/MAT/LAW69
Block Format Keyword This law is an extension of /MAT/LAW42 (OGDEN) and defines a hyperelastic and incompressible material specified using the Ogden, MooneyRivlin material models.
It is generally used to model incompressible rubbers, polymers, foams, and elastomers. Material parameters are computed from an engineering stressstrain curve from uniaxial tension and compression tests. It is used with shell and solid elements.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW69/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
law_ID  fct_ID_{blk}  v  Fscale_{blk}  N_pair  Icheck  
fct_ID_{1} 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material
identifier. (Integer, maximum 10 digits) 

unit_ID  Unit Identifier. (Integer, maximum 10 digits) 

mat_title  Material
title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial
density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
law_ID  Hyperelastic material
model type. 2 (Integer)


fct_ID_{blk}  Function which scales the
bulk coefficient as a function of the relative volume. 6 (Integer) 

v  Poisson's ratio. Default = 0.495 (Real) 

Fscale_{blk}  Scale factor for fct_ID_{blk}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
N_pair  Number of material
parameter (
$\mu $
_{p} and α_{p}) pairs in the
representation of the strain energydensity function
(W). The material parameters are calculated
from the givenstressstrain curve
(fct_ID_{1}). (N_pair ≤ 5) Default = 2 (Integer) 

Icheck  Validity check of material
parameters (
$\mu $
_{p} and α_{p}).
(Integer) Parameters fitting uses the compression and
tension test data:
Parameter fitting uses only the tension test data:


fct_ID_{1}  Function identifier for
the engineering stressstrain curve from uniaxial compression and
tension test. (Integer) 
Example (MooneyRivlin Formulation)
#RADIOSS STARTER
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW69/1/1
LAW69 rubber
# RHO_I
1E9
# LAW_ID FCT_ID NU FSCALE N_PAIR ICHECK
2 0 .495 0 2 0
# FCT_ID1
2
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/2
LAW69 e.strain e.stress
# X Y
0 0
.03 .30
.06 .55
.10 .80
.20 1.4
.30 2.0
.50 2.7
.70 3.4
1.0 4.0
#12345678910
#ENDDATA
/END
#12345678910
Example (Ogden Formulation)
#RADIOSS STARTER
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW69/1/1
LAW69 rubber
# RHO_I
1E9
# LAW_ID FCT_ID NU FSCALE N_PAIR ICHECK
1 0 .495 0 2 0
# FCT_ID1
2
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/2
LAW69 e.strain e.stress
# X Y
0 0
.03 .30
.06 .55
.10 .80
.20 1.4
.30 2.0
.50 2.7
.70 3.4
1.0 4.0
#12345678910
#ENDDATA
/END
#12345678910
Comments

Radioss currently accepts test data from the
following deformation schemes:
Uniaxial tension and compression.
The input stressstrain data (fct_ID_{1}) is engineering stress as a function of engineering strain. The engineering strain should be monotonically increasing ranging from a negative value in compression to a positive value in tension. In compression, the engineering strain should be greater than 1.0. If data in fct_ID_{1} is not complete (only tension data), then only tension will be considered.
 The strain energy density formulation
used depends on the law_ID.
 law_ID = 1 (Ogden
law):
(1) $$W=\sum _{p}\frac{{\mu}_{p}}{{\alpha}_{p}}\left({{\lambda}_{1}}^{{\alpha}_{p}}+{{\lambda}_{2}}^{{\alpha}_{p}}+{{\lambda}_{3}}^{{\alpha}_{p}}3\right)$$  law_ID = 2 (MooneyRivlin
law):
(2) $$W={C}_{10}\left({I}_{1}3\right)+{C}_{01}\left({I}_{2}3\right)$$
 law_ID = 1 (Ogden
law):
 After reading the stressstrain curve
(fct_ID_{1}), Radioss calculates the corresponding parameter
pairs using nonlinear leastsquare fitting.
 For classic Ogden law, the parameter pairs are $\mu $ _{p} and α_{p} (p=1,...5, max of N_pair is 5)
 For MooneyRivlin law, the parameter pairs are $\mu $ _{p} and α_{p} (p=1,2, N_pair always equals 2)
 To improve the quality of the
nonlinear least square fit, it is recommended that:
 The experimental data curve represents a smooth monotonically increasing function with uniform distribution of abscissa points. The number of data points in the experimental data curve should be greater than the number of parameter pairs (N_pair).
 If N_pair ≥ 3, the test data should cover at least 100% of the tensile strain and/or 50% of the compressive strain.
 N_pair should not be set to a very large value to avoid instabilities in the fitting procedure.
 Radioss Starter outputs the "averaged error of fitting" between input (experimental) and the stressstrain curve which is calculated from the strain energy density function (W) using the corresponding material parameters determined during the fitting process. The maximum "averaged error of fitting" should not exceed 10%.
 This material law is stable when
${\mu}_{p}{\alpha}_{p}>0$
(with p=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 a weaker condition (Icheck= 1: $\sum _{p}{\mu}_{p}{\alpha}_{p}>0$ ) is used. The latter is a necessary condition to enforce that the initial shear hyperelastic modulus ( $\mu $ ) is positive.
 Material incompressibility is
provided by using a penalty approach, which calculates the pressure proportional
to a change in density:
(3) $$P=K\cdot {\mathit{Fscale}}_{\mathit{blk}}\cdot {\text{f}}_{\mathit{blk}}\left(\text{J}\right)\cdot \left(\text{J}1\right)$$Where, f_{blk} is the function of fct_ID_{blk}
The proportionality coefficient (K) is the bulk coefficient which is generally a very high value. This provides a significantly high value for the pressureresistance when the incompressibility condition (J=1) is violated. The Jacobian (J) can be interpreted as the ratio of the current element volume with respect to the initial element volume.
fct_ID_{blk} provides additional control for the incompressibility (see Figure 1). It allows the scaling up of the bulk coefficient value based on the value of J. By default, the function identifier is zero and the value of the bulk scaling function is equal to 1. It is advisable to output and control the density distribution of LAW69 components to make sure that the density variation is small, that is the value of J is close to 1.  Poisson's ratio v
is used only for computing the bulk modulus (K).
For pure incompressible materials, $\upsilon =0.5$ . This value of Poisson's ratio implies an infinite value for the bulk modulus (K). Therefore, the recommended Poisson's ratio for incompressible materials is $\upsilon =0.495$ (default). Higher values of the Poisson's ratio may lead to a small time step value or divergence in case of implicit and explicit simulations.
 /VISC/PRONY can be used with this material law to include viscous effects.
 Further explanation about this law can be found in "NonLinear Elastic Deformations", by R.W Ogden, Ellis Horwood, 1984.