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 three planes of symmetry at x₁ = 0, x₂ = 0 and x₃ = 0. The first method starts with solving the two governing partial differential equations simultaneously, while the second method begins with solving one equation and ends with enforcing the other. The former follows the approach of Eshelby, Read and Shockley, whereas the latter is based on an extended version of Green's theorem and thus has similarities with Airy's stress function method. The two displacement methods lead to the same characteristic equation that is identical to the one obtained by Lekhnitskii using a stress formulation method. The general solutions resulting from the two displacement methods can be used to solve plane elasticity problems of orthotropic materials with displacement or mixed boundary conditions.