On some geometrical nonlinear theories for the plastic bending, stretching and buckling of sandwich plates

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Title:
On some geometrical nonlinear theories for the plastic bending, stretching and buckling of sandwich plates
Physical Description:
xv, 202 leaves. : ill. ; 28 cm.
Language:
English
Creator:
Monzel, Fred Jacob, 1931-
Publication Date:

Subjects

Subjects / Keywords:
Sandwich construction   ( lcsh )
Elastic plates and shells   ( lcsh )
Plasticity   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 198-200.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 13800979
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Full Text








ON SOME GEOMETRICAL NONLINEAR
THEORIES FOR THE PLASTIC BENDING,

STRETCHING AND BUCKLING

OF SANDWICH PLATES






By
FRED JACOB MONZEL


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA


April, 1967















ACKNOWLEDGMENTS


The author is grateful to Dr. I. K. Ebcioglu, chairman of his

supervisory committee, for suggesting this project and for his contin-

ual encouragement and constructive criticisms. Also, the author is

grateful to the other members of his supervisory committee, namely,

Dr. W. A. Nash, Dr. S. Y. Lu, and Dr. T. 0. Moore, for their personal

interest and guidance over the years.

Because of the financial assistance received through a NASA

traineeship to engage in graduate study for three years, the author

is grateful to the National Aeronautic and Space Administration.

In addition, the results presented in this dissertation were obtained

under the NSF Grant No. GK-640 of the National Science Foundation to

the University of Florida.

The author is also indebted to the Computing Center of the

University of Florida for providing the services of an IBM 709

electronic computer, without which the scope of this work would have

been curtailed.












TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . ii

LIST OF FIGURES . . vi

KEY TO SYMBOLS . . ... .. vii

ABSTRACT .. .... .......... xiv

SECTION
1. INTRODUCTION .. ..... .. 1

2. THE VARtATIONAL PRINCIPLE . .. 8

3. GENERAL NONLINEAR THEORY OF A SANDWICH PLATE 15

3.1. Description of the Sandwich Plate . 15
3.2. Displacement Functions . 20
3.3. Fundamental Equations . .. 24

3.3.1. Equations of motion 24
3.3.2. Boundary conditions 35
3.3.3. Strain-displacement equations .. 43
3.3.4. Stress-strain equations . 47

3.4. Buckling Problem . 60

3.4.1. Field equations ..... 65
3.4.2. Boundary conditions 72
3.4.3. Variation of strain ...... 78
3.4.4. Variation of stress 82

3.5. Modified Buckling Problem ...... 86

3.5.1. Field equations ... 86
3.5.2. Boundary conditions . 90
3.5.3. Variation of strain .. .... 92
3.5.4. Variation of stress ..... 93

4. SIMPLIFIED LARGE DEFLECTION THEORY . 94

4.1. Fundamental Equations . .. 95

4.1.1. Equations of motion . .. 95
4.1.2. Boundary conditions *.. . 100











TABLE OF CONTENTS (Continued)


4.1.3. Strain-displacement equations
4.1.4. Stress-strain equations .


4.2. Buckling Problem


4.2.1.
4.2.2.
4.2.3.
4.2.4.


. .


Field equations .
Boundary conditions
Variation of strain
Variation of stress


. .


4.3. Modified Buckling Problem


4.3.1.
4.3.2.
4.3.3.
4.3.4.


Field equations ..
Boundary conditions ..
Variation of strain .
Variation of stress .


5. SIMPLIFIED LARGE DEFLECTION THEORY FOR SANDWICH PLATE
WITH THIN FACINGS . .

5.1. Fundamental Equations . .


5.1.1.
5.1.2.
5.1.3.


Equations of motion .
Boundary conditions .
Strain-displacement equations


5.1.4. Stress-strain equations .


5.2. Buckling Problem .


5.2.1.
5.22.2
5.2.3.
5.2.4.


Field equations .
Boundary conditions
Variation of strain
Variation of stress


5.3. Modified Buckling Problem


5.3.1.
5.3.2.
5.3.3.
5.3.4.


Field equations
Boundary conditions
Variation of strain
Variation of stress


152
158.
154
155


156


6. NUMERICAL EXAMPLE . .. .


135
139
144
147


Field Equations .
Boundary Conditions .
Variation of Strain .


. a .

a a a .


. 161
. 163
. 164


6.1.
6.2.
6.3.


Page

103
105

106

106
108
108
110

110

111
111
111
111


112

114

116
123
126
128


. . .


* *


*
*
*
*


. o e










TABLE OF CONTENTS (Continued)
Page

6.4. Variation of Stress ......... 165
6.5. Solution .. .... .. 166
6.6. Results and Conclusions . 186

APPENDICES
A. COMPUTER PROGRAM .. .. . ... 190

B. MATERIAL PROPERTIES .. .... .. .... 195

BIBLIOGRAPHY . .... 198

BIOGRAPHICAL SKETCH .. .................. 201













LIST OF FIGURES


Figure Page

1. Plate coordinates .. ..... 16

2. Coordinates for sandwich plate . 18

3. Rectangular sandwich plate . ... .. 156

4. Critical buckling stress, comparison of compressible and
incompressible theories . .. 187

5. Face critical buckling stress vs. rigidity parameter for
square sandwich plate under uniaxial load 188

6. Stress-strain curve for 17-7 PR stainless steel in
compression . .. .... ... 196

7. Tangent and secant moduli for 17-7 PH stainless steel 197














KEY TO SYMBOLS


All the symbols in this work are defined when introduced;

however, for the convenience of the reader, the following is a list

of the symbols used with a brief explanation of their meaning.

Every effort has been made to use conventional notations of mechanics.

Because tensors have been used in the formulation of the equations,

the tensor conventions of reference [18] have been adopted; for example,

Latin indices are used to denote space tensors (three-dimensional),

while Greek indices denote subtensors. Also, in the following list,

a tensor is written as either a covariant or contravariant tensor, but

both forms may appear in the text. In addition, because different

theories are developed, some quantities may take on different values

in different sections; for these quantities, the explanation will cor-

respond to the case where the symbol is first introduced. However,

to remind the reader that in a particular section the quantity may

have been redefined, a double asterisk appears just prior to the

explanation.









*Numers in bracts designate eferenes n the Biblography.
Numbers in brackets designate references in the Bibliography.


vii









a
i
a

A A
O V 0 S


TA, A, EA
Ai8

AcnivJ

dA

b

bi
Bi3

Bo'ivj
B







i i i
c
c c e6


n n *

d

dij

D

DO

6e,3

E, Es, Et
Ei3

E
Eijkl
i
f
o


h
limit of integration, equal to 'h + *

stress resultant defined by Eq. (3.31)

portions of boundary surface upon which displacements
and stresses are prescribed, respectively

top, bottom and edge surfaces of plate, respectively

** tensor defined by Eq. (3.19)

tensor defined by Eq. (5.102)

differential area element
h
limit of integration, equal to 2

stress resultant defined by Eq. (3.32)

** tensor defined by Eq. (3.19)

tensor defined by Eq. (5.103)
i
variation of b see Eq. (3.117)
h
limit of integration, equal to 2

stress resultants defined by Eqs. (3.34), (3.33),
(3.49), respectively

variation of stress resultants, see Eqs. (3.119),
(3.118), (3.144), respectively

limit of integration, equal to "h -

coefficients, see Sect. 6.5

rigidity parameter, see Eq. (6.86)

** tensor defined by Eq. (3.25)

** variation of Ei3, see Eqs. (3.120) (3.122)

Young'% secant, and tangent moduli, respectively

** tensor defined by Eq. (3.25)

tensor defined by Eq. (3.103)

acceleration components


viii









body force components

core rigidity ratio, equal to

y


gij
G



G ,G
Gs G

Gx, Gy









k y
h

h, H
Ill 12 13



k, k1







iK


K



I1
0
Lx, L
x* y
m


Bi M
m
mi, Mr*


Mijkl

n

n.
o0j


metric tensor

shear modulus

secant and tangent shear moduli, respectively

shear moduli in A and 0 directions, respectively
4 2
layer thickness

parameters, see Eqs. (6.25) and (6.83)

strain invariants, see Eqs. (3.77)

stress invariants, see Eqs. (3.83)

material constants corresponding to the bulk modulus
for large and small strain, respectively

variation of K see Eq. (3.114)

material parameter that depends upon the amount of
plastic strain

elastic and plastic buckling coefficients, respec-
tively

stress resultant defined by Eq. (3.28)

tensor defined by Eq. (5.60)

stress resultant defined by Eq. (3.49)

length of plate in A1 and A2 directions, respectively
'h'E
parameter, equal to --
h"E
variation of M see Eq. (3.114)

stress resultants defined by Eqs. (3.31) and (3.28),
respectively

tensor defined by Eq. (3.163)

integer in Eq. (6.78)

components of external unit normal vector of
boundary surface


F
0
g










6Sn, 6n33


NO', N33

i i i
P np n P


P
o
6Pi, 6 P 6 P
n n


q

6q



r

R2

Roaiyj

6 s'
o
6sij

S

S
y
S
0


Si





6t

To



ui

6Ui


variations of NO$ and N33, respectively, see Eqs. (3.114)
and (3.116)

stress resultants defined by Eqs. (3.28) and (3.30),
respectively

stress resultants defined by Bqs. (3.34), (3.33),
(3.49), respectively

parameter, see Eq. (6.83)
i i i
variations of p p p,, respectively, see
n n "
Eqs. (3.119), (3.118), (3.144)

integer in Eq. (6.78)

variation of Q see Eq. (3.115)

stress resultant defined by Eq. (3.29)

core shear stiffness parameter, see Eq. (6.83)

external resultant force in 82 direction

tensor defined by Eq. (5.101)

variation of S
o
variation of stress tensor S
1 k
mean normal stress, equal to 3 Sk

uniaxial yield stress

quantity defined by Eq. (3.91) and corresponds to the
octahedral shear stress of classical theory

stress resultant defined by Eq. (3.49)

stress vector

stress tensor

variation of T see Eq. (3.115)

stress resultant defined by Eq. (3.29)

stress deviator

displacement of middle surface

variation of ui









v.

