Temperature-related effective properties and exact relations for thermo-magneto-electro-elastic fibrous composites

Based on the asymptotic homogenization method, a three-dimensional boundary value problem for two-phase periodic thermo-magneto-electro-elastic composites is presented. We consider composite materials consisting of identical parallel cylindrical fibers periodically distributed in a matrix. Both constituents are transversely isotropic and the unit-cell has rhombohedral shape. The derivation of the corresponding homogenized problem, effective coefficients and local problems is summarized. Exact relations are derived without imposing any restriction on the global behavior of such fibrous composite materials. In particular, proportionality relations linking the temperature-related effective coefficients are obtained. Simple analytical formulae are obtained for these temperature-related effective coefficients as a function of four effective elastic coefficients, and they show the existence of pyroelectric and pyromagnetic product properties in the composite material when these effects are not present at the constituents level. Moreover, the formulae are valid for any fiber cross-section geometry and periodicity. An analytical formula of the fiber area fraction for which the effective pyroelectric and pyromagnetic coefficients are maximal is given. Some numerical examples are considered in order to illustrate the relevance of the combination of the exact relations with homogenization methods. Comparisons with analytical and numerical results are included.