Séance Séminaire

Séminaire des Doctorant·e·s

Mathematical modeling of linear thermo-electromagnetoelastic materials

The partial differential equations of elasticity show that, when an elastic structure is subjected to a system of external loads, it undergoes a passive deformation, without being able to control its strain state. In the case of the so-called smart structures, the strain state is constantly under control with the help of sensors and actuators, usually made of piezoelectric and piezomagnetic materials, which are integrated within the structure- for this reason, their conception and use have undergone a major development over the past few years. The behavior of such structures, which we assume linear, is described mathematically by means of a mixed parabolic-hyperbolic system of evolution partial differential equations, namely, the 'classical' force balance equation for three-dimensional continua, Maxwell's equations and the linearized (with respect to temperature) energy balance equation. Existence and uniqueness of the solution to this system can be proven by virtue of a result of semigroup theory, i.e., a version of the Hille-Yosida theorem. Concerning the modeling part, I will try to give a physical insight into the phenomenologies involved in the problem and recall some elements of continuum thermomechanics; on the other hand, for the mathematical part, I will recall some basic elements of semigroup theory and provide a sketch of (part of) the proof regarding the existence and uniqueness of the solution to the system of PDE at hand. If time permits, I will also outline rapidly some further work that has been done so far, in particular, the nondimensionalization of the equations and its use to justify the quasi-static assumption on the electric and the magnetic fields. The presentation will not take into account technical details.