6Vi

V
0
dV
i
w

6W
i
x

Y1 -Y' 2 3

Z..




8


Bij

Vy

Yo


Yij

oVij' V1ij' 2Vij
6F..
61

o ij ij 62


8

6
i





60

0ot


displacement components

variation of vi

volume of undeformed body

differential volume element

stress resultant defined by Eq. (3.32)

variation of w see Eq. (3.117)

fixed rectangular Cartesian coordinates

amplitude of the middle surface displacement func-
tion, see Eq. (6.78)

parameters defined by Eqs. (6.83)

core thickness-length ratio, equal to-
L

wave-pattern aspect ratio, -I-
nL
x
coefficients, see Sect. 6.5

uniaxial strain at yield point

quantity defined by Eq. (3.91) and corresponds to
the octahedral shear strain of classical theory

** strain tensor

** portions of strain tensor as defined by Eq. (3.68)

** variation of strain tensor yij

** portions of strain tensor variation as defined
by Eq. (3.151)

variational symbol

Kronecker delta

parameter defined by Eq. (6.83)
'E
ratio of moduli, equal to --

geodesic normal coordinates

stress resultant defined by (3.145)

xi











Mv, U'p




ijkt


00
0

0,' 02
*


cp






8t.


o



^0
i )

[ij,k]


{j k

Subscripts

i, j,k,...


( )
cr

( )i
( )'i


Poisson's ratio

Poison's ratio in the transition and plastic
regions, respectively
S22
stress ratio of face layers, equal to n-2
22
tensor defined by Eq. (3.104)

mass density of the undeformed body

parameters defined by Eqs. (6.83)

strain energy density of the undeformed body

quantity defined by Eq. (5.104)

the metric form

displacement function, equal to v(9"',0),3

variation of r

constant defined by Eqs. (6.59) and (6.60)


material parameter for large strain
which corresponds to Gs of classical theory


material parameter for large strain which
corresponds to Gt of classical theory

Christoffel symbol of the first kind

Christoffel symbol of the second kind




Latin indices that take on the values 1, 2 and 3

Greek indices that take on the values 1 and 2

quantity associated with the critical buckling condition

covariant derivative-with respect to the 08 coordinates

partial derivative with respect to the 9 coordinates








Superscripts

i, J,k,...

SP ,y, ,

(')


( )

'( )

(-)

"'( )


Latin indices that take on the values I, 2 and 3

Greek indices that take on the values 1 and 2

quantity associated with the stable equilibrium
position of buckling

prescribed (given) quantity

upper layer quantity

middle layer quantity

lower layer quantity


xiii












Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


ON SOME GEOMETRICAL NONLINEAR THEORIES
FOR THE PLASTIC BENDING, STRETCHING
AND BUCKLING OF SANDWICH PLATES

By

Fred Jacob Monzel

'April, 1967


Chairman: Dr. Ibrahin K. Ebcioglu
Major Department: Engineering Science and Mechanics


On the basis of the modified Hellinger-Reissner variational

theorem of nonlinear elasticity, a general nonlinear theory is devel-

oped for sandwich plates that are stressed in the plastic region.

The plasticity equations correspond to the stress-strain equations of

Hencky, extended to large deformations. As a result, the material is

taken to be isotropic, and the loading is assumed to be both propor-

tional and active. The general theory is then extended to the buckling

of sandwich plates by applying the bifurcation theory.

In addition, by omitting certain quadratic terms from Green's

strain tensor, a simplified large deflection theory is developed for

the plastic bending, stretching and buckling of sandwich plates.

Furthermore, the theory is then extended to those sandwich plates which

have thin outer layers; in this case, the Kirchhoff-Love hypothesis

is applied. Finally, a numerical solution is obtained for the plastic









buckling of a simply supported sandwich plate with very thin outer

layers and a soft core under a uniaxial compressive load.


xv














1. .INTRODUCTION


By way of introduction, it probably should be mentioned first

that many man-made structures consist of flexural members, that is,

plates, shells and thin bars. For these structures to behave in a

predictable manner, the ability to analyze the flexural members is of

vital importance. In general, flexural members fail in one of two

ways, either they become overstressed or they deform excessively, in

particular, the buckling phenomenon. Often, two separate analyses are

required to investigate both possibilities. Well-established branches

of mechanics have been developed which deal with such analyses, for

example, strength of material, theory of elasticity, theory of plas-

ticity, theory of plates, theory of shells, theory of elastic stabil-

ity, etc.

In this work, one small class of flexural members will be con-

sidered, namely, a three-layered plate which is commonly called a

sandwich plate. Usually, the primary purpose of a sandwich plate is

to obtain a relatively rigid plate that weighs less than a solid (one

material) plate of the same rigidity; however, sandwich plates have

certainly been designed for other purposes.

The sandwich plate has found numerous applications in the air-

craft and aerospace industries; as a result, considerable attention and

work has been devoted to this type of plate in the last twenty years.








Because of their geometry, it is natural to develop a theory for sandwich

plates by extending the theory of ordinary plates. Unfortunately, how-

ever, there are many theories for ordinary plates; for example, there

are theories which depend upon the plate geometry (thick or thin), and

the amount of deformation (small or large deflections), and the state

of stress (elastic or plastic). Even more numerous theories can be

developed for sandwich plates because of its structure; for example,

consider the outer two layers which may be thick or thin, or different

thicknesses, or different materials, etc.

Because of this multitude of theories, there comes a time when

it is desirable to develop a general theory which encompasses the var-

ious specialized theories. However, by their nature, the equations in

a general theory are usually too complicated to be solved, and, there-

fore, recourse must be taken to the simpler theories for solutions.

One of the advantages of reducing a general theory to a special theory

is that one is made aware of the factors being neglected. In contra-

distinction, when the special theory is developed directly, one is

usually aware of only those factors being considered; note the dif-

ference. Because of the variety of methods available, a general the-

ory can be developed by different techniques, for example, from one of

the variational principles or directly from the three-dimensional

equations of nonlinear elasticity. Now, the fundamental equations of

the nonlinear theory of elasticity consist of the following: the

equations of motion, the boundary conditions (both stress and displace-

ment), the strain-displacement equations, and the stress-strain equa-

tions; these equations are the ingredients of a general plate theory.









Recently, a general theory for sandwich plates was given by Ebcioglu

[1], in which Hamilton's principle was employed.

Because of its importance to this work, it is desirable to

discuss some of the salient characteristics of Eblcoglu's work. The

equations were derived with reference to the undeformed and stress

free state of the plate, called the reference state; in this frame of

reference (material or Lagrangian), there are two types of stress

tensors, the Lagrangian stress tensor and the Kirchhoff stress tensor.

The Kirchhoff stress tensor has the advantage of being a symmetric

tensor and was used by Ebcioglu. From geometrical considerations, the

most general strain-displacement relations are the Green's strain ten-

sor (material coordinates) and the Almansi's strain tensor (spatial

or Eulerian coordinates), both of which are nonlinear; Ebcioglu used

Green's strain tensor. From a physical point of view, Ebcioglu took

very general stress-strain equations which include thermal strain and

are applicable to elastic anisotropic materials; the equations allow

for the possibility of a nonlinear material.

From the preceding paragraph, it becomes apparent that the

general theory can take different forms depending upon the type of

stress tensor applied and/or the frame of reference used. However, in

this work, an alternate form of the general theory will not be devel-

oped; instead, the theory will be developed from a different varia-

tional principle, namely, the modified Hellinger-Reissner theorem.

The modified Hellinger-Reissner theorem has recently been employed by

Habip [2] to develop a general theory for ordinary plates, and, later,

Huang [3] applied the theory to a sandwich shell.








It should be mentioned that for a general theory there are

no uniqueness theorems which state that the solution is the only pos-

sible solution; this is as it should be because it is well known that

there are different deformations possible for a given environment,

commonly known as the buckling phenomenon. Therefore, a special for-

mulation is required to predict the onset of buckling. To the author's

knowledge, the general theory for sandwich plates has not yet been

extended to the buckling problem; the buckling equations will be devel-

oped in this work.

Although the stress-strain equations employed by Ebcioglu may

be nonlinear, it is well known that for most structural materials (i.e.,

steel, aluminum, etc.) a nonlinear stress-strain equation implies that

the material has been stressed into the plastic range. Because the

theories of plasticity and elasticity are not the same, different gen-

eral theories should be developed which are applicable to both types of

plate problems. In this work, we are interested in developing a gen-

eral theory which is based on the equations of plasticity.

