131 114 17MB
English Pages [319] Year 1995
Table of contents :
Preface
1. Introduction
Kinematics of continua
2. Deformation
3. Displacement vectors and strain tensors
4. The linearized theory of strain
5. Position and timedependent tensor fields
Fundamental laws of coninuum mechanics
6. Internal force system of the continuum
7. Fundamental laws of continuum mechanics
8. Basic equations of wave generated in the continuum
9. Fundamentals of thermodynamics of the continuum
10. Constitutive relations
Special cases of continuum mechanics
11. The elastic body
12. Plastic bodies
13. Viscoelastic solid bodies
14. Fluids
Appendix
CONTINUUM MECHANICS
Continuum
Mechanics
Gyula Béda—Imre Kozak—)6zsef Verhas
Continuum
Mechanics
Cj
Akadémiai Kiadó, Budapest
This book is the revised English version of the Hungarian Kontinuummechanika
published by Miiszaki Kdnyvkiado, Budapest
Translated by L. Kapolyi (Mrs.)
Translation revised by D. Durban
ISBN 963 05 6758 X
(C Gyula BédaImre Kozak—Jozsef Verhas, 1995 © English translation L. Kapolyi (Mrs.), 1995
All rights reserved. No part of this book may be reproduced by any means or transmitted or translated into machine language without the written permission of the publisher.
Published by Akadémiai Kiado, H1117 Budapest, Prielle Kornéliau. 1935. Printed in Hungary by Akadémiai Kiadó és Nyomda
Kft., Budapest
Contents
Preface
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1. Introduction . . 1.1. On continuum 1.2. Definitions . 1.3. Notation . 1.4. Shifters . . 1.5.
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Twopoint tensors
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Kinematicsof continua . . . . . . . . . . . . . ..
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2. Deformation . . . . . . . . . . . e e e e e e e e e e e e e e e 21. Motion . . . . . . s 2.2 Deformations and strains . . . . . . . . . . . . . o0 000 .
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Polar decomposition. Rotation tensor. Rigidbody rotation
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Principal axes (eigenvectors), eigenvalues, scalar invariants of the deformation TENSOTS
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Deformation tensors in a geometrical sense
2.6.
Generalization of deformation and strain tensors
2.7.
Compatibility conditions . . . . . . . . . . . . ...
3. Displacement vectors and strain tensors 4. The linearized theory of strain
4.1. 4.2. 4.3.
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Strain tensors in a geometrical sense
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Notation, equations . . . . . . . . . . . . . o .00 e e e Components of the strain tensor in the Cartesian and cylindrical coordinate
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Compatibility of the strain field . . . . . . . . . . . . . ... ... .. Obtaining the rotation and displacement vector fields from the strain field . . . Compatibility conditions . . . . . . . . . .. kk
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Material representation of the material derivative . . . . . . . . . . . . . Kinematic definition of the deformation rate tensor . . . . . . . . . . . Physically objective time derivatives . . . . . . . . . . . . . . . . . ..
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Velocity vector field . . . . . . . . The velocity gradient tensor . . . . Additional position and timedependent Time derivatives of tensors . . . . .
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Material (substantial) time derivative of significant tensors
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Material time derivatives of integrals of tensor fields, related to tinuum . . . .. L L L L L
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Fundamental laws of continuum mechanics
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6. Internal force system of the continuum
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7. Fundamental laws of continuum mechanics . . . . . . . . . . . . . . . . . 7.1. Law of mass conservation (continuity equation) . . . . . . . . . . . . . .
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Equations of motion
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Equations of equilibrium, stress function tensor . . . . . . . . . . . . . . Principle of virtual power and virtual work . . . . . . . . . . . . . . . .
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Fundamental laws of microstructured continua
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8. Basic equations of wave generated in the continuum
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Classification of waves. Compatibility conditions along the surface carrying the
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JUMP . .o L L o e e e e k k k Dynamic compatibility conditions . . . . . . . . . . . . . . ... . . Constitutive compatibility conditions . . . . . . . . . . . . . . . . . .
9. Fundamentals of thermodynamics of the continunum 9.1. First law of thermodynamics . . . . . . . . .
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Second law of thermodynamics in a simplifiedcase
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Linear laws describing the velocity of material processes . . . . . . . . . . Dissipative processes. Gyarmati’s variational principles . . . . . . . . . . .
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10. Constitutive relations 10.1. Introduction . .
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How to set up the constitutive equation . . . . . . . . . . . . . . .. The more important constitutive equations . . . . . . . . . . . . . . . Elasticbody . . . . . . . . . . . . ... . l S Fluids. . . . . . . . . . . ..o
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Definitions. Linear theory of elasticity . . . . . . . . . . . . . . . .. . . . . . . . . . . . Primal and dual systems of the theory of elasticity Primal system of the linear theory of elasticity . . . . . . . . . . . ..
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Variables, field equations,
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primal system . . . . . . . Isotropic linear elasticbody . Anisotropic linear elasticbody Strainenergy density . . . .
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Energy equation in elastodynamic problems Energy equation in elastostatic problems . . Uniqueness of the solution . . . . . . . .
11.3.8. 11.3.9. 11.3.10. 11.3.11. 11.3.12. 11.3.13. 11.3.14. 11.3.15.
. . . . . . . . . . The Navierequation Special fields of static problems in the linear Principle of virtual work . . . . . . . . . Minimum principle of total potentialenergy . . . The Lagrangian variational principle The Ritzmethod . . . . . . . . . . . . Additional variational principles . . . . . Betti’'stheorem . . . . . . . . . . . . .
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Dual system of the linearized theory of elastostatics . . Stress boundary condition and the stress function tensor
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. . . . . . . . . .. tt Compatibility conditions Variables, field equations and boundary conditions of the dual system Uniqueness of the solution . . . . . . . . . . . . . . ... ... . . . . . . . . Minimum principle of total complementary energy . . . . . . . . . . . . . . principle The Castigliano variational Threedimensional elastic problems . . . . . . . . . . . . . . .
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boundary conditions and initial conditions of the
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PapkovichNeuber solution of the Navier equation (for elastostatic problems) Infinite elastic body under concentrated load at the origin (Kelvin’s problem) Influence functions of the isotropic linearly elasticbody . . . . . . . . . Solution using surface integrals . . . . . . . . . . . . . . . . . . .. Elastostatic plane problems . . . . . . . . . . . . . . ..o
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Plasticity theories . . . . . . . . Extreme value theorems of plasticity
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Introduction Simple fluids
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A1. Vectors and tensors (Introduction using invariant notation) . . . . . . . . . .
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A2, Determinants . . . . . . . . . . L L A3. Coordinate systems . . . . . . . . . . A3.1. The curvilinear coordinate system . . A3.2. Quantities characteristic of the geometry A3.3. Components of tensors . . . . . . A3.4. Relationships between characteristics of
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The matrix formalism
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Transformation of tensors
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A4. Notation of tensors by indexing . . A4.1. Definition of the tensor in general A4.2. Notation of tensorial operations by A4.3. Symmetric and isotropic tensors .
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Lo o e e e e e e . . . ... oo . . . . . . . . . . . . . . . of the coordinate system . . . . . . . . . . ... ... the coordinate system . . . .
Physical components of tensors
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AS. Secondorder tensors . . . . . . . . . k AS.1. Freguently used relationships . . . . . . . . . . . . . . . . . AS5.2. Theeigenvalue problem . . . . . . . . . . . . .
AS5.3.
Scalar invariants of symmetric secondorder tensors
A5.4.
The CayleyHamilton theorem
AS5.5.
Tensor polynomials
A5.6. AS5.7. A5.8.
Deviatoroftensors . . . . . Orthogonal tensors . . . . . Polar decomposition of tensors
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A8. Fundamental quantities in cylindrical and spherical coordinate systems A8.1. Cylindrical coordinate system . . . . . . . . . . . . . . ... A8.2. Spherical coordinate system . . . . . . . . . . . . ... . .
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A7. Integrals of tensor fields . . . . . A7.1. Integral transformation theorems
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Subject index
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Preface
Originally, this book has been written in the Hungarian language, with a view to give
a summary of the most important elements and laws of continuum mechanics, and to make widely accessible, and arouse interest in the fields discussed. Also, the authors intended to publish the most important results of Hungarian scientists in this field. Considering the problems discussed and the mathematics used, a rather ‘abstract’ domain of science, the Hungarian edition could come out only because Prof. Istvan Gyarmati had been standing by the authors firmly and his standpoint had won the
publisher (Múszaki Konyvkiadd) over to the cause. The publisher did a fine, highly professional, job with even the most special requirements met, a pioneering work
indeed. The present content of the book is based essentially on the Hungarian edition. However, it differs from the original edition necessarily in that a more comprehensive
description of the physically objective time derivates is presented, a wider range of the measures of strain is used, a possible variation of kinematic and dynamic equations of polar bodies is given and also the constitutive equation is slightly extended. Also
the structure of the work has been changed with a view to make it, in its comprehension, hopefully more understandable.
In writing this book, much valuable aid has been offered by our colleague, Laszlo Szabo, Ph.D., including comments and calculation check, for which we express our gratitude. It is our pleasant duty to thank Professor David Durban for revising the
manuscript and for his many valuable remarks. Acknowledgement is due to Mrs. L. Kapolyi who translated the manuscript into English. Without her contribution, the book would not be available in English language. Finally, we are indebted to the
Publishing House of the Hungarian Academy of Sciences for the forebearing as well as for the effort to make the book accessible to anybody interested abroad. The English version of this book was edited by Gy. Béda. The Authors
XI
1. Introduction
1.1. On continuum mechanics Mechanics as a field of physical sciences addresses bodies of idealized properties (socalled models) reflecting the characteristic response of the material to applied loads, on the basis of experience and observation. The continuous medium is one of the models widely used in physics and it has the following two, idealized, basic properties: 1. The geometrical space available is continuously filled by the material (with the molecular structure or microstructure neglected but not forgotten).
2. The relations describing the state and the changes of state of the material are expressed by tensor functions of the local coordinates (position vector) and time and they are, together with their derivatives of adequate number, continuous except for
internal surfaces of a finite number. Here and in the sequel of this book, the definition ‘tensor’ is used in a general sense unless speaking of scalars and vectors specifically.
(Accordingly, the scalar is a tensor of zero order while the vector is a tensor of first order and we speak also of tensors of second, third, or higher order.) Continuum mechanics is part of the science of mechanics, designed to study the
global mechanical movement (mechanical state, changes of state, e.g. deformation, velocity distribution, wave propagation) of gases, liquids and nonrigid solid bodies by means of the continuum model. This book uses continuum mechanics as understood
above in both a wider and narrower sense. Thermodynamical studies are extended also to nonmechanical states and changes of state and the studies are based on classic, nonrelativistic mechanics and thermodynamics using the threedimensional Euclidean geometry. (The knowledge of statistical mechanics and of the atomic and molecular structure of the material may often help to understand the description of a continua.) Tensors take a prominent part in the continuum hypotheses of physics, thus also in continuum mechanics. An outline of the tensor calculus used in the book is given in the Appendix.
Continuum mechanics can be divided into the following subdomains: — kinematics or the theory of motion, dealing with the displacement and deformation of continua and studying the time and space dependent tensor fields of continua in general, ! — basic principles, describing the general physical theorems and laws applied to continua independently of their material characteristics, — theory of constitutive relations detailing the behaviour of continua for different materials in response to external agents and/or to study the general principles and methods of constructing the constitutive equations,
— special continuummechanical theories for formulation and solution of boundarycondition and initialvalue problems of the different continua of idealized geom
etry under different and laws (e.g. fluid The kinematics continua while the
external impacts on the basis of the different methods, theorems mechanics, theory of elasticity, theory of plasticity, shell theory). of continuum mechanics and the basic principles apply to every subdomain of constitutive relations provides equations to deter
mine the properties resulting from the structure of the material only.
Note that both the general theory and the special theorems of continuum mechanics apply to idealized models and give therefore an approximate description of the actual mechanical processes taking place in nature. Applicability of the different theorems and acceptability of the solution obtained by them as an approximation should be judged, or at least considered empirically. Independently of applicability, distinctions are made between exact and approximate mathematical solutions of the equations of continuum mechanics (boundaryvalue and initialvalue problems).
1.2. Definitions Defined below are terms generally used in continuum mechanics: Continuum element (termed also mass element, mass point, material point, elementary mass, particle) is a small (infinitesimal) neighborhood of a point of the continuum, the mechanical state of which can be determined to an acceptably accuracy by quantities of finite number connected to one single point. It follows from the definition of a continuum and a continuum element that a suitably small neighborhood of any point of the continuum can be considered to be a continuum element and that the continuum can be divided in to continuum
elements, discernible from each other, in different ways of infinite number and at any instant. The division is assumed to take place by means of coordinate surfaces and the continuum element is assumed to be bounded similarly by coordinate surfaces.
The continuum itself is considered to be a complex of continuum elements at any instant, and by state of the continuum we understand the complex of the states of continuum elements. The state of the continuum can be given in general by means of
continuous tensor fields of finite number (spacetime functions) where the values of these functions assumed at one point determine the state of the continuum element, the small neighborhood of that point, at given instant.
The continuum is homogeneous with respect to some property provided this property is the same at any point. The homogeneity of the continuum is described by
the tensorposition vector function of constant value. The continuum is isotropic with respect to some property provided this property is identical in all directions.
The complex of geometrical and physical characteristics of the continua is called initial configuration at instant t—t,—0 while current (or deformed) configuration at any arbitrary instant. In initial configuration the continuum is assumed to be in its original, undeformed, state. 2
The mechanical movement of the continuum is studied in comparison with an arbitrary, socalled spatial coordinate system, assumed to be curvilinear for generalization’s sake but the use of the Cartesian coordinate system is not excluded by this assumption either.
A reason for use of a curvilinear coordinate system is, among others, the fact that the surface of solid bodies and the geometry of the liquid and gas tanks are seldom plane surfaces. In this case, the equations of boundaryvalue problems can be better formulated in a curvilinear coordinate system. Another reason is that even originally
plane surfaces become curved as a result of deformation, an important factor especially in the range of finite deformations. The points, lines, surfaces and domains (or line elements, surface elements, volume elements) of the reference coordinate system are called geometrical ones while those of the moving continuum are referred to as ones belonging to the continuum by
adding for them the attributive “continuum (or material) to make a distinction. The continuum points, lines, surfaces and domains move along with the continuum with
their shape and dimensions being changed. The variables of the tensor (position vector, time) functions of continuum mechanics are connected either to geometrical (stationary) or continuum points. Accord
ingly, there are different ways of description in continuum mechanics. In a domain, the complex
of values assumed
at all points
by a tensor (position vector,
time)
function defined there is a called a tensory field. The tensor field will be stationary if it is independent of time. In practice, two ways of description are used in general. In case of the Eulerian (spacelike or local) description, the tensor fields are connected to the geometrical points of the spatial coordinate system. In this case, we speak of Eulerian tensor fields, tensors
or coordinates. The value of the tensor field, assumed at given point and at given instant, can then be associated with the state of that continuum element where the continuum point coincides with given geometrical point in the current configuration.
