# Viscoelasticity

## Contents

## Viscoelasticity

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time dependent strain. Whereas elasticity is usually the result of bond stretching along crystallographic planes in an ordered solid, viscoelasticity is the result of the diffusion of atoms or molecules inside of an amorphous material

A basic viscoelastic material is modeled as a purely viscous damper and a purely elastic spring. The damper is modeled as as a Newtonian fluid and the spring with Hooke's law. If we connect these two elements in series we get a Maxwell material, and in parallel we get a Kelvin-Voigt material.

**Spring and Dashpot Models:**

Viscoelastic fluids: Strain rates add.
(Maxwell Model) |
Viscoelastic solids: Strains add.
(Kelvin-Voigt Model) |
And so on, with decreasing physical interpretation. |

## The Maxwell model

Maxwell mode: Add the strain rates: | |

For the elastic component: | |

For the viscous component: | |

The resulting differential equation is: |

## Viscoelasticity - Creep test

For a *creep* test: A stress is applied instantaneously and maintained. The strain is measured as a function of time"

## Viscoelasticity - Stress relaxation

For a *stress relaxation*, a rapid strain is applied and held. The problem is to determine the stress as a function of time:

## Viscoelasticity - Molecular relaxations

## Experimental methods

- Viscoelasticity
- Osmotic pressure
- Force microscopy: scanning, tunneling, atomic
- Dynamic scattering over time scales
- Static scattering – x-rays, neutrons, back-scattering
- NMR

## Canonical ideas

- Typical energies are kT
- Fluctuations and Brownian motion
- Wide length scales – use coarse grain models
- Hierarchical structures
- Physics dominates over chemistry (so far)

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## Example

One example of a viscoelastic material is a cell cytoskeleton; see "The Cell as a Material" for more information.

## Bibliography

- De Gennes, P.-G.; Badoz, J.
*Fragile objects*. Springer-Verlag: New York; 1996. - Einstein, A.
*Investigations on the theory of the Brownian motion*. Dover Publications: New York; 1956. - Elimelech, M.; Gregory, J.; Jia, X.; Williams, R.A.
*Particle deposition & aggregation*; Butterworth-Heinemann: Woburn, MA; 1995. - Hamley, I.W.
*Introduction to soft matter*. Revised ed.; John Wiley & Sons: New York; 2007. - Israelachvili, J.
*Intermolecular and surface forces*, 2nd ed.; Academic Press: New York; 1991. - Witten, T.; Pincus, P.
*Structured fluids: polymers, colloids, surfactant*s. Oxford University Press: New York; 2004.