Numerical solution to the problem of the interaction of partially insulated cracks in a two-component material subjected to a heat flux

The paper is devoted to a heat conduction problem for two-component materials (bimaterials) with a system of cracks, i.e. an interface crack and internal cracks in one of the materials. It is supposed that the cracks are partially thermal insulated, and the bimaterial is subjected to a heat flux applied at infinity and normal to the interface. Previously obtained system of singular integral equations is used, where the unknowns are the derivatives of the temperature jumps on the crack lines. The regular kernels of the equations contain the geometry of the problem, i.e. the coordinates of the crack centers, inclination angles of the cracks to the interface and crack lengths. The singular integral equations were solved numerically using the quadrature formulas based on the Chebyshev polynomials. Then, the local characteristics of the heat distribution near the crack tips, namely, the thermal intensity factors, were obtained. The thermal intensity factors are an analogue of the stress intensity factors in the fracture mechanics. The influence of the inclination angles of internal cracks and their location on the thermal intensity factors in the tips of the interface crack was analysed for different thermal conductivity coefficients of the constituent materials and for different parameters of thermal permeability of the crack surfaces. The following combinations of materials were used: $\mathrm{TiC/SiC}$, $\mathrm{Al_2O_3/SiC}$, $\mathrm{TiC/Al_2O_3}$, $\mathrm{TiC/ZrO_2}$. It was shown that internal cracks can either decrease or increase the thermal intensity factors at the interface crack tips in comparison to the results for a single interface crack. The variation of the thermal intensity factors in the vicinity of the interface crack causes essential non-homogeneity of the heat fluxes in these domains which could lead to high residual stresses at the interface.