/MAT/LAW62 (VISC_HYP)

Block Format Keyword This law describes the hyper visco-elastic material. This law is compatible with solid and shell elements. In general it is used to model polymers and elastomers.

Format

(1)	(3)	(4)	(5)	(7)	(8)
/MAT/LAW62/`mat_ID`/`unit_ID` or /MAT/VISC_HYP/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
$ν$	`N`	`M`	$μ_{max}$	`Flag_Visc`	`Form`

Define N parameters (5 per Line)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$μ_{1}$		$μ_{2}$		$μ_{3}$		$μ_{4}$		$μ_{5}$
$α_{1}$		$α_{2}$		$α_{3}$		$α_{4}$		$α_{5}$

Define M parameters (5 per Line)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$γ_{1}$		$γ_{2}$		$γ_{3}$		$γ_{4}$		$γ_{5}$
$τ_{1}$		$τ_{2}$		$τ_{3}$		$τ_{4}$		$τ_{5}$

Define N parameters (5 per Line)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$ν_{1}$		$ν_{2}$		$ν_{3}$		$ν_{4}$		$ν_{5}$

Definition

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
$ν$	Equivalent Poisson's ratio. Default = 0.0 (Real)
`N`	Law order - must be positive. (Integer)
`M`	Maxwell model order. = 0 Law is hyper elastic. (Integer)
$μ_{max}$	Maximum viscosity. Default = 10³⁰ (Real)	$[Pa \cdot s]$
`Flag_Visc`	Viscous formulation flag, used if M > 0. = 0 (Default) Viscous stress is accounted for in the deviatoric stress only and thus should only be used for incompressible materials with Poisson’s ratio close to 0.5. = 1 Viscous stress is accounted for in both the deviatoric and volumetric stress which enables the lateral expansion effect for the entered Poisson’s ratio.
`Form`	Initial visco-elastic modulus formulation used, if `M` > 0. = 0 (Default) Set to 1. = 1 Initial elastic modulus is the instantaneous rigidity. = 2 Initial elastic modulus is the long-term rigidity. (Integer)
$μ_{i}$	`i`^th parameter of the ground shear modulus. (Real)	$[Pa]$
$α_{i}$	`i`^th material parameter. (Real)
$γ_{i}$	`i`^th stiffness ratio. (Real)
$τ_{i}$	`i`^th time relaxation. (Real)	$[s]$
$ν_{i}$	`i`^th Poisson’s ratio. Used only if one of the `i`^th Poisson’s ratio is greater than 0. (Real)

Example (Hyper-elastic Rubber)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW62/1/1
LAW62 RUBBER 1
#              RHO_I
                1E-9                   
#                 Nu         N         M              mu_max Flag_Visc      Form
                .495         2         0                   0         1         0
#         mu_i
                   2                   1
#      alpha_i
                   2                  -2
#         nu_i
                .495                  .4
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW62/2/1
LAW62 RUBBER 2
#              RHO_I
                1E-9                   
#                 Nu         N         M              mu_max Flag_Visc      Form
                .495         2         2                   0         1         0
#         mu_i
                   2                   1
#      alpha_i
                   2                  -2
#         gamma_i
                  .2                  .3
#         tetha_i
                .007                 .05
#         nu_i
                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

Strain energy W is computed using the following equation:(1)
$W (λ_{1}, λ_{2}, λ_{3}) = \sum_{i = 1}^{N} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3 + \frac{1}{β_{i}} (J^{- α_{i} β_{i}} - 1))$
With
- $λ_{i}$ are eigenvalue of F (F is deformation gradient matrix),
- J is Jacobian determinant, with $J = \det F$ ,
- N is the order of law,
- $μ_{i}$ and $α_{i}$ are the material parameters:(2)
  $β_{i} = \frac{ν_{i}}{(1 - 2 ν_{i})}$
- $ν_{i}$ is the Poisson's ratio with $0 < ν_{i} < \frac{1}{2}$ .
  If $ν_{i}$ are 0 or not defined, equivalent Poison’s ratio $ν$ is used, instead.
  
  If $ν_{i}$ are defined, the equivalent Poison’s ratio is recomputed according to the shear and bulk modulus.
Coefficients ( $G_{i}, η_{i}$ ) are used to describe rate effects through the Maxwell model:

Figure 1.

The initial shear modulus is:(3)
$G_{0} = \sum_{i = 1}^{N} μ_{i}$

The sum of $μ_{i}$ should be greater than 0.(4)
$G_{0} = G_{\infty} + \sum_{i} G_{i}$

The stiffness ratio is:(5)
$γ_{\infty} = \frac{G_{\infty}}{G_{0}} = 1 - \sum_{i} γ_{i}$
(6)
$γ_{i} = \frac{G_{i}}{G_{0}}$

With, (7)
$γ_{i} \in [0, 1], \sum_{i} γ_{i} < 1$

and (8)
$G_{0} = G_{\infty} + \sum_{i} G_{i}$
is the ground shear modulus

The relaxation time, $τ_{i}$ must be positive:(9)
$τ_{i} = \frac{η_{i}}{G_{i}}$
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 is not the same depending on input values. Furthermore, in one case it corresponds to the long-term shear modulus, whereas in another case it corresponds to the initial shear modulus.