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

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

Rayleigh waves in anisotropic crystals rotating about the normal to a symmetry plane

The propagation of surface acoustic waves in a rotating anisotropic crystal is studied. The crystal is monoclinic and cut along a plane containing the normal to the symmetry plane; this normal is also the axis of rotation. The secular equation is obtained explicitly using the "method of the polarization vector", and it shows that the wave is dispersive and decelerates with increasing rotation rate. The case of orthorhombic symmetry is also treated. The surface wave speed is computed for 12 monoclinic and 8 rhombic crystals, and for a large range of the rotation rate/wave frequency ratio

Stoneley waves and interface stability of Bell materials in compression; Comparison with rubber

Two semi-infinite bodies made of prestressed, homogeneous, Bell-constrained, hyperelastic materials are perfectly bonded along a plane interface. The half-spaces have been subjected to finite pure homogeneous predeformations, with distinct stretch ratios but common principal axes, and such that the interface is a common principal plane of strain. Constant loads are applied at infinity to maintain the deformations and the influence of these loads on the propagation of small-amplitude interface (Stoneley) waves is examined. In particular, the secular equation is found and necessary and sufficient conditions to be satisfied by the stretch ratios to ensure the existence of such waves are given. As the loads vary, the Stoneley wave speed varies accordingly: the upper bound is the `limiting speed' (given explicitly), beyond which the wave amplitude cannot decay away from the interface; the lower bound is zero, where the interface might become unstable. The treatment parallels the one followed for the incompressible case and the differences due to the Bell constraint are highlighted. Finally, the analysis is specialized to specific strain energy densities and to the case where the bimaterial is uniformly deformed (that is when the stretch ratios for the upper half-space are equal to those for the lower half-space.) Numerical results are given for `simple hyperelastic Bell' materials and for `Bell's empirical model' materials, and compared to the results for neo-Hookean incompressible materials

Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds

Rayleigh waves are considered for crystals possessing at least one plane of symmetry. The secular equation is established explicitly for surface waves propagating in any direction of the plane of symmetry, using two different methods. This equation is a quartic for the squared wave speed in general, and a biquadratic for certain directions in certain crystals, where it may itself be solved explicitly. Examples of such materials and directions are found in the case of monoclinic crystals with the plane of symmetry at $x_3=0$. The cases of orthorhombic materials and of incompressible materials are also treated

The explicit secular equation for surface acoustic waves in monoclinic elastic crystals

The secular equation for surface acoustic waves propagating on a monoclinic elastic half-space is derived in a direct manner, using the method of first integrals. Although the motion is at first assumed to correspond to generalized plane strain, the analysis shows that only two components of the mechanical displacement and of the tractions on planes parallel to the free surface are nonzero. Using the Stroh formalism, a system of two second order differential equations is found for the remaining tractions. The secular equation is then obtained as a quartic for the squared wave speed. This explicit equation is consistent with that found in the orthorhombic case. The speed of subsonic surface waves is then computed for twelve specific monoclinic crystals

Surface waves in orthotropic incompressible materials

The secular equation for surface acoustic waves propagating on an orthotropic incompressible half-space is derived in a direct manner, using the method of first integrals