A composite von Karman plate in postbuckling is considered. Using the first Piola stress tensor and the d isplacement gradient tensor, a complementary energy variational theorem is proven. The proof is given in the case of symmetric lay -up. According to the theorem, at th e actual stress state of the plate the complementary energy (as a function of the internal forces and of the moments) reaches its stationary value. The stationary feature of the actual state is valid as compared to other states satisfying the static equili brium and the static boundary conditions. It is shown how the theorem may be generalized to the case of a non -symmetric lay-up. Keywords: composite plate, postbuckling, complementary energy, first Piola stress tensor. _______________________________ __________________________________________ © Sergey Selyugin, 2019 1. INTRODUCTION The behavior of the composite plates is wide ly explored since the end ing decades of the XX century. The c onsiderable attention is paid , in particular, to the postbuckling of the plates, as the phenomenon correspond s to the highest load -carrying capacity. The review s of Turvey and Marshall (1995), Falzon and Aliabadi (2008 ) describe the state-of-the-art in the field. The majority of papers deal s with the various numerical appro aches. All the approaches are based on the kinematic description of the structural behavior in the form of equations o r the kinematic (or mixed) variational principle s. In particular, the paper of Raju et al. (2015) is devoted to a special mixed variationa l approach for the so -called Variable Angle Tow (VAT) plates. The purely static (complementary energy) principle for composite plates in postbuckling is not used yet. It is known that the complementary energy approach is successful for the nonlinear tasks where it is possible to derive the com plementary energy functional of internal forces. For doing that (see Washizu 1975) it is necessary to express the stress es as functions of the nonlinear strains. Such a possibility does not exist in general for the structures. But in several particular cases the inversion of the stress -strain relations is possible. For example, i n the paper of Stumpf ( 1975) devoted to the case of the isotropic nonlinear von Karman plates , the inversion was made and a complementary theo rem was proven. In the present paper we derive a variational theorem for the complementary energy of the composite plates in postbuckling. The derivation is based on the use of the first Piola stress tensor together with the d isplacement gradient tensor . The plates obey the von Karman law. Due to postbuckling consideration the key attention is paid to the plates with the symmetric lay-up (in the case of the non -symmetric lay -up the buckling level becomes considerably lower). The way of the theorem generali zation to the non -symmetric lay -up case is indicated. The paper content is the following. Section 2 describes the assumptio ns and the definitions . Section 3 is devoted to the description of the potential energy theorem. In Section 4 , the derivation of a c omplementary energy theorem is presented . In Section 5 , the conclusions are indicated . 2. ASSUMPTIONS AND DEFI NITI ONS We consider a flat composite plate made of stacked orthotropic layers . The Cartesian coordinate system XYZ is used. The undef ormed (prior to the deformation) plate mid -plane is located in the X-Y plane . The Z axis is orthogonal to the X -Y plane. The plate mid-plane undef ormed surface is denoted S. The total undef ormed plate volume is V. The plate is loaded at the part 1C of the boundary contour C by the in -plane loads. The contour is supposed to be smooth. The boundary loads are assumed to be piecewise -smooth. At the remaining 2C part of the boundary contour the displacements are prescribed. In postbuck ling, the plate is deformed in accordance with the von Karman theory (Washizu 1975) . The plate deflection is assumed to be non -negligible as compared to the plate