Two displacement methods for in-plane deformations of orthotropic linear elastic materials

Two displacement formulation methods are presented for the plane strain and plane stress problems of orthotropic linear elastic materials having the t hree planes of symmetry at x(1) = 0, x(2) = 0 and x(3) = 0. The first metho d starts with solving the two governing partial differential equations simu ltaneously, while the second method begins with solving one equation and en ds with enforcing the other. The former follows the approach of Eshelby, Re ad and Shockley, whereas the latter is based on an extended version of Gree n's theorem and thus has similarities with Airy's stress function method. T he two displacement methods lead to the same characteristic equation that i s identical to the one obtained by Lekhnitskii using a stress formulation m ethod. The general solutions resulting from the two displacement methods ca n be used to solve plane elasticity problems of orthotropic materials with displacement or mixed boundary conditions.