Finite deformation of materials exhibiting curvilinear aeolotropy

Abstract

Using tensor notation, a general theory is developed for finite elastic deformations of compressible and incompressible materials which exhibit curvilinear aeolotropy. The theory is formulated for materials which are completely unsymmetrical, orthotropic or transversely isotropic with respect to the curvilinear co-ordinate system which is employed to define the aeolotropy. In applications, attention is confined to cylindrically symmetrical and spherically symmetrical problems, from which emerge as special cases the inflation, extension and torsion of a cylindrical tube, and the inflation of a spherical shell. In addition, the flexure of a cuboid of rectilinearly aeolotropic material is considered as a limiting case of the cylindrically symmetrical problem. The conditions for the tube or spherical shell to be everted, and for the curved faces of the deformed cuboid to be free from applied stress, are obtained in terms of a general strain-energy function in forms which are independent of symmetries in the material.