/MAT/LAW111
Block Format Keyword Describes the Marlow material model, which can be used to model hyper elastic behavior. This law is only compatible with solid elements.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW111/mat_ID/unit_ID or /MAT/MARLOW/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

Itype  fct_ID  Fscale  $\nu $ 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  (Optional) Unit identifier. (Integer, maximum 10 digits) 

mat_title  Material title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Itype  Test data type (stress strain curve).
(Integer) 

fct_ID  Function identifier defining engineer stress versus engineer
strain. (Integer) 

$\nu $  Poisson ratio. Default = 0.495 (Real) 

Fscale  Scale factor for ordinate (stress) in function
fct_ID. Default= 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Example (Aluminum)
#12345678910
/UNIT/1
unit_Mg_mm_s
Mg mm s
#12345678910
/MAT/MARLOW/1/1
Aluminium
# Init. dens.
1.0E9
# Itype fct_ID Fscale Nu
1 11 0 0.495
#12345678910
#12345678910
/FUNCT/11
eng. stress vs eng. strain (data from Treloar 1975)
# X Y
0.0 0.0
0.118558340245158 0.147781942125394
0.229807469073257 0.235370392667332
0.352872064166251 0.317098176147032
0.575267906053679 0.4127492327798
0.826025385319594 0.497843181600741
1.15247263605042 0.600395253711613
1.41741319317695 0.681964895195418
1.99461340483096 0.866102422989683
2.57183319597017 1.06544342104666
3.01188513821085 1.24856432172444
3.75483021047915 1.60560301197591
4.32044087300526 1.97102586370987
4.74886561326187 2.30619464551444
5.13008421468603 2.70807751093106
5.41191132474589 3.04925719469397
5.61340230088995 3.43730640075521
5.84795248474777 3.79023377564191
6.0210291095147 4.1479076695769
6.14916629739038 4.49627566011047
6.26551964740034 4.87506376966911
6.36059461533377 5.25621459595217
6.44855620568032 5.62216985907231
6.59373206402824 6.34826716084913
#12345678910
#enddata
Comments
 The Marlow energy density
is considered as: $$W=U\left(\overline{I}\right)+V\left(J\right)$$
with $\overline{I}={\overline{\lambda}}_{1}^{2}+{\overline{\lambda}}_{2}^{2}+{\overline{\lambda}}_{3}^{2}$ and $J={\lambda}_{1}{\lambda}_{2}{\lambda}_{3}$
Where, ${\overline{\lambda}}_{k}={J}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$3$}\right.}}{\lambda}_{k}$ the stretch in the directions $k$ =1,2,3.
$V$ is computed using bulk modulus $K$ (computed from test data function and Poisson's ratio).
$$V\left(J\right)=\frac{1}{2}K{\left(J1\right)}^{2}$$$U$ is determined using the data of uniaxial, biaxial or shear test (function fct_ID) as:
$$U\left(\overline{I}\right)={\displaystyle {\int}_{0}^{{\lambda}_{t}1}T\left({\lambda}_{t}1\right)}\partial \epsilon $$Where, $T$
 Stress for the engineering strain ${\lambda}_{t}1$ .
 ${\lambda}_{t}$
 Equivalent stretch corresponding to Uniaxial tension, equibiaxial or planar test.
 The Cauchy stress is
computed as:$$\sigma =\frac{2}{J}\frac{\partial U}{\partial \overline{I}}dev({b}^{*})+\frac{\partial V}{\partial J}I$$
Where, ${b}^{*}={J}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{1ex}{$3$}\right.}F{F}^{T}$ .
 /VISC/PRONY can be used with LAW111 to include viscous effects.