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 $F$:

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

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

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

then the material is called long memory material since the state of the constraints at time $t$ 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.