, measures on 7-k since, with regard for (1.1.3)- (1.1.5) and the relation

20

1. Volterra's Dislocations in Elastic Body. Large Deformation A+ = A _ . Q,

(1.3.7)

which is a consequence of (1.3.2), we have +

= QT (A-,) •

_

.Q

(1.3.8)

Therefore the components of Finger's measure are discontinuous through T but the eigenvalues (and so the invariants) of )~-* coincide with eigenvalues of A, i.e. they conserve continuity, but the principal axes of )~-* change their directions by jump. Properties of )~, the Almansi's strain measure, and i = !(E)~) the Almansi strain tensor are similar. 2 Let us consider the case of doubly connected domain that is when the domain is homeomorphic to a torus, in detail. In linear elasticity, if the field of strain tensor is continuous, satisfies the compatibility equations and there is a jump of displacements on the cut one says about the Volterra's dislocation or distortion (Lurie 1970; de Wit 1977). For doubly connected elastic body, we tell there is a Volterra's dislocation (or isolated defect) if the above b and q are not zero simultaneously. Similarly to linear elasticity (de Wit 1977), b and q are called the Burgers' and Frank's vectors, respectively; b and q are uniquely defined by the field of Cauchy strain tensor. Let us first establish some properties of the multiplicative integral over a closed contour that cannot be deformed into a point. Let a partition T convert the doubly connected domain into a simply connected one. Take two points ~4 and Af on T. In the simply connected domain consider the closed contour AJ_A4+Af+Af_A/I_ (cf. Fig.l.2), here j~/l_ refers A/I to one side of T whereas A/l+ to another. By virtue of Cartan's theorem and the properties (1.2.10) of multiplicative integral, we get A

A

/5'

(E+dr.II).

A

A

(E+dr.YI).

(E+dr-H). +

#

( E + d r . I I ) = E. (1.3.9)

From continuity of YI it follows A

A

N+ (E + d r . H) =

(E + d r . H) - K.

(1.3.10)

(E + d r . II). K.

(1.3.11)

+

From (1.3.10) and (1.3.9)we have A

A

~+ (E + d r . H) = K-*.

This means that the values of multiplicative integrals over different closed non-contractible contours are, in general, different but they are similar tensors

1.3 Extension of Weingarten's Theorem to Large Deformation

21

Fig. 1.2. of second order. The contours in question must be oriented in a similar manner; on change of direction of passage along the contour, the multiplicative integral value is substituted by the inverse tensor. We can keep the orientation of contour by the convention that it is through r from the side marked with "-" to another with "+". In general, values of multiplicative integral over closed domain depend on a choice of initial point in the contour, this has to be shown in the notation. For two initial points, A41 and A42, on the contour, we compare the values A

A

/M ( E + d r . H )

and

#~

1

Omitting the integrands we obtain A

This implies

(E+dr.H). 2

A

A

22

1. Volterra's Dislocations in Elastic Body. Large Deformation

A A

A

A

• 2

A

~

1

-1

•

2

(1.3.12)

.

1

1

1

So if we change the initial point of integration then the value of multiplicative integral over closed contour becomes a tensor which is similar to its initial value. Consider two non-contractible-into-point contours lying in a doubly connected domain. Let the contours have a common initial point of integration; then the values of multiplicative integral over these contours are the same.

To show this, it is sufficient to place a partition ~- through the initial point and apply (1.3.10)and (1.3.9). Now suppose there is a closed non-contractible-into-point contour and there is an initial point on the contour such that the multiplicative integral equals the identity tensor, then the value of the multiplicative integral over any noncontractible-into-point contour is the identity tensor. By (1.2.13), (1.3.2) we get

/i A

Q = Co 1.

A

/i A

(E+dr.II).

(E+dr.H).

(E+dr-H).C0.

(1.3.13)

o

It is easy to show that the term on the right of (1.3.13) does not depend on a choice of point A/I and an oriented closed non-contractible-into-point contour through ~/~. Formula (1.3.13) can be simplified if for ~4 we take a point A/10 at which the value of deformation gradient is given as follows A

Q = Co x. f ~

(E + d r . II). C0.

(1.3.14)

o

In terms of Q Frank vector can be expressed by solving equation (1.3.5) ( Zubov 1982) 2 q = (1 + trQ) -1Q×' (1.3.15) where Q× denotes the vector invariant of second order tensor Q, (QkSrkrs)× = QkSrk × rs. From formulae (1.3.13)-(1.3.15) the Frank vector of isolated defect is expressed in terms of given field of Cauchy strain tensor I. As is known (Lurie g where K is the 1961), the module of finite rotation vector q is equal to tg-~, value of rotation angle. The angle of rotation for Frank's vector is determined by (see (Zubov 1982)) A

1

cosK=~(trQ-1)=~

1

[ tr f

(E+dr-H)-I

]

(1.3.16)

1.3 Extension of Weingarten's Theorem to Large Deformation

23

In (1.3.16), we can take as a closed contour for multiplicative integration any oriented contour, which cannot be contracted into a point, with arbitrary initial point of integration. The Burgers' vector b is expressed in terms of strain tensor I by (1.2.6), (1.3.4) as follows A

b=

dr'.

(E + d r . I I ) . A + Ro. (E - Q).

(1.3.17)

o

The closed oriented non-contractible-into-point contour of ordinary integration in (1.3.17) must be through point A/t0; the contour of multiplicative integration in (1.3.17) must not intersect the partition T. If b -~ 0 and Q :~ E then displacements in a doubly connected domain are non-single-valued. From (1.3.2) it follows that the general solution to Cauchy problem (1.2.2), (1.2.5) in doubly connected domain is of the form C=C,.Q

n-m,

(1.3.18)

where C, is the value of deformation gradient given in a simply connected (i.e. cut) domain by (1.2.13) , n is an integer number of turns of the integration contour in positive direction, that is when a partition T is intersected from "+" to "-" bank, m is an integer number of turns of integration in negative direction. Let m - 0. In accordance with (1.2.6),(1.3.18), in a doubly connected domain the general expression o f / / i s R =

E

d r . C , . Qn + R0

0

+ ~(n)[b+ Ro. ( Q - E)]. (E + Q + . . . + Qn-1), 6(n)=~

O, n = 0 l 1, n # 0

Here the contour of ordinary integration must not intersect the partition T. On the ground of (1.3.14), (1.3.17), the condition of single-valuedness of displacements is A

/

(E + d r . H) = E,

/

A

dR'.

(E ÷ d r . H) = 0.

(1.3.19)

o

In (1.3.19) we can take any non-contractible-into-point contour. In an intermediate case when the Frank's vector is zero and Burgers' vector is not zero, the isolated defect is a translational dislocation. Now only the first condition from (1.3.19) holds, it expresses single-valuedness of rotation field A ( R ) in doubly connected volume. In general case, when q -~ 0, b =~ 0, a Volterra's dislocation contains a translational dislocation and disclination, here vector q characterizes the disclination component of Volterra's dislocation.

24

1. Volterra's Dislocations in Elastic Body. Large Deformation

Notice that Frank's vector is determined by the field of Cauchy strain tensor in the body solely that is why the condition q = 0 and the notion of translational dislocation have objective meaning whereas Burgers' vector, when q # 0, does not since (1.3.17) shows that b depends on a choice of the origin for radius-vector R. Dependence of Burgers' vector on the frame origin is clearly seen in formula (1.3.6) too. For a specific choice of the origin it is possible that q -~ 0, b = 0 but on another choice of the the origin this Burgers' vector can be non-zero. So the notion of "pure" disclination, i.e. when q # 0, b = 0, has no objective meaning. In linear elasticity, Frank's vector is expressed in terms of linear strain tensor e (see (1.1.27)) with use of ordinary curvilinear integral over a closed non-contractible-into-point contour (de Wit 1977) as follows q= /

d r . ( rot e) T.

(1.3.20)

Let us show that under condition of smallness of strains formulae (1.3.14), (1.3.15) imply (1.3.20). In the reference configuration we introduce Cartesian coordinates xm for a constant orthonormal basis ira. By (1.1.6), (1.1.27), (1.2.2), (1.2.3) for small strains we have

Gmr ~ (~mr+ 2emr,

~ n r ,~ ~nr ~

2enr,

e=

•

•

emr$m~r,

(1.3.21) n ,~ (Oemn/OX k -~-Oekn/OX m - - O e k m / O X n ) i k i m i n , A

~,~ being Kronecker symbol. Represent the integral J~Mo (E + d r . H) in the form (1.2.9), i.e. as the limit of integral products, and conserve only those terms which are linear with respect to tensor e and its derivatives; then the integral products are substituted by integral sums and the multiplicative integral becomes ordinary, that is A

/

(E+dr.YI)~E+h, (1.3.22)

o

h ~ f dxk (Oemn/Oxk+ Oekn/Oxm- Oekm/Oxn)imQ. Neglecting nonlinear terms, from (1.3.14), (1.3.15), (1.3.22) and C = E + grad u we get A

Q ~

(E + d r . H) ~ E + h,

1

q ~ ~h×.

(1.3.23)

o

Since tensor e is symmetric, on the ground of (1.3.23) we obtain the needed result

q . ~ / d x k (Oenk/OXm)imXin= /

dr • (rot e) T.

1.3 Extension of Weingarten's Theorem to Large Deformation

25

For small strains, transformation of the multiplicative integral (1.3.17) in a similar fashion, gives Burgers' vector formula of linear elasticity, b = f [ e + r × (rote)]. dr.

(1.3.24)

In linear elasticity the jump of distortion tensor grad u - Cpartition of doubly connected domain is given by (de Wit 1977) ( grad u)+ - ( grad u)_ - -E

× q.

E on a

(1.3.25)

Since q - const it follows that the increment of distortion tensor for a passage over closed contour is constant for all non-contractible-into-point contours. As it follows from the relation

(grad u)+ - (grad u)_ = ( Q - E ) + ( g r a d u ) _ . ( Q - E), obtained with use of (1.3.2), in nonlinear elasticity this property fails. Let us consider multi-connected bodies when it is necessary to use more than one partition to get a simply connected domain. A good illustration of multi-connected domain is a three-dimensional ball with drilled pipes as shown in Fig.l.3; the pipes must be either through the ball, that is they initiate and end on the body surface, or closed, i.e. their axes are closed curves inside the ball. In case of n-connected domain we have n - 1 Burgers' vectors and n - 1 Frank's vectors. Each pair of these is expressed in terms of strain tensor by formulae of the type of (1.3.14), (1.3.17), where each closed contour encloses only one pipe, i.e. intersects the only partition. Nature of non-uniqueness of the deformation gradient and rotation fields in multi-connected domain is determined by a number of intersections of multiplicative integration contour with partitions Tm and, besides, by an order of passage about the pipes. For example, in a three-connected domain for deformation gradient there exist the following solutions C = C,.Q1.Q2,

C = C,.Q2.Q~,

C = C,.Q~.Q2,

(1.3.26) C = C . . Q I Q 2 Q1,

c = c . . Q~I. Q1,

etc.

where Qk (k = 1, 2) are determined by A

Qk = C-~(M)" ~M (E + d r . H ) . C ( M ) and the closed contour must intersect a unique partition Tk, i.e. it must circumvent a unique pipe. For non-commutativity of orthogonal tensors Q1 and Q2, that is a consequence of non-commutativity of finite rotations, all expressions in (1.3.26) are different. Such character of multi-valuedness of A(r) in a multi-connected body does not appear in linear theory. The multi-valuedness of rotation field determines complex nature of nonuniqueness of displacements in multi-connected domain.

26

1. Volterra's Dislocations in Elastic Body. Large Deformation

Fig. 1.3.

Consider a sequence of multi-connected domains with decreasing pipe diameters; the limit passage brings us to a simply connected domain. But in general, on the curves which are limit position of pipes, continuity conditions of strain tensor and its derivatives will fail. In such a case, Cartan's theorem on uniqueness of solution of system (1.2.2) in a simply connected domain is not applicable and the fields of rotations and displacements are not single-valued, moreover, character of non-uniqueness is similar to one in multi-connected domain. These singular curves are called curves of dislocation and (or) disclination. In physics of solids, consideration of Volterra's dislocations as singular curves, that is of linear defects characterized by continuity and single-valuedness of strain tensor and its derivatives in the volume except the pipes of fairy small diameter about lines of dislocations, is of major interest. In physics of strength and plasticity, the theory of translational defects, when Frank's vectors are zero, is wide-spread but the rotation defects, disclinations, really exist in solids, liquid crystal and other structures (Vladimirov et al. 1986). For last decade disclination ideas were widely used in some areas of physics (Vladimirov et al. 1986); but in physics of disclinations only the relations of linear theory of Volterra dislocations were used, it is not correct since

1.3 Extension of Weingarten's Theorem to Large Deformation

27

the module of Frank's vector (the power of disclination) is not always small and thus it is more adequate to use the above non-linear relations. Assume that a multi-connected volume occupied by body in deformed state is given. Sometimes we need to determine the reference configuration of body when the field of Almansi strain tensor is determined as a function of Euler coordinates. From physical point of view, the case is even of more importance than the above since it is natural to consider defects and inner stresses which they generate in the actual configuration. This case can be considered in a manner we have used, the only difference is the reference and deformed configurations must interchange their roles. Let QM be spatial (Euler) coordinates, RM = OR/OQ u and r M = Or/OQ u vector bases in the space and the reference configuration, respectively, they are associated with the coordinates QM. The bases which are reciprocal to RM and r M are denoted by R N, r N, respectively. Now the volume V occupied by the body in deformed state is given as well as quantities RM and GUN -- l:lM. l:lN which are some given functions of Euler coordinates. We discuss briefly how to find the displacement field u(QN), or equivalently, r ( R ) , in a multi-connected volume V with given functions gMN(Q P) = r M ' r N being twice differentiable. First we compose a system to determine tensor C -1 = Grad r (Grad = RMO/OQ M) which is similar to (1.2.2). Having found C -1, we determine r(Q M) by quadratures. The limit values of deformation gradient on a partition of multi-connected domain are now related by C+ = Q*. C_,

Q , . Q,T = E,

(1.3.27)

and a jump of displacement vector is

u+-u_=-

( l + 4 q1* . q *

q* x ( r _ + q * x r _ ) + b ,

Q,T _ (4 + q*. q*)-1[(4 -- q*. q*)E + 2q'q* - 4E x q*] By hypothesis, Almansi and Finger strain measures, ,,k - gMNI~MI~ N and _ gMNI~MRN, are continuous and single-valued in the multi-connected domain V whereas tensors A and I change their principal directions by jump on intersections with the partitions which convert the volume into simply connected. The relation for the jump of displacement vector given by de Wit (1973a) is not applicable in this case too. In (de Wit, 1973a), an expression of the type (1.3.28) contains the continuous displacement vector R instead of the r_ of discontinuous vector r, that means that the jump of displacements is expressed in terms of continuous position vector in the configuration in which the body is multi-connected. This contradicts to correct statement (1.3.28) of Weingarten' s theorem, in which there participates the discontinuous position vector of the configuration which appears after relaxation of internal stresses in the cut body. .~-1

28

1. Volterra's Dislocations in Elastic Body. Large Deformation

1.4 Multi-valued Displacements Dislocations in Nonlinear Plane

and Volterra Elasticity

The plane deformation of continuum is described by the relations X l -- X l (Xl, x2),

X2 --- X 2 ( X l , x2),

X3 = x3,

(1.4.1)

where Xk and Xk are Cartesian coordinates in the reference and deformed body states, respectively. The frame unit vectors are denoted by ek (k - 1, 2, 3). In what follows, it is convenient to use complex coordinates (~, ~) and (z, 2) = Xl "~-ix2,

~ = Xl -- ixg, (1.4.2)

z = X1 + iX2,

-- X1

-

-

iX2.

We shall also use the following notations

~1 = ~,

~2 = ~,

~3 = z3,

z 1 = z,

z 2 = e,

z 3 = X3.

The main and reciprocal vector bases, fk and fn respectively, which are associated with coordinates (1.4.2), are defined by the expressions (Zubov 1983)

0 f l = -f 2 -

~ 0 (X~em),

~lf~ - ~1 ( e l - ie2),

f ~ = f ~ = e3,

f~

fk" f~ = 5k - 7~ = 2 f 2 = el-~-ie2,

1

(1.4.3)

e~ = f ~ + f ~ = -~(f~ + f~),

i f2 _ f l ) .

e2=i(fl-f2)=~(

The plane deformation (1.4.1) may be specified by a complex-valued function

z = z(¢, ~).

(1.4.4)

The deformation gradient corresponding to the transformation (1.4.4) has the form ~Z k C

__.

o~fnfk Oz f~

O~

Oz

0--~

u~

a~

O~f~

(1.4.5)

Under plane deformation, material fibers rotate about the axis e3, so that the rotation tensor takes a representation: A = (E - e3e3)cos X + (ele2 - e2el) sin X + e3e3. For (1.4.3), we obtain

1.4 Multi-valued Displacements and Volterra Dislocations

13]a.

A = e i X f l f I + e-iX~2~ J J2+

29 (1.4.6)

From (1.4.5) we find the Cauchy strain measure

0z02.

, I = A~= b--db7 ~ -

0z02.

(1.4.7)

0~0~'

Oz O~

A~ = A 1 = 0~ 0~" Complex components A~ of the tensor A are expressed in terms of its Cartesian components, G ~ = e~ • A.e~, by

AI=~1(Gll

+

A~ = ~1 (G11 -- 2iG12-- G22).

G22),

Let us set up the problem of finding the plane field of displacements when A(Xl,X2) is given. This problem is equivalent to one on finding the function z((, ~) from the nonlinear equation system (1.4.7) with given functions A~(¢, ~) which are continuously twice differentiable. For plane strain, the compatibility equations with respect to G,~ reduce to a single relation that means that the component R1212 of the Riemann-Christoffel tensor in the metric G ~ is zero,

(G1'G22 -G122)

( 02G22 02G12 02Gll ) Ox2 20XlOqX2+ 0x22

( 0G12 OG2~. 10qGllOG22 20x20x2

1 (0G22) 2)

+Gl1\ ~ 0~ +G22

2\0xl

OqGlloqG12 loqG11oqG22 1 (oqG11)2) OXl OqZ2 2 0Xl 0Xl 2 ~ (~X2 0612 0G12

-G12 20qXl Ox2

(1.4.8)

0612 oqG22 OqGll0G12 OXl OXl OX2 0X2

1 06110G22 Ox2

20xl

1 0Gll 0G22~ = O. 2 C~X2 0Xl

]

In complex variables, the relation is written as 1 2 ( 02A~ 1= -- A2A 1) 20(0~

(AIA2

(0AI 0A~ + hi

02A2

(~2

1 0A~ 0A~

02A1 0~ 2 )

1 (c3A~~ 2~

]

0Al 0a~ o~ o¢ 2o~ o~ 2 --~-] 0AI 0AI 0AI 0A~ 0AI 0A~ o~ o~ o~ o~ 1 OA~ OA~

1 0A21 _ 0A~) =0.

2 o~ o~

2 o~ o~

(1.4.9)

30

1. Volterra's Dislocations in Elastic Body. Large Deformation

For plane deformation, it is possible to represent the solution of the above problem in ordinary quadratures without use of the multiplicative integral. The polar decomposition of deformation gradient is U = h ~/2 = U~f~J'~ + f3f3

C = U . A,

(a, e = 1, 2).

(1.4.10)

Substituting (1.4.6)into (1.4.10), we obtain C - U l e i x f l f l -I- U 2 e - i x f l f 2 -F U l e i x f 2 f l + U2e-iXf2f2 -F f 3 f 3 .

(1.4.11)

On comparison of expressions (1.4.5) and (1.4.11), we get Oz • 0--~ = U~e'X"

0Z __ Ulei X

0-~

'

(1.4.12)

Using the formula (2.48) from (Zubov 1982, Chapter III), we can obtain the following explicit expressions of the stretch tensor components in terms of the components of A • g 1 = g 2 = l I v /2

AiA 2 _ A1A 2 1 2 + 2A]

= ~IV/ 2v/GllG22 - G12G21 + Gll + G22, 1

U12 = U 1 = ~ (U 1)

-1

1

A~ - ~ (U 1)

-1

(1.4.13)

(Gll - G22 - 2iG12).

If the field of rotation angles of the principal axes of deformation X((, ~) is known, then by (1.4.12) the function z((, () is determined by integration z = / eix (U1 d( + U1 d~). It is seen the integrability condition of the system (1.4.12) with respect to z((, ~) is C3 (U:eix) = ~0 (U~eiX) • (1.4.14)

This and its complex conjugate relation are the equations to determine the rotation field, X((, (); indeed, they can be transformed as follows OX o~ = ~(¢' ¢)'

OX o~ = ,7(¢, ~),

ou~ ~(~, ~) -- - i zx -~

u1

0~

ou 1

OU1)

,

(1.4.15)

A = ~~-Ul1~.

Using the representations (1.4.13), we can verify that the integrability condition for the system (1.4.15),

1.4 Multi-valued Displacements and Volterra Dislocations 0~ o~

31

0O o¢

coincides with the compatibility strain equation (1.4.9), to which the components of tensor A satisfy. If a value of the rotation angle, Xo = X:(~o,~o) is given at a point Ado with complex coordinates ~o, ~o, then the rotation field in a simply connected domain is uniquely determined by the system (1.4.15). Having found X(~, ~), we can uniquely evaluate the function z(~, ~) by integrating the system (1.4.12) with a given value z0 = z(~o, ~0). Let us now consider the case of doubly connected domain. Suppose that the integration contour consists of a curve connecting points Ado, M and nonintersecting the partition ~-, and a closed, non-contractible-into-point contour which revolves n times (n full turns) in positive direction. In a doubly connected domain the solution of equations (1.4.15) is multi-valued and takes the form X, = Xo +

X = X, + n K ,

(rl d~ + 0 d~), (1.4.16)

0 P

X(~o, ~o),

X0

K = j~ (r/d( + gld~),

where, (o , ~o are the complex coordinates of the point Ado . From (1.4.12), (1.4.16), it follows a multi-valued expression for the quantity z which describes the location of continuum particles in the deformed state z = Zo + einK

e ix" (U 1 d~ + U1 d~) 0

+ / e ix" (U11d~ + U~ d~) (1 + eiK + . . . + ei(n-1)K),

(1.4.17)

z0 = z(¢o, ¢o). Cutting the domain by T we transform it into simply connected so that single-valuedness of functions X and z is achieved but the limit values of these functions on the cut sides do not coincide. From (1.4.16), (1.4.17), it follows that the limit values on the opposite banks of the cut are related by X+ - X- = K.

(1.4.18)

z+ = eiKz_ +/3.

(1.4.19)

K = ~ (rl d~ + J 3 =

/

exp i

d~),

(1.4.20)

]

(r/d( + ~d~) + iXo (U 1 d(' + U~ d~') + Zo (1 - eiK). 0

In (1.4.20) the passage along the closed contour is from the cut side signed with "-" to another side with "+". Besides the contour must start and end at the point 34o.

32

1. Volterra's Dislocations in Elastic Body. Large Deformation

The formula (1.~. 19) shows us that in the deformed state the position of the two banks of the cut border differ by a finite plane motion of rigid body so that the real constant K is an angle of finite rotation and the complex constant defines the relative translational displacement of the cut banks.

Thus, the relations (1.4.18), (1.4.19) present the Weingarten's theorem of plane nonlinear theory of elasticity (Zubov & Karyakin 1987). We notice that this proof of Weingarten's theorem, which uses only the operation of ordinary integration over a curve, is independent of the method of Sect.l.3. The relation (1.4.19) implies the formula for the jump of displacement vector u: u+--u_ =

( 1 ) - 1 l +-~q.q

u = R - r,

q×

(

R = X~e~,

K q = 2 tg --x-ea, Z

1

l t _ +-~q x R _

)

+b,

r = z~e~,

(1.4.21)

b = Re ~el + Im ~e2.

By virtue of continuity of the integrand in a doubly connected domain, the first integral in (1.4.20) does not depend on the choice of closed contour which revolts about the hole only one time. It is not so for the integral /

dz = / e i:~* (U1 de + U1 d~),

(1.4.22)

since the limiting values (with K # 0) at opposite banks of the cut of its integrand are different. Hence the integral in (1.4.22), like a multiplicative integral over closed contour, depends on the choice of initial integration point, but its value is the same for all the closed curves ending at the initial point and enclosing the hole. By this reason, if the contour is through an arbitrary point 3/11, the multi-valued expression for z is more complex in comparison with (1.4.17): Z =

dz

Zo "Jr- e i n k

+ (1 +

e iK + . . .

-t-- e i ( n - 1 ) K )

dz

0

1

+ (eiK- i)

(1.4.23)

dz. I

Equation (1.4.23) implies a more general than (1.4.20) representation of Burgers' vector = Z0(1 -- e iK) -1-

d z -~- (e iK 1

1)

fo

dz.

(1.4.24)

1

It is easy to show that the expression (1.4.24) does not dependent on the choice of point Ad 1, i.e. it has the same value for all non-contractible-into-point contours. Indeed, let us consider a contour which starts and ends at a point Ad2. Cut the domain through 3/11 and Ad2 to make it simply connected. Consider the closed contour consisting of the contour 3/t[.M +, the line segment Ad+Ad +,

1.4 Multi-valued Displacements and Volterra Dislocations

33

M~

Fig. 1.4. the contour 3 d ~ 3 4 2, and the line segment 3d234 i- as shown in Fig.l.4. This contour lies within a simply connected region, therefore, ~ dz = 0 for this contour. Represent this integral as a sum of integrals over the above parts of the contour; with regard for continuity of U1, U1 in the doubly connected domain and relations (1.4.18) for values of X, at opposite cut banks, we obtain dz JM~

dz + eiK JM2

dz ~

dz = 0.

(1.4.25)

1

Summing the expressions (1.4.24) and (1.4.25), we get ~O

= z0(1 - e iK) -t-/3,12 dz + (e iK - 1)

fM

dz,

2

as was to be proved. For a plane region having more than one hole, the multi-valued solution to the system (1.4.7) is of more complex form. We consider arising peculiarities for a 3-connected domain shown in Fig.l.5. An arbitrary integration contour from point 3,10 to a current point A4 is homotopically equivalent to the following sequence of contours: the closed contour L1 revolts m l times in the positive

34

1. Volterra's Dislocations in Elastic Body. Large Deformation

M~

L1 L2

!

~

r2

M

M

Fig. 1.5.

direction; the closed contour L2 revolts nl times in the positive direction; then the contour L1 with m2 turns and L2 with n2 turns, etc., at last, the contours L1 and L2 with ms and ns turns and besides, the curve A40A4 lying within the cut domain. Values of mj, nj may also be negative and besides, for ms, ns, they may be zero. Conserving the earlier notation for single-valued (in the domain with cuts) branch of the integral (1.4.16), for three-connected domain we obtain the following multi-valued solution of the system (1.4.7):

1.5 Total Burgers and Frank Vectors

35

s

(Klm3 + K2nj),

X = X. + E j=l

KI=~

dx,

K 2 = j ~ L dx,

1

z = Zo + (1 +

+

2

e iK1 + . . .

+

e l(ml-1)K1)

+ ( e imlK1 -

1)

/2 o

[

(1 + e

+

...

•

dz fM~+ d.hd 1

dz

(1.4.26)

I

+

3,t + dz

J~-

+ ( e in~K~-l) + exp i

dz + . . .

(Klmj + K2nj) j=l

]/;

dz.

0

This implies the formulae for the jump of function z on cuts z(J ~+) = eiK~z(fi/[1-) + ~1, z(.£4 +) -- eiK2z(.M2) + ~'2, j31 = z0(1 - e i g l ) +

~ M~dz + (eiK1- 1) /~~0 dz,

]32 = z0(1 - e ig2) +

dz + (e iK~- 1) Jj~

dz. ~"

It is obvious that the above relations can be extended to the case of generalized plane deformation almost without changes; the latter is described by relations which are slightly different of (1.4.1): X1 = Xz(xl, x2),

X2 = X2(xl, x2),

X3 = Ax3,

A = const.

This deformation is realized in a fairy long prismatic body under the load being independent of the coordinate x3. The dislocation lines are parallel to the prism axis, i.e., to the axis x3 .

1.5 Total Burgers and Frank Vectors for a S y s t e m of Isolated Defects. Density of Continuously Distributed Dislocations and Disclinations The Burgers and Frank vectors introduced in Sect.l.3 are quantitative characteristics of the Volterra dislocation (an isolated defect). Under some assump-

36

I. Volterra's Dislocations in Elastic Body. Large Deformation

tions, similar quantitative characteristics may be introduced for a system of few isolated defects in total. First we consider a case when all the defects in a body are pure translational dislocations, i.e., the Frank vector of each defect is zero. The deformation gradient C now is a single-valued function of coordinates in a multiply connected domain occupied by an elastic body in the reference configuration. The problem to locate a position of body particle in the deformed state, i.e. to find R(r), by a given field of C(r) reduces to finding a vector-function by its total differential:

d R = d r . C.

(1.5.1)

The necessary and sufficient condition for integrability of equation (1.5.1) is rot C = 0.

(1.5.2)

On any cut of a set of cuts which converts the multiply connected domain into simply connected, the jump in It(r), determined by equation (1.5.1), is given by R+ - R_ = bk = constant where the Burgers' vector b of kth dislocation is expressed as an ordinary curvilinear integral over any simple closed contour 7k that encloses the axis of the kth defect solely,

bk--~k

dr.C.

(1.5.3)

Consider a closed contour 70 which encloses axes of two dislocations at once as shown in Fig.l.6 and draw a curve V through the points M0 and M1 • The integrals over the curve in two opposite directions are mutually canceled thus it holds the following equality o

,1

4,

A relation, similar to (1.5.4), is valid for any number of translational defects: N

k=l k

N

k=l

where the contour 70 encloses the axes of all N dislocations. Thus, in view of (1.5.5), the vector b0 = J~o d r . C may be called the total Burgers vector of a N-dislocations system. Following Nye (1953), grSner (1960), and Vakulenko (1991), we consider continuous distribution of dislocations using formula (1.5.5); the deformation gradient due to a continuous field of dislocations will be called the plastic distortion and denoted by C (p) . By the Stokes formula, we have

bO=~o

dr'C(P)=~on"

rotC(P)da

(1.5.6)

1.5 Total Burgers and Frank Vectors

_

_

37

|ll

Mo

Fig. 1.6.

By (1.5.6), the flux of the tensor c~ = rot C (p) through any surface a gives the total Burgers vector of dislocations intersecting the surface; that is why c~ is called the dislocation density tensor. The total (combined) distortion C is represented as a product of plastic and elastic distortions (KrSner 1960; Vakulenko 1991), C = C (p). C (e),

(1.5.7)

with the elastic distortion being related to stresses by constitutive equations of nonlinear elasticity. The compatibility equation (1.5.2) and the definition of dislocation density imply the following equations with respect to C (p) and c(e): r k X C (p). 0 c ( e )

Oqk . (C (e))

-1

+ ol -- O,

(1.5.8)

rot C (p) - c~ = 0. Combining (1.5.8) with the equilibrium equations and constitutive relations from the next Sect.2.1, we shall obtain the total system of equations to find out

38

1. Volterra's Dislocations in Elastic Body. Large Deformation

C(P)(q 8) and c(e)(q s) when a dislocation density c~(q8) is given. If the Frank vectors of isolated defects are not zero, finding the total Frank vector for a system of defects meets serious difficulties. Indeed, for a case of two Volterra dislocations (shown in Fig.l.6); Q1 denotes the orthogonal Frank's tensor for the first defect and Q2 for the second. By (1.3.14), we have A

Q1 = Co 1" l (E + d r . H). Co, J.y1 ^

(1.5.9)

Q2 = Co 1" i (E + d r . II). Co. Jr 2 Here the contour 71 consists of the curve 7 and the part of 70 between the points A41 and A4o, whereas the contour 72 consists of the part of 70 between A4o and A41 and the curve 7. The passage along 71 and 72 begins at A4o and is counter-clockwise. We now consider a multiplicative integral of the form of (1.3.14) over a closed contour 7o with the initial point A4o. By properties of multiplicative integrals, we have A

^

A

(~)-1

A

~

1

~

~ 2

~ 1

(1.5.10) 2

From (1.5.9), (1.5.10), it follows A

C°1" ~o (E + d r . H). Co = QI" Q2.

(1.5.11)

The left-hand side of (1.5.11), being a product of orthogonal tensors, is also an orthogonal tensor; we denote it by Q0, A

Qo = Co 1" ~o (E + d r . H)-Co.

(1.5.12)

From (1.5.11), (1.5.12), we have Qo = Q I Q2.

(1.5.13)

The tensor Q0 should be called the total Frank tensor for a system of two defects, since the rotation described by this tensor is, in view of (1.5.13), the superposition of the rotations due to tensors Q2 and Q1. Unfortunately, the attempts to extend the formula (1.5.13) to the case of more than two defects have failed. The matter is that a multiplicative integral

1.5 Total Burgers and Frank Vectors

39

over a contour, as shown in Sect.l.3, depends on the choice of initial point. Formula (1.5.13) was derived under the assumption that all the three contours, 71, 72 and 70, intersect at a common point AA0 that is taken as initial. But in case of three and more defects, in general, there is no such a common point so the multiplicative integral over a contour enclosing all the defects cannot be a product of integrals over contours each of which encloses a single defect. Thus, in general, a multiplicative integral of the form (1.3.14) over simple closed contour enclosing all the axes of the system of linear defects cannot be the total Frank tensor for this system. Moreover, a multiplicative integral over a closed contour cannot be transformed into an integral over a surface whose boundary is this contour. So there are objective obstacles to extend formulae of discrete distribution of defects to continuous ones in nonlinear theory (i.e. when the Frank's vectors are not small) and thus to introduce the notion of disclination density of physical meaning. A reason of these troubles is non-commutativity of finite rotations. In linear theory of disclinations, when the Frank's vector of an isolated linear defect is expressed by ordinary curvilinear integral, these difficulties are absent and the disclination-density tensor can be defined by the way due to de Wit (1977). The above troubles disappear in a particular case of nonlinear theory, namely for plane deformation of elastic body. In this case, the directions of Frank vectors of all defects are the same and each defect is uniquely characterized by a real number K (the Frank's angle) and a two-dimensional Burgers' vector (See Sect.l.4). From (1.4.20) it follows that the total Frank angle for a system of N Volterra dislocations is N

Ko=

z/d( + ~ d ( = 0 N

K~ s=l

(1.5.14)

=

s=l

s

Here, % is a contour enclosing only the sth defect, 7o is a contour enclosing all the N defects. The complex valued function z/((, ~) is expressed in terms of the field of strain tensor by equation (1.4.15). Proceeding to continuous distribution of disclinations in (1.5.14), let us transform the contour integral using the Green's formula. Replacing ~ by r/(p), where ~(P) is expressed in terms of plastic strains by (1.4.15), we obtain

Ko= Lo [2i/Z0v(P) (~(P) \ ~ 0( ) ]

da.

(1.5.15)

The integrand in (1.5.15) can be called the disclination density for plane

nonlinear elasticity.

2. S t r e s s e d S t a t e of N o n l i n e a r l y Elastic B o d i e s w i t h D i s l o c a t i o n s and D i s c l i n a t i o n s

Finding the Stressed State of Elastic Body with Given Characteristics of Isolated Defect; Set u p o f the Problem 2.1

Given the Burgers' and Frank's vectors, bk and qk, let us consider the problem of finding displacements and stresses in multiply connected nonlinear elastic body with Volterra dislocations. There are two lines of attack on the problem. On the first way, the displacement vector components are unknown variables satisfying the nonlinear equilibrium equations and force conditions on the boundary surface of multiply connected volume v . We are setting up the boundary value problem in a simply connected domain that is obtained by making cuts in v . On the cut surfaces, the displacements must satisfy the jump conditions (1.3.6). It is clear that we can find the vector/~ instead of the displacement vector u . For unique solvability of the problem in question, we need to determine values of the displacement vector and the rotation tensor at some point of the body. From continuity at the cuts of Cauchy strain measure, A , it follows continuity of Kirchhoff stress tensor, P . Thus by equations (1.1.11), (1.3.2), it follows a relation for the jump on a cut in Piola stress vector, n . D , n.D+

=n.D_.Qk,

(2.1.1)

where n is the normal vector to the cut surface Tk • On the second way, the boundary value problem is formulated in the multiply connected domain; now the components of Cauchy strain measure, G sk , which are determined by the compatibility conditions (1.2.4) and the equilibrium equations (1.1.12)1, are unknowns. If there are no body forces these equations are easily reduced to a form containing only unknown G sk • Indeed, multiply equation (1.1.12)1 by C , with regard for D = P . C , we obtain [div P . C)]. C -~ -- 0. By equation (1.2.2), we have

(2.1.2)

42

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations div P + r k. P . Hk = 0.

(2.1.3)

Since the tensors Hk are expressed in terms of Gmn and the Kirchhoff stress tensor for any material ( and thus for anisotropic elastic material) is expressed in terms of Cauchy strain measure, we can see that the only unknown variables in equations (2.1.3) are Gm~. Assume that the surface load is of tracking type, i.e. the vector f in (1.1.12)2 is of the form

f(r) = fo(r, A(r)) • C(r).

(2.1.4)

In view of equation (2.1.4), the boundary conditions (1.1.12) are n . P = f0,

(2.1.5)

so they are also expressed completely in terms of Gmn • Combining the integral relations (1.3.13), (1.3.17), which express the defect parameters, bk, qk , in terms of the tensor field A , with equations (1.2.4), (2.1.3) and the boundary conditions (2.1.5), we get the final set up of the problem of equilibrium of a body with isolated defects in Cauchy strain measure components. Let us consider the case when the Frank's tensor of each defect is zero; all the defects are translational dislocations. Now the strain gradient C is a singlevalued function in a multiply connected domain, and to specify the strained state, one can formulate the boundary value problem with the components of C taken as unknowns. The compatibility equations for the deformation gradient consist of 9 linear equations, rot C -- 0

(2.1.6)

Combining these with the equilibrium equations and boundary conditions (1.1.12) and, besides, with the integral relations

~ dr . C = bk,

(2.1.7)

k

we get the set up of the problem on translational dislocations in terms of the deformation gradient. In equations (2.1.7) % is a contour enclosing the line of kth dislocation with Burgers' vector bk . Dependence of intensities of loads k and f on the deformation gradient is arbitrary. In particular, the vectors k and f may be some given functions of Lagrangian coordinates (the dead load). Under dead load, the equilibrium equations (1.1.12)1 is the identity if D = rot ,I, + D*,

(2.1.8)

where ,I~ is a twice-differentiable tensor of second order, D* is a particular solution to the equations (1.1.12)1. As well as for the given reference configuration, we can consider a case of given deformed configuration with prescribed characteristics of defect and

2.1 Finding the Stressed State

43

compose equations with respect to components of Almansi strain tensor. Such a set up, when Eulerian coordinates are independent variables, is possible only for isotropic material because the Cauchy stress tensor is fully determined by Almansi strain tensor only in isotropic elastic solids, that is

tMN= FMN(gps) ,

tMN= nM. T. R N.

(2.1.9)

The functions gps(Q,M) must satisfy the compatibility equations, the equilibrium equations, DivT - 0,

(2.1.10)

the boundary conditions N . T = F and some relations which are similar to the equations (1.3.13), (1.3.17). If equations (2.1.9) can be uniquely inverted then the problem of Volterra dislocations can be stated in stresses. Now the equilibrium equations (2.1.10) may be identically satisfied with the tensor of stress functions, O = O T, Rot O = R M × 0 0 / i ) Q M.

T = Rot ( Rot o)T,

The tensor O must satisfy the compatibility equations, boundary conditions, and integral relations determining Burgers' and Frank's vectors of given defects. Consider now formulation of the system of equilibrium equations for a nonlinearly elastic body with continuously distributed field of translational dislocations when the density c~ is given in advance. Note that in the constitutive relation (1.1.20) for the Cauchy stress tensor, by (Vakulenko 1991), the deformation gradient must be replaced by the elastic distortion tensor, C (e) • det

C (e)) T

0A(e) ,

In view of equations takes the form

,

(2.1.11)

A(e) - C(e) • (c(e)) T

(1.1.19), (1.5.7), and (2.1.11),

0A(e)

the Piola stress tensor

•

On substituting equation (2.1.12) into equation (1.1.12)1, we obtain a vector equation that, being combined with equations (1.5.8), forms the full system for the unknowns C (p) and C (e).

44

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

2.2 Variational F o r m u l a t i o n of t h e P r o b l e m of Volterra D i s l o c a t i o n s in N o n l i n e a r Elasticity Let external forces be conservative; we apply Lagrange variational principle to an elastic body with Volterra dislocations, that is the elementary work of external forces in equation (1.1.9) is the variation of a functional ~, called the potential of external loads, with respect to the displacement vector,

~ pk . ~Rdv + ~ f . SRda = 5E.

(2.2.1)

Transform the multiply connected volume v into a simply connected one making a sufficient number of partitions and consider the potential energy functional for the elastic body, A = .~,,YYdv - t~.

(2.2.2)

The functional is defined on the set of displacement fields, which are twice differentiable in the simply connected domain and satisfy the jump conditions (1.3.6) with given parameters bk, qk on the cuts 7k. Since 5u = 5R we can consider the vector field R , specifying the location of body particles in the deformed configuration, as an argument of the functional ,4 instead of the displacement field, u. The jump conditions for the vector R are equations (1.3.4). With regard for (1.1.20), (2.2.1), we obtain the variation of functional A (f,4= jfvtr (D T. grad~R) d v - ~ p k . ~ R d v - ~ f . S R d a . Using the divergence theorem and the equation (1.3.4) in the stationary condition, 5A = 0 we obtain

- ~(div D + pk). 3Rdv + f ( n . D - f ) . ~ R d v

(2.2.3) +E/(nk" k

D _ . ~R_ - nk. D+. QT. ~R_)da = O.

k

Here, nk is the unit normal to the partition Tk which directs from the partition side marked by "-" to the "+"side. From the variational equation (2.2.3) it follows the differential equilibrium equations (1.1.12)I, the boundary conditions n. D = f on the body surface, and the dynamic conjugation conditions (2.1.1) on the cuts which transform the domain to be simply connected. Thus, the Euler equations for the potential energy functional are the equilibrium equations in displacements and the natural boundary conditions are the force conditions and dynamic conjugation conditions on the cuts. Now let us turn to the variational formulation of the problem of Volterra dislocations which is based on the concept of complementary energy. The specific

2.2 Variational Formulation

45

complementary energy of elastic body is introduced (Zubov 1970) as a function of Piola stress tensor. This function relates with the function of specific potential energy, )/V(C), by Legendre transformation

I;(D) = tr (D. CT(D)) - W(D),

C = 01)/0D.

(2.2.4)

To construct the function ~;(D), we need to invert the relationship for Piola stress tensor D(C) with respect to the deformation gradient, i.e., to express the deformation gradient as a function of Piola stress tensor. For an isotropic elastic body, the problem of representation of the deformation gradient in terms of Piola stress tensor has been fully studied in (Zubov 1976). This paper states in particular that there is a unique inverse of the relation D(C) if the rotation angles of material fibers are not too large. As to anisotropic material, the above problem remains to be unsolved. In this case, one can use a weaker formulation (de Veubeke 1972; Christoffersen 1973) of the complementary energy principle, in which the specific complementary energy is a function of two tensors: the Piola stress tensor and the rotation tensor A. In this case, the constitutive relation for elastic material takes the form

S=~1 ( p .

U + U . P) = ~1 (D. A T + A . D T) = 0)4;/OV.

(2.2.5)

So it follows tr (D. C T) - 14; = tr (S. U ) - 14;.

This enables us to represent the specific complementary energy V as a function of the tensor S called Jaumann stress tensor: l)(S) = tr (S. U(S)) - )4; [U(S)].

(2.2.6)

To construct the function (2.2.6), it is necessary to invert the dependence S(U) linking two symmetric tensors. This problem is much easier than that of defining the dependence C(D). In the weaker complementary energy principle, Jaumann tensor is assumed to be expressed, in accordance with equation (2.2.5), in terms of independent variations, Piola stress tensor, D, and the rotation tensor, A. By varying equation (2.2.6), we obtain (iV = tr (U. 5S) = t r ( U . A . 5D T) + tr [(A T. U . D ) . (A T. 5A)]

(2.2.7) = t r ( C . 5D T) + tr [(C T. D ) . (A T. 5A)].

