MSR/ressources/Viscoelasticity.md

Viscous materials, like water, resist both shear flow and strain linearly with
time when a stress is applied. Elastic materials strain when stretched and
immediately return to their original state once the stress is removed. The study of such materials is covered by [[Rheology]].

A viscoelastic substance dissipates energy when a load is applied, then removed.

In most of the analyses, extensional/compressional strains of the viscoelastic layer are not taken into account since the damping comes mostly from the shear strain.

The properties of a viscoelastic layer change with the frequency of excitation .

# Models
(see [[Rheology#Linear viscosity]] for the definition of $f$ and $g$, and for the sources)

## Perfectly elastic solid

Rheologic equation $$\varepsilon(t)=J\tau(t)$$

Where $J$ is the elastic compliance. $G=\frac{1}{J}$ is called the elastic modulus

the creep function $f(t)$ is then :
$$f(t)=J$$

## Newtonian viscous fluid

Rheologic equation (see [[Rheology#Dynamic viscosity]]) 
$$\frac{d\varepsilon(t)}{dt}=\frac{\tau(t)}{\eta}$$
initial conditions : $t<0$ ; $\tau(t)=0$ ; $\varepsilon(t)=0$, so
$$\varepsilon(t)=\frac{1}{\eta}\int_{0}^{t}\tau(t')dt'$$
under a constant stress $\tau_0$ :
$$\varepsilon(t)=\frac{\tau_0}{\eta}t$$
the creep function is obtained with a constant unit stress, so with $\tau_0=1$
$$f(t)=\frac{t}{\eta}$$
## Associations

### Parallel
- stress to the whole is **the sum of** the stresses on each branch
- strain to the whole is **equal to** the strains in each branch
### Series
- stress to the whole is **equal to** the stresses on each branch
- strain to the whole is **the sum of** the strains in each branch