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

Abstract

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 ¼ 0. The cases of orthorhombic materials and of incompressible materials are also treated.

Approximation of the flexural velocity branch in plates

“…(3) 1 , inferred from (1) 1 , is a better approximant than the estimate v 2 f ðxÞ % jxd=ð1 þ jxd=c 2 R Þ, formally constructed on the same grounds as (1) 1 but lacking the advantageous counterbalance of the long and short wave trends. Concerning the evaluation of c R involved in these and other approximations to be discussed, one may consult the review by Destrade (2003) with ample bibliography and also Achenbach (1973), Pham Chi Vinh andOgden (2004a,b).…”

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

“…The notation I 1 ; : : : ; I 5 is fairly standard for these invariants; see, for example, [Merodio and Ogden 2002;2003]. In the reference configuration B r these reduce to…”

“…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].…”

An approach for obtaining approximate formulas for the Rayleigh wave velocity

“…It is easy to verify that the system (17) with coefficients defined by (18) has precisely one solution, namely:…”

“…For these materials, the dispersion equation of the Rayleigh waves in the explicit form was found recently by Destrade [18], namely:…”