DEFORMATIONS OF AN ELASTIC, INTERNALLY CONSTRAINED MATERIAL .3. SMALLSUPERIMPOSED DEFORMATIONS AND WAVES

The general equations for small deformations superimposed on a large s tatic homogeneous deformation are derived for a class of materials for which the deformation is subject to an internal material constraint d escribed in experiments by James F. Bell on the finite deformation of a variety of annealed metals. Specific formulae are provided for the c ase when the underlying finitely deformed state is a pure homogeneous deformation. These equations are applied to study the effects of a sma ll, superimposed simple shear, a shear superimposed on an equibiaxial stretch, and a superimposed simple extension. Results similar to those known for incompressible materials are obtained for the problem of a small, superimposed torsional deformation, including Rivlin's universa l result relating the applied torque to the torsional stiffness. The d ifference, however, is that a Bell material cannot sustain any sort of finite isochoric deformation relative to its undeformed state, while an incompressible material supports only isochoric deformations. In ad dition, general universal relations are derived relating the increment al stress and stretch tensors to those of the underlying finitely defo rmed state. Small amplitude plane waves propagating on a finitely defo rmed state of pure homogeneous strain are studied. Several universal w ave relations similar to those found for incompressible materials are reported. It is shown that at most two plane waves may propagate in an y given direction in an isotropic Bell material. These are pure shear waves only when the directions of propagation are along the principal axes of the underlying static homogeneous deformation; otherwise, thes e shear waves are called quasi-shear waves. Longitudinal, incremental plane waves are not possible in any isotropic Bell material. When the material is hyperelastic it is shown that the squared wave speeds are real. Also, for propagation in a direction n it is seen that the two a mplitude vectors form an orthogonal triad with Fn, where F is the defo rmation gradient of the basic homogeneous deformation. Circularly pola rized waves are considered. It is shown that a necessary and sufficien t condition for their propagation in any given direction n is that the secular equation have a double root corresponding to n. When the unde rlying deformation is equibiaxial it is seen that the secular equation has two simple factors. The corresponding wave solutions are obtained . It is seen that circularly polarized waves may propagate along the d irection of the equibiaxial deformation and also along a circular cone whose axis is the axis of the equibiaxial deformation.