/MAT/LAW34 (BOLTZMAN)

Block Format Keyword This law describes the Boltzmann (visco-elastic) material. This law is applicable only for solid, shell, truss and integrated beam elements and can be used to model polymers and elastomers.

Format

(1)	(3)	(5)
/MAT/LAW34/`mat_ID`/`unit_ID` or /MAT/BOLTZMAN/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`K`
$G_{0}$	$G_{l}$	$β$
`P`₀	$Φ$	$γ_{0}$

Definition

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`	Bulk modulus. (Real)	$[Pa]$
$G_{0}$	Short time shear modulus. (Real)	$[Pa]$
$G_{l}$	Long time shear modulus. (Real)	$[Pa]$
$β$	Decay constant. (Real)	$[\frac{1}{s}]$
`P`₀	Initial air pressure. (Real)	$[Pa]$
$Φ$	Foam versus polymer density ratio. (Real)
$γ_{0}$	Initial volumetric strain. (Real)

Example (Plastic)

#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/BOLTZMAN/1/1
plastic
#              RHO_I
               2E-10
#                  K
               66.67
#                 G0                  Gl                Beta
                 100                 100               50000
#                 P0                 Phi              Gamma0
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

For closed cell foam material, the pressure may be augmented (only for solid):(1)
$P = - K ε_{k k} + P_{a i r}$

With (2)
$P_{a i r} = - \frac{P_{0} \cdot γ}{1 + γ - Φ}$
(3)
$γ = \frac{V}{V_{0}} - 1 + γ_{0}$
(4)
$ε_{k k} = \ln (\frac{V}{V_{0}})$

Where,

$γ$

Volumetric strain

$Φ$

Porosity

$P_{0}$

Initial air pressure

$γ_{0}$

Initial volumetric strain

$K$

Bulk modulus
Stress deviator for viscoelastic material:(5)
$s_{i j} = 2 \int_{0}^{t} ψ (t - τ) (\frac{\partial e_{i j} (τ)}{\partial τ}) d τ$

While the shear relaxation moduli $ψ (t)$ is:(6)
$Ψ (t) = G_{l} + G_{s} e^{- β t} = G_{l} + (G_{0} - G_{l}) e^{- β t}$

Where,

$G_{l}$

Long time shear modulus

$G_{s}$

Short time shear modulus

$β$

Decay constant with $β = \frac{1}{τ_{s}} = \frac{G_{s}}{η_{s}}$

When t = 0, then $ψ (t) = G_{0}$ and when $t = \infty$ , then $ψ (t) = G_{l}$ .