Thibaut Weller

Piezomagnetic tensors symmetries: an unifying tentative approach

Traditionally, one inquires as to what is the form of a physical property tensor invariant under the point group of a crystal. For example, one finds listings showing 16 distinct forms of the piezomagnetic tensor that can arise in a crystal. Alongside each of these 16 distinct forms is listed the point groups which give rise to that specific form (see Birss (1966) for example). This is a classification of the form of piezomagnetic tensors in crystals. We claim however that this classification is quite ambiguous in the sense that it hides the very difference between microscopic and macroscopic symmetries. The topic of the paper is the both a clarification of the difference between material and behaviour symmetries and a classification of piezomagnetic tensors in behaviour equivalences classes, found to be 15.

Quinze Ans Après...

We aim to present mathematical models of smart devices and smart structures. Smart devices are made of materials which present significant multiphysical couplings. They are integrated in smart structures which take technological advantages of some multiphysical effects. We first propose simplified but accurate models of thin plates or slender rods made of piezoelectric or electromagneto-elastic materials in both static and dynamic cases. Then we focus on smart structures such as piezoelectric patches bonded on a linearly elastic body and piezoelectric junctions between two linearly piezoelectric or elastic bodies.

Mathematical Modelings of Some Smart Materials and Structures

Models of smart materials and structures are derived through rigorous mathematical methods. We establish a classification of piezoelectric and piezomagnetic crystalline materials and propose simplified but accurate models of thin structures made of piezoelectric or electromagneto-elastic materials.

Asymptotic Modelling of Linearly Piezoelectric Plates

Here, we rigorously derive a theory of linearly piezoelectric plates by studying the limit behaviour of a three-dimensional flat body as its thickness tends to zero. In the static case, two limit models appear depending essentially on the nature and the magnitude of the electromechanical loading. In the dynamic case, under the realistic quasi-electrostatic approximation, the limit behaviour depends further more on the relative magnitudes of the density and of the thickness of the plate. The transient problems can be formulated in term of evolution equation in Hilbert spaces of possible states with finite electromechanical energy, so that the studies of these transient problems are easily deduced from the static case trough the Trotter theory of convergence of semi-groups of operators acting on variable spaces.

Mathematical Modeling of Thin Multiphysical Structures

Smart materials, which present significant multiphysical couplings, are now widely used for the conception of smart structures whose mathematical modelings are here presented in the case of thin plates or slender rods made of piezoelectric or electromagneto-elastic materials.