Ogden (hyperelastic model)
Encyclopedia
The Ogden material model is a hyperelastic material
model used to describe the non-linear stress
-strain behaviour of complex materials such as rubber
s, 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.
is expressed in terms of the principal stretches
, 
as:
where
, 
and 
are material constants. Under the assumption of incompressibility one can rewrite as
In general the shear modulus results from
With
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
(
, 
) or the Mooney-Rivlin material (
, 
, 
).
Using the Ogden material model, the three principal values of the Cauchy stresses can now be computed as
where use is made of
.
. The principal stresses are given by
The pressure
is determined from incompressibility and boundary condition 
, yielding:
The simplest of these hyperelastic models, is the Neo-Hookean solid
.
where
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 
is a linear combination of two invariants
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.
Hyperelastic material
A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress-strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.For many materials, linear elastic...
model used to describe the non-linear stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
-strain behaviour of complex materials such as rubber
Rubber
Natural rubber, also called India rubber or caoutchouc, is an elastomer that was originally derived from latex, a milky colloid produced by some plants. The plants would be ‘tapped’, that is, an incision made into the bark of the tree and the sticky, milk colored latex sap collected and refined...
s, polymers, and biological tissue
Biological tissue
Tissue is a cellular organizational level intermediate between cells and a complete organism. A tissue is an ensemble of cells, not necessarily identical, but from the same origin, that together carry out a specific function. These are called tissues because of their identical functioning...
. The model was developed by Ray W. Ogden in 1972. The Ogden model, like other hyperelastic material
Hyperelastic material
A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress-strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.For many materials, linear elastic...
models, assumes that the material behaviour can be described by means of a strain energy density function
Strain energy density function
A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient....
, 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 densityStrain energy density function
A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient....
is expressed in terms of the principal stretches
, as:where
, and are material constants. Under the assumption of incompressibility one can rewrite asIn general the shear modulus results from
With
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 solidNeo-Hookean solid
A Neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the...
(
, ) or the Mooney-Rivlin material (, , ).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 byThe pressure
is determined from incompressibility and boundary condition , yielding: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
Neo-Hookean solid
A Neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the...
.
where
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 is a linear combination of two invariantsThe 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.