In the field of plasticity, there are many theories, usually

classified as the incremental (or flow) theories -and the deformation

(or total strain) theories. The book by Hill [4] deals almost exclu-

sively with the incremental theories (in particular, the Reuss equa-

tions) because these theories describe the actual behavior of materials

better than the deformation theories. However, the deformation theories

are simpler mathematically and, therefore, are ofted used, particularly

Rencky's equations. It is well known that Hencky's equations can be

derived as a special case of Reuss' equations, namely, when the principal









axes of stress do not change and the stress ratios remain constant,

see Smith and Sidebottom [5]; this is a practical condition that occurs

quite often in plates. In Hill's [4] comments on Hencky's equations,

he mentions that the equations have not been extended to the case of

large strains (e.g., Green's strain tensor); however., as we shall see,

the foundations have been established for large strain theory.

The modified Hellinger-Reissner theorem assumes the existence

of a strain energy density from which an explicit form for the stress-

strain equations is obtained. However, a strain energy density implies

that the material is elastic, not plastic; but, according to Novozhilov

[6], Kachanov has proved that for an active deformation an elastic-

plastic body is indistinguishable from an ideally elastic body, both

with the same stress-strain curve. Therefore, the concept of a strain-

energy density can be employed to a restrictive class of plasticity

problems. Furthermore, for large strains (Green's strain tensor),

Novozhilov [6] has shown that equations corresponding to Hencky's equa-

tions can be obtained in terms of a strain energy density. The proce-

dure of Novozhilov will be used in this work; and, furthermore, explicit

expressions for the strain energy density and, hence, Hencky's equa-

tions will be derived.

The theory for the plastic buckling of plates is largely a

result of experience gained in the study of the plastic buckling of

columns. As is well known, two different philosophies had prevailed

for many years concerning the plastic buckling of columns, which were

based on the assumption of whether or not strain reversal occurs during

buckling. The two different approaches were unified by Shanley [7]









in 1947. As discussed by Gerard [8], von Karman concluded from

Shanley's work that the stability limit should be redefined-for plastic

buckling as "the smallest value of the axial load at which bifurcation

of the equilibrium positions can occur, regardless of whether or not

the transition to the bent position requires an increase of the axial

load." Therefore, the bifurcation theory for buckling will be employed

in this work; a formal statement of the theory is given in Sect. 3.4.

It is important to note that in the bifurcation theory the loading is

active.

Based on the above discussion, a general theory for sandwich

plates which includes buckling is developed. The theory is based on

stress-strain equations of the Hencky type, extended for large strains.

The theory is developed in Sects. 2 and 3 of this work. In addition,

some specialized formulations are considered.

In bending problems, the lateral displacements are usually

much larger than the displacement components in the plane of the plate.

On physical grounds, it has been argued that often certain quadratic

terms associated with the inplane displacements may be omitted from the

strain tensor. In Sect. 4, a modified theory is developed in which

only those quadratic terms of the lateral displacement are retained in

the strain tensor.

Because most sandwich plates are made with thin outer layers,

special consideration is given to this type of structure. As mentioned

by Fung [9], probably the most important discovery in plate theory was


e ovozhilo
See Novozhilov [6].









Kirchhoff's first assumption (stated in Sect. 5) which reduced the

equations to a form where practical solutions could readily be obtained.

Kirchhoff's assumption along with Love's assumption are basic hypotheses

in current thin plate theory. In Sect. 5, a theory for sandwich plates

with thin facings is developed which utilizes the Kirchhoff-Love hypo-

thesis along with the modified strain tensor described in the preceding

paragraph.

Finally, in Sect. 6, a sample problem is worked out in detail

and compared with the work of previous investigators.












2. THE VARIATIONAL PRINCIPLE


It is well known that in the nonlinear theory of elasticity
the basic equations can be obtained from variational principles,'in
particular, Reissner's theorem. Because of the early work of Hellinger
[10], the variational principle is often referred to as the Hellinger-
Reissner theorem, see Truesdell and Toupin [12]. Rather than work
with the theorem as stated by Reissner [13], an alternate form developed
by Washizu [14] is more convenient for our purposes, referred to as the
modified Hellinger-Reissner theorem. Habip [2] has used the modified
Hellinger-Reissner theorem to develop a general nonlinear plate theory;
in this work, the procedure used will be quite similar to that given
by Habip. The variational principle as given by Habip is
"the modified Hellinger-Reissner theorem asserts that the varia-
tional principle




+vr + ti .)l V
ov J 'i
+ i dA + ,(2.1)
+f4 ( ) dfo .)




According to Naghdi [11], Hellinger's work did not include
the boundary integral.









ij i
where Yi, S v, and S, are varied independently, is equiv-

alent to Cauchy's first law in V, to the stress boundary condi-

tion on the part A of the boundary, to the displacement boundary
os
condition on the remaining part A and to the stress-strain and
o v
strain-displacement relations in 0V, when the symmetries of yij

and S are both used."

All of the various quantities in the above theorem are referenced to

a body that is undeformed and stress free, known as the reference

state; for example, the symbol V denotes the volume of the undeformed

body, i.e., the volume in the reference state. With the understanding,

hereafter, that all of the quantities are with respect to the refer-

ence state, the meaning of the various symbols used in the statement

of the theorem is as follows:

A is the portion of the boundary of the body upon which the
displacement vectors are prescribed (given);

A is the remaining part of the boundary upon which the
os
stress vectors are prescribed;

f are components of acceleration;
o

F are components of body force per unit mass;

S are contravariant components of the stress tensor;

S are contravariant components of the stress vector;

v (vi) are contravariant (covariant) components of the
displacement vector;

V is the volume of the body;

Yij are covariant components of the strain tensor;

6 is the variational symbol used in calculus of variations;

po is the mass density of the body;

E is the strain energy density per unit volume;







is the covarLant derivative symbol used in tensor calculus;
is the tilde symbol which is used to denote prescribed
quantities.
When the indicated variation is performed, Eq. (2.1) can be put
into the form


{[s (,fr+ aj + ,o (F-/ "i)}% t, d


+ v iI)] 2b 61A


of
where Gass there






+ [S
where Gauss' theorem


J"/ v4


)] ~&s dv


=-


dv (so. l-. s. ',, dcA (2.3)

has been employed and where the strain energy density has been assumed
to be a function of the strain tensor, i.e.,




In Eq. (2.2), 8r is the Kronecker delta, and n. is the external unit
i I ]
normal vector of the boundary surface. Then from the generalization
and extension of the basic lemma of calculus of variations (hereafter,


(2.2)







referred to as the fundamental theorem of calculus of variations), as
indicated by Weinstock [15], Sq. (2.2) yields the fundamental equations
of the nonlinear theory of elasticity in terms of the reference state
as follows:

the equations of motion in V (Cauchy's first law),


[S )(1 +f( 7-/D ) = 0; (2.4

the boundary conditions on A ,



S -o, +* (2.5)

the boundary conditions on A ,



2. ir. = 0; (2.6)

the strain-displacement relations in oV,



= i (2v1i, + v 7f i ;) > ((2.7)
the stress-strain relations (in terms of the strain energy density)
in oV,


0^ I -I
= 11.) -(2.8)

These equations are geometrically nonlinear; and depending upon E ,
the equations can also be physically nonlinear. Later, in Sect. 3.3.4,








it will be shown that by a proper choice of E the equations are appli-

cable to a certain class of plasticity problems. Perhaps it should be

mentioned that in the literature yij is known as Green's strain tensor

and Si. is known as Kirchhoff's stress tensor.

Examination of the modified Hellinger-Reissner theorem shows

that by certain modifications, which are practically self-evident, equa-

tions corresponding to alternate theories can be obtained. For example,

in the classical theory of elasticity, the strain displacement equation

is






it is obvious that this expression can be obtained from the theorem by

omitting the quadratic term






from the third integral in Eq. (2.1). Thus, Eq. (2.1) would read







0 )d




which after carrying out the indicated variation becomes
which after carrying out the indicated variation becomes












S + ,o (orf 'i)] Sv dv
+ (, -S ,S.)6v. dA +- -sS' dA



Y


0~ 2 i)]Si "

The application of the fundamental theorem of calculus of variation,
indeed, yields the classical equations of elasticity, provided of
course, that S is now interpreted as the Eulerian stress tensor
and E is chosen to yield Hooke's law.

In the study of plates, theories have been developed which are
based on both the general nonlinear equations and the classical equa-
tions. In addition, intermediate theories have been developed, for
example, the work of Ebcioglu [16] in which a strain tensor was
employed as follows:



y+ + //3 L ,i zj)

where the comma denotes the partial derivative; a more restrictive
form of the above strain tensor is used in the well-known von Karman
theory. Because virtually all plate theories are based on simplify-
ing assumptions, it is sometimes difficult to establish a consistent









set of equations, particularly when the equations are developed by the
*
direct method which is often employed. In general, this difficulty is

overcome by using the modified Hellinger-Reissner theorem because by

introducing the simplifying assumptions in a systematic manner the

theorem automatically produces a consistent set of equations.








































In the direct method the basic equations are obtained by
considering the stress resultants acting on a plate element and then
applying the equations of equilibrium.














3. GENERAL NONLINEAR THEORY
OF A SANDWICH PLATE


3.1. Description of the Sandwich Plate


In this work, a sandwich plate is a three-layer, laminar

structure of constant thickness. The layers may be dissimilar mater-

ials of different thicknesses; however, the thickness of-each individ-

ual layer is constant. In addition, each layer is intimately fixed in

relation to the other so that the sandwich plate, as a whole, is a con-

tinuous medium; thus, the displacements are continuous throughout,

including across the interfaces. As a result of the application of
*
Gauss' theorem to obtain Eq. (2.2), the plate must be a regular region;

therefore, there are no cavities or holes through the plate. Also, it

is desirable to require that the edge surface is a portion of a right

cylindrical surface. Naturally, the terminology "plate" implies a

flat structure (as distinguished from a shell which has a curved sur-

facel the thickness of which is small as compared with its other dimen-

sions.
**
To properly describe the sandwich plate, a coordinate system

must be chosen; naturally, curvilinear coordinates should be chosen to

utilize the general form of the variational principle. However, to


See Kellogg (17].
**
All coordinate systems will be taken as right-handed.




