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

secpiezo

## Piezoelectricity

In the study of piezoelectricity ([#References|references]), on\index{piezo electricity} the form chosen for ${\displaystyle \sigma _{ij}}$  is:

${\displaystyle \sigma _{ij}=\lambda _{ijkl}u_{kl}+\gamma _{ijk}E_{k}}$

The tensor ${\displaystyle \gamma _{ijk}}$  traduces a coupling between electrical field variables ${\displaystyle E_{i}}$  and the deformation variables present in the expression of ${\displaystyle F}$ :

${\displaystyle F=F_{0}+\epsilon _{ij}E_{i}E_{j}+{\frac {1}{2}}\lambda _{ijkl}u_{ij}u_{kl}+\gamma _{ijk}E_{i}u_{jk}.}$

The expression of ${\displaystyle D_{i}}$  becomes:

${\displaystyle D_{i}=\left.{\frac {\partial F}{\partial E_{i}}}\right)_{u_{fixed},T_{fixed}}}$

so:

${\displaystyle D_{i}=\epsilon _{ij}E_{j}+\gamma _{ijk}u_{jk}}$

## 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:

${\displaystyle \sigma _{ij}=a_{ijkl}u_{kl}+b_{ijkl}{\frac {\partial u_{kl}}{\partial t}}}$

Material that obey such a law are called {\bf short memory materials} \index{memory} since the state of the constraints at time ${\displaystyle t}$  depends only on the deformation at this time and at times infinitely close to ${\displaystyle t}$  (as suggested by a Taylor development of the time derivative). Tensors ${\displaystyle a}$  and ${\displaystyle b}$  play respectively the role of elasticity and viscosity coefficients. If the strain-deformation relation is chosen to be:

eqmatmem

${\displaystyle \sigma _{ij}=a_{ijkl}u_{kl}+\int _{0}^{t}b_{ijkl}(t-s)u_{kl}(s)ds,}$

then the material is called long memory material since the state of the constraints at time ${\displaystyle t}$  depends on the deformation at time ${\displaystyle t}$  but also on deformations at times previous to ${\displaystyle t}$ . 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 ${\displaystyle \delta '(t)}$ . This allows to treat this case as a particular case of formula given by equation eqmatmem.