Nonisothermal diffuse interface model for electrical breakdown channel propagation

The paper is devoted to the derivation of the gradient (weakly nonlocal) diffuse interface model, which describes an electrical breakdown channel propagation under the application of an electric field. In contrast to earlier presented models, the derived one is nonisothermal and consists of Maxwell's equations in quasi(electro)stationary approximation, Allen—Cahn type equation which governs phase field evolution, and energy balance equation. The derivation of the model is provided based on the rational thermomechanics framework using M.Gurtin microstress and microforce theory and Coleman-Noll procedure to derive constitutive relations of the model. The derived model is thermodynamically consistent and satisfies entropy inequality in the respective form. The closed-form formulation of the model and complete set of constitutive relations are presented.