An arc-shaped crack in nonlinear fully coupled thermoelectric materials

We present a rigorous treatment to the problem of a circular-arc crack in an infinite thermoelectric solid subjected to a combined electrical and thermal loading. Formulating the problem in terms of the complex potentials and reducing it to the Hilbert arc problem, the solutions of quantities in both thermoelectric field and the associated thermoelastic field are presented in a closed form based on the electrically permeable and thermally insulated crack model. The results show that the fields of heat flux, energy flux, and stress exhibit the traditional square-root singularity at tips of arc crack. The applied electric current and energy flux generate both mode I and mode II stress intensity factors (SIFs), which are dependent on the loading direction, electric conductivity, thermal conductivity, central angle of the crack, and thermoelastic isotropy. Electrically induced SIF is a quadratic function of applied electric current density, and thermally induced SIF is a linear function of the imposed total energy flux. In addition, the effects of the loading direction and half crack angle on the thermal SIFs are also presented in a graphic form. This is the first theoretical paper to study the effect of the crack shape on the fracture of fully coupled thermoelectric materials by a rigorous inference of physics and mathematics.