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.  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

In the Ogden material model, the strain energy density is expressed in terms of the principal stretches $\,\!\lambda _{j}$ , $\,\!j=1,2,3$ as:

$W\left(\lambda _{1},\lambda _{2},\lambda _{3}\right)=\sum _{p=1}^{N}{\frac {\mu _{p}}{\alpha _{p}}}\left(\lambda _{1}^{\alpha _{p}}+\lambda _{2}^{\alpha _{p}}+\lambda _{3}^{\alpha _{p}}-3\right)$ where $N$ , $\,\!\mu _{p}$ and $\,\!\alpha _{p}$ are material constants. Under the assumption of incompressibility one can rewrite as

$W\left(\lambda _{1},\lambda _{2}\right)=\sum _{p=1}^{N}{\frac {\mu _{p}}{\alpha _{p}}}\left(\lambda _{1}^{\alpha _{p}}+\lambda _{2}^{\alpha _{p}}+\lambda _{1}^{-\alpha _{p}}\lambda _{2}^{-\alpha _{p}}-3\right)$ In general the shear modulus results from

$2\mu =\sum _{p=1}^{N}\mu _{p}\alpha _{p}.$ With $N=3$ 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($N=1$ , $\alpha =2$ ) or the Mooney-Rivlin material ($N=2$ , $\alpha _{1}=2$ , $\alpha _{2}=-2$ ).

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

$\sigma _{\alpha }=p+\lambda _{\alpha }{\frac {\partial W}{\partial \lambda _{\alpha }}}$ where use is made of $\,\!\sigma _{\alpha }=\lambda _{\alpha }P_{\alpha }$ .

Uniaxial tension

We now consider an incompressible material under uniaxial tension, with the stretch ratio given as $\lambda ={\frac {l}{l_{0}}}$ . The principal stresses are given by

$\sigma _{\alpha }=p+\sum _{p=1}^{N}\mu _{p}\lambda _{p}^{\alpha _{p}}$ The pressure $p$ is determined from incompressibility and boundary condition $\sigma _{2}=\sigma _{3}=0$ , yielding:

$\sigma _{\alpha }=\sum _{p=1}^{N}\left(\mu _{p}\lambda _{p}^{\alpha _{p}}-\mu _{p}\lambda _{p}^{{\frac {1}{2}}\alpha _{p}}\right)$ Relation to other models

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.