In the present paper the composite plates in postbuckling are explored. The dual extremal variational principles are created. The principles generaliz e the stationary ones obta ined earlier. The stability of the plate near the single -modal bifurcation point is proven. Some useful energy relations are established. It is also demonstrated that the total complementary energy may be considered as a measure of the compliance for the p ost-buckled composite plate. The latter measure leads to the same lay -up optimality conditions as obtained earlier via the maximization of the total potential energy. Basing on the complementary variational principle, a monotonic plate compliance minimizat ion approach is proposed . The approach allows determin ing the stiffest lay -up. Keywords : composite plate, postbuckling, variational principles, extrem um, duality, stability , compliance minimization _______________________________________________________ ____________________ © Sergey Selyugin, 2019 - 2 - 1. INTRODUCTION Postbuckling behavior of composite structures, in particular, plates, attracts attention of many researches. The basics of postbuckling are described in such books as Hutchinson and Koiter (1970) , Budiansky (1974) . Some aspects of the phenomenon are discussed in the books of Turvey and Marshall (1995), Falzon and Aliabadi (2008), Kassapoglou (2010) , Alfutov (2000), Shen (2016) . There are several reviews covering the subject, the non-exhaustive list of them contain s the papers of Bruneel (2008) and Ghiasi et al. (2009, 2010 ). A brief review of the original studies close to the subject of the present paper is presented below. In the paper of Stein (1983) the nonlinear large -deflection equations of von Karman are written for specially orthotropic plates. By assuming trigonometric functions in one direction, the plate equations are converted into ordinary nonlinear differential equations that are solved numerically using Newton's method. The postbuckling behavior is obtained for simply supported and clamped, long, rectangular, orthotropic plates covering the complete range of dimensions and material properties. In the important theoretical paper of Haslach (1986) the potential energy function in a neighbo rhood of the buckling point for a biaxially loaded, linear elastic, orthotropic, simply supported, rectangular plate is obtained from a perturbation analysis. The postbuckling behavior is proven to be stable. It is also possible to have a bimodal buckling point for certain combinations of loads; this point is also stable. In the paper of Zhang and Shen (1991) the buckling and postbuckling behavior of orthot ropic composite rectangular plates with simply supported edges in biaxial compression are investigated using a deflection -type perturbation technique. Using the technique, the - 3 - nonlinear generalized von Ka rman equations of orthotropic plates are solved. The effects of geometrical imperfections are also considered. The paper of Bisagni and Lanzi (2002) deals with the definition of a post -buckling optimi zation procedure for the desig n of composite stiffened panels subjected to compression loads . The optimiz ed structures are then characteri zed by a local skin buckling between the stiffeners and by a high ratio between the collapse load and the buckling load. An optimiz ation procedure is developed. It is based on a global approximation strategy, where the structure response is given by a system of neural networks and the finite element analysis. The optimiz ation procedure shows that a local skin buckling between the stiffeners allows a weight reduction equal to 18%. The aim of the paper of Rahman et al. (2011) is to study the postbuckling behavior of the optimized plates using a perturbation method . The perturbation approach is used to compute postbuckling coefficients, which are used to make a quick estimate of the postbuckling stiffness of the panel and to establish a reduced -order model. In the paper , the postbuckling analysis of variable stiffness plates is carried out using the reduced -order model, and the potential of the approach for incorporation within th e optimization process is demonstrated. In the paper of Mittelstedt et al. (2011), the postbuckling behavior of rectangular orthotropic laminated composite plates with initial imperfections under in -plane shear load is investigated . The considered plates are infinitely long . The boundary conditions are taken into account a t the longitudinal edges only. Using Timoshenko -type shape functions for both the initial bifurcational buckling analysis and the subsequent Marguerre -type postbuckling studies, the closed -form analytical solutions for the buckling loads and for the postbuckling state variables are derived. A comparison with the geometrically non -linear finite element computations shows that the derived analysis approaches are suitable for postbuckling stud ies in the load ranges not