/MAT/LAW82

Block Format Keyword This keyword defines the Ogden material. This law is compatible with solid and shell elements. In general it is used to model polymers and elastomers.

Format


(1)	(3)	(5)
/MAT/LAW82/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`N`	`v`
$μ_{1}$	$μ_{2}$	$μ_{3}$
. . . `N` values of $μ$ (five per line)
$α_{1}$	$α_{2}$	$α_{3}$
. . . `N` values of $α$ (five per line)
`D`₁	`D`₂	`D`₃
. . . `N` values of `D` (five per line)

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)	$[\frac{kg}{m^{3}}]$
`N`	Order of the Ogden model. (Integer, maximum 10 digits)
`v`	Poisson's ratio Default value depends on `D`₁ input. 2 (Real)
$μ_{i}$	`i`^th Parameter (`i`= 1,`N`). (Real)
$α_{i}$	`i`^th Parameter (`i` = 1,`N`). (Real)
`D`_i	`i`^th Parameter (`i` = 1,`N`). (Real)

Example (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/LAW82/1/1
LAW82 RUBBER
#              RHO_I
                1E-9                   0
#        N                            Nu
         2                          .495
#               Mu_i
                   2                   1
#            Alpha_i
                   2                  -2
#                D_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----|

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/LAW82/1/1
Rubber
#              RHO_I
                2E-9                   
#        N                            Nu
         3                             0
#               Mu_i
            1.061898            .0578289            .0159176
#            Alpha_i
             .428246             5.71269            -4.59726
#                D_i
                1E-4                   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

The strain energy density $W$ is computed using the following equation:
$W = \sum_{i = 1}^{N} \frac{2 μ_{i}}{{α_{i}}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3) + \sum_{i = 1}^{N} \frac{1}{D_{i}} {(J - 1)}^{2 i}$
with $\bar{λ} = J^{- \frac{1}{3}} λ$ and J = $λ$ ₁ $λ$ ₂ $λ$ ₃
Where, $λ_{i}$ is the i^th principal stretch.
The initial shear modulus:
$μ = \sum_{i = 1}^{N} μ_{i}$
The Bulk Modulus is calculated as $K = \frac{2}{D_{1}}$ , based on the following rules:
- If $ν = 0$ , then D₁ should be entered.
- If $ν \neq 0$ , D₁ input is ignored and will be recalculated and output in the Starter output using the formula:
  $D_{1} = \frac{3 (1 - 2 v)}{μ (1 + v)}$
- If $ν = 0$ and D₁ = 0, then a default value of $υ = 0.495$ is used and D₁ is calculated using Equation 3
To get a material without Poisson ratio effect, v should be defined with a small value (1e-10).
Further explanation about this law can be found in "Non-Linear Elastic Deformations", by R.W Ogden, Ellis Horwood, 1984.