Singularities in bi-materials: parametric study of an isotropic/anisotropic joint

Problems in fracture mechanics are frequently solved in terms of crack tip singularities. Geometries other than cracks also exhibit singular stresses at points such as corners, edges and interfacial joints. Corners occuring in monolayers or multilayered media have been studied under the assumption that each layer is isotropic. For general elastic plane problems, the present study extends the earlier results to anisotropy. For orthotropic joints, generalized Dundurs parameters are introduced. Isotropic results are a limiting case of the present analysis. In the vicinity of a singular point, the displacements and stresses may be expressed as a function of the polar coordinates r − θ by: u=hr δg(φ), σ=hr δ−1 F(φ) where h is the intensity factor and δ the singularity exponent (0 < δ < 1). The values of h and δ reduce to the stress intensity factor k and the complex exponent 0.5 + iε for the limiting case of cracks at the interface of dissimilar media. Using an anisotropic complex potential method, the present analysis gives δ as the solution of an eigenvalue problem and as the root of a nonlinear equation det A(δ) = 0 . It leads to a closed-form expression for g and F . The matrix A depends on the number of layers at the singular point, their relative elastic properties and the boundary conditions such as free surface or bonded interface close to the singularity corner. A closed-form expression is derived for A which depends on 3 generalized Dundurs parameters for a metal matrix composite isotropic metal interface joint. This compares to the 2 Dundurs parameters needed for joints with isotropic layers. The intensity factor h of any singularity is determined from a path independent integral, using an extraction function which is more singular than that defining the actual stress state.