On the Equivalence of the Stress and Displacement Methods in Planar Anisotropic Elasticity

Abstract

The stress formalism of Lekhnitskii and the displacement formalism of Eshelby, Read and Shockley (ERS) are firstly recapitulated and compared for problems of planar anisotropic elasticity. It is explicitly demonstrated that for such problems the two formalisms lead to the identical characteristic equation and thus are equivalent. An alternative displacement formulation method is then presented for the same problems using an extended version of Green’s theorem. This approach introduces a displacement function, which satisfies a fourth-order partial differential equation whose characteristic equation is identical with that obtained using the methods of Lekhnitskii and ERS. The new approach is proved to be equivalent to those of Lekhnitskii and ERS for problems of planar anisotropic elasticity. The present displacement function method exhibits similarities with the Airy stress function method.