/MAT/LAW81
Block Format Keyword This law is based on DrückerPrager yield criteria with cap. It has a strainhardening cap model based on the principles of Foster. Plasticity has an isotropic hardening.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW81/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
K_{0}  G_{0}  c_{0}  P_{b0}  
$\varphi $  $\psi $  
α  Eps_max  ${\epsilon}_{v0}^{p}$  
fct_ID_{K}  fct_ID_{G}  fct_ID_{C}  fct_ID_{Pb}  I_{soft}  
K_{w}  n_{0}  S_{0}  U_{0}  
Tol  ${\alpha}_{v}$ 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
K_{0}  Initial bulk
modulus. (Real) 
$\left[\text{Pa}\right]$ 
G_{0}  Initial shear
modulus. (Real) 
$\left[\text{Pa}\right]$ 
c_{0}  Initial material
cohesion. (Real) 
$\left[\text{Pa}\right]$ 
P_{b0}  Initial cap limit
pressure. (Real) 
$\left[\text{Pa}\right]$ 
$\varphi $  Friction
angle. (Real) 
$\left[\mathrm{deg}\right]$ 
$\psi $  Plastic flow
angle. (Real) 
$\left[\mathrm{deg}\right]$ 
α  Ratio of: $\alpha =\frac{{p}_{a}}{{p}_{b}}$ Default = 0.5 (Real) 

Eps_max  Maximum dilatancy
(negative number limiting
$\frac{\rho}{{\rho}_{0}}1$
). Default = 10^{20} (Real) 

${\epsilon}_{v0}^{p}$  Initial value of the
plastic volumetric strain. 3 (Real) 

fct_ID_{K}  (Optional) Function
identifier for the bulk modulus scale factor vs the plastic
volumetric strain. 4 (Integer) 

fct_ID_{G}  (Optional) Function
identifier for the shear modulus scale factor vs the plastic
volumetric strain. (Integer) 

fct_ID_{C}  (Optional) Function
identifier for the material cohesion scale factor vs the equivalent
plastic strain. (Integer) 

fct_ID_{Pb}  (Optional) Function
identifier for the cap limit pressure scale factor vs the plastic
volumetric strain. (Integer) 

I_{soft}  Cap softening flag.
(Integer) 

K_{w}  Pore bulk modulus
(water). (Real) 
$\left[\text{Pa}\right]$ 
n_{0}  Initial
porosity. (Real) 

S_{0}  Initial
saturation. (Real) 

U_{0}  Initial pore
pressure. (Real) 
$\left[\text{Pa}\right]$ 
Tol  Tolerance for cap shift
viscosity. Default = 1.0E4 (Real) 

${\alpha}_{v}$  Viscosity
factor. Default = 0.5 (Real) 
Example
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
kg m s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW81/1/1
LAW81
# RHO_I
1700
# K0 G0 c0 PB0
2.83E9 1.31E9 1 1
# BETA PSI
15 10
# ALPHA EPS_p_max EPS_0
.5 .02 .002
# Fct_IDK Fct_IDG Fct_IDc Fct_IDPb I_soft
0 0 3 4 1
# Kw n0 S0 U0
2.5E10 0.1 0.99 0.0
# Tol alpha_v
0.0001 0.5
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/3
Yield Hardening
# X Y
0 2000
.1 2002000
1 2002000
#12345678910
/FUNCT/4
Cap Hardening
# X Y
1 1000
0 1000
.001 30000
.0022 70000
.0024 80000
.004 100000
.0056 200000
.0078 800000
#12345678910
#ENDDATA
/END
#12345678910
Comments
 The yield surface is defined
as:
(1) $$F=q{\mathrm{r}}_{c}\left(p\right)\cdot \left(p\mathrm{tan}\varphi +c\right)=0$$Where,
${\mathrm{r}}_{c}\left(p\right)=1$ if $p\le {p}_{a}$
${\mathrm{r}}_{c}\left(p\right)=\sqrt{1{\left(\frac{p{p}_{a}}{{p}_{b}{p}_{a}}\right)}^{2}}$ if ${p}_{a}<p\le {p}_{b}$
Where, p
 Pressure
 q
 von Mises stress
 c
 Material cohesion
 P_{0}
 Pressure, where $\frac{\partial F}{\partial p}\left({p}_{0}\right)=0$
 p_{b}
 Cap limit pressure
In this material, yield surface and failure surface are the same.
 Plastic flow is governed by the
nonassociated flow potential G, as:
$G=qp\cdot \mathrm{tan}\psi =0$ if $p\le {p}_{a}$
$G=q\mathrm{tan}\psi \left(p\frac{{\left(p{p}_{a}\right)}^{2}}{2\left({p}_{0}{p}_{a}\right)}\right)=0$ if ${p}_{a}<p\le {p}_{0}$
$G=F$ if $p>{p}_{0}$ , the flow becomes associated on the cap.
 If cap softening is allowed, ${\epsilon}_{v}^{p}$ can decrease, therefore it is recommended to define the following curves on a relevant range. For example, if ${\epsilon}_{v0}^{p}=0$ , negative values.
 The initial values for bulk
modulus, shear modulus, material cohesion, and cap limit pressure can be scaled
by defining a function as the scale factor curve for each respective value. If
the function is not defined, then the value is considered constant. For
example:
If fct_ID_{K} = 0 then, $K={K}_{0}$
If fct_ID_{K} ≠ 0 then, $K={K}_{0}{\mathrm{f}}_{K}\left({\epsilon}_{v}^{p}\right)$ , with the function ${\mathrm{f}}_{K}$ defined in fct_ID_{K}
 The initial bulk modulus and shear
modulus can be calculated as:
${K}_{0}=\frac{{E}_{c}}{3\left(12\nu \right)}$ ; ${G}_{0}=\frac{{E}_{c}}{2\left(1+\nu \right)}$
With, $\nu $
 Poisson’s ratio
 ${E}_{c}$
 Young’s modulus of concrete
 The porosity is defined so that
it represents the volume fraction of voids, with respect to the total material
volume.
(2) $$n=\frac{{V}_{void}}{{V}_{total}}$$In the elastic case, the void volume does not change. However, in the plastic case, the porosity change is defined by:(3) $$n=1\left(1{n}_{0}\right){e}^{{\epsilon}_{v}^{p}{\epsilon}_{v0}^{p}}$$  Effect of pores filled with
water: The initial state of the pores is defined by the initial porosity, initial saturation, and initial pore pressure n_{0}, U_{0} and S_{0} which can be calculated as:
(4) $${n}_{0}=\frac{{V}_{void}}{{V}_{total}}\text{and}{S}_{0}=\frac{{V}_{water}}{{V}_{void}}$$If the ${\mu}_{0}>0$ then the entered value for S_{0} is not used and instead S_{0} is recalculated.
 The following user variables are
available for posttreatment:
USR1 is the equivalent plastic strain EPSPD
USR2 is the plastic volumetric strain EPSPV
USR3 is the cohesion c
USR4 is the cap limit pressure P_{b}
USR5 pore pressure U
USR6 porosity n
USR7 saturation S
USR8 cap shift ${u}^{*}$