Plane strain asymptotic fields of a crack growing along an elastic-elastic power-law creeping bi-material interface

Plane strain asymptotic stress and strain fields near the tip of a crack growing quasi-statically along the interface of a linear elastic and an incompressible elastic power law creeping material is studied. The solutions for the stress and strain fields are found to be in the separable forms, γ_ij, ∼ a ̇ BE_+r ^{1 (n-1)} γ ̃ij(θ,n,β) and ε{lunate}ij ∼ a ̇ BE_+r ^{1 (n-1)} ε{lunate} ̃ij(θ,n,β), whenn ≥ 3. Heren and B are the power law creeping exponent and coefficient respectively, E₊ and E_- are moduli of elasticity of the elastic-creeping and elastic materials, β = E₊ E_-, r is the radial distance from the moving tip and θ is the angle measured from the interface. The amplitude of the field is determined by the crack tip velocity, a, and material parameters. The angular functions \ ̃gs_ij(θ, n, β) and \ ̃ge_ij exhibit only two modes, one of which is close to pure mode I and the other one is close to pure mode II ; arbitrary mode mixity cannot be prescribed. For a given β, there is a corresponding n, close to n = 3, at which the two solutions coalesce into one. Below this value of n no real separable solution exists. We also studied the special case of a quasi-statically growing crack in a homogeneous elastic power law creeping material. We find that only pure modes I and II are possible near the tip, and the amplitude of the field is determined by a and material parameters. Thus the near tip field is independent of the far field loading conditions for both the bi-material and the homogeneous cases. For 1≤n< 3, the elastic strain dominates the creep strain so that the near tip stress field of the growing interface crack is given by the classical bimaterial complex K field and therefore has an oscillatory singularity. The complex K in this case depends not only on the specimen geometry and the applied load, but also on the crack growth history.