Some simple Cartesian solutions to plane non-homogeneous elasticity problems

The paper presents some simple solutions to plane stress non-homogeneous isotropic elasticity problems described in Cartesian coordinates. The general problem is formulated in terms of the usual Airy stress function allowing spatial variation in Young’s modulus while keeping Poisson’s ratio constant. The resulting general equation is lengthy and involves various derivatives of the stress function and first and second order derivatives of the modulus distribution. Two inverse schemes are presented which greatly simplify the general governing equation and allow simple exact solutions to be generated. The first scheme employs simple biharmonic polynomial forms for the stress function thereby automatically giving stress fields identical to the homogeneous case. The governing equation is reduced to a form involving only modulus gradation terms and can often be easily solved for the allowable modulus distribution. The resulting displacement fields are then determined by standard methods. A second related method uses special simplifying elastic modulus variation to greatly reduce the general equation thereby allowing simple integration to determine the solution.