In case of the Lagrangian representation, the tensor fields are connected to the geometrical points of the domain occupied by the initial configuration and we speak then of Langrangian tensor fields, tensors and coordinates. In this case, the value assumed by the tensor field at given point and at given instant can be associated with the state of that continuum element, where the continuum point coincided with a given geometrical point in the initial configuration. In case of solid bodies, two additional methods are used for description, namely a material description where tensor fields are connected to moving continuum points,
and a relative description to connect the tensor fields to the points of a domain occupied by some current configuration picked out at random. The latter is useful in studying the state of the continuum in the short interval before and after the time
selected at random. Common properties used in continuum mechanics are density functions describing the distribution of a given quantity according to the following assumption: the quantity is assumed over some volume, surface or linear domain (e.g. kinetic energy, resultant force, resultant couple) in a current configuration of the continuum that
agrees with the corresponding volume, surface or line integral of the density function. 3
1.3. Notation In the spatial coordinate system (Eulerian description), the spatial coordinates are
denoted by x', x%, x°, the covariant base vectors by g, g,, g; (see the Appendix) while the coordinate system itself is denoted by (x). With the Lagrangian representation, X', X? and X? denote the coordinates while G,, G,, G, denote the covariant
base vectors. The Lagrangian coordinates X', X?, X? may be identical with the coordinates x', x?, x* of the domain occupied by the initial configuration, illustrated in the spatial coordinate system, but they can be plotted also independently. The reference coordinate system, X' X? X?, is denoted by (X). R and r are used to denote the position vector in initial and current configuration, respectively. Figure 1.1 shows
the most
general case where
the reference coordinate
systems
XYZ
(and
X" X?X") and the spatial coordinate systems xyz (and xix?x?) differ one from each other. Definition of the curvilinear coordinate system xixóx? as compared with the Cartesian xyz coordinate system is given in para 3.1, while their geometrical characteristics are discussed in paras 3.2, 3.4 of\the Appendix.
Coordinates X', X2, X? and associated covariant base vectors G,, G,, G, are used in case of material description. Material coordinates X', X2, X? are conveniently assumed to be identical with the Lagrangian coordinates, defined in the initial configuration to be connected to, among moving later along with, the con
tinuum: X'=X!, X?=X2,
Í7— IÓ.
We shall use both an invariant (symbolic) notation notation are equally used for the tensor calculus.
and
an
index
(tensorial)
Scalar quantities (tensor of zero order) are printed in italics for both notations. Invariant notation:

— vector (tensor of first order): boldfaced, e.g. u, p, M; — — — —
X
tensor tensor scalar vector
of second and higher order: boldfaced and italicized, e.g. A, T, c; (dyadic, general) product: ab, Ab; product: ab, n  T; product: axb, nx A;
a)
Fig. 1.1. Coordinate systems a) Initial configuration: reference (Lagrangian) coordinates, X Y Z: Cartesian coordinates, X! X? X?: curvilinear coordinates b) Current configuration: spatial (Eulerian) coordinates, xyz: Cartesian coordinates, x! x? x?: curvilinear coordinates
— mixed product of three vectors: (abc); — twofold scalar product: a : b; — unit tensor: £. ! When indices are used for notation, the letters of invariant symbols are employed, printed in italics and indexed according to the following convention: — small letters for the components of Eulerian tensors, e.g., u,, A", C p95 — capitals for the components of Lagrangian tensors, e.g., u;, E" ; — if Roman letters are used as an index, they will assume the values of 1, 2, 3; — 1f the index is a Greek letter, it will assume only the values of 1, 2;
— according to the summation convention, an upper index and a lower index in a term, if identical, will assume all the possible values and the quantities so obtained will be added up, e.g.
a*b, =a'b,+a’b,+a’b,, H*l, =H'l,+
H*L,+ Hl,,
C. =C'+C; — identical upper and lower indices are called dummy indices while the other indices are free indices, e.g. k and o are dummy indices while / is a free index in the above expressions; within a formula, the symbol used as a dummy index can be
changed willingly, e.g. H*l,=H"l,,=H"1,; — notation and definition of the Kronecker symbol. 5l 6—
{1
Tf
k=1
or
az—f,
0
if
k#1
or
a#p;
— notation and definition of the permutation symbol: 1
€um, €m=
dX* mapping) in the coordinate system, reversed and shifted to point P,, of principal axes n,, n, (coordinate system of principal axes N,, N,, rotated and shifted to point P). Change of the surface elements and volume elements can be seen in Fig. 2.6. Accordingly, a surface element
dA,=dR,xdRy;
d4}=€,,,dX{dX}
at the initial configuration becomes surface element
dA
=dr;xdry;=dR, : (D" x D) : dRy
d4, —E
pqr
dXxí/dxj —E
pqr
(259)
1" d u dX74X71
of the current configuration during dR—dr mapping (motion). For further transfor
mations, recall the definition of the permutation tensor according to (A37), formula (211), as well as the formula —
S
Aa
exrmd = esqrd k@ d
A
y
Fig. 2.6. Change of surface element and volume elements
26
which follows from the definition of the Jacobi determinant (22) and of the deforma
tion gradient (26). Introducing the notation
9 ;, 7 \/;J—J,
: (260)
the relation between both surface elements is obtained in the form dA,,=.7(d“)Kp dA2 ,
i dA —J(D))" .dA, .
The equation
(261)
B
dV=JdV,
(262)
between a continuum volume element dVy=(dR;dR;;dR;;)) =€, dXT A X dX of the initial configuration, and a continuum volume element dV
= (dr;dry dryy) — €,,, dxf dxfdxiy
of the current configuration is obtained in a similar way. (262) can be written in a different form if the third scalar invariant of Green’s
deformation tensor is determined on the basis of (214), (260) and (AS526):
1 1 g En— G l Egl = E(dp[()2' KAE 5J2=J"2. Hence, with respect to also relationship Eyy — cn
(263)
, it can be written that
1 dV:
EIII
dVO;
dV:
dVo
.
(2'64)
v/ €
By specific dilatation we understand quantity dv gy=———1. —a,
265 (265)
According to (264), 8y=
EIII
—
l =
l
—
1 .
(2'66)
A/ €111
The most important relationships describing the geometric nature of the deformation and strain tensors are summed up in Table 2.
2.6. Generalization of deformation and strain tensors It can be seen from para 2.4 that the eigenvectors of the tensors of right stretch M, Green
and
Piola
deformation
E, E!
and
Lagrange
these eigenvectors be vectors N x and NY. If M, =),
strain H are identical.
Let
are the eigenvalues of M, 27
a
X
vo__%ías
TVD
D
5/
5959
m—
" g Ibdp
1105105 TX 0. govopygir"vso0
502
=P

.—l
‘AP AP

!
s
—
AP
—
"A4P—AP
pue


—
43
=¥P
—Z = 0 2 I
T
,. కロاعلس벼[4
0y.q.10
v
% 509
+EI
—p—Op—=1II"'T 4 —
Iy
0. oz4"0s00
T
 N="y
T
 37
M=
SP —= sp "
o.yag— N —
T
D
j917 0 N="4APr —96 1]1 =AP
(e+1)(e+1)
¥p.(@r
:
0 堕 0 AP=——="4P"F I
)
M\
L
(4
ロون댁نرسدи클иロ
G
110, 9H
(+pG+n
=0V
+El:m
%
7VP4PXr
V(
v
vmqmiu\/
T
IWobattg
REEaNIM\z
T
. . 1
— 147949 HT M= 794994 /1—1499  H 3¢
SP SP—SP
\u
a
asp=4p
4
SIOSU3]J UTRI)S PUB VUONBULIOJ9P JO 3INJBU ANJUMOA
94
Oagp=4p
"T AgBL
28
then, expressed in terms of 4,, the eigenvalues of E, E~'
and H will be A}, 47 ?
and 1/2(A; —1), respectively. Thus the tensors in question can be written also as
MzAgNANY, with H=1/2(Ax —1)NgN*,
E=JANNK,
E'=1;NN¥
or, in a more general form (Hill 1968), %:f(ÁÁ)NKNK§
Ny : NF=0dg.
(267),
Subscript K, of 4 is an accompanying index not included in the summation convention but always assuming the numeral value of index K of vector N, or N, included in the summation convention.
fin (267), is an arbitrary analytical function of a complex variable z and thus f(2) can be expanded as a power series of z. In the cases studied earlier / is an analytical
function of this type. Should, in addition, / satisfy conditions f(1)=0 and /" (1) =1, we will arrive at the Hill generalization given by H
=f (M) =f ()'L(.) NKNK
(2'67)2
as can be seen from paras A5.2, A5.4 and A5.5. A possible form of (267), is
H=H" = 2上 [MZ”—I]
(268)
n
(Seth 1964, Hill 1968, Sang, Duan 1991). If
n=1, then the Lagrange strain tensor is obtained, but a different strain measure
can be defined for each value of n. Thus if n—0, we will arrive at the Hencky strain (268), namely
limL[Mz"—l]:lnM, n—0
2h
the Hencky strain tensor is the logarithmic strain tensor. Because Ín z is analytical and on the basis of the Cayley—Hamilton theorem (para A5.4), the Hencky strain tensor can be written as
In M= a, I+ o, M+ o, M"
(269)
where a,, «,, a, are scalar functions of the scalar invariants of the stretch tensor M (Fitzgerald 1980). Also, the deformation
tensors
can
be
generalized
in form
given
in (267),.
However, now the possible general form is E®™=M?*"
(270)
instead of (268). f(z) is an arbitrary analytical function also in this case with, however, f(1)=1 and S'(1)=2n. Then, if n=1, the Green deformation tensor, while if 1— — 1, the Piola deformation tensor will be obtained.
The above calculation is possible also for the left stretch tensor L. Of course, we obtain in this case the general form of the deformation or strain measures, including 29
the Cauchy deformation tensor and the Euler strain tensor, respectively. Hence, the Hencky strain tensor can be defined also with the left stretch tensor L (Fitzgerald 1980).
2.7. Compatibility conditions As has been shown in the definition of the deformation tensors in para 2.2, the metric tensor Gy, of the initial configuration becomes E,,, the Green deformation tensor, during dR—dr mapping (motion). In case of dr—dR inverse mapping (motion),
there exist a similar relation between
the metric tensor g,, of the current con
figuration and c,,, the Cauchy deformation tensor.
The question arises now in regard to all deformation tensors introduced, e.g. Ey; and c,,, whether any secondorder symmetric tensor
Ex (X', X, X?)
or
cu(x ,X , X")
can actually be a deformation tensor. That is whether it can be derived from some continuous motion, in accordance with the discussion in para 2.2, or not. If so, this
is possible then the deformation field shall be called compatible field. The above question (compatibility of the deformation field) is of special importance if the deformation tensors themselves were considered to be the basic independent variables instead of starting from the motion according along the lines described
in para 2.2. The answer can be simply formulated. Fundamentally, the geometric space has been assumed to be an Euclidean space in both initial and current configuration. Therefore, the metric tensors G,, and g,, satisfy the RiemannChristoffel condition for the curvature tensor, defined according to formulae (A311) and (A332) of
the Appendix. Accordingly, the deformed metric tensors E,; and c,,, must be positive definites ( E,,  #0; c,, #0), and moreover, they must satisfy the Riemann— Christoffel condition for the curvature tensor. This condition is called for the compatibility (or integrability) condition for large deformations. In the mathematical formula from that condition, tensors K, or c,, shall be written instead of g,, for the Riemann—Christoffel curvature tensor according to (A332). The same applies to formula (A329) of the Christoffel symbols of the first
kind where the inverse of Ey; or c,, shall be written in place of g™". The compatibility condition for the Lagrange strain tensor Hy, of Euler strain tensor a,,, will be written in a similar way. In this case, the formulae Ex, =Gy
+2Hy,
or
c¢,,=g,,—2a,,
will be used, taking into consideration that both G, and g, satisfy the Riemann— Christoffel curvature tensor condition. As suggested by (A332), the RiemannChristoffel curvature tensor, Ry, is antisymmetric with respect to indices &, / and p, ¢: Rklpq = — lepg;
30
Rklpq = — Rqup
(271),
while it is symmetric with respect to exchange of indices k, / and p, g: Rklpq =
Rpaik
(2'71)2
It follows from the previous discussion that the tensorial equation R,,, =0 compatibility condition, implies essentially six different scalar equations: R1212 =R2323 =R3131 = R1223 = R2331 =R3112 =0.
as a
(2'72)
However, the six scalar equations thus obtained are not independent as they have
to satisfy the Bianchi identity: Rk/PG; mt Rqum; p t Rklmp; =
0.
(273)
31
3. Displacement vectors and strain tensors
Let u be the displacement of a continuum point P from point P, of the initial configuration to point P of the current configuration as shown in Fig. 3.1. The displacement vector can be expressed also in coordinate system (X) associated with
P,, and in coordinate system (x) in associated with P:
u=u(X",X2, X; D— u GT
(31),
u=u(x', x’, x% )=u,g’.
(31),
The gradient of the displacement while 14. , in current configuration. Since
vector
field is uy. , in initial configuration
r=R+u
(32)
as shown in Fig. 3.1, we have from (212) that or
OR
ou
ox* =& d'x= oxk ? axk — Gx t GT uy, x ;
É
(33)
while from (215),
, OR
9= o

21k _
A D
Or
ou 
o
e 8E sn
G4
Thus — from (3.3), 
from (3.4)
—
E
V eg"Egzgkt9""um, x
(35)
(d—l)Kp:Gch:gf_ngus;p
(3'6)
=Gyp+ug, 1 FUr kt GMNuM; KÜN; L
(37)
Cpa — 9pa — UWp; 9 — Ug; p TG Uy, plh 4
(38)
from (33) and (213), from (34) and (216), ] —
Hy = ョ (uK; pru
g+ GMNuM; KUÜN,; L)
(39)
from (37) and (220),
1
Qpg = E(up;q’*'uq;p_g
32
s
Up, pUs; 4
(310)
Fig. 3.1. The displacement vector
from (38) and (221), —
Ox —(x
te. m) (E I/Z)ÉI
(311)
from (35), (230) and (243) while 
(Q_l)pq=(gps_up;
s) (c—1/2);
(3'12)
from (36), (230) and (214). For later use let us study the relationship between the tangent unit vectors e, and e of the associated material curves C and c be studied (Fig. 3.2). Let S and s be
the arc lengths measured along curves C and c, respectively. According to definition, dR € L ds
IR L oxt
dXx* dxt L G, — ds ds
and
dr e=—. ds
The derived formula relating to the two unit vectors is obtained with the aid of the relation
o $( , du)ds ds dS dS/ ds according to Fig. 3.2 and by use of (252), (253) and the relation du — őu dS
dXxX"
0X"
ds
=uV)e,
Fig. 3.2, Unit vectors of associated material curves
33
namely, 1
e=
I+uV)e,.
1+e,
(313)
Relation (261) for associated material surface rewritten by (262)—(266) as well as (36), as dAp
(¢, being formula
the specific
= (1 + 8V) (g;( _ngus;
dilatation
by
a further).
elements
dA,
and
dA
can
be
p) dAl%
Transformation
we
d4,=(14¢,) dAzgy(ő7 —u™ , )
arrive
at the
(314),
or, using invariant notation,
for the two associated material surface elements.
There exists relationship 9
dV=
/5 Jar,
between volume element dV, and dV of the initial and current configuration, respectively, by (260) and (262), where J is the determinant of the deformation gradient (Jacobian determinant) and it can be calculated by means of formula
1 J= ge’“Mepq,(nggﬁuR; , (Gi+giu®. , ) (gu 974" ; m ) from (35) and relations
(A235).
