Some generalizations of the shear problem for isotropic incompressible materials

In the theoretical treatment of finite elastic deformation for isotropic, incompressible materials, a series of relations may be obtained, which, in the most general case, can be reduced to four partial differential equations for the determination of four unknown quantities. For these dependent variables we may conveniently choose the three components of displacement in any given system of coordinates, and an arbitrary parameter which arises from the assumption of incompressibility. The equations thus obtained are non-linear in form, and exact solutions subject to specified boundary conditions are usually not readily obtainable by orthodox methods of approach. The successful treatment of a number of problems by Rivlin (see references), and by other workers (1,2), has depended essentially upon the imposition of a restriction upon the form of the deformation, so that by an appropriate choice of initial and final reference frames in which to specify the configuration of the elastic body, partial differential equations can be avoided.