too far beyond bifurcation buckling . - 4 - The paper of Wu et al. (2014) studies the optimal design of the postbuckling behavior of rectangular laminated composite plates under axial compression. In the numerical optimization process the lamination parameters are used . Using perturbation analysis, an asymptotic closed -form solution is derived to model the postbuckling behaviors of the orthotropic laminated composite plates under the axial compress ion. The postbuckling design criteria o f the composite laminated p lates are based on the minimiza tion of either the maximum normal displacement or the end -shortening strain for a given compressive load. The optimization process is a numerical one. In the paper of Raju et al. (2015) a geometrica lly nonlinear analysis of the symmetric variable angle tow (VAT) composite von Karman plates under the in-plane shear is investigated. The differential quadrature method is used for the numerical analysis . The buckling and postbuckling behavior of the VAT plates under the positive and the negative shear is studied for different VAT fiber orientations, aspect ratios, combined axial compression ; and their performance is compared with that of straight fiber composites. It is shown that there can be enhanced sh ear buckling and postbuckling performance for both displacement -control and load -control and that the underpinning driving mechanics are different for each. The paper of Henrichsen et al. (2016) is a numerical one and presents a novel method for gradient b ased design optimization of the post -buckling performance of the structures. The post-buckling analysis is based on Koiter’s asymptotic method. To perform the gradient based optimization, the design sensitivities of the Koiter factors are derived and new d esign optimization formulations based on the Koiter factors are presented. The proposed optimization formulations are demonstrated on a composite square plate and a curved panel where the post -buckling stability is optimized. No attempts of the theoretical analysis of the deflection -load curve near the bifurcation point are made. - 5 - In the paper of Selyugin (2019 a) two variational principles for the von Karman composite plates in postbuckling are derived and presented. The principles are stationary ones. One of them is a kinematic principle, another one is a static principle. The principles lead to the results which may be not unique. The present paper deals with the generalization of the variational principles (described by Selyugin 2019 a) to extremum -type o nes. The generalization leads to the useful energy relations and allows proving the postbuckling stability near the bifurcation point. Also, using the complementary energy as a measure of the structural compliance and choosing the laminate lay -up by minimi zing the compliance, we come up to the same optimality conditions as in (Selyugin 2019b). In Section 2 the extremum -type variational principles for the composite plates in postbuckling are developed. Section 3 is devoted to the proof of postbuckling stab ility of the symmetrically laminated plates near the single -modal bifurcation point (see also Alfutov 2000). Section 4 deals with a discussion of some energy relations . Section 5 describes the use of the total complementary energy for choosing the lay -up of the considered laminates. Section 6 formulates the conclusions. 2. VARIATIONAL PRINCIPL ES IN COMPOSITE PLATE POSTBUCKLING This Section deals with the generaliz ation of the variational principles for post -buckled plates (Selyugin 2019 a) from the station ary-value type to the extremum type . The assumptions of the present paper are: the symmetric point -wise arbitrary lay-up, the single -modal bifurcation point (Ohsaki and Ikeda 2007 ), symmetric buckling , the Cartesian - 6 - orthogonal coordinates XYZ with respect ive displacements u, v, w; the plate with piece -wise- smooth boundary contour is located in the XY plane; the in-plane loading is applied with the value not much higher than the bifurcation load ; the boundary conditions are the simple support or clamped . We explore the plate near the bifurcation point using the Timoshenko approach (described by Alfutov 2000 , Section 7.2 ) and the classic laminated plate theory (Gibson 1994) . In the Alfutov’s description it is supposed that near the bifurcation point the stru cture is stressed but not deformed. In our opinion, the assumption is reasonable and is widely used. According to the Timoshenko approach the displacements of the plate points in their new (deviated) state are: 22 1 022 1 022 1 0 w w wwv v vvu u uu   (1) where α is the infinitesimal small parameter, ( 0 0 0,,wvu ) are the non -disturbed functions of the initial point locations, ( 1 11,,wvu ) and ( 2 2 2,,wvu ) are the finite functions of coordinates. Applying the Timoshenko appr oach, we obtain the displacement functions ),(),,(),,(2 2 1 yxvyxuyxw and the Airy function ),(yx satisfying