Observing
that, from
the analysis
of para
(315) 1.4, that the
1=1621=1g%gX1=19%! 1gX]
and
G
19k1=19"9," Guxl= —1g," g
apply to the shifters, we find that
1
gk
G — . g
— 1951
(316)
With the multiplications performed in (315) we obtain, with the aid of (316) and (A22), and considering also formulae (A528)—(A530) G
e e, gkglgi=061gk=6
e
34
KLM
P
A9
A4,
epqrgKngTu
T
M
_2
/; ; 모
g
U
15
e
e
KLM
KLM
G—
r, T T — ept]rg%gggTu ;Lu ;M_‘2
P
pX ,
, ,R
€pgrIRISITU
MU
S
—
;M—6
g
UII9
G g
UIlI
along with additional relations that have not been written here, similar to the second and third formula given above, where U, U, and Uy are the first, second and third scalar invariant of the gradient 1". , of the displacement vector field, respectively. Finally, we note that
G J=
百
(1+UI+UII+
UIII)
(3'17)
and dV:(l+8V)dV0=(1+UI+UII+UIII)dVO
(3'18)
are obtained from (315). Hence, the specific dilatation is given by 8V=UI+
UII+UIII‘
(3'19)
Relation (314), between the surface elements can be written in detail in the initial configuration as dAs(14
U+
Uy + Ugy) (ÖIIE — UL; K) dAg
where dAy is the coordinate of vector the dA—d4,g",
(320)
written in the G* base,
that is d4x=g%dA4,. According to (35)—(312), all the defined strain tensors as well as the rotation tensor can be expressed by means of the gradient of the displacement vector field. The
relationship between the material surface elements and volume elements is similar according to formulae (314), (318) and (320). The question arises, however, how the displacement vector field can be defined
with known strain tensors. The condition for definition of the displacement vector field, that is the compatibility or integrability condition, is discussed in general in para 2.7. The question will be discussed in detail within the framework of the linearized
theory of strain later in paras 4.84.12. To understand better the relationship between the strain tensors and the displace
ment gradient let the coordinates of the Green deformation tensor (37) be written in the unitbase coordinate system of the principal axes normal to each other (in initial configuration). Now, it follows from (257) and (220) that HKL=O
and
EKL:_GKL:O
if
K#L
1s the matrix of the Green deformation tensor and the right stretch tensor will be [Ex 1  En O
0
0
E22
O

IM
1— [ M, O
0
0
M22
0
.
(321)
35
With notation uy; =u,. ;introduced and relationship (248), between the deformation tensors utilized, the scalar coordinates of the tensors can be written on the basis of (37) as follows: E\=E
=(M,;)’=(M))>=1+u;,)*+ )
E,, — E.—(M,,Y —( MY =(1+ )’
+ (u3,)*
+ (1,7)* + (us,)’
(322), . ;
Ezz — E; :(M33)2 =(M3)2 (1 +u33)2 +(u13)2 l—(u23)2
and E12=E21=M12=M21=0=
=(1+u)up+ E
(3'23)1—3
(1 +uyp)uyy +uy
1432
— Ezz — M 3 — M1, — 0—
 (1 HU22)t3 + (1 4 33 ) 433 + 1y E31 =E13=M31
=M13=0=
=(1+uss)uy
+ (1 +uy)uyy
H Ua3142; ,
respectively. Formulae (323) essentially imply conditions for the gradient of the displacement
field because the two deformation tensors have been written in the coordinate system of the principal axes. The matrix of rotation tensor taking into consideration (246),: [Qki]=
[
Qg,
can
be produced
1+u,
Uy
Us
M,
M,
M,
Uy
14wy
103
M,
M,
M,
31
432
E
M,
on
the basis
of (311),
1.
(324)
— 1415 M,
Of course, conditions (223) apply to (324) as well. The matrix of the Lagrange strain tensor follows from (220) in the form [H KL] 
] H 11
0 0
0
0
H, 0
0 H,
!
(3'25)
where Hy=u,+
%[(U ”)2+(u21)2+(u31)2]
Hy,=uy+
%[(u12)2+(u22)2+(u32)2]
Hazz — uz3 t %[(u 1)+ 36
(23)* H(Gz3) ] .
(326),. 3
According to (254),, the strains in the direction of the principal axes can be calculated by means of formula £m=\/E_m—1;
m=1,23,
e.g. in the direction of axis 1, &=
\/(1+u11)2+u221+u321 —1.
As suggested by formulae (322),, (323), (324), the principal strains are less affected (root shall be extracted from of second powers) by mixed derivatives Ugr=ux,1;
K#L
than the rigidbody rotation (coordinates uy,; K # L being given linearly), if the
order of the mixed derivatives are smaller than 1. For this very reason, the case where the principal strains are small while the rigidbody rotations are large occurs quite frequently. Note, however, that in general, use of the strain tensor of the linearized theory of strain for calculation is inadmissible also in cases like this (see Section 4). In the case of u;;=—0.2; u,; =0.5 and u,; =0.1 according to e.g. (322), Ell
=(1
_0.2)2+0.52+0.12=0.9
€
=4/09—1=0.0513.
and At the same time, the result would be 81=H”=u11=
if the linearized
theory
were
taken
—0.2
as a basis
for calculation.
The
difference is
important. To sum up, classification can be made between the following cases with respect to strain and rotation:
— general case: large strain and large rotation — small strain, large rotation
— small strain, small rotation: the linearized theory of strain.
37
4. The linearized theory of strain
4.1. Introduction The formulae in Chapters 2 and 3 apply also to strain of arbitrary extent, socalled large strain. However, in practical engineering, the absolute value of strain and shear
below unity (e.g. in case of elastic deformation of structures or machine elements made of steel): le. .=
v.
[ ovdV=
 gadV.
v.
v.
Similarly, it is easy to show that the kinetic moment is the material time derivative of the impulse moment. A* is a surface located in the interior of the continuum, thus bounding part of the continuum. The density vector of the vector system arising along this surface, considered to be an external surface vector system with respect of the continuum, can be expressed by Cauchy’s stress tensor T and the couple stress tensor M. Thus the
equation of motion will be
[ eadV= [ fdV+ [ TdA 14
v"
A"
and
[ (rxaDodV%44
 exf+p)dV+ yt
[rxTdA+ A"
[ MdA. A"
With the surface integrals transformed into volume integrals using the Gauss—Ostro
gradski theorem (see formula (A7—4,) in the Appendix) we have  (ea—f—TV)dV=0
(79)
for the first ecluati?n, while after transformation of the integral and use of the equality
Gx T): V=rx(T 94
V)+rxTV, weget
frx(a—f—T  V)dV+
[ ddV
[ (u+txTV+M)dV
t
1%
4
for the second equation. The first integral on the lefthand side of the equation is zero because of (79), while the symbol  over r on the righthand side means that V (nabla) acts as a differentiating operations upon r only. All this considered, the second equation will be
f (@p—txTVMV)dV=0.
(710)
V*
Eguations (79) and (710) are satisfied for any continuously differentiable field functions, 0a—T
V". Therefore, for the case of
V +f
(711),
and ol=MV T"+pn.
(712),
(711) and (712) are the first and second Cauchy equations of motion, T" in the second Cauchy’s equation of motion, according to (712) is the vector invariant of the Cauchy stress tensor (cf. (A 411)). In indexed form, these equations can be written as follows:
ga* =14 +1*
(711),
for Cauchy’s first equation of motion and
m"
+ €1 +u*
(712),
for Cauchy’s second equation of motion. If 1, n are zero and the continuum is nonpolar, that is,
M is zero as well, then the
vector invariant of T will be zero, that is the Cauchy stress tensor T will be symmetric. Thus Cauchy’s second equation of motion takes the following form: T=
TT
OT
tkl:tlk
.
(7‘13)
If Mis different from zero and the continuum is polar, then Cauchy’s second equation of motion will be
m"
+€Fm g, =0
instead of (713). Nonpolar continua and problems where 1 and p are zero, are discussed below. Let us return to the identity (r x T)  V =txT  V +rx(T V). To show that the identity
is true,
let the
nabla
be,
with
arbitrary
curvilinear
coordinates,
V =
=g" aim (see (A 61) in the Appendix). Thus x
[(rxT) As is well known, 품 ' x
0 ]gmsíxTg'"irx x ox™
£g'". x
=g, (cf. (A 35),). Let the first term on the righthand side 95
of the identity be scalarly multiplied from the left by the unit tensor g"g,;:
[a e0 (rxT)]g'"=gpg,,(gmxgkr'"")ﬂx(rV). Furthermore, after rewriting,
@xT)V=g€, "
+rx(T V),
we find that the identity in question is xT)V=—T+rx(T
V).
Now it is easy to see how Cauchy’s second equation of motion has been derived and
how it has assumed its final form. Sometimes the first and second Piola—Kirchhoff stress tensors are used to express the equations of motions. This way of expression is used, however, in case of nonpolar
bodies only. So far the reference coordinates have been used to write the equations of motion. Let now the spatial coordinates be introduced as well.
Using the first Piola—KirchhofT stress tensor, denoted by S*X, the equations of motion are QOak=SkK:
K+
gﬂfk
o
or Skam’
In the first equation, and (17),):
S
=
Smek’
K :
S. , is the full covariant
=S
kE KSS + (S
G FT
derivative of S** (see (18),
STX
.
With the second Piola—Kirchhoff stress tensor, KX, taken into consideration and using also the expression of the full covariant derivative with respect to the spatial
coordinates X", we get Qoak :(KKka, K), L '*'(Frllexm, le, Lt F%ka, K)KkL + &fk o
for Cauchy’s first equation of motion, while KKL
—
KLM
is Cauchy’s second equation of motion. The above formulae can be derived by simple calculations. This is left to the reader.
Associated with the equations of motion is data related to the state of velocity, state of strain and the position of the condition. We speak also constraints, acting 96
stress determining the initial state of the continuum, as well as to continuum. That data, recorded at given instant, is called initial of boundary conditions reflecting the kinematic consequences of upon the continuum, and the external surface vector system. Thus,
if the surface bounding the continuum is A4, then the velocity and displacement of the surface points may be given over part of A that is for 4, and A4,, respectively. At
the same time, for another part of A, that is for 4,, it is the density p of the external surface vector system which is known. Hence, the boundary conditions are V, =V
(714)
ul, =1l
(715)
Tn, =p
(716)
and
where , dA dA
7.3. Eguations of eguilibrium, stress function tensor The equations of equilibrium are satisfied by the stress tensor field of the continuum at permanent rest. As follows from the Cauchy equations of motion according to
(711), and (713) in case of nonpolar continua, the equations of equilibrium will be
., +f*=0
and
H=r*,
(717)
respectively, if pa*=0. Suppose which
that a particular solution
¢ %I
of the system of equations (717), for
are satisfied, is known.
Now ¥ — %/ satisfied homogeneous equation (tkl__ tOkI); )= 0.
Calculation of such a particular solution :"" will be quite simple if the volume vector system /" has a potential, that is fk —
‘I’, pgpka
where W(x") is the scalar potential and thus A =gkt It is easy to understand that the equation of equilibrium according to (717) will be identically satisfied if a secondorder symmetric stress function tensor //. is in
troduced in the following way: tkl_ tOkl — €krm€lspf;s; mp 
(718)1
On the other hand, if written in invariant form, the stress function tensor will be
F=f,g'g’ and TT°=—V
xFx V.
(718),
Notice that /,, or F is differentiable as many times as required. 97
Note: The stress field generated by (718) will be complete only if the body is
bounded by one single closed surface. In this case, the stress field ¢*/(x)—¢%/(x) is called selfequilibrating. Consider sume that
now
(718) to be a
differential equation
for the field f,,(x) and
as
(719)
Jes=Is+17s
where //. (Xx) is some particular solution of equation (718), and f/;(x) is the general
solution of the homogeneous equation
ELTEPf
—0
(720)
resulting in an identically zero stress field. Equation (720) is analogous with SaintVenant’s compatibility equation 4
ab __
— €
cjakmablp
€
—
€kt mp — 0
which can be written according to (434). The general solution of the latter equation is a tensor field ¢,,(x) which can be generated by an arbitrary vector field (displacement field) u,(x). Accordingly, the general solution of equation (720) can be obtained by means of an arbitrary vector field /,(x) in the following way:
féz S0+l
(721)
Since no stresses result from the field f;;(x) according to (718), and (720), it can be seen from a comparison of (719) and (712) that any component of f,,(x),
(there are altogether three components denoted by subscript a, b) for which equation ) fab +
1 호 (la; b + lb; a)
(7'22)
has a solution (with respect to vector field / (x) for any three functions f7,), can be made egual to zero (according to what has been said in relation to notation in para
1.3, underlined subscripts assume only certain values from among the numbers 1, 2, 3). Hence, without losing generality, three components different from zero, suitably
selected from among the six independent components of f,,, will suffice. Let these components — as stress functions — be denoted by f,.=f, and let f,,=f,, be vanishing components of the stress tensor f,;, selected according to (722). Of course, the pairs ab, ba and rs, sr must fulfil all the possible index pairs of f,,. Note: From a somewhat different point of view, we can also say that any stress
function tensor f,, can be divided into the sums of two tensors from among which one agrees with the symmetric portion of the gradient of a vector field (and hence no stress field is resulting) while the other contains the three stress functions as nonzero components.
Since there are twenty different ways of selecting three elements from among six elements, at most 20 different structures (possibilities of selection of zero and nonzero
elements) of the stress function tensor are possible. 98
In the Cartesian xyz coordinate system, the following 17 ways offer themselves for selection of nonzero components (stress functions), with
Ul =] x e ex
J S [y
J Í [
)
(cf. formulae (471)(477)):
Jrs=FexsSops Jaz
(723)
Jrs=Faps JyzsSox Jys   : XVZ, Jes=FexsSyys
(724) (725)
Srs=FaxsSypss   X2,
(726)
Soys foxs   : XYZ2, Jrs=Fexs
(727)
Says Jozs   XYZ, Jrs=Fexs
(728)
Jrs=FaxsSyzsSoxs   XYZ.
(729)
The scalar relations according to of formulae (454) or, in the Cartesian course, when writing these, the zero twofold covariant derivation, should The stress functions according to
formula (713), can be written by application xyz coordinate system, by formulae (455). Of and nonzero components of f,; as well as the be taken into consideration. (723) are the Maxwell stress functions, while
those according to (724) are the Morera stress functions. In the Cartesian xyz coordinate system we have 0. . Ox—Ox —f;'y, 2z +fzz, o
0 _ Ty T Ty = _fzz, xy )
0. . . ay ". őy —Jzz, xx +fxx, 2z 9
0. . Tyz . Tyz 
_.f;cx, yz 5
9,—
Ux
_f:vy, s
OS :f;cx, yy +f:vy, xx
Tgx —
(7'30)
for the Maxwell stress functions, while 0
— aJ(:) —
Oy  O'}(,) = 0,—
for the Morera
2.fyz, yz ;  2f;x, zx 5
O'? —
2f;cy, xy 5
Txy  r)(r)y =f;iz, zx '*_,fzx, yz _.f;cy, 2z Tyz _T}?z =.f;x, xy +.f;cy, zx —f;:z, xx o Tx
T;?x :f;cy, yz +f:vz, xy —.fzx, yy
(7'3 1)
stress functions.
7.4. Principle of virtual power and virtual work To be specific in advance, before the principle of virtual power and virtual work is discussed, are some definitions. The velocity field will be called kinematically possible if it satisfies the kinematic
boundary conditions. A kinematically possible velocity field ¥ or, with contravariant 99
(or covariant) components, 6, satisfies equation (714). The symmetric portion of the covariant derivative of the kinematically possible velocity field is the kinematically possible deformation rate field. In invariant form, this is
ワュ몰നप十पന while in indexed form, with covariant components, A a
l , A 7 (Be : HŐj; 2) .
