Introduction to Mathematical Physics/Energy in continuous media/Other phenomena

__toc__ secpiezo Piezoelectricity In the study of piezoelectricity ([#References|references]), on\index{piezo electricity} the form chosen for is: The tensor traduces a coupling between electrical field variables and the deformation variables present in the expression of : The expression of becomes: so: Viscosity A material is called viscous \index{viscosity} each time the strains depend on the deformation speed. In the linear viscoelasticity theory ([#References|references]), the following strain-deformation relation is adopted: Material that obey such a law are called {\bf short memory materials} \index{memory} since the state of the constraints at time depends only on the deformation at this time and at times infinitely close to (as suggested by a Taylor development of the time derivative). Tensors and play respectively the role of elasticity and viscosity coefficients. If the strain-deformation relation is chosen to be: eqmatmem then the material is called long memory material since the state of the constraints at time depends on the deformation at time but also on deformations at times previous to . The first term represents an instantaneous elastic effect. The second term renders an account of the memory effects. Those materials belong ([#References references]) to a special class of materials called {\it rate type} for which there exist a linear relation between time derivatives of tensors deformation and strain tensors: By Laplace transform, it can be shown that it is equivalent to consider a law: where and where is a regular function. In the frame of distribution theory, time derivatives can be considered as convolutions by derivatives of Dirac distribution. For instance, time derivation can be expressed by the convolution by . This allows to treat this case as a particular case of formula given by equation eqmatmem.