Edge Stresses in a Composite Strip Subjected to Axial Temperature Gradients,Part I: Development of the Theoretical Solution

In Part I of this paper, a combination of Airy stress functions and direct displacement functions is utilized to obtain the plane elasticity solution for the stresses and displacements in a multilayer laminated anisotropic strip subjected to a temperature gradient that is arbitrarily symmetric in the longitudinal direction. The method of analysis utilized departs from previous works in that an eigenfunction solution is developed assuming a length coordinate expansion of the stresses and displacements, with an exponential variation in the thickness coordinate direction. This avoids the shortcoming of the nonorthogonal Fadle-Papkovitsch eigenfunctions in that an orthogonal series repesentation suitable for modeling of nonuniform as well as uniform temperature distributions is obtained. The resulting eigenfunctions satisfy the necessary equations of equilibrium, conditions of displacement compatibility, requirements on interlaminar stress and displacement continuity, traction free surface conditions, and shear traction free edge conditions. As interfacial conditions and strain compatibility are satisfied exactly, the approach overcomes the inter-facial approximations and average sense compatibility invoked in previous complementary virtual work approaches. The solution does not exactly satisfy the free edge normal traction requirement since only resultant force is enforced to zero; however, convergence for enforced zero transverse slope at the strip ends can be established, as the eigenfunctionsare orthogonal. Thus the solution is exact for these edge conditions.

In Part II of this paper, numerical results are presented for several examples and compared to those obtained from our own MSC/NASTRAN finite element analyses. The results indicate the shear and peel stress concentrations and axial stress distributions are in excellent agreement with the finite element analyses for the zero edge slope boundary condition. Also, good correlation was determined with finite element analysis results for the free edge boundary condition over the range of problems considered. This correlation with the finite element numerical results verifies the solution and indicates application of the solution as an approximation to free edge engineering problems is reasonable for abroad range of practical cases.