(732)
Similarly, the kinematically possible displacement field, ú,, satisfies the geometrical (kinematic) boundary conditions according to (715). Hence, űk l 4, —
űk
.
By a dynamically possible stress tensor field 7/ we understand the stress tensor which, in case of given volume force /" and acceleration a*, satisfies the Cauchy equations of motion, that is, the equations
. +ff—ga*=0
(733)
i= g
(734)
as well as the dynamic boundary condition according to (716), namely
?%Mthk. The
motion
of the continuum
will be called
(735) quasistatic
if, in (733),
oa" is
negligible as compared with :". , and /". In this case, (733) becomes :". ,+f*= while (734) remains unchanged. If also (735) is satisfied, then %' will be called a statically possible stress field. By a virtual velocity field we understand the difference between two arbitrary kinematically possible velocity fields. If the virtual velocity field is denoted by vf, then vF=0P V . (736) Over A,, the virtual velocity is zero, namely,
Ük
and
4, = Ve ,
U/El) 4,= Ui
and the difference between them lA
0.
(737)
Similarly, the difference between two arbitrary stress tensor fields £" is the virtual stress tensor, that is,
100
prkl_ fOk _ ()KL
(738)
The virtual stress tensor satisfies the following equations:
e
=0
(739)
Rkl
tlk
(740)
vE , 0.
(741)
On the basis of the definition of ?" and [t
as well as equations (733), (734),
(735) and (738), it is easy to understand that the equations listed above are correct. Equations (739), (740) and (741) of the virtual stress tensor are correct if, as is usual, the statically possible stress tensor is used to produce these equations.
To abridge writing, let f—pa=q Or f*—gpa*=g". Let the equation of motion according to (711), (using now g" in place of f*—pa*) be multiplied with velocity v, and integrated over the entire range
V of the continuum:
f :". 0. dV+  ¢*v, dV=0 Taking the identity
í
í 0),
o=
—
(742)
p 1
into consideration and transforming the integral  (¢'v,)., dV into a surface integV
ral by use of the GaussOstrogradski theorem, (742) can be rewritten to obtain, after suitable rearrangement,
[
v, dV=
V
[ g*v, dV+  t"v, d4,. V
(743)
A
Equation (743) can be further simplified by writing the velocity gradient v,., as the sum of a symmetric deformation rate tensor v,, and an antisymmetric spin tensor @, (56), (57), (58). Namely, in this case, t*'w,,=0 and thus, on the basis of (743),
j Mo, dV=  g*v, dV+  t*'v, d4, V
V
(744)
A
The principle of virtual power will be obtained if two arbitrary kinematically possible
velocity fields 657 and #{" are selected and used to write (744) as follows:
[ 6
dV= [0aV.  52 da, A
and
j oV
dV = jq 6, ) dV + j 1" 6, d4,,
respectively, the difference of both being _f "R — 6t dV = _í g*
Here 6
>
— 66" ) dV4 f (G. ) — 66 )) d4,.
— 0L — 08 is the virtual deformation rate tensor and 667 —6£) — v the vir
tual velocity. Thus
j tk dV— _fg oFdV + j tt d4,. 101
Taking into consideration that the virtual velocity vf is zero over A4,, the part of surface 4 bounding the continuum where the velocity is given and that, over 4,,
:" dA,= p* d A because of (716), the principle of virtual power can be formulated on the basis of the last equation, as follows:
[
o
dV=
V
[ g*vpdV+ V
 /DvÉdA .
(745)
A4,
The term on the lefthand side of (744) is the power of internal forces. Accordingly, (745) suggests that the virtual power of internal forces is equal to the virtual power of external volume and surface forces.
Equation (744) facilitates also the formulation of the principle of complementary virtual power. With the kinetically possible stress tensor fields 7 and [X" used to write (744), we have j [Pk
AV = j' g v, dV+
V
_í tOE p. dA;,
V
A
OT j
t(l)klkade
V
j
qude+
V
í
t(l)klvk
dAl,
A
the difference between both being I
l*klvk[dV—_—
V
j
t*klvk dA[,
Ay
taking into consideration that **d4,=0 over A,. According to (714), v, =7, over A,. Thus the principle of complementary virtual power can be formulated mathematically as follows: j
V
t*klvkldV:
I
A
t*klűk
dAl.
(7'46)
The above considerations and the transformations presented enable us to arrive
at the principle of virtual work and complementary virtual work as well. Let the eguation of motion (711), be taken as a
starting point. With this eguation multi
plied now by the displacement vector u,, and then integrated over the entire range V of the continuum, the following eguation
[
u dV [ ¢ u, dV=0
V
(747)
4
will be obtained instead of (742). After the transformations described above, (743)
is replaced by _í tkluk; l dV=
j
V
V
qkuk
dV+
j
tkluk
dAl
.
(7'48)
A
The difference between two arbitrary possible fields of displacement gradient u,. ,
1S 47. , (just as it is in case of the two possible displacement vectors, where the difference is the virtual displacement 47, the covariant derivative of which is the virtual displacement gradient). Of course, as in the previous case, ukl =#, and “kl 4,=0. 102
Let the symmetric part of u2 , be denoted by u. On
the basis of what
has been said above, the principle of virtual work can be formulated as
[ Mupdv=
 q*u}dv+  prupd4
V
V
(749)
Ap
because :" dA,=p* dA over A,. Also, the principle of complementary
virtual work can be derived in the way
described in relation to the principle of complementary virtual power, using the displacement vector u, in place of velocity v,. Accordingly, the principle of complementary virtual work is 5
t*kluk,dV:
í
V
t*klűk
dAl
(7'50)
Ay
since the value of displacement vector @, is given over A,, while !" dA,=p*d4 is given over A,, that is, here t*d4,=0,
A being the combination
of surfaces 4,
and A, (the surface bounding the continuum). If a functional distance can be defined
over the field «* or over
¢, and
the
field actually realized has always and neighbourhood that can be made sufficiently small, then the difference of the actual field and the possible field within its neighbourhood shall be considered preferred among the virtual fields, and called a variation of
u, Or t* and denoted by du, or 8¢*, respectively. In this case, (749) becomes
[ 8, dV= [ g*Su,dV+  5*du, dd V
V
(751)
Ap
where ötgy, is the symmetric part of du,., and 8uk; 1 (öuk); l.
Eguation (750) is now j.
uklﬁtkl
dV=
V
í
ukötkl
dA
.
(7'52)
Au
In case of the linearized state of strain, u,,=¢,, is the strain tensor. In this case,
a"" is usually used to denote the stress tensor [" and (751) can be written as j
V
O'klögk[dV:
j. qkSuk
dV+
dV=
űk öÖ'kl
V
fﬁkSuk
4,
dA
(7'53)
while (752) can be written as j
8k150'k’
V
j
dA[
!
(7'54)
Ay
The principle of virtual work can be written also in a different way. Assume as a starting point that the integral of the power over a given period [¢,, #,] is the work
during that period. Thus, with (745) integrated with respect to time ¢, t2
f [, h
v
!2
dvde=
[  gtvédVdtt
v
t2
  D"védAdi. t
(755)
Ap
As can be seen, the virtual deformation rate v} on the lefthand side of the equation has been replaced again by the virtual velocity gradient v} ,. This replacement 103
leaves the twice contracted product #*v¥ unaltered time derivative of the deformation gradient is
since
: kl _ 4k . The
material
and from this,
o
=Xk (XK
(756)
Using the displácement vector u r=R+u
and the derivatives thereof with respect to time, and with respect to X%, written in indexed form, we have Uk
:úk
Thus, (756) becomes
With the volume configuration,
and
vk ú
)'Ck, K:úk,
K
:
kX" , .
(757)
integral on the lefthand side of (755) transcribed
the integrals
with
respect
to time
and
calculated
over
for initial volume
F.
can be interchanged. A similar transformation applied to the righthand side of the equation, using also formula (261), results in t2
j
j‘
Vo
t1
t2
tklú*k,
KXK_
l"TdthO:
j
j
Vo
4
t2
+  [ J:"úsxt
jgkúÉdIdVo"'
drdAg.
(758)
0 ¢ Apl
Consider now the integrals with respect to time, one after the other: 12 j
t2 jtkIXK,
Iﬂ*k,
K
151
dt:
j‘
[(.TtkIXK,
,u*k,
K)‘_(;TtkIXK’
1).u*k’
K]
dt
.
t
The first integral on the righthand side is zero because u"" , is zero at time ¢, and :,, that is t2
t2
[ Tt X5 ik ,dt— — [G 01 , TÍ — to YXE , u" (dt. — (759), n
131
After a similar transformation, the first integral with respect to time on the righthand side of (758) becomes {2
i
[ Jatújdt——  TG +v', ,¢")utdr. t1
(759),
[3]
Here also, the virtual displacement 45—0, at time ¢, and ¢,, a fact utilized in (759),. Finally, after transformation (758), we get £2
of the second integral on the righthand t2
[ IXE júfdt— — [JGPv. , 419 — ". 6
104
t1
) X5 , ukdt .
side of
(759),
With
(759), , 3 substituted into (758) and after proper regrouping,
the principle
of virtual work reads £2
 £ (£ 99"
+ 150, Yu*,, , di=
1 12
{2
+j1
j (q.k+qus;s)u*dedt+
_í J.(p:k'i‘ﬁkvs;s_
V
141
Ap
— t" . n u" , dA dt ,
(760)
where j*=1"'n, and p*=1""n, over A,. Note that the other principles of virtual power and virtual work presented can also be written in the form of an integral over the period ¢,, t,, except for cases where the principles apply to a continuum at permanent rest.
Additional comments: By variation of fields, the literature understands sometimes, in a more general
sense, the difference between a possible field and the other possible fields within its neighbourhood. In this case, the literature uses the same term for virtual fields and variation of fields. It is also important to note that in case of virtual performance and/or virtual work, "" will be dynamically possible if the equation expressing the principle is true for any v2 or for uf. The same applies also to the case of supplementary virtual per
formance or supplementary virtual work. The virtual work principle according to (760) may affect the calculation of finite elastoplastic deformation by means of the finite element method (McMeeking and Rice 1975; Lee 1986). Otherwise, on the basis of what has been said above, the equation given below is
obtained after suitable mathematical transformation, using (711);: t"k
, HTW
k
__ .+ 42k =0,
where £"? is the Truesdell rate of Cauchy’s
stress tensor (the expression in the
parentheses of formula (5141)) and q'k=q°k+qus;s_qsvk;
s"
7.5. The first law of thermodynamics. The ClausiusDuhem ineguality According to the first law of thermodynamics, the time derivative of the kinetic energy T and internal energy U is equal to the power P of the external vector system acting upon the continuum and the power O of other than mechanical impacts:
T+U=P+Q. Considering part of the continuum of volume ¥V, bounded by any arbitrary closed surface 4, and understanding only thermal power by 0, the first law of thermodynamics is 105
( J%gvkvk_dV)
4 (JgudV)':
4
V
§tk’vde,+
JgrdV—
§h"dAk+
Jgkvde,
A
V
A
V
where u
internal energy density
r
thermal power of sources distributed along the volume
g"
volume body force density vector
h* thermal flux vector h*dA, thermal power flowing through surface d4,, negative in case of inflow. With the material differentiation with respect to time carried out, and the integrals transformed into volume integrals, we have j. (Qak—qk—tk’;[)vde+
j
V
4
Qd
dV=
j
(tklvk1+Qr_hk;
k)dV
V
Using the Cauchy equations of motion and taking into consideration that the above
equation applies to any arbitrary V, the local form of the first law of thermodynamics in the reference coordinate system reads gú:tklvk,lgr—hk;
In deriving
the formula,
it has been
k "
(7'61)
taken into consideration
that v,.,=v,,+wy,
and Mo, =t"v,+ " 0, =1"v,, because the stress tensor is symmetric while the spin tensor @, is antisymmetric. t*'v,, in (761) is the specific power of the internal vector system. The magnitude of the power
of the internal vector system
remains
unchanged
regardless
whether quantities determined with reference coordinates are used for calculation (as has been done so far) or whether quantities are determined with spatial coordinates. That is, /
I3 툴
det
(%)
. tklvk, =
KKL
HKL
(7'62)
where K"" is the second Piola—Kirchhoff stress tensor, while H,, is the Lagrangian strain tensor (220).
The stress tensor can be divided into two parts, that is , T characteristic of reversible change in state, and , T — the socalled dissipative part — characteristic of irreversible change in state: =g T,T. (763) g T can be derived from a scalar function a. This function a is called strain energy density or specific strain energy. According to the definition, the relationship between
the strain energy density and ,Tis: &) 106
v=0d
(764)
Should a be a function of x" ,, the above relationship can also be written as N where x"
T ED“=0x
0a_ o
(765)
’Kaxf',,(
is the deformation gradient or, with the Lagrangian strain function, )"
=0
oa
xk, le, L
OHy;
(766)
while with the Cauchy strain function, da
7 " 0c?,
(i =20c{
The reversible part of the second Piola~Kirchhoff stress tensor is da
K) =0 " öHer
(762)
Now, using (763) and (764), the first law of thermodynamics according to (761) can be written as G(Ú—Ú):(Dt)klvkl'*'gr—hk; k "
(768)
Let the product Ts be introduced in place of #—a in formula (768), then the first factor of the product, T, is the absolute temperature, while the second factor, s, is the
specific rate of entropy. s is called specific entropy. With the new quantities introduced, equation (758) becomes (Eringen 1962): oTs=0p0) " vy+or—h* , . S—
(769)
 osdV is the total entropy for which the above equation becomes V
S
!—
1
J—líf(Dt)klvkl'*'
1
797'—
1
!
Fhk]‘,k]dV_
§?hknde’
v
A
the rate of total entropy over volume V of the continuum, bounded by a closed surface A. In deriving that formula, the relation 1 1 —h* T ; ,k — 'L+'(Q
T,
/
)
where, as is quite understood on the basis of (939), the second term on the righthand side is zero, and the identities
os —
0s
—Ar
ÖL öD
¢
í—íL:;—;DT
(940)
are obtained. With this substituted into the condition of equilibrium, we obtain 0s T— —oT eT —:I
941 (941)
L.
With function s(u, L) selected in different ways, we arrive at different cases of hyperelasticity. Thus, e.g. the entropy function sz
[1
o, su.  [LD LI) — 20T
L :L
results in the theory of rubber elasticity. In case of small deformations, introduction of the notation 6= T, £¢=L— I seems
to be most reasonable. In this case, (941) takes the following form
o= ةيبgr睾० (e+1).
(942)
A series expansion of function s(u, L) with respect to the components of L results in the expression
s=S,+a,6—aef —a,(g: €) ,
(943)
where a,, a, and a, are constants independent of &£ With this substituted into (942), relationship
o= —0oT[a,I2a,,12a,¢]
 (¢+1)
is obtained. If we take now into consideration that, as a result of selection of a spatial configuration, 6=0 when £=0, then we will deduce the numeral value a,=0. Since the entropy function according to (943) has been obtained as a seconddegree
partial sum of Taylor’s series, the expression obtained for the stress is reliable, in 141
compliance with the structure of (942), only as far as linearity is concerned. With the terms of higher degrees omitted, Hooke’s law of classic elasticity theory is obtained for the stress tensor o=2uc+ e l , (944) where we have introduced Lamé’s elastic coefficients
A=20Ta,
and
pu=g¢Ta,.