thickness, but, at the same time, it is very small as compared to the other plate dimensions . Denote the displacements along the X, Y, Z axes at the arbitrary point within the plate z y x vvv ,, , respectively. The X, Y, Z displacements at the mid -surface are denoted u, v, w , respectively. Neglecting the terms of the low er order of the magnitude, the displacements at the arbitrary point within the plate are written as follows : x x zwuv, ; y y zwv v, , ; wvz (1) where (and everywhere further) the subscript after comma means the differentiation w.r.t. the corresponding coordinate. The non-zero components of the Green strain tensor are: 2 , ,21 xz xx xx v v e ; 2 , ,21 yz yy yy v v e ;  yzxz xy yx xy vv v v e, , , ,21 (2) 3. THE POTENTIAL ENERGY THEOREM FOR COMPOSIT E PLATES IN POSTBUCKLING The total potential energy variational principle for the considered configuration is written as follows ( see Washizu 1975 , Ashton and Whitney 1970 ):   11 Cy x SdCvNuN dS I  (3) where  y xN N, are the given force s per unit contour length at the undef ormed boundary 1C (the given values of the displace ments at the remaining part 2C of the contour will be marked further by a top bar also) , Π is the strain potential energy per unit undeformed plate area . The latter energy is derived from the ply strain energy density integrating throu gh the plate thickness. The ply strains of the laminate are xy y x ,, , respectively. The ply strain energy density ply in the undef ormed volume i s written (Gibson 1994): xyy xyx yx xy y x ply Q Q Q Q Q Q  26 16 122 662 222 11 2 2 221 21  (4) where the ply stresses xy y x ,, and strains xy y x ,, are related to each other as follows:              xyyx xyyx Q Q QQ Q QQ Q Q   266 26 1626 22 1216 12 11 (5) and ijQ, i,j=1,2,6, is the matrix of the ply elastic constants. The first Piola stress tensor components ij~ and the Kirchhoff stress tensor components ij are given by the relations:  ijji ijvv ,, ~  (6)  ijij ijee  (7) where π is the strain energy density in the undef ormed volume . In general, the tensor ij~ is non-symmetric. Calculating the components of ij~ for a ply, we obtain:  yzxz xy yx yz yy xz xx xx vv v vQ v vQ v vQ, , , , 162 , , 122 , , 1121 21 ~     (8)  yzxz xy yx yz yy xz xx xy vv v vQ v vQ v vQ, , , , 262 , , 222 , , 1221 21 ~     (9)  yzxz xy yx yz yy xz xx yx xy vv v vQ v vQ v vQ, , , , 662 , , 262 , , 1621 21 ~ ~      (10) Direct differentiatio n of (4) according to (7 ) gives the xx, yy, xy, yx components of the Kirchhoff tensor equal to (8) -(10), respectively. Using the known relations (see Washizu 1975 and the above note on equality of the in - plane components of the Kirchhoff and the first Piola stress tensors) yzxy xzxx xz v v, ,~ ~ ~  (11) yzyy xzyx yz v v, ,~ ~ ~  (12) we obtain the remaining non -zero components of the ij~ tensor. The components of the ij tensor differing from xx, yy, xy, yx are equal to zero. Integrating over z through the plate thickness, we introduce the stress (force) and the moment resultants ijN and ijM :  2 2~h hxx xx dz N (13)  2 2~h hyy yy dz N (14)  2 2~h hxy yx xy dz N N  (15)  2 2~h hxz xz dz N (16)  2 2~h hyz yz dz N (17)  2 2~h hxx xx dzz M (18)  2 2~h hyy yy dzz M (19)  2 2~h hxy yx xy dzz M M  (20) Substit uting (8) -(12) into (13) -(20) and remembering the definition of A, B, D stiffness matrices for laminates via ijQ, i,j=1,2,6 (Gibson 1994, Section 7.3), we obtain these forces and moments. Rewrite the formula for the total potential ener gy as       11 0 0 021 21 Cy x ST T TdCvNuN dS D Be eAe I   (21) where T means the transposition, the vectors ,0e are the mid -surface 2D -stains and the mid-surface curvatures: 00 2      zxyyyxx eee e ; 022222 2       zyxwywxw  (22) The first integral in (21) is obtained from (4) (with account of (1), (2)) after integration through the plate thickness. The variation of the total potential energy (3 , 21) is written as follows:         11 0 0 021 21 Cy x ST T TdCv Nu N dS D Be eAe I     (23) Using the known relation (Washizu 1975) ij ji ij