46

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

Piola stress tensor fields satisfying the equilibrium equations and the force boundary conditions will be called statically admissible. These fields, accordingly to equation (2.1.8), are expressed in terms of the tensor of stress functions, ,~, which should meet the boundary conditions on the surface of multiply connected body n . (rot • + D*) = f.

(2.2.8)

We have assumed that the load is dead, i.e. the vectors f and k are some given functions of Lagrangian coordinates. The expression (2.2.8) can be transformed into the following V. ( e . ¢ ) = f - n . D * ,

e =

-n

× E,

V ~ = (E - n n ) . grad ~, V . q =_ r ~ . O~/Oy ~

(2.2.9)

(a=l,2),

where y~ are the Gaussian coordinates on the surface a; r ~ is the reciprocal vector basis; V is the gradient operator on the surface a; e is the surface discriminant tensor (Zubov 1982). The tensor stress function satisfies equation (2.2.9) and the dynamic conjugation conditions (2.1.1) on the cuts converting the body into simply connected, that is ~7. (ek" O+) = V. (ek-O_). Qk,

ek = --nk × E.

(2.2.10)

The complementary energy functional for elastic body, A1, is defined on the set of differentiable orthogonal tensors and twice differentiable tensors of stress functions undergoing the restrictions (2.2.9) and (2.2.10). This functional is

AI[~I,, A ] = f V[D(~I,), A]dv+ E

bk "~k V " (ek " cI'+)da"

(2.2.11)

k

We will show that stationary points of .AI satisfies the compatibility equations (2.1.6), written in terms of the rotation tensor and stress function tensor, the symmetry of the tensor C. D T and the jump conditions (1.3.4) for the position vector R. Using equations (2.1.8) and (2.2.7), for the variation of ~41 we obtain: 5A1 = ~ [ t r ( C T. rot~O) + t r ( C T. D . A T. ~A)] dv (2.2.12)

+ E bk" ~k V . e k . 5"~+da k With regard for the identity tr rot (x. y) = tr (y. rot x) - tr (x T. rot yT)

2.2 Variational Formulation

47

integration by parts in (2.2.12) implies 5A1 = ~ [tr(bO T. rotC) + tr(C T. D- A T. hA)] dv - .~. tr (e. 5O. CT)da +~/ k

[tr(ek. ~ + . C T) - tr(ek. ~ _ - c T ) ]

da

.Irk

+~bk" k

f

V . e k . 5O+da.

J~'k

Let ~A1 = 0. First take 5~ = 5~+ = 5cI,_ = 0 on a and Tk (this is compatible with the restrictions (2.2.9), (2.2.10)). In view of the fundamental lemma of calculus of variations we derive the continuity equations (2.1.6). These equations mean that in the domain occupied by the body there is a vector field R, whose gradient is C(O,A). In a simply connected volume, I / i s a single-valued function of coordinates, which is defined up to an additive vector constant. Thus the stationary condition for A1 takes the form tr (C T. D. A T. 6 A ) d v - £ , V. (e. ~ . +~

R)da + j f R . ( V . e . 6O)da

J~~k[R_. (V. (ek. 5O_)) - R+. (V. (ek. 5¢+))] da

k

+~bk"

~k V . e k . ~ + d a = 0.

k

Here, a' is a closed surface constituted by a and both sides of all the cuts By the divergence theorem on a surface (Zubov 1982), the integral over a' vanishes. The integral over a becomes zero because of equation (2.2.9). From orthogonality of the tensor A it follows that A T. (~A is skew-symmetric. Hence the stationary state of the functional A1 implies the tensor C T. D to be symmetric that is equivalent, in view of equation (1.1.19), to symmetry of Cauchy stress tensor and provides the balance of moments of all the forces acting on any part of the body (Lurie 1970). With regard for equation (2.2.10), the variational equation becomes

Tk.

/(R+ k

- R _ . Qk - bk). [V. (ek. ~¢+))]da = 0.

JTk

Since ~ + on ~'k is arbitrary, we establish the jump condition (1.3.4). On the basis of the two fundamental variational principles for bodies with Volterra dislocations, discussed above, we can construct functionals for other stationary principles similar to those in (Zubov 1971). A limitation of above variational theorems is that the varied functions are defined in a simply connected volume obtained by making cuts but not in the

48

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

initial multiply connected domain. The additional boundary surfaces complicate the set up and solving the equilibrium problem and bring some elements of artificiality. In a specific case of plane or generalized plane deformation without disclinations, there is a variational principle without making the cuts, i.e., the principle in which the functions to be varied are defined and single-valued in multiply connected plane region. Under deformation of the form (1.4.1), for isotropic, orthotropic and some other materials, Piola stress tensor will have the representation 3 3 D = D{f~fl + D2flf2 + D~f2fl + D2f2f2 + Daf f3,

(2.2.13) m

D~¢ = D~(¢, ¢). Let body forces be absent; in terms of the complex stress components the equilibrium equations are

OD~/O-¢ + 0D~/0¢ = O.

(2.2.14)

On introduction of the complex function of stresses, ~(¢, ~), D{ = 00/0¢,

D~ = -0(I)/0¢,

(2.2.15)

equation (2.2.14) becomes identical. The force conditions on the boundary contour of the plane region, ¢(s), s being the length parameter, can be written as follows dCD{ de D~ = iF1 , d---~ - ~s

F 1 = f " af1,

(2.2.16)

where F 1 is the complex component of the vector of external dead load. To derive equation (2.2.16), we used a formula for the normal-to-contour vector,

o- -i :l + i :2 From equations (2.2.15), (2.2.16)we have d~ d--7 = iF1

(2.2.17)

on the domain boundary. The boundary of multiply-connected plane domain w consists of the outer contour, 70, and the inner contours, % (k - I, 2, ...m). In what follows we shall suppose, for conciseness, that the boundaries of holes are load-free, i.e., F 1 - 0 along %. Therefore, (I)IT k - - Olk~

2.2 Variational Formulation

49

where ak are complex constants. By equation (2.2.15), the stress function ¢ is defined by the stress field only up to a constant, so we may arbitrarily choose the outer-contour constant arising while we integrate equation (2.2.17). We choose it such that ¢[7o = 0 when F 1 = 0; hence ~1-~o = i

Flds.

(2.2.18)

If in the body there are only translational dislocations then the rotation angle X (see Sec.l.4) and Piola stress tensor are single-valued functions in multiply connected domain. Now the integral in (1.4.20) is zero for any closed contour. The complex parameters characterizing the Burgers' dislocation vectors are ~k = ~£ eix(U~d( + Uld~). J.yk

(2.2.19)

Instead of the contour % in equation (2.2.19), we can take any closed contour enclosing k-th hole only. If the relations (2.2.15) are considered to be equations to define the function • when the Piola tensor is given, then they reduce to the problem of finding the stress function with total differential given in advance. Using equation (2.2.16) we can easily show that • is single-valued in the multiply connected domain if and only if the principal vector of external load applied to each hole contour is zero. This condition holds in the case under consideration (of load-free holes). As in (2.2.6), we consider the specific complementary energy of an elastic material under plane deformation,

V = S~U~ - IV,

S~ = f a . S. f~

(a,/3 = 1,2),

to be a function of Jaumann stress tensor, whose complex components are expressed in terms of the rotation angle and the stress function as follows

S~ S~ =

=

Re

(e-iXOgP/O~), (2.2.20)

S~ = S~ =

-eixO~/O(.

On the basis of (2.2.20), we have

= U.SS~ " =

U~5(e -~xo 1o + eiXO-~/O-~)-U~5(e-ixO'blO~)-U~5(eiXO-~lO~).

Let us consider the following functional in which the rotation fields and stress functions, both single-valued in the multiply connected domain w, as well as the contour constants, a k , are varied independently: m

( / 3 k ~ k _ ~kak). A2[~, ~:,al,...,a~] = Of Vdw + ~i E k=l

(2.2.21)

The varied stress function must satisfy equation (2.2.18) and the conditions = ak on the contours 7k.

50

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

It is easy to verify that at the stationary state of the functional (2.2.21) it follows the compatibility equations (1.4.14) expressed in terms of the stress function and the rotation field, the symmetry property of Cauchy stress tensor Im ( V l e - i X O ~ / o q ¢

-

U2 e-iX o(~ / (~-~)=

0

and the integral relations (2.2.19).

2.3 The Semi-inverse M e t h o d for Solving Quasi-static Problems of Nonlinear Mechanics of Solids By semi-inverse methods in continuum mechanics one usually means the techniques for constructing specific solutions such that the system of resolving equations reduce to equations in a less number of independent variables. The semiinverse methods are of great importance when exact statements of boundary value problems are so complex that cannot be solved by other methods. It is well known a great role of the semi-inverse method, proposed by St. Venant to solve the problems on torsion and bending of prismatic beams in development of linear elasticity as applied science. By St. Venant, threedimensional problems of elasticity were reduced to two-dimensional ones for the Laplace equation, that enabled us to obtain efficient solutions for a number of problems of practical importance. In theory of finite strains of elasticity, the use of semi-inverse methods is primarily due to Rivlin, Ericksen, Green, Adkins, Lurie (Rivlin 1948; Ericksen 1977; Green, Adkins 1960; Lurie 1980). These authors found types of deformations for which the system of equilibrium equations reduces to one or more ordinary differential equations. For incompressible materials, the method results in a series of exact solutions to large deformation problems of practical importance. It should be noted that using semi-inverse methods it is hard to expect to satisfy boundary conditions exactly. On a part of the boundary we are forced to "weaken" them, i.e. to satisfy these conditions approximately. For example, in the classical St. Venant problem, the boundary conditions on the beam ends are satisfied only in integral sense within the accuracy in forces up to a force system being statically equivalent to zero. Although for our needs it is enough to consider only elastic bodies, we shall formulate the semi-inverse method for simple media with memory with regard for links between mechanical and thermal effects (Zubov 1981). Proposed here is the procedure for constructing such types of quasi-static deformations of bodies with arbitrary memory as those for which equilibrium equations have only one independent spatial variable. Unlike usual semi-inverse approach, which consists in specifying the deformation type based on some semi-intuitive and hardly formulated consideration and subsequent testing of adopted assumptions, we suggest a procedure to construct a class of needed deformations. This procedure is based on the analysis of equilibrium equations

2.3 The Semi-inverse Method

51

for continuum with rheology, which are written in a particular form using expansion of tensors in elements of dual Lagrange-Euler basis. In so doing, the invariance of the equations with respect to certain transformations, conditioned by flame-independence and material symmetry properties (isotropy, orthotropy, etc.) of continuum is essentially used. Consider the equation system describing a quasi-static process of deformation of thermomechanical continuum with memory. It consists of the equilibrium equations, the energy conservation law, and the constitutive equations: DivT + pk = 0,

Divh+ tr(T.6)+pq=p~,

(2.3.1)

2~: "-- C -1./~k. C -T.

P(t) = ¢[Zt(s)],

J(t)T(t) = cT(t) • P(t). C(t),

e(t) = ¢[Zt(s)], =

h(t)

=

cT(t)

--

C = grad R,

g = Grad0,

if(s) - f ( t - s),

•

ez[Zt(s)], (2.3.2)

0

J = det C,

s > O.

Here, T is the symmetric Cauchy stress tensor, P is the symmetric Kirchhoff stress tensor, p is the material density in the actual configuration, k is the body force vector, h is the heat flux vector, q is the mass density of volume heat sources, e, r/are the mass densities of the internal energy and entropy, respectively, Grad is the spatial gradient operator, grad is the gradient operator in the reference configuration metric, R is the radius-vector of a particle in the actual configuration, O is the absolute temperature, g is the temperature gradient, A is the Cauchy deformation measure, e is the rate-of-strain tensor, t is the time, if(s) is the history of f up to time t (this history also includes the actual value of a function). The over-dot denotes the material derivative with respect to time. The relation (2.3.2) is the general form of constitutive equations for simple materials with memory which is consistent with the requirement of frameindifference (Truesdell 1977). The operators ¢, ¢, ~, w, defined on the histories of the deformation gradient, the temperature, and the temperature gradient, must satisfy the restrictions following from Clausius-Duhem inequality that is the most commonly used formulation of the second law of thermodynamics in continuum mechanics (Truesdell 1977), p//>_ Div (O-lh) + O-ipq.

(2.3.3)

52

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

On introducing the Piola stress tensor, D = J C - T . T, and vector d J C - T . h, the equations (1.2.1) can be written as div D + pok = 0,

div d + tr (D. c T ) + Poq = P0~,

(2.3.4)

P0 being the density in the reference configuration. Let Yk (k = 1, 2, 3) be orthogonal curvilinear coordinates in the reference configuration of material body (Lagrange's coordinates), Y~ (s = 1,2,3) arbitrary orthogonal curvilinear coordinates in the space (Euler's coordinates). The Lame coefficients of these coordinates are denoted by am(yk) and An(Y~), respectively, and the orthonormal vector bases associated with the above coordinates are im(yk) and ln(Ys), respectively. We assume that the vector bases im and In are of the like orientation (the left or the right simultaneously). We shall use the following decompositions of the above vectors and tensors k = knln,

l = grad 0 = C . g = lmim,

h = hnln, D = dmni,~ln,

C = Cmni~ln,

d = dmim,

A = Amnimin,

P = P,~nimin,

(2.a.5)

T = Tmnlmln.

From equation (2.3.5) it follows that Tn~ = J-1CmnDm~ = J-1CmnP~kCks,

J = ICm l,

(2.3.6) Amn = CmkCnk,

Dm~ = P,~nCn~,

hn = J-1Cmndm.

Although C ¢ (Cmn)" imIn, the identity t r ( D T. 12) = Dm,(Cmn)" holds. The proof is based on the symmetry of tensor C T. D and orthonormality of bases ik and In. Let functions Y~ = Y~(yk,t) describe motion of continuum. There is the representation

Cks = a-klAs(OY~/Oyk)

(no summing over

s and k!).

(2.3.7)

= ala2a3.

(2.3.8)

The energy balance equation takes the form

1 0 (x/"gds ~ + Dkn(Ckn).Zc POq = PO~, x/~ Oys as ,/

~

A special form of the equilibrium equations, found in (Zubov 1981), appeared to be convenient to construct semi-inverse solutions. These equations

2.3 The Semi-inverse Method

53

are formulated with respect to Piola tensor components in the dual LagrangeEuler basis and have the form

1 0 x/~ Oys

(.) Dsm

as

+

AmAk

OYk

Dsm- ~Dsk OYm

)

+pokm = 0.

(2.3.9)

In (2.3.8), (2.3.9), we use summation over indices s, k, n from 1 to 3 and no summation over m. The easiest way to derive equation (2.3.9) is to use, with regard for (2.3.7), the virtual work principle t"

f tr (D T. 3C)dv - / v pok. 3Rdv = 0.

(2.3.10)

In equation (2.3.10), dR = 0 on the body boundary. We shall consider gyrotropic material media. The fact that material is gyrotropic imposes the following restrictions on the material response operator in equation (2.3.2)

¢[z:(~)] = o . ¢[z'(~)]. o ~,

¢[z:(~)] = ¢[z'(~)], w[Z~(s)] = w[Zt(s)],

~[z,'(~)] = o . ~[z'(~)],

(2.3.11)

z:(~) = { o . A'(~). 0 ~, 0~(~), O. V(~)}, where O is an arbitrary properly orthogonal tensor, which is constant (i.e., independent of t and s). Write now the component representation of the constitutive relation in orthonormal basis

P~.

=

¢~.( A t~,

0t

,~).

(2.3.12)

The property of gyrotropness (2.3.11) means that this component representation does not depend on the choice of orthonormal basis of the like orientation. In other words, the form of operators Cmn, i.e., dependence of components of tensor P on pre-history of the Cauchy strain measure and the temperature gradient, is the same in all orthonormal bases of the like orientation. It follows, in particular, that, in spite of dependence of vector basis im on the coordinates Yk, the relation of components in this bases of P with components of the tensor A t and the vector I t is the same in all points of homogeneous body. A similar statement is true for operators ¢, w, ~. In what follows, the coaxiality theorem will be of major importance. Theorem. Assume that in some motion of gyrotropic medium, for a fixed

particle, there exists a unit vector i having properties: a) i and I are coUinear vectors for all t, s; b) i is an eigenvector of tensor At(s) for all t, s. Then at this particle the vector d(t) is coUinear to i, and i is an eigenvector of P(t). To prove this, in (2.3.11) we take a proper orthogonal tensor O of the form 0 = 0 T = 2 i i - E.

(2.3.13)

54

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations Under the conditions of theorem, we have O . At(s) • O T = At(s). From equation (2.3.11), it now follows the relations O . P(t) = P ( t ) . O,

O - d ( t ) = d(t).

(2.3.14)

Substituting equation (2.3.13) into these and multiplying the first equality in (2.3.14) by the vector i, we obtain ( i . P . i ) i = i.P, that is i is an eigenvector of tensor P. The second equality in equations (2.3.14) gives us i(i. d) = d. Q.E.D. In the system of equations (2.3.8), (2.3.9), the Lagrangian coordinates, Ys , and the time, t, are independent variables whereas Eulerian coordinates, Yn(Ys, t), and the temperature, O(y~, t), are unknown functions. We will seek the continuum motions for which the number of independent spatial variables reduces to one, i.e., all the quantities in equations (2.3.8), (2.3.9) depend on the only spatial coordinate yl and the time. We may satisfy this condition if the coordinate frames y~, Yn are such that the corresponding Lame coefficients of the systems depend on a single coordinate, am = am (Yl), An = An (]I1), respectively. Using the Lame relations, we may show that the only coordinates which satisfy these requirements are Cartesian and circular cylindrical coordinates and those which differ from them in transformation of the form: y~ = ~(yl), y~ = c2y2, y~ = caYa, c2 , Ca being constants. In so far, it is necessary that the given values of km, q do not depend on y~, Ya. For the Lame coefficients An to be independent of y2, Ya in the Lagrangian representation, it should be taken Y1 = c~(yl, t). By (2.3.7), it follows that C,1 = 0 (# = 2, 3). Let the magnitudes of Cmn and 9 be functions of the coordinate yl and time t only. From gyrotropy of medium, for homogeneous body, it follows that the quantities Pm~, din, e, 77 (and thus all the values defined by equations (2.3.6)) will also depend on the coordinate yl and the time only. From equation (2.3.7) it follows that Cta = 0 and Cmn are independent of y2, ya if and only if the body motion is of the form Y1 = o~(yl, t),

Y2 ~- ~(yl, t) + p2(t)y2 + T2(t)y3,

(2.3.15) Y3 - "Y(yl, t)+ p3(t)y2 + T3(t)y 3.

The functions in (2.3.15) must satisfy the condition J > 0. Under above conditions, both sides of inequality (2.3.3) do not depend on y2, Ya. Thus, the above assumptions do not contradict the Clausius-Duhem inequality. For motions of the form (2.3.15), the system of resolving equations (2.3.8), (2.3.9) reduces to a system of four equations for four functions of a single spatial variable and the time: a, /3, 7, 9. As boundary conditions on two surfaces yl = constant, we can specify, say, the temperature and stress components Tim (m = 1, 2, 3) being functions of the time only. On the rest part of body surface, the boundary conditions may be satisfied in the integral (averaged) sense, by matching the functions pi(t), ri(t).

2.3 The Semi-inverse Method

55

Note that the Lagrangian and Eulerian coordinates may be of different type. For example, we can choose the Cartesian coordinates to be Lagrangian whereas the cylindrical coordinates to be Eulerian, or vice versa. Therefore, the representations (2.3.15), in spite of their simplicity in writing, contain a rather wide set of types of deformation. The semi-inverse method also enables us to find out the continuum strains which satisfy identically some part of the equilibrium equations. For the motions (2.3.15), let il be an eigenvector of A'(s). In view of C,1 = 0 (# = 2, 3), this implies the equalities C12C22 ÷ C13C23 = 0, C12C32 ÷ Ci3C33 = 0, which hold if C12 = C13 = 0. The general representation of deformation satisfying the above conditions has the form of equation (2.3.15), with ~ and ~ depending on the time only. By coaxiality theorem and relations (2.3.6), we have P1, = T1, = D1, = d, = h , = 0 (m = 2, 3). If k2 = k3 = 0 and m = 2, 3, the equilibrium equations (2.3.9) are identical. The system of resolving equations consists of the only equilibrium equation and the heat influx equation (2.3.8). On the surfaces yl = constant, it may be specified the uniform normal pressures, being the time functions, as well as the temperature or the heat flux. Let us list some important cases of deformation described by equations (2.3.15) with ~ = ~ = 0: inflation, torsion and extension of a hollow cylinder; bending of a rectangular block into a sector of a circular cylinder or into a complete circular-cylindrical tube; straightening of a sector of circular cylinder into a rectangular block; cylindrical bending of a hollow-cylindrical sector; shearing of a sector of a circular tube along the axis; eversion of a cylinder; etc. Using expressions (2.3.15) we can describe combinations of all these deformations. In problems for circular cylinder, indefiniteness in deformation of the form (2.3.15) is such that we can satisfy end force conditions in St. Venant sense. In pure mechanical theory of viscoelasticity with no regard for thermal e f fects, the constitutive equations for stresses do not include the temperature, and no consideration is given to the heat influx equation. In this case for isotropic homogeneous bodies, under the condition k2 = 0, it can be proved that the second equilibrium equation (m = 2), is identically fulfilled in the motions of the type (2.3.15) with ~ = P2 = r3 = 0 or ~ = P3 = r2 = 0. If ~ = P2 = r3 = 0 or "~ = P3 = r2 = 0, the third equilibrium equation is satisfied when k3 = 0. These semi-inverse solutions were found by systematic trials, based on search of tensors C such that one of ik is an eigenvector for At(s), using essentially the coaxiality theorem. In a similar way it can be proved that the above semi-inverse solutions with a single independent spatial variable remain valid for orthotropic bodies with predominant directions il, i2, i3 and arbitrary non-homogeneity with respect to coordinate yl.

56

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

2.4 Combination of Screw Dislocation and Wedge Disclination in Nonlinear Elastic Cylinder Isolated defects such as screw dislocations, wedge disclinations or their combination are rather simple for investigation in the framework of nonlinear elasticity. Let r, ~, z be the cylindrical coordinates in the reference configuration of elastic body; R, ~, Z be the cylindrical coordinates in space. The unit vectors tangential to the coordinate lines in reference configuration and in space will be by denoted er, e,, ez and eR, e¢, ez , respectively. Consider the following deformation of continuum

R = R(r),

¢ = ~ + Cz,

Z = (27r)-~b~ + az,

(2.4.1)

where ~, ¢, b, a are constants and a > 0. This deformation relates to the class of continuum motions considered in Sec.2.3. For this class, the system of resolving equations reduces to an ordinary differential equation. When ~ > 1, relations (2.4.1) describe the deformation appearing if one has removed the sector 27r~-1 _< ~ _< 27r from a circular hollow cylinder, rotated the section ~ = 27r~-1 about the cylinder axis up to coincidence with the plane ~ - 0, then made translational displacement of this section by b~ -1 along the cylinder axis, and at last fastened tightly the planes ~ = 0 and = 27r~-1. Moreover, the cylinder is subjected to torsion with angle of twist ¢, axial extension (or compression), and inhomogeneous radial deformation defined by a function R(r). If ~ > 1, we have: 0 _< ~ _ 27r~-1, 0 _< (I) _ 27r. This means that in the reference configuration the body occupies a simply connected domain but in the deformed configuration the domain is doubly connected. The case 0 < ~ < 1 corresponds to addition of a material wedge into the cylinder which was cut by the half-plane ~ - 0. In this case, 0 _ ~ < 27r, 0 _ • _< 2r~, i.e., the body without additional material occupies the doubly connected domain in the reference configuration but in the deformed state the domain is simply connected. Using the formulae of Sec.2.3, let us find the deformation gradient associated with transformation (2.4.1)

~R C = R'ereR + ~ e ~ e ¢

+

b

e~ez + CReze¢ + C~ezez,

det C =

RR'( be) r

~

"

R' _- dR dr"

(2.4.2)

(2.4.3)

The deformation measures, defined by equation (2.4.2), are as follows

2.4 Combination of Screw Dislocation and Wedge Disclination

A = R

+

( ab

/2

erer +

~¢R2

+

n2

R2

b2 /

- ~ + 47r2r2

+

57

e~e~

+

+

r

A_1

= R ,2 e R e R +

~2

r2

+ R2¢~

e,~e,~

(2.4.4) +

(e~ez+eze¢)+

2rr2+aCR

47r~r2 + a 2 ezez,

A = (R')-2eReR + (27ra~ -- b¢)2R ~

-

aCR+27rr 2 ( e c e z + e z e ~ ) +

c~2 + 47r~r9

¢2R2+

r2

e~e~

ezez •

The invariants of the Cauchy strain measure, A, are R2

b2

I1 -- tr A = R '2 + n 2--~ + ¢2R2 + 47r2r2 + a 2, /2 - ~1 tr ( tr 2A - tr h 2) (2.4.5) =

an-

be ~

2 R 2 R '2 a 2 b2 + ¢ 2 R 2 + n 2 ~-~ + + 41r2r2 -~

R'2R2( /3-= d e t A =

r2

,

be) 2 an-~

.

If n > 1 the vectors eR, e,~ are single-valued and continuous in doubly connected domain occupied by the deformed body. Hence from equation (2.4.4) it follows that A -1 keeps continuity when it passes through the surface (I) a-lCZ which is the final position of the plane ~ = 0 after deformation whereas the deformation gradient, C, takes a jump, C+=Q1.C_,

C_=limC

(~ --, Ca-~z + 0),

C+ -- lim C (~ --+ 27r + ~2o/-1Z - 0), (2.4.6) Q1 = Q~-T = (E - ezez)cos 27r(1 - n -1)

+ezez - ez x Esin21r(1

- n-l).

The relation (2.4.6) is tested by using equations (2.4.1), (2.4.2), and the apparent representations e~ = eR cos(~ -- ~) + e¢ sin(qo -- ~),

58

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations e~ = --eR sin(~ -- ~) + e~ cos(~o -- ~).

The formulae (2.4.4), (2.4.6) show that the Volterra dislocation occurs in the cylinder. In this case, by equation (1.3.28), the Frank and Burgers vectors take the form

bl "- bn-lez,

ql = - 2 t g ~ ( 1 - ~-l)ez.

Here, the origin lies on the cylinder axis. If 0 < k < 1, the vectors er, e~ are single-valued and continuous in the doubly connected domain occupied by the body in reference configuration. From equation (2.4.4) it follows that the Cauchy strain measure, A, is continuous and single-valued in this doubly connected domain. Thus, if 0 < n < 1, it is the case of Volterra dislocation characterized by the jump equation (1.3.6) with the parameters

b = b~-lez,

q = - 2 t g ~ ( 1 - n)ez.

In linear theory of isolated defects (de Wit 1977), the Volterra dislocation, whose Frank vector is zero and the Burgers' vector is parallel to the dislocation axis, is called the screw dislocation, and the defect with Burgers vector of zero magnitude and the Frank vector being parallel to the defect axis is called the wedge disclination. By analogy, the isolated defect under consideration can be called the combination of screw dislocation and wedge disclination. Negative values of the parameter n correspond to generation of screw dislocation and wedge disclination in the cylinder which is initially inverted. From the constitutive relations of isotropic or orthotropic material, for deformation of the form (2.4.1) it follows that the Cauchy and Piola stress tensors are

T = aR(r)eReR + a¢(r)e~e¢ + T¢z(r)(ecez + eze¢) + az(r)ezez,

D = DrR(r)e~en + D~(r)e~e~ + D~z(r)e~ez

(2.4.7)

+Dz~(r)eze¢ + Dzz(r)ezez. The first two of the equilibrium equations in the form (2.3.9) are satisfied identically, whereas the third equation is a nonlinear ordinary differential equation with respect to the function R(r), namely,

dDrR dr

DrR- ~D~ r

-

CDz¢

=

O.

In terms of physical components of Cauchy stress tensor, the equation (2.4.8)

daR

aR -- a¢

dR

R

=0.

(2.4.9)

2.4 Combination of Screw Dislocation and Wedge Disclination

59

Boundary conditions for equation (2.4.8) are the load-free conditions on the lateral surfaces of the cylinder, DrR = 0. For a whole cylinder (without a hole), the condition on the inner surface is replaced by the requirement R(0) = 0. It can be shown that stresses in any cross-section of the cylinder (Z = const) are statically equivalent to a longitudinal force applied at the center of crosssection and to a twisting torque. By matching the parameters a and ¢ we can make these force factors to be zero. Thus, the problem of determining stresses caused by an isolated defect, being a combination of screw dislocation and wedge disclination, was reduced to a boundary value problem for an ordinary differential equation. For incompressible material, the Cauchy stress tensor is expressed in terms of strains up to an arbitrary spherical tensor. Now the function R(r) is found implicitly from the incompressibility condition, det C =. 1, and the spherical tensor is defined by equilibrium equations (2.4.8) or (2.4.9) with quadratures. The solution (2.4.1), describing formation of a screw dislocation and wedge disclination in an elastic circular cylinder, relates with medium deformation such that the system of equilibrium equations contains a single independent spatial variable. Let us now study a dislocation nonlinear problem wherein application of the semi-inverse approach reduces to a three-dimensional problem to the twodimensional form, i.e., to a system of differential equations and boundary conditions in two independent variables. Consider an elastic body that is a prismatic rod, i.e. noncircular cylinder, in the reference configuration (undeformed state). Cartesian coordinates xl, x2 refer to the plane of prism cross-section whose center of inertia is the origin, x3 refers to the rod axis. The coordinate basis vectors shall be denoted by ek (k = 1, 2, 3). Xn (n = 1, 2, 3) stand for Cartesian coordinates of points in the deformed body, they refer to the same directions ek and origin. Using the complex coordinates and complex bases introduced in Section 1.4, let us define deformation of the prismatic rod as follows z -- z0(~, ~)e i¢x3,

b X3 = c~x3 + ~ arg (~+ w(~, ~),

(2.4.10)

where ¢, a, and b are constants, z0 is a complex-valued function and w is a real single-valued function. Geometrically, the representations (2.4.10) mean that in the prismatic body there is a screw dislocation whose axis directs along the x3axis and the Burgers vector is equal be~; the bar cross-section, at a distance x~ from the origin, being deformed (plane deformation, given by the function z0) , rotates about the rod axis through the angle ¢x3, translationally moves along the axis for distance (a-1)x3, and warps (the form of warping is described by the function w). The deformation gradient C corresponding to the transformation (2.4.10) is defined by

60

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

c = c~f-/~, C1

C~

--2

c~=~, C1

0Z0 _i¢x3

= C 2 = --~-e

-- -C3 ~ --

C~

i¢zoeiCXa,

--2

:

C

1 :

OZo • _

o¢

e ~¢~3,

--a Ow = C2 = -~

(2.4.11)

ib 4~r("

The constitutive relations for isotropic material with respect to the Piola stress tensor may be written as follows

D = a~ (Iq)C + a~(Iq)A. C

+ o/3(/q)C -T,

(2.4.12) 11 = tr A,

1 tr 2A - tr A2), I~ = ~(

Ia = det A,

where aj (j = 1, 2, 3) are scalar functions of the invariants Iq (q = 1, 2, 3). For complex components of Cauchy strain measure, by equations (2.4.11), we have

All = A22 = OZo O-2-d Jr-~2

o¢ o¢ 1 (OzoO~o

A12=A21=2 A13 =

A23 =

A31 =

~oOzo)

0( 0~ { 0( 0-~

1

A32 -

~i¢

(0-20

+7~,

i)Zo'~

Zo--~- - -20 0¢ ,] + a'7,

(2.4.13)

Aaa = ¢~Zo~o + a 9,

Ank = f , . A . f k,

Ow "/= O~

ib 4r~

The invariants Iq are as follows

11 -- 4A12 + Aaa,

(Ozo O~o

12 = 4[(A~2) 2 + A12A3a- AliA22- A~aA2a],

oozo) + 2¢ Im (7-2o-~ ozo- "yZoozo 0( ] "

(2.4.14)

0( 0-~

The relations (2.4.13), (2.4.14)indicate that under deformation (2.4.10), the tensor A and its invariants do not depend on the coordinate x3. The tensors A . C and C involved in the constitutive relations (2.4.12) take the form

2.4 Combination of Screw Dislocation and Wedge Disclination

61

X/~3C-T= Ls fm f s,

A . C = K~nfm f ~,

--~. [OZo (OZo___ 1 2 ) O~o ,,0-2o0Zo K1 = K2 = L o¢ \ o¢ ~ +-~-£(-~ + 2~-~+ -~¢ zo-~o

(

zo)

+ ~ 2 ~ - ~ + i¢~zo

- ~¢Zo-~

, =_K1rozo(.ozo o oozo = Loc: ~ - ~ + - -o¢ -- £

e *~,

1¢~zo~o)

+ 2~+

( zo ) Lro 0¢ zo( o i,zoOZO ~ + iCzo--~ + lCZo-~-

+ ~ 27--0-~- + iCazo - ~¢z o--~j e'¢x3, K~ - K3

(2.4.15)

OZo ] + 2~V-b-~- + iCzo(¢~zo~o + £ ) ei¢~,

K3--3

( OzoOzo

)1

(0-2o

OZo)

= K~ = ~ 2 o¢ o~ + 4~2 + ~ i¢,:,,, zo--5-( - -~o-y(

(azo 0~o 0~oa~o

)

O'~o7) /(33 = 4a'y~ + iCzo (0-2o7 \--~ + -0~ 0Zo Ozo_~ -i*~o -~- + -O-~7) + a(¢2ZoZo + a2), = L2 = c~-~ - iCzo"/ e*~

L~= L-~I = ( iCzo~-C~--~je Ozo'~ '¢xa, L~

-2

( Ozo

Ozo

•

= L3 = 7"~- - V"~') e'¢~3,

L~ -3

= r~ =

1. ( 0zo 0-20 ~,¢ _~°~ + zo-b-(], 0zo 0-20 0-200Zo

The representations D 1

1 - p.(~, ~)e i~,

2 D.2 -- p.(~', ~)e -i~x3

D~ = p~(¢,-~), (~ =

1,2,3).

(2.4.16)

62

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

follow from the equations (2.4.11), (2.4.12), (2.4.15), where quantities p~ are independent of the axial coordinate x3. With regard for (2.4.16), the equilibrium equations are

Op~ Op1 i 1 O~ F - - ~ + ~¢P3 = O,

Op~ ~- Op~ O~ --~ = O.

(2.4.17)

With given constants b, a, and ¢, the relations (2.4.17) make up a system of equations with respect to the complex and real functions, z0 and w respectively, in two variables. To these equations it is necessary to associate boundary conditions on the lateral surface of the prism which, for the free load surface, are 0~ O~ ~ = O, ~88/.,2 -- ~88b,1

0~3 0¢ ~3 = O, ~8/j2 -- ~88b,1

(2.4.18)

where ~(s) describes the contour of cross-section of the bar, s is the arc-length of the contour. If z0 and w is a solution to the boundary value problem (2.4.17), (2.4.18) it is easy to see that the functions z0eiw and w + a with w and a being real constants also satisfy the conditions (2.4.17), (2.4.18). This ambiguity can be eliminated if we require Re

(~Z0 - l ° z ° O~

1) da = O,

f~ wda = O,

(2.4.19)

a being a cross-section of the undeformed bar. Under the conditions (2.4.19), we can expect uniqueness of solution to the problem (2.4.17), (2.4.18), at least for not too large strains. Assume that the bar cross-section is centrally symmetric, i.e. remains the same under rotation about the rod axis through 180 degrees (an example is a cross-section of the form of the letter "Z"). Cross-sections with two axes of symmetry are obviously in this class too. Make the following changes of independent variables and unknown functions in the system (2.4.17) and boundary conditions (2.4.18):

~' - -~, ~ " = -~,

-~' -

-~,-

-~" = ~,

Zo' = -z0, z0" = -zo,

w' = w, ~" = - w .

(2.4.20) (2.4.21)

There is the following Theorem. For homogeneous isotropic elastic body, under transformation (2.~.20), the boundary value problem (2.~.17)-(2.~.19) in a centrally symmetric

2.4 Combination of Screw Dislocation and Wedge Disclination

63

domain is invariant. If the domain a possesses two axes of symmetry coincident with the axes x l and x2 , the boundary value problem is invariant under transformations (2.~.20), (2.~.21). The theorem is proved by substituting equations (2.4.20), (2.4.21) into the boundary value problem. Let z0 = h(¢,~), w = m(~,~) be a solution to the boundary value problem (2.4.17)-(2.4.19). By the theorem, the functions Zo = - h ( - ~ , - ~ ) , w = m ( - ~ , - ~ ) satisfy the same boundary value problem. Uniqueness of the solution implies h(~, ~) = - h ( - ~ , - ~ ) , m(~, ~) = m ( - ~ , - ~ ) . So, if the cross-section is centrally symmetric, the solution possesses the property

~ o ( - ~ , - ~ ) = -zo(¢, ~), ~ ( - 4 , - ~ )

= w(¢, ~).

(2.4.22)

For bisymmetric cross-section, in addition to (2.4.22), there is

zo(¢, ¢) = ~o(¢, ¢), ~(¢, ;) = -w(¢, ¢),

(2.4.23)

From the equations (2.4.11)-(2.4.15), (2.4.22)it follows

p~(-¢,-¢) =p~(¢,¢), _

m

p ~ ( - ~ , - ¢ ) = -p~(¢,¢), (2.4.24) p~ (_if, _~) = _pa (~.,~)

(o~,/3= 1, 2),

p~(-¢, -~) = p](¢, ~). For the bisymmetric cross-section, besides, there holds

p~(~, ~) = -p~ (~, ~), --

Ot

-

-

3

--

3

m

P3(~, ~) = Pa(~, ~).

(2.4.25)

The principal vector F and principal moment M of forces acting in a rod section, xa = const, are

F=~fa'Dda,

M=-

R = Xmem = zkfk,

f fa'Dx(R-Ro)da,

(2.4.26)

RO = X3(0, 0, xa)f3,

R being the radius-vector of a point of deformed body; Ro is the radius-vector of the reference point which is the center of inertia of the deformed bar section. Based on equations (2.4.10), (2.4.16), and (2.4.26), we have

64

2. Stressed State of Nonlinearly Elastic Bodies with Dislocations

F = f~/~1,~i¢x3 ~2,a-iCx3 3 re3 ~ f l + ~'3 ~ f2 + Pafa) da,

M=

if

-~ l ( z o p ~ - -z0p1)

- zoP3) e -'~3 f2] da,

+ (w • 1 - z0p ) e

fl

(2.4.27)

= w(~, ~) - w(O, 0).

Using equations (2.4.22), (2.4.24) and (2.4.27) we obtain F.fZ=M.f~=0

(/3=1,2).

(2.4.28)

The relations (2.4.28) mean that under deformation of the form (2.4.10), the stresses in a cross-section of the cylinder can be reduced to a longitudinal force F3 and a torque M3. These quantities remain constant in all prism crosssections; on solving the boundary value problem (2.4.17)-(2.4.19), they become known functions of the parameters ¢, a in accordance with formulae

F3 = F . f3 = f~ p~da,

M3 = M . f3 = - f~ Im (zop~)da.

(2.4.29)

Thus, the assumptions (2.4.10) for a cylinder with centrally symmetric crosssection enable us, by solving two dimensional boundary value problem, to satisfy the equilibrium equations and boundary conditions on the lateral surface exactly, whereas the boundary conditions at the bar ends only in the integral sense, matching the constants ~, a.

3. E x a c t Solutions Stressed to the P r o b l e m s on Volterra D i s l o c a t i o n s in N o n l i n e a r l y Elastic B o d i e s

3.1 The Wedge Disclination in Nonlinearly Elastic Body Consider a particular case of deformations (2.4.1) R = R(r),

45 = ~ ,

Z = z,

(3.1.1)

where ~ is a positive constant. When ~ > 1, the relations (3.1.1) describe deformation due to the removal of sector 2 ~ -t < ~ <_ 2~ from a circular hollow cylinder and the rotation of the cross-section ~ = 2 ~ -t about the cylinder axis up to coincidence with section plane ~ = 0; the rotation is accompanied with radial displacements of cylinder points. If ~ < 1, then an absolutely rigid wedge of angle 2~(1 - ~) is inserted (without friction) into the cylinder which was cut by the half-plane ~ = 0; this results in the rotation of section ~ - 2~ about the cylinder axis. Alternatively, the case of ~ < 1 may be interpreted as if an elastic wedge of the same material were inserted into the cut cylinder. The above deformation of the cylinder may be classified as the generation of wedge disclination with Frank vector co-directed with the cylinder axis. It should be noted that we now consider finite rotations of the cut banks, whereas small ones were considered in (de Wit 1977). The length of Frank vector is equal to 2tgTr(1 - i,~ -1) if n > 1 and 2 t g ~ ( 1 - n) if t~ < 1. Physically realizable deformation of continuum must satisfy the condition det C > 0. In the question det C = nR'R/r so the function R(r) must meet the condition R' > 0. The constitutive relation for isotropic elastic, compressible material can be written in the following equivalent forms T = a l E + a2)~-1 + a3)~-2, D

= o~1 A

+ a2C + a 3 U . C,

( det C ) T

(3.1.2) = C T.

D,

where T is the Cauchy stress tensor; D is the nonsymmetric Piola stress tensor; am and a,,~ (m = 1, 2, 3) are some functions of the strain invariants. For the

66

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

independent invariants, one can take the quantities sk = tr [(U

E)k],

-

U

(k = 1, 2, 3).

-----t 1/2

Based on equations (2.4.4), (3.1.2), for deformations (3.1.1), we get D = D r R e r e R + D~o¢e~oe¢ + D z z e z e z , D~R = a l ( s k ) + a 2 ( s k ) R ' + a3(sk)(R') 2,

(3.1.3)

= a (sk) + a2(sk) aR- +

2

01,1(Sk) "["O 2(Sk) "["Ot3(Sk);

Dzz =

T = O'ReReR -[- a ~ e ~ e ~ + a z e z e z ,

r

1

erR- --~DrR,

(3.1.4)

r

0"¢ = -~D~o~,

a z = a R R , D~z.

In (3.1.4) aR, a¢, a z are physical components of the Cauchy stress tensor (the true stresses) in the orthonormal basis of Eulerian coordinates. We should substitute the relations following from (2.4.4), sl = R ' + - -

- 2,

s2 = ( R ' - 1) 2 +

- 1

f

f

(3.1.5)

3

S3 = ( R ' -

1) 3 +

-

1

r

into the expressions for functions o/re(s1, 82, 83) in (3.1.3). The equilibrium equation (2.4.8) of the disclination problem takes the form (a prime denotes differentiation with respect to r) D'~R +

D r R --

aDvo =

0.

(3.1.6)

r

For h a r m o n i c type material, also called s e m i - l i n e a r m a t e r i a l (Lurie 1980), the specific potential energy, ~/Y, and coefficients am, are

w=-~1 As~ + #s2

A, # = const;

(3.1.7) O/1 = 2 ~

us1

1 -- 2U

1

a2 = 2#,

'

a3 = O,

u=

2(A + #)

.

For semi-linear material, from (3.1.3), (3.1.5), (3.1.7), we have DrR =

1

(1- u)R' + ~ --

2# D~oe = 1 - 2u

1

r

[

uR' +

2#u D z z = 1 - 2u

R' +

r r

]

- 1

2 .

