On dissipative effects in thermo-electrically coupled systems: Hill–Mandel-type homogenisation, asymptotic expansions and two-scale convergence

Accurately predicting the macroscopic behaviour of heterogeneous materials, particularly in thermo-electrically coupled systems, remains a challenging problem in materials science and engineering. Against this background, this study presents a comprehensive framework for the homogenisation of electrical conductors subject to strongly coupled thermo-electrical processes in (quasi-)stationary settings, with particular focus lying on Joule heating and temperature-dependent electrical conductivities. The proposed methodology combines analytical homogenisation, asymptotic expansions and Hill–Mandel-type multiscale techniques. In doing so, both, a natural physical interpretation and a rigorous mathematical justification of the governing set of effective macroscopic field equations are provided. Representative boundary value problems in two- and three dimensional settings are eventually studied to validate the effectiveness of these methods in capturing the complex behaviour of heterogeneous thermo-electric materials.