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

secpiezo

PiezoelectricityEdit

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:

 

ViscosityEdit

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.

Remark: Those materials belong ([#References

Remark:

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.