Propagating and Static Exponential Solutions in a Deformed Mooney - Rivlin Material

Abstract

A homogeneous isotropic incompressible elastic material whose mechanical behavior is modeled through use of the Mooney-Rivlin form of the stored energy function is maintained in a state of finite static homogeneous biaxial deformation. Superimposed on the finite deformation is an infinitesimal, possibly time dependent, deformation. For purely homogeneous waves the slowness surface consists of two coaxial spheroids. Propagating exponential solutions (PES) are inhomogeneous plane wave solutions for which the planes of constant phase are not the same as the planes of constant amplitude. All PES are obtained systematically through the use of a formulation involving bivectors in which a directional bivector C and its attendant directional ellipse is prescribed. Of particular interest is the case when C is isotropic because it yields solutions with a displacement field independent of the material constants and of the parameters describing the homogeneous static biaxial deformation. Static exponential solutions (SES) are similar to PES but are time independent. All SES are obtained systematically. It is seen that for such solutions the directional ellipse of C must be similar and similarly situated to an elliptical section of either sheet of the slowness surface.