The Affine Transformation for Orthotropic Plane-Stress and Plane-Strain Problems

It is demonstrated that a single affine transformation of the type x = ax′, y = by′ immediately extends the solution of any isotropic plane-stress or plane-strain problem to the solution of an orthotropic plane problem where the orthotropic material is characterized by three independent constants. Since orthotropy, defined as elastic symmetry with respect to two orthogonal axes, implies four independent elastic constants, the affine transformation introduces a restriction upon the orthotropic shear modulus. The orthotropic shear modulus differs from that used by previous investigators. This difference alters the equation which the orthotropic stress function must satisfy and, therefore, directly affects the solution to every plane-stress or plane-strain problem. Some arguments are advanced to favor the shear modulus, as here defined, whenever orthotropy must be restricted to three elastic constants. The two solutions of the orthotropic half plane subjected to a normal concentrated load are contrasted to illustrate the effect of the two definitions of orthotropic shear modulus.