The Ogden material model is a hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers and biological tissue. The model relies on the fact that the material behaviour can be described by means of a strain energy density function, whence the stress-strain relationships can be derived. The Ogden model is often used to model rubberlike materials such as polymers, and biological materials. These materials can generally be considered to be isotropic, incompressible and strain rate independent.
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Hyperelastic Materials
For many materials the standard linear-elastic material models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic, incompressible and generally independent of strain rate. Many biological tissues can also be modelled as these so-called hyperelastic materials.
Strain-Energy Potential
In continuum mechanics it is postulated that the stress-strain relation of an elastic material follows from a strain-energy potential, which should represent the physical properties of the material under consideration. Consequently there exists a huge number of strain-energy functions and corresponding constitutive theories. The strain energy density function <math>W</math> of an isotropic hyperelastic material can be expressed in terms of the invariants of the Cauchy-Green strain tensor (<math>\mathbf{C}=\mathbf{F}^T\mathbf{F}</math>), <math> W(\mathbf{C}) = W\left(I_1^C, I_2^C, I_3^C \right) </math> with <math> I_1^C = tr(\mathbf{C}) </math> <math> I_2^C = \frac{1}{2}\left( tr(\mathbf{C}^2) -tr(\mathbf{C})^2 \right) </math> <math> I_3^C=det(\mathbf{C}) </math> The strain-energy function of an isotropic material can equivalently be expressed as a function of the eigenvalues of the right Cauchy-Green stretch tensor (<math>\mathbf{U}</math>), <math> W(\mathbf{C})=W(\lambda_1^2,\lambda_2^2,\lambda_3^2) </math> where use is made of the fact that the eigenvalues of tensor <math>\mathbf{C}</math>, <math>\lambda_{\alpha}^2</math>, are the squares of the eigenvalues of tensor <math>\mathbf{U}</math>, <math>\,\!\lambda_{\alpha}</math>, <math>\,\!\alpha = 1, 2, 3</math>. The eigenvalues <math>\,\!\lambda_{\alpha}</math> and the invariants are related in the following manner: <math> I_1^C = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 </math> <math> I_2^C = \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 </math> <math> I_3^C = \lambda_1^2\lambda_2^2\lambda_3^2 </math>
Stress-strain relationship
After establishing a suitable strain energy density function <math>W</math>, the 1st Piola-Kirchoff stress tensor can be calculated as <math> P_{iJ}(\mathbf{F}) = \frac{\partial W}{\partial F_{iJ}}(\mathbf{F}). </math> To ensure incompressibility of an elastic material, the strain-energy function can be written in the following form: <math> W = W(\mathbf{F}) - p(det(F)-1) </math> where the hydrostatic pressure <math>p</math> functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola-Kirchoff stress now becomes <math> \mathbf{P}=p\mathbf{F}^{-T}+\frac{\partial W}{\partial \mathbf{F}} </math>. This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy Stress tensor.
Material Models
Linear Elastic Model
The simplest model describing the constitutive equations for linear-elastic behaviour is Hooke's law, relating the linearized strain <math>\boldsymbol{\epsilon}</math> and corresponding stresses <math>\boldsymbol{\sigma}</math>, <math> \boldsymbol{\sigma}=\mathbb{C}\boldsymbol{\epsilon} </math> where the tensor <math>\mathbb{C}</math> is a function of Young's modulus <math>E</math> and Poisson's ratio <math>\nu</math>. For this material model the strain-energy density function is written as a function of the linearized strain: <math> W(\boldsymbol{\epsilon}) = \frac{\lambda}{2}(tr(\boldsymbol{\epsilon}))^2 + \mu tr(\boldsymbol{\epsilon}^2) </math> where <math>\lambda</math> and <math>\mu</math> are the Lamé constants. The stress-strain relationship can then be written as <math> \boldsymbol{\sigma} = \lambda tr(\boldsymbol{\epsilon})\mathbf{I} + 2\mu\boldsymbol{\epsilon} </math>
Hyperelastic Models
For rubbery 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.
Ogden Material Model
In the Ogden material model, the strain energy density is now expressed in terms of the principal stretches <math>\,\!\lambda_{\alpha}</math>, <math>\,\!\alpha=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 material (<math>N=2</math>, <math>\alpha_1=2</math>, <math>\alpha_2=-2</math>). Using the Ogden material model, the three principal values of the Cauchy stresses can now be computed as <math> \sigma_{\alpha} = p + \lambda_{\alpha}\frac{\partial W}{\partial \lambda_{\alpha}} </math> where use is made of <math>\,\!\sigma_{\alpha} = \lambda_{\alpha}P_{\alpha}</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_{\alpha} = p + \sum_{p=1}^N \mu_{p} \lambda_{p}^{\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_{\alpha} = \sum_{p=1}^N\left( \mu_{p} \lambda_{p}^{\alpha_p} - \mu_{p}\lambda_{p}^{\frac{1}{2}\alpha_p} \right) </math>
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.
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].
http://mechanik.tu-berlin.de/weinberg/Lehre/fem2/Chapter4.pdf