16



take advantage of the geometry of a plate, the curvilinear coordinates

can be specialized without introducing a loss in generality to the

theory; these specialized coordinates and the rectangular Cartesian

coordinates to which they are referred are shown in Fig. 1.


3
X


Fig. 1. Plate coordinates


Here, the 83 coordinate curve is rectilinear and orthogonal to the

1 and 82 coordinate curves; furthermore, the functional relationships

between the 0 and x coordinates are


3 3
= x


+ const.,


X3= e3


4 const.,


S = "9(x ),


and


CK aO





17



where the convention is now introduced that the Greek indices (S,B,v,...)

take only the values 1,2. Note that all planes parallel to the upper

and lower faces of the plate are described by equations of the type


3
=3 const.

By introducing a translation of coordinates, any plane parallel to the

faces can be written as


_3 3
0 = const. = 0. (3.1)


In plate theory, the usual procedure is to write the equations in terms

of coordinates such that the middle plane is described by an equation

like Eq. (3.1). Therefore, it is convenient to choose the coordinate

system for the sandwich plate as shown in Fig. 2 where the middle planes

of the upper, middle, and lower layers are, respectively, described by

the equations


3 3 3
= 0, = 0, = 0; (3.2)


and are related by the equations


S3 ( h

(3.3)
"3 3



where 'h, h, "h are the thicknesses of the upper, middle, and lower

layers, respectively. Hereafter, the notation of a prime, a bar, and








X3


X2


Fig. 2. Coordinates for sandwich plate







a double prime will be used to distinguish quantities associated with
the upper, middle, and lower layers, respectively.
For the above coordinate system, the metric form


4) = c1 dO'c'i9
becomes


^ a l a de p + (610'3
where, from the definition of the metric tensor,

2 dxr JX

(3.4)
o(3 = 0 = .3 0.

As a result it is easy to show that all the Christoffel symbols with
the index 3 become zero, i.e.,


[.3,j] = [3i,jJ = [ 3j,3] = 0,
(3.5)

{jA} ={3j' iJ 0
also

o(3 33
= ( ,, 0 ( = / 0, (3.6)








Because of the above characteristics,













where the comma denotes the ordinary partial derivative with respect

to the 9 coordinates.

The coordinate system is the Euclidean space case of the

geodesic normal coordinate system discussed by Synge and Schild [18];
ii
as a result, when a space tensor, say S is split up into the
V yS&,4 S 3 333 (.
components S S S the components are known as subtensors,

subvectors and subinvariants, respectively.


3.2. Displacement Functions


Throughout the region of the plate, the displacement is assumed

to be a sufficiently well-behnved function that can be expanded in a

Taylor series, at least to the first two terms. For our purposes, it
i 3 3
is desirable to expand v in terms of 83 about the plane 93 = 0, i.e.,



v'(OJ) = v'(+0,o) o3L v '( ,o),j.



Some authors use the terminology surface tensors, e.g.,
see Green and Zerna [19].







By introducing the notation


], ( ) = nJ (0 ,6) ,

S(00?) (6i~0) ,

the displacement can be written as


2 (6J) = a('f"}) 6 (8). (3.8)

The function ui represents the displacement of a particle in the
middle surface. As mentioned by Novozhilov [6], the function *
characterizes the direction of a fiber in the strained state which
was initially normal to the middle plane. Note that fibers which were
initially normal to the middle plane are not necessarily normal to the
middle surface after deformation, as assumed in many plate theories.
By extending the above to a sandwich plate, the displacement
of particles in each of the three layers can be written as


v ') = (B) + ^f (}) ,
1 -3


(e ) ( g')- e .'(s ), (3.9)

"z,-* C9j) # a2( ()

By use of Eqs. (3.3), the equations can be rewritten in terms of the
coordinates of the middle layer, i.e.,












.ir.(. ) ( -3 i'


(3.10)


These equations can be related further by making use of the fact that
the displacements must be equal at the interfaces (continuity condi-
tions)


=7!-

-t
-wZ


-at3


(3.11)


-3
, at


Substitution of Eqs. (3.10) into (3.11) gives


L'=L I[
i lzi +


't = Lz


- I I


4 vp


v2 I- i[
u.< : = 'i -j

^Vt-1


Therefore, the displacements can be written as either


i i


(3.12)


(3.13)


v7.


h
2


r
2L


2 c~;v'- ;1;

1 2 Yh













-33

1 -- + .14)



or


= {- (+'f -"h,')l


2/ 2
(3.15)
U / + i
h '





In applying the variational principle, either Eqs. (3.14) or

Eqs. (3.15) can be substituted into Eq. (2.2). If the former are

used, the resulting expressions will be in terms of the quantities


-i r i /, I


or, if Eqs. (3.15) are employed, the results will be in terms of


I C If I I


In either case, there will be twelve unknown displacement functions

to be determined. Regardless of which set of variables one chooses









to work with, the resulting equations can be reformulated in terms of

the other variables through Eqs. (3.12) or (3.13); this will be demon-

strated later for a simple case. Because Eqs. (3.14) are more compact

than Eq. (3.15), the equations will be formulated in terms of the

variables
















3.3. Fundamental Equations


3.3.1. Equations of motion

Because the variational principle leads to a rather lengthy

equation, i.e., Eq. (2.2), which is awkward to handle in its entirety,

it is desirable to split up the equation if possible. Examination of

Eqs. (2.2) and (2.4) shows that the equation of motion is a direct

consequence of the first integral of Eq. (2.2). Therefore, the equa-

tion of motion can be obtained from the equation





S( r (3.16)
When written in terms of the subtensors, as discussed in Sect. 3.1,

Eq. (3.16) becomes











=S 3(65Y +// s"v '<,= *,p,, s r-/)
0 6




+[,~au s0
(3.17)
For future use, it is convenient to rewrite Eq. (3.17) as follows:


where


L3A I+ ", B33 p. '0}-




S3)3)
_i w
^V 3^-/


(3.18)


(3.19)


C3 t3 (
Bi3- S~j61ly


5j/) 33(6 t,3) .


The total volume V of the sandwich plate consists of the sum of the
0
upper layer volume 'V, the middle layer volume V, and the lower layer
o
volume "V. Therefore, for a sandwich plate, Eq. (3.18) becomes
o








+ [-"-/ 0 d


!i (~l~l *, 6 ..

0~'/f +--.N 2

V
\,w^v
^{[^-**}
0"AP,i


+""j i, 61i0.


(3.20)


Note that


d '= d dA,


Jd7V= d3d A,


dA 6-14= DE


Then, with Eqs. (3.21), the integrals of Eq. (3.20) can be combined
to obtain the equation


b
[ + ... )
[1 +.)


tc


( '4Y


) (h- 1-+ t ) /]
(% p Y3


(,4"P/a


F
d


+ .. ) i ]


j1)


(3.22)


where the limits are


a-h


2


S- h
2 '


2


(3.23)


c h
2


where


(3.21)


dv = d-dA


d~".


6v +


_jg3 -
d'4=0


...)


+[ [ /






Expressions for the displacements were established in Sect. 3.2; thus,
substitution of Eqs. (3.14) into Eqs. (3.22) gives


'o "/, + + ''E ,3 ")_





51) + ...]J +... ?j/
-f+ -) /

(3.24)
b (c




a.


3 -"
+_ ( h) / f d = 'O 3
+ [ -fat 00 "
[ -1




28

where






+ + Al l )
"' /i 'i] f: 4 +




=+ ( + + ),

'b s = 6, 3 ,-3



(3.25)








+ s(6 *F )6

^h,- (/ -
i '" gJ"1^^ }






Note that
L



S3
b


,3 d 3
, 3d8


a


- ( )


Therefore, Eq. (3.24) becomes


f [ vf
O~A


+b
^c


+(


( + f
^''-7d
re


[ 5" C


cL
k= b


,/1")) 1cL
C
+Lf')1+'
-I- *%


Ik < 3-d 6 oc


- )- 3
-J, d/ -J


C


- f4E


3 o3b
9 E )c'


2


)C


-3 -3 r/oAi i,-
oI --)L1 ~Fd


+ ( o3 3)


C


' d"" -J


43L


(3.26)


d3
d


+ (


A (y
E.v


+ (


.. Q3


0(3 b
E c


rC)s"ia'...I dti3


h a
2


!


( -: 42)"/ + '-] 3-







S(63


a
+ f 3be"//


S( 33


b
d


+ 33 ),


1. (733)


f-


33 C2


A
2


-3 33
t( E ,


-3
9


( ; 33


+


af


a
-b


(3
9 -


3 33 a
^'^hb


+C


-i(C(


I;


( 3)bI }


d


-3
94


S33 3
EJ6


C/
0-


I 33 -3
Ea'6


t, )
3, 33C);


" d3


-
2


+f


+-t--]d 3


+;


3 l3


37


- ..


3


33 -3
E dd


C


(y3L


2


= 0.