'

(3.1.8)

3.1 The Wedge Disclination in Nonlinearly Elastic Body

67

Now equation (3.1.6) takes the form R' r

R"q

~2R 1= . r2 (1 - / ] ) r

(3.1.9)

For a solid cylinder (0 _ r < r0), the unique solution to equation (3.1.9), satisfying the conditions R(0) = 0, and DrR(ro) = 0, is (Zubov 1986b) r 1 p=--, /3= ro ( 1 - v)(1 + ~)' DrR = 2 # ~ (pn-1 _ 1), D ~ = 2#/3 ( ~ f - 1 _ 1),

R = r0~ [(1 - 2/])p ~ + p],

D , z = 2#/](1-/]-1) (2~(1 + K;)-lp~ - 1 - 1),

I,AE,=I[

I = ~

~

(

(1 - 2/])2/32 Np~-i + 1 - 2/]

lJ

+ ~1[~ 2(1 - 2/]) 2/32 (p~-I + 1 - 2/])1 I(E_A)=

1 [

i=~

~

1 [ +~

]

- 1 erer (3.1.10)

(1-/])2(1+~)2 1-

((1-2/])~p

~-1+1)2

eReR

(1-/])2(l+n) 2 ] 1 - ~2 ( ( 1 - 2/])pn-1 + 1) 2 e¢e¢,

2 # ( f - 1 - 1) gR = (1 -- 2/])p ~-1 + 1'

2#(~p~-1 _ 1) a¢ = (1 - 2/])~p ~-i + 1'

2#/](1 -/])(1 + E ) ( i g p n - 1 - 1 - ~) az = [ ( 1 - 2/])Epn-1 -}- 1] [ ( 1 - 2/])p ~-1 + 1] ~" In linear elasticity, the stress field due to an isolated wedge disclination is given by (de Wit 1977; Lurie 1970) - ~

aR-- 1--/]

In p,

a~ = ~

1--/]

1

2#/]----:-(lnp + ~)

aZ= l_/]

,

(In p + 1)

'

(3.1.11)

~ = ~- 1

By these formulae, there is a logarithmic singularity in the stresses at the disclination axis. In (Wesolowski 1981) the problem of wedge disclination in a solid cylinder is considered in the framework of finite deformation theory under assumption of non-compressibility of the material. In that case R = g-1/2r and therefore the invariants of strain tensors are constant, and the stresses possess a logarithmic singularity as p ~ 0.

68

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

For the solution (3.1.10) with ~ > 1, the stresses and the strains remain bounded as p + 0, i.e. there is no singularity on the disclination axis. If k < 1, by (3.1.10), the components and invariants of the Catchy strain tensor infinitely increase as p tends to 0. In particular, the local relative variation of volume, det C - 1, increases proportionally to p2(~-1). However, for t¢ < 1, the true stresses aR, a¢, and Oz, as is seen from equations (3.1.10), are bounded as p - + 0. Thus in the framework of geometrical nonlinearity and compressibility of the material, the stress field of an isolated rectilinear disclination has no singularity on the disclination axis. If p ~ 0, for small magnitudes of Frank vector, the exact solution of equations (3.1.10), may be expanded in powers of 5 = ~ - 1. In this expansion, the terms of 1st order in ~ coincide with the solution by linear theory (equations (3.1.11)). But the stresses by linear elasticity, contrary to ones of equations (3.1.10), increase proportionally to In p as p --+ 0. Since the expansion of p~ in powers of a is not valid at p = 0, on the disclination axis, linear theory cannot give an approximation of nonlinear solution even for small a. On the basis of equations (3.1.7), (3.1.10), we can obtain an expression of potential energy per unit length of the axis of a cylinder with disclination, gt = w

fo TM

co#r~(1- ~)2 l/Yr dr = 2(1 - tJ)(1 + ~)2,

w=

{ 2rt, 27r~-1,

~ < 1 ~> 1

(3.1 13) "

Linear theory of elasticity, in view of equations (3.1.11), gives another value for the disclination energy:

rr#r2o52

A = 4(1 - u)"

(3.1.14)

Comparing (3.1.13) with (3.1.14), we can see that linear theory gives a higher value of energy of isolated disclination if ~ > 1 and a lower one if ~ < 1. For a hollow cylinder (rl _< r _< r0), the solution satisfying equation (3.1.9) with boundary conditions DrR(rl)= DrR(ro)= 0 is R = r0/3 [(1 - 2~)Mp '~ + Np -'~ + p], M = 1 - p~+l 1 p~ ,

N=

p2~ I ( 1 - P ~p O-i~~ ) I

_

(3.1.15) ,

Pl

rl

=

--.

_

r 0

From equations (3.1.8), (3.1.15), it follows

DrR = 2p~3 (Mp ~-1 - Np -'~+1 -1), D~o¢ = 2#~3 (Mp '~-1- Np -~-12p~

Dzz = 1-2z,,

(

2~ M f _ l _ i )

1+

~-1),

(3.1.16)

"

The physical components of stresses (the true stresses) can be found with use of equations (3.1.4), (3.1.15), (3.1.16). The solution to the disclination problem

3.1 The Wedge Disclination in Nonlinearly Elastic Body

69

for solid cylinder, found above, can be obtained from the solution (3.1.15), (3.1.16) by the limit passage as Pl ~ 0. Absence of singularity in stresses and strains at r = 0 for semi-linear material with ~ > 1 may bring one to an idea that the point r = 0 were not singular at all; that contradicts the general theory of Sections 1.2, 1.3, since, by that theory, displacements in a simply connected domain are single-valued for smooth fields of strain tensor. But in reality there is no contradiction. Indeed, since, by (3.1.10), the tensor of affine strain, H, involved in the problem of finding the displacements by given strain tensor, has a singularity of the type r -1 as r ~ 0 for any k ¢ 1. Consider another model of compressible elastic body, a simplified version of Blatz-Ko material. The elastic potential for this material is (Lurie 1980) -- ~ # ( 1 2 1 3 1 -

5),

2V/~3 -

# - const.

(3.1.17) / 2 = ~ (1t r

2A - t r A 2)

/3 =

det A.

From equations (3.1.17) it follows the constitutive relation for the Piola stress tensor D = #I31 [ l i E - A -4- (13/2- 12)A-l] • C. (3.1.18) For small strains, the material (3.1.18) obeys Hooke's law with shear mod1 ulus # and Poisson's ratio being ~. Components of the Piola stress tensor for deformation (3.1.1) are DrR = # [toRt - 1 -

(dR/dr)-a],

D~¢ = # [ d R ~ d r - r3/(~R)3] ,

(3.1.19)

Dzz = # [ ~ R r - l ( d R / d r ) - 1].

Substituting (3.1.19) into the equilibrium equation (3.1.6), we obtain a nonlinear equation for function R(r),

3d2R/dr 2 - r - l d R / d r + r2n-2R-3(dR/dr)4 = 0

(3.1.20)

and boundary conditions, expressing the fact that the lateral surfaces of cylinder are load-free,

~R/r = (dR~dr) -3

at

r -- T0, r -- T 1.

(3.1.21)

On introduction of dimensionless quantities P = Rico,

~

=

T/TO,

Pl

=

rl/ro,

the boundary value problem (3.1.20), (3.1.21) takes the form (a prime denotes the differentiation with respect to p)

70

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

3 P " - p-lp, + a-2p-3p2(p,)4 = O, ,¢p-lp(p) = [p,(p)]-3

at

p = 1, p = Pl.

(3.1.22)

(3.1.23)

By the change P(p) = exp f(~), ~ = ln p, following (Karyakin 198S), we reduce equation (3.1.22) to

3f" + ~-2(f,)4 + 3(f,)2 _ 4f'= O.

(3.1.24)

Another change, f'(~) = y(~), reduces equation (3.1.24) to y , = _(3n2)-ly [y3 + n 2 ( 3 y _ 4)].

(3.1.25)

There are three solutions to equation (3.1.25):

1) y = 0 ; 2) y = a = const, where a is a real root of the equation

~ + ~(3~ - 4) = o;

(3.1.26)

3) a solution determined by integration

f y-, [v3 + ~(3y - 4)]-1 dv = - ( 3 ~ ) - ~ .

(3.1.27)

Let us examine each of solutions. If y = 0, then f(~) = const; therefore

P(p) = const; this obviously makes no physical sense, and hence we drop the solution y = 0. There is the only real root of equation (3.1.26)

it is a monotone increasing function of n, and, besides, sign (a - 1) = sign (n - 1).

(3.1.28)

For small values of ~ = n - 1, there is a representation a=l+

3

_

~(~2 + . . .

(3.1.29)

The solution of the initial equation (3.1.22), that corresponds to the case of y = c~, takes the form

P(p) = exp(~ In p + C1) = Cp '~, (3.1.30) C = exp C1, C being an arbitrary positive constant. On taking the indefinite integral in equation (3.1.27) and performing elementary treatment, we get

3.1 The Wedge Disclination in Nonlinearly Elastic Body co(y)

Ol

In I1 - a/y] - 5 In I1 +

-- 0~-1~ 2

n2y_3

71

(3y - 4)1

4

(3.1.31)

__ -1(012 _[_ N2)~ _[_ C2" - 50/

co(y) =

4 2 (6n 2 - 5 a )(12n ~ + 3c~2)-1/~x

× arctg [(2y + a 2 + n2)(12n 2 + 3a2)-1/2] , 6'2 being a constant. In what follows, we shall restrict ourselves to the case of solid cylinder (p~ = 0). To satisfy the condition P(0) = 0, it is necessary to define behavior of solution (3.1.27) as ~ --~ - c o . So we transform equation (3.1.31) to the form expco(y)ll -

c~/yl-~/~[1 + ~y-a(3y _

4)[-~/3 = (3.1.32)

--

C3e x p

ol-l(oL2 ..t.. N2)~

;

C3 = exp 6'2 > 0. On introducing wl(Y) = expco(y),

K = ol-~2/a(4t~2) -a/3' I K;-2y3 -a/3

~(Y) = 11 - y/-l-~/~l I - ~y- ~

l

- 1,

we can rewrite equation (3.1.32) in other way

Kwl (y)[l + ~(y)]y~-~(~2+~2)= C3 exp [~c~-i(~2 + ~2)~]. On boundedness of function

w(y), we

get

lim y(~) = 0 as ~ --~ - c o , lim COl(Y) = col (0) ( co as y ~ 0. W i t h regard for the asymptotic relation e(y) ~ ( ~ / ~

1

+ ~)y

as y -~ 0,

we obtain an asymptotic formula,

y,'-,C4exp(~~)

as ~ --, - c o ,

C4 = C3[KWl(0)] -a(a2+a2)-~, which implies an asymptotic representation of f(~),

72

3. Exact Solutions Stressed to the Problems on Volterra Dislocations f(~) ,'~ Cs exp

~

+ C~,

Cs = ~C4

with arbitrary constants C5, (76. From equation (3.1.33), it follows a representation of hood of point p = 0

P(p) = A exp

[B (p4/3 +

P(p)

(3.1.33)

in a neighbor-

o(p4/3))],

(3.1.34)

A, B being constants. The formula (3.1.34) characterizes the solution to (3.1.27) in a neighborhood of disclination axis. From the condition P(0) = 0 we obtain A = 0; so, by (3.1.34), P(p) = 0 that is physically meaningless. Hence we must drop the solution (3.1.27). Thus it remains the only solution to equation (3.1.25)" y = a. On finding the constant C in (3.1.30) by condition (3.1.23) at p = 1, we obtain an exact solution to the disclination problem in a solid cylinder made of the Blatz-Ko material: P(p) (~a3)-'/4p ~, (3.1.35) =

or, turning back to the dimensional variables, R(T)

(t~o~3)-1/4r 01 - c ~ r o~.

The Cauchy strain measure in a cylinder with disclination is A "-- (Olt~-l)i/4p2(a-1)erer

+

(0~-1t~)3/4p2(C~-l)e~e~p"~-eze.z.

This, with regard for equation (3.1.28), shows that the strains have no singularity on the disclination axis if ~ >_ 1 but if ~ < 1 they unboundedly increase as r tends to 0. As has been stated above, a like behavior characterizes the semi-linear material. Based on equations (3.1.4), (3.1.19), (3.1.35), for the principal stresses, we obtain (3.1.37)

From the relations (3.1.37) it is seen that the stresses have no singularity on the disclination axis when ~ <_ 1; they possess a power singularity when ~ > 1. This differs from that of semi-linear material for which the stresses are bounded near the axis for all values of ~. Therefore, the character of singularity of stress field on the defect axis essentially depends on the properties of nonlinear elastic material. In case of Blatz-Ko material, the disclination energy per unit length of cylinder is

A

=

with constant w determined by equation (3.1.13).

1],

3.2 The Stress Field due to Screw Dislocation 3.2

The

Stress

due to S c r e w

Field

in the Nonlinear Elastic

73

Dislocation

Body

Let us consider the problem of screw dislocation in a circular solid cylinder. First, we assume that elastic material is incompressible, i.e. there holds the condition of conservation of volume of any part of the body, det C = 1

(3.2.1)

Setting ~ = 1, a = 1, ¢ = 0 in equations (2.4.1), by condition (3.2.1), we obtain R~R = r that gives R = r for the solid cylinder. According to equation (2.4.4), the Finger and Almansi measures as well as the strain invariants become A -1 = eReR + e~e~ + 2 - ~ ( e e e z + ezee) + A = eReR +

(

1 + 47r2R2 e z e z ,

1 + 47r2R~., eee~ - ~ R ( e e e z b2 I1 = I2 = 3 + 47r2R-------~,

+ ezee) + ezez,

h = 1.

Note that b coincides with the magnitude of Burgers vector if ~ = 1. For the Mooney material (Lurie 1980), the specific potential strain energy, YY, and the constitutive relation take the form, respectively, W = C1 ( I 1 - 3 ) + C2(I2 - 3), T = 2C1)~- 1 _ 2C2)~ - pE, where C1, (72 are some constants, p is the pressure independent of the strains. In case of Mooney material, based on equations (2.4.9), (3.2.2), we have b2 R ~ - R O'R

--

6 2 - -

~

4r 2 R ~ R 2

2(C1 + C 2 ) b

2 ,

T4~Z

=

=

--

1)

+ N

R2

2rR

(3.2.3) '

R 02

,

Ro being the cylinder radius. The solution (3.2.3) shows that the tangential stress TZ¢ has a singularity of R -I type on the dislocation axis (like in linear elasticity (Landau &: Lifshits 1965)), and the normal stresses, aR, a~, az, behave like R -2 as R --+ 0. This implies, in particular, that the resultant longitudinal force in a cross-section of the cylinder, F3 = 2~ ~0 R° a z R dR,

74

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

has no finite value. This fact, also confirmed for the neo-Hookean material (that is when 6'2 = 0), is not of physical meaning. So we have to admit that the model of Mooney material is not acceptable to analyze the stress field near the dislocation axis. But for some constitutive relations (even with the condition of incompressibility of material) it is possible to avoid the paradox on longitudinal force and eliminate some singularities arising in classic linear theory of screw dislocation. Consider the Bartenev-Khazanovich material model (Chernykh 1986), given by the relations l/Y= 2# (trA-1/2 - 3 ) ,

# = const (3.2.4)

T = 2#A -1/2 - pE, # being the shear modulus, )k-1/2 being the positive-definite square root of Finger strain measure. Using equation (3.2.2), we obtain

1

[

A-1/2 = eneR + v/167r2R2 + b2 47rRe~e¢ 87r2R2 + b2 + b(eeez + e z e e ) + 27rR ezez

(3.2.5) .

From equations (2.4.9), (3.2.4), (3.2.5), we find the stresses 2#b ~'ze = v/167r2R 2 + b2,

ae=2#

1 + V/1 + b2/(167r2R2) aR = 2# In 1 + v/1 +

v/167r2R 2 + b 2 - 1 (

az=2#

+aR,

(3.2.6)

87r2R2+ b2 ) 2rRv/167r2R2+b 2 - 1 +aR.

As is shown by formulae (3.2.6), the tangential stress, TZe, remains bounded on the dislocation axis, R = 0, whereas, in linear elasticity (Landau & Lifshits 1965), it is R-l-singular as R ~ 0. The normal stresses aR and ae have a In R singularity whereas a z behaves like R -1 near the dislocation axis. The latter implies that the resultant longitudinal force over a cylinder cross-section is finite. If we retain the first-order terms with respect to b, then the solution (3.2.6) reduces to that of linear elasticity (Landau & Lifshits 1965) #b TZe = 27rR'

fir

- - f f ~ =" f f Z - - 0 .

For small magnitudes of Burgers vector, the solution to nonlinear problem of screw dislocation can be expanded in powers of b, rZ¢,

=

.b( 1

2-~

-

b ) + O(b4).

327r2R 2

(3.2.7)

3.2 The Stress Field due to Screw Dislocation

75

It follows that the range of applicability of linear theory is restricted to the range b2/R 2 ~ I. In particular, the linear theory fails near the dislocation axis. Moreover, the character of the stress singularity near the dislocation axis, by the linear theory, is also improper since the expansion of the exact nonlinear solution (3.2.6) in powers of b breaks down at R - 0.

Using equations (3.2.4), (3.2.5), we can find the energy of screw dislocation per unit length of cylinder of radius Ro: A = 2~

/o

+ i6~21n

~ V R d R = 4~tt

Ro+

Ro

b2/(16~ 2) - R] +

R~+

~+b2/(16~ 2) - 1 6 ~ 2 1 n ~

b]

=

(3.2.s)

-- #b2 Ro + O(b4). 8---~(1 + 2 In 8~) + ~#b2 In -~Note that linear elasticity does not enable us to find out the energy of screw dislocation in a solid cylinder because of divergence of the integral at the lower limit, R = 0, (Landau ~ Lifshits 1965). For the Mooney material, as is seen from equation (3.2.2), the solution of nonlinear problem of screw dislocation results in the infinite energy. To examine the stress field far from dislocation axis, let the cylinder radius, R0, tend to infinity in equations (3.2.6), this means that we are considering a screw dislocation in unbounded elastic body. It is easy to see that the principal part of stress asymptotics as R -, cx~ is TZ¢ "~

pb(2~R) -1,

a~ ~ -#b2(32~2R2) -1,

aR ~

#b2(32r2R2) -1,

az ~

7#b2(32r2R2) -1.

It follows that on increasing the distance far from the axis, the magnitude of normal stresses decreases faster than that of tangential stresses. For sufficiently large R, the tangential stresses coincide with ones by the linear theory and the normal stresses are negligible in comparison with the tangential ones. Hence the linear and nonlinear theories give the same result if the distance from the dislocation axis is sufficiently large. The exact solution of the problem of screw dislocation, we have obtained in the nonlinear framework, shows that the nature of solution near the dislocation axis alters qualitatively on taking into consideration the geometrical nonlinearity and thus enables us to avoid some troubles and contradictions of linear elasticity. At the same time it should be mentioned that for incompressible material, the behavior of stresses near the dislocation axis depends essentially on the choice of the model of material. Along with the Bartenev-Khazanovich material, there are other models for incompressible bodies characterized by the finite energy of screw dislocation in

a solid cylinder. For example, we can show that it is finite for the material with the three-constant elastic potential (Ogden 1972)

76

3. Exact Solutions Stressed to the Problems on Volterra Dislocations 1 ]/Y = ~#m -2 [(1 + 13)tr A-m + (1 - / 3 ) tr Am - 6],

0 < m < 1,

(3.2.9)

#, m, 13 = const The model of incompressible elastic body, 14~ = 1 A ( a + 2)-1F ~+2

2

W=

h = ~1 ln,k_ 1,

AF"h - pE,

1 + a > 0,

(3.2.10)

F = V/2trh 2,

(3.2.11)

h being the logarithmic measure of deformation, is used to describe a metal under nonlinear deformation (Nadai 1954). This model can be considered as one of the modifications of the finite deformation theory of plasticity for strengthening body. For the material (3.2.10), the screw dislocation energy in a solid cylinder of unit length is finite, the tangential stress is bounded as R ~ 0 but the normal stresses behave like (ln R) ~+2. In (Teodosiu 1982), the screw-dislocation problem in the nonlinear framework was solved by the successive-approximation (due to Signorini) method, based on the assumption that the solution depends analytically on a small parameter being the magnitude of Burgers vector. The method is not correct near the dislocation axis, since in accordance with equation (3.2.6), the exact solution of nonlinear problem is not analytical in b at b = 0. The first approximation of Teodosiu's solution is a solution in linear elasticity framework so it conserves incorrectness of the linear approach. If the assumption of material incompressibility breaks down, the problem becomes considerably more complex, as the function R ( r ) in the case must be found from the nonlinear equilibrium equation (2.4.8) but not from the incompressibility condition. Following M.I.Karyakin (1989), we consider the screw dislocation problem for the Blatz-Ko material which is described by relations (3.1.17), (3.1.18). With regard for equations (2.4.1), (3.1.18), with ~ = a = 1, ¢ = 0, the Piola stress components involved in equation (2.4.7) take the form DrR = # [ r - l R D ~ z = # e r r -2,

(R')-3],

D ~ = # [(R') - r ( r 2 + a2)R-3],

Dz~ = #a [(R 2 + r 2 + a2)R -3 - r - ' R ' ] ,

Dzz = # [r-IRR '-

( R 2 + a2)R-2] ,

(3.2.12)

b a = ~2 or

Substituting equation (3.2.12) into equation (2.4.8), we get the differential equation with respect to the function R(r): 3R" - r - l R ' + (r 2 + a 2 ) R - 3 R '4 = O.

(3.2.13)

Let r0 be the radius of solid circular cylinder, whose axis coincides with the dislocation axis in the reference configuration (i.e., before the generation of

3.2 The Stress Field due to Screw Dislocation

77

dislocation). On the load-flee lateral surface of the cylinder we have DrR -- 0 that, by equation (3.2.12), implies r o l R ( r o ) = [R'(r0)] -3.

(3.2.14)

At first glance, it seems to be quite natural that the second boundary condition on the function R ( r ) has to be stated as follows: n(0) = 0.

(3.2.15)

Let us show that the equation (3.2.13) has no solutions satisfying the condition (3.2.15). Indeed, the equation (3.2.13), on replacement R ( r ) = exp f(r),

(3.2.16)

3f" 4- 3f '2 - r - i f ' 4- (r 2 4- a2)f '4 = O.

(3.2.17)

reduces to the form

The boundary condition (3.2.15) is replaced by lim f ( r ) = - c ~ .

(3.2.18)

r---,0

From equation (3.2.17), denoting f ' ( r ) by y(r), we obtain 3y' 4- 3y 2 - r - l y 4- (r 2 4- a2)y 4 = 0.

(3.2.19)

In view of equation (3.2.18), the behavior of y(r) as r ~ 0 is described by the relation lim y(r) = c~, (3.2.20) r--,0

moreover this function has a nonintegrable singularity at r = 0, namely, its order is r -1 or more. Indeed, setting = c , (1 +

x ( r ) --* 0 as r --, 0

and integrating, we get f(r) = -

y(r) dr + C2 = -

C l r -~ dr -

C ~ x ( r ) r -~ dr + C2, (3.2.21)

where C~, 6"2, and p are constants. If c~ < 1, the expression (3.2.21) has a finite limit as r ~ 0 that contradicts to equation (3.2.18). Therefore, c~ _> 1. Bearing equation (3.2.20) in mind in the analysis of asymptotic behavior of y(r) for small r, we can neglect 3y 2 in comparison with a2y 4, as well as r 2 with a2; these imply 3y', - r - l y , 4- a 2 y,4 = O, (3.2.22) where y , ( r ) is the approximation of y(r) as r ~ O. The boundary condition for equation (3.2.22) is the requirement that

78

3. Exact Solutions Stressed to the Problems on Volterra Dislocations lim y.(r) = c~.

(3.2.23)

r--*0

The general solution to equation (3.2.22) is =

1 (a2r + Cr_l)_l/3

C = const.

On satisfying the condition (3.2.23), we obtain

l(a2r)-1/3 Thus, as r ~ 0, we have

y(r) ~ ~/2a-2r -1/3. It follows that the order of singularity of y(r) at r = 0 is less than 1; thus f ( r ) has no singularity at zero, i.e., it does not satisfy condition (3.2.18). This contradiction shows that the equation (3.2.17) has no solution satisfying condition (3.2.18). The lack of solutions to (3.2.13) which vanish at r = 0 means that that screw dislocation generation in a solid cylinder made of the Blatz-Ko material causes appearing a cylindrical cavity about the cylinder axis. Note that axial pores really exhibit in thread-like crystals (Berezhkova 1969), this can be associated with the existence of dislocations. In (Cottrell 1969), it was shown a possibility of appearance of a hollow dislocation core. Such impossibility to satisfy the condition (3.2.15) makes us to change the set up of the problem and to consider the solid cylinder in the reference configuration as the limit state of hollow cylinder as the cavity radius rl tends to zero. The cavity surface is load-free so the boundary condition in the new set up is lim D~R(rx) = 0 (3.2.24) r l ---*0

or, in view of equations (3.2.12), (3.2.16), lim e x p [ 4 f ( r l ) ] f ' 3 ( r l ) r ~ l =

1.

(3.2.25)

rl---*0

As was shown in the previous Sect. 3.1, for the problem of disclination in semi-linear material, conditions (3.2.15) and (3.2.24) are equivalent, that is not true for the problem of screw dislocation. We are seeking a solution to equation (3.2.17), which satisfies (3.2.25), of the form

f ( r ) = A + f i r ~ + f2 ra2 + . . . ,

0 < ol1 < ol2 < ... (3.2.26)

A = In R1,

R 1 - R(0),

R1 being the cavity radius, fk, ak being some constants. Substituting the solution (3.2.26) into (3.2.25), we get c~1 = 4/3,

fl = ~exp

(4) -~A

.

(3.2.27)

3.3 Solving the Wedge Disclination Problem

79

The rest constants fk, ak (k = 2, 3,...) are determined sequentially by equation (3.2.17). Within the accuracy of r4-order, we have f(r) = A +

fl/'4/3

-

1 -~f27"8/3 -- 64 ,.2 ¢4~.10/3 405 ~

J1"

•

(3.2.28)

The constant A (and thus the cavity radius) is determined by the condition (3.2.14) on the external surface of the cylinder.To do this, we will use numerical solving the equation (3.2.13) as follows. Given the dislocation parameter, a, choose a sufficiently small positive number, ~ in such a way that, within a specified range of accuracy, the neglected expansion terms in (3.2.28) have no effect in the value of f(~). Then using (3.2.28), (3.2.16), we express the values of R(~) and R'(~), used as the initial conditions in Cauchy problem for equation (3.2.13), in terms of A. The boundary condition (3.2.14), within the given accuracy, determines the choice of A. The numerical results showed that for the range 0,002rr0 _ b _ 0, 2ur0, the cavity radius, R(0), is proportional in a high accuracy to the Burgers vector magnitude, b: R(0) = 0, 2557r-lb. Using equations (3.2.12), (3.2.16), (3.2.28), as well as the relations expressing the true stresses in terms of Piola stress components, aR -- r R - 1 D r R ,

a¢ = ( R ' ) - I D v ¢ ,

az = a(RR')-lD~z

7-¢z - ( R ' ) - I D ~ z ,

+ r(RR')-lDzz,

we obtain the following limiting relations as r ~ 0: fir ~

O,

a ¢ ---+ O,

O'Z ~

#,

T¢Z ~

O.

The screw dislocation energy per unit length of the cylinder is A = 2~-

fo

W r dr,

where the elastic potential, 14~, is defined by expressions (3.1.17), (2.4.5) with ¢ = c~ = ~ = 1. Equations (3.2.16) and (3.2.28) show that 14~(r) has a singularity of the r -2/a order as r --. 0 so that the energy .,4 is finite. Thus for the Blatz-Ko compressible material, the nonlinear set up of screw dislocation problem eliminates singularities in all the stress components and makes the screw dislocation energy to be finite.

3.3 Solving the W e d g e Disclination P r o b l e m with U s e of the

Compatibility

Equations

In Sect. 2.1, we pointed out the method to study the stressed state of elastic bodies with isolated defects; it was based on finding the strain tensor components by the equilibrium and compatibility equations with use of integral

80

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

relations determining the defect characteristics. Following (Zubov & Karyakin 1987), let us illustrate the method with an example of the disclination problem having solved in Sect. 3.1 in another way. Consider the plane deformation; using the cylindrical coordinates in the reference configuration, r, ~, z, we shall seek the Cauchy strain measure in the form A = Gll(r)erer + r-2G22(r)e~oe~o+ ezez. (3.3.1) The compatibility equation (1.4.8) takes now the form

G,xG22d2G22 ------5dr

1dG22 (G22dG11 2 dr

dr

dG22) ~- G l l ( J r

-- 0.

(3.3.2)

Using positive definiteness of tensor A, let us introduce positive functions A(r), B(r)such that

G~(r) = A2(r),

G22(r)= B2(r).

(3.3.3)

By these, equation (3.3.2) becomes d2B A~ dr 2

dB dA =0. dr dr

Integrating this equation and denoting the integration constant by In h (h > 0), we obtain IdB/dr] = hA; in this, we can omit the symbol of modulus but then cannot pre-define the sign of h (it may be positive or negative). The case of ,~ < 0 ( or dB/dr < 0) corresponds to the deformation of eversion of a ring and so will not be discussed here. Thus in what follows, we shall omit the modulus symbols, considering h to be positive, dB

dr

= hA.

(3.3.4)

Relation (1.4.20) for the Frank angle now takes the form

K =

Ull dr

~~

- 1

d~.

(3.3.5)

Here, Ull and U22 are the components of the stretch tensor, U, in the orthonormal basis of the cylindrical coordinates. In view of equation (3.3.3), Ull(r) = A(r), U22(r) = r-lB(r). By (3.3.4), relation (3.3.5) reduces to g = 2 r ( h - 1). Thus given the Frank angle, we find out the constant h: 27r+K h= ~ . 27r

(3.3.6)

Now the rotation field (Sect. 1.4) is X. = ( h - 1)~o. We assumed that the cut was along the line ~ = 0, besides, X(A40) = 0 at Ado on the cut.

3.3 Solving the Wedge Disclination Problem

81

In the above, we did not consider the solution of equation (3.3.2) which was described by the relation d B / d r = O. For this solution, K = - 2 r , thus the Frank vector, q, by equation (1.4.21), is equal to zero. This particular case corresponds to the choice of ~ = 0 in equation (3.3.4). Rearranging expression (1.4.20) with respect to parameter ~, we obtain the relation = [(1 - exp(ig)][z0 - ~-lB(r0)] which demonstrates that the representation (3.3.1) enables us to solve the wedge disclination problem whereas this representation is not sufficient to solve the translational dislocation problem. Indeed if ~ =fi 0, setting K = 0 we obtain that ~ - 0 too, that m e a n s t h a t there is no defect in the body. For a semi-linear material with elastic potential (3.1.7), the Kirchhoff stress tensor corresponding to deformation (3.3.1) takes the form P = [A + 2# + A r - I A - 1 B -

2(A + # ) A - 1 ] e r e r

+ [A + 2# + A r A B - 1 - 2(A + # ) r B - 1 ] e ~ e ~ + A(A + r - l B -

(3.3.7)

2)ezez.

Substituting these into the equilibrium equation (2.1.3), we find out that the second and third equations of (2.1.3) are satisfied identically and the first, on some rearrangement, now is (A+2,)\

~mdm ~+r-

1m2_r_2sdS~ dr]

+2r-'(A+#)

(as) ~r -m

=0.

(3.3.8)

Solving the system (3.3.4), (3.3.8), we can write A ( r ) = C l r ~-1 + C2r-~-I -J-

1 (1 + ~)(1 - v)'

B ( r ) = C l r ~ - C2r -~ + (1 + ~)(1 - v)r,

A v = 2(A + #)"

Here, C1 and 6'2 are arbitrary constants, v is the Poisson's ratio. The constant is expressed in terms of the Frank vector by relation (3.3.6). To determine the constants, let us consider the boundary conditions. We assume that the ring surfaces (r = rl and r = r0 ) are load-free, therefore the boundary conditions may be written as (A + 2#)A(r) + r - l A B ( r ) - 2(A + #) = 0, when r = rl, r0. The constants are Ci -

1-2v

~

1-v

1+~

r~+l_r~+l r2~-r~ ~ '

(3.3.9)

r~-1 _ r~-lr~+lr~+l C 2 = l _ 1v l + ~ ~

~o ~ _ r 2~

•

82

3. Exact Solutions Stressed to the Problems on Volterra Dislocations Let the disk be solid, that is rl = 0. From equations (3.3.9), it follows Ci =

1 - 2tJ

1-v The expressions for stretches are

Vll -U22 --

1

1 - 2L,

1+~

~

~pn-1

+

1-~ 1+~

2~,

'-=,

c =o.

1

(1+~)(1-~)'

t¢ p ~ - l +

1-~ 1+~

p=~.

(1+~)(1-~)'

r0

By equations (1.1.3), (1.1.19), the principal stresses, ak (which are the eigenvalues of the Cauchy stress tensor, T) coincide with the corresponding eigenvalues of tensor J - 1 U . P . U, defined by formulae (3.3.7), (3.3.9); on application of these formulae, we find out the relations that coincide with (3.1.10), 2#(p ~-1 - 1) 0"1 = (1 - 2~)p ~-1 + 1'

2#(top ~-1 - 1) 0"2 = (1 - 2~)~p ~-1 + 1'

2#v(1 - ~)(1 + ~)(2~p ~ - 1 - 1 - ~) 0"3 = [(1 - 2~)~p ~-1 + 1] [(1 - 2L,)p ~-1 + 1] ~" Let us now study the case of ~ - 0 in equation (3.3.4), in which the Frank vector is zero. Now B ( r ) = Bo = const, /3 = - 2 n i B 0 and thus the Burgers vector is not zero. The equilibrium equation (3.3.8) reduces to d A / d r + A / r = (1 - / / ) - I T -1 so that

A(r) = ( 1 - L,B o / r ) ( 1 - ~)-1 wherein we have taken into account that the surfaces r = r0, r = rl are load-free. The principal stresses are 0-1 ~ 0,

0"2

=

2#(B0 - r)(r - ~Bo).

(3.3.10)

The representation (3.3.1) describes straightening of the cut cylinder into a rectangular block; the cut surfaces rotate through the angle -27r, and in view of equation (1.4.21), their relative displacement is determined by the vector 2~Boe2. The forces on the cut surfaces of the cylinder to straighten it, can be reduced to the resultant Q, Q =

0"2(r)A(r) dr = 2#(1 - u) -1 (rl - ro + Bo ln(ro/rl)) 1

and the bending moment per the unit length of cylinder.

3.4 Conjugate Solutions in Nonlinear Elasticity

83

The constant B0 can be restricted by the condition Q = 0, so B0 = ( r 0 On this choice of B0, for straightening of the cut cylinder it is sufficient to apply only a bending moment to the cut banks.

r~) ln-l(ro/rl).

3.4 Conjugate Solutions in Nonlinear Elasticity and Their Application to the Disclination Problem This section establishes existence of solutions of nonlinear elastostatics which are conjugate in a certain sense (Zubov 1992). Namely, it will be proved that each solution of the equilibrium equations for homogeneous isotropic elastic body generates a new solution expressed in terms of the former and related, in general, to another material. This enables us to extend the set of known exact solutions in nonlinear elasticity and to obtain, in particular, some new solutions to the nonlinear dislocation and disclination problems. Starting from the elastic-body constitutive relation (1.1.20), let us assume that the specific potential energy, ~V, is a function of A -I being the inverse to the Cauchy strain measure. Let us obtain the following representation for Cauchy stress tensor T = - 2 ( det F)F-(d~4]/dA-1) • FT;

F

=

C -1

=

Gradr,

C = grad R,

r = R =

xsis,

Xnin,

Grad = grad =

(3.4.1)

ikO/OXk,

isO/Oxs,

where xk are Cartesian coordinates of the particle in the undeformed (reference) configuration, i.e. Lagrangian coordinates, ik are the frame unit vectors, Xn are Cartesian coordinates of the particle in the deformed state, i.e. Eulerian coordinates, R is the radius-vector of the particle after deformation, grad is the gradient operator in the reference configuration, F is the inverse deformation gradient, r is the radius-vector of the particle in the reference configuration, at last, Grad is the gradient operator in Eulerian coordinates. Taking into account that A -1 = F T. F, we transform (3.4.1) into T

=

W'E-

(0VV/0F). F T, (3.4.2)

W ' = (det F)~Y = j - l ~ y , ]/Y~ being the potential strain energy per unit volume of deformed body. We assume that the mass forces are not applied. Take Eulerian coordinates for independent variables, then the continuum equilibrium equations, in view of equation (1.1.18), are Div W = 0, (3.4.3) Div being the divergence operator in Eulerian coordinates.

84

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

For a homogeneous body, the specific energy W' depends on the coordinates Xk only in terms of tensor F(Xk), i.e. the dependence of W' on Xk is not explicit. Using equation (3.4.2), let us now find Div T:

is. 0 [ W ' E - (OW'/OF) . F T]/OXs --- ( ( ~ ) ' / ( ~ F m n ) ( C ~ g m n / ( ~ X s ) i

s -- Fkn(~((~Vt/c~Fsn)/(~Xsi

k

(3.4.4)

Finn = ira. F . Q. Since Fsn = OXn/OX~, we have OF~n/OXk = OFkn/OX~, hence (OW'/OF

)(OFm

/aX

(3.4.5)

)L =

By (3.4.4), (3.4.5), the equilibrium equations take the form Div K = 0,

K = OW'/OF,

F = Grad r.

(3.4.6)

This system can be evidently reduced to equations in unknowns x8 for which Xk are independent variables. Compare equations (3.4.6) with the equations of nonlinear elastostatics in Lagrangian coordinates (See Sect. 1.1), div D = 0,

D = OW/OC,

C = grad R,

(3.4.7)

where D = J C - T . T is Piola's stress tensor, div is the divergence operator in Lagrangian coordinates. The systems (3.4.6) and (3.4.7) are very similar, on passing from one to another, the roles of the reference and the deformed configurations are mutually interchangable. Let W and W* be the densities of potential strain energies for two elastic materials, respectively. We prove the following statement. If there hold the relations W = #(A),

4) being solution satisfies The and the is valid.

W* = v / det A #(A -1),

(3.4.8)

a scalar isotropic function of a tensor argument, and R = f ( r ) is a to the system (3.~. 7) for the first material, then the function r = f (R) the equilibrium equations (3.~.6) for the second material. proof is based on the analogy between the systems (3.4.6) and (3.4.7) fact that • is an isotropic function, that means that formula (1.1.22) In view of material isotropy, we obtain ,P(A -~) = # ( A T. A -1. A) = #(A),

(3.4.9)

where A is the rotation tensor defined in equation (1.1.3), X is the Almansi strain measure. Furthermore, according equations (1.1.4), (3.4.8), (3.4.9), we have W = ~(A) = ~ ( C . C ~) = ~(C),

~)'* = J-1W* = qb()~) - ~ ( F . F T) = ~(F).

3.4 Conjugate Solutions in Nonlinear Elasticity

85

Hence the elastic potential W is expressed in terms of tensor C as well as the elastic potential W ~* in terms of F. Therefore we have obtained that the systems (3.4.6) and (3.4.7) are completely identical, Q.E.D. So each solution R = f(r) to the equilibrium equations for homogeneous isotropic body generates a new solution R* = g(r), expressed in terms of the former and, in general, related to another material. This pair of solutions will be called the conjugate solutions. The function g is the inverse to f thus the conjugate solutions describe mutually inverse deformations of elastic body. The two elastic materials, the specific energies of which are related by condition (3.4.8), will be called the conjugate materials. According to equation (3.4.8), the function of specific strain energy W* is derived from the function W by replacing the argument A by A -1 and multiplying the latter by v / det A. This conjugation operation possesses the reciprocity property, (W*)* = W. As is known (Lurie 1980), a scalar isotropic function of a symmetrical tensor A is a function of three scalar arguments, namely of the invariants of A, I1 = tr A,

1 tr 2A - tr A 2) , /2 = ~(

Ia = j2 = det A.

Therefore #(A) = ¢(I1,/2, h). If the condition ¢(I1,/2,/3) + a = V~3¢(12131, Ii131,I31),

(3.4.10)

a being an arbitrary constant, holds for all possible positive I1,/2, I3, then W* = 142 up to an additive constant whose exact value is not important in further consideration. In this case, the conjugate solutions relate to the same material. An isotropic material with the specific energy satisfying equation (3.4.10) may be called the self-conjugate material. Note that for solving the system (3.4.6) it is not necessary to use Cartesian coordinates Xk. The operators Div and Grad are of invariant nature and so we can apply them in arbitrary curvilinear Eulerian coordinates used as the independent variables. In an incompressible elastic body, the Cauchy stress tensor is defined by strains up to an additive spherical tensor; therefore, the constitutive relation (3.4.2) for incompressible material changes as follows T = - ( 0 W / 0 F ) . F T - pE,

det F = 1,

W' = W,

(3.4.11)

where p is the pressure which cannot be expressed in terms of deformation. The theorem on conjugate solutions is true for incompressible homogeneous body too. To prove this, we set K = 0 W / 0 F + (W + p)F -T,

(3.4.12)

so, in view of equation (3.4.11), it follows that T = WE-

K . F T.

(3.4.13)

86

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

With regard for (3.4.3), the equilibrium equation for a homogeneous body (3.4.13) takes the form tr [(OW/OF - K ) . OFT/Ox~]i~ -- (Div K ) . F T = 0. In view of equation (3.4.12) and the incompressibility condition, we have tr [(OW/OF - K ) . OFT/Oxs]i~ = = - ( W + p)tr (F -1. O F / O X s ) i s = - ( W

+ p) Grad ( det F) = 0.

Therefore, the equilibrium equation system for incompressible homogeneous body in Eulerian coordinates becomes Div ( O W / O F - qF -T) = O,

F = Grad r,

det F = 1,

(3.4.14)

q = -W-p.

The system of equations for incompressible body in Lagrangian coordinates, which is similar to equation (3.4.7), becomes as follows div ( O W / O C - p C -T) = O,

C = grad R,

det C = 1.

(3.4.15)

For the identity of systems (3.4.14) and (3.4.15), to any solution R = f ( r ) , p = h ( r ) of equations (3.4.15) for isotropic incompressible material with specific energy W = ~(A) = v)(C), there corresponds a solution r = f ( R ) , q = h ( R ) of equations (3.4.14) for the material with specific energy W* = ~(A -1) = ~(F). Setting I3 = 1 in equation (3.4.10), we obtain the self-conjugacy condition for incompressible material, (3.4.16)

¢(I1, h ) + a = ¢(/2,/1)

Under the plane deformation of incompressible body, A1 = A~ 1, Aa = 1, An being an eigenvalue of A. Thus I1 = / 2 . Therefore, under the plane deformation, any incompressible isotropic material is self-conjugate, i.e., the conjugate solutions are two different displacement fields in the same elastic body. This result is a new proof of A d k i n s ' theorem on reciprocal solutions in nonlinear plane elasticity (Green & Adkins 1960); the above theorem generalizes Adkins' theorem to the case of three-dimensional problem of nonlinear elastostatics as well as to the case of compressible material. Let us apply the theory of conjugate solutions to the wedge disclination problem for a circular cylinder. Consider an elastic compressible material that is conjugate to an harmonic (semi-linear) material (see equation (3.1.7)). The specific energy W' for this new material, in view of equation (3.4.8), takes the form W'

= 1A tr 2(U -1 - E ) + # 2

where A, tt are some constants.

t r ( V -1 -

E) 2,

U -~

=

(F T. F) ~/2,

(3.4.17)

3.4 Conjugate Solutions in Nonlinear Elasticity

87

Consider the disclination problem, taking the Eulerian cylindrical coordinates R, ~, Z, in contrast to Sect. 3.1, as the independent variables. Following the semi-inverse approach, set the deformation as follows r = r(R),

~ = ~-1~,

z = Z.

(3.4.18)

Geometric meaning of the positive constant a is the same as in Sections 2.4 and 3.1. By (3.4.1), the inverse deformation gradient, corresponding to equation (3.4.18), is dr r~ - 1 F = -~eRer + R e~e~ + ezez. (3.4.19) Using equations (3.4.6), (3.4.7), and (3.4.9), we find the stress tensor K involved in the equilibrium equations

K= A ~-~+ R + a

+2.

+)~

~--~-1

eRer

R

eCev

1

(3.4.20)

( dr rK;-1 ) ~--~+---R----2 eze~.

