/MAT/LAW81

Block Format Keyword This law is based on Drücker-Prager yield criteria with cap. It has a strain-hardening cap model based on the principles of Foster. Plasticity has an isotropic hardening.

Failure surface is limited to the standard linear Drücker-Prager relation, with symmetry around the pressure axis. This law is LAG, ALE and EULER compatible.

Format

(1)	(2)	(3)	(4)	(5)	(7)
/MAT/LAW81/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`K`₀		`G`₀		`c`₀	`P`_b0
$ϕ$		$ψ$
α		`Eps_max`		$ε_{v 0}^{p}$
`fct_ID`_K	`fct_ID`_G	`fct_ID`_C	`fct_ID`_Pb	`I`_soft
`K`_w		`n`₀		`S`₀	`U`₀
`Tol`		$α_{v}$

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`K`₀	Initial bulk modulus. (Real)	$[Pa]$
`G`₀	Initial shear modulus. (Real)	$[Pa]$
`c`₀	Initial material cohesion. (Real)	$[Pa]$
`P`_b0	Initial cap limit pressure. (Real)	$[Pa]$
$ϕ$	Friction angle. (Real)	$[\deg]$
$ψ$	Plastic flow angle. (Real)	$[\deg]$
α	Ratio of: $α = \frac{p_{a}}{p_{b}}$ Default = 0.5 (Real)
`Eps_max`	Maximum dilatancy (negative number limiting $\frac{ρ}{ρ_{0}} - 1$ ). Default = -10²⁰ (Real)
$ε_{v 0}^{p}$	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. = 0 (Default) Cap softening is allowed. = 1 Imposes that $ε_{v}^{p}$ and `P`_b cannot decrease. (Integer)
`K`_w	Pore bulk modulus (water). (Real)	$[Pa]$
`n`₀	Initial porosity. (Real)
`S`₀	Initial saturation. (Real)
`U`₀	Initial pore pressure. (Real)	$[Pa]$
`Tol`	Tolerance for cap shift viscosity. Default = 1.0E-4 (Real)
$α_{v}$	Viscosity factor. Default = 0.5 (Real)

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                   m                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  3. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/3
Yield Hardening
#                  X                   Y
                   0                2000                                                            
                  .1             2002000                                                            
                   1             2002000                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/4
Cap Hardening
#                  X                   Y
                  -1                1000                                                            
                   0                1000                                                            
                .001               30000                                                            
               .0022               70000                                                            
               .0024               80000                                                            
                .004              100000                                                            
               .0056              200000                                                            
               .0078              800000                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The yield surface is defined as:(1)
$F = q - r_{c} (p) \cdot (p \tan ϕ + c) = 0$

Where,

$r_{c} (p) = 1$ if $p \leq p_{a}$

$r_{c} (p) = \sqrt{1 - {(\frac{p - p_{a}}{p_{b} - p_{a}})}^{2}}$ if $p_{a} < p \leq p_{b}$

Where,

p

Pressure

q

von Mises stress

c

Material cohesion

P₀

Pressure, where $\frac{\partial F}{\partial p} (p_{0}) = 0$

p_b

Cap limit pressure

Figure 1.

In this material, yield surface and failure surface are the same.
Plastic flow is governed by the non-associated flow potential G, as:
$G = q - p \cdot \tan ψ = 0$ if $p \leq p_{a}$

$G = q - \tan ψ (p - \frac{{(p - p_{a})}^{2}}{2 (p_{0} - p_{a})}) = 0$ if $p_{a} < p \leq p_{0}$

$G = F$ if $p > p_{0}$ , the flow becomes associated on the cap.
If cap softening is allowed, $ε_{v}^{p}$ can decrease, therefore it is recommended to define the following curves on a relevant range. For example, if $ε_{v 0}^{p} = 0$ , 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, $K = K_{0} f_{K} (ε_{v}^{p})$ , with the function $f_{K}$ defined in fct_ID_K
The initial bulk modulus and shear modulus can be calculated as:
$K_{0} = \frac{E_{c}}{3 (1 - 2 ν)}$ ; $G_{0} = \frac{E_{c}}{2 (1 + ν)}$

With,

$ν$

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)
$n = \frac{V_{v o i d}}{V_{t o t a l}}$

In the elastic case, the void volume does not change. However, in the plastic case, the porosity change is defined by:(3)
$n = 1 - (1 - n_{0}) e^{ε_{v}^{p} - ε_{v 0}^{p}}$
Effect of pores filled with water:
The initial state of the pores is defined by the initial porosity, initial saturation, and initial pore pressure n₀, U₀ and S₀ which can be calculated as:(4)
$n_{0} = \frac{V_{v o i d}}{V_{t o t a l}} and S_{0} = \frac{V_{w a t e r}}{V_{v o i d}}$

If the $μ_{0} > 0$ then the entered value for S₀ is not used and instead S₀ is recalculated.
The following user variables are available for post-treatment:
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 $u^{*}$