A note on some general integrals arising in plane finite elasticity

Abstract

In a series of papers by Hill and co-workers, a number of exact integrals are derived for plane strain, plane stress, and axially symmetric deformations of an isotropic, incompressible Varga elastic material. Here we show for more general plane strain deformations with transverse stretch that the angle ψ appearing in these integrals is, in fact, the local rigid-body rotation angle in the polar decomposition of the deformation gradient. The result is independent of any internal constraint and of the particular constitutive description of the material response for which the general first integrals were obtained. This physical identification of ψ will greatly assist in terms of exploiting these integrals for the subsequent solution of practical problems. It is also shown that the general integrals do not include a non-homogeneous, local simple shear deformation.