Modeling of smart materials with thermal effects: Dynamic and quasi-static evolution

Abstract

We present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille–Yosida theory and of the Faedo–Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations. We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter — the ratio of the largest propagation speed for an elastic wave in the body to the speed of light — tends to zero.