On local and global universal relations in elasticity

In the nonlinear theory of isotropic elastic materials, there are relations between force-like quantities and deformation-like quantities which are independent of material properties. Such relations have been called "universal" relations [1]. The best known example is that of Rivlin for simple shear [2]. It relates components of normal and shear stress and the amount of shear. Another important example arises in the problem of the combined extension and torsion of a circular cylinder. Rivlin [3] showed that for small angles of twist superposed on finite extension, the ratio of the twisting moment to the angle of twist depends only on the axial force and the extension ratio. In this note relations of the first type, involving components of the stress and stretch tensors, will be referred to as "local". Relations of the second type, involving resultant forces or moments, will be referred to as "global". A general method for constructing local universal relations has been presented by Hayes and Knops [4]. In section 2, this method is used to derive a universal relation for the homogenous deformation consisting of simple shear superposed on triaxial extension. This is the local deformation which each material element undergoes in the non-homogeneous deformation of combined inflation, extension and torsion of a circular cylinder. Rivlin's global universal relation for this problem is normally obtained by first deriving expressions for the resultant force and moment in terms of the strain energy density and stretch tensor components, and then comparing them. In section 3, it is shown that Rivlin's global universal relation can be derived from the local universal relation of section 2. Finally, in section 4, global universal relations are established for small amplitude axial or circumferential shearing superposed on finite extension and inflation of a circular cylinder.