# 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 $\sigma_{ij}$ is:

$\sigma_{ij}=\lambda_{ijkl}u_{kl}+\gamma_{ijk}E_k$

The tensor $\gamma_{ijk}$ traduces a coupling between electrical field variables $E_i$ and the deformation variables present in the expression of $F$:

$F=F_0+\epsilon_{ij}E_iE_j+\frac{1}{2}\lambda_{ijkl}u_{ij}u_{kl}+ \gamma_{ijk}E_iu_{jk}.$

The expression of $D_i$ becomes:

$D_i=\left.\frac{\partial F}{\partial E_i}\right)_{u_{fixed},T_{fixed}}$

so:

$D_i=\epsilon_{ij}E_j+\gamma_{ijk}u_{jk}$

## 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:

$\sigma_{ij}=a_{ijkl}u_{kl}+b_{ijkl}\frac{\partial u_{kl}}{\partial t}$

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

eqmatmem

$\sigma_{ij}=a_{ijkl}u_{kl}+\int_0^tb_{ijkl}(t-s)u_{kl}(s)ds,$

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