Rayleigh waves and surface stability for Bell materials in compression; comparison with rubber

TL;DR

This paper investigates Rayleigh wave stability on Bell materials under compression, deriving exact equations and comparing the stability limits with rubber-like materials, revealing differences in compressive capacity before buckling.

Contribution

It provides the exact secular equation for Rayleigh waves on Bell materials and compares their stability limits with those of rubber-like materials under compression.

Findings

Bell materials are less compressible than rubber-like materials before instability.

Exact secular equations for surface waves on Bell materials are established.

Comparison shows different bifurcation thresholds between Bell and rubber-like materials.

Abstract

The stability of a Bell-constrained half-space in compression is studied. To this end, the propagation of Rayleigh waves on the surface of the material when it is maintained in a static state of triaxial prestrain is considered. The prestrain is such that the free surface of the half-space is a principal plane of deformation. The exact secular equation is established for surface waves traveling in a principal direction of strain with attenuation along the principal direction normal to the free plane. As the half-space is put under increasing compressive loads, the speed of the wave eventually tends to zero and the bifurcation criterion, or stability equation, is reached. Then the analysis is specialized to specific forms of strain energy functions and prestrain, and comparisons are made with results previously obtained in the case of incompressible neo-Hookean or Mooney-Rivlin materials. It is found that these rubber-like incompressible materials may be compressed more than "Bell empirical model" materials, but not as much as "Bell simple hyperelastic" materials, before the critical stretches, solutions to the bifurcation criterion, are reached. In passing, some classes of incompressible materials which possess a relative-universal bifurcation criterion are presented.