# Viscoelastic

## Definition

A substance that displays behavior that is both viscous and elastic is said to be **viscoelastic**. In this sense, viscoelastic materials are said to be a combination of the ideal (elastic) Hookean solid and the (viscous) Newtonian liquid, along with a time a dependence.

## Hookean Solid

A Hookean solid is one that displays perfectly elastic behavior. This corresponds to the fact that an applied shear stress produces a shear strain in response. Recall that the shear stress (<math>\sigma</math>) is given by the applied force over the area, namely <math>\sigma = F/A</math>, and the shear strain (<math>e</math>) is given by <math>e = \Delta x/y</math>. See Figure 1 for clarification.

For a Hookean solid, we simply have the shear stress proportional to the applied stress by a proportionality constant called the shear modulus (<math>G</math>), <math>\sigma = Ge</math>.

## Newtonian Liquid

In the case of a Newtonian liquid, the shear stress is proportional to the first time derivative of the shear strain by a constant called the viscosity (<math>\eta</math>), <math>\sigma = \eta \dot{e}</math>.

## Example

Since viscoelastic behavior comes in various forms,