# Advanced elasticity/Ogden material

The **Ogden material model** is a w:hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers, w:polymers, and w:biological tissue. The model was developed by w:Ray W. Ogden in 1972. ^{[1]} The Ogden model, like other w:hyperelastic material models, assumes that the material behaviour can be described by means of a strain energy density function, from which the stress-strain relationships can be derived. These materials can generally be considered to be w:isotropic, w:incompressible and w:strain rate independent.

## Ogden Material Model[edit]

In the Ogden material model, the strain energy density is expressed in terms of the principal stretches , as:

where , and are material constants. Under the assumption of incompressibility one can rewrite as

In general the shear modulus results from

With and by fitting the material parameters, the material behaviour of rubbers can be described very accurately. For particular values of material constants, the Ogden model will reduce to either the w:Neo-Hookean solid(, ) or the Mooney-Rivlin material (, , ).

Using the Ogden material model, the three principal values of the Cauchy stresses can now be computed as

where use is made of .

### Uniaxial tension[edit]

We now consider an incompressible material under uniaxial tension, with the stretch ratio given as . The principal stresses are given by

The pressure is determined from incompressibility and boundary condition , yielding:

### Relation to other models[edit]

There exist many models to describe hyperelastic behaviour, each starting from a given strain-energy density function. In practice, however, the Ogden material model has become the reference material law for describing the behaviour of natural rubbers as it combines accuracy with computational simplicity.

## References[edit]

- ↑ Ogden, R. W., (1972).
*Large Deformation Isotropic Elasticity - On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids*, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 326, No. 1567 (Feb. 1, 1972), pp. 565-584.

- F. Cirak: Lecture Notes for
*5R14: Non-linear solid mechanics*, University of Cambridge. - R.W. Ogden:
*Non-Linear Elastic Deformations*, ISBN 0-486-69648-0 - K. Weinberg: Lecture Notes for
*Zur Methode der finiten Elemente in der Mechanik II: Nichtlineare Probleme*, TU Berlin [in English].

http://mechanik.tu-berlin.de/weinberg/Lehre/fem2/Chapter4.pdf