# 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, with a response to shear stress dependent on time (i.e. viscoelastic substances react in a time-dependent manner to a constant applied shear stress).

## Hookean Solid

A Hookean solid is a solid that displays perfectly elastic behavior. This corresponds to the fact that an applied shear stress produces a constant 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>.

## Examples

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 shear stress is initially applied at time <math>t = 0</math> and is held constant. The material initially acts in an elastic (Hookean solid) manner (i.e. there is a constant strain response to a constant shear stress) until the relaxation time <math>\tau</math>, after which it reacts to the applied stress in the same way a Newtonian liquid would (i.e. there is a linearly increasing shear strain response to a constant shear stress). 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.

Another (more fun) example can be found here: Viscoelastic Example. In this video, a large amount of cornstarch and water have been mixed together to form a non-Newtonian liquid. As demonstrated, a short applied stress (quickly running across the material) results in an elastic response, whereas a long applied stress (standing on the material) results in a viscous response.

## References

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