# Difference between revisions of "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 $\sigma = \eta (\dot{e}) \dot{e}$.

## 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 $t = 0$ 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 $\tau$, 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.

Figure 2, taken from reference [1]

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).