/MAT/LAW33 (FOAM_PLAS)

Block Format Keyword This law models a viscous-elastic foam material with unloading/reloading like plastic behavior. This law is applicable only for solid elements and is typically used to model low density, closed cell polyurethane foams such as impact limiters.

Format

(1)	(3)	(4)	(5)
/MAT/LAW33/`mat_ID`/`unit_ID` or /MAT/FOAM_PLAS/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$
`E`	`K`_a	`fct_ID`_f	`Fscale`_crv
`P`₀	$Φ$		$γ_{0}$
`A`	`B`		`C`

Read only if K_a = 1

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`E`₁		`E`₂		`E`_t		$η^{*}$		$η_{0}$

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)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$
`K`_a	Analysis type flag. = 0 The skeletal behavior before yield is elastic. = 1 The skeletal behavior before yield is visco-elastic. (Integer)
`fct_ID`_f	Yield stress vs. volumetric strain curve function identifier. (Integer)
`Fscale`_crv	Scale factor for ordinate (stress) for `fct_ID`_f. Default = 1.0 (Real)	$[Pa]$
`P`₀	Initial air pressure. 5 (Real)	$[Pa]$
$Φ$	Ratio of foam to polymer density. (Real)
$γ_{0}$	Initial volumetric strain. (Real)
`A`	Yield parameter. (Real)
`B`	Yield parameter. (Real)
`C`	Yield parameter. (Real)
`E`₁	Coefficient for Young's modulus update. (Real)	$[Pa \cdot s]$
`E`₂	Coefficient for Young's modulus update. (Real)	$[Pa]$
`E`_t	Tangent modulus. (Real)	$[Pa]$
$η^{*}$	Viscosity coefficient in pure compression. (Real)
$η_{0}$	Viscosity coefficient in pure shear. (Real)

Example (Foam)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/FOAM_PLAS/1/1
Foam
#              RHO_I
               2E-10
#                  E        Ka  func_IDf          Fscalecurv
                 200         1         0                   1
#                 P0                 Phi             Gamma_0
                   0                   0                   0
#                  A                   B                   C
                1E30                   0                   0
#                 E1                  E2                  Et            eta_comp           eta_shear
                   0                   0                   2                2E27                1E27
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

If the limiting yield curve is not defined, then the material follows a Maxwell-Kelvin-Voight visco-elastic model.

Figure 1.
If the limiting yield curve is defined, then the material initially follows the visco-elastic law until it intersects the defined yield curve, which limits the visco-elastic stress in tension and compression. The material does not experience plasticity, but instead behaves in a visco hyperelastic way.
If the yield function fct_ID_f = 0, then(1)
$σ_{y} = A + B (1 + C γ)$

Where, $γ$ is the volumetric strain:(2)
$γ = \frac{V}{V_{0}} - 1 + γ_{0} = \frac{ρ_{0}}{ρ} - 1 + γ_{0} = \frac{- μ}{1 + μ} + γ_{0}$
If the yield function fct_ID_f ≠ 0, then $σ$ vs. $γ$ is read from input of the curve idenfier fct_ID_f. The curve can be defined for tensile ( $γ > 0$ ) and compression ( $- 1 < γ < 0$ ). The input stress should be positive for both tension and compression.

Figure 2.
The optional air pressure, as a function of the volumetric strain can be added to the structural pressure. Pressure is applied only on the spherical part of the stress tensor.(3)
$P_{a i r} = - \frac{P_{0} \cdot γ}{1 + γ - Φ}$
Young’s modulus is used as the initial slope for unloading. It can be constant or variable, based on the strain rate.(4)
$E = \max (E, E_{1} \dot{ε} + E_{2})$
The unloading or the loading direction change (tensile <-> compression) is following the current elastic modulus, like an isotropic elastic-plastic material or highly viscous foam material. However, there is no plastic strain accumulation.

Figure 3.