(3.27)


E d

-oD)


] di3


d6-J 3


fb%


-3

<+ ...] i


S ,'-3
i:~~ )l 19


- l] d


3 33C






SNext, stress resultants are defined as follows:


'N^, Vfi~, "Nj
;IV Ci~p /V N"P
W, 'v-r
(P # ce#


T T
TT


6 C
-3 6 I 4 '3 I
' f 3


dJ ; (3.28
d6 (3.28)


3



5


d3


(3.29)


(3.30)


7Lb
C t
--i 1 }.
"Fr; \ J{


Si3
[ = ,


b

L

, /


-3 i3
0 E.
3 3
- 7 j3
8L3~;


=I


1r
u-r


.+ -#
'2 /P


{ = (-,) _
oh 3j
-3
2


I'-
Sc -
-
c


ic
,C1
-L
C


-3
fI


(3.31)


-3

(3.32)


(3.33)


H+ 2 l / zL
uiui + --


I'
T,


I'/M







,/ II
[/ 7'
Si


,iL
pL
P'


I/


li}

/'I
0)'


(3.34)


33, 33


t


oj






By applying the fundamental theorem of calculus of variations to
Eq. (3.27), and by using Eqa. (3.28) (3.34), the equations of motion
are finally obtained, and can be put in the following form:


+{ & ) + ,) N ( h ) I ,
((kP-lc~'~2 N, )'Y "Yel


+ h- ( '- /3 ',) + 'Py


[J / -t J

#, + a., +
a +a +


/ e(,v, "v ) i,


(3.35)


- 0(74(/j


zU K Z'}I(3

h ~ j"-qp


= '- 33 ^ 0? if ) +

-2 a < 9


(3.36)


17 //An~d h~B~~%
(I 2Sc ~B~ j;yS"). NIPd~c6~tlLd~)t 2 (/t~l 2







;4 -(Z' )


) W


- A .i+ Wj


<8~


C C


(3.37)


UBj/


_t o//
+ 4 ^ ,-


)'2/A /3


a- ( p/I l


2


(3.38)


'{I ) (, ) +
.~zh jyB~~paj.M BI )o


/+P3 ) #d?/(


1+51)31


+ 3t3 # 3


lC~h


- ,,~ or'
--a-- +/fl
2


2
-,,- i
, ('M +


+ (i)P


(3.39)


4+ F ,


f ~" j~


r/J)


;Y 3j) 1~0(


ci~


-nl /2


i3dj6d,


( 1 'Y ) -


h ~2d t ~md
2


(, + f


N/,)Pe


u l))-


S79iP


SV 33


+cr%-


-' r C
9 16 [+


k /) '73,,e
` M
N )~


-,


S 3 -a,+ ,


i~)










I;
z


(P3-p ) =


/~l


17,


~i,)


T+ (4-3)(I+?


3


2-h
- IV32()


p


~P~lt51/3


X)


- 3, Y )]


It
a3


/b3


I3
I#C3


I,


-'L 3 .


) 2\'


^2


2


-2


- 4


^3


I
2


,3)


(3.40)


(Ii


[I 3 ; e


f


S
2


3 /
eZ3 13 )3


C3


(3.41)


N f -Pl)]t u3


-th 3d. 4jLS,
2 V1


kS~,RB"I~J,,


;;/B", IjV3,~/


tcTP~


- A") (/


rh ~P~jlirlf~


-t
Z" iU3~~)


(040-


~4P~,, ~)


~~pa~_ h ~1/3~~+ ~B~


(~I'"~~


S( 2 /' r W


~
irP"I ~V3, d ilp


-1/33)(,


-~dLLC3,d~


h
z


(3.42)










3.3.2. Boundary conditions

It is obvious that the boundary conditions, i.e., Eqs. (2.5)
and (2.6), are a direct result of the second and third integrals,

respectively, of Eq. (2.2). Therefore, as in the preceding section,

the boundary conditions can be obtained from the equations









o r2r
A L ^i ~S() 6/ )A =A (3.43)


The total boundary A of the sandwich plate consists of the top sur-

face fA, bottom surface OA, and the edge surface A. In addition, the

edge surface consists of portions of the upper, middle, and lower

layers; thus,






01 = A-A1 A). (3.45)





Let us first establish the stress boundary conditions which are

obtained from Eq. (3.43). In terms of subtensors, Eq. (3.43) can be

written as









In
0tJS~(d ~ii~M 5i3a


A S


fu~F/


03 /


33 2 e
^3 %Y 2,3J


(i 7V3,3


-,13 33 (


3 T 3


dA =0,


-/


A,-1 _I C4)


f1P ,. 335


-41


S)SJ =dA


where Eqs. (3.19) have been employed. Because of the geometry and
coordinate system, observe that:


on T kS 7


on As
a '


I


II



= 0


S,

^


and on A )
Alr


n3 = 1. o

SIn 0-
mo 7 = -/. O
o J = 63 &

/F o 0 3 = 0
a 0 3 a 3


+ I


%3
64


-o
0(M


-, P3
o S


- 3


2.7


o S


(3.46)


O


,S ^Y








Also, note that the edge integrals can be written as


(-**) dA


(.


(.


OAi


4dA d


(.


Therefore, Eq. (3.46) can now be written as


if


/a ) I


- 4


- 833) j


JdA


'3
+ (51*


/, f -3
3]\d6


-42I
o/f3


A0/)6'3


S3.0.
(3.47)


Next, by substituting the displacement functions as given by Eqs. (3.14),


Eq. (3.47) becomes


( --) da ds


E5


5


-3


-3
a)il6AU


1+ 3
Si5,


,I
"s3
"<


" 633) & v


A") b 33

s)^+(^


('C


"/I


-3
(s*


-^ I .3
i3


,


e3


:3)
^y3)^


4(


y,
1


Y))

















it( sf
'-3



"/ *(5


- k33) [ 6 hZ


//


)S'~v3]


) [') 6 C'


'I
,--


a' of /J o(


C
-3
+ ( +t


2~d8)


Ni~


+ a -1( -
t~~is- h\ ~u3 'B'3
z/Jb -


'-3 of 36


-j 3
I d,12,


where Eqs. (3.25) have been employed.


.2


-3
f( -


S 3


/7


6) ]


dA,


h
2


3 ;


3
(6s


2


- m


C3
C i


= 0


(3.48)


i q3 )3
1E + h ) 6~ 6


- h g+

6^+


t- ~E
~~"')j6Z


-3
+(Bt


0(
S
;k


3
l3-)62

0 )3B7,u








Now, let us introduce the following definitions:


S0 o 2

/ 2


1 5 Z s
01 ,o 01 20


Then, with the above equations and Eqs. (3.28) (3.34), the boundary
conditions are obtained from Eq. (3.48) by applying the fundamental
theorem of calculus of variations. The stress boundary conditions
are as follows:


on the top surface,


/ I /
,21 ,/


2


I L


on the bottom surface,


, + ,


/I // L
,C, +, = -O


on the edge surface,


(3.49)

L I/ I
c S
C!b3 ICi;d4


S
II
5,


(3.50)


=0,


(3.51)


" -3
-3 A T
63 AT ^|
S$ b


I/ L
bJ ,,
"(vt







o +., ,C / 3 ,i, ),

h / {I = .- (('v R PC "w" ( ^f {^ l,)
+ h 5
+ "##, W ;t)(


(3.52)


-h /' /)Y/ (/( V' Y / 7,


(3.53)
S) ( V /, + Z / J


(3.54)
)


3 3 1/3 O (
+0 + = 0/ (N % If


+ NI ," t


(3.55)


- y 3 y


(""" ~"" I~/


( h ) 3 ,,o7 +Uj s ) ?


(3.56)


1-
0


2h


+h h
S2


+(114+ 2 /V y i ,o


)- = a{[h (;e-v ^]s /J


+ 2 7
)k "v4') t K 9 p y,


SC^ iw,' /2


{(/ -


ii;v /3d -' pd /8_ h
2 -h ~J/


( +AW P f2-7/
11 -, ( 6 w-t l


=0


Lt ^ f h


;"~ ~P,










S(f& i k%,


/U,


(3.57)


+ ~U*d


(3.58)


(3.59)


Now, let us establish the displacement boundary conditions
which are obtained from Eq. (3.44), i.e.,


= 0.


S J


- 3 i 3
- (6 5)


-41
0(3


- (" vM


2


(I u )


2


a3


(,u3 )


3
0


13




[A)24


Nv ) M- ]


+[ ( -/v // ') '


VI]-,, +


// ) 0s,3 Y


= f
0 /I (ta


V ')( Z- 21


i~d~


/13Y- / t k/ e'l


iiia d-I- 2h


Al )(M3'l


wV3


fiTr


I 7t-L 7/;Z








But, from Eq. (3.45), the expanded form is







/. f2 .




/--
V f

z dV 0 (3.60)
for the three layers. By again applying the fundamental theorem of

calculus of variations, the displacement boundary conditions become:


on the top surface,



2^- ir,IZ = ; 0(3.61)

on the bottom surface,



S- 7 = (3.62)

and on the edge surface,



#/ l/# .
,r- f. = o.



S V 7S 01 ,(3.63)



V -- = 0.








