/MAT/LAW42 (OGDEN)
Block Format Keyword This keyword defines a hyperelastic, viscous, and incompressible material specified using the Ogden, MooneyRivlin material models.
This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. This material can be used with shell and solid elements.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW42/mat_ID/unit_ID or /MAT/OGDEN/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
$\nu $  ${\sigma}_{\mathit{cut}}$  fct_ID_{blk}  Fscale_{blk}  M  I_{form}  
${\mu}_{1}$  ${\mu}_{2}$  ${\mu}_{3}$  ${\mu}_{4}$  ${\mu}_{5}$  
Blank Format  
${\alpha}_{1}$  ${\alpha}_{2}$  ${\alpha}_{3}$  ${\alpha}_{4}$  ${\alpha}_{5}$  
Blank Format 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

G_{1}  G_{2}  G_{3}  G_{4}  G_{5}  
. . . M values of G (five per line)  
${\tau}_{1}$  ${\tau}_{2}$  ${\tau}_{3}$  ${\tau}_{4}$  ${\tau}_{5}$  
. . . M values of $\tau $ (five per line) 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
$\nu $  Poisson's ratio. 3 Default = 0.495 (Real) 

${\sigma}_{\mathit{cut}}$  Cutoff stress in tension. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
fct_ID_{blk}  Function identifier which scales the bulk coefficient as a function of the relative
volume. 6 (Integer) 

Fscale_{blk}  Bulk
function scale factor. Default = 1.0 (Real) 

M  Number of viscous terms in the Prony series (order of the Maxwell
model). (Integer) 

I_{form}  Incompressibility for shell elements
formulation flag. This flag is not effective for solid elements.


${\mu}_{1}$  First parameter of the shear hyperelastic modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{2}$  Second parameter of the shear hyperelastic modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{3}$  Third parameter of the shear hyperelastic modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{4}$  Fourth parameter of the shear hyperelastic modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{5}$  Fifth parameter of the shear hyperelastic modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\alpha}_{1}$  First material exponent. (Real) 

${\alpha}_{2}$  Second material exponent. (Real) 

${\alpha}_{3}$  Third material exponent. (Real) 

${\alpha}_{4}$  Fourth material exponent. (Real) 

${\alpha}_{5}$  Fifth material exponent. (Real) 

G_{i}  i_{th} multiplier of the Prony
viscous term. 7 (Real) 
$\left[\text{Pa}\right]$ 
${\tau}_{i}$  i_{th} time relaxation of the Prony
viscous term. (Real) 
$\left[\text{s}\right]$ 
Example (Rubber)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
kg mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/OGDEN/1/1
rubber
# RHO_I
1E6
# Nu sigma_cut funIDbulk Fscale_bulk M Iform
.495 0 0 0 0 0
# Mu_1 Mu_2 Mu_3 Mu_4 Mu_5
2e3 1e3 0 0 0
# blank card
# alpha_1 alpha_2 alpha_3 alpha_4 alpha_5
2 2 0 0 0
# blank card
#12345678910
#enddata
/END
#12345678910
Comments
 This
material model defines a hyperelastic, viscous, and incompressible material specified using
the Ogden, NeoHookean, or MooneyRivlin material models. This law is generally used to
model incompressible rubbers, polymers, foams, and elastomers. This material can be used
with shell and solid elements.LAW42 uses the following strain energy density representation of the Ogden material model.
(1) $$W\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3}\right)={\displaystyle \sum _{p=1}^{5}\frac{{\mu}_{p}}{{\alpha}_{p}}\left({\overline{\lambda}}_{1}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{2}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{3}{}^{{\alpha}_{p}}3\right)}+\frac{K}{2}{\left(J1\right)}^{2}$$Where, $W$
 Strain energy density
 ${\lambda}_{i}$
 i^{th} principal engineering stretch
 $J$
 Relative volume defined as: $J={\lambda}_{1}\cdot {\lambda}_{2}\cdot {\lambda}_{3}=\frac{{\rho}_{0}}{\rho}$
 ${\overline{\lambda}}_{i}={J}^{\frac{1}{3}}{\lambda}_{i}$
 Deviatoric stretch
 ${\alpha}_{p}$ and ${\mu}_{p}$
 Material constants coefficient pairs.
The initial shear modulus $\mu $ and bulk modulus ( $K$ ) are given by:(2) $$\mu =\frac{{\displaystyle \sum _{p=1}^{5}{\mu}_{p}\cdot {\alpha}_{p}}}{2}$$and(3) $$K=\mu \cdot \frac{2\left(1+\nu \right)}{3\left(12\nu \right)}$$Where, $\nu $ is the Poisson's ratio and is only used for computing the bulk modulus.
 Parameters
${\alpha}_{p}$
and
${\mu}_{p}$
must be chosen so that initial shear modulus is
$\mu >0$
.
For material stability, it is required that each material constant pair ${\mu}_{p}\cdot {\alpha}_{p}>0$ .
 Poisson's ratio
$\nu $
is used only for computing the bulk modulus, Equation 3.
For pure incompressible materials, $\nu =0.5$ . This value leads to a finite bulk modulus ( $K$ ). Therefore, the recommended maximum Poisson's ratio for incompressible materials is $\nu =0.495$ .
Higher values of the Poisson's ratio may lead to a small time step value or divergence in case of implicit and explicit simulations.
 A particular
case of the Ogden material model is the incompressible MooneyRivlin model, which can be
represented using the following equation for the strain energy density
function:
(4) $$W={C}_{10}\left({I}_{1}3\right)+{C}_{01}\left({I}_{2}3\right)$$Where, ${I}_{1}$ and ${I}_{2}$ are the first and second invariants of the right CauchyGreen Tensor.
This representation can be derived from the Ogden strain energy density function when:(5) $${\mu}_{1}=2\cdot {C}_{10}$$(6) $${\mu}_{2}=2\cdot {C}_{01}$$(7) $${\alpha}_{1}=2$$(8) $${\alpha}_{2}=2$$  A simple case of the Ogden material model
is the NeoHookean model represented using the following equation for the strain energy
density function:
(9) $$W={C}_{10}\left({I}_{1}3\right)$$Where, ${I}_{1}$
 First invariants of the right CauchyGreen Tensor
 ${C}_{10}$
 Material constant
This representation can be derived from the LAW42 Ogden strain energy density function when:(10) $${\mu}_{1}=2\cdot {C}_{10}$$(11) $${\alpha}_{1}=2$$and(12) $${\mu}_{2}={\alpha}_{2}=0$$  In cases when the bulk modulus of a material is not large enough to prevent compression, LAW42 allows the input of a function (fct_ID_{blk}) which scale the bulk modulus as a function of the relative volume so that the bulk modulus can be increased to maintain incompressibility.
 Viscous (rate) effects are modeled in
LAW42 using a Maxwell model, which can be described in a simplified manner as a system of n
springs with stiffness'
${G}_{i}$
and dampers
${\eta}_{i}$
:
The Maxwell model is represented using Prony series inputs ( ${G}_{i},{\tau}_{i}$ ). The hyperelastic initial shear modulus $\mu $ is the same as the longterm shear modulus ${G}_{\infty}$ in the Maxwell model, and ${\tau}_{i}$ is the relaxation time:
(13) $${\tau}_{i}=\frac{{\eta}_{i}}{{G}_{i}}$$The ${G}_{i}$ and ${\tau}_{i}$ values must be positive.
 /VISC/PRONY can be used with this material law to include viscous effects.