A slip wave solution in anti-plane elasticity

Several interfacial wave solutions are known in elasticity theory. In anti-plane elasticity, the Love wave (Love, 1911) exists in bonded contact of an elastic layer with a dissimilar elastic half-space. In in-plane elasticity, the slip wave (Achenbach and Epstein, 1967) and Stoneley wave (Stoneley, 1924) solutions are well known. The slip wave (also known as the generalized Rayleigh wave) occurs for frictionless contact of two dissimilar elastic half-spaces while the Stoneley wave occurs for bonded contact of dissimilar elastic half-spaces. It is shown here that another interfacial wave solution exists in anti-plane elasticity, namely, a slip wave for antiplane sliding of an elastic layer on an elastic half-space.

Consider an isotropic elastic layer of thickness h sliding on an isotropic elastic halfspace at a steady rate V o as in Fig. 1. A shear stress τ o is applied at the boundary such that it is at the friction threshold, τ o = fσ o , where σ o is the compressive normal stress at the boundary and f is the constant friction coefficient. The shear modulus, density and shear wave speed of the layer are denoted by µ , ρ and c s , respectively, and corresponding properties of the half space are denoted by µ’, ρ’ and c’ s .

A Cartesian coordinate system is located so that the interface between the solids is at

x 2 = 0 and the layer slides in the x 3 direction (Fig. 1). The elastic fields are assumed to be independent of the x 3 coordinate. We obtain the elastodynamic relation between the anti-plane slip and stress perturbations at the interface. If u i (x 1 , x 2 ,t) , i = 1,2,3 denote the displacements, an anti-plane displacement field is given by

(1)

Let τ ij (x 1 , x 2 ,t), i, j = 1,2,3 denote the stresses. The only non-zero stresses corresponding to the displacement field in Eq. ( 1) are τ 13 = τ 31 and τ 23 = τ 32 . They are given by

the latter being the traction component on planes normal to the x 2 direction.

The equation of motion for the layer is

(3)

Substituting for the stresses from Eq. ( 2), one gets the 2D wave equation

where c s = µ /ρ . Similarly, the equation of motion of the elastic half-space in the region x 2 < 0 is

where c’ s = µ’ /ρ’ is the shear wave speed of the half space.

Consider slip at the interface of the form

where the first term represents steady state slip at a rate V o and the second term represents a perturbation of amplitude D(k, p) in a single Fourier mode of wavenumber k . The variable p is the time response of the perturbation. The corresponding traction component of stress at the interface can be written as

where T (k, p) is the amplitude of the shear stress perturbation.

Following Ranjith (2009), it can be shown that the amplitudes of the shear stress and slip perturbations at the interface satisfy

where

Interfacial wave solutions:

For a given k , a pole of F(k, p) indicates a stress perturbation with no associated slip perturbation. The pole is associated with the Love wave solution, which is well studied. The phase velocity of the Love wave speed depends on the wavenumber k .

The wave always exists for any k and µ /µ’ as long as c s < c’ s .

Using the notation c = ±ip / k for the phase velocity, the dispersion relation for the Love wave can be written as

This dispersion relation is multi-valued due to the multi-valued nature of the inverse tangent function. To show explicitly the multi-valued nature of the dispersion relation, it can be rewritten as

where arctan denotes the principal value of the inverse tangent that lies between 0 and π , and n ≥ 0 is an integer.

A zero of F(k, p) indicates a slip perturbation with no associated stress perturbation.

The zero corresponds to the slip wave solution. For generic k , zeroes occur when c s ≤ c and they are determined by the condition that

for an integer n ≥ 0 . Similar to the Love wave, the slip wave is also a dispersive wave since its phase velocity depends on the wavenumber. The slip wave exists irrespective of whether c s ’ ≤ c s or c s ≤ c s ’ and hence exists for any pair of bi-materials. It may be noted that the phase velocity of the slip wave does not depend on the properties of the half-space.

By studying Equations ( 11) and ( 12), it is clear that for the same mode (value of n ), the phase velocity of the slip wave is less than that of the Love wave. The ordering of the phase velocities of the two interfacial waves may also be seen using the Rayleigh quotient (Rice et al., 2001). Standing waves may be constructed by a superposition of exp(ik(xct)) and exp(ik(x + ct)) type solutions: these have a frequency | k | c .

Since the displacement field for the Love wave (for a given mode n ) is admissible for the slip wave, it is of a higher frequency. Hence the phase velocity of the Love wave is higher than that of the slip wave for the same mode.

Further, the interleaving of the slip wave and Love wave phase velocities is also seen. 12)) then it is clear that

It may be noted that the result

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