/MAT/LAW95 (BERGSTROM_BOYCE)
Block Format Keyword This law is a constitutive model for predicting the nonlinear time dependency of elastomer like materials.
It uses a polynomial material model for the hyperelastic material response and the BergstromBoyce material model ^{1} to represent the nonlinear viscoelastic time dependent material response. This law is only compatible with solid elements.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW95/mat_ID/unit_ID or /MAT/BERGSTROM_BOYCE/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
C_{10}  C_{01}  C_{20}  C_{11}  C_{02}  
C_{30}  C_{21}  C_{12}  C_{03}  s_{b}  
D_{1}  D_{2}  D_{3}  
A  C  M  $\xi $  Tau_ref 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
C_{10}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{01}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{20}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{11}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{02}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{30}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{21}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{12}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
C_{03}  Material parameter for hyperelastic
model. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
S_{b}  Stress scaling factor for network B. Default = 0.0 (Real) 

D_{1}  Volumetric material parameter 1, for bulk modulus computation. $K=\frac{2}{{D}_{1}}$ Default = 0.0 (Real) 
$\left[\frac{1}{\text{P}\text{a}}\right]$ 
D_{2}  Volumetric material parameter 2. Default = 0.0 (Real) 
$\left[\frac{1}{\text{P}\text{a}}\right]$ 
D_{3}  Volumetric material parameter 3. Default = 0.0 (Real) 
$\left[\frac{1}{\text{P}\text{a}}\right]$ 
A  Effective creep strain rate. Default = 0.0 (Postive Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
C  Exponent characterizing the creep strain
dependence of the effective creep strain rate in
network B (1 < C < 0). Default = 0.7 (Real) 

M  Positive exponent (
$M\ge 1.0$
) characterizing the
effective stress dependence of the effective creep
strain rate in network B. Default = 1.0 (Real) 

$\xi $  Constant for regularization of the creep strain
rate near undeformed state. Default = 0.01 (Real) 

Tau_ref  Reference stress for the Effective creep strain
rate in secondary network. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Example
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
kg mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW95/1/1
BERGSTROM
# RHO_I
1.42E6
# C10 C01 C20 C11 C22
0.2019 0. 4.43E5
# C30 C21 C12 C03 Sb
1.295E4 0. 0. 0. 2.0
# D1 D2 D3
2.1839E3 8.68E5 1.794E5
# A EXPC EXPM KSI Tau_ref
1.0E1 0.7 5 0.01
#12345678910
#ENDDATA
#12345678910
Comments
 The response of the material can be represented using two parallel networks A and B. Network A is the equilibrium network with a nonlinear hyperelastic component. In network B, a nonlinear hyperelastic component is in series with a nonlinear viscoelastic flow element, and hence the timedependent network.
 The same polynomial strain energy
potential is used for the hyperelastic components
in both networks. In network B, this potential is
scaled by a factor
S_{b}.
The strain energy density is then written for the
hyperelastic component of the
network:
(1) $${W}_{A}={\displaystyle \sum _{i+j=1}^{3}{C}_{ij}{\left({\overline{I}}_{1}3\right)}^{i}\cdot {\left({\overline{I}}_{2}3\right)}^{j}}+{\displaystyle \sum _{i=1}^{3}\frac{1}{{D}_{i}}{\left(J1\right)}^{2i}}$$and(2) $${W}_{B}={S}_{b}\cdot {W}_{A}$$Where, ${\overline{I}}_{1}={\overline{\lambda}}_{1}^{2}+{\overline{\lambda}}_{2}^{2}+{\overline{\lambda}}_{3}^{2}$
 ${\overline{I}}_{2}={\overline{\lambda}}_{1}^{2}+{\overline{\lambda}}_{2}^{2}+{\overline{\lambda}}_{3}^{2}$
 ${\overline{\lambda}}_{i}={J}^{\frac{1}{3}}{\lambda}_{i}$
 For special value of
${C}_{ij}$
, the polynomial model
can be reduced to the following material
models:
 Yeoh: j=0
Where, C_{10}, C_{20}, C_{30} are not zero
 MooneyRivlin: i+j =1
Where, C_{10} and C_{01} are not zero, and D_{2} =D_{3}=0
 NeoHookean:
Only C_{10} and D_{1} are not zero
 Yeoh: j=0
 The initial shear modulus and the bulk
modulus are computed as:
(3) $$\mu =2\left({S}_{b}+1\right)\left({C}_{10}+{C}_{01}\right)$$and(4) $$K=\frac{2}{{D}_{1}}\left(1+{S}_{b}\right)$$  If D_{1}= 0, an incompressible material is considered.
 If $A$ =0, then only the hyperelastic polynomial material model is used with no viscoelastic time dependent response.
 The effective creep strain rate in
network
$B$
is given by the
expression:
(5) $${\dot{\epsilon}}_{B}^{v}=A{\left(\tilde{\lambda}1+\xi \right)}^{C}{\left(\frac{{\overline{\sigma}}_{B}}{{\tau}_{ref}}\right)}^{M}$$Where, $\tilde{\lambda}=\sqrt{\frac{{\overline{I}}_{1}}{3}}$ and ${\overline{\sigma}}_{B}$
 Effective stress in Network B
 $\xi $ , $M$ , $C$ and ${\tau}_{ref}$
 Input material parameters