Herein we derive an explicit first-order perturbation formula for the phase velocity of Rayleigh waves that propagate along the free surface of a homogeneous pre-stressed half-space, the incremental elasticity tensor L of which has a principal part C that is orthorhombic (with two of the axes of two-fold rotational symmetry being normal to the free surface and being the propagation direction, respectively) or transversely isotropic (with the ∞-axis being normal to the free surface). The remaining (perturbative) part A of the incremental elasticity tensor can be arbitrarily anisotropic. As an application of the perturbation formula, we consider the case where C is transversely isotropic and A is given by a contraction of the sixth-order acoustoelastic tensor with the pre-stress, and we examine the possibility of and the problems and issues about determination of the pre-stress by boundary measurement of phase velocities of Rayleigh waves.
TanumaKManC-S. Propagation of Rayleigh waves in anisotropic media and an inverse problem in characterization of initial stress. In: Mantic̆V (ed.) Mathematical Methods and Models in Composites. London: Imperial College Press, in press.
2.
TanumaKManC-S. Perturbation formula for phase velocity of Rayleigh waves in prestressed anisotropic media. J Elasticity2006; 85: 21–37.
3.
DattaSKLedbetterHM. Elastic constants of fiber-reinforced boron-aluminum: observation and theory. Int J Solids Struct1983; 19: 885–894.
4.
WuT-TChaiJ-F. Propagation of surface waves in anisotropic solids: theoretical calculation and experiment. Ultrasonics1994; 32: 21–29.
5.
NayfehAH. Wave Propagation in Layered Anisotropic Media with Applications to Composites. Amsterdam: North-Holland, 1995.
6.
RoseJL. Ultrasonic Waves in Solid Media. Cambridge: Cambridge University Press, 1999.
7.
DattaSK. Wave propagation in composite plates and shells. In: KellyAZwebenC (eds) Comprehensive Composite Materials. Amsterdam: Elsevier, 2000, 511–558.
8.
DattaSKShahAH. Elastic Waves in Composite Media and Structures: with Applications to Ultrasonic Nondestructive Evaluation, Boca Raton: CRC Press, 2009.
9.
LotheJBarnettDM. On the existence of surface-wave solutions for anisotropic elastic half-spaces with free surface. J Appl Phys1976; 47: 428–433.
10.
ChadwickPSmithGD. Foundations of the theory of surface waves in anisotropic elastic materials. In: YihC-S (ed.) Advances in Applied Mechanics. New York: Academic Press, 1977, vol. 17, 303–376.
11.
BarnettDMLotheJ. Free surface (Rayleigh) waves in anisotropic elastic half-spaces: the surface impedance method. Proc R Soc London A1985; 402: 135–152.
12.
TingTCT. Anisotropic Elasticity, Theory and Applications. New York: Oxford University Press, 1996.
13.
TanumaK. Stroh formalism and Rayleigh waves, J Elasticity2007; 89: 5–154.
14.
FuYBMielkeA. A new identity for the surface-impedance matrix and its application to the determination of surface-wave speeds. Proc R Soc London A2002; 458: 2523–2543.
15.
MielkeAFuYB. Uniqueness of the surface-wave speed: a proof that is independent of the Stroh formalism. Math Mech Solids2004; 9: 5–15.
16.
FuYB. An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain. Int J Non-Linear Mech2005; 40: 229–239.
17.
EdmondsonRTFuYB. Stroh formulation for a generally constrained and pre-stressed elastic material. Int J Non-Linear Mech2009; 44: 530–537.
18.
NakamuraG. Existence and propagation of Rayleigh waves and pulses. In: WuJJTingTCTBarnettDM (eds) Modern Theory of Anisotropic Elasticity and Applications. Philadephia: SIAM Proceedings, 1991, 215–231.
19.
KawashitaMNakamuraG. The poles of the resolvent for the exterior Neumann problem of anisotropic elasticity. SIAM J Math Anal2000; 31: 701–725.
20.
ManC-SNakamuraGTanumaKWangS. Dispersion of Rayleigh waves in vertically-inhomogeneous prestressed elastic media. (preprint)
21.
