/MAT/LAW62 (VISC_HYP)
Block Format Keyword This law describes the hyper viscoelastic material. This law is compatible with solid and shell elements. In general it is used to model polymers and elastomers.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW62/mat_ID/unit_ID or /MAT/VISC_HYP/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
$\nu $  N  M  ${\mu}_{max}$  Flag_Visc 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\mu}_{1}$  ${\mu}_{2}$  ${\mu}_{3}$  ${\mu}_{4}$  ${\mu}_{5}$  
${\alpha}_{1}$  ${\alpha}_{2}$  ${\alpha}_{3}$  ${\alpha}_{4}$  ${\alpha}_{5}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\gamma}_{1}$  ${\gamma}_{2}$  ${\gamma}_{3}$  ${\gamma}_{4}$  ${\gamma}_{5}$  
${\tau}_{1}$  ${\tau}_{2}$  ${\tau}_{3}$  ${\tau}_{4}$  ${\tau}_{5}$ 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
$\nu $  Poisson's ratio. Default = 0.0 (Real) 

N  Law order  must be
positive. (Integer) 

M  Maxwell model order.
(Integer) 

${\mu}_{max}$  Maximum
viscosity. Default = 10^{30} (Real) 
$\left[\text{Pa}\cdot \text{s}\right]$ 
Flag_Visc  Viscous formulation flag,
used if M > 0.


${\mu}_{i}$  i^{th} parameter of the ground shear
modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\alpha}_{i}$  i^{th} material
parameter. (Real) 

${\gamma}_{i}$  i^{th} stiffness
ratio. (Real) 

${\tau}_{i}$  i^{th} time
relaxation. (Real) 
$\left[\text{s}\right]$ 
Example (Hyperelastic Rubber)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW62/1/1
LAW62 RUBBER
# RHO_I
1E9
# Nu N M mu_max Flag_Visc
.495 2 0 1000 1
# mu_i
2 1
# alpha_i
2 2
# gamma_i
# tetha_i
#12345678910
#ENDDATA
/END
#12345678910
Comments
 Strain energy
W is computed using the following equation:
(1) $$W({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})={\displaystyle \sum _{i=1}^{N}\frac{2{\mu}_{i}}{{\alpha}_{i}{}^{2}}\left({{\lambda}_{1}}^{{\alpha}_{i}}+{{\lambda}_{2}}^{{\alpha}_{i}}+{{\lambda}_{3}}^{{\alpha}_{i}}3+\frac{1}{\beta}({J}^{{\alpha}_{i}\beta}1)\right)}$$With ${\lambda}_{i}$ are eigenvalue of F (F is deformation gradient matrix),
 J is Jacobian determinant, with $\text{J}=\mathrm{det}\mathbf{F}$ ,
 N is the order of law,

${\mu}_{i}$
and
${\alpha}_{i}$
are the material
parameters:
(2) $$\beta =\frac{\nu}{\left(12\nu \right)}$$  $\nu \ne 0$ and $\nu \ne 1/2$
 $\nu $ is the Poisson's ratio.
 Coefficients (
${G}_{i},{\eta}_{i}$
) are used to describe rate effects through the
Maxwell model:
The initial shear modulus is:
(3) $${G}_{0}={\displaystyle \sum _{i=1}^{N}{\mu}_{i}}$$The sum of ${\mu}_{i}$ should be greater than 0.(4) $${G}_{0}={G}_{\infty}+{\displaystyle \sum _{i}{G}_{i}}$$The stiffness ratio is:(5) $${\gamma}_{\infty}=\frac{{G}_{\infty}}{{G}_{0}}=1{\displaystyle \sum _{i}{\gamma}_{i}}$$(6) $${\gamma}_{i}=\frac{{G}_{i}}{{G}_{0}}$$With,(7) $${\gamma}_{i}\in \left[0,1\right],{\displaystyle \sum _{i}{\gamma}_{i}}<1$$and(8) is the ground shear modulus$${G}_{0}={G}_{\infty}+{\displaystyle \sum _{i}{G}_{i}}$$The relative time, ${\tau}_{i}$ must be positive:(9) $${\tau}_{i}=\frac{{\eta}_{i}}{{G}_{i}}$$  Rate effects are modeled using a convolution integral using Prony series. This is an extension of small strain theory to large strain. Strain rate effect applies only to the deviatoric stress. The full expression of the deviatoric viscous stress can be found in the Radioss Theory Manual.
 There are several differences between /MAT/LAW42 (OGDEN) and /MAT/LAW62. Special care should be taken that the ground shear modulus expression depending on input values is not the same. Also, it corresponds to the longterm shear modulus in one case, whereas to the initial shear modulus in another case.