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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. GK640 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. Straindisplacement equations .. 43 3.3.4. Stressstrain 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. Straindisplacement equations 4.1.4. Stressstrain 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 . Straindisplacement equations 5.1.4. Stressstrain 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. Stressstrain curve for 177 PR stainless steel in compression . .. .... ... 196 7. Tangent and secant moduli for 177 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 (threedimensional), 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 thicknesslength ratio, equal to L wavepattern 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 n2 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 derivativewith 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 HellingerReissner 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 stressstrain 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 KirchhoffLove 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 manmade 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. Wellestablished 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 threelayered 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 threedimensional 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 straindisplacement equations, and the stressstrain 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 straindisplacement 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 stressstrain 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 HellingerReissner theorem. The modified HellingerReissner 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 stressstrain equations employed by Ebcioglu may be nonlinear, it is well known that for most structural materials (i.e., steel, aluminum, etc.) a nonlinear stressstrain 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 HellingerReissner 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 stressstrain 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 redefinedfor 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 stressstrain 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 KirchhoffLove 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 HellingerReissner theorem. Habip [2] has used the modified HellingerReissner 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 HellingerReissner 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 stressstrain and o v straindisplacement 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 straindisplacement relations in oV, = i (2v1i, + v 7f i ;) > ((2.7) the stressstrain 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 HellingerReissner theorem shows that by certain modifications, which are practically selfevident, 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 wellknown 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 HellingerReissner 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 threelayer, laminar structure of constant thickness. The layers may be dissimilar mater ials of different thicknesses; however, the thickness ofeach 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 righthanded. 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 wellbehnved 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 ^Vt1 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 614= 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 ah 2 S h 2 ' 2 (3.23) c h 2 where (3.21) dv = ddA 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 < 3d 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 d6J 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, 'vr (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 ur .+ # '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 , ((kPlc~'~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 (P3p ) = /~l 17, ~i,) T+ (43)(I+? 3 2h  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 = AA1 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 (;ev ^]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 f27/ 11 , ( 6 wt 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 dI 2h Al )(M3'l wV3 fiTr I 7tL 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. Straindisplacement equations The straindisplacement 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 straindisplacement 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 straindisplacement 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 straindisplacement 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. Stressstrain equations As before, the stressstrain 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, stressstrain 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 HellingerReissner 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 d91T3 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 stressstrain 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 stressstrain 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 '  (3k2 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 stressstrain 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) (3k2z ! ) 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 stressstrain equations become 3j= , where S9 jk) 49 (Jk2 (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 stressstrain 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 stressstrain 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 stressstrain 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 +
