Analysis of edge stresses in composite laminates under combined thermo-mechanical loading, using a complementary energy approach

An approximate method is presented to investigate Mgy the interlaminar stresses near the free edges of composite N laminate plates that are subjected to a combined NI~ thermo-mechanical loading. The method is based upon N~ admissible function representations of stresses which account Q!~) for the effects of both the global mismatches and the local ,~!k/ mismatches in two of the elastic properties, the Poisson's t(kl ratio and the coefficient of mutual influence. For this purpose, (x~, x 2, x3) new thermo-mechanical mismatch terms are defined to reflect an effective deformation under the combined (Xx, X2,Z) thermo-mechanical loading. Closed form solutions of all the stress components are sought by minimizing the z {k) complementary energy with respect to the unknown functions, mech in the stress representations, of the width coordinate. These eH unknown functions are determined by solving five ordinary ~]' differential equations along with a set of free edge boundary #o, conditions, which allow complex as well as real roots for their exponential decaying rates. The resulting solutions satisfy the ~~ stress equilibrium and all of the boundary conditions exactly, qt2.z but compatibility is met in a weak form. Numerical examples t]12,1{k) are given for several typical laminates, and are compared with .(k) '112,1E previous results obtained by finite element and other approximate methods. It is found that the present approximate Ir method yields interlaminar stress results in an efficient, fast /A.~,~ and yet reliable way. It is also concluded that unlike some previous approximate methods, the current method is (k/ //12,22,12,33 numerically robust and stable. List of symbols A,B, D b A T (k) A T I~) ( x3) e(k) 01,2,3,12 h Laminate stiffness matrices as defined by (27) Half laminate width Temperature difference at the top of the k-th ply Temperature difference in the k-th ply given as a function of through-thickness coordinate x 3 Thermal strains of the k-th ply in the laminate axes Laminate thickness Applied uniform bending load (moment/unit length) Communicated by S. N. Atluri, 30 August 1994 T. Kim, S, N. Afluri Computational Mechanics Center Georgia Institute of Technology, Atlanta, Georgia 30332-0356 USA Correspondence to: T. Kim This work was supported by a grant from the Federal Aviation Administration, to the Center of Excellence for Computational Modeling of Aircraft Structures at Georgia Institute of Technology. V12 (k) VI2 v(k) 12E a~j 6~j c Applied thermal moments (moment/unit length) Total number of plies Applied uniaxial load (force/unit length) Applied thermal loads (force/unit length) Ply stiffness for the k-th ply in the laminate axes Ply compliance for the k-th ply in the laminate axes Thickness of the k-th ply Local coordinate system with the out-of-plane coordinate x 3 defined at the bottom of each ply Global coordinate system with the out-of-plane coordinate z defined at the mid-plane of the laminate Global out-of-plane coordinate for the top of the k-th ply Applied mechanical axial strain Applied total (including thermal) axial strain Total in-plane strain vector at arbitrary plane as defined by (32) Total in-plane strain vector at the midplane Laminate coefficient of mutual influence Coefficient of mutual influence for the k-th ply Equivalent coefficient of mutual influence for the k-th ply as defined by (23) Total curvature vector Coefficients of thermal expansion for a uni-ply in its material principal axes Coefficients of thermal expansion for the k-th ply in the laminate axes Laminate Poisson's ratio Poisson's ratio for the k-th ply Equivalent Poisson's ratio for the k-th ply as defined by (22) Total stress components in tensor notation Far-field stress components Companion stress components