The relationships introduced above apply, in a strict sense, to the state of rest only.
In some practical cases, application of these relationships to a moving medium is also permissible, but we must be aware of the fact that the stress tensor is affected by
dissipative processes taking place in the course of motion. Stresses arising in the course of motion can be determined on the basis of Onsager’s linear laws. For the sake of simplicity, let us restrict ourselves to an isothermal medium. Now, it can be seen from the shape of the energy dissipation function, according to (935), that the thermodynamic currents describing the velocity of the processes are components of the tensor V, while the thermodynamic forces associated with these currents the components of the tensor
[
os
T+oTZ
87 8L
LI.
]
According to Onsagers linear laws, there is a homogeneous linear functional connec
tion between the forces and currents which, expressed for forces and taking also the isotropy of the medium into consideration, can be written directly in the invariant form
T+QT§—2L=217 V4, VI.
(945)
The material coefficients 4 and #, can be derived from the conduction coefficients denoted by £. in the formula of Onsager’s linear laws given in (923) and they indicate the viscosity or volume viscosity of the medium in question, respectively. The constitutive equation (945) describes the behaviour of the viscoelastic medium called
the Kelvin body in rheology. The constitutive equation of viscous liquids will be obtained as a special case of this, provided we assume that the specific entropy of the medium depends on the determinant of tensor L, exclusively. With this latter assump
tion adopted, and (945) substituted into Cauchy’s equation of motion, the Navier— Stokes equations are obtained.
142
10. Constitutive
relations
10.1. Introduction Essentially, equations (73), (711),, (713) and (761) represent altogether 8 scalar equations while in the equations, mass density g, particle velocity v*, stress tensor
t?1, internal energy density u, temperature T and heat flux density vector A" stand for 18 unknown functions of scalar value. With (773) also added to the equations, the number of unknowns increases to 20. Hence, the number of unknown functions exceeds the number of equations, to be written, by at least 11. Accordingly, at least
11 additional equations are required for the mathematical formulation of continuum mechanical and thermodynamical processes. These additional equations are called constitutive relations.
In this work, a group of constitutive relations is called a constitutive equation. Essentially, the constitutive equation expresses the mechanical interaction, that is if the thermodynamic processes are disregarded then, in the knowledge of the appropriate
initial and boundary conditions, the equations describing the constitutive equation will permit the problems in relation to movement of the continuum to be formulated mathematically, using the mass conservation law and the equations of motion. The constitutive relations must comply with (73), (711),, (713), (761) and (773), they must not infringe the law described by any of these equations. This
restriction offers a range still wide enough to permit the constitutive relations and within this, the constitutive equation, to be written. Hereinafter we deal with the constitutive equation alone. Before the constitutive
equation is written, two questions in particular shall be answered. In the first place, what variables are contained in the constitutive equation while in the second place, with what consideration can the shape of functions or functionals constituting the constitutive equation be selected? All the observations and experiments required to determine the constitutive equation for a given material can take place only with the knowledge of answers to the above question. Any real material has mechanical properties of a wide variety. It is impractical to take each of these properties into consideration in the constitutive equation. Therefore, only some properties of the material, most typical of given motion, are selected and the continuum is considered to have these properties only. The material
of a continuum with properties thus reduced is called an ideal material, while the continuum itself is called a body. The constitutive equation is the equation continuum with reduced properties.
of a
Often we speak of a homogeneous body. This means that the same constitutive equation applies to every point of the continuum. 143
A body will be called isotropic if the constitutive equation is not connected to a privileged direction at any point of the continuum that is the material shows identical mechanical properties in every direction.
The variables occurring in the constitutive equation are usually selected from among the socalled physically well as of the reference system. sidered to be the initial state in different bodies, be determined. constitutive equation assume a
objective quantities, independent of the observer as However, it is necessary that a definite state, conthe formulation of the constituve equations for the This is the state where the functions appearing in the definite constant value at every point of the con
tinuum, that is the field determined by the functions is homogeneous. The value of the functions is usually zero or, in case of tensors, constant and proportional to the metric tensor. Thus, e.g. the velocity field v* is zero, the temperature field T, is constant, the stress field is a metric tensor multiplied by a constant and the deformation tensor
is zero. This state is considered to be the natural state of the body. It is not that the constitutive equation cannot be described merely by mechanical. variables in the strict sense of the word even in the simplest case. Introduction of at least 2 but at most 4 additional variables is required even in for a continuum
undergoing small deformation (Béda 1989). This means that the group of constitutive relations that can be called constitutive equation can be selected only approximately
and sometimes rather arbitrarily.
10.2. How to set up the constitutive equation Different theories of setting up the constitutive equation are found in the literature. A brief description of the most important considerations serving as a basis for the
most typical theories is given below. Most of the theories are based on the first (761) and second (773) laws of thermodynamics.
Derivation of the constitutive equation and constitutive relation
based on (761) and (773), respectively, has been shown in the last section of Chapter 9. As a starting point, that the entropy density s has been assumed to be a function of the internal energy
density
4 and
the deformation
gradient
D. With
entropy
production density o, transformed using the first law of thermodynamics, the equations desired have been obtained as the conditions of thermodynamic equilibrium. Considerations based on the laws of thermodynamics assume a definite formula for the second law, and also that the entropy density s is a function of either the internal energy density u, or of the free energy density H=u—Ts, or of the free enthalpy density g=u—Ts— LKP ?
Hpo, as well as a function of suitably selected
Qo
variables describing the mechanical motion. If the suitably selected thermodynamic and mechanical variables are independent, the constitutive equation and/or the constitutive relations can be written uniquely. In a more general sense, the constitutive
relations can be included in suitably selected dissipation potentials from which, after mathematical formulation of Gyarmati’s principle, the constitutive relations, and 144
moreover also the parabolic partial differential equations of the irreversible transport
processes, are obtained. Ziegler’s more special principle of maximum dissipation, and the GlandsdorflPrigogine local potential method, which apply, however, only to the case of constitutive relation of linear constant coefficient are also widely used (Lebon 1971; Wisnewski—Stanisewski—Symanik 1977). According to another consideration, axioms, called ColemanTruesdellNoll
axioms can be set up to formulate the constitutive equation. With the number of axioms increased, not only the constitutive equation but also the constitutive relations can be set up. The assumption taken as a starting point is that stress tensor T, heat flux density h, internal energy density 4 and entropy density s are constitutive functions or functionals of the deformation gradient D, temperature 7 and temperature gradient V 7. That is, the constitutive functions are functions or functionals
of the same variables, provided this is not inconsistent with the equations of motion and the laws of thermodynamics. If these functions or functionals are in agreement with what applies to the fields in general, also in case of a homogeneous strain field or temperature field, then we will speak of a simple material. A simple material is usually the point in the formulation of the constitutive equation, on the basis of the
consideration discussed now. From among the axioms, discussed below, are only those of fundamental importance with respect to formulation of the constitutive equation. In doing so, mathematical expressions are used which are more general than those used so far, in that also functionals are used in addition to functions. Functions
and functionals in combination are denoted by scripttype capital letters. Included among independent variables are also reference coordinate X" and time ¢, that is
the body is considered to be inhomogeneous and rheonomic. For the sake of invariant representation, particle X designated by the coordinates is used below in place of
coordinates X". Let x (X, ) be the function of motion of particle X of the continuum. According to the first axiom, the stress field T is determined by the function of motion (X, 2). The stress field T depends not only on the instantaneous value of x but also on the
quantity x(X, 1) — oo will be zero on the base plane, coordinate line x* will be normal the base plane and coordinates
x" themselves
will correspond
to the distance
(+
or
—)
measured
from the base plane. In a coordinate system like this, the matrix of the metric tensor is
911 912 0 [9
g 9
— [ 492 922 0[;
0
[9"]—)]97
01
0
0
97
0
(11216)
01
and the Christoffel symbols of the second kind (para 3.2 in the Appendix) are zero: r,; 0;
T, —0.
(11217)
The body is assumed to be isotropic and linearly elastic. 11.6.1. Plane strain
Plane strain is the case where the following conditions apply to the displacement
vector t:
u,=0
and
u, ,=0.
(11218)
In case of plane strain, the sections of the body which are parallel to .S experience identical deformation, the verticals normal to S are displaced as rigid lines, and the distance between the planes parallel to S remains unchanged. The nonzero displacement coordinates depend on position coordinates x', x* only: u,=u,(x", x?). Coordinates and the first scalar invariant of the strain tensor are
pZX X) = %(’ua;ﬁuﬁ; ;
=t =0,
(11219) 221
and ,
glzgkleklzgapgap:gapua; p=E1s >
(11220)
respectively. Subscript S stands for the socalled planar invariant, that is part of the entire invariant ¢ , formed of strain components of subscript 1, 2, called planar components. (In case of plane strain, ¢ =¢g.) The components and the first scalar invariant of the stress tensor are obtained on
the basis of Hooke’s law: aaﬂ = 2G
—— 2
(11229)
6" .+ fP=0.
(11230)
According to (1128), the strain energy density is 1 Uu=
a
Eaaﬂgaﬂ=G[8
v ﬂsaﬁ'i‘
É(SIS)Z].
(11'231)
The Navier equation can be written on the 'basis of (11 47) as
ua
g t S0.
(11232)
Associated with the dual system of plane strain, as a definition of plane strain, the conditions are
Eup=E,5(x", X) 5
Ea=65,=0.
(11233)
The forthcoming applies to elastostatic problems only. Let the stress functions f,, (as configuration variables) be selected for the dual system as follows: .fll(xla
x2),
f22(x19
x2),
fEB(xla
x2)=U(x19
x2)'
The other nonstress function components of the stress function tensor
f,; are a
priori zero: f;,=/,3=/f3; =0. For the twodimensional stress components satisfying the equilibrium equation (11245) on the basis of (11122), we obtain that
0
LEE
Y,
(11234)
In the formula, 0""? can be simply produced as a particular solution of the eguilibrium equation (11230) if the volume force system /" is conservative. Namely, if
Y — Y(x!, x?) is the potential of the force system, ff=¥
,9*
and
"
=Y
g*.
(11235)
For the other stress components, O.u3=o.3a=0
and
0.33=€3xp€3ﬂvf;tﬂ;
u3
are obtained from (11122). The stress function f;, and f,, occur in 07 only. Since 223
o can be determined also by means of (11223), a simple formula indeed, the stress functions f,, and A, are not needed for plane strain problems. The twodimensional constitutive equation corresponding to (10108) is given by equation (11224). Taking also (11234) and (11235) into consideration, the relation 9 — É
1
— É
(Gys — VO1s gyS) —
(gyagsp — Vgapgys)ea
x
5E
U. a F (1 — 2) Y 9,s
(11236)
can be written. As follows from assumptions (11233), the following components from among the components of the incompatibility tensor 4" are identically zero: 4" — 4 — 912 — Z4Őő—y?!—0. Hence, the only nonidentically zero compatibility field equation according to (11109) is obtained for rs=33: n =€,
,,=0.
(11237)
Finally, with (11236), (11237) substituted and identities of the type gaﬂeaﬁeﬂ,w:gx).
(11238)
as well as the interchangeability of the secondorder covariant differentiation taken
into consideration, the following field equation is obtained for the stress function U(x', x*) as a basic variable of the dual system: gulguv
U;xluv
+
]1_
2v
gﬂv
T;uv
:0,
(l
1_239)1
—v or, using an invariant notation,
44U+ 1—
—v
qw 0.
(11239),
In the case without load on volume the biharmonic function U(x!, x?) satisfying the equation
A44U=0 is the socalled Airy stress function. Under conditions (11233), equations (11234), (11236), (11237) are twodimen
sional field equations for stress function U(x!, x?) as a basic variable of the dual system in elastostatic plane strain problems. Since, in case of plane strain, all the six SaintVenant compatibility equations and
thus also the compatibility boundary condition are satisfied, the solution of the differential equation (11239) generates a compatible strain field. The solution of (11239) is obtained as the sum of an arbitrary biharmonic function (see para 11.5.1) and a particular solution. The boundary conditions in the primal system are ua:űa;
224
X € Gous
naaaB:őB;
X€YJos>
(11_240)
Fig. 11.12. Notation for calculation of the stress function along the boundary curve
where #, is the prescribed displacement for section g,, of the boundary curve, ő " is the prescribed traction for section g,, of the boundary curve and g, +go, =9,In a dual system,
n,o¥=n€€MY
+¥nf=¢*;,
xeg,
(11241)
if the surface load ő"? is prescribed for the boundary curve g, along full length and the volume force system is conservative. 
On the basis of (11241), the gradient of U and U itself can be determined along the boundary curve. Let s be the arc length measured along the boundary curve according to Fig. 11.12, let be dR t:t"gll:
百
(11242)1
the tangential unit vector of the curve and let be n=txg;;
n,=€,,"
(11242),
the normal unit vector directed out of domain S, according to what has been said so far. With n, substituted into (11241),
(€U ), 1= ad;(eﬁ” U,)=6—¥n’
(11243)
is obtained along the boundary curve after reduction. Integration from the zero point of the curve results in the gradient of U: U,=U,)+
Öí'CW(őB—TnB) ds,
(11244),
“or, using the invariant notation, VU=(VU),+e;x
É(ö—'l'n)ds.
(11244),
The derivative of U along the curve: dU =t (V)= (V U)o—e, [tx j(ö—'l'n) ds !.
225
Then, after integration, U along the boundary curve is obtained as
U=U,+(R—R,) : (VU),—e, [R x [ (&—¥n)ds— [ Rx(&— ¥n) ds], (11245) 0
0
also with respect to (11244). According to (11244), V U will be single valued along the boundary curve g, if
$ (6—¥n)ds=0,
(11246),
go
and U will be single valued if, in addition to (11246),,
$ Rx (66— ¥n)ds=0.
(11246),
go
Equations identical with (11244), (11246), (11247) can be written for every boundary curve if S is multiply connected. The resultant of the volume force system can be calculated on the basis of (11235).
In case of a simply connected twodimensional domain S, j fdA—
[ VYdA—
$ Pnds.
S
S
90
(11247)
Hence, a surface force system o on the lateral boundary, and volume force system f, acting upon the body in combination result in an equilibrium system, an evidence expressed by (11246), ,.
Note. Both the primal system and the dual system form a twodimensional boundary value problem for the configuration variables and plane variables resulting from them. The other variables of the threedimensional continuummechanical problem result either from the defining condition (#;=0 or ¢g,;=0) or can be calculated separately (e.g. 0,3—0). From among the latter variables, the relation 033 = V015 =Vg,56™*
(11248)
is most remarkable.
11.6.2. State of plane stress We speak of a state of plane stress if the following condition applies to stress tensor Op
Gy =03, =0.
(11249)
In the state of plane stress, the surfaces of the body parallel to S are stressless and, according to Hooke’s law, N
1 É
£3=83,=0;
226
(aaB—
v I——Iv aISgaB) ; ]
v
€3— — 5G mőls,
(11250)
where 013—9"0,4—0;. The
relationship
between
component
(11251)
€37;
and
the
first
scalar
invariant
€) = €15 t €33 1S obtained from the equation
2G
033 — €33 t
y
€r — €33 T
1—2v
v
1—2v
CErs H 3) —0
according to Hooke’s law v
v
€33 —
1—v
€s — — — — £ ] — 2v
..
(11252)
The inverse of Hooke’s law: v ' E 0.4—2G = 1—v2 K! — V)&t veISgaﬂ)] .
(11253)
In the Cartesian xyz coordinate system: E 0,— Ú(exlvey);
E 0, — v
(ve, t €, ).