all boundary conditions of the problem. We accept approximately that for small but finite plate deflections the plate state is described by the following functions (ξ is the small parameter) : ),( ),(),( ),(),( ),(),( ),( 222221 yx yxyxv yxvyxu yxuyxw yxw   (2) - 7 - In this case the change of the total plate potential energy E due to the deviations from the original plane equilibrium state reads: 44 2 222 12 6612 212 262 212 2212 212 16 212 212 122 212 11 2 0 4 44 2 21 W dvquqyxwDyxw ywDywDyxw xwDyw xwDxwD dxdy EEE y xS                              (3) where 0E is the total plate potential energy independent of ξ; D is the plate bending matrix; y xqq, are stress flows at the boundary contour Γ aroun d the plate S. The value of 4W is given by the expression :        Sxy xyy xyx y yx x A A A A A A dxdy W2'' 66'''' 26'''' 162'' 22'''' 122'' 11 4 4 4 4 221 (4) where A is the plate 2D -strain stiffness matrix,              yw xw xv yuyw xvxw xu xyyx 1 1 2 2 ''2 1 2 ''2 1 2 '' 212121  (5) Since the functions ,,,2 2 1 vuw of (x,y) are known as the results of the linear stability plate problem, the postbuckling behavior near the bifurcation point is determined by the parameter ξ only. It is obvious that the quantity 4W (as an energy quantity ) is always positively definite ; this leads to a conclusion that after passing through the bifurcation point (under the assumptions made) the total plate potent ial energy is a convex function of the displacements . - 8 - Basing on this conclusion, we may state that the kinematic variational pr inciple for plate postbuckling (see Selyugin 2019 a) is a minimal variational principle. Moreover, the result (the displa cements ) coming from the principle is a unique one. The relation between ξ and the load above the bifurcation point may be found from the condition of the stationary total potential energy. For the examp le of uniaxial ly compressed by the force P plate, calculating the derivative of ( 3) w.r.t. ξ, we have  0 1 443   W P Pcr (6) where the quantity β and the critical compression force crP read:   dvquqy x 2 2 2 (7)                         2 12 6612 212 262 212 2212 212 16 212 212 122 212 11 4 44 2 yxwDyxw ywD ywDyxw xwD yw xwD xwD dxdy PS cr (8) Here y xqq, is the external load distribution for P=1. Making ξ 0 we obtain the buckling load. For ξ not equal to zero we have crPP K2 (9) where crPPP and 412 WVK . 1V is given by the fo rmula:                         2 12 6612 212 262 212 2212 212 16 212 212 122 212 11 1 4 44 2 yxwDyxw ywD ywDyxw xwD yw xwD xwD dxdy V S (10) - 9 - It follows from the latter relation that in the case of 1 crPP the applied load P is a quadratic function of the maximal deflection maxw . The functio nal of the kinematic variational principle is written as follows:          121 21 0 0 dv Nu N dS D eAe Iy x ST T   (11) where the mid -plane ( z=0) vector -columns are 00 2      zxyyyxx eee e , 022222 2       zyxwywxw  (12) and e.. with subscripts are the corresponding strain components. The x and y force flows applied at the part 1 of the contour are  y xN N, . Further we keep the notations of (Selyugin 2019 a). Now we consider the static variational principle for the post -buckled plates. As it was shown in (Selyugin 2019 a), the prin ciple is a stationary one; the solutions resulting from it may be non -unique. In the paper of Stumpf (1979) it is shown that the total complementary energy principle (for non-linear stable von Karman plate) dual to the minimal total potential energy principle is of maximum type. This conclusion follows from the convexity (under some assump tions) of the total plate potential energy. The complementary energy principle functional in the case of the composite plate (Selyugin 2019 a) is written as follows:  dSwMwNvNuN dS In z y x Sc c  2,    (13) - 10 - where 2 is the part of the boundary contour with prescribed displacements wvu,, ; vM is the bending moment acting along the border tangential line , nw, is the derivative of the deflecti on along the direction normal to the contour . Hereinafter the subscript after comma means the differentiation w.r.t. the proper coordinate. T he complementary energy density per unit undeformed plate area is given by the formula:   xy xy yy yy xx xx y yz x xzy x x y xy y y yy x x xx c w M w M w M wN wNww v uN w v N w uN , , , , ,, , , ,2 , ,2 , , 221 21 2121 21 21 21 21         (14) Hereinafter N and M with subscripts means the first Piola stress tensor components (multiplied to z in the case of M) integrated over the plate thickness. The f ollowing relations are also used above: )(1 2 , ij xy yy xxyz xy xz yy x NN NNNN NNw  (15) )(2 2 , ij xy yy xxxz xy yz xx y NN NNNN NNw  (16) With the below notations (T means the transposition)  T xy yy xx NNN N ,, (17)  T xy yy xx M M M M , , (18) after some elementary but cumbers ome transformations, we write (1 4) in the form : ij yz ij xzT T c N N N N MDM NAN2 11 1 21 21 21 21      (19) where the s uperscript ( -1) means the invers ion of the matrix. The solution of the latter principle is a maxima l one and is unique. The both principles give the upper and lower bounds of the solution values for both variational principles. - 11 - Applying the results of Stumpf (1979) to our case, we obtain that near the bifurcation point the dual extremum -type variationa l principles (for the total potential energy and for the total complementary energy) occur. The result of every principle is unique. 