Ogden's law is applied to slightly compressible materials as rubber or elastomer
foams undergoing large deformation with an elastic behavior. The detailed theory for
Odgen material models can be found in 1. The strain energy
is expressed in a general form as a function of
:
Where,
, ith principal stretch
, with
being the ith principal engineering
strain
is relative volume with:
is the deviatoric stretch
and
are the material constants.
is order of Ogden model and defines the number of
coefficients pairs
.
This law is very general due to the choice of coefficient pair
.
- If
=1, then one pair
of material constants is needed andin this
case if
then it becomes a Neo-hookean material
model.
- If
=2 then two pairs
of material constants are needed and in this
case if
and
then it becomes a Mooney-Rivlin material
model
For uniform dilitation:
The strain energy function can be decomposed into deviatoric part
and spherical part
:
With:
The stress
corresponding to this strain energy is given
by:
which can be written as:
Since
and
for i=j and
for i≠j
Equation 7 is
simplified to:
For which the deviator of the Cauchy stress tensor
, and the pressure
would be:
Only the deviatoric stress above is retained, and the pressure is computed
independently:
Where,
a user-defined function related to the bulk modulus
in LAW42 and LAW69:
For an imcompressible material
,
and no pressure in material.
With
being the initial shear modulus, and
the Poisson's ratio.
Note: For an incompressible
material you have
. However,
is a good compromise to avoid too small time
steps in explicit codes.
A particular case of the Ogden material model is the Mooney-Rivlin material law which
has two basic assumptions:
- The rubber is incompressible and isotropic in unstrained state
- The strain energy expression depends on the invariants of Cauchy tensor
The three invariants of the Cauchy-Green tensor are:
For incompressible material:
The Mooney-Rivlin law gives the closed expression of strain energy
as:
with:
-
-
-
The model can be generalized for a compressible material.
LAW69, Ogden Material Law (Using Test Data as Input)
This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and
incompressible material specified using the Ogden or Mooney-Rivlin material models.
Unlike LAW42 where the material parameters are input this law computes the material
parameters from an input engineering stress-strain curve from a uniaxial tension and
compression tests. This material can be used with shell and solid elements.
The strain energy density formulation used depends on the
law_ID.
- law_ID =1, Ogden law (Same as LAW42):
-
- law_ID =2, Mooney-Rivlin law
-
Curve FittingAfter reading the
stress-strain curve (fct_ID1), Radioss calculates the corresponding material parameter
pairs using a nonlinear least-square fitting algorithm. For classic Ogden law,
(law_ID =1), the calculated material parameter pairs are
and
where the value of
is defined via the N_pair
input. The maximum value of N_pair = 5 with a default value
of 2.
For the Mooney-Rivlin law (law_ID =2), the material parameter
and
are calculated remembering that
and
for the LAW42 Ogden law can be calculated using this
conversion.
,
,
and
.
The minimum test data input should be a uniaxial tension engineering stress strain
curve. If uniaxial compression data is available, the engineering strain should
increate monotonically from a negative value in compression to a positive value in
tension. In compression, the engineering strain should not be less than -1.0 since
-100% strain is physically not possible.
This material law is stable when (with
=1,…5) is satisfied for parameter pairs for all
loading conditions. By default, Radioss tries to fit the
curve by accounting for these conditions
(Icheck=2). If a proper fit
cannot be found, then Radioss uses a weaker condition
(Icheck=1:) which ensures that
the initial shear hyperelastic modulus (
) is positive.
Once the material parameters are calculated by the Radioss Starter in LAW69, all the calculations done by LAW69 in the simulation are the
same as LAW42.