# 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 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 ($\sigma$) is given by the applied force over the area, namely $\sigma = F/A$, and the shear strain ($e$) is given by $e = \Delta x/y$. See Figure 1 for clarification.

Figure 1, taken from reference [1]

For a Hookean solid, we simply have the shear stress proportional to the applied stress by a proportionality constant called the shear modulus ($G$), $\sigma = Ge$. 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 ($\eta$), $\sigma = \eta \dot{e}$.

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

## 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 $t = 0$. Until the relaxation time $\tau$, 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 (just like a Hookean solid would), and for the viscous response, the strain changes linearly with time (just like a Newtonian liquid).

Figure 2, taken from reference [1]

## References

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