Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state dependent law with velocity-weakening behavior in the steady state. The stability to spatial perturbations of the form exp(ikx_1), where k is the wavenumber and x_1 is the coordinate along the interface is studied. The critical wavenumber magnitude, |k|_cr, above which there is stability and the corresponding phase velocity, c, of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of |k|_cr and c on the unperturbed sliding velocity, V_o, are provided for various bi-material combinations of practical interest.

💡 Research Summary

The paper investigates the dynamic stability of steady anti‑plane sliding along a planar interface separating two dissimilar anisotropic linear elastic solids. Both solids are assumed to possess a plane of symmetry normal to the slip direction, which eliminates in‑plane displacements and normal‑stress variations on the interface, thereby reducing the problem to a single out‑of‑plane displacement field u₃(x₁,x₂,t). The authors adopt a rate‑and‑state friction law with velocity‑weakening steady‑state behavior, characterized by the parameters a, b, L and the normal stress σ₀.

A rigorous elastodynamic formulation is developed. Starting from the constitutive relations for each material (characterized by three independent stiffnesses C₄₄, C₄₅, C₅₅ and density ρ), the governing equation for the perturbation field is obtained, and Laplace (in time) and Fourier (in space) transforms are applied. This yields a second‑order ordinary differential equation in the depth coordinate x₂ whose solution is an exponential e^{αx₂}. The decay exponent α satisfies a quadratic equation whose physically admissible root leads to the definition of an effective shear modulus μ = C₄₄C₅₅ − C₄₅² and an associated shear‑wave speed c₁ = μ/√(C₄₄ρ). An analogous set (μ′, c₁′) is defined for the lower half‑space.

The traction‑displacement relation at the interface for each half‑space takes the compact form

 T̂(k,p) = −μ|k|