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



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:





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:



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.

Remark: Those materials belong ([#References


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.