Large Elastic Deformations of Homogeneous Anisotropic Materials

Using the natural state theory of elasticity, one can write down stress-strain relations valid for any perfectly elastic material which is initially homogeneous. Various equivalent forms of these relations, together with references as to their origin, are listed in a recent paper by Truesdell [1]. The problem of determining what restrictions are placed on these relations by symmetries which exist in materials is non-trivial and has been solved completely only for isotropic materials, though Birch [2] and Murnaghan [3] have determined these conditions for various types of crystals for the case where the strain energy is a polynomial of degree two or three in the strains. Here this problem is completely solved for materials possessing transverse isotropy. We obtain some general solutions to the equations of equilibrium and motion for incompressible materials of this type. These are similar to those previously obtained by Rivlin [4] and Green & Shield [5] for isotropic materials. We also derive stressstrain relations for materials subject to constraints of a rather general nature. In addition, we give the general solution for homogeneous deformation of any perfectly elastic material which is initially homogeneous.