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

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

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

The expression of $$D_i$$ becomes:

so:

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:

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:

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.