ijv vv, , , ,~  (24) (hereinafter the repetitive indices mean the summation on them) and substituting (1) into it, after integration over the plate thickness, we obtain the variation of the strain energy Π per unit undef ormed area : xy xy yy yy xx xx y yz x xz x y xy y yy x xx w M w M w M wN wN v u N uN uN, , , , , , , , , 2      (25) Further we denote y xnn, the components of the normal vector to the boundary contour. Using them, we introduce the x, y, z stress resultants  z y x N N N , , at the boundary contour , and the moment results there also :       y y x xyy y xy x yyx y xx x xyz y xz x zyy y xy x yyx y xx x x Mn Mn MMn Mn MMn Mn MNn Nn NNn Nn NNn Nn N (26) In the further description we use the x and y components of the normal y xnn, to the undef ormed boundary contour in the relations ( where s is the direction along the contour) snnnysnnnx x yy x  (27) where n, s mean the normal and the tangential direction to the contour. Integating (2 3) by parts and using the Gauss divergence theorem, we obtain the variation of the total potential energy functional :             2 ,1 ,, , , , , , , , , 212 dCwMwNv Nu NdCwMwNv N Nu N Ndsw N N M M Mv N Nu N N I Cn z y xCn v z y y x xSyyz xxz yyyy xyxy xxxx yyy xxy yxy xxx                   (28) On the part 2C of the boundary contour we have: 0,nwMwv u  (29) where the latter term (the product of the normal moment and the normal derivative) covers both the simple support and the clamped conditions. Then the stationarity of I w.r.t. the displacements leads to the equilibrium equations at the plate (in S): 0 200 , , , , ,, ,, ,  yyz xxz yyyy xyxy xxxxyyy xxyyxy xxx N N M M MN NN N (30) where, according to (11), (12), (16), (17) y yy x yx xyzy xy x xx xxz wN wN NwN wN N , , ,, , ,  (31) The third equation of ( 30), after substituting of (31), leads to the known z-equilibrium equation of the von Karman plate: 0 222 2 22            ywNxwNy ywNxwNx yM yxM xM yy xy xy xxyy xy xx (32) The linear boundary conditions on 1C are: 0000     MNN NN N zy yx x (33) 4. A COMPLEMENTARY ENER GY THEOREM Now , after indicating the total potential energy theorem, we proceed with the complementary energy theorem derivation. The complementary energy density c and the strain energy density ply for a ply satisfy the Legendre t ransformation (if the displacement gradient tensor jiv, can be expressed through the first Piola stress tensor): ji ply kllk c v v, ,~  (34) Using (1), we have:  y yz x xz xy x y xy yy y yy xx x xx kllk w w zw v u zw u zw u v, , , , , , , , ,~ ~2~ ~ ~ ~     (35) The relation (4) may be re -written as:  y x xx x y xy y yy y yy x xx x xx ply ww zw v u w zw v w zw u, , , , ,2 , , ,2 , , , 2~ 21 21 ~ 21 21 ~ 21           (36) Subtracting (36) from ( 35), we have the ply complementary energy density:  y yz x xz y x xx x y xyy yy y yy x xx x xx c w w ww zw v uw zw v w zw u , , , , , , ,2 , , ,2 , , , ~ ~2~ 2121 ~ 21 21 ~ 21            (37) Integrating ( 37) over z through the plate thickness, we obtain the complementary energy per unit undef ormed mid -plane area c:   xy xy yy yy xx xxy yz x xz y x x y xy y y yy x x xx c w M w M w MwN wN ww v uN w v N w uN , , ,, , , , , ,2 , ,2 , , 221 21 2121 21 21 21 21         (38) Integrating (1 1), (12) over the plate thickness and inver ting the relations , we have: )(1 2 , ij xy yy xxyz xy xz yy x NN NNNN NNw  (39) )(2 2 , ij xy yy xxxz xy yz xx y NN NNNN NNw  (40) Now we define the vectors :  T xy yy xx NNN N ,, (41)  T xy yy xx M M M M , , (42) For simplification of the transformations, we consider further the plates with the symmetric lay -up only. This means that the B stiffness matrix is composed of zeros. Such an approach lo oks reasonable for postbuckling analysis, as otherwise the buckling level may come down considerably, decreasing the total load -carrying capacity. The derivation of the theorem for the case B≠0 is straightforward. It may be done using the obvious equality:  M BDNeB BDA1 01    (43) and obtaining the Green strains , the d isplacement gradient components, and