/MAT/LAW95 (BERGSTROM_BOYCE)

It uses a polynomial material model for the hyperelastic material response and the Bergstrom-Boyce material model ¹ to represent the nonlinear viscoelastic time dependent material response. This law is only compatible with solid elements.

Format

Definitions

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW95/1/1
BERGSTROM 
#              RHO_I        
             1.42E-6 		
#                C10                 C01                 C20                 C11                 C22
              0.2019                  0.             4.43E-5
#                C30                 C21                 C12                 C03                  Sb
            1.295E-4                  0.                  0.                  0.                 2.0
#                 D1                  D2                  D3
           2.1839E-3             8.68E-5           -1.794E-5
#                  A                EXPC                EXPM                 KSI
              1.0E-1                -0.7                   5                0.01
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

1. 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 time-dependent network.

   
   
   Figure 1. Time-dependent Network

2. 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)
   $$W_A={\displaystyle \sum _{i+j=1}^3{C}_{ij}{\left({\overline{I}}_1-3\right)}^i\cdot {\left({\overline{I}}_2-3\right)}^j}+{\displaystyle \sum _{i=1}^3\frac{1}{D_i}{\left(J-1\right)}^{2i}}$$
   and(2)

   $$W_B=S_b\cdot W_A$$
   Where,

   • ${\overline{I}}_1={\overline{\lambda }}_1^2+{\overline{\lambda }}_2^2+{\overline{\lambda }}_3^2$
   • ${\overline{I}}_2={\overline{\lambda }}_1^{-2}+{\overline{\lambda }}_2^{-2}+{\overline{\lambda }}_3^{-2}$
   • ${\overline{\lambda }}_i=J^{-\frac{1}{3}}{\lambda }_i$
3. For special value of $C_{ij}$ , the polynomial model can be reduced to the following material models:
   • Yeoh: j=0

     Where, C₁₀, C₂₀, C₃₀ are not zero

   • Mooney-Rivlin: i+j =1

     Where, C₁₀ and C₀₁ are not zero, and D₂ =D₃=0

   • Neo-Hookean:

     Only C₁₀ and D₁ are not zero

4. 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)

   $$K=\frac{2}{D_1}\left(1+S_b\right)$$
5. If D₁= 0, an incompressible material is considered.
6. If $A$ =0, then only the hyperelastic polynomial material model is used with no viscoelastic time dependent response.
7. The effective creep strain rate in network B is given by the expression:(5)
   $${\dot{\epsilon }}_B^v=A{\left(\tilde{\lambda }-1+\xi \right)}^C{\overline{\sigma }}_B{}^M$$
   Where,

   $\tilde{\lambda }=\sqrt{\frac{{\overline{I}}_1}{3}}$

   Effective stress in Network B

   $\xi$ , $M$ and $C$

   Input material parameters