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

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secpiezo

Piezoelectricity[edit]

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[edit]

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.