3. STABILITY IN COMPOSITE PLATE POSTBUCKLING In this Section we consider the stability of composite plate in postbuckling ne ar the single -modal bifurcation point. As it is said in previous Section, the total potential energy of the post -buckled composite plate is a convex functional (under the assumptions made). Due to that the second variation of the energy w.r.t. the displac ements is positive ly defined and the plate behavior is stable near the single -modal bifurcation point . 4. DISCUSSION ON ENERGY VALUE S Using the above -indicated duality and extremum -type of the variational principles, we may write the following equality for the resulting variational functionals:  SdS W~ ~ (20) where ~ is the total plate potential energy, W is the second int egral (the external work potential) in (11), and 2 22 22 21 21 xy yy xxxy yz xz xy yy xxxx yz xy yy xxyy xzN NNNNN N NNNNN NNNN (21) The rhs of (20) corresponds to the complementary energy functional (13). The value of Ω in the postbuckling in the case of two -directional com pression is obviously negative. Then t he relations (20) and ( 21) are transformed to - 12 - 0~2W (22) As we see, in the considere d case due to the non -linearity the total strain potential energy is larger than one half of the external work potential. In the infinitesimal case without the postbuckling we have an equality in (2 2) instead of a non-equality. 5. LAY -UP OPTIMALITY VIA CO MPLEMENTARY ENERGY APP ROACH We discuss here the following. The total complementary energy functional (1 3) of the post-buckled composite plate may be considered as a measure of the structural compliance. We formulate the following optimization problem: to ch oose the ply orientation angles of the symmetric lay -up for minimizing the structural compliance. The complementary energy density is given by ( 19). Keeping in mind the known equality on the differentiation of the inverted square symmetric matri x (say, G) as the function of the argument g (Korn and Korn, 2003) : 1 1 1    GdgdGG Gdgd (23) we make the necessary transformations (similar to the approach of Sely ugin 2019b) for performing the proper variations of the forces, the moments , and the ply angles . After that , we obtain the same optimality conditions for the ply orientations as indicated in (Selyugin 2019 b). 6. CREATION OF THE PLY ANGLE OPTIMIZATION A PPROA CH Basing on the complementary energy principle, the following compliance minimization approach is proposed ( leading to a stiffest lay -up): - 13 - 1. The process starts from an initial structure with given ply angle distribution and given in-plane loading “above” bu ckling. 2. For the ply angle distribution the forces yz xz yy xy xx NNNNN ,,,, and the moments yy xy xx M M M , , are calculated at every analysis point (say, at an FE center or a node ). 3. The optimality conditions (Selyugin 2019b ) give the direction of the complementary functional increase. For the “ frozen ” forces and moments of the step 2 the maximum of (13) in the direction is numerically determined (the two last terms in (19) are not necessary to consider , as they are independent o n ply angles) . A new ply angle distribution is the result of the step 3. 4. New forces yz xz yy xy xx NNNNN ,,,, and moments yy xy xx M M M , , are calculated. 5. Convergence check (say, the closeness of the ply angle distributions for the steps 2 and 3 ) 6. If the convergence is reac hed, then we go to the step 2. Otherwise we stop the process saying that the numerical solution is found. Observing the above steps, one may say that the total complementary energy decreases monotonically during the steps of the algorithm. Hence, the stru ctural stiffness increases monotonically also. 7. CONCLUSIONS Near the single -modal bifurcation point:  The dual variational principles of the paper are of the extremum type ;  The plate is stable in postbuckling; - 14 -  Due to the geometric non-linearity, the value of the total strain potential energy is larger than the one-half of the external work potential ;  The use of the total complementary energy as a measure of the minimized structural compliance leads to the same lay -up optimalit y conditions of the first or der, like the use of the total potential energy value as a measure of the structural stiffness .  The monotonic compliance minimization approach is proposed. The approach leads to a stiffest lay -up. ACKNOWLEDGEMENT The author is thankful to colleagues for fruitful discussions.