Equilibrium equations (3.4.6), in view of equation (3.4.20), become d2r dR 2

1 dr R dR

r (aR) 2

2(A + #) 1 - a =0. A + 2# aR

(3.4.21)

On replacement of ~-I for a in equation (3.4.21), we obtain an equation similar to equation (3.1.9) of the disclination problem for semi-linear material; the only difference is that r and R interchange mutually their positions. This agrees with the theory of conjugate solutions. Having restricted ourselves to the case of solid cylinder, we find a solution of equation (3.4.21) satisfying the condition r(0) = 0, t~

r = C R 1/~ +

/2--

( 1 - u)(1 + a)

R)

(3.4.22)

+

The constant C is determined by the load-free condition on the cylinder surface R = Ro, e R " T = 0, that is c -

- 2

)(1 +

4 ( 1 - u)

.t ~0

*

(3.4.23)

In view of equations (3.4.2), (3.4.6), (3.4.20), (3.4.22), and (3.4.23), the Cauchy stress tensor is

88

3. Exact Solutions Stressed to the Problems on Volterra Dislocations T

= O R e R e R -Jr- o4~ec~e4~ -Jr- O z e z e z ;

CrR-- 1--P

[

[1

1 ]

E

'

oe = 1 K ~, 1 _ ~ -1 # [(1-2~,)(1-t-~)~ ( R ) az = (1 - v) 2 8n 2 ~

)1/~-1

( 1 - 2v2)(1 + ~ ) ( R 2~ R00

(3.4.24)

2(1/~-1) -

2t~

+ v(1 - 2 v ) - (1 + ~)2

+1]

As is shown by formulae (3.4.24), when ~ < 1, the stresses remain bounded as R -~ 0, i.e. they have no singularity on the disclination axis. If ~ > 1, the stresses increase unboundedly as R --, 0. This is different of the stress behavior in the disclination problem for semi-linear material (See Sect. 3.1), in which the stresses are bounded for all values of ~.

3.5 The Edge Dislocation in a Nonlinearly Elastic M e d i u m The problem of rectilinear edge dislocation in nonlinear statement is much more complex than the nonlinear problems on screw dislocation or wedge disclination that have been considered above in this Chapter; this is caused by the fact that the latter problems are one-dimensional in a certain sense, for they reduce to ordinary differential equations. For the edge dislocation, the fields of stresses and strains depend on two coordinates referred to the plane which is orthogonal to the dislocation axis, so they are defined by a system of nonlinear partial differential equations. It is important to find out an exact solution to the problem since any approximate method cannot give correct distribution of stresses near the dislocation line. We shall find out a solution to the two-dimensional problem of edge dislocation in an unbounded non-linearly elastic medium with use of a representation of solution for nonlinear equilibrium equations in terms of analytic functions of complex variable. Such representations are known for the harmonic material model, considered in Sections 3.1, 3.2, and 3.3. The presentation of this section follows the lines of (Zubov & Nikitin 1994). We shall describe the plane deformation of continuum by the complexvalued function (1.4.4). The equilibrium equations in complex coordinates take the form of a single complex relation, OD{ OD{ 0¢ ~- - ~ = 0,

D~ = f ~ . D . f~,

(3.5.1)

where D~ is a component of Piola's stress tensor in the complex basis introduced in Sect. 1.4. From the constitutive relation for the harmonic (semi-linear)

3.5 The Edge Dislocation in a Nonlinearly Elastic Medium

89

material, D = [~( tr U - 3 ) - 2~t]A + 2#C; C = grad R,

U = ( C . cT) 1/2

(3.5.2)

A -- U - 1 . C

A, # = const, with regard for equations (1.4.5),(1.4.6),(1.4.11), and (1.4.12), we find OZ

•

OZ

= 2(~ + ~ ) FOz / _ 2(A + #)eiX;

(3.5.3)

Oz D~ = 2#-~. Substituting expressions (3.5.3) into equation (3.5.1), we obtain

(

or

0 o-~

D~+

oz)

2.~

=0

(3.5.4)

From equation (3.5.4), it follows that Oz = 2(A + 2~)~--~ Oz _ 2(A + p:)eix D~ + 2#~-~ is an analytic function of complex variable ¢; we denote it by 2#~'2(0, then OZ -- 2()~ + # ) e ix -- 2#(p'2(~'). o¢,

2()~ + 2 # ) - ~

(3.5.5)

Assuming the quantity

°( /

OZ

= 2(~ + 2, )

OqZ -

2(~ + ~)

to be positive, on the basis of equations (1.4.12), (3.5.5), we can write a

• = 2#¢p'2(~'1, ~O z ) eix (3.5.6) • @x =

~'(¢) ~'(0

Using equations (3.5.5), (3.5.6) we obtain

90

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

Oz = it ~,2(()+ ~ + , ~'(~) o( A+2# A + 2#~,(()

(3.5.7)

By integrating equation (3.5.7) with respect to (, we have

z(¢, ¢)= a +# 2~ f ~,9(~)d~ + -

~+' (~(~) +¢-~), + 2, ~,(~)

(3.5.s)

where ¢(~) is an analytic function. Differentiating expression (3.5.8) with respect to ~, we get 0(0--~z=

AA+# . 2# (~(()~"(() +~,9(() _ ¢,(())

(3.5.9)

The formulae (3.5.7)-(3.5.9) present us the general solution to the equilibrium equations of nonlinear plane elasticity for harmonic material. This solution, expressed in terms of two arbitrary analytic functions ~(~), ¢(~), was derived earlier in the papers (John 1960; Lurie 1972) in other ways. Let us formulate the force boundary conditions on the boundary of plane domain, specified by a curve equation ~ = a(s), s being the arc length, in terms of complex potentials. Using the decomposition of external load vector, r/, normalto-boundary-curve vector n, and the Piola's stress tensor, D, in components of the complex basis, ----- 771fl -~- 772f2,

nl

. da = -l~ss,

n2

n -- n i l 1

d~ = i~--~s,

-~- n 2 f 2 ,

D = D~

f~

f~,

with regard for relations (3.5.3), (3.5.7), and (3.5.9), we reduce the boundary condition n . D = ~7 to n ~ [ ~'~(~)

~,~(~) - ¢'(~)]

~p'(a) ] - n2[~(a)~"(a ) ~

(3.5.10) A+2#

1 = 2"(A + ,) ~" On comparison of two decompositions of the Cauchy stress tensor,

T = t~e~ez + tssese3 = t~f ~f~ + tss fs fs we find out its complex components, expressed in terms of Cartesian ones, 1 (tll -~-t22), (3.5.11) t~ = -t 1 = 1 (tll -- t22 - 2it12).

3.5 The Edge Dislocation in a Nonlinearly Elastic Medium

91

From equation (1.1.19), relating the Cauchy and Piola stress tensors, we obtain

Jt I

Oz - i

Oz

-~D 1 -~- -~D~,

=

J=detC=

Oz 2

~

(3.5.12)

Ozl~

-

~--~ .

From equations (3.5.3), (3.5.11), and (3.5.12), it follows the representations of Cartesian components of Cauchy stress tensor in the plane problem for a harmonic material,

2a Oz t~ + t2~ = 7

- ~ - 4~,

2a Oz -l Oz Oz tll - t22 + 2it12 = ~ ~-~

(3.5.13)

O{0~'

taa=

:

~ o,- 1

)

,

a=2(A+2#)

~°z o,

.

Let us apply the above relations to solving the problem of edge dislocation in the domain D which is a plane with a circular hole of radius R1. The center of the hole is the coordinate origin. In this case, the Frank vector of the defect is zero, so the problem under consideration, according to the theory discussed in Sect. 1.4, can be reduced to finding the analytic in D functions satisfying the following conditions:

1) the functions Oz/O~ and Oz/O~, determined by formulas (3.5. 7), (3.5.9), are single-valued in variables ~, ~; 2) for any closed contour l, enclosing the circle I~1 = R1, there holds the relation Oz d¢ + ~-~ Oz d~ = bl + ib2, (3.5.14) ~-~ where the constants bl, b2 represent the Burgers' vector components in the Cartesian basis; 3) the boundary condition (3.5.10) is fulfilled on the circle of radius R1 when

771=0; For the circle, n 1 = a/R1 and n 2 = ~/R1. Dividing both of the sides of equation (3.5.10) by a, with regard for a~ = R~, we obtain

~'~(a)

¢p'(a) ~,(a)

&2 [~(a)~"(a) R~ ~'~(a)

¢~(a)] _ 0 '

lal _ R1"

d Extracting the total derivative ~aa' we rewrite the boundary condition as follows

92

3. Exact Solutions Stressed to the Problems on Volterra Dislocations ~~d ()

~-°-+

~°'=(a) - da ~o'(a)

~.2

(o)

R-712¢' .

0,

.

I<~1. R, .

(3.515)

-4) the stresses tan, defined by :formulas (3.5.13), tend to zero as ]~1 -+ oo; The displacement field in the problem under consideration is defined up to a rigid-body displacement; this enables us to impose an additional constraint which, without loss of generality, simplifies the form of solution. 5) the rotation X, determined by the formula eix = [OZlO~l-l(Oz/O~), tends to zero as I~] --+ co. Let us determine the form of functions ~o(~) and ¢(¢) following from the conditions 1, 2, 4, and 5. Rewrite eqiation (3.5.7) in other way Oz = (,I ~o,~(¢)1 + ~ + ~)eiarg~'=(¢) (~ + 2~)~

(3.5.16)

From this and single-valuedness of Oz/O( it follows that ~o'2(~) is a holomorphic function in D and thus it can be represented as a power series convergent in D: al a2 ~'=(¢)- ao + 7 + ~ + ' " = ~(~)' I¢1 > R1. (3.5.17) The condition 4 results in laol = 1; taking condition 5 into account, we get Co= 1. Integrating by parts, we find the expression for ~o(~)

(3.5.18) where u = x ~ { ) . First we obtain a representation for ~o(~) far from the dislocation axis. As the rotation X vanishes at infinity, we conclude that for sufficiently large I¢1, the absolute value of X does not exceed r/2. Since arg ~0'2(¢) = X, we obtain that Re~o'2(~) > 0 when I¢1 > Ro, Ro being a sufficiently large number. Thus, for Izl > Ro , if we fix one of the two branches of the square root then ~o'(~) = X / ~ is a single-valued function, and so it can be expanded into a series convergent in the domain I¢1 > R0, Cl

c2

(3.5.19)

~'(~1 = ~0 + 7 + ~ +

Comparing equation (3.5.19) with equation (3.5.17), we find that Co = 1 and 2cl = al. Integrating equation (3.5.19) we obtain dl d2 ~o(¢) = ¢ + Clln~+-~- + ~ + . . . ,

I¢l > Ro.

Single-valuedness of Oz/O¢ implies that ¢'(~) has the form

~'(~) - - ~(~°"'(~--------LJ e~ In ~ + qs, ((~), V"(0

I(I > Ro,

(3.5.20)

3.5 The Edge Dislocation in a Nonlinearly Elastic Medium

93

where q5,(¢) is a holomorphic in the domain [¢1 > R0 function. Integrating this representation, we have C1

¢(¢) = ~k~'~'-----7In ¢ + k In ¢ + ¢,(¢),

1¢1 > R0.

(3.5.21)

where ¢. is a holomorphic function in [¢[ > R0, k is a coefficient which has not been defined. Coefficients a I and k are found from equation (3.5.14) and the condition, following from (3.5.15), that the principal vector of forces acting on an arbitrary closed contour is zero. Equation (3.5.14), in terms of complex potentials, becomes

Oz

=

# A+2#

~,2(¢) d~ +

Oz de =

A + 2 # \~,(¢)

+ ¢(¢)

l

= bl + ib2,

where symbol []z denotes the jump in the expression in square brackets on passing along the contour l. With regard for representations (3.5.17) and (3.5.21), we get # 2Trial A + 2#

A + # 27rik = bl + ib2. A + 2#

(3.5.22)

The force balance condition for a closed contour gives us

~,2(¢) de -

~'(¢)

z

or

27ria, + 2rik = 0. On solving the system

of

(3.5.23)

equations (3.5.21), (3.5.23), we obtain bl + ib2 , 2ri

al = ~

k=

bl - ib2 2ri

Thus, functions ~(~), ¢(~), when ICI > Ro, have the following representations bl + ib2 ~(~) = ~ + A In ¢ + ~v,(~), A= 47ri ' _ (3.5.24) ¢(¢) = ~,(~--~In ~ - 2A In ~ + ¢,(¢), where ~,(¢), ¢,(() are holomorphic functions when [¢1 > Ro • Taking into account equations (3.5.18) and (3.5.24), we shall seek ~v(¢) and ¢(() in the form

94

3. Exact Solutions Stressed to the Problems on Volterra Dislocations qD(~) -- ~V/~I(~)-+- A l n ~2(n(ff)),

A ¢(~) = qo'(4)In Y2(~(4))- 2Aln4 +¢0,

(3.5.25)

where 12(~) - Y2(~({)) is an analytic function, and ~1(~) and ¢0(~) are holomorphic functions in D. It can be verified that functions ~, ¢ in the form of (3.5.25) satisfy the conditions 1, 2, 4, and 5 if there hold a2

2A

a3

~i({) = 1 + --~ + ~ + ~ + . . .

(3.5.26)

d [ A In Y2(()] = - ¢4(0 , d( 2V/ai(()

(3.5.27)

=

1

By (3.5.27), if ~1(() is known, we can find Y2(() by solving the differential equation (3.5.27). To determine ~ (~) and ¢0(~), we use the condition 3. Following the Cauchy integral method (Muskhelishvili 1966), we apply the Cauchy integral operator

1 f ~o(a)da 7 being a circle of radius Ri, to the boundary condition (3.5.15) and its complexconjugate expression. As is known, the Cauchy operator,/C, possesses the following properties 0 if ~o(() is holomorphic when [(I < R1; ~o(oo)- ~o(~) if ~o(~) is holomorphic when I~[ > R1;

K:[~o(()] =

=

1 f

qo(a) da

Taking these into account, as well as the representations (3.5.25), (3.5.26), we obtain the following nonlinear functional equations ~o'2(~) - 1

2~ + ~ 1i f~ [Alnl2(a) + ~(a)] ~o'(a)(a da -

¢~(~) = ~

fi~ln Y2(a) + qo(a) ~d(a)(a - ()2

~)2 = 0 ,

~2"

(3.5.28)

(3.5.29)

In physics, of most interest is the problem of edge dislocation in a solid, i.e., in the complex plane with a punctured point, ~ = 0, which is the limiting state of the domain D as Ri ~ 0. Therefore, it is sufficient to solve the functional equations (3.5.28) and (3.5.29) as RI ~ 0.

3.5 The Edge Dislocation in a Nonlinearly Elastic Medium Set

2A ~;1(~)-- 1 + - 7 . ¢,

From equations (3.5.27) and

(3.5.30)

95

(3.5.30)

we find

~?(~) = 1 + V/~'(¢).

(3.5.31)

1 - 4~;1(~) Substituting these into equations (3.5.25) and (3.5.28), we see that the lefthand part of the last equality tends to zero as R1 + 0. Thus, the functions ~1(~) and ~2(~), defined by relations (3.5.30), (3.5.31), constitute a solution to equation (3.5.28) as R1 + 0. The substitution of the above hi (~) and g2(¢) in equation (3.5.29) and the passage to the limit as R1 + 0 imply ¢~({) = 0, that is ¢0 = const. According to equation (3.5.8), the addition of a constant to the function ¢(~) corresponds to superposition of a rigid translational motion onto the body deformation. Therefore, we can set ~b0(~) = 0. On the basis of equations (3.5.8), (3.5.25), (3.5.30), and (3.5.31), we obtain an exact solution to the problem of edge dislocation in the infinite elastic body, _~ ~~ + ~ ~~(~) ,

_Oz = ~ ~ ( ~ )

o¢

~ + 2~

Oz = _

O(

:~ + 2~,~(~)

~ +_._

_

[

_ A ¢ ~ ( ¢_____-2)

(A + 2#)~(~ + 2A)

~(~)

21AI~ In 1 + ~(~) + A ( ~ + 4A)] n(~)

]

1 - n(~)

m

ul + iu2 = z(~, 4) - ~" = 2iA arg

2AA A+2#

f A + # (.~(~) 2A A + 2 # ~___-.~- { + __~ In 1 - ~ ( ( )

~(~) =

1~

_

(3.5.32) in [ ¢ l +

'

~,

where ul, u2 are the projections of the displacement vector of medium points to the axes x l, x2. Note that the choice of a branch of square root in equations (3.5.31), (3.5.25), (3.5.32) is arbitrary since the replacement of a branch is equivalent to the replacement of ~o(~) by -~(¢), that does not change the stress-strain state. From equation (3.5.8) it also follows that the solution of the problem does not depend on the choice of logarithmic branch in expressions for ~(~) and

¢(~). The stress field in a nonlinear medium with edge dislocation, determined by equations (3.5.13), (3.5.32), is

96

3. Exact Solutions Stressed to the Problems on Volterra Dislocations 4#(A -+- 2#) (#l~(~.)l 2 + A + #)I~(~)[ 2 - 4#,

tll - t22 + 2itl= = t33=

4#(A + 2.) ~i~i~ ~=(~)E(~' ~)'

2#(A + 2.) (t~(()l= - 1) J1(¢, ()

J1(¢, ~) = (,1~(¢)1 ~ + ~ + , ) ~ - I E ( ¢ ,

-

(3.5.33)

A+# [ A¢~(g) 21Al~ln 1 +

~(¢' ~ ) - ~(~ + 2~)

~(¢)

~(~)

~)1 ~ , ~((~)

1 - ~(~')

+ A ( ~ + 4A)].

Finding the asymptotics of (3.5.33) as I{I -+ oo, we obtain the stress field far from the dislocation axis: t l l + t22 "-" 1 - v

~- +

tll - t22 + 2it12 ,-~

'

~' = 2(,~ + tt)'

2~ A~- A;

1 - ~,

~2

(3.5.34) •

The expressions (3.5.34) coincide with ones of the edge dislocation problem in linear elasticity (Teodosiu 1982). Thus, the linear and nonlinear theories offer identical results far from the dislocation. On the other hand, the nonlinearity significantly changes the stress field near the dislocation axis. In particular, the limiting values of stresses as I~1 --+ 0, defined by (3.5.33), do not depend on the path of approach to the dislocation axis and prove to be finite, tll (0) -- --~/2-1(1 q- COS 2W),

t22(0) = - # v - l ( 1 + cos 2w), (3.5.35)

t12(0 ) = --#V -1 sin 2w,

t~(0) = 0,

where w is the slope angle of the Burgers vector towards the xFaxis. Therefore, the nonlinear approach eliminates the stress singularity at the dislocation axis, whereas linear elasticity brings the I¢1-1 singularity (Teodosiu 1982). Fig.3.1 to 3.3 show the stress distributions near the dislocation axis when bl = 0. On Fig.3.1, it is shown the line of constant stress tll corresponding to the value tll = 0 in the exact nonlinear solution obtained above. Fig.3.2 illustrates the same isoline for the linear solution. The diagonal size of Fig.3.1 and Fig.3.2 is ~b2. 1 Fig.3.3 represents the dependence of dimensionless stress t22/# on the coordinate Xl when x2 = 0; the dotted line corresponds to the linear theory solution; the scale for x 1-axis is 2b2. Thus, as was notedabove, nonlinearity significantly changes the stress distribution pattern near the dislocation axis. For small magnitudes of the Burgers vector when ~ -~ 0, the solution (3.5.32), (3.5.33) can be expanded into a series in powers of A. The first-order terms in that expansion coincide with the solution obtained in the framework

3.5 The Edge Dislocation in a Nonlinearly Elastic Medium

97

X2

X Y 0

Fig. 3.1.

X2

-t-

Xl

-t-

Fig. 3.2.

98

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

1oo

2o

"-2--T,

-4

2

4

-10g

Fig. 3.3.

of linear theory. However the expansion noted does not valid when ( - 0, and, in the immediate vicinity of the dislocation axis; the linear solution cannot approximate the exact solution even for fairy small IAI. The semi-linear material elastic potential under the plane deformation reduces to YY = 2(A + #)

(Oz)

~-~ - 1

2

Oz2

+ 2 , ~-~

(3.5.36)

From equations (3.5.32), (3.5.36) it follows that ~Y has a singularity of the order I~1-2 as ( ~ 0. So for the model of semi-linear material, the energy per unit dislocation length is found to be infinite, as well as in linear elasticity.

3.6

A Cavitation

near

the Line of Dislocation

or Disclination It has been established in Sect. 3.2 that for some models of compressible elastic material, a screw dislocation in a cylinder cannot exist without appearance of a cavity. In this section, following (Eremeyev & Zubov & Karyakin & Tchernega 1992), under the assumption of material incompressibility, we shall study the generation of cavities near the line of a wedge disclination or screw dislocation. The first investigation of discontinuoussolutionsdescribing appearance of cavitations, in the non-linear theory of elasticity was due to J.M. Ball (1982), he studied the radial-symmetrical deformation of an elastic ball. A possibility of generation of a cavity about the wedge disclination axis is shown in (Mikhaylin & Romanov 1986) by molecular dynamics methods.

3.6 A Cavitation near the Line of Dislocation or Disclination

99

Proceeding as in Sect. 2.4, consider the following transformation of the reference (undeformed) configuration into the actual (deformed) configuration: R = R(r),

4) = ~ ,

Z = c~z,

(3.6.1)

where r, ~, z and R, qb, Z are cylindrical coordinates in the reference and actual configuration, respectively. The transformation (3.6.1) describes the deformed state of a cylinder with wedge disclination. The case of ~ < 1 corresponds to the cylinder deformation arising on cutting the cylinder by the half-plane ~ = 0 and inserting a wedge of the angle of 27r(~- 1) into the cut; the case of ~ > 1 corresponds to the deformation occurring on removal of a sector 27r~-1 < ~ < 27r from the cylinder with fixing the cut edges together; simultaneously it is possible axial extension (~ > 1) or shortening (c~ < 1). From the incompressibility condition it follows that (3.6.2)

R ( r ) = v/r 2 + A l v i n ,

where A is an integration constant. Solving equilibrium problems for a solid cylinder, one normally assumes that R(0) = 0, which, together with equation (3.6.2), implies R ( r ) = r / x / ~ . Such solutions were called "regular" (PodioGuidugli & Vergaga Cagarelli & Virga 1986). We shall suggest another formulation of the boundary condition at r = 0 which leads, in some cases, to the appearance of another solution, a "singular" one. The equilibrium equations of elastic medium are div D = 0,

(3.6.3)

where div is the divergence operator in the reference configuration, D is the Piola stress tensor, S = OI/V/OC = (01/V/0U). A,

D = - p C - T + S,

(3.6.4) C = grad R,

U = ( C . cT) 1/2,

A = U -1 • C,

p is the pressure function to be determined, 142 = W( J1, J2) is the specific potential energy function that depends on the first and second invariants of the stretch tensor, U, the C is the deformation gradient. The constitutional equations of non-linearly elastic material are of the Bartenev-Khazanovich type (Lurie 1980), 142 = 2#( tr U - 3). The equilibrium equations (3.6.3), in view of equations (3.6.1), (3.6.4), reduce to dp _ 2#I-~ d-T -

1 + A

First let us consider the case of a hollow cylinder whose inner and outer radius are rl and r0, respectively. The inner and outer lateral surfaces of the

100

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

cylinder are load-free, that is n . D = 0, n being the outward normal to the undeformed lateral surface. Introducing the dimensionless quantities p = r/ro,

q = p/2#,

7 2 = A/r~,

Pl = r l / r o ,

we represent the boundary value problem of equilibrium of a hollow cylinder with wedge disclination as follows d___qq= 1 - ~ 1 dR ~ V/p2 + ~2' q(pl) =

Pl

(3.6.5)

1

(3.6.6)

jpl + 1

q(1) = ~

1

V/1 + 72.

(3.6.7)

In case of a solid cylinder, the boundary condition on the cylinder axis can be obtained by passing to the limit as pl --+ 0 in the boundary condition (3.6.6) for the hollow cylinder, it is q(0) =0. (3.6.8) Thus the boundary value problem of equilibrium of a solid cylinder with wedge disclination is given by equations (3.6.5), (3.6.6), and (3.6.8). Its solution takes the form 1- ~ [P dp q(P) v/7~ .,,., v/'p2 + .),2 (3.6.9) Substituting the solution (3.6.9) into the boundary obtain the equation to determine constant ~,:

condition (3.6.7), we

1

l-n=

.

(3.6.10)

V/1 -+-~2 In ((1 + V/I -I- ")'2) "/-1 ) Note that the right hand side of equation (3.6.10) is always positive, therefore the equation has a solution only when ~ < I. Thus, the formation of cavity at the axis is possible only for a negative disclination. The constant a, describing extension or shortening of the cylinder, can be found from an integral boundary condition on the cylinder edges, this condition represents the equality to zero of the axial stretching force and so it is OL2/3 =

~7 2 X/-~V/1 + 7 ~ 1+ ~ +

Studying the process of generation of cavitations in solids, we need to take into account the energy for the generation and expansion of the cavity. As in the fracture theory (Cherepanov 1974), we shall relate the cavitation energy to the surface energy of cavity boundary. The necessity of consideration of surface

3.6 A Cavitation near the Line of Dislocation or Disclination

101

energy to study singular solutions of non-linear elasticity was also noticed in (Podio-Guidugli & Vergaga Caffarelli & Virga 1986). We take the surface-energy functional in the form

Y2 = w f~ d~, where w is the constant coefficient of surface tension, Z is the cavity surface in the deformed state. Numerical results showed that if the surface energy is accounted for, a cavitation may occur only if the value of the parameter ~ = 1 - ~ exceeds a certain number depending on w. A most effect of the surface energy occurs for values of disclination parameter ~ such that the cavity radius is sufficiently small in comparison with the cylinder radius.

Thus, it is established that for the Bartenev-Khazanovich material, the problem of equilibrium of non-linearly elastic cylinder with disclination has two solutions, a "regular" one, that is when the deformed cylinder remains solid, and a "singular" one, when a cavity appears about the dislocation axis in the deformed state. To define the "preferability" of these solutions with co = 0 and a = 1, the related elastic energies of the cylinder were compared. Computations showed that the cylinder energy without cavity exceeds that of the cylinder with a cavity although their difference is comparatively small. However this means that the generation of cavity is energetically preferable and so the "regular" solution to the equilibrium problem for a cylinder with wedge disclination is unstable with respect to finite disturbances. It can be shown that cavitation in cylinder with wedge disclination is also possible for a more general constitutional law. We can study the problem of cavitation in a non-linearly elastic body with another linear defect, a screw dislocation, in a similar manner. The formation of screw dislocation in a cylinder (by making a cut along the line ~o = 0 and shifting one cut border with respect to another by a vector b parallel to the cylinder axis) is described by the following transformation (see Sect. 2.4) R = R(r),

~, = ~o,

Z = a~o + ~ z ,

(3.6.11)

where a = [bl/2rc is the dislocation parameter. Let the longitudinal stretch of a cylinder be equal to unit. Taking into account that the length variation does not bring new peculiarities in the solution of the problem. From the incompressibility condition, we get R(r) = v/r 2 + A. (3.6.12) As a model of non-linearly elastic material, we select the constitutive relation proposed by K.F.Chernykh and I.M.Shubina (Chernykh 1986): ]/Y = # ((1 +/3)J1 + (1 -/3)J2) ; J1 = t r U ,

(3.6.13)

J2 = t rU-1.

In view of equations (3.6.11)-(3.6.13), the equilibrium equations can be represented as

102

3. Exact Solutions Stressed to the Problems on Volterra Dislocations

dr

R

v/(R+r)2 +a 2 "

The boundary conditions for a solid cylinder (obtained, as before, from the boundary conditions for the hollow cylinder by passing to the limit) take the form q(0) = 0.5(1 -/3) V/¢2 _~2, q(1) =

1

1 ((1 +/3) + ( 1 - ~)~/e2 + (1 + v/l:~::~2) 2)

v/1 + ~ 2

wh~r~ q = p / 2 , , ~ = a/~o, "Y = 4 - ~ / ~ o = R(O)/~o. The constant ~/is a root of the equation In ( l + v / l + 7 2 ) ( 1 + V / I + ~ 2 / V 2 ) VII + 72 + V/~2 + (1 + y/~ 2 + 72) 2

_

1

=0.

V/1 + .),2

Numerical results for the solution of this equation for/3 = 1 showed that, within a wide range of change of the dislocation parameter, one can take

R(0) ~ 0, 231~0,

~ e (0, 0001; 0, 1),

(3.6.14)

with sufficient accuracy. As in the case of wedge disclination, the elastic energy of the cylinder with cavity (for 0, 0001 < s < 0, 1) is for 5-10°£ less than that for the solid cylinder. A relation of the form (3.6.14) holds if the variation of the cylinder length (a -~ 1) is accounted for. The axial shortening of the cylinder implies some increase in cavity radius.

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

4.1 General Relations in Nonlinear Theory of Asymmetric Elasticity Classical continuum mechanics, in particular, classical elasticity, is based on the model of simple material (Truesdell 1977); the free energy density and stresses at a particle of simple material are completely determined by the values of deformation gradient, C, and temperature, ~, in the particle; besides, the Cauchy stress tensor, T, is symmetric. The model of simple material perfectly describes behavior of medium in many cases however there are situations when we have to consider micro-nonhomogeneous structure of material; to these, polycrystal grained materials, polymers, composites, suspensions, liquid crystals, geophysical structures, and some others assume to attract ideas of micro-nonhomogeneity. To describe mathematically the physical-mechanical properties of above medium, continuum theories dealing with couple stresses and rotational interaction of particles are used. Such a model of continuum medium is called the Cosserat's continuum (by two brothers Cosserat published the fundamental work (Cosserat E. & Cosserat F. 1909) on the theory of materials having the couple stresses). The linear theory of couple stress elasticity has been abundantly addressed in the literature; a particular mention should be made of the works (Aero & Kuvshinski 1960; Koiter 1964; Pal'mov 1964). Few papers are devoted to the couple stress theory of medium subject to large deformations (Toupin 1964; Shkutin 1980; Zhilin 1982; Zubov 1990; Zubov & Karyakin 1990; Eremeyev & Zubov 1990). The most essential distinction between the results of couple stress elasticity and those of classical theory occurs when the stressed state of the body changes drastically, that is in some vicinity of stress concentrators such as corners, crack edges, dislocation and disclination lines and other defects. Thus the study of nonlinear effects of couple stresses in theory of dislocations and disclinations is of interest. The model of nonlinearly elastic Cosserat continuum suggests that any continuum particle has all the rigid body degrees of freedom. A position of particle in the deformed state is specified by the radius-vector R, while the particle orientation is determined by a proper orthogonal tensor H called the microrotation tensor. Following the principle of local action in continuum

104

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

mechanics, we suggest the function of specific (per reference configuration unit volume) potential strain energy of elastic continuum of the form W = )/V(R, grad R, H, grad H),

grad - rSO/Oq s, (4.1.1)

rk = Or/Oq k,

r s . rk = 5~,

S, k = 1, 2, 3,

where q~ are Lagrangian coordinates, r is the particle radius-vector in the reference (undeformed) configuration. By the principle of material frame indifference (Truesdell 1977), the strain energy density of elastic body, W, is invariant under rigid body motions of medium. The invariance of kV under translations results in the independence of kV (in equation (4.1.1)) of the argument R. The invariance under observer frame rotations implies that

W[( grad R ) . O, H . O, ( grad H ) . O] = W( grad R, H, grad H) for any orthogonal (that is (4.1.2), we obtain

O T :

0 -1

(4.1.2)

) tensor O. Setting O - H T in equation

W - W[( grad n ) . H T, (grad H ) . HT].

(4.1.3)

In equation (4.1.3), it was taken into account that H . H T = E, where E is the unit tensor. The relation (4.1.3) is a necessary consequence of the equality (4.1.2); it is easy to verify that it is sufficient for the invariance of the energy under rigid body motions. Taking into account the skew symmetry of tensors ( O H / O q k ) . H T (k = 1, 2, 3) we can represent the third order tensor ( O S / O q k ) . H T in terms of the second order tensor (rather, pseudotensor) L as follows

grad H . H T - - L x E, From

L = - ~ r l k[(OH/Oqk) . HT]x .

(4.1.4)

now on, T x will denote the vector invariant of a 2nd order tensor T,

Tx = (TskrSrk)x = Tskr ~ × r k. By equations (4.1.3), (4.1.4), the elastic potential, ~/Y, at a given material particle, relates with the deformation of a neighborhood of the particle by two 2nd order tensors: the strain measure, Y = (grad R ) . H T , and the bending strain t e n s o r , L. From equation (4.1.4), using the well known representation (see (Zubov 1982)) of a proper orthogonal tensor in terms of finite rotation vector,

H - S+~- S_ = S_. S+~,

1 S+ = E :k ~E x 0,

(4.1.5)

we obtain the expression for the bending strain tensor in terms of the microrotation vector O:

4.1 General Relations in Nonlinear Theory of Asymmetric Elasticity 1 L = 4(4 + 02)-1(grad ~). (E + ~E x {?),

~2 = e . e .

105 (4.1.6)

To derive equilibrium equations and boundary conditions for the nonlinearly elastic Cosserat continuum, let us invoke the Lagrange's variational equation 5 ~ }4]dv - 5'g = 0,

(4.1.7)

where v is the body volume in the reference configuration, 5'8 is the elementary work of external loads which is not, in general, a variation of a functional. Based on equations (4.1.3), (4.1.4), and (4.1.6), we have 5W = tr [D T. (grad 5R) + D T. (grad R × ¢) + G T. ( grad ¢ ) ] , ¢ = 4 ( 4 + 0 2 ) -1

(lo) 50+2

D = (01/V/0Y). H,

x~0

(4.1.8) ,

G = (01/V/0L). H.

(4.1.9)

Integrating by parts and using the divergence theorem, by equation (4.1.8), we get

(4.1.10)

- f(div o ) . eRdv -

[div G + (( ad

Cd .

The equality (4.1.10) determines an expression for elementary work of external forces,

(4.1.11) In equations (4.1.10), (4.1.11), s is the body surface with the normal n in the reference configuration, k and l' are the mass densities of volume-distributed forces and couples, respectively, ~o0 and tt 0 are the intensities of force and couple loads distributed over a, respectively; p0 is the material density in the reference configuration. From the variational equation (4.1.7) and the expression (4.1.11), it follows the equilibrium equations and the static boundary conditions,

div D + pok = 0,

div G + [( grad R) T. D]× + pol'= O,

n . D = ~o0,

n . G = tt o.

(4.1.12)

(4.1.13)

106

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

Since the requirement ¢ = 0 is equivalent to the condition 50 = kinematic boundary conditions turns into the displacement vector, u = and the microrotation vector, 0, to be given on the body surface. With for Piola's identity (1.1.17), the equilibrium conditions (4.1.12), (4.1.13) written in the geometry of deformed state as Div T + pk = 0,

Div M + Tx + pl' = O,

N . T = ~,

T - j - l ( g r a d R ) T . D,

N . M = tt,

0, the R - r, regard can be

(4.1.14) (4.1.15)

M = j-1 (grad R) T. CJ, (4.1.16)

J = det ( grad R), where T and M are the tensors of stresses and couple stresses, which are similar to the Cauchy stress tensor in the elasticity theory of simple materials, ~ and tt are the load per unit area of E in the deformed state, N is the unit normal to E, p is the material density in the deformed configuration. The tensors D and G are analogous to the Piola stress tensor in classical nonlinear elasticity. The first of the equations in (4.1.14) states the balance of all the forces acting on an arbitrary part of the body, and the second represents the couple balance. Unlike the simple material, now the Cauchy stress tensor, T, is not symmetric in the Cosserat continuum. The general equations of nonlinear theory of elasticity stated above for the bodies with couple stresses were obtained in (Toupin 1964) by other methods. The elastic potential for gyrotropic medium, W, is a gyrotropic function of Y and L, i.e. 14; satisfies

W ( O T. Y - O , O T. Y . O) = W(Y,L),

(4.1.17)

where 0 is an arbitrary proper orthogonal tensor. The tensors Y and L are similar to the Cauchy strain measure (see Sect. I.I). Interchanging the reference and deformed configurations, i.e. making the substitutions R --, r, grad -~ Grad, 0 -~ -e, we get tensors which are analogous to the Almansi strain measure (see Section I.I) in the classical nonlinear theory of elasticity,

y = ( G r a d r ) • H,

l=

4

4 + 82 (Grad0)

•

(E - 1 - E ) 2

x0

(4.1.18)

1 = 1RM(OHT/OQ u . U)x. z

The tensors y and 1 are called the second strain measure and second tensor of bending strain, respectively. It follows from equations (4.1.5), (4.1.6), and (4.1.18) that

4.1 General Relations in Nonlinear Theory of Asymmetric Elasticity

y-1 = H T " y . H,

y - 1 . l = - H T. L . H .

107

(4.1.19)

Setting O = H in equation (4.1.17), based on equation (4.1.19), we arrive at )/Y = )/Y(y, l) for gyrotropic material. In a similar manner, it can be shown that the stress tensor, T, and the couple stress tensor, M, in gyrotropic Cosserat continuum, depend on the medium deformation in terms of the second strain measure, y, and the second tensor of bending strain, 1. Constitutive relations (4.1.9) are somewhat modified when some constraints are imposed on the material, that is some restrictions on medium deformation, i.e. on the strain measures Y and L. In particular, the scalar constraint satisfying the condition of material frame indifference takes the form

7(Y, L) = O, and the vector constraint may be written as H T. s(Y, L) = 0. The constitutive relation for constrained material is obtained by introducing Lagrange multiplier (Lurie 1980)" D . H T = 0W1/0Y,

G . H T = OW1/OL,

(4.1.20)

where

)/V1 =

FY - pT(Y, L) for a scalar constraint, )/V + r/. H T. 8(V, L) for a vector constraint.

(4.1.21)

When both of the constraints are imposed, we must include the both additional terms into l/V1. In equation (4.1.21), p is an indifferent scalar, r / i s an indifferent vector. With the notation of r/. n T by q, the expression for W1, in case of vector constraint, can be written in the form

l/V1 = 14/+ q. s(Y, L). One of common constraints in nonlinear elasticity is the incompressibility condition which means that the volume of any part of body is unchangeable, 7(Y) = det Y - 1 = 0.

(4.1.22)

We can find an example of a vector constraint in linear mechanics of model of Cosserat pseudocontinuum (Nowacki 1975), when the microrotation is identified with the macrorotation of particle. Generalizing this to the case of finite strain, let us identify the microrotation tensor, H, with the elastic rotation tensor, A, from the polar decomposition of deformation gradient (see Sect. I.I), grad R = U. A, where U is a symmetrical, positive definite tensor. Thus, Y -- U and hence Y - yT. Therefore,

108

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

Y× = 0.

(4.1.23)

In case of small deformation, linearizing the tensors Y and L with respect to 0 and grad u, u = R - r being the displacement vector, we arrive at the strain tensor e and the bending-torsion tensor grad 0 which are used in linear couple-stress theory of elasticity, Y ~ E + e,

e = grad u + E x 0,

L ~ grad 0.

(4.1.24)

Linearizing the constraint equation (4.1.23), with regard for equation above, we will obtain the well known relation of the theory of Cosserat pseudocontinuum (Nowacki 1975), 0 = fi1 rot u. From equations (4.1.20) - (4.1.23), it follows that the constitutive relations for nonlinear elastic, incompressible Cosserat pseudocontinuum are T = - p E + (grad R) T. (0~V/0Y - q × E ) . H, (4.1.25) M = (grad R) T. O~Y/OL. H, where the scalar p and vector q are not expressed in terms of strains Y and L.

4.2 Weingarten's Theorem and Volterra Dislocations in Couple Stress M e d i u m U n d e r g o i n g Large Strains Consider the problem of determining the displacement and microrotation fields of the Cosserat continuum when the fields of tensors y and 1 are given as twice differentiable functions of Eulerian coordinates QN. From equations (4.1.5), (4.1.18), we have

OHT/OQ g = HN" H T,

I-IN = - E x (RN" 1).

(4.2.1)

The necessary and sufficient conditions for solvability of these equations with respect to H, are the following nine independent relations:

OrIs/OQ N - OIIN/aQ s = IIN" 1-Is - 1-Is" IIw.

(4.2.2)

The solution to equations (4.2.1), as in Sect. 1.3, may be written with use of multiplicative integral, A

H(M) =

(E + d R . H ) . H T,

H = RNHN,

(4.2.3)

J~4o

where A/[o is a point of the domain V, in which the initial value of tensor H0 is specified: H(A/t0) - H0, jr4 being a current point. In a simply connected domain V, the value of H(A/[) does not depend on the choice of a curve joining A/10 with A/[. Having found the H by formula (4.2.3), we can define the location of body particles in reference configuration by integration the equation (4.1.18),

4.2 Weingarten's Theorem and Volterra Dislocations r(A//) =

d R . (y. H T) + r(A40).

109 (4.2.4)

o

The necessary and sufficient conditions for the integral in (4.2.4) to be pathindependent in a simply connected domain are (4.2.5)

R g × (i)y/OQ N) + R g x y . I-IN = O.

The conditions (4.2.2), (4.2.5), consisting of 18 scalar relations, are the strain compatibility relations of nonlinear theory of couple stress elasticity. Similar equations for the strain tensors Y - E and L, given as functions of Lagrangian coordinates, were obtained in (Shkutin 1980). If the domain V occupied with the elastic body in deformed state is multiply connected, the displacements, u = R - r, and the microrotations determined by the formulas (4.2.3), (4.2.4), in general, will not be single-valued; their nonuniqueness can be remedied by making cuts which convert the domain into a simply connected one. But now the vectors r and 0 may have a jump on intersection of each cut. It can be shown by the method of Sect. 1.3 that the jump is described by the following formulas

H+ = f~. H_,

0+=4_w.0_

= 2(1 + tr~)-l[2×,

w+O_+

xw

(4.2.6)

r+ = f~. r_ + b,

where i2 is a proper orthogonal tensor that is constant for all points of each cut, w and b are constant vectors. The formulas (4.2.6) mean that if we cut a nonlinear-elastic Cosserat body, occupying a multiply connected domain in the stressed state, in which the Almansi-type strain measures y, I are continuous (as well as the stress tensor, T, and couple stress tensor, M), then in the unstressed state, the opposite cut borders differ in their positions by a rigid body motion. A similar statement for nonlinear elastic medium without couple stresses was established in Sect. 1.3. In the case of doubly connected domain, i2 and b are expressed in terms of strain tensor fields y, 1 by formulas similar to ones of Sect. 1.3, A

i2 T = H0. f

(E + d R . H ) . HoT, A4o

A

b = J d R ' . y(R').

(4.2.7)

(E + d R . II). H T + r0. (E - aT). A4o

Thus, we have established that in nonlinearly elastic bodies with couple stresses, defects in the form of Volterra dislocations can occur. As in Sect. 1.3, the defect parameters b and w will be called the Burgers' and Frank's vectors, respectively. The system of equations to determine the stressed state of

110

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

nonlinear-elastic Cosserat medium with Volterra dislocation with given characteristics b and w consists of the equilibrium equations (4.1.14), in which the tensors T, M are expressed in terms of y and 1, the compatibility equations (4.2.1), (4.2.5), and the relations (4.2.7). In a similar manner, one can consider the problem of finding of displacements and microrotations in a multiply connected domain, which is occupied by Cosserat medium in the undeformed configuration if the fields of tensors Y and L are given as continuous and twice differentiable functions of Lagrangian coordinates. Restricting ourselves to the case of plane strain described by the relations Xl - X,(x,, x2),

X2 - X 2 ( x ~ , x 2 ) ,

X3 - x3,

(4.2.8)

where xk, X k are the coordinates of medium points in the Cartesian basis {ek } before and after the deformation, respectively, we can simplify the set up of the problem on the stresses due to isolated defect, in particular, we can'obtain expressions for its characteristics in terms of ordinary contour integrals. Let us introduce the complex coordinates

= x l + ix2,

~ = x l - ix2,

z = X l + iX2,

Z = X l - iX2.

The plane deformation (4.2.8) is described by a complex-valued function Z -- Z(~, ~),

X 3 -- X3.

(4.2.9)

In the multiply connected domain occupied by the body in undeformed state, the tensors Y and L are given by L = L1 (~, ~ ) f l f 3 + L2(¢, ~) f 2f 3 , Y - Y~(~,~)f~f~

3 + f f3,

(~, ~ = 1, 2,

(4.2.10) (4.2.11)

where f~, f~ are the complex bases (1.4.3) associated with the complex coordinates ~, ~, f 3 = f 3 = e3. We shall seek H in the form H = exp(ix)flfl + exp(-ix)f2f2

+ f 3 f 3.

