/MAT/LAW111

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`
$ρ_{i}$

Function input

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`Itype`	`fct_ID`	`Fscale`		$ν$

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`Itype`	Test data type (stress strain curve). = 1 (Default) Uniaxial data test. = 2 Equibiaxial data test. = 3 Planar data test. (Integer)
`fct_ID`	Function identifier defining engineer stress versus engineer strain. (Integer)
$ν$	Poisson ratio. Default = 0.495 (Real)
`Fscale`	Scale factor for ordinate (stress) in function `fct_ID`. Default= 1.0 (Real)	$[Pa]$

Example (Aluminum)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit_Mg_mm_s
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/MARLOW/1/1
Aluminium
#        Init. dens.         
              1.0E-9
#    Itype    fct_ID  	          Fscale                  Nu
         1        11                   0               0.495
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|   
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/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  	 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|  
#enddata

Comments

The Marlow energy density is considered as: (1)
$W = U (\bar{I}) + V (J)$

with $\bar{I} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2}$ and $J = λ_{1} λ_{2} λ_{3}$

Where, ${\bar{λ}}_{k} = J^{- \frac{1}{3}} λ_{k}$ 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).(2)
$V (J) = \frac{1}{2} K {(J - 1)}^{2}$

$U$ is determined using the data of uniaxial, biaxial or shear test (function fct_ID) as: (3)
$U (\bar{I}) = \int_{0}^{λ_{t} - 1} T (λ_{t} - 1) \partial ε$

Where,

$T$

Stress for the engineering strain $λ_{t} - 1$ .

$λ_{t}$

Equivalent stretch corresponding to Uniaxial tension, equibiaxial or planar test.
The Cauchy stress is computed as:(4)
$σ = \frac{2}{J} \frac{\partial U}{\partial \bar{I}} d e v (b^{*}) + \frac{\partial V}{\partial J} I$

Where, $b^{*} = J^{- \frac{2}{3}} F F^{T}$ .
/VISC/PRONY can be used with LAW111 to include viscous effects.