/MAT/LAW34 (BOLTZMAN)
Block Format Keyword This law describes the Boltzmann (viscoelastic) material. This law is applicable only for solid elements and can be used to model polymers and elastomers.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW34/mat_ID/unit_ID or /MAT/BOLTZMAN/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
K  
${G}_{0}$  ${G}_{l}$  $\beta $  
P_{0}  $\mathrm{\Phi}$  ${\gamma}_{0}$ 
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) 

${\rho}_{i}$  Initial
density (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
K  Bulk
modulus. (Real) 
$\left[\text{Pa}\right]$ 
${G}_{0}$  Short time shear
modulus. (Real) 
$\left[\text{Pa}\right]$ 
${G}_{l}$  Long time shear
modulus. (Real) 
$\left[\text{Pa}\right]$ 
$\beta $  Decay
constant. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
P_{0}  Initial air
pressure. (Real) 
$\left[\text{Pa}\right]$ 
$\mathrm{\Phi}$  Foam vs. polymer
density ratio. (Real) 

${\gamma}_{0}$  Initial volumetric
strain. (Real) 
Example (Plastic)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/BOLTZMAN/1/1
plastic
# RHO_I
2E10
# K
66.67
# G0 Gl Beta
100 100 50000
# P0 Phi Gamma0
0 0 0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 For closed cell foam
material, the pressure may be augmented:
(1) $$P=K{\epsilon}_{kk}+{P}_{air}$$With(2) $${P}_{air}=\frac{{P}_{0}\cdot \gamma}{1+\gamma \mathrm{\Phi}}$$(3) $${\epsilon}_{kk}=\mathrm{ln}\left(\frac{V}{{V}_{0}}\right)$$Where, $\gamma $
 Volumetric strain
 $\text{\Phi}$
 Porosity
 ${P}_{0}$
 Initial air pressure
 ${\gamma}_{0}$
 Initial volumetric strain
 $K$
 Bulk modulus
 Stress deviator for
viscoelastic material:
(4) $${s}_{ij}=2{\displaystyle {\int}_{0}^{t}\psi \left(t\tau \right)}\left(\frac{\partial {e}_{ij}\left(\tau \right)}{\partial \tau}\right)d\tau $$While the shear relaxation moduli $\psi (t)$ is:(5) $$\mathrm{\Psi}(t)={G}_{l}+{G}_{s}{e}^{\beta t}={G}_{l}+({G}_{0}{G}_{l}){e}^{\beta t}$$Where, ${G}_{l}$
 Long time shear module
 ${G}_{s}$
 Short time shear module
 $\beta $
 Decay constant with $\beta =\frac{1}{{\tau}_{s}}=\frac{{G}_{s}}{{\eta}_{s}}$
When t = 0, then $\psi (t)={G}_{0}$ and then, $\psi (t)={G}_{l}$ .