 Ogden (hyperelastic model)

The Ogden material model is a hyperelastic material model used to describe the nonlinear stressstrain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Ray W. Ogden in 1972.^{[1]} 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 stressstrain 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 is expressed in terms of the principal stretches , as:
where N, and are material constants. Under the assumption of incompressibility one can rewrite as
In general the shear modulus results from
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 NeoHookean solid (N = 1, α = 2) or the MooneyRivlin material (N = 2, α_{1} = 2, α_{2} = − 2).
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
We now consider an incompressible material under uniaxial tension, with the stretch ratio given as . The principal stresses are given by
The pressure p is determined from incompressibility and boundary condition σ_{2} = σ_{3} = 0, yielding:
Other Hyperelastic Models
For rubber and biological materials, more sophisticated models are necessary. Such materials may exhibit a nonlinear stressstrain behaviour at modest strains, or are elastic up to huge strains. These complex nonlinear stressstrain behaviours need to be accommodated by specifically tailored strainenergy density functions.
The simplest of these hyperelastic models, is the NeoHookean solid.
where μ is the shear modulus, which can be determined by experiments. From experiments it is known that for rubbery materials under moderate straining up to 3070%, the NeoHookean model usually fits the material behaviour with sufficient accuracy. To model rubber at high strains, the oneparametric NeoHookean model is replaced by more general models, such as the MooneyRivlin solid where the strain energy W is a linear combination of two invariants
The MooneyRivlin 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
 ^ 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. 565584.
 F. Cirak: Lecture Notes for 5R14: Nonlinear solid mechanics, University of Cambridge.
 R.W. Ogden: NonLinear Elastic Deformations, ISBN 0486696480
 K. Weinberg: Lecture Notes for Zur Methode der finiten Elemente in der Mechanik II: Nichtlineare Probleme, TU Berlin [in English].
http://mechanik.tuberlin.de/weinberg/Lehre/fem2/Chapter4.pdf
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