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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 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 stressstrain 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.
Hyperelastic MaterialsFor many materials the standard linearelastic material models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stressstrain relationship can be defined as nonlinearly elastic, isotropic, incompressible and generally independent of strain rate. Many biological tissues can also be modelled as these socalled hyperelastic materials.
StrainEnergy PotentialIn continuum mechanics it is postulated that the stressstrain relation of an elastic material follows from a strainenergy potential, which should represent the physical properties of the material under consideration. Consequently there exists a huge number of strainenergy functions and corresponding constitutive theories. The strain energy density function W of an isotropic hyperelastic material can be expressed in terms of the invariants of the CauchyGreen strain tensor (),
with
The strainenergy function of an isotropic material can equivalently be expressed as a function of the eigenvalues of the right CauchyGreen stretch tensor (),
where use is made of the fact that the eigenvalues of tensor , , are the squares of the eigenvalues of tensor , , . The eigenvalues and the invariants are related in the following manner:
Stressstrain relationshipAfter establishing a suitable strain energy density function W, the 1^{st} PiolaKirchoff stress tensor can be calculated as
To ensure incompressibility of an elastic material, the strainenergy function can be written in the following form:
where the hydrostatic pressure p functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1^{st} PiolaKirchoff stress now becomes . This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy Stress tensor.
Material ModelsLinear Elastic ModelThe simplest model describing the constitutive equations for linearelastic behaviour is Hooke's law, relating the linearized strain and corresponding stresses ,
where the tensor is a function of Young's modulus E and Poisson's ratio ν. For this material model the strainenergy density function is written as a function of the linearized strain:
where λ and μ are the Lamé constants. The stressstrain relationship can then be written as
Hyperelastic ModelsFor rubbery 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.
Ogden Material ModelIn the Ogden material model, the strain energy density is now 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 tensionWe 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:
Relation to other modelsThere exist many models to describe hyperelastic behaviour, each starting from a given strainenergy 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


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Ogden_(hyperelastic_model)". A list of authors is available in Wikipedia. 