/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)
/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`

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)
$γ_{1}$		$γ_{2}$		$γ_{3}$		$γ_{4}$		$γ_{5}$
$τ_{1}$		$τ_{2}$		$τ_{3}$		$τ_{4}$		$τ_{5}$

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
$ν$	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.
$μ_{i}$	`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)	$[s]$

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
#              RHO_I
                1E-9                   
#                 Nu         N         M              mu_max Flag_Visc
                .495         2         0                1000         1
#         mu_i
                   2                   1
#      alpha_i
                   2                  -2
#         gamma_i

#         tetha_i

#---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)
$W (λ_{1}, λ_{2}, λ_{3}) = \sum_{i = 1}^{N} \frac{2 μ_{i}}{α_{i}^{2}} ({λ_{1}}^{α_{i}} + {λ_{2}}^{α_{i}} + {λ_{3}}^{α_{i}} - 3 + \frac{1}{β} (J^{- α_{i} β} - 1))$
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)
  $β = \frac{ν}{(1 - 2 ν)}$
- $ν \neq 0$ and $ν \neq 1 / 2$
- $ν$ is the Poisson's ratio.
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 relative 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 depending on input values is not the same. Also, it corresponds to the long-term shear modulus in one case, whereas to the initial shear modulus in another case.