Steady-state interfacial cracks in bi-material elastic lattices

Fracture mechanics serves both engineering and science in various ways, such as studies of material integrity and physics of earthquakes. Its main object is to analyse crack nucleation and growth depending on features of a particular application. It is common to study cracks in homogeneous materials, however analysis of cracks in bi-materials is important as well, especially in modelling of frictional motion between solids at macro-scale and inter-granular fracture in polycrystallines at micro-scale. The analysis of fracture in dissimilar materials is the main topic of this research. We present the analytical model of steady-state cracks in bi-material square lattices and show its connection with associated macro-level fracture problem.  We consider a semi-infinite crack propagating along the interface between two mass-spring square lattices of different properties. Assuming the linear interaction between lattice masses, we can apply integral transforms and obtain the matrix Wiener-Hopf problem from original equations of motion. In this particular case, the kernel matrix is triangular which significantly simplifies the factorisation procedure and even makes possible to reduce to the scalar Wiener-Hopf problem. The discreteness of the problem, however, does not allow to derive factorisation analytically and numerical factorisation was performed. We show that the problem discreteness reveals microscopic radiation in form of decaying elastic waves emanating from a crack tip. These waves are invisible at macro-scale but their energy contributes to the global energy dissipation during the fracture process. We also demonstrate effects of the material properties mismatch and link the microscopic parameters with the macro-level fracture characteristics.