the curvatures as the functions of the forces and the moments. Now, s ubstituting (3 9)-(42) into ( 38), we obtain the final expression for the complementary energy density per undeformed plate area: )(21)(21 21 21 2 11 1 ij yz ij xzT T c N N N N MDM NAN       (44) The functions 2 1, in (38), (39) are incompatible in the case of an arbitrary stress state. They become the corresponding deflection derivatives for the actual state only. For creating the complementary energy functional cI we substitute into (3) the relation (34), integrated over the plate thickness. Then, integrating by parts and using the Gauss divergence theorem , we obtain the functional I as follows:              2 ,1 ,, , , , ,, , , , 212 2 dCwMwNv Nu NdCwMwNv N Nu N NdSw N N M M Mv N Nu N N wNvNuN dS I Cn z y xCn v z y y x xyyz xxz yyyy xyxy xxxxSyyy xxy yxy xxx Cxz y x Sc                  (45) Below the geometric boundary conditions on 2C are used, namely, the values of the displacements are prescribed there and uu ; vv ; ww ; 0,nvwM (46) If the equilibrium equations ( 30), (31) and the static boundary conditions ( 33) are valid for the variations of the stress state, then the complementary variational functional cI in (4 5) is written in the following form ( c comes from (4 4)):  dSwMwNvNuN dS I Cn z y x Sc c 2,    (47) The variation of c reads: xy xyc yy yyc xx xxcyz yzc xz xzc xy xyc yy yyc xx xxc c MMMMMMNNNNNNNNNN          (48) It is necessary to prove that for the stress states satisfying the static equilibrium conditions (30), (31) and the static boundary conditions ( 33) the actual stress state correspond s to a stationary point. Observing (4 5), one may sa y that the displacements there may be considered as the Lagrange multipliers , when he/she looks for a stationary value of (47). We augment (4 7) by the items corresponding to the equilibrium equations and the static boundary conditions with the Lagrange mu ltipliers z y x ,, . After that, u sing the Gauss divergence theorem and integrating by parts, we obtain:      021 21 2 , ,, , ,, 2 , 1 , , 21,2 2 ,2 11 1 2                   dCM w Nw Nv NudS M M MN N NN N MDM NAN I Cv n nz z z y y x xS xy xyz yy yyz xx xxzyz yz xz xz xy xy yxyy yy xx xxT T c        (49) If we interpret the Lagrange multipliers as the displacements , we obtain the geometric boundary conditions on 2C : ux ; vy ; wz ; 0)(z vM or nz nzw, , (50) Also, in S we have the compatibility conditio ns for 2 1, as functions of yx xz xy yy xx NNNNN ,,,, : xz, 1 ; yz, 2 (51) and the force (moment) – displacement (curvature) relations :            21 , ,1 661 261 162 2 ,1 261 221 122 1 ,1 161 121 11 2121  xy yx xy yy xxyy xy yy xxxx xy yy xx NA NA NANA NA NANA NA NA (52)          xyz xy yy xxyyz xy yy xxxxz xy yy xx MD MD MDMD MD MDMD MD MD ,1 661 261 16,1 261 221 12,1 161 121 11  (53) where 1 1, ij ijDA mean the elements of the inverse matrices to A, D . The rhs of the relations (52) looks similar to the rhs of (2) for z=0. The xyz yyz xxz , , , , , may be treated as the plate curvatures. Thus, the relations ( 51)-(53) are equivalent to the known strain compatibility condition for the von Karman plates: 22 2222 02 202 202 2yw xw yxw yxe xe ye xy yy xx       (54) where 0 0 0 ,,xy yy xx eee are the Green strains at z=0. It is supposed in the both sides of (54) that :   zyx wvu  (55) We have proven that the stationary value of the complementary energy corresponds to the solution of the composite plate in postbuckling. 5. CONCLUSIONS  The complementary energy theorem for the composite plates in postbuckling is proven  According to the theorem, the actual stress state of the plate corresponds to a stationary value of the complementary energy functional in the case that the resultants of the first Piola stress tensor satisfy the static equilibrium and the static boundary conditions.  The theorem is proven for the symmetric plate lay -ups (having better buckling resistance than other ones) . The way of the proof of the theo rem for a non -symmetric lay-up is clearly indicated.