(4.2.12)

In this general representation of rotation tensor under plane strain, X is the particle finite rotation angle to be determined. Substituting equations (4.2.10), (4.2.12) into equation (4.1.4), we get OZ/O¢ = L1,

OX/O~ = L2.

(4.2.13)

We can write the solvability condition (4.2.13) with respect to X as OL1/O~ = OL~/O¢.

(4.2.14)

4.2 Weingarten's Theorem and Volterra Dislocations

111

Comparing the expression for the deformation gradient, grad R, answering the transformation (4.2.9), with g r a d / t = Y . H, derived from the definition of Y, with regard for equations (4.2.11), (4.2.12), we find Oz/O¢ = v l exp(~z),

exp(~x).

Oz/O2 = v7

In view of equation (4.2.13), the solvability condition for these equations takes the form

d y l l d z - d y l l d z + iL,Y)

-

L2Y

1

=

O.

(4.2.15)

Thus the equations (4.2.14), (4.2.15) are the compatibility equations under plane medium deformation which are equivalent to three real-valued equations. The further treatment is made in the same manner as in Sect. 1.4. In the case of doubly connected domain, we get the following expressions k:, =

X = X., + n K ,

(Lld~ + L2d~) + X0, o

K =

/

Lld~ + L2d~-

(4.2.16) z = zo + einK

d x" (Y1ld~ + Y2ld~) o

+ (1 + ~ + ... + e~°-~'~) ~ eiX.

(Y?<+

J

where X0 and z0 are given values of the function X and z at a point A/10 of the domain, n is a constant for multi-valuedness of the problem. The integrals with variable upper limit are evaluated over the curves which do not enclose the domain hole; the closed contour is through J~/10 and passes around the hole one time. On the cut converting the domain into a simply connected one, the limiting values of functions X and z are related by k:+ - X- = K,

z+ = z_ exp(iK) +/7,

r /7 = z0[1 - e x p ( i K ) + t exp(ix,)(Ylld( + Y21d~ •

(4.2.17) (4.2.18)

The relation (4.2.17) represents the statement of Weingarten's theorem in the case of nonlinear couple-stress theory of plane elasticity; it can be rewritten in the real-valued form u+-u_=4(4+w2)-lwx b = Re fie1 + Im fie2,

(R_+ ~wx R_)+b, • = 2 tg (K/2)e3.

(4.2.19)

112

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

4.3 N o n l i n e a r P r o b l e m s of S c r e w D i s l o c a t i o n a n d W e d g e D i s c l i n a t i o n w i t h R e g a r d for C o u p l e S t r e s s e s As in the case of simple material (see Sect. 2.1), the problem to find out the stressed state of couple-stress body with Volterra dislocation can be solved in two ways. In the first one, we take the displacement vector, u, and the microrotation vector, e, for unknowns; they must satisfy the relations (4.2.6) on the cuts converting the domain into a simply connected one, and the equilibrium equations (4.1.4) in which the stress and couple-stress tensors are expressed in terms of u and e. The second approach states the problem in a multiply connected domain, the strain tensors y and 1 (or Y and L), as unknowns, must satisfy the equilibrium equations (4.1.14), the compatibility equations (4.2.2), (4.2.5), and the integral relations of the form (4.2.7) specifying the dislocation and disclination parameters. As an example of application of the second approach, let us consider the plane problem of defect in an elastic ring. We shall seek tensors Y and L in the form

Y = Y~(r)e~e~ + Y,(r)e,e, + eze~, (4.3.1)

L = Lr(r)e~ez + L~(r)e~ez, where r, ~, z are cylindrical coordinates in the reference configuration, e r , ev ,ez are the unit basis vectors related to the coordinates. The compatibility equation (4.2.14) now becomes

dL~/dr + L~/r = 0, whence L~ = cir. To define the constant c, we invoke the relation (4.2.16) with a given constant K,

Lv = K/2~r.

(4.3.2)

In view of equations (4.3.1), (4.3.2), the complex-valued compatibility equation (4.2.15) is equivalent to

L~(r) = O,

dYv/dr + Yv/r - aY~/r = 0'

a = 1 + (2r)-~K.

(4.3.3)

Using the equation (4.2.13), we find the rotation field of medium ~(. = (t~- 1)~. Thus by equation (4.2.12), the microrotation tensor is defined. Calculating the parameter ~ from equation (4.2.18), we obtain = [1 - exp(iK)][Zo

-

g-lY~o(r0)].

This is similar to the formula obtained in Sect. 3.3 and hence the representation (4.3.1) describes the deformed state of wedge disclinated ring. Let us now study the stressed state of a body with disclination for "physically linear" material with the specific potential energy of the form

4.3 Nonlinear Problems of Screw Dislocation and Wedge Disclination

W=~1 [A tr 2 (Y +(#

113

E) + (# + a ) t r ((Y - E). (Y - E) T) -

a)tr

(Y -

E) 2 + 5

(4.3.4)

tr 2L

+ ('y + r/)tr (L. L T) + ('~ - r/)tr L2], where A, #, a, 5, -)', and r/are constants. Using equations (4.1.9) (4.3.1), and ((4.3.4), one can verify that the couple equilibrium equation in (4.1.12) with l' = 0 is satisfied identically, and the force equilibrium equation for unloaded body takes the form (A + 2#)dYr/dr + AdY~o/dr + [A(1 - ~c)+ 2#]r-lyr (4.3.5) +[A(1 - ~ ) - 2#~]r-lY~ = 2T-1 (z~ -~"#)(1 -- ~). We assume that the ring boundaries, r = rl and r = r0 are free of load. This implies the following boundary conditions: ()~ -[- 2#)Mr -]- ~r~o -- 2()~ -~- #)

at

r ~- rl, r 0.

(4.3.6)

The case of solid disk (rx = 0) is of most interest. Solving the boundary problem (4.3.3), (4.3.5), and (4.3.6) and then passing to the limit as r I ~ 0, we obtain Yr = (1 - u)-1(1 - 2u)~(1 + E)-lp~-I _[_ (1 + ~)-1(1 --/2) -1, V~ = ( 1 - / ] ) - 1 ( 1

- 2u)~(1 + ~ ) - l f - ,

A

+ ~(1 + ~)-1(1 - v) -1,

(4.3.7)

r

P=E

On the basis of equations (4.3.4) and {4.3.7), we find the components of Cauchy stress tensor

tRR = 2 # ( f - ' -- 1 ) [ ( 1 - 2v)p ~-' + 1] -1 , t ¢ ¢ = 2#(~p ~ - 1 - 1 ) [ ( 1 - 2u)K;p~-1 + 1] -1 . These expressions have no singularity at the disclination axis and coincide with principal stresses in elastic medium without couple stresses (see Sect. 3.3). The non-vanishing components of the couple-stress tensor take the form m~z = ('y + r / ) ( ~ - 1)(rYe) -1,

(4.3.8) raze = (7-

r])(K;- 1)(rrrY~o) -1.

Relations (4.3.7), (4.3.8) show that, as p ~ 0, the couple stress m c z has a singularity of the order p-~ if ~ > 1 and of the order p-~ if ~ < 1, whereas

114

4. Isolated Defects in Nonlinearly Elastic Bodieswith Couple Stresses

the singularity of m z ¢ is of the order p-1 if a > 1 and of the order pl-2~ if < 1. The linearization of equations (4.3.8) with respect to parameter ( a - 1), when p > 0, results in some formulas, known from linear couple-stress theory (Nowacki 1974), according to which the singularity of couple stresses is of the order p-1 as p ~ 0 for all a =fi 0. Using the equations (4.3.2), (4.3.4), and (4.3.7), we can easily establish that, with regard for couple stresses, the potential energy per unit length of the cylinder with disclination, is

jfr

TM

A = g(a)

rI4;rdr,

1

g(K;) -

27r,~ < 1 27rt~-1, ~ > 1 '

It has a logarithmic singularity as rl -o 0. To study a combination of screw dislocation and wedge disclination, let us apply the semi-inverse approach which is similar to that of Sect. 2.4. We shall seek the solution of the equilibrium equation of couple-stress continuum in the following form

R = R(r),

~ = ~

+ Cz,

Z = ((27r)-lb~ + a z , (4.3.9)

H = e r e n + cos x ( r ) ( e ~ e ¢ + ezeZ) - sin x ( r ) ( e z e ¢ - e ~ e z ) ,

where ~, ¢, b, and a are constants and R ( r ) , x ( r ) are unknown functions. The strain measures Y and L, corresponding to (4.3.1), are

Y = R'erer +

cos X + ~

sin X e~e~

+ (2-~r cos X - --~R r sin X ) e ~ e z + ( ¢ R c o s X + a sinx)e~e~ + (a cos X - CRsin x)e~e~,

(4.3.10)

L = x'e,.e,. + - s i n )~e~e~ + r - 1 (/~ COS X - 1)e~ez r

+ ¢ sin Xeze~ + ¢ cos xezez. As in Sect. 2.4, it can be shown that the relations (4.3.10) describe the cylinder deformation resulting from generation of a screw dislocation with Burgers vector b = b ~ - l e z and a disclination with Frank vector w = - 2 t g T r ( 1 - ~ - l ) e z , when ~ >_ 1, and w = - 2 tg 7r(1- ~)ez, when 0 < ~ < 1. The constant ¢ enables us to take into account the cylinder torsion and the constant a characterizes an axial extension or shortening. Let the expansions of tensors of stresses and couple stresses be as follows T

= tRReReR

W tRceRe¢

W ...

4.3 Nonlinear Problems of Screw Dislocation and Wedge Disclination M = mRReReR • mRceRe~

115

• ...

Write the equilibrium equation (1.3.14) in cylindrical coordinates for unloaded body

OtnR OR

t R R -- t ¢ ¢

÷

OtR¢ OR

R

(~t@R

(~tZn

+ RO~ + O--Z

=0,

fRO -+-tOR 0 t ¢ ¢ i)tz¢ , + ROO + OZ = 0 R

i)tRZ

tRZ

+ -K +

OtCZ

+

(4.3

11)

OtZZ OZ = O,

OmRR m n n - m~¢ + i)mCR + OmZR + tcz -- tz¢ = 0 OR I R RO~ OZ OmR¢ mR¢ + mcR i)m¢¢ Omz¢ OR I R + RO~ + OZ + t z n - t n z = O '

(4.3.12)

Omnz mnz Omcz Omzz i)-----R + --R - + Ri)@ ÷ i) Z + tR¢ -- tcR -- O.

For gyrotropic couple-stress medium, it follows from equation (4.1.17) that the tensors T* = 0W/0Y,

(4.3.13)

M* = 0~Y/0L

are gyrotropic functions of tensor arguments Y and L, i.e. they obey the requirements T*(O T . Y . O , O T. L. O) - O T. T * ( Y , L ) . O, M*(O T . Y . O , O T. L. O) - O T. M * ( Y , L ) . O, with any proper orthogonal tensor O. Setting O = O1 - 2e~e~ - E in equation (4.3.14) and taking into account that, in view of equation (4.3.10), O T. Y. O1 = Y, O T. L. O1 - L, we obtain 1 "

T*

--

T*

" O1,

O1" M* = M*.

(4.3.15)

O1.

From equation (4.3.15), we have er" T* = (er. T*. e~)e~,

er" M* = (e~. M*. er)e~.

(4.3.16)

The expressions (4.3.16), combined with equations (4.1.9), (4.1.16), and (4.3.9), result in

tR¢ -- t~R -- tRz

-- tzR

- - O,

mn¢

-- m~R

= mRz

= mzR

- - O.

116

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

If the body is homogeneous or inhomogeneous (in elastic properties) only in the radial coordinate r direction, then non-zero components of tensor T and M do not depend on coordinates (I), Z. Therefore, the second and third from the force equilibrium equations (4.3.11), as well as the second and third couple equilibrium equations from (4.3.12), are satisfied identically. The rest two equilibrium equations are transformed into a system of ordinary differential equations with respect to functions R(r) and x(r). In general case, these 2nd-order equations are essentially nonlinear and can be studied only by numerical methods. It is possible to establish an exact solution in the closed form for incompressible Cosserat pseudocontinuum determined by the relations (4.1.25). We restrict ourselves to the problem of screw dislocation in the unbounded space, neglecting the twisting and axial extension, i.e. when a = 1, ¢ = 0, a = 1. In case of incompressible Cosserat pseudocontinuum, the functions R(r) and x(r) are immediately determined by constraint equations (4.1.22)and (4.1.23),

R(r) = x/'r2 + A,

x(r) = arctg

a , r + x/'r 2 + A

b a = --. 27r

Assuming continuity of medium near the dislocation axis, i.e. R(0) = 0, we get A = 0; whence it follows that a

R = r,

X = arctg -2r- .

(4.3.17)

Consider a material with the energy density

W=2#tr(Y-E)+~tr

5

2L + 7 +2 r/tr (L. L T) + 7 - 2 r/tr L 2.

(4.3.18)

If there are no couple stresses (/i = 3' = r] = 0, Y = y T _ A1/2 ), the expression (4.3.18) coincides with the Bartenev-Khazanovich potential for incompressible material; the problem of screw dislocation for that material was investigated in Sect. 3.2. From equations (4.1.25) and (4.3.18), we obtain T = - p E + (grad R) T. (2#H - q × H),

M = ( grad R) T. (5( tr L)E + (7 + r/)L + (7 - rl)LT) " H.

(4.3.19)

(4.3.20)

Substitution of equations (4.3.19), (4.3.20) into the equilibrium equations (4.3.11), (4.3.12)yields

4.3 Nonlinear Problems of Screw Dislocation and Wedge Disclination 01o Or

=

2#

4#

~

r

x/4r 2 "+ a 2

+ (5 + 2~)

117

4a 2

r(4r 2 + a2) 2'

Op=OP=o, O~ Oz 4a q~ = - ( ~ + 27)(4r2 + a2)3/2,

(4.3.21) % = O,

q~ = O,

q = qre~ + q~e~o + qzez.

Using equations (4.3.20), (4.3.21), we get

mR

R --

2a(5 + 27)

a5 r x/4r 2 + a 2

m¢¢=

4r 2 + a 2 ,

(4.3.22)

a(5 + 2"7) 2a5 r v/4r2 + a 2 - 4r 2 + a2.

Keeping only the 1st-order terms in a in the solution (4.3.22), we get the solution to the screw dislocation problem in the framework of Cosserat pseudocontinuum linear theory, mORR =

-ya

o

r2 ,

m~

"ya

= -~.

On comparison of equations (4.3.22) with (4.3.23), we see that accounting for nonlinearity reduces the degree of singularity of couple stresses near disclination axis (mRR and m¢¢ are proportional to r -1 as r --, 0). Far from the axis, the difference between linear and nonlinear theories is negligible,

m RR

~ m

0

RR ,

m,~,~

~

m

o

,~

as

r ~

(:x:).

It can be easily shown that the couple stresses, m c z , m z ¢ , and m z z , which are absent in linear theory, decrease proportionally to r -3 as r ~ oc, that is one order more than for m R R and m¢¢. The shear stresses determined by equation (4.3.15) have the form t c z = 2# sin X + q,- cos X,

,zo =

-cosx-sinx r

) - q r (a s i n x - c o s x ) •

(4.3.24)

On the basis of equations (4.3.17), (4.3.21), (4.3.24), we conclude that, on approach to the dislocation axis, the stress t c z tends to the finite limit 2#, as in Sect. 3.2, but the stress tz~ increases proportionally to r -1 , as in linear elasticity. The dislocation energy per unit length, calculated from equations (4.3.10), (4.3.17), and (4.3.18), has a logarithmic singularity on the defect axis.

118

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

4.4 Volterra Dislocations in Nonlinearly Elastic B o d i e s with

Microstructure

In the Cosserat continuum model we used above, every particle of continuum has rigid body degrees of freedom. The orientation of a point of the medium may be mathematically represented by three unit vectors remaining mutually orthogonal during deformation of the medium. In a more general model of continuum, called the medium with microstructure, the direction vectors, specifying the particle orientation, may change in length and rotate relative to each other. Linear theory of medium with microstructure was developed by R.D. Mindlin (Mindlin 1964), the fundamentals of nonlinear theory were formulated by R.A. Toupin (Toupin 1964). Let Pk (k = 1, 2, 3) be a triple of non-coplanar direction vectors of a given particle in the current configuration of medium; in the reference configuration they are denoted by Pk. Let the vector basis pS be reciprocal for Pk in the sense that p S . p k -- (~; form a nonsingular 2nd-order tensor by F = pkPk,

(4.4.1)

we shall call it the microdistortion tensor. Assuming that the frames Pk and Pk are of the same orientation, we have det F > 0. Using the polar decomposition of microdistortion tensor F = B . H,

(4.4.2)

B being a symmetric positive definite tensor, H being a properly orthogonal tensor, we can easily see that any point of the medium with microstructure is characterized not only by its position in space, R, and rotation, H, but also by the homogeneous deformation described by B. Thus, a particle of continuum with microstructure has 12 degrees of freedom. Following the principle of local action and the reasons of Sect. 4.1, we take the function of potential strain energy of elastic continuum with microstructure in the form )/Y = W(R, grad R, F, grad F).

(4.4.3)

Under rigid-body motions of the medium, the 14; must be invariant with respect to the choice of reference system. The invariance with respect to translational motions of the observer results in independence of 14; on the argument R. The requirement of invariance with respect to rotations of the observer's reference system leads us to the condition

}V(( gr ad R ) . O, F - O, ( grad F ) . O) = )N ( grad R, F, grad F) for any orthogonal tensor O. Set O = H T in equation (4.4.2) then

(4.4.4)

4.4 Volterra Dislocations in Nonlinearly Elastic Bodies with Microstructure

W = W ( C . H T, B, ( grad F ) . HT),

grad//:.

C=

119

(4.4.5)

By (4.4.2), it follows that C . H T - C . F T. B -1,

B - (F. FT) 1/2,

(grad F ) - H T = (grad F ) . F -1. B whence we conclude that the tensors ---- C" F T,

F" F T,

K = ( g r a d F ) . F -1

(4.4.6)

may be thought of as arguments for )4; instead of equation (4.4.5). Taking into consideration the equality F . F T -- ~ T . A-1. ~,

A _-- C . C T,

(4.4.7)

A being the Cauchy strain measure, we can see that the set of tensors A,

(I) = A. ~I/-T = C . F -1,

K

(4.4.8)

offers an equivalent alternative to that of equation (4.4.6). Thus we have established that the necessary condition for the energy of an elastic body with microstructure to be invariant under rigid motions is its representation in the form

)4; = W(A, (I,, K).

(4.4.9)

The tensors A, (I), and K do not vary under the change of reference system, therefore, the representation (4.4.9) is also sufficient for the energy to be invariant under rigid-body motions. In what follows, the 2nd order tensor (I, will be called the relative strain measure and the 3rd order tensor K the bending strain measure. Similarly to the other models of elastic media, the elastic potential W in equation (4.4.9) may be dependent upon a number of constant (that is they don't change in deformation process) parameters, tensors or vectors which, however, can vary on change of reference configuration. Moreover, for inhomogeneous bodies, there is the explicit dependence of the elastic potential on Lagrangian coordinates. The equilibrium equations and boundary conditions for elastic body with microstructure can be obtained from the virtual work principle which is also referred to as the Lagrange's variational equation f ],, Wdv - 6'E = 0,

(4.4.10)

5~A being the elementary work of external load. On the basis of equation (4.4.9) we have

120

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses 1

5W = ~ tr (r*. 3A) + tr (a*. 5oT) + ~r* • 5K, (4.4.11) v* = (~.,)T = 20W/0A,

a* = 0 W / 0 O ,

~r* = 0 W / 0 K .

the symbol "," stands for the contraction of two 3rd-order tensors,

X , Y = XmnPYmnp. Later on, we shall use the following operation with respect to a 3rd-order tensor X = Xmnpr~rnr p and three 2nd-order tensors Tk (k = 1, 2, 3): (T1, T2, T3)AX - Xmnp(Tl" rm)(T2 • rn)(T3 " rP).

(4.4.12)

As is seen, the result is a 3rd-order tensor. Introducing the tensors d = ~ C1-

1 • 5A . C-T, .

."~ .C -I 5O F, (4.4.13)

F = F -1. 5F,

c = (C -1, F -1, FT)A~K,

and integrating in (4.4.10) over the body volume in the deformed configuration, we obtain v [ t r (~-. d) + tr ( a . ,.yT) .~_ 71" * C] dV = 5'g,

T

"--"

T

T

--

J - l e T . r*. C,

a - j - l e T , a* • F -T,

= J-I(CT, FT, F-1)A1r *,

(4.4.14)

J = detC.

From equations (4.4.5)-(4.4.7) and (4.4.13), it follows 1 d = ~ [Grad 5u + ( Grad "7 = Grad ~u - F,

5u)T],

c = Grad F,

(4.4.15)

Grad = C -1. grad. Here, 5u is the field of virtual displacements of medium, F is the field of virtual microdistortions, Grad is the spatial gradient operator. With regard for equation (4.4.15),the equation (4.4.14) can be transformed as follows

-/v

[(Div (1- + or)). 6u + tr ((Div lr + ~rT) •

dV (4.4.16)

+ ~ [ N . (7- + o'). 5u + tr ( N . lr.

dr

= 6'E.

4.4 Volterra Dislocations in Nonlinearly Elastic Bodies with Microstructure

121

Here, E is the boundary surface in the deformed state, N is the unit normal to E. The left-hand side expression in (4.4.16) defines the function of elementary work of external loads,

6'£ = /v Pk "6udV + /v Ptr (~" FT)dV (4.4.17)

+ f Pt" SudE + f ptr(¢ "FT)dE, where p is the medium density in the deformed configuration, k is the external mass load, t is the external surface force. Physical meaning of the 2nd order tensors ¢' and ¢ will be later clarified. In view of equation (4.4.17), from variational equation (4.4.16) it follows the equilibrium equations, Div (v + a) +

pk = 0,

D i v ~r

+

O "T -~- p~t - - 0

(4.4.18)

and the dynamic boundary conditions, N . (r + a) = t,

N . T r = ~j.

(4.4.19)

To understand physical meaning of ~' and ~, as well as of the 3rd order tensor lr, we simulate a particle of medium with microstructure as a volume V0, filled with ordinary medium, i.e. with a simple material. Suppose that this volume is homogeneously deformed and calculate the work of body forces of density q, 5'$ = .f,~ q. ~udV. ~

v U

Under the homogeneous deformation, the field of virtual displacements has the form ~u = ~u0 + R . L0,

(4.4.20)

where the vector 5Uo and tensor Lo do not depend on the coordinates. On the basis of equation (4.4.20), we get

~'C = 5Uo. /go qdV + tr (.~.. L0), (4.4.21)

= fvo qRdV. The tensor .~. is called the force tensor of the load system. Its vector invariant, taken with opposite sign, is coincident with the principal moment of force -"

system

f -~× = Jvo R x qdV. The formula (4.4.21) shows that the principal vector of force system and the force tensor do work on translational motions of the volume and on homogeneous

122

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

deformation respectively. Decomposing the tensor L0 into the symmetric and skew-symmetric components, we can see that the principal moment does work _[_~T) on a rigid rotation, and the symmetric part of the force tensor, 5l(p~. does work on the pure deformation of the material volume. It is clear that the principal vector and force tensor are unchangeable if the force system distributed over V0 is replaced with a point force, acting at the point R = 0 and having the value of fyo qdV, and the concentrated force couple of value of -.=.× at the same point as well as with three mutually orthogonal double forces directed towards the principal axes of the tensor ~ + By the double force (or the force dipole), we mean a pair of forces of the same magnitude, with the opposite directions and common line of action, moreover, the points of force application come arbitrarily close to each other, and the magnitude of each force increases in such a way that the product of the distance between points of application by the force magnitude remains constant. The concept of a force dipole is discussed from a different standpoint in (KrSner 1965; Lurie 1970). The collection of concentrated pair and three mutually-orthogonal force dipoles will be called the generalized double force. It is characterized by a non-symmetric 2nd order tensor. Now it becomes apparent that the tensor ~' in (4.4.17) doing work on a virtual microdistortion is the mass density of external generalized double force distributed continuously over the volume, whereas the tensor ~ is the surface density (per unit area) of generalized double force; the vectors - ~ " and -~× are the densities of external couple loads distributed over the mass and surface, respectively. The 3rd order tensor lr at a point of the body defines the generalized double forces, acting on any area element through the point, it is called the tensor of double stresses. The couple stress pseudotensor M, defined in Sect. 4.1, is derived from the tensor ~r by M = - r r ~ n P / ~ ( P ~ x P~).

(4.4.22)

The tensor er is said to be the relative stress tensor, and the symmetric tensor r is simply called the stress tensor. The tensors ~', a, and lr characterize those contact interactions between body parts in the deformed state which do not depend on the choice of reference configuration. Therefore, the tensors ~', or, lr are analogous to the Cauchy stress tensor in the theory of elasticity of simple materials, whereas the tensors ~'*, a*, and lr* introduced in equation (4.4.11), are analogous of the Kirchhoff stress tensor. Notice that the equilibrium equations (4.1.14) in mechanics of Cosserat continuum can be obtained from equations (4.4.8) when one takes into account the equation (4.4.22) and r× = 0, DivT+pk=0,

DivM+T×-p~'× T = ' r + o'.

=0,

4.4 Volterra Dislocations in Nonlinearly Elastic Bodies with Microstructure

123

The integral conditions of equilibrium of an arbitrary part of the body with microstructure are of interest, they are equivalent to equations (4.4.8) and have the form f f ]_ (~ + t R ) d E + ]_. (p~' + p k R ) d V = ].. rdV,

1"

JE

(4.4.23)

,IV,

JV.

where V, is an arbitrary volume of the body, E, is its surface. The equation (4.4.23) means that the force tensor of the force system acting on an arbitrary body part is equal to the integral (total) value of the symmetrical stress tensor. Let us now study Volterra dislocations in elastic bodies with microstructure. The starting point to develop the theory of isolated defects is the problem of finding the displacement and microrotation fields for multiply connected body by given fields of those tensor strain measures which are arguments of the specific potential strain energy, W. So assume that an elastic body with microstructure in the reference configuration occupies a multiply connected domain and the tensors A, O, and K are single-valued continuously differentiable functions of Lagrangian coordinates, q~, in the domain. Moreover, we consider twice differentiable tensor fields A and O. By the relation F . F T -- • -1. A. (I~ - T

(4.4.24)

following from (4.4.8), the microstrain measure, F. F T, is also defined, if A and are given. At first glance it seems that the microdistortion tensor could be found from a system of equations, following from the definition (4.4.6) of the bending strain measure, K, grad F = F.K.

(4.4.25)

But this way of solving is wrong since on substitution of the tensor F, found by (4.4.25) into (4.4.24) we get contradiction with the given right-hand side of (4.4.24). A correct way to find out the microdistortion field is to compose a system of equations for the microrotation tensor, H. On the basis of equations (4.4.2), (4.4.25), we have

0H _ B_~. (K~. B - ~-~q~). 0B H, Oq~

K~ = r~. K.

(4.4.26)

I

From orthogonality of H it follows skew-symmetry of tensors (0H/0q~) •H T. Therefore the necessary condition for solvability of the system (4.4.26) is

B-~ • (K~. B -

0B T 0B B_ I ~&) + (B. K~ - ~-~&) • =0

(s = I, 2, 3).

(4.4.27)

These are 18 scalar equations with respect to components of strain measures A, O, K. If the conditions (4.4.27) hold, then the system (4.4.26) can be rewritten as

124

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

OH Oqs

=Ys'H,

Ys

I(B. K T

_

1B-~ ( K s . B -

= -~

"

OB ). B -1 aq o

0B -~qS)

(4.4.28)

(s = 1, 2, 3).

Once the tensors A, (I), and K have been given, the skew-symmetric 2nd order tensors Yk are known. The necessary and sufficient conditions for the system (4.4.28) to be soluble are OYk Oqs

OYs Oqk

= Y ~ ' Y k - Yk'Y~

(k, s = 1, 2, 3),

(4.4.29)

that, in view of skew-symmetry of the tensors Ys , consist of 9 independent scalar relations. The relations (4.4.27), (4.4.29) together are the necessary and sufficient conditions for the system (4.4.26) to be soluble with respect to the microrotation tensor, when the fields of tensors A, (I), K are given. The system (4.4.28) is analogous to the system (4.2.1) that determines the microrotation tensor in Cosserat continuum mechanics; its solution can also be written in terms of a multiplicative integral. As for Cosserat continuum, on each of cuts converting the multiply connected domain into a simply connected one, there holds H+ = H_-{2,

(4.4.30)

where {2 is a properly orthogonal tensor being constant for all the points of the cut. Once the microrotation field has been found, the displacement field of a medium, u = R - r, is determined by integrating the relation grad R = (I). B . H,

(4.4.31)

that follows from equation (4.4.8). For the equation (4.4.31) to be resolved with respect to R, it is necessary and sufficient the conditions rot ((I). B) + r k x (I)- B . Yk = 0,

(4.4.32)

to be fulfilled. The relations (4.4.32) are equivalent to 9 scalar equations and, together with equations (4.4.27) and (4.4.29), form the complete system of strain compatibility equations in nonlinear theory of elastic media with microstructure. As in Sect. 4.2, using equation (4.4.30) we can prove that the jump in R on a cut of multiply connected domain is R + = R _ • $1 + b,

b = const,

(4.4.33)

4.5 Energy Integrals

125

which means that the positions of borders of a cut in the deformed state differ by a rigid-body motion. Thus, we have proven that in nonlinearly elastic bodies with microstvucture, defects in the form of Volterra dislocations may occur i.e., there are such states of multiply connected body in which the specific potential energy and tensor strain measures are single-valued and continuous , whereas the displacements and microdistortion have a specific multi-valuedness. The Burgers and Frank vectors for the Volterra dislocation in a body with microstructure are expressed in terms of the field of strain measures by formulas that are analogous to equations (4.2.7) and involve curvilinear multiplicative integrals. Solving the problem of finding of strained state of bodies with isolated defect, we should associate these integral relations with the compatibility equations (4.4.27), (4.4.29), and (4.4.32) and the equilibrium equations (4.4.18). Note that if there are no external body loads k and t~~ then the technique, employed in the derivation of equations (1.5.3) in case of simple elastic materials, can be applied to transform the equation (4.4.18) in such a way that Lagrangian coordinates q8 are independent variables and the measures A, ~, K are unknown functions. The tensors A, O, K are the strain measures of Cauchy-Green type. By interchanging the deformed and reference configurations, i.e. replacing R, grad, and F by r, Grad, F -1 in (4.4.6)-(4.4.8), respectively, we get the tensors being the strain measures of Almansi's type:

A = (Grad r). (Grad r)T,

• ' = ( Grad r ) . F, (4.4.34)

K' = ( Grad F - l ) • F. Using the method similar to one of Sect. 4.1, we may show that W, the specific potential energy of an isotropic elastic material with microstructure, can be considered as a function of tensors ,k, O~, and K~; the stress tensors r , rr, and lr will be isotropic functions of A, O', K' as well. The above strain measures of Almansi type arise naturally in the problem of isolated defects when the domain occupied by elastic body in the deformed state is multiply connected. In this case, it is necessary to consider the problem on finding the displacement and microrotation fields when the strain measures A, ~', and K' are given as single-valued, twice continuously differentiable functions of Eulerian coordinates. This problem can be solved in a way which is similar to one for the above problem on finding the displacements and microrotations by tensor fields A, O, and K specified as the functions of Lagrangian coordinates.

4.5 Energy Integrals Characterizing the Force of A c t i o n on a Defect in Micropolar Media Let an elastic body, described by the Cosserat continuum model, be in equilibrium under a specified external load. Consider a surface integral J, defined on

126

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

actual fields of displacements and microrotations, i.e. the fields satisfying the equilibrium equations (4.1.12) with no body forces and couples,

J = f~ n . Tda,

T = WE-

D . ( G r a d R ) T - G . H T. L T.

(4.5.1)

0

Let us prove that the integral in (4.5.1) is equal to zero for any closed piecewise smooth surface a0, which is the boundary of volume v occupied by homogeneous material, with the outer normal n; the tensors Y and L are continuously differentiable functions of coordinates. As the tensor H is orthogonal, the tensors H T- OH/Oxm (m = i, 2, 3) are skew-symmetric and may be represented in terms of vectors ~m as follows

0H - - E × ~m, Oxm

H T.

(4.5.2)

where Xm are Cartesian coordinates in the reference configuration. From equations (4.1.4)and (4.5.2)we obtain

L -- istCs. H T.

(4.5.3)

Using equation (4.5.3) and the identity

Ot~m

O~s

Ox~

Ox~

m

t~ m

X

~s

following from equation (4.5.2), we can prove that OL • H = grad ~m. Oxm

(4.5.4)

For a homogeneous body, the specific potential energy W being a function of tensors Y and L does not depend on the coordinates x8 explicitly. Hence based on equations (4.1.4), (4.1.9), (4.5.2), and (4.5.4), we obtain

Ox~ = tr

-~

"~x-~x~+

--~

"~--~x~ +

(4.5.5)

tr G T. grad ~ . This can be transformed into

OW

Ox8 = - [(gradR) T. D ] × . t¢8 + div

(

OR D • ~-~x~ (4.5.6)

+div ( G . e~) - ( d i v D ) . OR _ (div G) • t%.

4.5 Energy Integrals

127

In view of the equilibrium equations (4.1.12), from equation (4.5.6) we have gradW-

div [D. ( g r a d R ) T + G . H T. LT],

whence, according to equation (4.5.1), it follows that div T = 0.

(4.5.7)

Using the constitutive relations (4.1.9), we can rewrite the expression (4.5.1) for tensor T in another way

OW

T = W(Y, L)E - aY • yT

OW

_

~ .

aY

L T.

(4.5.8)

The relation (4.5.8) shows that if the specific energy W is a twice continuously differentiable function of its tensor arguments and the tensor fields Y and L are single-valued and continuously differentiable in the domain v then tensor T is a single-valued continuously differentiable function of the coordinates xs in the v. In case of multiply connected domain v, the displacement and microrotation fields, as stated in Sect. 4.2, may be non-single-valued. On application of the Gauss-Ostrogradski formula to equation (4.5.7), we get J = 0, which proves the statement. The equation (4.5.7) has the form of conservative law and, as well as the expression (4.5.1) for the tensor T, it could also be deduced using Noether's theorem (Olver 1986) with regard for invariance of energy functional for homogeneous body with respect to translations of Cartesian coordinates of the reference configuration. The above direct proof of the conservative law (4.5.7) followed (Zubov 1990). If the conditions of the above theorem do not hold in a domain v' C v, then the integral J over the boundary surface of v', in general, differs from zero. In this case, by virtue of the theorem, the integral value does not depend on the choice of a closed surface a~ enclosing v'. The sub-domain v' may have inclusions, cavities, nonhomogeneities, dislocational and disclinational loops, singular points of the tensor fields Y, L, and other defects. The integral J is analogous to Eshelby's integral (Eshelby 1956) in nonlinear theory of simple elastic materials. For the plane problem of nonlinear couplestress elasticity, the integral J becomes an ordinary contour integral ? d = ~ n . Tds, J-yo

(4.5.9)

where n is the normal to the fiat closed curve 70 • If tensor T is continuously differentiable inside the contour 70, the integral ill equation (4.5.9) is zero. But if there is a defect inside a simple closed contour 70, the integral from equation (4.5.9) has the same value for all closed curves enclosing the defect. In a special case such as a defect, we refer to the rectilinear Volterra dislocation with axis being orthogonal to the plane, in which the contour 70 lies.

128

4. Isolated Defects in Nonlinearly Elastic Bodies with Couple Stresses

Consider a rectilinear infinitely thin crack with load-free borders that are parallel to a unit vector h. It can be proved now that the integral h . J, J being expressed by the formula (4.5.9), has the same value for all contours, which enclose one of the crack ends and begin and end at the opposite borders. The material property of homogeneity in the only direction of the crack are sufficient for the integral h . J to be independent of the choice of a contour. The invariant integral h . J is similar to the Cherepanov-Rice integral in plane elasticity (Cherepanov 1967; Rice 1968), moreover, they coincide if the term due couple stresses in the expression for T, equation (4.5.1) is deleted. The significance of the above invariant integrals is due to a fact that they define the increment of potential energy in the elastic body appearing under the change of defect location in the body. To prove this, consider a defect which is a cavity in the bulk of a homogeneous body, the cavity surface a' is free of loads; the body is in equilibrium under the action of conservative loads distributed over the outer surface a. Consider another body with the only difference from the former that the cavity changed its position by translation ~l, where l is a unit vector, and s is a small parameter. Calculate the potential energy increment f A = / ~ ~Ydv - E

(4.5.10)

due to this change of cavity position. In equation (4.5.10), v is the body volume in the reference configuration, $ is the external load potential, written as an integral over a. The increment of body potential energy, due to the change of cavity position, can be represented by two terms within first order small quantities with respect to ~. In calculation of the first component, it is taken into account only the variation of the body boundary, the values of displacements and microrotations at each point are considered to be fixed. The second component corresponds to a change of the solution under consideration due to the change of body boundary. If the outer body boundary is fixed whereas the cavity moves then the first component of the energy increment, according to (Germain 1983), takes the form

~

, n ' l / Y d a . ~l,

where n ~ is the outward normal to a ~. The second component, ~d ,41~=0, is the variation of the functional A when the boundary of v is fixed, it becomes zero by virtue of the fact that the body state is balanced, i.e. the field of displacements and microrotations is a solution to the boundary problem of equilibrium, and this solution, according to Lagrange's principle, provides the stationary value to the functional .A. Finally, the potential energy increment due to the change of the cavity location is AA = - ~ l . ~ , n W d a + O(~2),

(4.5.11)

4.5 Energy Integrals

129

where n is the outward normal with respect to the cavity. With regard for invariance of the integral in equation (4.5.1) and the conditions n.D - n.G - 0, which holds on load-free cavity, we arrive at the equality

nl/Vda=~ n. Tda, I

(4.5.12)

0

where a0 is any closed surface in v, enclosing the cavity. From equations (4.5.1), (4.5.11), and (4.5.12)it follows the relation

(4.5.13)

AA = - e l . J + O(e2),

that enables us to identify the vector integral J with the resistance force arising due to the defect motions. In a similar manner it can be also proved the validity of equation (4.5.13) for any defect in the body when the outer boundary a is free of load. It is obvious that the formula (4.5.13) is also applicable to the case of couple-stress theory of plane elasticity, when the integral J is given by the expression (4.5.9). In particular, if the defect is a rectilinear Volterra dislocation, the integral (4.5.9) defines the force acting on the dislocation in nonlinearly elastic medium with couple stresses. As to the integral h . J, it characterizes, as well as Cherepanov-Rice integral in the linear elasticity, the energy release under crack motion in a body with couple stresses. The invariant energy integral can be also constructed for nonlinearly elastic medium with microstructure with the model described in Sect. 4.4. In the case, omitting the proof, we state the Eshelby-type integral

J = j f n . ( ~ Y E - r * . A - c r * . ¢ T - I r *oK)da, o

In these, the notations of Sect. 4.4 are used.

~r*oK -

u~tnKstni~is.

(4.5.14)

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

5.1 General S t a t e m e n t s of Nonlinear Theory of Elastic Shells A material body whose mass is concentrated near a surface is called the shell. Numerous thin-walled structures in engineering can be considered as the shells. We can also consider such two-dimensional physical systems as membranes, films, surface crystals etc. to be the shells. There are different approaches to develop the shell theory. The detailed survey of the state-of-the-art of nonlinear theory of elastic shells was given by W. Pietraszkiewicz (Pietraszkiewicz 1989). In this section we shall present a mathematical model of nonlinearly elastic shell of Love type (Koiter 1966; Pietraszkiewicz 1977; Zubov 1982), considering the shell as a two-dimensional material continuum, i.e. as a material surface possessing certain properties. Let a and ~F be the surfaces corresponding the reference and deformed configurations of the continuum, respectively. The position of a point of a is defined by the radius-vector p(ql, q2), qa (a = 1,2) being Gaussian coordinates on a. The unit vector normal to a is denoted by n, and the vectors of the main and reciprocal bases on a are denoted by p~ and p~, respectively, Pa -- cgP/c3q a,

P~ " Pa --

~,

Pa " n = p~ . n = O.

From now on, Greek indices take the values of {1,2}. The coefficients of the first and second quadratic forms for the surface a are determined by the following expressions (Sokolnikoff 1971)

g~

-

p ~ . p~,

b~

=

b~

=

-P~

•

Oq~"

(5.11)

Considering the coordinates q~ to be Lagrangian coordinates of the material surface, let us specify the point position on the surface ~F by the radius-vector p(ql, q2), that is the position in the deformed configuration of the material point whose position was p(ql, q2) in the reference configuration. We denote the normal vector to ~F by N and the basis vectors by P~ and P~, then

132

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells P , = OP/Oq ~,

P~ . P z = ~ ,

P~ . N = P z . N = O,

(5.1.2)

ON

G~Z = P ~ . PZ,

B ~ = BZ, = - P , .

Oq~ '

G "~ = P " - P Z ,

where G ~ and B ~ are the coefficients of the first and second quadratic forms of the deformed surface, Z, referred to the coordinates q~. Derivatives of the basis vectors are as follows, OP~ OqZ = F ~ P ~ + B ~ N , Op ~ Oq~ = - F ~ P ~ + B ~ N ,

(5.1.3) B~ = B ~ G ~

As is well known (Sokolnikoff 1971) the Christoffel symbols, F ~ involved in equation (5.1.3) can be expressed in terms of coefficients of the first quadratic form F~ = I ( OG~ O G 3 ~ OG~3 ) ~G ~* + (5.1.4) Oq~ Oq~ Oq~ " Derivatives of the vectors p~, p~ are expressed by formulae, which are analogous to equations (5.1.3), in terms of the Christoffel symbols, "~,~, of a and the coefficients b~. For the symbols ~ , the relations (5.1.4), expressing them in terms of the metric g ~ in non-deformed shell configuration, are valid. In what follows, we shall use two-dimensional gradient operators (nablaoperators) on the surfaces a and Z: 0~ VO = p ~ ~ Oq~ '

00 V'O = P ~ ~ Oq~ '

(5.1.5)

where ,I,(q 1, q2) is a tensor of arbitrary order on the three-dimensional Euclidean space. The energy, that is stored due to shell deformation, is defined by not only extensions of the surface Z but also by its bending, i.e. its curvature changes. To take the effect of bending into consideration, let us assume that for an elastic shell, the specific potential strain energy, 14;o (per unit area of the surface a), depends on the motion of a neighborhood of the point of two-dimensional continuum through not only the first but also the second displacement gradients, Wo = 1/Yo(VP, V V P ) .

(5.1.6)

The condition that the specific energy is independent of superposition of rigid body motions imposes the following constraint on 1/Yo: Wo [(VP)- O, ( V V P ) . O] = 1/Yo(VP, V V P ) ,

(5.1.7)

where O is an arbitrary orthogonal tensor. Let us consider the distortion tensor of the deformed surface (Zubov 1982) Co = V P + n N = p~P~ + n N .

(5.1.8)

5.1 General Statements of Nonlinear Theory of Elastic Shells

133

The polar decomposition of nonsingular tensor Co implies Co = (Uo + n n ) . Ao,

Uo + n n -- (Co. CoT ) 1/2

Co" CoT = ( V P ) . ( V P ) T + n n = G ~ p ~ p ~ + n n

(5.1.9)

Uo = [ ( V P ) . ( v P ) T ] 1/2 , where Ao is a proper orthogonal tensor, Uo is a symmetric positive definite two-dimensional tensor. By a two-dimensional tensor of 2nd order we mean a linear operator acting in two-dimensional subspace of the three-dimensional Euclidean vector space. For the tensor Uo, this two-dimensional subspace is the plane that is orthogonal to the vector n, so that Uo.n=n.