(11254)
The eguilibrium eguations are, in compliance with (11249),
o, +ff=0;, f7—0.
(11255)
With the stress functions / =f,,=0; 733 — U and coordinates f,,=f,; =13, =0, other than the stress function, of the stress tensor,
¢ —g™=€EMY,,
M =¢%=0.
(11256)
Thus the equilibrium equation (11255), and condition (11249) are satisfied. ¢°** is the same as in (11235). On the lateral boundary A" of the body, a surface force system m 0" — 6! is acting .
Also, the compatibility equation #**=0 will be satisfied if
g"g" U, g H(1—v)19" ¥,,,=0,
(11257),
or, using invariant notation,
AAU+(1—v)4¥P =0
(11257),
(¥ being the potential of the volume force system while 4 is a twodimensional differential operator in this particular case).
The other compatibility equation will not be satisfied if the state of plane stress is considered to be a (twodimensional) problem depending on (x', x?) coordinates only. This will be quite understood if the strain components
(11250) assumed
to
depend on x, y only, e.g. in the Cartesian xyz coordinate system are substituted into SaintVenant compatibility equations (455). In this case, the equation 0%, ox*>
0%, _ 0y®>
0%, _ 0xdy 227
should be satisfied, that is as follows from (11250), ¢, + 0, =05 could be at best a linear function of coordinates x, y. Apart from some exceptional cases, this latter
condition never exists, that is the compatibility conditions cannot be satisfied in case of a state of plane stress independent of coordinate x. However, it is possible to prove that all the SaintVenant compatibility equations
will be satisfied with stress function f;;=U*(x', x%, x") depending on all three position coordinates, if S is the middle plane of the body and f=0:
US(XX, x) = U, XI— L1 —— ()2 4U(x", x?),
(11258),
AAU(x', X)— 0.
(11258),
2
where
1+v

To make sure that this is true, let, e.g. in the Cartesian xyz coordinate system, stresses 0,, 0,, T,, be produced
by stress function (11258), then substituted into
the Beltrami—Michell compatibility equations (11106). If the thickness 25 of the body in direction x? is small, the term containing x? in (11258) will be negligible as compared with the first term. Therefore, the state of plane stressof thin bodies is usually treated as a twodimensional problem. However, the solution obtained in this way is an approximate solution, a fact to be remembered.
11.6.3. Generalized state of plane stress We speak of a generalized state of plane stress if the following conditions apply to
stress tensor 0(X" , x%, x"), using the notation according to Fig. 11.11 (S in the middle plane): O'k3=0'3k=0;
033:0;
aaﬂ(—x3) :Úap(x3)§
xEA,,
A”;
OBa(_xa):
(11'259)
—0'3a,(X3);
xeV.
(11'260)
In the generalized state of plane stress, the peripheral boundaries A" and A" are unloaded, o3, is zero throughout the body, the components 0,4 are even functions while
the
components
g4,
are
odd
functions
of x?.
The
surface
force
system
n, 67 =g is acting upon the lateral boundary A" while the force system of density f* is acting upon volume V. At the same time, the following conditions must exist: G (—x*)=6%(x%;
(=x)=f(%);
&*(—x*)=6%(x%,
(=x)=().
"
(11261)
(11262)
A thin body of thickness 22 in the generalized state of plane stress is called a disk
(or plate strained in its own plane), while the appropriate boundary value problems are called disk problems. As follows from the definition, the threedimensional problem of the generalized state of plane stress can be reduced to a twodimensional problem, using the overlined average value, related to the thickness of the body, of all the quantities included,
integrating all equations across the thickness of the body and dividing by 22. 228
Taking also conditions (11259)—(11262) into consideration, the average values (means of integrals) are b
, 1 %=5Jqﬂﬁ
b
. ss o71
_ [/d9ad 03 F=0y,=0,
b
(11263
b
.
b
A
€=
5
.
J E dXx7;
A1
iy= 2p
—b
b
J u, dx" ,
(11264),,,
—b
7é6 — —1 Jn[ 42 =2
x g3
( 11265 )
b
fB:ZI_bedea; 720,
(11266)
b
=L 2b [s0dx®; 67=0.
(11267)
b
Taking (11263) into consideration, we get _
l
€B—
/.
v
E;(oaﬂ_
oL
I—_Halsgaű);
1
£3=8;,=0; a3 = &34
v
3— — 2G — —0ys, 33 l_*_vo'ls
( 11268 )
from Hooke’s law, where 6IS=gaﬁ&aﬂ=&I'
(11'269)
The kinematic equations are now modified, as follows:
9 2Gh . 1
Ea3=
1
(11270)
+b
E(I:'Z_bua]_b
1
"'üB;a);
533:
Since £,;=0 and u,(b)=u,(—b), thus 4. —const.—0. = —14(b) results in b&,; =u,(b).
The equilibrium equation with optional index
+b
l:guíiíl_b
(11'271)
Similarly, condition u;(—b)=
=1, 2 becomes
G+ f=0
(11272)
with respect to (11259), while the equilibrium equation with optional index b=3
is
identically satisfied because of conditions (11260) and (11262).
It
ff=¥ ,9
and
¢ =yg* 229
are the volume force system, resulting from potential ¥ (x', x?), and the particular solution of the equilibrium equation associated with it, than O.—aﬁ=€axﬁ€,813
U;x).+
'I/gaﬁ
will be the general solution of the equilibrium equation (11272), where
U(x!, x?)
is a stress function. To produce the average value of the incompatibility coordinates 4", let formulae (454) and conditions (11260), as well as the relations
eup(—X%) =,p(x%); Bz4(— X) = —834(X") 5
233(—X)" —833(X7) ; £3,(+b) —834(—b) =0
based on Hooke’s law, be used. We get
_
1 Oe,,
[*2
llzgízi
97
_
_
+ €z33; 22 ;
_
1 ősy
[9.
gnzzzgíl;
_
F €33; 115
=81, 22 H82; 11 — 2812; 125
_
1 ően
98 ==&y 1n— —b őx —
[??
.
_
5 gi”=g7'=0.
(11273)
Speaking of the equations 7°°=0 (complying with the SaintVenant compatibility conditions), we can say that equation for 477 must be true for the compatibility of the strain components of average value. While limits are set to, or information is supplied about the distribution of the strain components ¢,, , &, and g;, (or stress components 0,4) along the thickness. Or, to be more exact, the values of the derivatives of the components with respect to x> along the peripheral boundaries,
from the equations 77'' =#72*=#'2=0. Equations
47 —72?! —0 are identically satis
fied. Accordingly, the twodimensional field equations in elastostatic problems representing the generalized state of stress of the bodies under conditions (11259)—(11262)
are
1 €8— E(üa; B F Üp; a) ; 2a
_
v
.
¢ =2G (3a5+ Eﬁs%s)
E
= Ú[(l
(11274), —
—
—V)&,5+ Vérs9 pl ;
0 , tHf0
(11274),
(11274),
for the configuration variable #,(x', x?) in the primal system, while
(11275),
öFEBEB Y aT Pg",
Lőlsgys) 85— i(őys— 14v 2G : %[(gwgﬂs— %Hgaﬂgw) EE 230
4 — 'ngs],
(11275),
7l EE s
=0
(11275),
for the configuration variable U(x', x?) in dual system. Substituting (11275), into (11275), results in the following equation stress function U(x', x*) as a configuration variable of the dual system:
99" U, i+ (1 —V)g"' Y. ,,, =0,
for the
(11276),
or, using invariant notation, AAU+
(1—v)9"7 Y. ,,=0.
(11276),
Notes 1. Irrespective of the difference between the average value and actual value, the
twodimensional field equation of the generalized state of stress will be obtained in v both primal and dual system if is written in place of v in the field equations of
l+v plane strain (G remaining unchanged) and vice versa. The field equations of plane
strain will be obtained from the field equations of the generalized state of plane stress if
is written in place of v (G remaining unchanged). Because of this considerable —V similarity, it is enough to find solutions of one state only, with the solution of the other state resulting from this solution accordingly.
2. Irrespective of the difference between the average value and actual value again, the twodimensional field equations of the dual system of the state of plane stress and generalized state of plane stress are identical. However, the solution of the equation
system is only an approximate solution of the problem in the first case (for the actual values in case of the state of plane stress), while the solution is exact in the second case (for average values in case of the generalized state of plane stress). 3. The general solution U(x', x?) of equations (11239) and (11276) in the dual system consists of a biharmonic function as a solution of the homogeneous equation and a particular solution. The particular solution resulting from a volume
force system f can be lumped together with the particular solution ¢°#(x', x?) for the stress field. Of course, the boundary conditions specified shall be satisfied by the
sum of both solutions. However, the solution of the homogeneous equation can be considered
as a solution satisfying modified
boundary
conditions.
And
since the
particular solution associated with the force system can usually be produced quite simply and accordingly, the boundary conditions can be modified, the particular solution is usually considered to be known or the volume force system to be negligible
in the problems.
231
12. Plastic bodies
The behaviour of a body experiencing deformation under load, but not recovering its original shape and size after removal of the load, is called plasticity. To be more exact, we say that the body has undergone plastic strain under load. Therefore, the plastic
strain experienced can be identified only subsequently, after the load has been removed. It is therefore difficult to write the fundamental equations of a plastic body since the constitutive equation depends on whether the body undergoes elastic or plastic deformation or, possibly, deformation of a different type. To cope with this
problem, assume that there exists a condition of plastic strain. If this condition is satisfied, the body will experience plastic strain. This condition is usually formulated
for the state of stress. In the knowledge of the yield condition, it is possible to detect early plastic deformation in the course of the application of the load. More fundamental equations of the theory of plasticity are known depending on the type of equation used as a constitutive equation. Of course, by “constitutive equation” we understand now also the selected formula from among the possible formulae of yield conditions.
The usual fundamental equations of the theory of plasticity assuming that the body undergoes small deformations are described below. The problems of the theory of plasticity usually imply also the problem of relief. In case of a moving plastic body, loading and relief shall be investigated locally in the interior of the body. Relief (unloading) is considered to be a deformation process
where the portion of the continuum involved in relief behaves as an elastic body. Hereinafter the body is assumed to be homogeneous, relief, also scleronomous.
isotropic and, in case of
12.1. Yield conditions The yield condition (1032), is, in case of an isotropic body, a function of the scalar invariants of the stress tensor. Since the principal stresses are invariant scalars themselves, it is possible that only the three principal stresses are interlinked by the
yield condition. Experience shows that in hydrostatic fields, the body undergoes elastic deformation even under a very high pressure. Taking this into consideration,
the stress deviator can be used as a variable of the yield condition in place of the stress tensor. Hence, S(s15 S) 232
=0
(121)
or
S(s15 82, 53) =0
(122)
can be written instead of (1032), if the stress deviator s is used in place of the stress tensor 0.
Function / always implies plasticity threshold k depending on whether K is written or not. Recall the definition of the stress deviator: S=0—
—I‘O'II,
3
where s, , 5, and s, in (122) are the eigenvalues of the stress deviator.
In (121),
5;=0 according to (A540) and thus it has not been indicated. The simplest yield condition of type (121) is the Mises yield condition according to which
—sy—k*=0.
(123)
Here, s/, is the second scalar invariant of the stress deviator, that is
1 SII —
ESMSM
.
Accordingly, (123) can be written in detail as %qus”"—k2=0.
(124)
Let the formula
T — %s
51
(125)
be introduced for the intensity of tangential stresses 7. Now the Mises yield condition
becomes T:
k
"
(12'6)
where k is a material characteristic equal to the yield stress te in the crosssection
of a thinwalled pipe in case of free and pure twist. According to the Mises yield condition, the yield stress 7z is closely related to the yield 0, found
for the same
material in the tensile test, or, to be more precise, x/3— Tp =0F .
On the analogy with the intensity of the tangential stress 7, let also the intensity of angular changes T be introduced: r=2
/%epqe""
,
(127)
1 ! ) ! ; ! where e,, —€,, — — €19 1S the strain deviator derivable from the strain tensor ¢, . 3 233
In addition to the intensity of angular changes, other typical parameters are also used in some cases, like the Odqvist parameter denoted by g, calculated as
E
(pe)" a
Of J2dGe)",d(e)," ,
(128)
where (,e)?, is the plastic deformation part of the strain deviator. Another yield condition of type (122) is the Tresca—Mohr—Guest yield condition, §1—8=0,—03=0¢F.
(12'9)
In fact, the formula of the Tresca—Mohr—Guest yield condition is max(lcl—0'3l,
On
the basis of (1210),
193—03]1,
0'1_0 05,4
(1252)
exists. This means that if load is applied to the body, ds,, and the normal of to s ps the yield surface will include acute angle. That is, the projection of the normal, points outwards from the fivedimensional space bounded by the yield surface, on the vector is positive. A. A. Iliushin (see para 10.2) uses the following equation as a comprehensive formula for the theories of plasticity on the basis of his isotropity postulate:
do,=Nde,—(N—P) 219
0,0! 7
n
& ,
(1253)
where a, 7, 4—1,2, . . ., 5; N and P are functions of definite variables, namely, of variable s or p,. The variable s has been defined in para 10.2, as follows: [1
s= [
/6" dt, (a—1,2,...,5)
(1254)
to
241
and the definition of p, is e" de, D=
!
(1255)
e"e, ds
According to Iliushin, another possible formula for the constitutive equation n
de,= — dő,  (l — l)í'ld—őaa,
/
N
P
N/
(1256)
o,
where «, y, #=1,2, ...,5; and N and P being a function of S and p,, respectively. The differential of S is
(ds)? — do, do"
(1257)
p,= 99. — ,
(1258)
and 0,0!? ds There exists the proportionality (1219), already mentioned between the first scalar invariants of the stress tensor and strain tensor in both cases. In case of formulae (1253) and (1256), the theory of elastoplastic deformation can be formulated after suitable reduction as well as calculation of the differential of oo,
0,—
e,
e'e, Accordingly,
N= Fle,eh)
and
P=F(e,e")
e,e! in both cases. The PrandtlReuss equation will be obtained from (1256) if N=2G
and
P= ﬂ, 3G+2H
where
H
:
F(leae)
,
3 26 @) Or do H— — , d pe
where
o=
/—0,0"
[3. e" e= , ge
The differential of pe, is d
242
(Pe)a
=de,— €,
do — . 2G
a
and
The equations of Prager’s theory of plastic flow will be obtained if N=2G
and
P=
;
2G
.
\/;ag(o)+l Thus the Prager constitutive equation, e.g. equation of type (1235), is
do, =26 dea—[l—
!
og(o)+1]
n
]'ímde " 60,0"
A9
(1259)
The function g(o) can be determined on the basis of the tensile diagram by writing (1259) for the case of uniaxial tension (Iliushin 1963). It is quite easy to rewrite (1253) and (1256), using the usual notation, if we write
stress deviator s,, in place of the stress vector g,, the strain deviator e,, in place of strain vector e,, the quantity proportional to the intensity of tangential stress \/5 T in place of o, and the quantity proportional to the intensity of angular changes \/5 r
in place of e. It is possible to write the fundamental equations in such a way that the equations
of motion and the geometrical equations are completed with the constitutive equation based on the socalled encochrone theory. Only the mechanical effects are taken into consideration also in this case. The endochrone theory (Valanis 1971) can be formulated after introduction of some symbols. Let
dé? — P, dép; deg,
(1260)
where P, is positive definite and, in case of an isotropic homogeneous body, P, . =K 64,05+ where K, + %Kzgo
and
K, (0404 + 01 0yp)
(1261)
£, 20, but the equality cannot exist in both cases simul
taneously, K, and K, being material constants. Furthermore, let the symbols ¢ and z be introduced. parameters, then
dí —aa dő?4 p*de?,
If x and
f are material
(1262)
where t is the time and { can be calculated from (1260). Finally, z is some function of €, that is z=2z({) for which the condition
dfl—(CC) >0,
(e€(0, o)
(1263)
is satisfied. All this considered, the constitutive equations of the endochrone theory are
Spg=2 Jy(z—z') %ez%idz'
(1264)
20
243
and
o= J K(Gz—z) 0 — az .
