The Ogden material model is a hyperelastic material model used to describe the non-linear stress–strain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Raymond 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 stress–strain relationships can be derived.
In the Ogden material model, the strain energy density is expressed in terms of the principal stretches
λj
j=1,2,3
W\left(λ1,λ2,λ3\right)=
N | |
\sum | |
p=1 |
\mup | |
\alphap |
\left(
\alphap | |
λ | |
1 |
+
\alphap | |
λ | |
2 |
+
\alphap | |
λ | |
3 |
-3\right)
where
N
\mup
\alphap
W\left(λ1,λ2\right)=
N | |
\sum | |
p=1 |
\mup | |
\alphap |
\left(
\alphap | |
λ | |
1 |
+
\alphap | |
λ | |
2 |
+
-\alphap | |
λ | |
1 |
-\alphap | |
λ | |
2 |
-3\right)
In general the shear modulus results from
2\mu=
N | |
\sum | |
p=1 |
\mup\alphap.
With
N=3
N=1
\alpha=2
N=2
\alpha1=2
\alpha2=-2
λ1λ2λ3=1
Using the Ogden material model, the three principal values of the Cauchy stresses can now be computed as
\sigmaj=-p+λj
\partialW | |
\partialλj |
=-p+
N | |
\sum | |
p=1 |
\mup
\alphap | |
λ | |
j |
We now consider an incompressible material under uniaxial tension, with the stretch ratio given as
λ= | l |
l0 |
l
{l0}
p
\sigma2=\sigma3=0
\sigma1=
N\mu | |
\sum | |
p |
\alphap | |
\left(λ |
-
| |||||
λ |
\right)
Considering an incompressible material under eqi-biaxial tension, with
λ1=λ2=
l | |
l0 |
p
\sigma3=0
\sigma1=\sigma2=
N\mu | |
\sum | |
p |
\alphap | |
\left(λ |
-
-2\alphap | |
λ |
\right)
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.
W(C)= | \mu |
2 |
C-3) | |
(I | |
1 |
where
\mu
W
W(C)= | \mu1 |
2 |
C | |
\left(I | |
1 |
-3\right)-
\mu2 | |
2 |
C | |
\left(I | |
2 |
-3\right)
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.