# 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.

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 initially acts in an elastic (Hookean solid) manner (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).