(1265)
0z
If the body under hydrostatic pressure remains elastic, then K(z) is constant and the first scalar invariant o; of the stress tensor is proportional to the first scalar invariant ¢ of the strain tensor in the way already described. In this case, the relation 0'1 =
KSI
(12‘66)
can be written in place of (1265). Let
W(z)=poe". Thus, (1264) becomes
S9 = 2Ho J e ="
den(Z).
(1267)
The differential equation
de, = %5 Ppa
2[10
p4
dz+ ——ds 2,”0
(1268)
pa
follows directly from (1267). The first and the second terms on the righthand side of the equation are, according to the conception adopted so far, the plastic part and the elastic part of the strain deviator, respectively. Next thing to do is to select function z({), of course with (1263) kept in mind. z may be a linear function of { but, alternatively, the differential equation
d¢
. 1bi or in a more general equation,
Wr4 F(Ő
may exist (Valanis 1971). The eguations described here and the constitutive equations recommended additionally in the literature can be written by means of a fourthorder tensor 27 (Szabó 1988, 1989). If dois the increment of the stress tensor while deis the increment of the strain tensor, the comprehensive formula will be
de=2%:de, where, in case of an isotropic homogeneous body,
97 = —ZíílKII—ZG( 1+b
244
a
l+a
.
5 )í
1+b/s:s
with T =I,— —l—II 3 when I, is the fourthorder unit tensor. Here, G is the modulus of elasticity for shear while K the volumetric modulus of
elasticity. An appropriate selection of a and b permits the constitutive equations already mentioned to be derived (Szabó 1988). Completed with a term C d¢, the constitutive equation written above contains also additional equations recommended in the literature (Szabó 1988, 1990). With the acceleration wave assumed equation F,( )=0 satisfies the equation
OF,
50%
0F,
0
s
to exist (see para
10.2), the constitutive
(o, 7, 9=1,...,6)
— and (5,4=1,2,3,9)
(Béda 1987) which, if taken as a starting point, permits the correctness and generality
of the comprehensive incremental constitutive equation to be proved (Béda 1988), (Béda and Szabó 1990). Accordingly, the comprehensive constitutive equation recommended by Szabó can be written in a general case, as follows:
Ly, do" 4de, — B, dt . 12.3. Extreme value theorems of plasticity Before formulation of the extreme value theorems, let first of all the performance of the internal forces be calculated. As has been seen in para 7.4, the power density of
the internal forces is defined in a general sense by the expression :"?v,,. Staying with linearized strain we write 0"?v,, =07%¢,, for this. Considering now the stress deviator 1 s s"" and the straini deviator i €, e,,, we have
oté
1 , l , (s"” + 3 a,g"") (em + 3 s,gpq>.
After multiplication and reduction, the power density of the internal forces is
P18,
$" épy F O1€) .
(1269)
Multiplying (1269) by d¢, dw — s"? de, , + 0, dg
(1270)
which is the density of the elementary work of internal forces. With the appropriate notation used in (745), writing expression 0¢,, — dé,, + %öál 9pa in place of the virtual deformation rate, a variation of ¢, , pq> the variational principle for the velocity
field will be
f a*18é,,dV=  7főv. dV  p*bv,d4 V
V
Ap
245
that is
f s%86,dV+  0,86 dV= [ f*dv, dV+ [ p*dv,dA. 4
4
V
(1271)
4,
After rewriting of (746), with the variation of the stress field, 60"" written in place of the complementary variational principle, we get after suitable rearrangement the variational principle
f 6,055 AV + J érő0, dV— J 5, L+ (L — ÉLIH)Izo. However, if a symmetric secondorder tensor has three different eigenvalues, the characteristic equation of the tensor will be a cubic equation unless the coefficients of the above equation are zero. Accordingly,
s
=0,
D
_
s
=0
and eT
ős 0Ly,
which results in the relationship
55 . 2 ov 254
T
.
that is well known in thermodynamics considering that 1; is proportional to the specific volume 27. In the state of local equilibrium, the formula
s=s(u, 2) can be written for the specific entropy of fluids and gases. This means at the same time that in the description of deformations of fluids and gases, all the spatial configurations associated with the same specific volume are equivalent. In the case of flowing fluids, the classic literature of hydrodynamics knows only ideal fluids and viscous fluids (Newtonian fluid) for the most part, whereas fluids of
quite a number are used in practice the behaviour of which cannot be formulated by the classic framework. Materials of such a behaviour are called collectively nonNewtonian liquids.
14.2. Simple fluids Let now the state of stress developing in flowing fluids be studied. One of the most efficient models from among the known models is the simple fluid.
In general, a medium is called simple if the value of the stress tensor is determined by the history
of the deformation
rate
tensor.
This
means
that
the state
of stress
developing in simple fluids is determined by what has happened at a given point before given instant. Hence, for determination of the stress tensor in some simple fluid at a time ¢, the knowledge of the value of the deformation rate tensor at every preceding
instant ¢’ is necessary. We can say that the instantaneous state of stress depends on function D(t") defined over interval t £:. This relationship expressed by a functional can be formulated briefly, as follows:
T()= 7 {D(1)}.
(142)
Here we focus first of all on the consequences of deformation, leaving other processes, like e.g. heat conduction, out of consideration. However, the above relationship will
become absolutely correct if we restrict ourselves to investigation of, e.g., isothermal processes. — The actual structure of functional (142) cannot be given on the whole, as it requires that the general model be further particularized. However, some general properties of the function permit considerable consequences to be drawn. Consider first the state of rest. If function D(r ) is invariable over interval £" < ¢, then (142) must approach Pascal’s law:
T {D(t)}=—p@)L
(143)
rát
Of course, it is a requirement that the state of stress according to (142) be independent of the subjective elements of description of the motion, in particular, of the selected spatial configuration and of the coordinate system used. Use of the in
dependence of selection of the spatial configuration requires caution since the specific 255
volume of the fluid has a real physical meaning; only spatial configurations associated
with the same specific volume are equivalent. With a view to avoid complications, hereinafter we restrict ourselves to investigation of incompressible fluids. On the basis of equivalent spatial configurations, there exists the relationship
T {D()}= T {D() U} for any tensor U of determinant
(144)
+ 1 or —1. Equation (144) is independent of the
coordinate system, that is
7 {Q() : D( ) )— Q) : Z. {D()} 07(0),
(145)
an equality that must exist for any time dependent orthogonal tensor Q(z). Since we focus on the stress tensor associated with time ¢, it seems reasonable to change over from variable :" to variable s defined by the relationship
and to produce a deformation rate tensor D(¢)= D(t—s) in the form D(t—s)=D/!(s) : D(1). Tensor D/(s) is called the relative deformation
rate tensor. In case of incompres
sible fluids, the determinant of the deformation rate tensor associated with any spatial configuration that can be brought about realistically is one, that is D(¢) is unimodular
st
o
in every case. Making use of this fact as well as of (144) and selecting U=D"'(¢) we get T= J {D@)}= 7 {D/(s)}. (146) ]
For the reguirement according to (145), we can write
(Z_ (000) : D/ : 07(0))— 000) : 09: {D/(5)}Q(0) using the relative deformation an arbitrary orthogonal tensor the structure of the functional. for a dynamical description of
rate tensor depending However, the class of
(147)
and, once more, (1414). In (147), Q(s) is on s. At first sight, (147) says little about hereinafter it will be found as still enough socalled viscometric flows, a range rather
narrow from a kinematical point of view, but very significant for practice. (For the definition of viscometric flow see para 14.4.)
14.3. Shear flow Simple shear flow is the simplest viscometric flow. Simple shear flow is the case where a Cartesian rectangular coordinate system xyz can be found in which the velocity field of the fluid is described by formulae v,=xny, 256
v,=0,=0,
(148)
x being constant and called velocity of shear. For any selected point of the material, (148) can be written in the form of the differential equation system
dx _,, dr
V. $ .
" dr
de
the general solution of which being xX(t )— xet +c,
y( )Z,
z(t)=cs,
where c,, ¢, and c; are integration constants. x, y, z at time ¢, the formula
Speaking now of a particle at point
x(t)=x('—)y()+x(1), y(@)=y(t), z(£)=z(?) can be written as a solution, becoming
x(8)=x(t)—xay(1), y(9)=y(1), z(s)=z(2) after introduction of notation £ — [—s. The matrix of the relative deformation rate tensor is l —xs O [D!(s)]=0 1 0]. (149) 0 0 1 For the sake of what follows, it seems reasonable to introduce the notation
D/ (s)=I—%sM where
M=ji;
M]=0
0 1 0 0 ol. 0 0 0
Let (1410) be substituted into (146), then T=9
(1410)
’
{I—xsM}
0=s
to determine the stress tensor.
Since the shape of the function serving as an independent variable of the functional depends on the tensor » M alone, the functional is reduced to one function, the domain and set of values of which being constituted by the secondorder tensors:
T=9 ÍI—xsM) =T(xM).
(1411)
0=s
Reasonably, only tensore Q(s), independent of s, should be used to write the requirement according to (147). Using (1411), the requirement is
T(xQM Q") =Q T(xM) Q.
(1412) 257
This equality means at the same time that every tensor Q for which the equality
QMQ'=M is true leaves also the stress tensor invariant. This requirement is satisfied by tensor Q, of given matrix
10 0 [@]=]0 1 01. 0 0 1
The invariance of tensor T leads us to the equality t3=1,3=0, that means that function
T (> M) can take the shape
[TGeM)]
=
th(%)
tp(x)
0
1,(%)
1,(0)
O
0
0
155(%)
(1413)
only. With tensors ]
I9:J—)
0
0
0
1
0
0
0
1
and l
[@]=(0
0
—1 0
0
0 0
1
substituted into (1412) we get
t 09 =tn(—=%),
1) =tn(—%)
ty3 (%) =t33(—%),
t(—%) =—1,0%).
(1414)
Expressions widely used in the literature will be obtained if the notation 0,(%) =1,,(3) —133(%)
0,(%) = 155(%) —133(%) T(%) =1;5(%) is introduced. Functions
Functions o,(%), 0.(x) and t(») are called viscometric functions.
0,(x) and
o,(x%) are called the normal
stress difference while function
T(x) is called shear stress. The viscometric functions determine the stress tensor to the extent of a scalar
pressure:
T=—pl+0,60)M M"+0,6)M"  M+1(%) M+M") .
(1415)
On the other hand, it can be seen on the basis of (1114) that, from among the viscometric functions, the shear stress function 7(x) is an odd function of the shearing velocity while functions o; (x) and 0,(x) are even functions. 258
Viscometric functions describe the behaviour of nonNewtonian liquids and give account of the Poynting and Kelvin effect. Their shape can be determined by means of measurements. Availability of viscometric functions types of material on the basis of measurements, similarly values of viscosity in case of Newtonian liquids, would calculations for a much wider range of flows than shear
tabulated for the different for the table of numerical permit the construction of flow alone.
14.4. Viscometric flow The viscometric functions presented above can be applied not only to simple shear
flow but to any flow where the relative deformation rate tensor is described by formula
D! (s) =Q(s)  [I—x»sM],
(1416)
where Q is an arbitrary orthogonal tensor while M is the dyadic product of two unit vectors normal to each other. This is a straightforward result of (147). A flow of this type is called viscometric flow, as flows of this type are taking place in the usual viscometers. Investigated below are the most important types of viscometric flow.
14.4.1. Capillary flow In case of stationary flow taking place in a long pipe of circular crosssection, the
conditions of symmetry lead us to believe that the flow rate is parallel to the centreline of the pipe everywhere, and that its magnitude depends on the distance measured from the centreline alone. Let a Cartesian rectangular coordinate system be selected
in such a way that its origin will lie on the centreline of the pipe with axis z pointing at the direction of flow. The velocity field can be described by formula v=u(R)k , where R?—x?4y? . The position of the different material points in the function of variable s=¢—1' is given by equations
x(s)=x(1), y(s)=y(), z(s) =z(¢) —su(R). The matrix of the relative deformation rate tensor is
1 0 (D]
=
0 1
— sdu R
dR
0 0
ZLy du=] R
dR
259
This can be rewritten as
Dis) =Is ÉM dR where
M=k (ii+ X j) . R
R
Introducing the notation du
X—  L£ 3R
( 1417 )
the formula according to (1416) is obtained for the deformation rate tensor. Hence, the stress tensor is defined according to (1415). Í
14.4.2. Flow around a cylinder In Couette’s viscosimeter, the liquid flows between the mantle of two coaxially arranged cylinders rotating at a different angular velocity each. The flow taking place
in Couette’s viscosimeters is called Couette flow. Let a Cartestan rectangular coordinate system be selected in such a way that its axis z will be the common
centreline of the cylinders. Assume that the flow pattern
is described by velocity field v=w(R)kxr+ U(R)k, (R*=x%*+y?. This is a more general pattern than that taking rotational viscometers and therefore flow of this type is called marginal case, U(R) =0, of helical flow is called Couette flow. relative deformation rate tensor, it seems reasonable to change coordinates determined by the transformation formulae x=Rcos @,
place in the usual helical flow. The To determine the over to cylindrical
y=Rsin o.
Now the velocity field defined leads to a system of differential equations
R _o,
9. 0R,
ds
ds
Z. ds
UR
for which the solution
R(5)R(t),
0(9)—9(t)—so(R),
Z(s)=z(t)—sU(R) is obtained, whence on the basis of the equalities
dr=eg(s) dR(s)+e,(s) do(s)+e,(s) dz(s)=
=ex(s) dR(2) +e,(s) [d,— e’ +a* e’ =0, aalel+a3282+((133—/1)63
=0.
(A5'14)2
The solution of the equation system will be other than zero (nontrivial) only if the determinant . . . D, —
=D3(A)
1 __eklme
3¢
=d"
=a
1—151
 =
— AÓn) 40£) (@' —A0f) (@ — ": "Eper(A
(A515) 287
calculated from the coefficients, socalled characteristic polynomial, is zero. Polynomial D, =D,(4) is a thirddegree function of 4 and the roots of characteristic equation
D;(A) = a*,—16f =0
(A516)
are the eigenvalues of tensor a*, thatis 4,; s=1, 2, 3.
