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The '''Ogden material model''' is a [[hyperelastic material]] model used to describe the non-linear [[stress (physics)|stress]]-[[strain (physics)|strain]] behaviour of complex materials such as [[rubber]]s, [[polymers]], and [[biological tissue]].  The model was developed by [[Ray W. Ogden]] in 1972.<ref>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.</ref> 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.
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== Ogden Material Model ==
 
In the Ogden material model, the [[strain energy density function|strain energy density]] is expressed in terms of the principal stretches <math>\,\!\lambda_j</math>, <math>\,\!j=1,2,3</math> as:
:<math>
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)
</math>
 
where <math>N</math>, <math>\,\!\mu_p</math> and <math>\,\!\alpha_p</math> are material constants. Under the assumption of incompressibility one can rewrite as
 
<math>
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)</math>
 
In general the shear modulus results from
 
<math>
2\mu = \sum_{p=1}^{N} \mu_p \alpha_{p}.
</math>
 
With <math>N=3</math> 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]] (<math>N=1</math>, <math>\alpha = 2</math>) or the [[Mooney-Rivlin solid|Mooney-Rivlin material]] (<math>N=2</math>, <math>\alpha_1=2</math>, <math>\alpha_2=-2</math>, with the constraint condition <math>\lambda_1\lambda_2\lambda_3=1</math>).
 
Using the Ogden material model, the three principal values of the Cauchy stresses can now be computed as
 
<math>
\sigma_{j} = p + \lambda_{j}\frac{\partial W}{\partial \lambda_{j}}
</math>
 
where use is made of <math>\,\!\sigma_{j} = \lambda_{j}P_{j}</math>.
 
=== Uniaxial tension ===
 
We now consider an incompressible material under uniaxial tension, with the stretch ratio given as <math>\lambda=\frac{l}{l_0}</math>. The principal stresses are given by
 
<math>
\sigma_{j} = p + \sum_{p=1}^N \mu_{p} \lambda_{j}^{\alpha_p}
</math>
 
The pressure <math>p</math> is determined from incompressibility and boundary condition <math>\sigma_2=\sigma_3=0</math>, yielding:
 
<math>
\sigma_{j} = \sum_{p=1}^N\left( \mu_{p} \lambda_{j}^{\alpha_p} - \mu_{p}\lambda_{j}^{-\frac{1}{2}\alpha_p}  \right)
</math>
 
== Other Hyperelastic Models ==
 
For rubber and biological materials, more sophisticated models are necessary. Such materials may exhibit a non-linear stress-strain behaviour at modest strains, or are elastic up to huge strains. These complex non-linear stress-strain behaviours need to be accommodated by specifically tailored strain-energy density functions.
 
The simplest of these hyperelastic models, is the [[Neo-Hookean solid]].
 
<math>
W(\mathbf{C})=\frac{\mu}{2}(I_1^C-3)
</math>
 
where <math>\mu</math> is the shear modulus, which can be determined by experiments. From experiments it is known that for rubbery materials under moderate straining up to 30-70%, the Neo-Hookean model usually fits the material behaviour with sufficient accuracy. To model rubber at high strains, the one-parametric Neo-Hookean model is replaced by more general models, such as the [[Mooney-Rivlin solid]] where the strain energy <math>W</math> is a linear combination of two invariants
 
<math>
W(\mathbf{C})=\frac{\mu_1}{2}\left(I_1^C -3 \right) -\frac{\mu_2}{2}\left(I_2^C - 3\right)
</math>
 
The Mooney-Rivlin material was originally also developed for rubber, but is today often applied to model (incompressible) biological tissue. For modeling rubbery and biological materials at even higher strains, the more sophisticated '''Ogden material model''' has been developed.
 
== References ==
<references/>
* 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]. <br/>http://mechanik.tu-berlin.de/weinberg/Lehre/fem2/Chapter4.pdf
 
[[Category:Continuum mechanics]]
[[Category:Solid mechanics]]

Latest revision as of 00:46, 13 January 2015

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