Uo=0

(5.1.10)

By tensor Uo 1, the inverse to the two-dimensional tensor Uo, we mean a two-dimensional (i.e. satisfying equation (5.1.10)) tensor such that U:~ • Uo = U o . U o ~ = g, g = E-

(5.1.11)

n n = go~p°~p ~,

where E is the three-dimensional identity tensor, g is the two-dimensional identity tensor in the plane spanned by the vectors Pl, P2. From equations (5.1.9)(5.1.11) we get V P = Uo. Ao; (5.1.12) Ao = Uo 1- V P + n N ,

A T = ( V P ) T. Co I -+- N n .

(5.1.13)

Using the formulae (5.1.1)-(5.1.3), we obtain the expression for the 3rd order tensor V V P : V V P = B ~ ¢ p ~ p ~ N + b~p~nP,~ + (F~Z - "7~z)p~p~P~, b~ =_ b~zg ~.

(5.1.14)

Let O = A T in the constraint (5.1.7); consider the relations following from equations (5.1.12), (5.1.13) (VP).

A T =

Uo,

(VVP). A T = B~p~p~n + b~G~o~np ~ . Uo'

(5.1.15)

+ (F~ --")/~)G,f.~pap~p"Y"U~-1. The quantities p~, n, b~, associated with the reference configuration of a shell, are invariant under deformation, hence they are constant during deformation and we can drop them from the arguments of Wo. On the basis of equations (5.1.7), (5.1.15), we arrive at the representation

134

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells (5.1.16)

we have taken into account that the components of the tensor Uo in the basis pa are expressed in terms of Ga~. Since the arguments of 142o in equation (5.1.16) are invariant under superposition of rigid-body motion of the surface, the representation (5.1.16) is not only necessary but also sufficient for the specific energy of the shell to be independent of the choice of the reference system. Let us impose an additional natural constraint on ~2o. Namely, we require that in a particular case, when the shell is a fiat thin plate which is deformed without bending, the constitutive shell relations must coincide with those of the plane theory of elasticity of simple material. In the latter, the specific energy is a function of the Gaz only and does not depend on the derivatives of Gaz with respect to coordinates q~. Since for the plane deformation of the plate Ban = 0, with regard for (5.1.4), we can see, that the above restriction is fulfilled if and only if there is no dependence of the specific energy (5.1.16) on Christoffel symbols F~Z. Thus we finally arrive at a Love-type model for elastic shell, the specific energy of which depends on deformation in terms of coefficients of the first and second quadratic forms of ~F, Wo = Wo(G

,,

(5.1.17)

As was above-mentioned, the right-hand side of equation (5.1.17) can depend on some constant tensors-parameters, i.e., on those that are invariant under deformation, but which specify the dependence of the specific energy on the choice of the shell reference configuration; their examples are the coefficients of quadratic forms g~, b~z for the undeformed surface, a, as well as tensors specifying anisotropy in mechanical properties of the shell material. It is apparent that the tensors of tangential and bending strains Io = I~zp~p ~, 1

Ko = K~zp~p ~, (5.1.18) K ~ = B ~ - b~z

may be thought as of arguments for Wo, instead of equation (5.1.17). To derive the equilibrium equations and the boundary conditions in the theory of elastic shells, let us invoke the virtual work principle

f~ Wo da - 5't7 = 0,

(5.1.19)

where ~'E is the elementary work of external loads. According to equations (5. I. 17), (5. I. 18), we represent the variation of the shell elastic potential, Wo, in the form

5Wo = ~1 / ~ v " Z S G ~ - ~/~#c~Bo,~ '

(5.1.20)

5.1 General Statements of Nonlinear Theory of Elastic Shells ~/~ X

0Wo

{ 1, a =/3

u~Z= 20G~------~= OI~z'

OWo

X = ,, 2, a :~/3

_X./--G#~z = 0Wo _ 014]o , OB~ G -

GllG22

-

G~2,

135

(5.1.21)

OK~

9 = 911922 -

922 •

The factor v ~ / g , introduced for further convenience, is the ratio of surface element area of the shell in the reference configuration to one in the deformed configuration U'7-_

dS = ~/G da.

(5.1.22)

On the basis of equations (5.1.20)-(5.1.22), we have (5.1.23) If we decompose the vector of virtual displacements, ~P, of a point of the surface S, ~ P = v'rP.y + w N (5.1.24) then 5 G ~ = Gz.~V~v "~+ G~.yVzv "Y - 2 B ~ w = 2 ~ ,

(5.1.25) 5B

, =

+

+

(V,v

-

=

V, being the symbol of covariant derivative in the metric G,~. Let us introduce the symmetric tensors (Zubov 1982) G = G,zP~P ~ = E~o = ~,~zP~ P~,

NN,

B = B ~ z P ~ P ~,

~o = n~zP~PZ

and take into account the relation ~N = -(V'w + B. ~P)

(5.1.26)

then, from equations (5.1.5), (5.1.25), we obtain eo=~l [V'~P + (V'SP) T] + ~I ( N ~ N + ~NN) (5.1.27) e~o = - ( V ' ~ N ) . G + B. (V'SP) T. Now the expression (5.1.23) can be represented as follows 5 f Wo da = ~ tr (v • eo - tt. ~o) dS,

(5.1.28)

136

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells v

= u~ZP~Pz,

~ = #~PaPz.

This was derived basing on the idea of a shell as of a two-dimensional material continuum. It turns out that this expression can also be obtained from other considerations when one treats the shell as a three-dimensional body made of a simple elastic material. In this case, the physical meaning of tensors u and becomes clear. A real shell is a three-dimensional body that, in the deformed configuration, is bounded by the surfaces Z1 and Z2 at the distances of H1 and //2 from the surface Z in opposite directions to Z', respectively, and by a ruled surface generated in motion of the straight line, co-directed to the normal N , along the boundary contour of Z. The thickness of the shell, Hi +/-/2, is assumed to be small in comparison with the minimum of radii of curvature of Z. The surface Z need not be a median one with respect to ~'1 and Z2, and the shell thickness may be variable, i.e. the magnitudes of H1 and H2 may be some functions of Gaussian coordinates ql, q2 on Z. The position of a point of this three-dimensional body will be specified by coordinates ql, q2, and Z, where Z is the distance of the point from the surface Z (measured along the surface normal). Clearly, - H i _ Z
(5.1.30)

This law of distribution of displacements along the transverse coordinate Z is referred to as Kirchhoff-Love kinematics hypothesis (Love 1927). The threedimensional gradient operator in the coordinates ql, q2, Z has the form Grad = ( G - ZB) -1. V' + N O / O Z , whence, on the basis of equations

(5.1.30), (5.~.27),

it follows that

Grad ~R + ( Grad 6 R ) T = ( G - Z B ) - 1 . leo - Z ( ~ o - B . 60)]

(5.1.31)

+ [Co - Z(~¢o - B . So)]. (G - ZB) -a. Substituting equation (5.1.31) into the equation (5.1.29) and integrating with respect to Z, with regard to the equality

5.1 General Statements of Nonlinear Theory of Elastic Shells

137

dV = det (G - ZB) d Z dZ, we obtain the relation (5.1.28)I in which the tensors ~, and tt are expressed in terms of Cauchy stress tensor in the shell by formulae 1

v = L,T = v' + ( S . tL')T ~" =

S ( Z ) . T . G dZ,

tL = ~(tt' + tt' =

H1

][~IT)

S(Z). T . G Z dZ,

(5.1.32)

H1

S(Z) = (G - ZB) -1 det (G - ZB). These formulae make clear the mechanical meaning of two-dimensional tensors L, and tt, called the tensors of membrane forces and bending couples, respectively. Using equation (5.1.27) and integrating by parts, we transform the expression (5.1.28) as follows f~ 1/Yoda = ~

tr [(v - # . B ) . (V'~P) T + # . (V'~N) T] d Z

= - ~ [[V'. (v - tt" B)]. 5P + (V'. tt)" 5N] d Z + ~r [ M . ( v - / z .

B). 5P+ M.,.

V " t t -= P~"

(5.1.33)

5N] dS,

Ott Oq~ "

where F is the boundary contour of Z, dS is the arc element on the F, M is the unit outward normal to F which satisfies the condition M - N - 0. In view of equation (5.1.33), the expression for elementary work of external loads in equation (5.1.19) should be written in the form (5.1.34) It is apparent that the vector F represents the surface density of the force distributed over Z, p is the linear density of the force over the shell boundary. The vectors t~ and d must satisfy the condition t~. N = d . N = 0 since N . (iN - 0. It is easy to see that the vector t~ × N is the surface density of the couple distributed over Z, and d × N is the linear density of couple loads on the contour F. According to equation (5.1.26), the variation 5 N is not independent of virtual displacement 5P. So in the variational equation (5.1.19), it is necessary to transform the expressions (5.1.33) and (5.1.34) by integrating by parts once more in both of the surface and contour integrals. Assuming for simplicity that the contour F is smooth, we then obtain the following variational equation

138

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells f

- , / ~ [v'. , , - v'. (~,. B ) - S . (V'. ~,) + N V ' . [C. (V'. ~,)] + F + N V ' . ~ - B. ~]. ~P dE + .~ -

(M

. Iz . M

+-~(M

[[M. (v- 2~. B)+ d. B-p. G]. ~iP -

M

. d)

i)

(N

. dP)

. tz . t) - p . N -

+ (N

. dP)[M

(5.1.35)

. (V' . /z)

(d. t) + M . ~] d S = O.

Here, t is the unit tangent vector to the contour F. From the equation (5.1.35) it follows the shell equilibrium equation which we write down putting = 0, i.e. neglecting the couple load on the surface, V'. ( v - / z .

B ) - S . (V'./z) + N V ' . [G. (V'./z)] + F = 0.

(5.1.36)

The vector equation (5.1.36) is equivalent to the three scalar equations,

V~VZ# sz + B~Z (.~Z - Z~# ~z) + F = 0; F = F. N,

(5.1.37)

F z = F . P~.

Note that the equilibrium equations (5.1.36) or (5.1.37) can be obtained from pure static considerations (Zubov 1982), treating the balance of arbitrary part of the shell between the surfaces Z = -H1, Z = /-/2. With regard for equations (5.1.37), the variational equation (5.1.35) becomes

+

MZV~#"Z + ~ -

(MzG#~e) - P -

0

( M , M z # sz - M s d s) ~--~(N. P)

pZ=p.

PZ

p=p.N,

(5.1.38)

~

dS = 0,

dS=d.P

s

°

The unit normal and tangential to F vectors, M = M s P s and t = t s P s = G P s, respectively, can be expressed (Zubov 1982) in terms of the unit vectors of normal, m = m s p s, and tangent, Vo = T s p s , to the boundary contour 3' of the undeformed surface a, Ms = ; ~ ) ~ - l m s ,

t~ = A-1TZ'

A = dSds = V/'rs'rzGa~ : ~ / ( G / g ) m s m ~ G s ~

(5.1.39)

5.1 General Statements of Nonlinear Theory of Elastic Shells

139

ds being the arc element of 7. If the boundary contour F is not fixed, then the quantities

5P = G . P + N ( N . 5P)

O(N.SP)/OM

and

are arbitrary continuous functions of coordinate S on F. Arbitrariness of 5 P means that points of the contour can move in any direction, whereas the arbitrariness of O(N. (~P)/OM implies possibility of free rotation of the normal vector to the surface about tangent-to-contour direction. By the fundamental lemma of calculus of variations, from the equations (5.1.38) and (5.1.39) we obtain four boundary conditions which link contour values of the internal forces and couple tensors with given contour loads, 2B~# ~ ) : ~ ( p ~ - B~d ~)

~ / ~ m ~ (u ~ -

(13 = 1, 2),

/~~

m'~m~#`~ = )~m'~d~'

m~V~+~

d (,V g

(5.1.40)

/

d ()_lT,~d,~) In the case of a shell made of isotropic elastic material, the specific energy, ~Yo, will be an isotropie function of the following two-dimensional tensors (Zubov 1982)

b,~p'~p ~,

G,~p'~p ~,

B,~p'~p ~,

i.e. a function of the nine joint invariants of the above tensors 9 ~ b~,

9 ~ Ga~,

9 ~z B ~ ,

(bl1522- 522)/ 9,

G / 9, (5.1.41)

(B11B22 - B22)/9,

b~G,~,

b~B,~,

9~g'r~G,rrB~.

From the equations (5.1.21) and (5.1.41), we can get the following general representation of constitutional law of isotropic elastic shell,

~,~ =aog a~ + al b~ + a2Ga~ +aaB ~ , # ~ = cog~ + Clb~ + c2Ga~ -k- c3Ba~, ba/3 = g~g~b,~,

(5.1.42)

B ~ = G ~ G ~B~a,

where a~, c~ (i - 0, 1, 2, 3) are functions of the invariants defined by (5.1.41). If an isotropic shell is sufficiently thin and elongation and relative shears, due to deformation of the surface S, are small, the specific energy function for the shell can be taken in the form (Koiter 1966)

140

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells Wo =

Eh

[/2 tr 2Io + (1 -/2) tr Io2] 2(1 122) Eh a + [/2 tr 2Ko + (1 -/2) tr Ko2], 24(1 /22)

(5.1.43)

where E and/2 are the Young modulus and Poisson's ratio of the shell material, respectively, h is the shell thickness, and the tensors Io and Ko are defined by the relations (5.1.18).

5.2 Volterra Dislocations in Classical Nonlinear T h e o r y of Elastic Shells We state the problem of finding the displacement field, u = P - p , on the surface of a Love-type shell by given components of tensors of tangential and bending strains which are defined by formulas (5.1.18). Specifying of these quantities is clearly equivalent to specifying of the quadratic form coefficients G~z and B ~ of the deformed surface. In what follows, we shall assume that the functions p~ (q~), n(q'Y), gaz(q'~), and G ~ ( q ~) are continuous and have continuous derivatives up to the second order in the domain a, and the functions b~(q~), B ~ ( q ~) are continuously differentiable. Using the expression for distortion tensor of the shell, equation (5.1.8), and formulas (5.1.1)-(5.1.3), we obtain the following relations (Zubov 1989a) 0Co = II~. Co, (5.2.1) Oq~ (5.2.2) The 2nd order tensors Ha (a = 1, 2) are fully defined if the functions I~(q~), K ~ (q~) are specified. The relations (5.2.1) are the system of equations to determine the distortion tensor Co by given tensors of tangential and bending strains. The necessary and sufficient conditions of solvability of the system (5.2.1) are 0H~ Oq~

0H~ - YI~. YI~ - II~. YI~. Oq~

(5.2.3)

The equation contains three independent scalar relations which are equivalent to Gauss-Codazzi equations, well known in differential geometry, with respect to G~Z and B ~ . The Gauss-Codazzi conditions for the functions g~, b~ are now fulfilled identically since the surface a is given. Having defined the distortion tensor, with regard for (5.1.8), we can find the radius-vector P of the deformed surface Z 0P Oq~ = p,~ . Co.

(5.2.4)

5.2 Volterra Dislocations in Classical Nonlinear Theory of Elastic Shells

141

The conditions of integrability of the system (5.2.4) with respect to P hold by virtue of equation (5.2.1). Indeed, these conditions have the form O(p~ . Co) Oq~

O(p~ . Co) Oq~

and, in view of equation (5.2.1), are equivalent to the relations p~.II~ = p~.II~, which follows from the symmetry properties, = b o,

= 8

o,

=

= rL.

A solution to the system (5.2.1), as in Sect. 1.2, can be written with use of curvilinear multiplicative integral A

Co(M) =

(E + d o - H o ) . Co(M0),

(5.2.5)

o

dp = dq~p~,

IIo = p~II~,

where 3d0 and 3d are the initial and current points, respectively, of the integration curve. The system (5.2.4) can be solved by ordinary integrating P =

do" Co + P(Ad0).

(5.2.6)

o

If the surface a is simply connected, the formulae (5.2.5), (5.2.6) define a single-valued (i.e. independent of the choice of integration path), twice continuously differentiable field of displacements of shell surface. In the case of multiconnected surface, the distortion tensor field and vector displacement field may be multi-valued. This non-uniqueness in values can be removed by cutting the surface along the curves Ik, which turn the multi-connected surface a into a simply connected one; then the tensor Co and vector P , in general, have jumps on intersecting the cuts Ik. It can be proved that the values Co+ of the distortion tensor on the opposite borders of the cut are related by the formula Co+ = Co_. (I)k,

(5.2.7)

where (I)k is a 2nd order tensor that is constant on each of the cuts lk. In view of continuity of the quantities G ~ , from the polar decomposition of the surface distortion tensor (5.1.9), we obtain that the tensors (I)k are proper orthogonal, ~k • ~ " = E,

det ~k = 1.

According to equation (5.2.4), on the cut lk, we have OP+/Os = r . Co+,

(5.2.8)

where s is the arc length, r is the unit tangent vector to the curve lk. From equations (5.2.7) and (5.2.8)it follows

142

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells O P + / O s = ( O P _ / O s ) . Ok.

Integrating this over the curve Ik, we obtain P + = P _ • Ok + ak,

(5.2.9)

where ak are constant vectors. Formula (5.2.9) means that in the deformed shell state, the position of one cut border differs from the position of another border by a finite displacement of absolutely rigid body. Using the representation of the orthogonal tensor Ok in terms of finite rotation vectors ~bk, we arrive at the formula for the displacement jump 1 U+ -- U_ =

)-1 Ck X

l +-4 ~k . fbk

(1

P _ +-~fb k x P _

)

+ak.

(5.2.10)

Topological structure of multi-connected surface may be very complex. In what follows, we shall consider a multi-connected surface a that is homeomorphic to a circle containing holes, the boundary of a hole is a simple closed curve. For a doubly connected shell (k = 1), the parameters ~1, a l are expressed in terms of tensor fields of tangential and bending strains by f

A

.

~1 = Col(M0) • t

(E + d p . Ho). Co(M0),

JM

^

a~ =

/ f2 dp'.

0

(5.2.11)

(E + d p . Ho). Co(A40) + p(A/f0). (E - ~ ) .

0

Using the properties of curvilinear multiplicative integral, one can prove that the quantities ~bl, al, defining the displacement jump, do not depend on the choice of a cut 11 which converts the domain into a simply connected one. If the vectors a l, ~bl are not zero simultaneously, we shall say that the doubly connected shell contains a Volterra dislocation. When ~b1 - 0, it is a translational dislocation. In general, the Volterra dislocation contains translational dislocation and disclination, the latter is characterized by the vector ~b1. If there are Volterra dislocations in a multi-connected shell then its displacement field is multi-valued. A multi-valued solution to equations (5.2.1) in a doubly connected domain has the form Co

Co,

. (I~ n1- m

,

where Co. is the value of distortion tensor given by formula (5.2.5) in the simply connected (i.e., being cut) domain, n denotes the number of positive (i.e. intersecting the cut from "+"- to "-"-border of the cut) complete revolutions of integration curve whereas m the number of negative (that is from "-" to "+") ones. When m = 0, the general expression of vector P in a doubly connected domain becomes

5.2 Volterra Dislocations in Classical Nonlinear Theory of Elastic Shells

143

P =

d p . Co,. ,I,~ + P o + o -[- (~(n)[al -[- Po" (,I)~ - E)]. (E + (I)1 -b...-[-- (I)? -1), Vo = V(Mo),

-

o,

n -- 0

( 1, n-riO. Here, the path of ordinary integration must not intersect the cut 11. Note that non-uniqueness of values of distortion tensor, by virtue of single-valuedness of tangential strain tensor, is entirely due to multi-valuedness of the rotation field, Ao (qa), of the deforming surface. If it is necessary to cut the shell more than one time to convert it into a simply connected one, the orthogonal tensors ~k and Burgers vectors ak are determined by equations of the type of (5.1.11), in which the closed contour must intersect the only cut Ik. On substituting the constitutive relations for Love-type nonlinearly elastic shell, equations (5.1.21), into the equilibrium equations (5.1.37), in which F ~ and F are given, the latter, combined with the compatibility conditions (5.2.3), constitute a complete system of equations with respect to tangential (Ia~) and bending (K~) strains. For multi-connected domain, to these, we should supplement the integral relations (5.2.11) defining the characteristics of isolated defects. When ak - ~bk - 0, the relations (5.2.11) express the requirement for displacements to be single-valued. Force boundary conditions on the edge of shell, written in the form of equation (5.1.40), are the relations with respect to I~, K~ and their derivatives; they do not contain the displacements. Thus, the problem of equilibrium of nonlinearly elastic shell with Volterra dislocation is reduced to the boundary value problem in the unknowns I~z and K~z. It is possible that the domain ~F, occupied by a shell in the deformed state, is not simply connected. Now the radius-vector P is continuous through the cuts converting the ~F into a simply connected one. In a similar manner, it can be proved that for single-valued and sufficiently smooth tensor fields of Ia~ and K~Z, on the cuts, the distortion tensor and the radius-vector defining the point position on the surface a in the deformed shell, have, in general, a jump of the form Co+ - ~k" Co_, p+ = p _ . ~T + dk, where ~k are constant, proper orthogonal tensors and dk are constant vectors. For a homogeneous and isotropic shell with Volterra dislocation, the problem of equilibrium can also be stated as a boundary value problem with respect to Ia~, Kay; the surface ~F and its boundary must be given. An important particular case of Volterra dislocation is the translational dislocation, the Frank vector ~b of which is zero. If Frank vectors are zero for all Volterra dislocations in multi-connected shell, then the rotation field and the field of the distortion tensor are single-valued in the multi-connected domain. Now one can state a problem to find out the shell displacement field by a given,

144

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

single-valued and continuously differentiable field of distortion tensor Co. This problem can be solved much easier than that on determining the displacements by given tangential and bending strains, because it does not require to use the multiplicative integrals. According to equation (5.2.4), the radius-vector of the deformed surface, P, is expressed as follows

P(J~J) = P0 +

dp. Co.

(5.2.12)

0

The necessary and sufficient condition for the integral in equation (5.2.12) to be independent of integration path in a simply connected domain is

~

dp. Co = 0

for any closed contour -y. Using the formula of transformation integral into a surface one, proved in (Zubov 1982),

~

dp. Co = - ~

(5.2.13) of a contour

V. (n x Co)da,

(5.2.14)

in view of arbitrariness of a part of the surface a, whose boundary is -y, based on the equation (5.2.13), we arrive at the following vector compatibility equation with respect to distortion tensor:

V. (e. C o ) = 0 , (5.2.15) e=-nxE. where e is the discriminant tensor of the surface a. The jump in displacement field on the cuts of multi-connected shell is now described by formula (5.2.10) with ~bk = 0. The Burgers vectors ak are expressed in terms of tensor field of Co by P

ak - ¢

dp. Co.

(5.2.16)

Jr k where ~/k is the contour enclosing the kth hole.

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells The nonlinear theory of Cosserat-type shells is similar to the Cosserat continuum theory discussed in Sect. 4.1. Contrary to the Love-type shell theory described in Sect. 5.1, each particle of two-dimensional Cosserat continuum in the deformed state is specified by its position P(qa) in the space and by a proper orthogonal tensor H(qa), i.e. the particle has the rigid-body degrees of freedom (Zhilin 1982; Altenbach & Zhilin 1988). The quantities q~ (ol = I, 2)

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells

145

are Gaussian coordinates on the surface a, that defines the reference configuration of the shell, as well as on the surface Z of the deformed configuration. Using the principle of material frame-indifference, we can represent the specific (per unit area of the surface a) potential strain energy of Cosserat-type shell, 14;o, as a function of two strain measures, Yo = ( V P ) • H T,

L o = ~p~ (0~qH a • H T ) x,

(5.3.1)

Wo = Wo(Yo, Lo). As in Sect 5 1, V = p~ o refers to the surface gradient on a, p~ is the vector basis on a. According to equation (5.3.1), the tensors Yo and Lo possess the following property n . Y o = n . Lo = 0 , (5.3.2) •

.

where n is the normal to a. The equilibrium equations and static boundary conditions for nonlinearly elastic Cosserat shell are derived from the principle of virtual work; they are V . ( r * . H) + f = 0,

V . (~r*. H) + (VP T.'r*. H)× + l = 0;

OWo

zr*

0Yo'

~'*. H,

(5.3.3)

0Lo '

m . r * . H[~ = f l , r = IG(Vp)T.

OWo m . ~r*. HI~ = 11;

~r = I G ( V p ) T .

It" • H.

(5.3.4) (5.3.5)

Here, r and ~r are the tensors of forces and couples in the shell, they are analogous to the Cauchy stress tensor in theory of elasticity of simple materials, ~'* and vr* are the tensors of forces and couples that are similar to Kirchhoff stress tensor, f and l are the force and couple loads per unit area of the surface a, f l and 11 are the force and couple contour loads, m is the normal to boundary contour 7 of a ( m . n = 0). For the Cosserat shell, the tensor of forces, r , and that of couples, ~r, satisfy the condition N . r = N . lr = 0, while the tensors ~'* and vr* satisfy the condition n . ~'* = n . zr* = 0. To introduce the concept of isolated defect in the theory of Cosserat-type shell, consider the problem of determining the field of displacements and rotations for a two-dimensional continuum by given fields of strain tensors Yo and Lo which are specified as continuously differentiable functions of coordinates q~.

From equations (5.3.1) we have OH

c3q~

= ~.

H,

~

= _ ~ T = - E x (p~. Lo)

(a = 1, 2).

(5.3.6)

The equations (5.3.6) are soluble with respect to H if and only if there hold

Oq~

i)qZ

= V~. V , -

V,. V.

(a, Z = 1,2);

(5.3.7)

146

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

this is equivalent to 3 scalar equations. The solution to equations (5.3.6) is A

H(M) =

E

(E + dp. @). H(M0),

(5.3.8)

o

= p~.

Having defined H by equation (5.3.8), we find out the point positions of the deformed shell surface from equation (5.3.1) by integrating

P(M) =

dp. (Yo" H T) + P(M0).

(5.3.9)

o

The necessary and sufficient conditions for the integral in equation (5.3.9) to be independent of integration path in a simply connected domain are n.

0Yo O~ × ~ +

p~

×Yo'~

]

=0.

(5.3.10)

The conditions (5.3.7), (5.3.10) consisting of six scalar relations, are the equations for compatibility of deformation in the nonlinear theory of Cosserattype shells. Using the equations (5.3.8), (5.3.9), we can prove that the following formulae H + = H _ • q~,

P+

= P_

• ,I~ + a .

(5.3.11)

A

v = ~ - ~ = I-I~(Mo).

(E + do. ~ ) . I-I(Mo), o

^

-

(5.3.12)

d # . Yo(#).

(E + do. ~ ) . H(Mo) o

+ V(Mo). (E- ~) hold on the cut which converts the doubly connected domain a into a simply connected one. The expressions (5.3.11) show that in Cosserat shells undergoing large deformation, defects in the form of Volterra dislocations may exist, i.e. there are such states in which strain tensors, as well as the tensors of forces and couples, r* and 7r* , are single-valued and continuous in the multi-connected domain whereas the displacements have jumps on the cuts which convert the domain into simply connected one; a jump correspond to a rigid-body relative shift of cut borders. The problem of equilibrium of Cosserat shell with an isolated dislocation or disclination can be reduced to a boundary value problem with respect to unknowns Yo and Lo. Indeed, on right-multiplying the equilibrium equation (5.3.3) by the tensor H T and taking equation (5.3.6) into account, we obtain

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells

147

V . 7"* + p a . 7", . ~a + if" HT -- 0, (5.3.13) V . l r * + p " . ~r*. ~I,, + (YT • r*)× + I . H T = O . Suppose that the load is tracking, i.e. the vectors f and l depend surface deformation as follows

on the

f ( q " ) = f o [Yo(q~), Lo(q"), q"]" H(q"), l(q ~) = lo [Yo(q"), Lo(q~), q"]. H(q~).

Since the tensors r* and 7r* are expressed in terms of Yo and Lo and f . H T = f0, 1. H T = /0, the equilibrium conditions (5.3.13) are the equations with respect to the strain tensors Yo, Lo. Combining the compatibility conditions (5.3.7), (5.3.10), the integral relations (5.3.12), and the boundary conditions (5.3.4) with the above equations and assuming that the contour loads also have tracking nature, we obtain a boundary value problem with Yo, Lo as unknowns. The theory of dislocations and disclinations in elastic shells can be significantly simplified if the strains and rotations are small, this means that we can turn to the linear theory of Cosserat's shells. Assuming the rotations and strains to be small, from equation (5.3.1), within the terms of the first order of smallness, we have Yo ~ g + e,

H ~ E - E x ~a,

e = V u + g × ~o,

Lo ~ ~;

(5.3.14)

~ = V~o; (5.3.15)

u = P-

p,

g = E-

nn.

Here, u is the displacement vector of shell surface particles, ~o is the linear rotation vector, e is the linear strain tensor, ~ is the linear tensor of bending strains. Now we do not distinguish the geometry of the deformed surface, ~w, and that of the reference configuration, a; thus the tensors of forces and couples, 7- and ~r, in equations (5.3.5) are identified with tensors 7-* and lr*, respectively. Therefore, the equilibrium equations in forces and couples and the constitutive relations in the geometrically linear theory of Cosserat's shells become V . 7- + f = O, 7- =

0e '

V . ~r + r× + t = O,

~r = 0~ '

Wo = Wo(e, ~).

(5.3.16) (5.3.17)

By elimination of displacement and rotation vectors from the relations (5.3.15), we obtain the compatibility equations of linear theory of Cosserat's shells V . (e. ~) = 0, V . (e. e) + (e. ~)x = 0, (5.3.18) e=-E×n. These equations can also be obtained by means of linearization of nonlinear compatibility equations (5.3.7), (5.3.10), with use of equations (5.3.14).

148

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

The problem of determining of the fields of u and ~ under given tensor fields e and ~ which are continuously differentiable on a, can be solved with use of relations (5.3.15). We have ~o = ~oo +

dp.~,

~Oo = ~o(M0);

(5.3.19)

o

du - d p .

~ + ~ × dp.

(5.3.20)

Substituting equation (5.3.19) into equation (5.3.20) and integrating the equation obtained, we get the u in the form of double integral. Reversing the order of integration, we can reduce the double integral to an ordinary one and obtain the following shell displacements field u ( p ) = Uo + ~o × ( P -

Po) +

d o " [e' + ~' × ( p -

if)I,

o

Uo = u(.Mo),

P0 = p(.Mo),

(5.3.21)

The prime in equations (5.3.21) marks the integration variable. It is apparent that the necessary and sufficient conditions for the integrals in equations (5.3.19), (5.3.21) to be independent of integration path in a simply connected domain coincide with the strain compatibility equations (5.3.18). Suppose now that the surface a contains holes, i.e., it is multi-connected. Consider a closed contour 7k enclosing one of the holes. Setting p -- P0 in equations (5.3.19), (5.3.21), we find the increments of vectors ~ and u on going around this contour U+ -- U_ = ak + Ck X P0,

~+ - ~ - = Ck,

(5.3.22)

Due to the fact that in linear shell theory, the Frank and Burgers vectors of isolated defects, Ok and ak, are expressed, in accordance with equation (5.3.23), by ordinary curvilinear integrals, using the results for a discrete set of Volterra dislocations we can get corresponding results for continuous distribution of dislocations and disclinations in a shell, following the method by de Wit (de Wit 1973a) for three-dimensional linearly elastic continuum. If the diameters of the holes in the multi-connected surface a tend to zero then we get the shell with a discrete set of isolated defects whose axes are straight and orthogonal to the surface a at the points. Using equation (5.3.23) we can find the total Frank vector for the system of defects over some part ao of the surface a, (5.3.24) k

k

k

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells

149

By well-known properties of elementary curvilinear integrals and singlevaluedness of tensor field ~, the sum of integrals in equation (5.3.24) can be replaced by a single integral over the closed contour % enclosing all the defects in the domain a0,

~bo = ~

dp. ~.

(5.3.25)

Let us now assume that the defects are continuously distributed over a, and the resulting Frank vector of all the defects contained in the domain a0 with the boundary 70 is given by formula (5.3.25). Now we consider the tensor field t¢ to be continuous and differentiable in a; this enables us to transform the contour integral in equation (5.3.25) into an integral over a0. It should be remembered that on the passage to continuous distribution of defects, there is qualitative change in physical meaning of strain tensors, in particular of the bending strain tensor ~. In the continual theory of dislocations (de Wit 1977), the strain component caused by distributed defects is called irreversible or plastic, it is denoted by ~(P) . Hence the total Frank vector of distributed defects over ao is represented by ¢o = ]] n . ( V × t¢(p)) da = ~ V. (e. t¢(p)) da. 0

(5.3.26)

o

As the area a0 is arbitrary, the formula (5.3.26) gives ground to refer to the integrand as to the disclination density, ~,

/3 = V. (e. to(P)).

(5.3.27)

Note that the disclination density in shells is a vector quantity whereas in a three-dimensional medium (de Wit 1973a) it is tensor-valued. Comparing equation (5.3.18) with equation (5.3.27) we see that for ~ :fi 0 the tensor ~(P) does not satisfy the compatibility deformation equations. This means that there is no vector field ~(P) such that V~ (p) = ~(P). To introduce the concept of dislocation density in the shell, consider the case when there is no disclination, i.e. the disclination density ~ is zero. Then, V × ~(P) = 0, and so there exists a vector ¢p(P) such that V~ (p) = ~(P). Thus, there is a plastic distortion defined, in view of equation (5.3.15), by the relation

(Vu) (p) = 6 (p) - g × ~(P).

(5.3.28)

If the Frank vector of each of isolated defects on the area a0 is equal to zero, the Burgers vector for the defect is defined by the formula

ak=~~

dp.(Vu)=~k

dp.(e-gx~a),

that is simpler than equation (5.3.23), and the total Burgers vector of all the defects in the domain a0 is expressed by

150

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

ao--Eak--Ej( k

k

dP" (¢ - g x ~) -- f o dP" (~ - g x ~).

(5.3.29)

k

Passing to continuous distribution of translational dislocations, let us transform the contour integral in equation (5.3.29) into a surface one; the distortion due to dislocations is considered, as above, to be plastic. This leads us, in view of equation (5.3.28), to the expression for resulting Burgers vector, a0

"-- ~ O~do,

o~-- V" (~(P)-g

× ~(P)).

(5.3.30)

o

Using the identity V. (e × ¢)

=

-(e-V¢)×,

which can be directly verified for any differentiable on a field ¢, from equation (5.3.30) we obtain a = V. (e. e (p)) + (e. to(P))×. (5.3.31) On the ground of equation (5.3.30), the vector quantity c~ should be called the density of translational dislocations. We also take the expression (5.3.31) as the definition of density of shell dislocations in the general case, i.e. when the density of disclinations/3 ~ 0. Using the definitions of densities of dislocations and disclinations, equations (5.3.31) and (5.3.27), we find out the resulting Burgers vector for a system of defects continuously distributed over the area a0; we now exclude the simplifying assumption on absence of disclinations. By the general formula (5.3.23), we have = ~ dp. (¢(P)- ~(P) x p) J-yo

ao /,

= [ Ja

(53

V. (e. 6 (p) - e. ~(P) × p) da. o

Using equation (5.3.27), transform the second term in the integrand of (5.3.32) V. (e. t¢(p) × p) =/3 × p - (e. to(P))×. Hence, by (5.3.31), a0

= [

(a + p x/3) da.

(5.3.33)

Ja o

Formula (5.3.33) shows that, under general continuous distribution of defects, the resulting Burgers vector is defined not only by density of dislocations but also by that of disclinations. A similar statement is valid in the threedimensional continual theory of disclinations (de Wit 1977). Note that the distinguish of continual theory of dislocations and disclinations in shells from the relevant three-dimensional theory (de Wit 1977) is not only in that the densities c~ and fl are vector-valued but also in that the vector fields c~ and/3 in the theory of shells do not tied by any differential equations.

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells

151

To state the boundary value problem of equilibrium of a shell with given densities of dislocations and disclinations, we need to remember that the total deformation ¢ in the geometrically linear theory, as well as the total bending deformation ~, consists of two components: elastic and plastic, _.. ~(e) _~_ $(p),

/~ ._ /~(e) .~_/~(p).

(5.3.34)

The elastic strains are state parameters of a shell, they relate to internal forces ~" and couples Ir by formulas (5.3.17),

OWo 7" "-- (~¢(e)'

OWo 71" "-- (~/~(e) '

(5.3.35)

Wo = Wo(6Ie/, ~¢e/). The total strains e and ~ satisfy the compatibility equations (5.3.18). With regard for equations (5.3.27) and (5.3.31), it follows the equations for elastic deformation V. (e./~(e)) _[_/~ __ 0, (5.3.36) V" (e. e (e)) + (e. ~(~)) × + c~ = 0.

Expressing the tensors of efforts and couples in terms of elastic strains from the governing relations (5.3.35) and substituting them into the equilibrium equations (5.3.16), we get the equations which, together with equations (5.3.36), constitute the complete system of equations to determine the tensor fields of ~:(e) and ~(e) under given external loads f, l and specified densities of dislocations and disclinations, c~ and j3. The equations (5.3.26), which may be called the incompatibility equations of deformation, copy the equilibrium equations (5.3.16) in forces and couples up to the notation. It is clear that the system of equations (5.3.16) becomes the (5.3.36) under the substitutions r ~

e./~(e),

11"

~

e.

$(e) (5.3.37)

and vice versa. The identity of the equilibrium equations in forces and couples with no external loads and the compatibility equation for strains is well known in linear shell theory, it is a so-called static-geometrical analogy (Goldenveizer 1976). We generalized this analogy to the case of non-zero surface load; now densities of disclinations, /3, and dislocations, c~, correspond to the intensities of force load, f, and distributed couple load, l, respectively. The system of resolving equations to determine the stress state of a shell with distributed defects can be constructed by other means, using the stress functions. To do this, note that the homogeneous equilibrium equations (5.3.16) are identical to the deformation compatibility equations (5.3.18) whose general solution is given by formulas

152

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

(5.3.15). Therefore, the general solution of the equilibrium equations in forces and couples may be represented as follows r = e. VI7 + 7"0,

lr = e. (VA + g × ~7) + lr0,

(5.3.38)

where ~7, X are some vector fields twice differentiable on a; ~'0, lr0 are a particular solution to equations (5.3.16) with given loads f , I. Vectors ~ and X will be called the stress functions of shell theory. Assume that the governing relations (5.3.35) can be uniquely inverted with respect to e (e) and ~(e), i.e. the elastic strains are expressed in terms of efforts and couples. Using the Legendre transformation, we can represent the inverted governing relations as follows

-- 0~" '

01r;

(5.3.39)

~2o(~', 7r) = tr (~.T. ¢(e) + lr T " ~(e)) _ Wo,

~o being the specific complementary energy of the shell. If we substitute the expressions for forces and moments in terms of the stress functions (5.3.38) into the governing relations (5.3.39), the elastic deformations 6(~) and ~(~) become expressed in terms of the stress functions. The substitution of these relations into the incompatibility equations (5.3.36) results in a system of equations in the stress functions ~7 and A. To state the boundary conditions at the shell edge, first we turn to the equilibrium problem with no distributed defects, i.e., with c~ - /3 = 0. Now the elastic strains are compatible and there exist the elastic displacement, u, and elastic rotation, ~. One of the wide-spread types of boundary conditions in shell theory is the kinematic conditions when, on some part of the boundary, 7~, there are given some displacements and rotations,

ul~ = u'(s),

~l~

= ~'(s),

(5.3.40)

u', ~' being given functions of arc-length of boundary contour. Based on equations (5.3.15), from equations (5.3.40) we obtain

t . ef~, = Ou'/Os + t × ~',

(5.3.41)

t = m . e = - m × n, where t is the unit tangent vector to 71. Conditions (5.3.41) impose the restrictions on the boundary values of tensors of strains and bending deformations and thus are called the deformation conditions. The deformation boundary conditions were first stated by Chernykh (Chernykh 1957) in the framework of linear theory of Love-type shells. Generally speaking, the deformation conditions (5.3.41) are not equivalent to the kinematic conditions (5.3.40). However

5.3 Isolated and Continuously Distributed Defects in Cosserat-type Shells

153

if the part of boundary, "/1, is connected, then the kinematic conditions, as easy to see, can be reconstructed, if we know the deformation conditions, up to a rigid body displacement. Since a rigid body displacement has no effect on the stressed state of the shell, we can consider the boundary conditions (5.3.40) to be equivalent to equations (5.3.41) when the portion 71 of the boundary contour ~/is connected. Elastic displacements and rotations do not exist in a shell with continuously distributed defects; so the kinematic boundary conditions are meaningless. Nevertheless, the deformation conditions may also be applied to the case of incompatible deformations. Thus, on a connected part 71 of the boundary of a, the deformation conditions can be stated as follows t " ~:(e)171 --

V(S), t " ~(e)]~1 = w(s) v(s), w(s). It is apparent

(5.3.42)

with some specified functions that the conditions (5.3.42), by equations (5.3.38), (5.3.39), may be formulated in terms of the stress functions ~7, A. Another common type of boundary conditions in shell theory is to define external loads on the contour: m . 1 - ] ~ - f l (s),

m.

7r],,/2 -- ll (S).

(5.3.43)

Now we also assume that the part of the boundary, 72, is connected, moreover, it supplements the curve 71 to the full boundary 7. Using the relations (5.3.38), the conditions (5.3.43) can be rewritten in terms of stress functions

d--'~Jv~ =

I1 -

m

"r0,

~

~-- t x ~

)

1"),2 - - l l

-

-

m

. l r0.

(5.3.44)

Let us find out the general solution to equation (5.3.44) which is a system of ordinary differential equations with respect to ~7(s), A(s):

,/(s) = V/o(S) + Ao,

A(s) = Ao(S)+ A1 + Ao x ( p - Po),

where A0, A1 are constant vectors, ~70(s), A0(s) is a particular solution of the system (5.3.44). The constants A0, A1 do not effect on values of tensors r and lr; thus for a connected portion 72 of the boundary, the conditions (5.3.44) may be considered to be equivalent to rll~ = rl0,

Xl~ = X0.

(5.3.45)

We can give a variational set up for the boundary problem of equilibrium of a shell with continuously distributed dislocations and disclinations. Consider the functional $2 defined on the set of twice differentiable stress functions which satisfy the conditions (5.3.45), ~? =

[ V(rl,A)de + [(a.rl + j3" A)de

Ja

Ja

(5.3.46)

f

+ [ (v.rl+w. J.y1

A)ds.

154

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

It is easy to verify that the stationary conditions for $2 are given by the incompatibility equations, written in terms of the stress functions, and the deformation boundary conditions (5.3.42).

5.4 Invariant Contour Integrals in the Nonlinear Theory of Plates and Shells We now consider energy integrals that define the resistance force arising due to the change in position of a defect in nonlinearly elastic plates and shells. The system of equilibrium equations for Love-type shells, presented in Sect. 5.1, is rather complicated that involves considerable difficulties to construct the invariant energy integrals. However there is a simple form of nonlinear shell equations, presented in the paper (Zubov 1989b), that enables us to derive a conservation law resulting in an expression for invariant contour integral. Let us consider a new tensor of membrane forces, Uo, and a new tensor of bending couples, tto, related to the tensors v and tt from Sect. 5.1 by

(5.4.1) /-to =

P~ =

• ~.

According to equation (5.1.17), the specific shell energy depends on the surface deformation in terms of tensors

G~p~p ~ and B,~p~p p. By using (5.1.2) and (5.1.5), it is easy to verify the relations

B,~p"p ~ = - ( V N ) . ( V P ) T

Go,~p°'p~ = ( V P ) . (VP) T,

(5.4.2)

thus we can consider the specific energy Wo to be a function of tensors VP and VN, Wo = Wo(VP, V N ) . (5.4.3)

By (5.4.1) and (5.4.2), the constitutive relations for elastic shell, equations (5.1.21), are transformed into

OWo

0Wo V o - OVP - -

]

~o

=

OVN

o

(5.4.4)

It can be verified that the equilibrium shell equations in forces and couples (5.1.36) can be written in the following divergent form V ' . [u - ~ . B + ( V ' - ~ ) . G N ] + f = 0.

(5.4.5)

5.4 Invariant Contour Integrals in the Nonlinear Theory of Plates and Shells

155

For any differentiable tensor field • defined on the surface a and satisfying the condition N . • = 0, the following identity (5.4.6) is valid (see (Zubov 1982)). By equations (5.4.1) and (5.4.6), the equation (5.4.5) takes another form, it is the equilibrium equations with respect to tensors Vo and tto,

v.

