/MAT/LAW72 (HILL_MMC)
Block Format Keyword This law describes the anisotropic Hill material with a modified Mohr fracture criteria. This law is available for shell and solid.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW72/mat_ID/unit_ID or /MAT/HILL_MMC/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  v  
${\sigma}_{y}^{0}$  ${\epsilon}_{p}^{0}$  n  F  G  
H  N  L  M  
C_{1}  C_{2}  C_{3}  m  D_{c} 
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) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E  Initial Young's modulus. (Real) 
$\left[\text{Pa}\right]$ 
v  Poisson's ratio. (Real) 

${\sigma}_{y}^{0}$  Initial yield stress. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{p}^{0}$  Initial plastic strain. (Real) 

n  Exponent for the isotropic function for the swift hardening: ${\sigma}_{y}={\sigma}_{y}^{0}{\left({\epsilon}_{p}+{\epsilon}_{p}^{0}\right)}^{n}$ It is also used as an exponent in the MMC failure equations. 2 (Real) 

F, G, H, L, M, N  Six
HILL Materials anisotropic
parameters. (float) 

C_{1}  First parameter for MMC fracture model. (Real) 

C_{2}  Second parameter for MMC fracture model. (Real) 
$\left[\text{Pa}\right]$ 
C_{3}  Third parameter for MMC fracture model. (Real) 

m  Exponent for the softening function. 3 (Real) 

D_{c}  Critical damage.
(Real) 
Example (Metal)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW72/1/1
Metal
# RHO_I
0.0028
# E nu
200E+3 0.3
# Sig0 Eps0 n F G
1276 1.63E3 0.265 0.5 0.5
# H N L M
0.5 1.5 0 0
# C1 C2 C3 m Dc
0.12 720 1.095 0.5 1.1
#12345678910
#ENDDATA
/END
#12345678910
Comments
 3D
equivalent Hill stress:
(1) $$f=\sqrt{F{\left({\sigma}_{\mathit{yy}}{\sigma}_{\mathit{zz}}\right)}^{2}+G{\left({\sigma}_{\mathit{zz}}{\sigma}_{\mathit{xx}}\right)}^{2}+H{\left({\sigma}_{\mathit{xx}}{\sigma}_{\mathit{yy}}\right)}^{2}+\text{\hspace{0.17em}}\text{}2L{\sigma}_{\mathit{yz}}^{2}+2M{\sigma}_{\mathit{zx}}^{2}+2N{\sigma}_{\mathit{xy}}^{2}}$$For shell element, take:(2) $$f=\sqrt{F{\sigma}_{\mathit{yy}}^{2}+G{\sigma}_{\mathit{xx}}^{2}+H{\left({\sigma}_{\mathit{xx}}{\sigma}_{\mathit{yy}}\right)}^{2}+2N{\sigma}_{\mathit{xy}}^{2}}$$  MMC
fracture criteria:
(3) $$D=\underset{0}{\overset{{\epsilon}_{p}}{\int}}\frac{d{\epsilon}_{p}}{{\epsilon}_{f}\left(\theta ,\eta \right)}$$ Anisotropic 3D model
(4) $${\epsilon}_{f}\left(\theta ,\eta \right)={\left\{\frac{{\sigma}_{y}^{0}}{{C}_{2}}\left[{C}_{3}+\frac{\sqrt{3}}{2\sqrt{3}}\left(1{C}_{3}\right)\left(\mathrm{sec}\left(\frac{\theta \pi}{6}\right)1\right)\right]\left[\sqrt{\frac{1+{C}_{1}^{2}}{3}\mathrm{cos}\left(\frac{\theta \pi}{6}\right)+{C}_{1}\left(\eta +\frac{1}{3}\right)\mathrm{sin}\left(\frac{\theta \pi}{6}\right)}\right]\right\}}^{\frac{1}{n}}$$While,
$\theta $ is Lode angle $\theta =1\frac{2}{\pi}\mathit{ar}\hspace{0.17em}\mathrm{cos}\hspace{0.17em}\zeta $
with Lode angle parameter $\zeta =\frac{27}{2}\frac{{J}_{3}}{{\sigma}_{\mathit{VM}}^{3}}$
${J}_{3}$ is the third invariant of the deviatoric stress.
$\eta $ is stress triaxiality with $\eta =\frac{\frac{1}{3}\left({\sigma}_{\mathit{xx}}+{\sigma}_{\mathit{yy}}+{\sigma}_{\mathit{zz}}\right)}{{\sigma}_{\mathit{VM}}}$
 2D Anisotropic Model
(5) $${\epsilon}_{f}\left(\theta ,\eta \right)={\left\{\frac{{\sigma}_{y}^{0}}{{C}_{2}}{f}_{3}\left[\left(\sqrt{\frac{1+{C}_{1}^{2}}{3}}{f}_{1}\right)+{C}_{1}\left(\eta +\frac{{f}_{2}}{3}\right)\right]\right\}}^{\frac{1}{n}}$$with,(6) $${f}_{1}=cos\left\{\frac{1}{3}\text{arcsin}\left[\frac{27}{2}\eta \left({\eta}^{2}\frac{1}{3}\right)\right]\right\}$$(7) $${f}_{2}=sin\left\{\frac{1}{3}\text{arcsin}\left[\frac{27}{2}\eta \left({\eta}^{2}\frac{1}{3}\right)\right]\right\}$$(8) $${f}_{3}={C}_{3}+\frac{\sqrt{3}}{2\sqrt{3}}\left(1{C}_{3}\right)\left(\frac{1}{{f}_{1}}1\right)$$
 Anisotropic 3D model
 Fracture and damage
with MMC fracture criteria:
 When D = 1: fracture initiate
 By 1 < D <
D_{c}: the yield stress is multiplied by
softening function
$\beta $
to reduce the deformation resistance.
${\sigma}_{y}=\beta {\sigma}_{y}^{0}{\left({\epsilon}_{p}+{\epsilon}_{p}^{0}\right)}^{n}$ with $\beta ={\left(\frac{{D}_{c}D}{{D}_{c}1}\right)}^{m}0<\beta <1$
 If D ≥ D_{c}, the element is deleted.
 The exponent m is used to describe the softening behavior. It is
recommended to use m > 0.
If 0 < m < 1, then the softening curve is convex.
If m > 1, then the softening curve is concave. The softening is between ${\epsilon}_{f}$ and ${D}_{c}\cdot {\epsilon}_{f}$ . Once the plastic strain is reached ${D}_{c}\cdot {\epsilon}_{f}$ (in this case $D>{D}_{c}$ ), then the element is deleted.
 It is possible to
display user variables in animation files (with Engine /ANIM/Eltyp/Restype) and in Time history file (with Starter /TH/SHEL and /TH/BRIC):
 USER1: Plastic strain
 USER2: Damage value