The incompressible limit in linear anisotropic elasticity, with applications to surface waves and elastostatics

Abstract

Incompressibility is established for three-dimensional and two-dimensional deformations of an anisotropic linearly elastic material, as conditions to be satisfied by the elastic compliances. These conditions make it straightforward to derive results for incompressible materials from those established for the compressible materials. As an illustration, the explicit secular equation is obtained for surface waves in incompressible monoclinic materials with the symmetry plane at x 3 = 0. This equation also covers the case of incompressible orthotropic materials.The displacements and stresses for surface waves are often expressed in terms of the elastic stiffnesses, which can be unbounded in the incompressible limit. An alternative formalism in terms of the elastic compliances presented recently by Ting is employed so that surface wave solutions in the incompressible limit can be obtained. A different formalism, also by Ting, is employed to study the solutions to two-dimensional elastostatic problems.In the special case of incompressible monoclinic material with the symmetry plane at x 3 = 0, one of the three Barnett-Lothe tensors S vanishes while the other two tensors H and L are the inverse of each other. Moreover, H and L are diagonal with the first two diagonal elements being identical. An interesting physical phenomenon deduced from this property is that there is no interpenetration of the interface crack surface in an incompressible bimaterial. When only the inplane deformation is considered, it is shown that the image force due to a line dislocation in a half-space or in a bimaterial depends only on the magnitude, not on the direction, of the Burgers vector.

Stroh formulation for a generally constrained and pre-stressed elastic material

“…Over the past five decades, such constrained material models have frequently been used in the study of acceleration waves, shock waves, and body waves [17][18][19][20][21][22][23][24][25]. They have also been used in the study of surface waves [26][27][28][29][30][31][32][33][34][35][36][37]. In this paper, we shall consider a generally constrained and prestressed elastic material.…”

“…After writing down the governing equations in Section 2, we derive in Section 3 the Stroh formulation for the problem under consideration, and in Section 4 an integral representation and a Riccati equation for the surface-impedance tensor. The integral representation is possible due to the fact that our Stroh formulation is written entirely in terms of tensors, unlike those of Chadwick [31] and Destrade et al [36] when the material is incompressible. All degenerate cases are also considered separately.…”

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

“…Recently, Destrade et al (2002) proved that in linear anisotropic elasticity, the constraint of incompressibility implied that certain relationships must be satisfied for some compliances. In the present context, the following relationships must hold, …”

“…This equation is valid for the propagation of a Rayleigh wave in any direction of a symmetry plane for crystals possessing one plane of symmetry and of course for crystals of a higher order symmetry class, such as orthorhombic symmetry. It is also easy to take an eventual incompressibility of the elastic halfspace into account (Destrade et al, 2002). Finally, after a rotation about the x 3 -axis, the quartic secular equation may reduce to a biquadratic which can then be solved explicitly.…”

Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

“…Surface waves in elastic solids were first studied by Lord Rayleigh [1885] for an isotropic elastic solid. The extension of surface wave analysis and other wave propagation problems to anisotropic elastic materials has been the subject of many studies; see, for example, [Musgrave 1959;Anderson 1961;Stoneley 1963;Chadwick and Smith 1977;Royer and Dieulesaint 1984;Barnett and Lothe 1985;Mozhaev 1995;Nair and Sotiropoulos 1997;Destrade 2001a;2001b;Destrade et al 2002;Ting 2002a;2002c;2002b;Destrade 2003;Ogden and Vinh 2004]. For problems involving surface waves in a finitely deformed pre-stressed elastic solid (strain-induced anisotropy) we refer to [Hayes and Rivlin 1961;Flavin 1963;Chadwick and Jarvis 1979;Dowaikh and Ogden 1990;1991;Norris and Sinha 1995 (concerning a solid/fluid interface) ;Chadwick 1997;Prikazchikov and Rogerson 2004 (concerning prestressed transversely isotropic solids); Destrade et al 2005;Edmondson and Fu 2009]; see also [Song and Fu 2007].…”