3.3.3. Strain-displacement equations

The strain-displacement relation, Eq. (2.7), is a direct result

of the fourth integral of Eq. (2.2); therefore, consider the equation




(3.64)
The notions employed in Sect. 3.3.1 are also applicable here. First,

Eq. (3.64) is written in terms of the three layers of the sandwich

plate, i.e.,


at'





Jy

o
^L


V


/I $


(3.65)


Although the integrals can be combined, there is no advantage to do

so in this case because 6'Sij, 8 ij, 6"SiJ are independent

quantities. Therefore, let us consider just the first integral of

Eq. (3.65) and rewrite it by splitting up the tensor as before;

then, we obtain


W / 5 V
i{ 2:lT i^/,)]85^1/--0.









-- i il 'v ,^]^
0






t 2' (2 /,, + if3 J,3 +





Now, the fundamental theorem of the calculus of variations
can be applied, and the strain-displacement relations for the top
layer are



+ ,I+'v j i 'f k?,.),


(3.67)



I ( 2 Y,3 t,3 o ,
33 7(2 ~,3 t / "







Note that


S',< ^V i


therefore, 'yi = 3a as expected.

By substituting the displacement functions Eqs. (3.14) into
Eqs. (3.67), the strain-displacement equations can be written in
the form


I
'=
/ 3

(33


/ -3 32 '

/ I3
3o( oe43 / 43

S3
zab


(3.68)


13O( i13,p


+


2


+U//3 'fS-)


+ 3i ( % c( a3, J (3,.69)


where


"ir6~i/d,


;J% j~13


"iP' L~ai,'~%


( v~/B u/r~


i Vliid tirid) iZ %


4_ e








(


/7~ ~'1K
2 h 7I T'


-- / /
+3 %3c *I3,



3


I)


f L


+2( 3,
b2((i,d


- 3,) V1 3]


/ / I // t i3 3
2 Y ( ^r^1 3 f3;


(3.72)
(3.72)


S3, ') (3.73)


Y3 13


2*


(3.74)


/


2dlp


0 o3


(3.70)


(3.71)


73


o3


il j /,( --


I
z(


iir~ jy~ j~


1C~,d t 1L % ~~


Sp) ,


r
21


'u~


2/ (Z;Uj t iUYjy~t










The strain-displacement equations for the middle and lower

layers can be obtained in a similar manner. However, because the equa-

tions are rather lengthy, it is sufficient to observe that the equation

for 'v.j can be obtained from the above by substituting a double

prime for each prime and a h for each + h; similarly, the equations

for v.. are obtained from the above expression for 'v.i by substi-

tuting a dash for each prime and by setting h = 0.




3.3.4. Stress-strain equations

As before, the stress-strain equation Eq. (2.8) is a direct

consequence of the fifth integral of Eq. (2.2); therefore, consider

the equation




2 *


In precisely the same manner as before, the integral can be written

in terms of the three layers; and, because A' i, 8.ij. 6y i are

independent quantities, the fundamental theorem of calculus of varia-

tion yields the equations





S--- (3.76)


2 '-


VI~ Li(d









for the upper, middle, and lower layers, respectively. Now, in this

work, stress-strain equations in the plastic range are desired. In

addition, the results must be consistent with the theory of plasticity

for small strains. The following method of obtaining the plasticity

equations for finite strain is basically due to Novozhilov [6].

In the modified Hellinger-Reissner theorem, it was assumed that

the strain energy density is a function of Green's strain tensor, i.e.,





If it is further assumed that the material is isotropic as in most

theories of plasticity, then the strain energy density can be written

in terms of the invariants of the strain tensor





where






/ / k Y _

(3.77)


SY Y Y *


zf


*48








Then, the first of Eqs. (3.76) can be written as


where the primes
Eqs. (3.77),

IT,


12


T3


Substituting Eqs


have


been
been


+
dropped fz
dropped f


r cc N"d ,13

4 d91-T3 yt
or convenience. Next, from


(3.78)


',j '


(3.79)


Y /


'2


. (3.79) into Eq. (3.78) gives


J'K= 2- 4j. 4td



d3


(3.80)
CI )


Similar to classical theory, a deviatoric stress tensor can
be defined in terms of the Kirchhoff stress tensor as follows:


T J




- J k


SI'


I -
-,57j


s =r,








where

k


Also, in most theories of plasticity a yield criterion is assumed.
We assume a yield criterion of the form


T- Tj 2K (3.81)
J
T t

where K is a material parameter depending upon the amount of strain;

note that Eq. (3.81) reduces to von Mises' yield criterion for small

strain theory.

When the yield criterion Eq. (3.81) is rewritten in terms of
the Kirchhoff stress tensor, we obtain



s = Z .3.82)
J 3 k I

Similar to the strain invariants, there are also stress invariants,

the first two of which are

k
(3.83)

/ e-,Sk 1 4" P k 2


* In terms of the stress invariants, the yield criterion Sq. (3.82)

becomes


2 2 K (3.84)









Also, note that through Eq. (3.80), the yield criterion can be written

in terms of the strain energy density and the strain invariants; thus,









Now, in the deformation theory of plasticity, it is assumed that the






strain tensor is proportional to the stress tensor, see Rachanov [20];
(3.85)
In addition, substitution of tq. (3.80) into Eq. (3.83) gives



,-- = (3.86)

Now, in the deformation theory of plasticity, it is assumed that the
strain tensor is proportional to the stress tensor, see Kachanov [20];

therefore, expressions of strain invariants are proportional to corre-

sponding expressions of stress invariants. Eqs. (3.85) and (3.86)

can be made consistent with this proportionality requirement if





and thus become

1 2 2 2T _.z-

2 2 ) r (3.89)

S -- -2, (.9
Jl 3 T ~ X'cL j2









Solving Eq. (3.88) for E /Bt2 yields






2 2
or

2 (3.90)


where


(3,5 S

(3.91)

( 3 -



The numerical factors in Eqs. (3.91) have been chosen so that in

classical theory S and o reduce to the octahedral shear stress

and strain, respectively. Now, Eq. (3.89) can be written as



-3 ds, i

or


2,I =) ,.(3.92)


Thus, the expressions for ia*/bI have been determined, and the

stress-strain equations can now be established by two different pro-,

cedures.








For the first procedure, Eqs. (3.87), (3.90) and (3.92) are

substituted into Eq. (3.85) to obtain



S3 3

or


5 = 2 + 1 -iSo /7 (3.93)


As already noted, J1 is proportional to Il, i.e.,



j= 34k (3.94)

where k is a physical constant of the material and corresponds to

the bulk modulus in classical theory. Then, substitution of Eqs. (3.77)

and (3.94) into Eq. (3.93) gives



= 2 3k 2 )'. (3.95)

In addition, the inverse relation can be obtained from Eq. (3.95) by

using Eqs. (3.77) and (3.94); thus, the expression for the strain

tensor in terms of the stress tensor is


3 )(3.96)


Eq. (3.93) is Hencky's stress-strain equation for finite theory as

given by Gleyzal [21].








For the second procedure, it is desirable to have an explicit
expression for E (Vy ) in order to apply the equations resulting from
the variational procedure, i.e., Eqs. (3.76). Consider the expressions
for ~E /bli i.e., Eqs. (3.87), (3.90) and (3.92), which are, respectively,


--6,
233

--= 2. -

Cd12I + r



Fromn Eq. (3.94), the last equation can be written as


(3k (3.97)

Then integration of Eqs. (3.87), (3.90) and (3.97) gives, respectively,







Z*T (3+ Sout



Together, the above equations imply that


Zj2So2^i(3k + Y)f ,
^1 '







or, from Eqs. (3.77),


S b kjd'
y ak~


S(3k


(3.98)


02,~, '


which is the desired result. Naturally, Eq. (3.98) can be put in many
different forms, such as


kr fs 5


-2 kr s
-2 lY y


(k 3
(3.99)


ki


S r A kl [ / sI
d'^ A^j^


(3.100)
L J(
(3.100)


Observe that from Eq. (3.99),


kr is
3 y


(uA h S+


As '


- (3k-2


0~~YJ


+ / rb


Sl '
0, / ^


dZ*"L


JY H


*8


= -
p


fi
J1
V


Because


4j
ik


+ (3k


' ( YA


+f G
s.,


b"








the equation


0V V


becomes


S
2 -
YO,


which is precisely Eq. (3.95).


Later, it will be desirable to have the stress-strain equations
in the forms


= E


i2fA


which will now be developed.


Because of the symmetry of


y -= / Li k
t 2 (


ii
V


i"k)


k SI / r ks
k= (


Is kr
+? 9


Therefore, Eq. (3.95) can be written as


* ;.


y 3P


-iv


and


ykl


S


YJ A,









( Jk -
(8. 5

(3k -2 s
bo


dA jk
+3 Y


) (9r *ks
Ay y 3


i jk)


(3k-2z !


) 9'/ rs


Similarly, because of the symmetry of S Eq. (3.96) can be written as


,/


3


Sk ji


/ 3 -
L 3k 2 2


Thus, the stress-strain equations become

3j= ,


where


S9 jk)
49


(Jk-2


(3.103)


/I


is kr)
Ik^


Jrs


3 S.
0


ik ij
9 9


I 9


61


-,-
3


ii ) 'Sk


& *1


(3.101)

(3.102)


Vka


/ ik 9









r( k jk
5. ^ 9


9 9" 3 3k


/
'2


5./(3.10
(3.104)


Note the symmetry relations


e'ki


= E


.jikA


= E


- =


ky iJ'
klKJ


*i&.j


(3.105)


Because of the preceding work, it is desirable to have the
stress-strain equations in terms of subtensors. For example,


3= E3S
rs


2E 3 /'
/3

( 4 O


(333 /
+ E 33


S = E3


3333


y,


But because of Eqs. (3.103),


a (3
Eo 0


=5y


< 333
= E


33/3


Therefore,


f V33 )y
33 )


.Aw


I/
==


(3.106)


-(3 Y3


(3.107a)


2 E33S


= 1.