DestradeM. Explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J Acoust Soc Am2001; 109: 1398–1402.
22.
TingTCT. Explicit secular equations for surface waves in monoclinic materials with the symmetry plane at x1 = 0,x2 = 0 or x3 = 0. Proc R Soc London A2002; 458: 1017–1031.
23.
DestradeM. Rayleigh waves in symmetry planes of crystals; explicit secular equations and some explicit wave speeds, Mech Mater2003; 35: 931–939.
24.
FuYBBrookesDW. An explicit expression for the surface-impedance tensor of a compressible monoclinic material in a state of plane strain. IMA J Appl Math2006; 71: 434–445.
25.
TingTCT. An explicit secular equation for surface waves in an elastic material of general anisotropy. Q J Mech Appl Math2002; 55: 297–311.
26.
TingTCT. Explicit secular equations for surface waves in an anisotropic elastic half-space; from Rayleigh to today. In: GoldsteinRVMauginGA (eds) Surface Waves in Anisotropic and Laminated Bodies and Defects Detection. Norwell: Kluwer Academic Publishers, 2004, 95–116.
27.
ChadwickPJarvisDA. Surface waves in a pre-stressed elastic body. Proc R Soc London A1979; 366: 517–536.
28.
TanumaKManC-S. Angular dependence of Rayleigh-wave velocity in prestressed polycrystalline media with monoclinic texture. J Elasticity2002; 69: 181–214.
29.
TanumaKManC-S. Perturbation formulas for polarization ratio and phase shift of Rayleigh waves in prestressed anisotropic media. J Elasticity2008; 92: 1–33.
30.
OgdenRWPhamCV. On Rayleigh waves in incompressible orthotropic elastic solids. J Acoust Soc Am2004; 115: 530–533.
31.
DowaikhMAOgdenRW. On surface wave and deformations in a pre-stressed incompressible elastic solid. IMA J Appl Math1990; 44: 261–284.
32.
ChadwickP. The application of the Stroh formalism to prestressed elastic media. Math Mech Solids1997; 2: 379–403.
33.
PrikazchikovDARogersonGA. On surface wave propagation in incompressible, transversely isotropic, pre-stressed elastic half-spaces. Int J Eng Sci2004; 42: 967–986.
34.
DestradeMOtténioMPichuginAVRogersonGA. Non-principal surface waves in deformed incompressible materials. Int J Eng Sci2005; 43: 1092–1106.
35.
PhamCV. On formulas for the velocity of Rayleigh waves in prestrained incompressible elastic solids. J Appl Mech2010; 77: 021006 (9 pages).
36.
OgdenRWSinghB. Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited. J Mech Mater Struct2011; 6: 453–477.
37.
SongYQFuYB. A note on perturbation formulae for the surface-wave speed due to perturbations in material properties. J Elasticity2007; 88: 187–192.
38.
HogerA. On the determination of residual stress in an elastic body. J Elasticity1986; 16: 303–324.
39.
ManC-SLuWY. Towards an acoustoelastic theory for measurement of residual stress. J Elasticity1987; 17: 159–182.
40.
ManC-SCarlsonDE. On the traction problem of dead loading in linear elasticity with initial stress. Arch Ration Mech Anal1994; 128: 223–247.
41.
ManC-S. Hartig’s law and linear elasticity with initial stress. Inverse Problems1998; 14: 313–319.
42.
ChadwickP. The existence of pure surface modes in elastic materials with orthorhombic symmetry. J Sound Vib1976; 47: 39–52.
43.
RoyerDDieulesaintE. Rayleigh wave velocity and displacement in orthorhombic, tetragonal, hexagonal, and cubic crystals. J Acoust Soc Am1984; 76: 1438–1444.
44.
ManC-S. Effects of crystallographic texture on the acoustoelastic coefficients of polycrystals. Nondestr Test Eval1999; 15: 191–214.
ManC-SHuangM. A representation theorem for material tensors of weakly-textured polycrystals and its applications in elasticity. J Elasticity2012; 160: 1–42.
