Analytic solutions of thermoelectric materials containing a circular hole with a straight crack

Abstract

The two-dimensional problem of thermoelectric materials under the action of uniform electric current and uniform total energy flux is studied by means of the complex variable function method and the conformal mapping technique. The model of infinite thin plate containing a circular hole with a straight crack is considered, and the boundary conditions of the inner surface are assumed to be electrically and thermally impermeable. The analytic solutions of the electric current density and total energy flux are derived. In addition, the electric current density intensity factor (EIF), the total energy flux density intensity factor (UIF) and the stress intensity factor (SIF) near the crack tip are obtained, which have a great significance for the engineering application of thermoelectric materials. It is found that, on the one hand, the EIF and UIF are in proportion to the applied far-field electric current loads and the total energy flux loads, respectively. On the other hand, all these field's intensity factors are closely related to the geometrical characteristics of the inner boundary (i.e. the circular hole radius R and crack length L). When the radius tends to zero, the degenerated results are identical to the pre-existing solutions in case of the Griffith crack. Numerical results are also made to mainly discuss the effects of radius and crack length on these nondimensionalized field's intensity factors.