S33 33(5 3333
3 3 E Y 3E (3.107b)


where, from Eq. (3.103),









_3 3 S, a'
S= (3.108)


3333 50 -k.


From the above, the subtensor form for Sq. (3.102) is obvious.

In this work, it will be assumed that the above stress-strain

equations apply to all three layers of the sandwich plate; however,

it is not assumed that the material of each layer is the same. Natur-

ally, as indicated by Eqs. (3.76), different stress-strain equations

could be applied to each layer which, indeed, would be necessary if,

for example, the middle layer is assumed orthotropic as is often the

case.









3.4. Buckling Problem


In this work, the bifurcation theory will be used to formulate

the buckling problem. The statement of the theorem, quoting from

Novozhilov [6], is

The moment of appearance of a possible bifurcation in the solu-
tion corresponds to the critical load. Hence, two positions of
equilibrium corresponding to an infinitesimal increment in the
critical load differ from one another by an infinitesimal
amount.

Note that the buckling criteria is associated with two equilibrium

positions; therefore, all of the preceding equations are applicable

to both positions providing, of course, that the inertia terms f
o
are omitted. To distinguish between the two positions, a dot will be

used. For example, the displacements corresponding to the position

which becomes unstable will be denoted by vi, and the other (stable)

position will be denoted by v.. Following the procedure of

Novozhilov [6], these displacements can be functionally related as

follows:



;= t7J V. (3.109)


where 6Vi is the infinitesimal change that occurs. Furthermore, we

will assume that the functions V. are finite and that 6 is an infin-

itesimal quantity which is independent of the coordinates. Naturally,

it is assumed that similar relations apply to all quantities; for

example, the stresses corresponding to the two equilibrium positions

would be related by the equation





61




S"A= 6A (3.110)


Corresponding to Eqs. (3.9), the displacements for the sandwich
plate in the stable equilibrium position would be


I* I 3,e

3

-- L" t (3.111)



S-; ^. B .
= 6 3 #


By applying the interface continuity conditions as before, and by
introducing the notations




(3.112)



we obtain the following expressions:


I (/ f ) +t & -

-3
S= ; (3.113)


V.I =6.-3'~


which have the same form as Eqs. (3.14).









Now, consider the stress resultants as defined by Eqs. (3.28) -

(3.34). For the stable equilibrium position, similar expressions hold;

for example,






By applying Eq. (3.110), an alternate form is



I I4O(/6= ( I I + 6 -3 -3


which from Eq. (3.28) is


3



Ndw, since the relationship between 'M and 'M which is consist-

ent with Eqs. (3.109) and (3.111) is






we find that




6,
as might be expected. The variation of the other stress resultants

are calculated in the same identical manner; the results are as

follows:









I O // <

/? r '}
/A1 J W


- { 3( 3 1 ) 1
(3.114)


(3.115)
(3.115)


33 1 33) 3
3,





W \ ur Ur ,


3 3
Ir d/

In


{'Z


; {
-


^P
IL


-L3

Se 3
e


-3


b -b b -
Ur ur LJ --


de
-3 i3
6e

9~3 e


}


(3.116)




(3.117)


S3 '
-3

i b
(3.118)
/W C
2 C.
d71- e4


(3.119)


The quantities 6e'


are calculated by substituting Eqs. (3.110)


and (3.112) into Eq. (3.25); thus,


(rVV4%,


I ~o0(/


, i3


C
i


/I&


L


P, C
fc1 I-
=/c


L 'I


2 /
4 4 4


>







-(b.V3~ 62a~3)[K6 f tL/ ,LL l


+


-3
+ ( k/r + & //r) +


1 I .


(P33


33)
+ s^)(6i


+ L
+

E 3


2
2.


- /j3


fh


- higher order terms,


/ :3
+ be


+ higher order terms.


Therefore,


+ L/, 7f (/,


-3
9-


2


I '


/ 33 /
(3.120)


Si3
E


(~i


KL)


+ 5Y7S i/,


~~7)


-3
86


Sl33 (3 3
#f 6^, (S3#


72YI


(~u%


i3 S 3 t L1
b'd [66 1_b -t-al


jULld)


O /9 ( 6 f-
14.1 3


'33 / z


S = i3


s~'"(6;


e i3
be


I t


'/


-3 ,
)tB Sd









and similarly

Si3= 3( 3' /1&3yj)+6f5 (cf')


Si'3


-3 y -33p 3
t 6J f /d ) 3 f


-. '3 i
=// OA6 [ Y


(3.121)


-3 y ,






3.4.1. Field equations

With the inertia terms


)c 1-3 33 \33 ,

S(3.122)
(3.122)


fi
o


omitted, Eqs. (3.35) (3.42)


represent the equilibrium equations for the unstable position. By

replacing all of the un-dotted quantities ('NC u etc.) by

dotted quantities ('N u etc.), the equations then correspond to

the stable equilibrium position. Thus, by using the form of Eq. (3.109),

the first equilibrium equation, i.e., Eq. (3.35), for the stable posi-

tion can be written as















(- ] ( g#Ir + 1






z N 4 ^ In S{ "') -t Ri




S+ V f0 (3.123)


Since Eq. (3.35) is still a valid equation that applies to the posi-
tion that becomes unstable, it can be subtracted from the above equa-
tion. Also, higher order terms such as


6^<^7.~(l


are neglected. Therefore, Eq. (3.123) reduces to








AlfPY tP Y) \ t. (A i ( b6 #tX)(f ILf)
At N -tN I// I t tI


i (^ 61 't7


Vv/


This same procedure is applied to the remaining equilibrium equations,
i.e., Eqs. (3.36) (3.42); then by cancelling the 6, the buckling


equations are obtained as follows:


^(W ) .-Al

t /irY- h /)-t/


fl,( II +
*(s^-Hs^)'^~7a~ "


1(4


t -y 6 +


4"6~d


+ 5 1


+ P6 7 &' P = 0.


, p 1 ,
4-) o


+ S~B








f(eY j + (m In )(e i,)


--p"


i+(^A2


4A) ;~~/?~


+


/,~3i
f~P q2+


jjc~,
K


/fl2
34z


--itf : (N N


p;P4


IL


Vt)


(, Ps-


#) 1//
^' ) 9' l


I! "


fl/s


33 -
(1541) P


- 0;


'~/6*


t JDS Pff2~


j (3.124)


I;
Ut
I;


h
2


( s ,+ -Z /y ) +t


h
/,,
-, -


(se/p


(P-


~I""~P~tn~j"'l ~%


V1) + 0'14,


'}) Y /


(//


// f" 1
Nt ^


-p y


~Plijd+


cw" ji~P u) i MPql U % + iS


ts Y, / A\ gy^*}/ -e /; \
(, -/Y/ ')4,; Jf^srU r/J


z
,~ylB:~1PY)t


/ CY T h
(-w 2


l(:'y + -i / ) '/


V -_


- 3 3


Ird -d
P )+ C


(3.125)


~~i;yP""'








Anl(Y(


+ 7;g/)


(J)^-h1 I
z N-,IV 4i


/ 33


y '/)J

IL(/


1 v33) '


//) '^"


x I];//


.tl L


It
-


ii(1)


#/ H33 '
'/- m ) f + f P i C


= O;


-+1ez
tU ly 4


(An


2


2


1~~~'>


+ (


17
2'


'P"


+ 6'


(3.126)


h
2


2/1


+[ (2


,>13PY


h
2.


Sh /" '
--zW f ^ r


_ A
2


ji,/j]


(3.127)


S+Fui


114 _


I' 1 l, /


+


2h iiiPf; d
-t


-g'jb'
cY


- it'~r ~


b'r u%


( 0/


( ,I'


(;i~


j N


(1/"


/ ,, [ )+ z t ,


-t /,L


/ I' ,]y /,


2
) M^'-i


- i"i u %


r


( "I/


( / -


33) y









S 2[ (A// ,,,- A'), ] ,

+/ (Wr ) t ^P^,


) I t


/("Y


+ P3
+#


+ )3


- /1 y h & 3,\
-' (n 1W ) 17,1


ii3


+' (/,6 t )


/' I (/' )


73 -3
+P+p


"P3


= 0


(3.128)


(;6 IMP4-T" -A +rIL3,(


+ ~-y v)


o-. )


Sp6K) i,


7. tL3,a


+ / 3 />3 ) 3
ID- )*8


,1f


146


=


(3.129)


' P' A ') U,,, t (,,,+ / +y "j ,r


, Y r) JP3,


* i Vp


ii j;y'i- ;yB~IMPIJ U,~
i


+ ( N R ]


I -


2 M


V^1 I
V J


h-
2h', Z


N


hw i
2


+ I+


( fp -3"-)i" -


(./, ") 3








{( 4 ) s, + 3,


fK ] F3,)


, /


- ~#


-~Y~B)


!- 13)
' t


- /,/r)] ( / V3")ls
I3
Np_~,


*3 3C


= 0;


(3.130)


_ 1-
2 Fb(LV3,i1Iv#Mp


l eI 3,Y /


(/' + ) L ')( U3,Y
2


(,+ ,3 ) p y [^), (f y, -[ '-
+i~hi P~~h t h


IIE


II
-/'


- N') P ,
3)( 3
33)(-) / + 7 0. c(3.
2


131)


) )+[(-2 h' /"


S )-h
)( + V/.3


2[,>, h .- h "]

- b"iuj~y


t (t Bi, ~SJ


( 44'e+ I At(


- f 4L;, J)


(i"




72


3.4.2. Boundary conditions
Here, the general procedure employed is the same as that used in
the previous section. All the boundary condition equations of Sect. 3.3.2
are applicable to both equilibrium positions.

