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

Figure 2, taken from reference [1]

## References

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