With the definite eigenvalues 4, ; s=1, 2, 3 substituted into equation (A514), a homogeneous algebraic equation is obtained for determination of the eigenvectors (e, ) ; s=1, 2, 3 associated with the eigenvalues in question. Since the coordinates of the eigenvector are not determined uniquely in this way, the requirement that the eigenvector be a unit vector, that is the equation
(e.)" gya(e:)? =1; is usually added to it. According to (A344),,
s5=1,2,3
the transformation
(A517)
of tensor d*, =a*,—Adf
in case of
coordinate transformation x" =x*(x'!, x'%, x ?); J#0 is dpq
:dkl'rpr'g[
and thus the transformation of the characteristic polynomial (A515) will be DI3
=DI3(A’)
=
d,pql
=
,dlekpoqll
5
whence, with respect to the multiplication rule of determinants as well as to (A339), the identity Dls(l) —
ldkll tól"
lo'ál —
ldkll :Da('l)
follows or, in other words, the eigenvalues (main values) and eigenvectors (principal axes) are invariant quantities independent of the selected coordinate system. The discussion below relates to symmetric secondorder tensors. In this case, the
roots of the characteristic equation (the eigenvalues) are real numbers, simple, double or triple roots. Simple roots (eigenvalues):
Let 4, and 4, be two different eigenvalues of a symmetric tensor A, while ¢, , e, are the eigenvectors associated with them, respectively. Now,
from the equations
written on the basis of (A513), ez'A'el
:/llez'el,
el
=/1201
'A'02
'e2,
the equation (A'l —/12)61
: 62
=0
(A5'18)
is obtained, taking also the symmetry of A into consideration. And since A, #4, according to the starting assumption, e,  e, =0 is obtained as a result, that is
in case of a symmetric tensor, the eigenvectors associated with the different eigenvalues are normal to each other. Eigenvalues are usually numbered
according to the inequalities
4 44, 41, and
eigenvectors e, , e, , €; are expected to constitute a righthand side vector triad. 288
In the coordinate system of the eigenvectors (principal axes),
A
lagl =[a™)=1[a,]1=[a™]=]0
0
0
0
4
0
(A519)
0 A
is the matrix of the tensor, produced by the tensorial producis 3
Az
) Aee,. s=1
In the coordinate system of principal axes, the powers of secondorder symmetric tensors take a simple shape (n being an arbitrary positive integer or, in case nondegenerate tensors, an arbitrary positive or negative integer or, in case of positive definite tensors, an arbitrary positive or negative integer or a fraction):
3
A  Y (Q)ee; [@)l=
)"
0
0
0 () 0 0 0 ()
(In this case, the matrix of the tensor is identically independent of the index position.) It follows from what has been said above, and also from (A513) that the eigenvalues of A" are the nth powers of eigenvalues A, of tensor A, (4.)" : s=1, 2, 3,
and that the eigenvectors of the two tensors are identical. Double roots: Let A, =4,>4,4.
Since 4, #4; and 4, 443 , equation (AS518) results in e, : ez =0 and e,
e, =0
with the serial numbers of the eigenvalues interchanged as required, that is eigenvectors ¢, and e, are normal to e;. And since A, =4, , equation (AS518) also suggests that e, and e, may be arbitrary (in the plane normal to e;). Hence, in case of double roots, any unit vector in the plane normal to the eigenvector
associated with the simple root can be considered an eigenvector. Taking this into consideration, the eigenvectors can be plotted in this case in such a way that vectors €, , €, , €; will constitute a righthand vector triad mutually normal to each other. Triple roots: 4, =4, =4;. In this case, as suggested by (A518), the eigenvectors as compared with each other may be arbitrary that is, any unit vector can be considered an eigenvector. In spite of
this fact, three righthand eigenvectors e, , e, , e; , mutually normal to each other, are usually plotted also in this case.
In case of triple roots, the matrix of eigenvectors (principal axes) is 4 [6] — ] 0 0
the tensor in the coordinate system of the 0 0 4 0 0 A
[—4.[ő;]
(A520);
that is, with respect to the tensor itself, the equation
A=A I
(A520),
is satisfied.
289
A5.3. Scalar invariants of symmetric secondorder tensors The coefficient of term A’ in characteristic equation
1 D3 (j') =  A
— ;l'gk!l —
百
eklm epqr(akp  )'gkp) (alq  /lglq) (amr  }'gmr) =0
(A521)
of tensor a,, will be
—lgul=—9g
(A522)
if written on the basis of (A516) while —lő l — 1 if written according to (A516).
Dividing with the coefficient of term A°, the characteristic equation can be written as ÁB—AIÁZ'*'AHÁ—AIH:O
in every case, where A;, An, Ar
(A5'23)
define the first, second and third scalar invariant
of the tensor, respectively. Their invariance upon coordinate transformation results from the invariance of the eigenvalues A,; s— 1, 2, 3 as has been proved previously.
According to formulae (A521), (A522), (A423) as well as by the rule of calculation of determinants, AI —
1
m
—ekl
Z— :
_ AII 
1 319
e
1
— 9
4
r
e"
319
kim
(akpglqgmr+gkpa1qgmr+gkpglqamr)=
(A5'24)
an
912
4913
91
42
913
412
92
9nt9n
X02
9n+9a
92
b3l7;
A)
932
933
032
933
9322
4033
e"
r
931
911
912
931
3
— (akp alqgmr+akpglqamr+gkpalq amr) 
a,
42
4913
A
912
43
412
43
[9
402
gzjtjtbi
92
ant9a
an
ayl, (A525)
az;
0432
933
9322
033
032
033
M)
911 931
Ay = Le""”e”"’a,qr,a,qa,,,,= 1 lA l. 319
(A526)
g
We will arrive at a similar formulae if (A521) is written with tensor components
with superscripts. For example,
A = 3!L, =g
290
(a¥g"g™ +g"alg™ +g" g1 a™) =
all
gIZ
g13
a21
g22
g23
a3l
g32
g33
+
gll
a12
gl3
g21
a22
923
g3l
a32
g33
+
gll
g12
al3
gZI
g22
a23
g3l
g32
a33
.
(A5_27)
In case of tensor components
with mixed
incides,
(A515)
can be taken as a
starting point. Taking also (A322), into consideration, the equation Ay=ad%,
(A528)
is obtained, and at the same time AII
—
L1

2'
Lkim e
epqm
af,a‘,
(A529),
can be written, which, considering the rule of production of adjoints, means that the
sum of subdeterminants associated with the elements of matrix with the components of tensor a*,, in the main diagonal is 2 a, 3 a,
A11=
2 az 3 43
1 a, 3 a,
1 az 3 as
1 a, 2 a,
1 a, 2 a;,
[¢*], produced

(A5'29)2
Finally, we have A
I

1 íe
kim
epqra
P
A9 aia
AT
— ,=
la
k
Il'
(A5'3O)
Expressed by tensor components written in the coordinate system of eigenvectors (principal axes), the relations Ay =M+
+ 4
Au=A+LA4+44;
A=Ay
.
(A5'31)1, 2,3
are obtained on the basis of (A519). The following significant relationships can be written between the scalar invariants introduced:
A= %(Alz—akpa”k),
(A532)
Ay — ‘6‘(—21‘113 +64,4y +2a",a",a%).
(A533)
1
Equation (A532) is obtained by means of (A529), and (A322), while (A533) is obtained by means of (A530), (A28) and (A532). The first scalar invariants of the powers A%, A%, ..., A" of a secondorder tensor A, defined according to (A512), , can be written on the basis of (A528) and (A512),: (4 2)1 = (az)kk =da kpapk ; (A3)I =(a3)kk
=akpapqaqk9
(A534) (A5'35)
and so forth. The invariant nature of (A534) and (A535) follows also from (A532) and (A533). It is also obvious on the basis of the definition of the scalar invariants that the scalar invariants and, accordingly, also the eigenvalues, of tensor A (independently of
whether it is symmetric or not) and transposed tensor AT are the same. 291
A secondorder tensor A will be positive definite if it is symmetric and if the inequality v  A  v— 0 is true for any vector v#0. In this case, all the three eigenvalues of the tensor and also its third scalar invariant are positive.
A5.4. The Cayley—Hamilton theorem According to the CayleyHamilton theorem, any secondorder symmetric tensor A satisfies its characteristic equation.
Namely,
according
(A5'12),
to the powers
of the secondorder tensor defined under
(a3)1l :(11)3, (43)22:().2)3,
(613)33 =(A3)3,
Or (‘12)1 1= (11)2,
(‘12)22 = (12)2:
(42)33 = (13)2,
will be obtained for the cases n=2 or n=3, respectively, if the tensor itself is considered to have been written in the coordinate system of eigenvectors, while zero is obtained for coordinates of different indices. Now, taking also formula (A523) of
the characteristic equation into consideration, equation 3 A—
3
Z
()vs)3eses=
Z
[Al(ls)3_AIIj's+AIII]eses
s=1
s=1
can be written which, after rearrangement, gives the CayleyHamilton theorem: AS—AIAZ"*'AH
A—AIHI:
0.
(A5‘36)
A5.5. Tensor polynomials By means of powers (A512) of a secondorder tensor A, tensor polynomials can also
be defined:
H=Y o,A".
(A537)
s=1
Using the Cayley—Hamilton theorem, any secondorder symmetric tensor polyno
mial can be expressed by means of powers A°=1I, A'=A4 and A’ of the tensor. And indeed, according to (A536), A=A, A" — Ay A=Ay, then, with this multiplied scalarly by A and substituted,
A=A
A — A
A"  Am A= (A} — A
AP+ (A
— A A
A+ A Am T
can be written. With the process continued, this leads to proof of the statement:
H=p8,A*+8,A+B,E.
(A538)
Tensor polynomials appear also beyond the scope discussed so far. Namely, an
analytical function f(z), if known, can always be extended also over a tensor domain 292
provided a tensor A is written in place of z. Since f(z) is analytical function, it is
possible to expand it into a series. Now,
fe)= ¥, uz’ is the power series of function f(z) within the convergence domain z < KR. In case of the nonsingular tensor A 00
f(A) =)
o,A°.
s=0 00
That means
that tensor function f is defined
by
) o,A" under
the condition
of
s=0
max. (A4;], 14.1, 45]) 5'g,g’ and H (if :k is the sign of an arbitrary multiplication between tensors B and H),
(B x H)V=(BxHyg =(B,*H+BxH,g = =
(bíj;phklm
_'_bíjhklm;p)
(gzgj)
*
(gkglgm)gp
,
(A6'3)l
or, in case of notation of indices exclusively,
(b’ 3 h*,,)., =h'. , A MŐ aa FB, % h*,,. , :
(A63),
A6.3. Divergence and rotation of tensors Divergence (rotation) of tensors is produced by contraction of the covariant deriva
tive (covariant derivative multiplied with the permutation tensor tensorially) with respect to the index of the covariant derivative and some index of the tensor (or double contraction with respect to the index of the covariant derivative and some indexof the tensor as well as two indices of the permutation tensor). For example, divergence in respect to the first index or rotation with respect to the third index of 298
. tensor /" k
:. hklm;
k
:blm
or
hklm;pemps
:Ckls
:
(A6'4)
respectively. The divergence of vector v=v*g,
(taking also (A331) into consideration) and
the left divergence of tensor A=a"'g,g, can be written also in the following form:
vk, = \—}g—(ﬁ o),
(A6.5)
and 1
VA= — (/g d“g), ,
(A66)
77
g
respectively.
A6.4. Multiple covariant derivative Also the gradient of gradient (A61)
D=HV =h",, , B8BE of tensor
H considered above can be produced:
DV = (HV)V = a—ﬂ(h"zm;,,gkg’g ghg! — a
m
m ,8c8'g"e’g?. m
(A67)
The double covariant derivative or, as a result of differentiation as many times as
as required, any covariant derivative of tensor h*,, (or any tensor) is obtained in the same way:
Interchange of the order of differentiation results in the formula hklm; pa  hklm; =
hslmRkqu
+ hksm Rslpa + hklsRsmpq
[J
(A6'8)
where R",,, is the RiemannChristoffel curvature tensor defined under (A311). The same curvature tensor expressed with a subscript is shown by formula (A332). It can be seen in (A68) that the order of covariant differentiation can be interchanged only if R" ,,, —0, which is at the same time the necessary and sufficient
condition that the space is Eucledian.
A6.5. The Laplace operator The Laplace differential operator applied to a scalar function ¥=¥(x'x?x?) is essentially a contraction between the indices of the double covariant derivative:
AP=(PV  V)=V, ,g".
(A69) 299
Since YV
= T, kgk :gkl 'P, k815
also formula 1
A¥ = —— (/g 9") ,
(A610)
9 Vo
obtained by use of (A65), exists.
A7. Integrals of tensor fields Integrals of tensor field H=H(x', x?, x?) for some curve G, surface A and volume V are
JH*
dr= JH*
G
dx?
ígpdl;
(A71)
G
Í H:xkdA [ Hx nd4;
[ HdV,
A
V
A
(A72)
where 4 is the parameter of the curve of equation x" — x"(47), n is the surface normal,
3k is the sign of multiplication between tensor H and some vector. In the curvilinear coordinate system, also the basic vectors depend on the position coordinates and therefore they cannot be displaced from behind the integral sign.
A7.1. Integral transformation theorems Gauss—Ostrogradski theorem If V is the volume region, A the boundary surface of it and in an arbitrary coordinate system, equation
f V % HdV
 n % HdA,
V
A
H an arbitrary tensor field
(A73)
Or
j H:x
VdV
[ H*nd4
V
(A74)
A
will exist by virtue of the GaussOstrogradski theorem. In the simple case where a vector c is written in place of H and sign k is assumed to stand for scalar multiplication,
f VedV= V
ncdd,
f e. dV— [1n,c"dA V
300
(A75),
A
A
(A75),
are obtained for the Gauss—Ostrogradski theorem. The integrals of type
f B 67 AV 4
can be transformed by means of identity hk!m; kblm — (hkzmblm); k —hklmblm; k
and by formula (A75), if Ck — hklmblm
is substituted into (A75),. Relationship I
hklm;
kblm
dV=
V
j
nkhklmblm
dA—
A
í
hklmblm;
k dV
(A7'6)
4
called the rule of partial (product) integration of volume integrals is then obtained. In the twodimensional case where H— H(x", x"), A is a simply connected two
dimensional region and 4 is the boundary curve of this region while n is the external normal of g, the GaussOstrogradski theorem takes the following shape:
Or
 V :x HdA
$n :k Hds,
4
a
f H:k
V d4dA— $ H % nds.
A
(A77)
a
Stokes’ theorem Let A be a simply connected twosided smooth surface of arbitrary geometry and let a be the boundary curve of this surface, consisting of smooth curve sections of a finite number. Furthermore, let n be the unit vector of normal direction to surface A4, s the
arc length measuring along curve ¢, e the tangent unit vector of the curve, and v=eXn
a unit vector directed away from the surface. According to Stokes’ theorem, the equation
or
$e :x Hds=
§ (nx V) % HdA,
q
A
$ H :x eds— [ H :x (nx V)dA.
(A78),
(A78),
is true for any tensor H defined over surface A in case of any arbitrary coordinate
system.
301
A8. Fundamental quantities in cylindrical and spherical coordinate systems A8.1. Cylindrical coordinate system x'=R,
x*=¢,
x’=z
(Fig. A2).
Metric tensor:
1
0
0
0
0
1
1
0 1
0
0
1
lgd={0 £ 0, [g"]=0 : 0}, 0
g=(x")’=R>. Nonzero Christoffel symbols of the second kind: 1 rlzzzí,
rzzl:_R
Differential operators:
öx "
OR "
R őp "
o0z "
ayov Y LR 2(R22), 1 22 , 2E. 0R\ OR R? 0¢* — Z? 2
2
Matrices of vector and tensor coordinates written with physical coordinates: For displacement vector u, the physical coordinates are [u(k)]
T
=
[uRu(puz]
:
Up [u
k
1
] =
Eu(p
u
3
[u 1] =
uR
Ruq;
2
xÉzPzconst
x3
Fig. A.2. Cylindrical coordinate system
For stress tensor @, if the physical coordinates are
[°'