# 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 (elastic) Hookean solid and the (viscous) Newtonian liquid, along with a time 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>. This type of solid obeys Hooke's law for any magnitude of applied stress.

## 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>.

One deviation from a Newtonian liquid is a liquid that has a viscosity that is dependent on shear rate, such that <math>\sigma = \eta (\dot{e}) \dot{e}</math>.

## Example

Since viscoelastic behavior comes in various forms that (in general) need to be treated individually, it is instructive to look at a simple example.

In Figure 2, a constant stress is being applied at time <math>t = 0</math>. Until the relaxation time <math>\tau</math>, the material acts in an elastic way, but after the relaxation time it acts in a liquid fashion. In this sense, the relaxation time for a viscoelastic material under a particular applied stress separates when the material acts like a solid, and when it acts like a liquid. Notice that for the elastic response, the strain is constant (Hookean solid behavior), and for the viscous response, the strain changes linearly with time (Newtonian liquid behavior).

## References

[1] R. Jones, "Soft Condensed Matter," Oxford University Press Inc., New York (2002).