+ ( v . t,o). (V'p)N] + f = 0, (5.4.7) f = ~/~F,

where f represents the intensity of load per unit area in the reference configuration of the shell. The vector equation (5.4.7) is similar to the equilibrium equation (1.1.12) of three-dimensional elasticity of simple materials, whereas the tensors of forces, vo, and bending couples,/to, are analogous to the Piola stress tensor. The dynamic boundary conditions (5.1.40) can be written in terms of tensors vo and ~o as follows 0 0 (A_ldsN), m . [•o + (V. tto). (V'p)N] + -~s(#mSA-1N) = no + -~s

#raM : d M , #ms - m . tto " t,

fl = d S / d s ,

#mM -- m . tto " M ,

do = d M M + dst.

(5.4.8) As in Sect. 5.1, M is the normal to the boundary F of surface Z ( M - N = 0), t is the unit tangential vector to F, and P0 and do x N are the intensities (per unit length of 7) of force and couple loads on the shell edge, respectively. Let us suppose that the external surface load, if, per unit area of a, consists of two parts: f = f0 + k(q 1, q2) N , fo = const being an uniform dead load and k N being a tracking normal load. For the nonlinearly elastic plate, that is a is plane, consider the following contour integral J = ~o [m(l/Yo - f0" P ) V p T =_ (VP) T,

m . (vo. V P T + pro" VNT)] ds,

(5.4.9)

V N T =_ (VN) T.

Now we prove T h e o r e m . Let 70 be a closed piecewise-smooth contour bordering the domain ao, in which the plate material is homogeneous, and the tensors V P and V I ~ are continuously differentiable functions of the coordinates ql, q2. Then the integral J is equal to zero.

156

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

Proof. Using the divergence theorem, we can transform the integral (5.4.9) as follows [VWo - f0" v p T - - V. (Vo. V P T + Vo. VNT)] da

J = ~

(5.4.10)

o

For a plate made of a homogeneous material, the specific energy 14~o depends on the coordinates ql, q2 only in terms of tensors VP and VN. (Note that it is not valid if a is not plane, even for homogeneous material.) Taking into account the equalities

N . V P T = O,

V N T = - ( V ' p ) . ( V V P ) . N, (5.4.11)

( V N ) . (V. Do) = V P . {V. [(V. Do)" (V'p)N]} and the relation

VWo = p~. Vo. ~ ( V P Oq~

T) +

"Do"

(VNT),

resulting from (5.4.4) and plate homogeneity, we obtain from (5.4.10) that J = - ~ {V. [vo + (V. Do)" (V'p)N] + f0}" v p T da.

(5.4.12)

o

From the equilibrium equations (5.4.7), relation (5.4.12) and first equality in (5.4.11), it follows that

J = ~ k N . V P T da = O, o

which concludes the proof. If the theorem conditions fail in some domain a' C a0, the integral J over a closed contour "f enclosing the # does not vanish in general, but its value is independent of the choice of ~/ enclosing #. The sub-domain a ~ may include non-homogeneities, inclusions, dislocations, disclinations, and other defects. The vector integral J is similar to Eshelby integral (Eshelby 1956) in three-dimensional elasticity. Another form of J-integral, following from (5.1.21) and (5.4.1), is of interest,

OWo

OWo

oI~ G~ + 2 0K~ B ~ r~ds.

(5.4.13)

Applying the equality

m.Do.vNT

-

O#mSA-1N O(VP) T Os "vPT--#mMN" O-----M--

( i)~ ) ~ - M " (V'V)

we obtain another representation of invariant integral in nonlinear plate theory

5.4 Invariant Contour Integrals in the Nonlinear Theory of Plates and Shells

J=

, m(Wo - fo. P) -

[

m . Uo +

Os

1

157

"v p T + (5.4.14)

+ #mMN" O(VP)T } OM ds. Using the expressions (5.4.14) and the boundary conditions in the form of equations (5.4.8), we may prove in a manner similar to one of Sect. 4.5 that the potential energy increment due to the change of defect position in the plate with load-free boundary is expressed by a formula of the form (4.5.13). This means that the integral (5.4.10) represents the resistance force arising on moving a defect in the plate. Let us suppose that in the plate there is an infinitely thin crack parallel to a unit vector h. The crack borders are load-free, i.e., the conditions (5.4.8) are fulfilled on the crack when P0 = do = 0. Now, on the basis of the theorem, the expression (5.4.14) and the equality N . V P T = 0, we can prove that the integral J - h . J has a value which is constant for all contours enclosing one of the crack ends and starting and finishing at the opposite crack borders. For integral J to be independent of the choice of contour it is sufficient the homogeneity of material only in the crack direction. Moreover, J-integral invariance occurs not only for plane plate but also for cylindrical shell of arbitrary cross-section if the crack is parallel to the cylinder generator. This is valid also for a cutout of finite width with parallel borders. The integral J is similar to Cherepanov-Rice integral (Cherepanov 1967; Rice 1968). Let P = p + u in (5.4.14), u being the displacement vector of the surface a; taking into account the equilibrium condition of a part of shell a' which follows from (5.4.8),

~ , f da + J(~, { m . [Vo + (V. tto) . (V'p)N] +

O#msA - 1 N } Os ds = O,

(the tracking load is absent, k = 0), we obtain the following representation of the energy integral

J =

~{, m(Wo - fo" u) +

#raMN "

0(Vu)r OM (5.4.15)

-

m .Vo + m . (V'o) T. (V. tto)N +

i)s

This expression of the invariant contour integral would be appropriate for use if the strains are small, in particular, in geometrically linear theory of plates and shells. Let the boundary "7 of a plate or a shell consists of two parts, "~ = "h ~ V2; the part ")'1is fixed or hinged and a dead force load acts on V2. Then the potential energy of elastic shell, with k = 0, is given by

158

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

It may be proved that the change of energy due to the crack propagation by a distance of d~ can be written as dA = - J d ~ . Since the invariant energy integral, in accordance with (5.4.16), the energy may be used as a criterion of crack bending, as in the case of plane stress

(5.4.16)

J, in nonlinear shell theory characterizes, release when the crack grows, the integral propagation in plates and shells under state (Broberg 1971).

5.5 T h e t w o P r o b l e m s : a W e d g e D i s c l i n a t i o n in a Shell of R e v o l u t i o n and a S c r e w D i s l o c a t i o n in a Cylindrical Shell Now we consider two particular problems for Volterra dislocations in nonlinearly elastic shells of the Love type. In these problems, the general system of equilibrium equations for shells can be reduced to a system of ordinary differential equations, that significantly simplifies finding of the stress state of the shell. Assume that the surface a of undeformed Love-type shell is a surface of revolution given by equation r = r ( x ) , x = ql being the distance along the shell axis of rotation, r being the distance from this axis. The second Gaussian coordinate on a is the polar angle ~0, q2 = ~o. So the position of a point of shell surface in the reference configuration is characterized by its spatial cylindrical coordinates r, ~o, x. Denote by R, ~, X the cylindrical coordinates of a surface point after deformation and consider the following deformation of the shell X = X(x),

• = ~0,

R = R(x),

~ = const.

(5.5.1)

Relation (5.5.1) describes the generation of a disclination by cutting out the sector 2r/3 -1 < ~0 _ 2r (in the case of ~ > 1) and connecting the cut borders by the rotation about shell axis. In case of 0 < fl < 1, a sector with angle of 2 r ( 1 - / 3 ) is inserted into the shell which is cut by the half-plane ~o = 0. If the origin of radius-vector P is on the axis of rotation, then the parameters of Volterra dislocation, in view of equation (5.2.10) are as follows al : 0, ~1 : 2 k t g r ( 1 - ~ - 1 )

when fl > 1;

al = 0, ~b1 = 2 k t g T r ( 1 - ~)

when/3 < 1;

where k is the unit vector directed along the rotational axis. This process of disclination generation is accompanied with axisymmetric deformation of the shell. The surface of the shell remains a surface of revolution after generation of disclination. The negative values of/3 correspond to the disclination generation in a pre-everted shell of revolution.

5.5 The two Problems

159

The method used here to specify the surface deformation is a particular case of general method to describe finite strains of a thin shell by means of coordinates in the reference and actual configurations (Zubov 1983b). Let us denote the basis vectors corresponding to the coordinates x, ~o, r and X, 4~, R by k, e~, er and k, e~, eR, respectively; we have e¢ = e~ cos(q5 - ~) - er sin(q5 - ~), eR = e~ s i n ( ¢ - ~) + e~ cos(4~- ~).

In what follows, we shall suppose that the vectors k, e~, e~ as well as k, e¢, eR make up a right-hand frame. The radius-vector of a point of a shell of

revolution, a, is represented as (5.5.2)

p = x k + re~.

By using the notations of Sect. 5.1, from equation (5.5.2) we find (the prime stands for the derivative with respect to x) Op

Op P2 = i)~O = re~,

P l - - O X - - r'er -}- k ,

gll = 1 + r ~2,

g12 =

Pl × P2 'n

= r 2,

(5.5.3)

--

r// -

g2~

er - r ' k

"-"

IPl x P~.l bll -

0,

b12 = 0,

V/1 + r '2'

x/'l + r '~' r

b22 =

v/1 + r '2

•

For the surface ~P, which is also a surface of revolution, by analogy with equations (5.5.2), we have P = Xk + ReR. (5.5.4) Differentiating equation (5.5.4), with regard for equations (5.5.1), we obtain O P = X ' k + R' P 1 = Ox eR, G l l = R '2 + X '2, N=

OeR O~ = ~Re~,

P2 = R ~ G12 = O,

Pl x P2

R"X'--

v / R '2

X"R'

+ X '2 ,

B12

G22 = ~2R2,

(5.5.5)

_- X ' e r - R ' k

IP1 × P21 Bll

0~o

x / R '2 + X '2' 3RX' =

0,

B22 =

-

x/R,2 + X '2

From equations (5.1.41), (5.1.42), (5.5.5), for an isotropic homogeneous shell, it follows that the forces and couples, v ~ and p ~ , are functions of the only coordinate x. In view of equation (5.1.4), the Christoffel symbols from the covariant derivatives also depend on the only coordinate x. Moreover, from

160

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

equations (5.1.42), (5.5.5) it follows that /212 - - # 1 2 ._. 0 ; this means that for F 2 = 0, the second equilibrium equation from equations (5.1.37) is the identity. Thus, if the external loads F 1, F do not depend on ~ then the system (5.1.37) turns into a system of nonlinear ordinary differential equations with respect to R(x), X ( x ) . The relations (5.1.40) on the shell edges, in which the boundary load components p2 and d 2 are equal to zero, while pl, p, and d I are independent of ~, are the boundary conditions for this system. For a circular cylindrical shell (r(x) = const) subject to uniform normal pressure (F 1 = 0, F = const), the above system has a simple solution, namely, R = Ro, X = ax, where a, R0 are constant. With regard for equation (5.5.5), these constants are defined by a given external pressure and given longitudinal force acting in shell section. Consider now the shell whose surface in the reference configuration is a cylinder with arbitrary section and the axis parallel to a unit vector k. We shall assume that the border of orthogonal-to-the-axis cross-section of the cylinder is a smooth closed curve described by an equation of the form r = r(~) in polar coordinates. In the cylindrical coordinates, we specify the shell deformation by X - ax + a~ + U 1 ((~), =

+ Cz +

(5.5.6)

R =

Here, c~, a, ¢ are constants and Uk(~) (k -- 1, 2, 3) are twice-differentiable Jr-periodic functions. It is easy to see that the relations (5.5.6) describe the generation of a screw dislocation with Burgers vector of magnitude 27~ak in the shell. The screw dislocation generation is accompanied with axial extension of the shell, its torsion, characterized by the constant ¢, and by deformation of the cross-section contour. The position of a point of surfaces a and E is defined by radius-vectors p and P , respectively,

P = Xk + ReR.

p = xk + rer,

(5.5.7)

On the basis of equations (5.5.6), (5.5.7), we obtain (the prime stands for the derivative with respect to ~) Pl = k ,

=

re~, !

!

P2 = (a + u'l)k + u3eR + R(1 + uj)e¢ ,

P 1 -- o[.k -it- C R e w ,

gll

P2 = rl er +

1,

g12 = O,

g22 = Tl2 "~- r2~

5.6 Equilibrium of a Membrane with Disclination Gll -

ct 2 + ¢ 2 R 2 ,

= (a +

161

G12 = a(a + ul) + ¢R2(1 + u;), /

2

+

2

1+

+

/2

(sss)

G = G11G22 - G~2 = = Ol2U;2 nt- -2,~2 ~ /1; U ,2 3 +

[a(1

+ tt;)

-

~)(a -t- tt ')]2R2; 1

- r ' e~ + rer n--

V ' r 2 + r '2

,

v/--GN = -au~aee + [aR(1 + u;) - CR(a + u/1)]eR + CRu~ak. Using the formulas (5.1.2)-(5.1.4) and (5.5.8), we may easily derive the coefficients of the second quadratic forms b,~, B,~ and the Christoffel symbols -),~z, F ~ , which we do not present here. We note that these quantities, as well as g,z and G,z, depend on the only coordinate ~. By this, for an isotropic homogeneous shell, the forces v "z and couples #"z are also independent of the coordinate x. Under the condition that the loads F ~, F are independent of x, the equilibrium equations (5.1.37) are a system of ordinary differential equations with respect to unknown functions uk(~) (k = 1, 2, 3). The periodicity condition plays the role of boundary conditions for these equations. The strained state of a homogeneous isotropic cylindrical shell under deformation of the form of (5.5.6) is the same in all its orthogonal cross-sections. For a circular cylinder shell (r = const = r0) under uniform normal pressure (F ~ = 0, F = const), the system under consideration has a solution, namely, Ul((fl) =

0,

tt2(~) = 0,

U3(~) -- const = R0.

(5.5.9)

Indeed, as follows from equation (5.5.8), under the deformation (5.5.9), the quantities g,~, G,~, b~, and B,~ are constant, so the forces and couples, v "~ and # ~ , are also constant. Since the Christoffel symbols F ~ are zero according to equation (5.1.4), the first two equilibrium equations from (5.1.37) become identities when F ~ = 0, whereas the left-hand side of the third equilibrium equation becomes constant. To determine the three constants R0, a, ¢, we have the following relations: the third equilibrium equation, the specification of the torque in the shell cross-section, and the specification of axial force acting in the cylinder cross-section. For a shell of finite length without cross-cut-load p, the last boundary condition in (5.1.40) is fulfilled identically. The third boundary condition in (5.1.40), in general, cannot be satisfied by the solution (5.5.9) when d ~ = 0. This means that for realization of the deformation (5.5.6), (5.5.9) in a circular cylindrical shell it is necessary to apply a bending couple uniformly distributed on the circles of cylinder edges.

5.6 E q u i l i b r i u m of a M e m b r a n e w i t h D i s c l i n a t i o n Consider a ring-shaped thin (h in thickness) plate, rl _ r <_ r0, 0 <_ ~ <_ 27r, where r, ~ are the polar coordinates in the reference configuration. Let R, 4~,

162

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

Z be the cylindrical coordinates on the surface of the flat ring after deformation caused by generation of a wedge disclination. Recall that the thin plate is considered as a material surface, i.e., two-dimensional continuum. By assuming that on generation of disclination the plate remains a surface of revolution, we shall seek the plate deformation in the following form

(5.6.1) = const.

As in Sect. 3.1, the case of t~ < 1 (negative disclination) corresponds to the ring deformation that appears after cutting it along the line ~ = 0 and inserting a wedge of the angle of 27r(1 - ~) into the cut. The elimination of the sector 27r~-1 <_ ~ _ 27r from the ring with pasting of the edges corresponds to the case of ~ > 1 (positive disclination). To solve the problem, we shall use the equilibrium equations for the free-load plate in the form of equation (5.4.7) with respect to the force tensor, Uo, and bending couple tensor, tto,

V . S = O,

S - Vo + ( V . pro). ( V ' p ) N .

(5.6.2)

In notations of Sect. 5.1, we take the function of specific potential strain energy of nonlinearly elastic plate in the form close to equation (5.1.43), 1

~Vo = ~ E1 [u tr (Uo - g) + (1 - u) tr (Uo - g)2] + 1 + ~E2 Iv tr 2Ko + (1 - u) tr K2o], Eh

E l = 1 - u 2'

(5.6.3)

Eh 3

E 2 = 1 2 ( 1 - u ) 2'

where E is Young's modulus, v is Poisson's ratio of the plate material. The tensots Vo and tto corresponding to deformation (5.6.1) are determined by equations (5.4.4), (5.6.3) and have the form

Vo = VrRere~R + v, oe, e¢ + v~ze,.k,

(5.6.4) tto = #rRereR + #~o~e~oe~ + # ~ z e r k .

Their components are expressed in terms of functions R ( r ) , Z ( r ) as follows

5.6 Equilibrium of a Membrane with Disclination

#rR

"--

r] q-

r2

E2~R ( r g-~

163

,

~2Z'R~ vr/+

r2

),

Z' /"rR "-- E1

R'-~

~nn'

~# = <

77n' (n'#rR + Z'#~z)

(v + 1)n']

~v~

,/7

(5.6.5)

¢'¢7

+ ~/V-(~

+ 1)

Rye. #,

Z'

¢ = R '2 + Z '2,

77= R' Z" - Z' R",

a prime stands for the derivative with respect to r. By (5.6.2), (5.6.4), the tensor S has the following representation

(5.6.6)

S = &Re,.eR + &,~e~oe¢, + &ze,.k.

Substituting equation (5.6.6) into (5.6.2) we obtain

s'R +

~ r R -- ~

= o,

(5.6.7)

7"

S'~z +

~rZ r

= 0.

(5.6.s)

Assume that the plate edges are load-free then the boundary conditions take the form (5.6.9) SrR = 0 at r = r l , ro~

Srz=O a t r = r l ,

r0,

(5.6.10)

#~R=0 atr=rl,

ro.

(5.6.11)

From equation (5.6.8), with regard for equation (5.6.10), we obtain immediately Srz(r) - O. (5.6.12) The relations (5.6.7) and (5.6.12) make up a system tial equations with respect to functions R(r), Z(r) with (5.6.9), (5.6.11). We now restrict ourselves to the case of a membrane with negligible flexural rigidity E2. Setting (5.6.3), we have tto = 0; S = Vo.

of ordinary differenboundary conditions coupleless plate, i.e., E2 = 0 in equations

In view of equations (5.6.13), the equation (5.6.12) takes the form

1

Z' 1 + v/R,2. + Z '2

( u R- - - u - l ) r

1 =0.

(5.6.13)

164

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

There are two versions: 1. Z' = 0. Counting this in expressions (5.6.5) for vrR and v ~ and substituting them into equations (5.6.7) and (5.6.9), we arrive at the boundary value problem to determine the function R(r), RIt _.~_~R' m T

~2R = ( 1 - ~ ) ( 1 - u) T

(5.6.14)

R R / + ut~-- = 1 + u, at r = rl, r0; T

its solution can be represented as follows =

r0

(1 + ~) [(1 - u)C,

p~

+ (1 + u)C2p -'~ + (1 + u)p],

(5.6.15)

where C1

= 1 - p~+l ; 1 - p~

C2 = p ~ _ p~+l 1 p~

r

r1

P = --;

Pl = --.

TO

TO

In particular, for a solid disk (pl = 0), the corresponding principal membrane forces UR, u¢, which are expressed, in view of equation (5.4.1), in terms of the tensor Vo by the relations VR = e R " v / - g / G ( V P ) T " Uo " e R , u¢ = e ~ . v / - g / G ( V p ) T . u o . e ~ with regard for equations (5.6.5), take the form uR=E1

(1--u2)(p ~ - 1 - 1 ) (1 - u)p '~-1 + (1 + u)'

uo=E1

(1-u2)(~p ~-'-1) (1 - u)~p ~-' + (1 + u)"

It may be verified that on the change of constants (u ~ u / ( 1 - u)), used in the passage from the plane stress state to the plane deformation (Lurie 1970), we arrive to the solution obtained in Sect. 3.1 for the problem of disclination in a cylinder made of a semilinear material. As in Sect. 3.1, the principal forces in a solid disk has no singularity at the disclination axis. 2. 1 +

u~R/r-

u-

1

- 0, then U~R = 0 and, in view of equation (1.2.6), v/R '2 + Z ,2 u~¢ = 0 too. To determine the functions R(r), Z ( r ) we get the equations R

R

v / R '2 + Z '2 + u n - - = 1 + u,

u v / R '2 + Z '2 + n - - = 1 + u,

7"

7"

so it follows -lr

= -,

=

+ const.

(5.6.16)

This solution exists only for positive disclination (~ > 1); the corresponding bent plate form is a frustum of a cone. An important feature of the solution (5.6.15) is its universality in a sense that it does not depend on elastic properties of the membrane material. Indeed,

5.6 Equilibrium of a Membrane with Disclination

165

it is easy to verify that formulas (5.6.16) describe isometric transformation of the plate into a cone, i.e. there holds the relation Uo - g. Since the surface metric remains the same, the membrane forces are equal to zero for any material, that guarantees fulfillment of the membrane equilibrium equations. Note that this isometric solution (5.6.16) could be expected in advance as a ring with a cut sector represents the development of a frustum of a cone. So, in the case of ~ > 1, the boundary value problem of membrane equilibrium, equations (5.6.7)-(5.6.11), has two axi-symmetric solutions: the plane solution (5.6.15) and the universal isometric solution (5.6.16) associated with loss of plane form of the membrane. In energy, the isometric solution is, as obvious, more preferable since its strain energy is zero. We shall attempt to construct an isometric solution for a plate with disclination in the case of t~ < 1. This solution, unlike equation (5.6.15), will be no longer axisymmetric. Consider a transformation of the form R = R(r, ~), • = ~ , Z = Z(r, ~). The condition of its isometry reduces to Uo = g.

(5.6.17)

The problem to find out the functions R(r, ~) and Z(r, ~) satisfying the condition (5.6.17) involves no dimensional parameters; this determines the function form, R(r, ~) = r/(~), Z(r, ~) = rg(~). (5.6.18) With regard for equations (5.6.18), the relation (5.6.17) can be reduced to three scalar equations with respect to functions f(~), g(~), f2 + g2 = 1,

f

dg + g~--~ = 0,

( d r ) 2 ~2f2 (dg) 2 ~ + + ~ = 1. (5.6.19)

The second of equations (5.6.19) readily follows from the first so it can be omitted; then, expressing g(~) from the first equation and substituting it into the third one, we arrive at the equation

which has three solutions: 1) f2 _ 1, this does not satisfy the system (5.6.19); 2) f2 _ ~-2 is the solution that describes rolling up of the plate into a cone (~ > 1), it was discussed above; 3) f = sn (~ + C). (5.6.20) The module k of Jakobi's elliptic function sn (~, k) in equation (5.6.20) is the disclination parameter ~ (~ < 1). Now, g 2 ( ~ ) = cn (~ + C), C being a constant that can be found from the continuity condition for the surface of deformed plate; the quantities R and Z must be continuous, 2~-periodic functions of the coordinate ~. so

166

5. Nonlinear Theory of Dislocations and Disclinations in Elastic Shells

Fig. 5.1.

0) =

(5.6.21)

o) =

(5.6.22)

Consider the case of small negative disclination (a < 1); (1 - a) is small in the sense that the period of sn ~p with respect to module a will be greater than 47r/~, K(n) > ~/~. (5.6.23)

(K(n) is the full elliptic integral of first type (Bateman & Erdelyi 1967). The range n E (0.9858, 1) satisfies the requirement (5.6.23).) For the constant C, by (5.6.21) we obtain sn C = sn (C + 21r/n) hence C = K - ~/~. The requirement (5.6.23) is a condition for the function f(~) (and hence R(r, ~)) to be positive on the interval [0, 27r/~]. The relation (5.6.22) holds if g(~) = Icn (~p+C)l which is a continuous function with equal values at the ends of interval [0, 2~/~] and satisfies the system (5.6.19) in all points except ~ = 0 and ~ = 7r/~, at which the derivative of g(~) is discontinuous. The surface corresponding to the transformation (5.6.18) has two edges (at 45 = 0 and ¢ = ~) that frequently occur in analysis of problems of deformation of coupleless shells in the framework of surface bending theory (Pogorelov 1967). Fig.5.1 shows this surface in case of ~ = 0.99. It should be noted that the problem considered has infinitely many solutions since the continuity conditions in the form of equations (5.6.21), (5.6.22) for a surface in the deformed state may be stated not only at zero (45 = 0 and 4~ = 27r) but also at any value of ~ (a and a + 2~). Therefore, the positions of edges in this surface are not fixed, and thus any two lines ~5 = a and ~5 = a + 7r, a E [0, 2~) can be taken as these edges. The results of this section are based on the paper of M.I. Karyakin (Karyakin 1992).

6. S t a b i l i t y o f E q u i l i b r i u m a n d W a v e P r o p a g a t i o n in B o d i e s w i t h I n h e r e n t S t r e s s e s

6.1 The Linearized Equilibrium and Motion Equations for Elastic Bodies with Inherent Stresses Assume that we know some equilibrium stress-strain state of an elastic body, caused by given external loads, the body has Volterra dislocations with given Burgers and Frank vectors, or continuously distributed dislocations with given density. Hence the stress state of the body consists of stresses due to external loads and inherent (internal) stresses due to presence of defects. This equilibrium state will be called initial or basic; it is characterized by the radius-vector of a particle, R°(r), r being the particle position in reference configuration. Besides the basic deformed state, under the same external conditions, consider another body state; this means that all the parameters determining the magnitude of external loads, as well as those determining the defect intensities, remain the same for the new (perturbed) solution characterized by the particle radius-vector

R(~) = R°(~) + ~(~).

(6.1.1)

For simplicity, let us assume that all the external loads on the body depend on a parameter 7) called the loading parameter. The intensities of mass and surface forces (k and f in equation (1.1.12), respectively) need not to be given functions of Lagrangian coordinates qS; they can be given in terms of the vector R and strain gradient C. This dependence is assumed to be such that the load is conservative; this means that the elementary work of external load is the variation of some functional E

f k[r, R(r), C(r)]. 3Rdv /,

(6.1.2)

+/~ f[r, R(r), C(r)]. ~Rda = 3E. Substituting equation (6.1.1) into equation (1.1.12) we obtain a nonlinear boundary value problem for the vector-function w(r),

168

6. Stability of Equilibrium and Wave Propagation

divD'+pok'=-K n.D'-f'=F D'=

in on

v,

(6.1.3)

a,

(6.1.4)

d D ( grad Ro + ~ grad w)

, ¢--0 !

I" = Af( d¢

,R

'

gradR °

I

.~=o

.

The differential expressions for D', k', and f" are linear with respect to w. The expressions for K and F in equations (6.1.3), (6.1.4) contain no terms which are linear in w, this implies that if we set w = cw0, then their expansions in powers of ¢ begin with ¢2. The basic solution R ° depends on the loading parameter P, hence, the T' is involved in the set up the boundary value problem (6.1.3), (6.1.4). The assumption that the basic state R ° and new one R = R ° + w correspond to the same external conditions means in mathematical terms that the boundary value problem (6.1.3), (6.1.4) has the trivial solution w = 0. Omitting nonlinear terms in (6.1.3), (6.1.4), we get a homogeneous linear boundary value problem divD'+pk'=0 n.D'-f'=0

in

v,

(6.1.5)

on

a.

(6.1.6)

For some of values of parameter P, called the critical loads, the problem (6.1.5), (6.1.6) may have non-trivial solutions. Let P0 be an eigenvalue of the boundary value problem (6.1.5), (6.1.6). In general, to an eigenvalue P0 it may correspond several eigenfunctions (buckling modes). Occurrence of non-trivial solutions of the problem (6.1.5), (6.1.6) implies that, under the same external conditions for P = P0, the original boundary value problem of equilibrium, along with the basic solution R °, has another solution R ° + w, close to the basic one. In other words, when P = Po, the bifurcation of solutions to the nonlinear boundary value problem of elastic body equilibrium occurs. Since an eigenfunction of the linear problem (6.1.5), (6.1.6) is determined up to a constant factor, the magnitude of buckling form, as well as the number of solutions branching off at the bifurcation point P = P0, cannot be defined within the linearized problem (6.1.5), (6.1.6). To do this, we should investigate the nonlinear problem (6.1.3), (6.1.4). The procedure for finding equilibrium positions of three-dimensional elastic bodies under loads which are close to critical ones is based on the L iapunov-Schmidt method, it is presented in the papers (Zelenin & Zubov 1985; 1987; 1988). In what follows we shall restrict ourselves only to the linearized boundary value problem whose solution enables us to define critical values of the loading parameters. It is easy to show that for P = P0, the second variation of potential energy of elastic body calculated for the basic equilibrium state is not positive definite, that means that the basic solution is unstable. This static method for investigation of stability of elastic bodies goes back to Euler. It is well known

6.1 The Linearized Equilibrium and Motion Equations

169

(Bolotin 1961; Ziegler 1968) that the static method for studying the stability of elastic systems yields results which agree with those of a more general dynamic method if external forces are conservative.

According to equation (1.1.20), the linearized Piola stress tensor D', involved in equations (6.1.5), (6.1.6), has the following representation in Cartesian coordinates 02W OWn Dst = OCstOCmn OX,n"

(6.1.7)

The equations (6.1.5), (6.1.6) provide the set up of the linearized boundary value problem of body equilibrium in the reference configuration coordinates, i.e. for the domain v occupied by the body in undeformed state. Sometimes it is more convenient to set up the linear problem in perturbations for the domain V° corresponding to the basic deformed body state, described by the basic solution R°; now the independent variables are the coordinates of the basic deformed configuration, such as Cartesian coordinates X ° = ik" R °. This set up of the linearized problem can be obtained from equations (6.1.5), (6.1.6) by using the Piola identity (1.1.17) and the formula of the transformation of elementary oriented area (see equation (1.1.16)), it is

DivO+pk'=O N °'O=.f''

in on

(6.1.8)

V °,

(6.1.9)

E °,

O = ( det C °)- 1C OT " D ° ' .f,= da ~E-Sf,

(6.1.10)

0 Div O - i k . -d-p-6 X k0

where C O is the strain gradient corresponding to the passage from the reference configuration to basic deformed, p is the material density in the initial deformed configuration, N O is the unit normal to the boundary surface E0 of the domain V 0. As was noted in Section I.I, the Piola stress tensor D, unlike the Cauchy stress tensor, depends on the choice of reference configuration. Bearing this in mind, we can notice that the tensor e is the linearized (perturbed) Piola stress tensor if the basic deformed configuration is taken for the reference one. Linearizing the equation (1.1.19), which relates Piola and Cauchy stress tensors, and using equation (6.1.10), one can obtain the representation of the tensor O in terms of perturbed values of the Cauchy stress tensor (Lurie 1980)

O = T ° + (Divw)T ° - (Gradw) T . T O (6.1.11) T ° = dT(c° d~

+ ~C °. Grad w) e=0

Expression (6.1.11) can be used to define the linear function e ( Grad w) for specific models of elastic material.

170

6. Stability of Equilibrium and Wave Propagation

The linearized equilibrium equations obtained above describe small strains of a pre-stressed elastic body, the vector field w ( R °) is the field of small displacements of body particles referred to the basic deformed state. To solve small oscillation problems for pre-stressed bodies or problems on propagation of small waves in the same bodies we need to supplement the equations with inertia terms. Thus, the linearized dynamic equations of an elastic pre-stressed medium with no external mass forces have the form Div 0

~2W

= p Ot~ .

(6.1.12)

If for spatial variables one takes certain orthogonal curvilinear coordinates = 1, 2, 3) introduced in the basic deformed configuration, then the dynamic equations (6.1.12) in the component form become

Ym (m

1 O( A1A2A3) A1A2A3 0II"8 O~m A---"~ 1 (OAm_ OAk ) 02win + A~Ak OYk'Ok~ oymOkk = p Ot2 ,

(6.1.13) m=1,2,3.

Here, Am are the Lame coefficients of the coordinate system, 0sin, wm are the physical components of tensor (9 and vector w, respectively, in orthogonal coordinates; we mean the summation over the indices k, s from 1 to 3 but not for m .

6.2

Stability of

Elastic

Cylinder

with

Disclination

Following (Zelenin & Zubov 1989), we shall investigate the effect of internal stresses due to wedge disclination on buckling of a hollow circular cylinder with lateral surface subjected to uniform hydrostatic pressure. Plane forms of loss of stability (i.e. when all points of a cylinder cross-section remain in its plane) are discussed here. We shall also consider the case of internal stresses caused by disclination and the inversion of the cylinder. To define a pre-critical state of the cylinder we have to solve a more general problem than in Section 3.1, in the latter the body surface is assumed to be load-free. So, let us consider equilibrium of an elastic circular cylinder with internal stresses caused by a wedge disclination and external uniform pressure of intensity q. Let r, ~, z and e r , e~, ez be the cylindrical coordinates and related unit basis vectors, respectively, in the reference configuration of elastic body, whereas R, ¢, Z and eR, e¢, ez be the cylindrical coordinates and unit basis vectors respectively in a current configuration of elastic body,. The mass-body-free equilibrium equations and the boundary conditions are divD=0,

(6.2.1)

6.2 Stability of Elastic Cylinder with Disclination e~. DI~=~ 1 = 0,

e~.

D[~=~o=

C = grad R,

- q J e r . (cT) -i,

171 (6.2.2)

J = det C,

where to, ri are the inner and outer radii of the cylinder, R is the radius-vector of a point of the deformed body, D is the non-symmetric Piola stress tensor. To describe elastic properties of the body, we take the semi-linear material model D = 2# [VSl(1 - 2u) -1 - 1] U - i . C + 2#C, (6.2.3) U=(c.cT)

~/2,

s~=trU-3,

where U is the left stretch tensor, #, L, are constants. For small strains, #, u are referred to as Young's modulus and Poisson's ratio, respectively. As in Section 3.1, an initial deformation, corresponding to pre-critical equilibrium state to be investigated for stability, is sought in the form R = f°(r),

~ b - n~,

Z-

z.

(6.2.4)

From now on, the symbol "degree" marks the quantities relating to the initial state. The case of n > 1 corresponds to deformation arising on removal of the sector 27r~-1 _ ~ < 2zr from a hollow circular cylinder and rotation of the section ~ = 27r~-i about the cylinder axis up to coincidence with the section plane ~ -- 0. Thus, in the reference configuration, the body occupies a simply connected domain (0 <_ ~ _< 27r~-i) but it is doubly connected in the deformed state (0 <_ (I) <_ 2r). The case of 0 < ~ < 1 corresponds to the deformation that occurs when a wedge of the angle of 27r(1 - n) is inserted in the cylinder, made of the same material, which cut by the half-plane ~ -- 0; now the body is doubly connected in both reference and deformed configurations (0 < ~ < 27r, 0 _ (I) < 27r). The case of ~ = - 1 corresponds to deformation of inversion of the cylinder. The case of ~ < 0 describes the deformation that arises due to generation of disclination in the inverted cylinder. Let us introduce the dimensionless quantities R ° = f°/ro, p = q/(2#), = r/ro and, from now on, we use all the relations in dimensionless form. The strain gradient answering the transformation (6.2.1) is C O --

R°'ere.R Jr- -nR° - e ~ o e , ~ -4- ezez,

R °' = dR° , d~

det C O = R °' nR° > 0.

(6.2.5)

( 6.2.6)

From equation (1.1.6) it follows that R °' > 0 if ~ > 0, hence

U 0 = R 0,erer + ~~R° e~e~

+ ezez,

si = R °' + ~~R° - 2.

(6.2.7)

The equilibrium equation (6.2.1) and boundary conditions (6.2.2), with regard for equations (6.2.3), (6.2.5), and (6.2.7), become

172

6. Stability of Equilibrium and Wave Propagation

R °'

n2R °

R°" -t

(1

-/2)-1(1

772 -

(1 - 212) -1 [(1 - u)n°'-t

-

n)

77

t~RO

/2nR 0

1 -~-P l ~

---- 0

,

(6.2.8)

(?7 = k, 1).

(6.2.9)

Here k = rl/ro and Pl = 0 at 77 = k; Pl = P at r / = 11 Solving the boundary value problem (6.2.8), (6.2.9), we obtain n 0 : ~ (C177 ~ ~- 627~ -a -~- ?~), ~ -- [(1-- /2) (1-+- t~)] - 1 ,

(6.2.10) C1--(1-

2u)pu ( 1 - k ~+1) ( P a - p2k2a) -1 ,

C2 -- (k2ap2 - p3 ka+l) (p3 - p2k2a) -1 ,

P2 = 1 - p,

(6.2.11)

P3 = 1 + p(1 - 2/2).

From equations (6.2.10), (6.2.11) it follows that the inner radius of the deformed cylinder R°(k) ~ 0 as p ~ 1; so we assume that p < 1. For n < 0, from equation (6.2.6), it follows that R °' < 0; thus instead of (6.2.7), we get

nR °

U 0 - -R°'erer

- --e~e~

~ ezez,

(6.2.12) ~R °

81 --- - R O'

77

2.

The boundary value problem (6.2.1), (6.2.2), with regard for equations (6.2.3), (6.2.5), and (6.2.12), takes the form R °" -t

(1 2u) -1 [(1 -

-

n °'

n2n°

u)R°' + unR°

r/2

=

(1 -/2)-1(n_ 1)

+ 11 + P l -nR° = 0

,

(,] = k, 1).

(6.2.13) (6.2.14)

Note that pl = 0 at ~ = 1 and pl = p at ~ = k in equation (6.2.14) since the load acts on the outer surface of inverted cylinder. Solving the equation (6.2.13) and the boundary conditions (6.2.14) simultaneously, we obtain R ° = ~(Hlr/~ + H2r/-~ - r/),

(6.2.15)

H~ = (1 - 2v)p2(1 - k'~+l)(pak2'¢-p2) -~, (6.2.16) H2 = (p3k 2~ - p2k~+l)(p3k2'~ Using the Lopitale's rule as n = - 1 , we get

-

p2) -1.

6.2 Stability of Elastic Cylinder with Disclination

173

R ° = w(Gl~7-1 + G~? - r] In r/),

co= ( l - u ) -1,

a~ = (~ - 2")p~ 1~ k(p~ - p~k-~) -1,

(6.2.17)

G2 = p2 In k(p2 - pak-2) -x. From equations (6.2.15)-(6.2.17) it follows that the inner radius of the deformed cylinder R°(1) ~ 0 at p --, 1. We shall seek plane equilibrium forms that close to the initial deformed state, i.e. we set

= ~

R = R ° ( n ) + p(~, ~),

+ ¢(v, ~),

z = z,

(6.2.18) R-

ReR + Zez.

On substituting equation (6.2.12)into equations (6.2.10), (6.2.2) and multiplying the both sides of equations (6.2.1), (6.2.2) by ( 1 - 2u)[2#(1- u)] -1 and (2m) -1 , respectively, we obtain the nonlinear boundary value problem for the unknowns p(~, ~), ¢(77, ~)

(6.2.19)

l(x, D)w(x) = f (w, x),

b(x, D)w(x) = F(w, x),

x = (77,~),

w = (p, ¢),

l(x,D) - (lij(x,D)),

(6.2.20)

(~7 = k, 1),

f = (fl, f2),

F = (F1, F2), (i,j = 1,2),

b(x,D) =_ (b#(x,D))

lll(~, D) = O~ + ,7-~BO~ + ~-10~ - ,~,7 -~, 112(x, D) = ~7-1R°BlO102 + [r/-l~(1 - u) - 1 - 2r]-2~R °] 02, 121(z,D) = ~-IBOIO2 + ~-2 [ 2 a - t~B1- B B I ( g 122(x, D ) = R°BO~ + ~-2R°O~

+B [ Ro' + -,Ro +

1/]

1)]02,

(6.2.21)

174

6. Stability of Equilibrium and Wave Propagation fl = n(Oql~b)2 + 2U-2tcpoq2¢ + r/-2n(02~b) 2 +5(1 - v) -1 [r/-I(MQ -1 - 1) + 01(MQ -1) - NQ-IOI.d2] 1)02¢,

--~7- 102(NQ-1) + ~7-1Ca-1 c32N1 - ~7-I ( M Q - 1

f2 = (i(1 - v) -1 [01(NQ -1) - c31(N1Ca1) + rl-I&(MQ -1) +r/-l(1 - 02gp)gQ -1 - r]- 1C 3-1 NI(1 - n) + (MQ- 1 - 1) (~1~) ] -2olpo,¢

- poise -

-lpOl¢ -

+

bll(X,D) =/]1((91-~- 71-lt~)

- ?']-lgp2 ,

b12(x, D) = 7/-1R°(/21 - p2)oq2, b~l(X, D)

=

-Vlr/-1Boq2 +

rl-1p202, (6.2.22)

bgo.(x, D) = VlR°BO1, F1 =/i(1 - 2 v ) - I ( M Q

-1 - 1) - (/21 - - p 2 ) M 2 ,

F~. = 5(1 - 2 v ) - I ( N Q -1 - N I C 3 1) - / ] i N 2 , /21 -- (1 - v)(1 - 2/./) -1,

B = 1 - B1,

B1 = 5(1 - / 2 ) - 1 C 3 1 ,

Ca = R ° ' + n r / - 1 R 0,

M = Ca + MI + M2,

N = NI + N2,

M1 - c31p + t~r/-lp + ?7-1R002~),

N1 = R°01¢ - ~-102p,

M 2 ----r/-lp02¢,

N2 = p01¢,

Q = Ca [1 + 2Cal(M1 + M2)+ Ca2(M1 + M2) 2 + Ca2N 2] 1/2,

= P2=

1 (K; > 0) --1 (n < 0) '

1 (7=k,n>0andr/=l,n<0) 1-p (77=k,n<0and77=1, 0

(6.2.23) n>0)'

0

The differential expressions for l and b in the left-hand side of equations (6.2.19), (6.2.20) are linear, whereas f and F include no linear components. As parameters, the boundary value problem (6.2.19), (6.2.20) involves the dimensionless pressure p and the quantity n that characterizes the Frank vector magnitude of disclination.

6.2 Stability of Elastic Cylinder with Disclination

175

Let us find the values of parameters p and ~ when bifurcation of equilibrium of the cylinder occurs. To do this, consider the linearized problem that is derived from equations (6.2.19), (6.2.20) when f = F = 0. Because of periodicity of solution in the coordinate ~, we shall seek the eigenfunctions of the linearized problem of the form

oo p = ~ p~ (r/) cos n2~, n=O

oo ~b = ~ ¢~(r/) sin ~2~, n=O

¢0(r]) = 0,

(6.2.-24)

where ~2 = [RIn. Substituting equations (6.2.24)into equations (6.2.19), (6.2.20) and making the terms with the same modes to be zero, we derive the boundary value problems to determine Pn, Cn: p~ + r/-lp~ - r/-2n2p0 = 0,

¢0 = 0, (6.2.25)

~p'o + ~-~,~(~ - ;~)po = o

(~ = 1, k),

p~ + T]-lpn/ -- T]-2(t~ 2 "b Blq,2)pn -'b t~2TI-1R 0 Bl~)n/

-[-/~2 [(~T]-I( 1 -- / ] ) - 2T]-2R0~] Cn -- 0,

¢.

--

B - 1 -?72

2-

/~2~n

+(R°) -1 2R°' q- r/-1R° + r/-l( ~ - 1)R°B1] ¢~n (6.2.26)

-~-I (R°)-I B-1BI ~2p~ -r/-2(a°)-l~;2 .~p" + ~ - ~ ( , 1

[ B - l g + ~ ; - B l ( g - 1)] Pn = O, - ;=)Pn + ~ - I R ° ~ ( ~

ulBR°~b~n-k rl-lK;2(uB1

-

P2)Pn =

- P~)¢~ = 0, 0

(r] = 1, k).