First, consider the stress boundary conditions on the top sur-
face which for the unstable position are given by Eqs. (3.51). For the
stable position, Eqs. (3.51) become


ZC


I 2
- iP


(3.132)


= 0.


From Eqs. (3.49) and (3.110),


.zszJ

/

-Ei


I ~-,


+ 6 ALI


F i2 6e


(3.133)


4


+ (Si,


I 4
/ L


20*
Z,



Zj^


AT 1


A r


2 i










where


2* ^a AT -t h


S' 3-3 h
P = ATr + h.
22

By substituting Eqs. (3.133) into Eq. (3.132) and then subtracting

Eq. (3.51), we obtain





For convenience, the 6 can be cancelled; the stress boundary condi-

tions for the buckling problem become


for the top surface,




2 F x 2 4 2 (3.134)

for the bottom surface,


I I i i / L '
,P P 0, ,C e 0, (3.135)

and for the edge surface (neglecting the higher order terms),







,i +
o o 0


+[j (P-r )


) 1] 0,/,


!zt/ *1


'A


'](Q


j-[()z


// 1 V
N16y)


( /A-


2E
2


-p
2


Nm') /


(/" 2 /


2 (n/W +


S(3.37)
(3.137)


-- -f -+Lt )V
(i3 v v ) I
of Y


'

3.36)

(3.136)


( /A/,
o I


0


om


(~16~oc~


2


-Q {f# +


+(^ ( )(t W


tlLdlJ)+


( wy


I' /


.+ ]


~p"


(" "
Al= --1F


+ i (C
+ ( n -I


-f i


# ) /I W


p /
4 ;


6 ?et


t NP'3


(In In'


f,4)


)t/fnl"'


/I p~ -d
f~j4i);m .,P?,~PrlU i,


1K""1~~


' /4') //








2Al kh


0aiP


+ [(A/4-h4' A'] 7/(


h
2


-/i,)


cM"Nt b2


, 3 3
Is + 1
0 0


= 4
=o?6


p /VI


,I


2 o


+
2


+ 2('- ( y)


*(4 -S-


'It


o o


(3.138)


+ (m +


- oY WM h" -
2^ )(bt. 4 I


(3.139)


,i6 t/ +Yw )U A
P/ fil 0,/d' ~'("e


t i"Aiv,


11A
S/B)3
ii~


2 / )


//,,


+ (/W p+


.9.


))


(3.140)


2[ ()l' yn] 7 /, ,


P")IU~


Ol i"


+ h ""' ] ,


ii'~h


%"+ P h /v'^ k' % } ,


(Alt N N 0)


dA t/ /,,p 15 -
7 (& I ) -, / i


N^e f ,,Pt


y1) /'-3, +


3,


P
+4 ~4~3


if "I, )








( 13 i ) ( -' h) [
a aC2 -/'o/ (L2 /M J


2 I(n--)


[(,/5vp%.


/I t -cf


-2 / ,,


2)


+[(20 h ^ 3


# + +


+ K:(/,+~


_ p 3 + i i


S(3. 11)
(3.141)


2 / 2 A)


f (',I, ,, -/%/)(, Z3 + 2


X( N
^^-r/ I 1/hN r
'LI/ -+A^

( hIAIAB/ n^,Xb


S3
a


/ +

3',K


(3.142)


J


= 0
o /


'c(P: -/5P :)(1Z,(
A^In

fI/ g


In:' I -m 3)


2h 3r
K sY


2


,(i82


(3.143)


3,/


2


[/8~3


,3
0


=


" 3
1


- 0 7 J,


2 h f


; ~')1,3


c;;,P', Zh~prj


T16,s


3)Y3, ) f ( '


(L;P-


S 4


N1)(


-3 h -


/ A N
















I i
/o { 7 L
6n


' I i 1- 1 i
4t 7 -3
it i AT
14 *4
^-J


S2+h


2 "


Si( -L a< b

ot J 3 I
o Io -o





In the equations, the quantities 6-s are the variations of the

scribed stress vectors that occur during buckling.

Next consider the displacement boundary conditions which

the unstable equilibrium position are given by Eqs. (3.61) (3.<

For the stable position, Eq. (3.61) becomes


(3.144)


3



(3.145)

pre-


for

53).


t, I,',
?fr- 2r =0
L L


or


Subtraction of Eq. (3.61) from Eq. (3.146) yields



V 'V = o.


After cancelling the 6's, the displacement boundary conditions for

the buckling problem become:


where


( S + .)- ( + V.) o.


(3.146)


-





78



on the top surface,



V V. = 0 (3.147)


on the bottom surface,



V Vi = j (3.148)

and on the edge surface,






V o, (3.149)



Vj V= 0.


In the equations, the quantities 6Vi are the variations of the pre-

scribed displacements that occur during buckling.


3.43. Variation of strain

Naturally, when the displacements undergo the incremental

changes 6Vi, the strain tensor also experiences an incremental change.

To calculate the incremental change of the strain tensor, first observe

that Eqs. (3.67) apply to the unstable position, and the same equations

apply to the stable position when all the quantities are dotted. For

example, in the stable position, the first of Eqs. (3.67) becomes










= (3.150)


Substituting Eq. (3.109) into Eq. (3.150) gives




( i -+ a i/) (V t v,)i,
I'-








or, by using Eq. (3.67),


k< / + P + hhigher order terms


where


P / i1 6 1J,, t),4 ,/I





Naturally, the 6 can be cancelled. Also, through Eqs. (3.14) and
(3.113), r can be written in terms of the displacement functions
U T) 'I, etc. In this manner, we obtain the following equations:









3-3
S. ),


13


2


)]jli 34


(jW;3,14


+h;-


43

P733


= P'
ad 3

3.

o' 33


where


(3.151)


I
0/-7
Or(


6/0(1


(h( /sp- 2 ,l


+t[ 1/


[+ ,3


+I-,73)


-+ T
1 3.


3, )


/
- ~k3,ci


(3.152)


2h -
^('p^^
-Wl'^i


( l!


+ X( e


S2 ( j- 'W )1


tf(13


-Mif








#/g [d,


*i~~JU '+


B,


I
+


/


2


i~i


I


#A/jZ3 p


4aZt7(7Y-


4I'f


I;


)]'3, t


I ,


2( 2


o 3


J3;


(3.153)




(3.154)


3,


(4-


+ i-e t


(P3)


/
2


I31


oP 33


(3.155)


/


03


I3


I
Ivy


(3.156)


(3.157)


!


h


~pli,


f ,7e rl


-+ 1
2 4 e/


A 2u (


( P3d-


+-t P


k~iiz,


h
f2(~,B


I
~Y~te


2(F3


if 14


% ,I~ j17% t f~~ %-i~% )I ii/


~tfz


- I ) 3


ly


~~~


) I









The above equations can also be applied to the lower layer by replacing

the primes by double primes and the + h/2's by h/2's. Similarly,

for the middle layer, the primes are to be replaced by bars and the

h/2's are set equal to zero.

3.4.4. Variation of stress
ii
To calculate the variation of the stress tensor 6s i due to

the variation of the strain tensor 6r.., one can start with Eq. (3.101),

i.e.,





the incremental operation on this equation yields



[=, Ii=/ (3.158.)

It follows from Eq. (3.103) that


ijkl ik j l jk ;2 2 V s


Because


(/So) O,


Oy)-


and from Eqs. (3.91)













'10

Eq. (3.158) can be written as






ik ; ji ;k Iki)( l
E 2
(3.159)
The above is an expression for the stress variation in terms of the

strain variation and the strain tensor. However, for most buckling

problems, it is more convenient to have the expression in terms of the

stress tensor rather than the strain tensor. From Eqs. (3.102) and

(3.104), it follows that



(3.160)

where




(3.161)
By substituting Eqs. (3.160) and (3.161) into Eq. (3.159), the equa-

tion for the stress variation can be put into the form



,4 (3.162)


where










/ = + .()(ii Si ( )(3 ).
(3.163)
i ikl
Note that Mi has the symmetry characteristics

ij k_ .k jik! kk<,f
,//' =/ k = M i (3.164)

In classical theory, the quantities

So S,

are defined as the tangent shear modulus and secant shear modulus,
respectively, and can be obtained from a simple stress-strain curve,
see Teodosiadis, Langhaar, and Smith [22].
Naturally, the stress variation tensor can be written in terms
of subtensors



M. 333 3
d A4 + 2, J/ iM b, (3.165)3


33 33^ 3383 3533 i
U 1 + 2#4 M


where, from Eqs. (3.106) and (3.163),








S 6 s 6)
" s )


h t3_ s3 = _(

3S2


b So
Ty


- p -


M1, M33 3 R


o'3 (/3.

A 333
M
3333
M


43 Y/3

= A

3333
= E +


E 2


2
Z
35;
2
3S
a
2
/ !


( 6S0


c9'3 (i3
35S


(3.166)


So.


L.,


33
8-


yo
a7


(3-
5 -


3)


6e& S/ S A
jls- ,-5^


S4"
V7

)(^- W 33)s" 2)