Advanced elasticity/Ogden material

The Ogden material model is a hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Ray W. Ogden in 1972. The Ogden model, like other 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 isotropic, incompressible and 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 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.