Solving equation (6.2.25) we obtain that a non-zero solution p0 exists only when p > 1. For p is always less than 1, the case of n = 0 is not considered. We also eliminate from consideration the case of n = 1 which corresponds to rigid motions of the cylinder. For n > 1 the boundary value problem (6.2.26) is reduced to a system of ordinary differential equations of first order so it can be numerically solved by the orthogonal sweep method (Godunov 1961). Numerical investigation of solution of the problem (6.2.26) in the case when u = 0.3 and n > 1 has shown that eigenvalues po(k) may be simple or double, depending on the values of k and n. Fig.6.1 and Fig.6.2 illustrate the eigenvalue curves p0(~), ~ > 0 for k = 0.1 and k = 0.44 and different values of n. To double eigenvalues there correspond the points of intersection of the curve for n = 2 with the curves for n = 3, 4, .... Note that, although the shape of the curves depends on the value of u, their points of intersection with the x-axis do not depend on n and monotonically converge to a certain value as n increases

176

6. Stability of Equilibrium and Wave Propagation

P n=150

i

e~/f

•

0.25

0.5

0.?5

1

1.25

Fig. 6.1.

n:150

/// ,/" Ill 0

0.5

1

1.5

2

Fig. 6.2.

infinitely. In practice, the numerical investigation gave the values of n up to 150. This means that in case of deformation which is caused by the only disclination (p = 0) , the eigenvalues are always simple and do not depend on n. Similar results were also obtained for other values of parameter k, related with the relative thickness t of the cylinder wall by t = 1 - k. The critical pressure p, is the minimum of the set p0(n, k, ~) for all values of n with k, ~ being fixed. Fig. 6.1 and 6.2 show that p, is achieved at n - 2

6.2 Stability of Elastic Cylinder with Disclination

p • 18 -2

177

i

n=12 15.8 12.5

58

.,~_

//!

18.8

?.5

/// 158 _

_

,-.5 i H 8

_

dLi 8.25

8.5

Fig. 6.3.

P/*e A

J

8.5

-12

-8

-4

8

4

8

Fig. 6.4.

or n = ce depending on k, n. Thus, an elastic body with disclination can have forms of loss of stability characterized by infinite number of waves. Fig.6.3 represents the eigenvalues curves po(k) for inverted shells (t~ = - 1 ) at a number of n; the abscissa of intersections of these curves with the t-axis is independent of ~,. On Fig. 6.3 it is seen that the minimum thickness of the cylinder when there is the loss of stability of inverted cylinder corresponds to the value of k = k, ~ 0.53 as n = oe; this means that sufficiently thick cylinders (k < k, ) are unstable in the inverted state. Thus, when n < 0, one should

178

6. Stability of Equilibrium and Wave Propagation

consider thickness for which k > k,. The stability areas in the plane of the parameters p, n for k = 0.9 are shaded on Fig.6.4, as well as on Fig.6.1-6.3.

6.3 T h e Effect of Screw D i s l o c a t i o n on the Stability and Wave P r o p a g a t i o n in Elastic Cylinder Let us study space forms of loss of stability of a circular elastic cylinder with screw dislocation, subjected to axial compression. As a basic (sub-critical) solution of the equilibrium equations, it is taken the deformed state of a nonlinearly elastic, solid circular cylinder with a screw dislocation characterized by Burgers vector of magnitude of 2~a. The cylinder material is described by the model of incompressible Bartenev-Khazanovich body (Lurie 1980) with constitutional relation T = 2 # A -1/2 - qE,

(6.3.1)

where T is the Cauchy stress tensor, A-1 is the Finger strain measure, E is the unit tensor, # is the material constant that, for small deformation, coincides with the shear modulus, q is the pressure in an incompressible material, which cannot be defined by given strains. The deformation arising under axial shortening (or extension) and screw-dislocation generation is given by relations of Section 2.4,

R -- c-1/2r,

• = ~,

Z = a~ + cz,

(6.3.2)

where r, ~, z and R, (I), Z are the cylindrical coordinates of body points before and after deformation, respectively, c is the axial shortening factor (c > 1 for elongation). An expression for Finger strain measure corresponding to equations (6.3.2) is

~-1

_

_

c-leReR q_ c-lece¢

+ar-lc-1/2(e~ez + eze~) + (c2 + a2r-2)ezez, hence )~-1/2

__

C-1/2eReR -t- (C-1 + cl/2)rKece¢

+ac-1/2K(ecez + eze¢) + [a 2 + (c2 +

c1/2)r 2] K r - l e z e z ,

(6.3.3)

K = [a2 + (c + c-1/2)2r 2] -1/2. Here, eR, eo, ez form the orthonormal basis for cylindrical coordinates in the deformed configuration. From the relations (6.3.1), (6.3.3) and the equilibrium equations, it follows that the tangential stresses 7RZ and ~R~ are zero, and all other components of tensor T in the basis eR, e,I,, ez depend on the only coordinate R.

6.3 The Effect of Screw Dislocation on the Stability

179

For unloaded lateral surface of cylinder, the stress expressions, which are defined by the equation Div T = 0, are

(c(c--~c-1/2)2R2)-111/2 [1 + a 2 (c(c + C-1/2)2R2)-1] 1/2'

1 + [1 + a 2 crR =

2#c - l p In

1+

a,I, = 2#(C + O'Z -- 2 # R - l c

-1/2

c-1/2)RF

q- a R --

[a 2 -4- (c 2 -4- c l / 2 ) c R 2 ]

2#C -1/2,

F --b OR --

2#c -1/2,

TCZ = 2#ac-1/2F,

F = [a ~ + (~ + ~-~Z~)~R ~] -1/~, where R0 is the radius of deformed cylinder. Stresses acting in any cross-section of the cylinder are reduced to a longitudinal force, P, and a torque, M, which are functions of the parameters c and a, as follows P(a, c) = 27r

/0

az(R)RdR

(6.3.4)

Numerical analysis shows that for the Bartenev-Khazanovich material with P = O, to any value of a there corresponds a value of c < 1, i.e., the generation of dislocation results in an axial shortening of the cylinder with no longitudinal

force. The isochoric deformation that appears on generation of dislocation in a hollow circular cylinder, is described by the formulae c(R 2 - R~) = r 2 - r~,

¢b = ¢,

Z = a~ + cz,

(6.3.5)

where r0 and R0 are the outer radii of the cylinder before and after deformation, respectively. Finding the Finger strain measure by equations (6.3.5), we obtain A -1 = c-2r2R-2eReR + R2r-2e¢e~ + aRr -2 ( e c e z + e z e ¢ )

+ (C2 + a2r -2)

ezez,

hence A -~/2 = r ( c R ) - l e R e R + r - I R ( R + cr)Le~e~ + r - l a R L (e~ez + e z e ¢ ) + r-~(a 2 + c R r + c2r2)Lezez,

L = [a ~ + (R + ~)~]-'/~.

(6.3.6)

180

6. Stability of Equilibrium and Wave Propagation For a hollow cylinder, the stresses and function q are

aR = 2#

f

[ r - I R ( R + c r ) L - r(cR) -1] RdR,

a¢ = 2# [r-IR(R + cr)L - r(cR) -1] + aR, az = 2# [r-l(a 2 + cRr + c2r2)L - r(cR) -1] + aR, TCZ = 2 # r - l a R L , q = --aR + 2#r(cR) -1. As for the solid cylinder, a hollow cylinder under axial shift with no longitudinal force becomes shorter in the axial direction. The constant R0 is determined by the load-free condition on the inner surface of the cylinder. To clarify the problem of cylinder stability, we assume that, together with its above sub-critical equilibrium state under the same external forces, there is an infinitesimally close equilibrium state which differs from the original one by an additional infinitely small strain, it is defined by the particle radius-vector, R + ew, where R is the radius-vector in the sub-critical state, w is the vector of additional displacement, e is a small parameter. The perturbed equilibrium state of the incompressible body, in view of equation (6.1.8), is described by Div O = 0,

Div w = 0,

(6.3.7)

where the second equation from (6.3.7) represents the linearized incompressibility condition. On the basis of equation (6.1.9), the boundary conditions on the lateral surface of solid cylinder are eR'O=O

at R = R 0 .

(6.3.8)

Component representation of the displacement vector w and the tensor O in basis eR, e¢, ez is W = UeR + v e ¢ + We.z,

0 = ORReReR + OROeRe.¢ + ....

The equations of neutral equilibrium (6.3.7) and boundary conditions (6.3.8) in cylindrical coordinates become

OORR ORR -- 0¢¢

1 ~ R R OOR¢, OR@+ Ocn 1 I ~ OR R R OR

OORz

ORZ

O0¢R t 0¢ 00¢,¢, ~ 0¢

1 O0¢Z

o--ff + - - f f ~ n o¢

OOzR -0, OZ OOze -0, OZ

OOzz

~ oz

=o,

(6.3.9)

6.3 The Effect of Screw Dislocation on the Stability

ORR -- ORO "-- ORZ = 0

at

181

R = R0.

(6.3.10)

= 0.

(6.3.11)

The incompressibility condition is written as

Ou

u

10v

Ow

0---R + R + R0-~ + ~

From equations (6.3.1) and (6.1.11), for the solid cylinder, we obtain the representation for components of O in terms of components of w,

~ - (q*)°, q, -Ou

(2#)_IoRR =

(2#)-loRo = ( - [ch + (cl/2 -t- c-1) r] F -t- R-lq*) ( °qu - v Ov

-Jr-C-1 (Th -Jr-C1/2 [a 2 -i- (C2 -~- C1/2) r2]) F 0----R

c3u Ow _ac-lrr__= _ c-larr O---R' OZ Ov Ou ) Ow ( 2 . ) - I O R z = -c-1/2aF - ~ - v + ( c h ) - l r - ~ + c - ~ a r r OR Ou - (c-1/2r [crh + a 2 + (c 2 + c 1/2) r 2] - q*) - ~ ,

(ou )

(2~)-10OR -- C-1/2 [h -1t- @3/2 _~_c)T] F

+c-ll2a(h + cr)r -I-

~

- v

Ou OZ

(6.3.12)

Ov (C-1 [C3/2rh "1- (C -Jr- C-1/2) T2] F -!- q*) ~

-

Ow c-larro--~,

-

(2,)-1000 = (c-1/2a2h -3 nc R-lq *) tt -t- ~Ov )

Ow

-

-

a2r¢ -1 h_ 3- OZ

Ov +ac-1/2h-3 [(c2 + cl/2)r2+ a2] OZ - a t @1/2 ..[_c-1) h_ 3 ~_.~ ( q , 0_W

(2~)-10~Z = --

--aT

).,

( ov)

(C1/2 -'[-"C-1) h -3 ?.t --[-~-~

[Th -3 (cl/2h 2 nt- c-la 2)

-

-

q*] "Ov ~ nt- C- l12h_ ag r 2 0Ow O

+c_l/2h_3ar2 (cl/2 + c_1) Ow OZ'

182

6. Stability of Equilibrium and Wave Propagation (2#)-10zR = ar(h + c~)r

(ou ) ~

-

-Jr-c-ll2r -1 (c 1/2 (a 2 4- c2r 2) [h -#- (c 4- c -1/2) r] Ow

-~- [(ca) -1 (r -- c112h) Jr- q*] ~

-

Ou a2r) F O---Z

-

Ov

- c-larV O---R'

(0v)

(2.)-1Oz, = r-lh-~ ~ [a~ + (~ + c1/~)~]

~+ ~

_.~..rh_3 [a2 _[_ (C2 _.~- C1/2) /.2]20V OZ -

-

(h -3 (c + C-1/2) [a 2 + (c 2 + C1/2) /,2]

_

_

R-tq,) Ow

Ow -ac-1/2h-3 [a2 -+- (c2 k-cl/2)r2] OZ'

(0v)

(2#)-lOzz = c-1/2h-aa 2 u + - ~

Ov Ow -+- (c-la2rh -3 + q*) -Ow +ah -3 (C1/2 + C-1) /'~--~ ~

-

-

(q*)°.

In the formulas (6.3.12), the following notations are used:

B = (c + c-1/2) 2,

q* = q/(2#),

F = [a 2 + Br 2 + (c + c-t/2)rh]-1,

h = (a 2 + Br 2) 1/2.

The quantity (q*)° is the perturbation of pressure q* in the incompressible body due to perturbation of the equilibrium state; (q*)', as well as w, is an unknown function of coordinates R, (I), Z. On substituting the expressions for components of O (see equations (6.3.12))into equations (6.3.9), we obtain a system of four equations with respect to four functions: u, v, w, and (q*)'. This system admits solutions of the form

=

u(n)cos(n~+ ~z),

v

=

V(R)sin(n(I) + aZ), (6.3.13)

w

=

W(R)sin(nO + aZ),

(q*)" = - Q ( R ) c o s ( ~

+ ~z),

where n = 0, 1, 2, 3, ..., a is a real number. Under this assumption, the components ORR, Ocz, Ozz, 0¢¢, and 0z~ of the tensor O become the products of functions of coordinate R by cos(n~ + aZ). The rest components of O are the products of functions of R by sin(nO + aZ). So the variables • and Z in the equations (6.3.9), (6.3.11) and boundary conditions (6.3.10) can be separated,

6.3 The Effect of Screw Dislocation on the Stability

183

so it appears a boundary value problem for a system of four ordinary differential equations with respect to U, V, W, Q. If we set the parameter c~ to be equal m ~ / l (m = 1,2,3,...), the solutions of the form (6.3.13) enable us to satisfy certain boundary conditions, as discussed in (Zubov & Moiseenko 1981), at the ends of the cylinder of the length l in the undeformed state. Change the variable: r = rop. The boundary conditions (6.3.10) on the lateral surface p = 1 take the form

U' + roQ = O, ( - n ' ~ + a * a * / c ) U - "yV + T V ' - a*c-1/2W ' -- 0 (na*c -1/2 - a * S c - U 2 ) U + a*c-1/2V - a*c-1/2V' + 5 W ~ = O,

= c-,/~ [B + (a*) ~]'/~ + (a,)~ + ~ + cV~, = c - m v / ( a * ) ~ + B + c 1/2 + c -~,

(6.3.14)

a* = a/~0,

Ol* = pOOl.

Here the prime denotes the derivative with respect to p. This linear system can be reduced to a system of six equations of first order, in the matrix form it is

Y' + A(p)Y = 0,

Y = (U, V, V', W, W', Q*),

Q* = Qro.

(6.3.15)

Here, Y is the column vector of unknown functions, A(p) is the matrix of system coefficients. To solve the system (6.3.15) in the case of solid cylinder, we need to get three additional boundary conditions at the point p = 0. To do this, we use boundedness of functions U, V, W, Q* and their derivatives at p = 0 which implies the expansions U(p) = Uo + u l p + u2p 2 + u3(p), V ( p ) = vo + v , p + v~p ~ + v~(p),

(6.3.16)

w(p) = ~0 + ~lp + ~p~ + ~(~), Q* (p) = qo + qlp + q2p 2 + q3(p).

Here, ua (p), va (p), wa (p), and qa (P), are the remainders of Taylor's expansions. Expand the coefficients of equations (6.3.16) into Taylor's series about p = 0 and substitute the (6.3.16) into equations (6.3.9). On equating of the coefficients of the same powers in p we get some systems of equations for the expansion coefficients of equations (6.3.16) as follows: for n = 0 U(0) = V(0) = 0;

184

6. Stability of Equilibrium and Wave Propagation

2[a*[-1W'(0)- a'Q*(0) = 0 , f o r n = 1, u(0) + v ( 0 ) = 0,

(6.3.17)

2(~*)21a*lU(0) + cl/2V'(0) + ~*W(0) = 0, 1 1/2 W'(O) + -~a*c[-2 + a'a* + (a*)2(a*) 2] U(0) = 0;

f o r n > 1, u ( 0 ) = y ( 0 ) = Q*(0) = 0.

The relations (6.3.17) make up the needed boundary conditions at the cylinder axis. Following the method of (Zubov & Moiseenko 1981), we write down the system of differential equations (6.3.15) in the finite-difference form, Y~+I = ( E - A~H)Y~,

p~=iH,

NH= I

A, = A(p,),

(i=0,1,...,N-1),

where H is the grid size, E is the unit matrix. On this basis we obtain the system of algebraic equations

YN = BN-1...B1BoYo,

Bi = E - AiH

with respect to the values: U(0), V(0), V'(0), W(0), W'(0), Q*(0), U(1), V(1), V'(1), W(1), W'(1), and Q*(1). Any six values from this list can be taken for unknowns. The rest ones are expressed in terms of the selection with use of boundary conditions (6.3.17) and (6.3.14). As the system of algebraic equations is homogeneous with respect to the unknowns, the zero determinant of this system gives a relationship for the parameters c and a* which determines buckling of the cylinder. Using the formulas (6.3.4), we can transform this relationship into one with respect to parameters P and a*. Fig.6.5 shows the dependence of critical value of dimensionless compressive force, P* = P/(4~#r2o), on the relative thickness of the cylinder a* when n = 1. If there is no longitudinal force, the critical relation between parameters a* and a* is plotted in Fig.6.6. Fig.6.7 illustrates the dependence of critical magnitude of Burgers vector on the cylinder length for different values of n in the case of hollow cylinder with radius ratio rl/ro - 0.5, rl being the inner radius of the undeformed cylinder. Following (Zubov & Moiseenko 1984), consider now the dynamic problem of small strains of a pre-stressed infinite hollow cylinder, the initial stresses in which are due to screw dislocation. Determine first the initial stress state

6.3 The Effect of Screw Dislocation on the Stability

185

-P~.IO

0.4

a~:O ~

t

a~=O.02~

/

0.3

0.2

a~-O,O_.....8 a~=O.l

0.1

0

0.1

0.2

0.3

N

0.4

Fig. 6.5.

a

.

/

0

0.5

1

1.5

2

N

Fig. 6.6.

of a cylinder due to deformation (6.3.5). We use the constitutive relation for isotropic incompressible material T = X1 (I1,/2, r)A -~ - X2(I~,/2, r)X - qE,

(6.3.18)

where XI, X2, being some functions of strain invariants, determine the material. Explicit r-dependence of these functions enables us to take into account possible material non-homogeneity in the radial coordinate. With regard for expression

186

6. Stability of Equilibrium and Wave Propagation

a

N

£.$

Fig. 6.7.

(2.4.4) for the Finger strain measure, from the equilibrium equation (2.4.9) we find the initial stresses

aR = -X~2(R)

~z=~(R)

~~

~, (R) r2

R2

c2R~

r2

c2R ~

~R~ -~(R)

R '

(1 ~

~

+o~,

aR a ~ z = x,(R) 7 + ~ ( R ) ~R' X.~(R) = X~ [I, (R), I2(R), r(R)]

(i = 1, 2).

The equations invoke the load-free condition on the outer surface of the cylinder, R = R0. For a hollow cylinder, the constant R0 is not given in advance and must be determined from the boundary condition aR(R1) -- 0, R1 being the inner radius in the deformed state. By equation (6.3.5), R1 is expressed in terms of the inner radius rl of the undeformed cylinder,

~(R~- R0~) = r~- ~. Then the axial extension factor, c, can be determined from the condition that the longitudinal force acting in any section of the cylinder is zero, P - 0.

6.3 The Effect of Screw Dislocation on the Stability

187

Let us assume that on the initial state of the cylinder, some small motions are superimposed; these motions are defined by the displacement vector w(R, ~, Z, t) that must satisfy the linearized equations (6.1.13) as well as the linearized incompressibility condition (6.3.11). The boundary conditions (6.3.10), then, must be satisfied at two points: R - R0 and R - R1. As in the stability problem, because of material incompressibility, the linearized pressure q°, along with the vector w, is an unknown function in the equations (6.1.13) with boundary conditions (6.3.10). This linearized dynamic problem for a cylinder admits solutions in the form of dispersive waves propagating along the cylinder axis,

U

U(R)

V

V(R) e i(n¢+kZ-wt).

W

W(R)

q"

Q(R)

(6.3.20)

where k is the wave number, w is the oscillation frequency, n is an integer. The integer-valued n is the parameter of wave generation in the circumferential coordinate, it determines behavior of propagating elastic waves in the cylinder. So, taking n = 0, one obtains longitudinal-torsional waves. The case of n = 1 corresponds to rod bending-torsional waves of deformation. Other values of n correspond to waves of more complex structure when the cross-section of the cylinder ceases to be circular.

The substitution of equation (6.3.20) into the motion equations and boundary conditions yields a linear, homogeneous boundary value problem with respect to functions U, V, W, and Q. Non-trivial solutions of this problem exist only for a specified dispersion relation,some dependence between the wave number k and frequency w. This boundary value problem for ordinary differential equations was solved numerically by a method described in the paper ( Zubov & Moiseenko 1984). The computations were made for a cylinder, made of neoHookean material (XI - P = const, X2 = 0, where p is the shear modulus), with the radius ratio rl/r2 : 0.8. Fig.6.8 illustrates the numerical results in the case of n : I. The first three dispersion curves are plotted. On the abscissa, the dimensionless wave number, kR0, is plotted, and the ordinate is the dimensionless phase velocity, Co : V/2p/#w/k, where p is the material density. Numbers 1 to 3 mark the curves corresponding to certain values of the parameter ~ : a/ro that characterizes the Burgers vector magnitude (~ : 0, 0.3, 0.5 respectively). As is seen from these plots, the increase in Burgers vector magnitude implies decrease in the phase velocity of bending-torsional waves propagating along the screw-dislocation line.

188

6. Stability of Equilibrium and Wave Propagation Co

1.8

-'~" ~ - _ _

_______ .

i

m.

0

1.2

2.4

kR o

Fig. 6.8.

6.4 Buckling of Thin Elastic Plate with Disclination

Following the paper (Karyakin 1992), let us study stability of plane stressed state of a thin elastic ring-shaped plate containing a wedge disclination. The plane stressed state was found in Section 5.6, it was described by formulas (5.6.15). The plane forms of loss of stability for a ring with disclination were investigated in Section 6.2; they are of no interest for a thin plate because first there appear bending modes of buckling for which the plate points leave the plane. For this reason, we shall study bending instability of a thin plate with disclination. Solving the problem of stability of a plate, we have to consider bending stiffness, i.e. it should be used the function of specific potential energy of the plate deformation in the form of equation (5.6.3). First we consider the stability of a plate with respect to axi-symmetric disturbances. We linearize the equilibrium equations in a neighborhood of the solution determined by formulas (5.6.15). First we set

P = R(r)eR + sw(r)ez.

(6.4.1)

Substituting equation (6.4.1)into equations (5.6.4), (5.6.5) and then the result into equation (5.6.2), within the quantities of first order in s, we arrive at the linear equation system with v, #0 as unknowns:

6.4 Buckling of Thin Elastic Plate with Disclination •l g

189

-5

ho_1.5

0.5

e

e.2

e.4

Ql

F i g . 6.9.

v'= P0 =

1 ( va2

~-7

-1)-/

+-r V

R/l)

-R-- \ r 2 R -

r 2

R'+

R/

1 ?-

#o,

(u+l)

v

(6.4.2)

1 #o,

= W ! , #o = #rR/E2,

l = El/E2.

The function R(r) in equations (6.4.2) is determined by equation (5.6.15). The boundary conditions for the system (6.4.2) expresses the fact that the plate edges r = rl and r = ro are load-free: #rR

-- 0

r -- rl, to.

at

(6.4.3)

The critical values of the parameter a will be such t h a t the homogeneous boundary-value problem (6.4.2), (6.4.3) has non-trivial solutions. Numerical computations show that for 0 < a < 1 there are no non-trivial solutions, i.e., the plate does not lose stability in an axi-symmetrical manner. The plots of dependence of critical value of the parameter 5 = a - 1 on the radius of the ring hole (Pl = rl/ro) for different thickness (h0 = h/ro) in the case of a > 1 are shown in Fig.6.9. Note that 5 is proportional to h02 to a high degree of accuracy. Modes of loss of stability represent near-conic surfaces. To analyze non-axi-symmetric forms of stability loss, we take w = w(r, ~). Denoting linearized quantities by a dot, we get

s"

ds

= de

w = w(r, )ez

(P + ~w) ~=0

Using equations (5.6.2), (5.6.3), and (5.4.4), we obtain

190

6. Stability of Equilibrium and Wave Propagation

s" = ~(~, ¢ ) ~ z

w'l(

1

~1 = , ~ . ~ + ~

s2=

R ~-F~-

( = -E~

(6.4.4)

1)

+ 7 ~''~

'

. ~ + -/t~r + r "~R + -)~'~r

VrR = E1

[~

a.

p'~, + r p ~ . - 7 ~

u~--~ + ~

grR = - E ~

+ ~(~, ¢ ) ~ z ,

R'w" + U~Rw"r

u-1

u - ~ R - R"

+ ~

-~R-

,

)

+uR'-u-1 r

,

(6.4.5)

w' + u~w,~r2 vR"

'

,

w' + --~-w,~

,

rR'

,

~r,~ -- - E ~ ( 1 - ,,,)7~((m, ~ - R % ) )

, ~

:

--~,.,~.

In equations (6.4.4), (6.4.5)we use the notation: f ' : Of/Or, f,~ : OrlOn. The linearized equilibrium equation V . S" = 0 now takes the form 1

s~' + - s l + s2,~ - 0 .

(6.4.6)

7-

Substituting equations (6.4.5) into equation (6.4.6) we obtain the system of four equations with respect to functions w, v - w',/2rR, and Sl WI

v' = - - ~u , ~

-

-

1 R" + -y(

" [ a2R #rR = ( v 2 - 1 ) 7 - +

"

V~

1 1

"~R)Vr - -fie #~R' (1 - u) R'~] ~-~r2] E2w,vv

~4 R 2 ] (~,~ - 1 ) E ~ N - ~,~Rj v + E~(1 -

+

r2R '

R'

,)-~v~ (6.4.7)

r

~1 = E~(~- ")h--~r~

~'~

1 + E2 ( 1 - u) ~ w,vvvv +/{72(1 - u)~-~r2

R"+

r2 ] ~7 +

1 1 R ,r 2 flr R,cpcp -- -Sl.r

-

v,vv

6.4 Buckling of Thin Elastic Plate with Disclination

e

o.J.

e.3

~ ~ - m . m . .

~

191

Q~

~

1.1g- 3

-gl.5

]'lloZ

\

-1.5

•1 0 - 5

Fig. 6.10.

The boundary conditions for free edge of the plate, in view of equation (5.4.8), have the form

o(1)

er " S F ~

#rO N

=0,

#r R = O.

After the linearization, we obtain (at r = rl, r0) 1

!

sl - E2(1 - u ) ~ (w~o~oR- w,~o~oR') = O,

(6.4.8)

~ r R --" O.

We shall seek nontrivial solutions to the boundary value problem (6.4.7), (6.4.8) in the form w(r, ~) = w n ( r ) s i n ( n ~

+ ~0),

n = 1, 2, ...

(6.4.9)

Substituting equation (6.4.9) into equations (6.4.7), (6.4.8) we arrive at a linear homogeneous boundary value problem for a system of ordinary differential equations. The values of ~, for which this system has nontrivial solutions were found numerically, the results, in particular, showed that if ~ > 1 the loss of stability occurres in an axi-symmetric manner, the critical values of ~ for nonaxi-symmetric forms (n < 1) are of higher order than for axi-symmetric forms. Dependence of ~ on the hole radius for ~ < 1 is depicted in Fig.6.10. Here non-axi-symmetric stability loss occurs with n = 2. A comparison of Fig.6.9 with Fig.6.10 shows that the disclinations which are opposite in sign have practically the same effect on the plate stability.

References

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Nye J.F. (1953): Acta Metallurgica 1, pp. 153-162. Novozhilov V. V. (1958): Theory of Elasticity (in Russian). Sudpromgiz, Leningrad. Nowacki W. (1974): Arch. Mech. 26, 1, pp. 3-11. Nowacki W. (1975): Theory of Elasticity (Russian translation). Mir, Moskva. Ogden R.W. (1972): Proc. Roy. Soc. London A326, pp. 565-584. Olver P.J. (1986): Applications of Lie Groups to Differential Equations. SpringerVerlag, New York, Berlin, Heidelberg, Tokyo. Pal'mov V. A. (1964): Prik. Mat. Mekh. 28, 3, pp. 401-408. Pietraszkiewicz W. (1977): Introduction to the Nonlinear Theory of Shells. RuhrUniversitat, Inst. Mech., Bochum. Pietraszkiewicz W. (1989): Advances in Mechanics 12, 1, pp. 52-130. Podio-Guidugli P., Vergaga Caffarelli G., Virga E.G. (1986): Journ. Elasticity 16, 1, pp. 75-96. Pogorelov A.V. (1967): Geometrical Method in Nonlinear Theory of Elastic Shells (in Russian). Nauka, Moskva. Rice J.R. (1968): Trans. ASME, Journ. Appl. Mech. 35, pp. 379-386. Rivlin R.S. (1948): Philos. Trans. Roy. Soc. London A240, pp. 459-508. Shamina V. A. (1974): Izv. AN SSSR. MTT (Mechanics of Solids). 1, pp. 14-22. Shkutin L.I. (1980): Zhurnal PMTF. 6, pp. 111-117. Sokolnikoff I.S. (1971): Tensor Analysis (Russian translation). Nauka, Moskva. Somigliana C. (1914): Atti Accad. Lincei Rend. t.23, pp. 463-472. Teodosiu C. (1982): Elastic Models of Crystal Defects. Springer-Verlag, Berlin, Heidelberg, New York. Toupin R.A. (1964): Archive for Rational Mechanics and Analysis 17, 2, pp. 85-112. Truesdell C. (1977): A First Course in Rational Continuum Mechanics. Academic Press, New York. Truesdell C., Noll W. (1965): The Non-Linear Field Theories of Mechanics. Handbuch der Physik III/3, Springer-Verlag, Berlin. Vakulenko A.A. (1991): The Connection Between Micro- and Macroproperties in Elastostatic Media. Itogi Nauki i Tekhniki. VINITI, Ser. Mechanics of Deformable Solids (in Russian). 22. Vladimirov V. I., Romanov A.E. (1986): Disclinations in Crystals (in Russian). Nauka, Leningrad. Volterra V. (1907): Ann. Sci. Ecole Norm. Supp. t.24, pp. 401-517. Weingarten G. (1901): Atti Accad. Lincei Rend. t.10, pp. 57-60. Wesolowski Z. (1981): Dynamic Problems of Nonlinear Elasticity (Russian translation). Naukova Dumka, Kiev. Zelenin A.A., Zubov L.M. (1985): Izv. AN SSSR. MTT (Mechanics of Solids). 5, pp. 76-82. Zelenin A.A., Zubov L.M. (1987): Prikl. Mat. Mekh. 51, 2, pp. 275-282. Zelenin A.A., Zubov L.M. (1988): Prikl. Mat. Mekh. 52, 4, pp. 642-650. Zelenin A.A., Zubov L.M. (1989): Izv. AN SSSR. MTT (Mechanics of Solids). 1, pp. 101-108. Zhilin P.A. (1982): Trudy Leningrad. Politekhn. In-ta, 386, pp. 29-46. Ziegler H. (1968): Principles of Structural Stability. Waltham, Mass.: Bleisdell. Zubov L.M. (1970): Pricl. Mat. Mekh. 34, 2, pp. 241-245. Zubov L.M. (1971): Pricl. Mat. Mekh. 35, 3, pp. 406-410. Zubov L.M. (1976): Pricl. Mat. Mekh. 40, 6, pp. 1070-1077.

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Notations

Symbol

description and number of page

A

rotation tensor 9 potentional energy functional 44 complementary energy functional 46 Lame coefficients for orthogonal curvilinear Euler's coordinates 52 Lame coefficients for orthogonal curvilinear Lagrange's coordinates 52 some functions of strain invariants 65 Burger's vectors for elastic shell 143 tensor connected with affine deformation tensor 15 positive definite tensor in polar decomposition of microdistorsion tensor 118 coefficients of second quadratic form of a surface in deformed configuration 132 Burgers' vector 20 coefficients of second quadratic form of a surface in reference configuration 131 deformation gradient 9 distortion tensor of deformed surface 132 axial shortening factor of elastic cylinder 178 Piola's stress tensor 12 space divergence 12 modified heat flux vector 51 coupleIoads on boundary of shell 137 divergence in reference configuration ii identity tensor II potential of external loads 44 linear strain tensor 14, 108 discriminant tensor of surface 46 Cartesian vector basis 27 orthonormal vector basis of cylindrical coordinates 56 density of internal energy 51 inverse deformation gradient 83

A A1 Am am am ak

B B

B~z b

b~ C Co C

D

Div d d div E

E e e

ek er~ e~o, ez e

F

198 F F

.f fk, fn G G

Gmn

gmn

9aZ g

Grad grad H h h I Io 11,/2 ,/3 lm im i J J J K K Ko K k k L Lo 1 l l' M

Notations microdistorsion tensor 118 intensity of surface load per unit square in deformed configuration 12 longitudinal force in cross-section of cylinder 64 intensity of surface load per unit square in reference cofiguration 11 vector bases associated with complex coordinates 28 history of function f(t) 51 Piola's tensor of couple stresses 106 first fundamental tensor of surface in deformed configuration 135 metric in deformed configuration I0 coefficients of first quadratic form for material surface in deformed configuration 132 metric in reference configuration i0 coefficients of first quadratic form for material surface in reference configuration 131 first fundamental tensor of surface in reference configuration 133 ratio of surface element area shell in deformed configuration to one in reference configuration 135 operation of space gradient Ii operation of gradient in reference configuration 9 microrotation tensor 103 logarithmic measure of deformation 76 heat flux vector 51 Cauchy's strain tensor 13 tensor of tangential strains in shell 134 invariants of Cauchy's measure of deformation 57 orthonormal vector basis 52 orthonormal vector basis 24, 52 Almansi's strain tensor 13 vector invariant integral 125 determinant of deformation gradient 12 invariant contour integral 156 modified stress tensor 84 bending strain measure in medium with microstrucrure 119 tensor of bending strains in shell 134 Frank's angle 22 mass force 11 unit vector 158 bending strain tensor 104 bending strain tensor in Cosserat-type shell 145 second measure of bending strain 106 couple load per unit area of shell 145 mass density of volume distributed couples 105 tensor of couple stresses 106

Notations

M

M3 M m

N

A; n

O P P P

q QM q qS

R R~ RM R /, /'s T

Rot rot

S S S S

T t t t U U

Uo V Y V V

W Wo W W

Xk

199

normal to boundary, contour of shell in deformed configuration 137 torque 64 point of Euclidean space 15 normal to the boundary contour of shell in reference configuration 138 normal to a surface in deformed configuration 11 point of Euclidean space 17 normal to a surface in reference configuration 11 orthogonal tensor 10 Kirchhoff's stress tensor 12 linear density of force over shell boundary 137 pressure in incompressible material Which is not expressed in term of strain 85 orthogonal tensor connected with Frank's vector 18 Euler's coordinates 9 Frank's vector 20 Lagrange's coordinates 9 radius-vector of particle in deformed configuration 9 vector basis of deformed configuration 9 vector basis in space 9 radial coordinate in deformed configuration 56 radius-vector of particle in reference configuration 9 vector basis of reference configuration 9 radial coordinate in reference configuration 56 curl in deformed configuration 43 curl in reference configuration 15 J aumann stress tensor 45 special force tensor in shell 161 length of curve after deformation 137 length of curve before deformation 138 Cauchy's stress tensor 12 unit tangent vector to the contour of shell in deformed configuration 138 parameter of curve 15 time 51 left stretch tensor 9 displacement vector 13 stretch tensor in shell 133 right stretch tensor 9 specific complementary energy 45 body volume in deformed configuration 11 body volume in reference configuration 11 specific potential energy of deformation 10 specific potential strain energy of shell 132 perturbed vector of displacement 167 function of warping 59 Cartesian coordinates in deformed body state 27

200 X8

Y

Y Yk

Z Z Z

OL O~

9 F F n

Vk V V

g g

go

(9 (9 0 0 0 ~o

A

Notations Cartesian coordinates in reference body state 27 strain measure in Cosserat continuum 104 orthogonal curvilinear coordinates in space 52 second strain measure in Cosserat continuum 106 orthogonal curvilinear coordinates in reference configuration 52 axial coordinate in deformed configuration 56 distance of point from surface 137 complex coordinates in deformed body 27 axial coordinate in reference configuration 56 dislocation density tensor 39 dislocation density vector in shell 150 coefficient of axial extension in elastic cylinder 56 constant of Cosserat-type material 113 disclination density in shell 149 complex Burgers' vector 31 constant in constitutive relation of elastic material 101 virtual microdistorsion 120 Christoffel's symbols of second kind in deformed metric 14 boudary contour of shell in deformed configuration 137 Christoffel's symbols of second kind in undeformed metric 14 closed contour in reference configuration 36 boundary contour of shell in reference configuration 138 constant of Cosserat-type material 113 Kronecker's symbol 9 symbol of variation 11 constant of Cosserat-type material 113 rate-of-strain tensor 51 linear strain tensor in Cosserat's shells 147 tensor of virtual strains in shells 135 complex coordinates in reference configuration 27 mass density of entropy 51 constant of Cosserat-type material 113 dimensionless radial coordinate 171 stress function in shell theory 151 tensor of stress functions in Euler's coordinates 43 perturbed Piola's stress tensor 169 microrotation vector 104 temperature 51 length of microrotation vector 105 linear tensor of bending strains in Cosserat-type shells 147 tensor of virtual bending strains in shells 135 parameter of wedge disclination 56 Caushy's strain measure I0 Almansi's strain measure I0 stress function in shell theory 151

Notations

A A

# V //

II 7r 7;"

P P~ P P P0 P P~ Z Z ~T O"

(~R~ ~ ~T

~Z

7" T To 7"RZ ~ TZ¢~ 7"R,~

~k

X

¢

¢ f~

201

Lame's elastic constant 66 stretch of material curve 138 couple load per unit area 106 tensor of bending couples in elastic shells 137 Lame's elastic constant 66 tensor of membrane forces in elastic shells 137 Poison's ratio 66 force tensor of load system 121 surface density of generalized double force 122 mass density of generalized double force 122 affine deformation tensor 14 tensor of double stresses 122 tensor of couples in Cosserat's shell 145 radius-vector of point of surface in deformed configuration 131 vector basis on surface in deformed configuration 131 material density in deformed state 12 dimensionless radial coordinate 67 material density in non-deformed state II radius-vector of point of surface in reference configuration 131 vector basis of surface in reference configuration 131 boundary surface of elastic body in deformed configuration 11 surface of shell in deformed configuration 132 boundary surface of elastic body in reference configuration II surface of shell in reference configuration 131 normal stresses in cylindrical coordinates 58 relative stress tensor in medium with microstructure 122 symmetric stress tensor in medium with microstructure 122 tensor of forces in Cosserat's shell 145 unit vector of tangent to boundary contour of undeformed shell 138 tangential stresses in cylindrical coordinates 58 tensor of stress functions in Lagrange's coordinates 46 relative strain meausure 119 complex functions of stresses 48 angular cylindrical coordinate in space 56 Frank's tensors in elastic shell 141 angular cylindrical coordinate in reference configuration 56 Frank's vectors in elastic shell 142 linear rotation vector in Cosserat's shell 147 complex potential of nonlinear elasticity89 rotation under plane deformation 28 angle of twist of cylinder 56 complex potential of nonlinear elasticity90 virtual rotation in Cosserat's continuum 105 Frank's tensor in elastic Cosserat's body 109 Frank's vector in elastic Cosserat's body 109

202

Notations coefficient of surface tension 100 oscillation frequency 187

Index

Adkins' theorem, 86 affine deformation tensor, 14 Almansi's measure of deformation, I0 Almansi's strain tensor, 13 angle of twist, 56 axial pores, 78 Bartenev-Khazanovich material, 74 basic deformed state, 167 bending strain measure, 119 bending strain tensor, 104 bending-torsional waves, 187 bifurcation of solutions, 168 Blatz-Ko material, 69 buckling modes, 168 Burgers' vector, 20

Cartan's theorem, 16 Cauchy integral operator, 94 Cauchy stress tensor, 12 Cauchy's measure of deformation, i0 Cauchy's strain tensor, 13 Cherepanov-Rice integral, 128 Christoffel's symbols of second kind, 14 Clausius-Duhem inequality, 51 coaxiality theorem, 53 compatibility equations for deformation gradient, 42 complementary energy functional, 46 complex coordinates, 27 complex function of stresses, 48 complex potentials, 90 condition of compatibility of strains, 15 conjugate materials, 85 conjugate solutions, 85 conservative law, 127 constrained material, 107 Cosserat pseudocontinuum, 108

Cosserat's continuum, 103 Cosserat-type shell, 145 couple stresses, 106 covariant derivative, 15 critical loads, 168 cross product of vectors, 12 curves of dislocation and (or) disclination, 26 curvilinear multiplicative integral, 16 cylindrical cavity, 78 deformation conditions, 152 deformation gradient, 9 differential equilibrium equations, 11 disclination density, 39 disclination energy, 68 discontinuous solutions, 98 discriminant tensor, 46 dislocation density tensor, 37 dispersion relation, 187 dispersive waves, 187 displacements, 13 distortion tensor, 14 distortion tensor of deformed surface, 132 double force, 122 dual Lagrange-Euler basis, 50 edge dislocation, 88 elastic distortion, 37 elastic shell of Love type, 131 energy of screw dislocation, 75 entropy, 51 Eshelby's integral, 127 Euler's coordinates, 9 eversion of ring, 80 Finger's measure of deformation, 10

204

Index

force dipole, 122 force tensor, 121 frame-independence, 51 Frank's angle, 39 Frank's tensor, 37 Frank's vector, 20 free energy, 9 Gauss-Codazzi equations, 140 G aussian coordinates, 46 generalized double force, 122 gradient - in reference configuration, 11 - in space, 11 on surface, 132 gyrotropic material media, 53 -

harmonic type material, 66 heat flux vector, 51 homogeneous body, 84 incompatibility equations of deformation, 151 incompressible Cosserat pseudocontinuum, 108 incompressible material, 59 inner product, 9 internal energy, 51 invariants of Cauchy strain measure, 57 inversion of cylinder, 171 isolated defect, 20 isometric transformation, 164 isotropic function, 13 isotropic material, 13 Jacobian identity, 15 Jaumann stress tensor, 45 kinematic conditions, 152 Kirchhoff's stress tensor, 12 Kirchhoff-Love kinematics hypothesis, 136 Kronecker's symbol, 9 Lagrange's coordinates, 9 Lagrange's variational equation, 11 Lame coefficients, 52 left stretch tensor, 9 Liapunov-Schmidt method, 168 linear defects, 26

linear strain tensor, 14 linear theory of Cosserat's shells, 147 linearized Piola stress tensor, 169 loading parameter, 167 longitudinal force, 64 longitudinal-torsional waves, 187 material surface, 131 material symmetry, 51 material with three-constant elastic potential, 75 metric of deformed configuration, 14 microdistortion tensor, 118 microrotation tensor, 103 microrotation vector, 104 Mooney material, 73 multiplicative integral, 15 nabla operator, 18 non-deformed metric, 15 orthogonal sweep method, 175 orthogonal tensor, 10 physically linear material, 112 Piola's identity, 12 Piola's stress tensor, 12 plastic distortion, 37 polar decomposition, 9 potentiM energy functional, 44 potential energy of deformation, 10 potential of external loads, 44 power of disclination, 26 principal moment, 121 principal vector, 121 principle of local action, 103 principle of material frame indifference, 104 pure deformation, 9 radius-vector - i n deformed configuration, 9 in reference configuration, 9 relative strain measure, 119 relative stress tensor, 122 right stretch tensor, 9 rotation tensor, 9 -

screw dislocation, 58 second strain measure, 106

Index second tensor of bending strain, 106 self-conjugate material, 85 semi-linear material, 66 specific complementary energy, 44 static method for investigation of stability, 168 static-geometrical analogy, 151 stress boundary conditions, 11 stress functions of shell theory, 151 surface energy, 100 tensor - of bending couples, 137 of bending strains, 134 - of membrane forces, 137 of tangential strains, 134 tensor of double stresses, 122 tensor of stress functions, 46 tensor product, 9 torque, 64 total Burgers vector, 37 total Frank vector, 37 translational dislocation, 23 two-dimensional tensor, 133 -

-

universal isometric solution, 164 vector basis - in space, 9 - of deformed configuration, 9 of reference configuration, 9 vector invariant of second order tensor, 22 vector of finite rotation, 19 Volterra's dislocation, 20 -

warps, 59 wedge disclination, 58 Weingarten's theorem, 19

205