/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

Definition

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.

   
   
   

2. The same polynomial strain energy potential is used for the hyperelastic components in both networks. In network B, this potential is scaled by factor S_b. The strain energy density of the hyperelastic component of the network can have 2 different forms for the volumetric part depending on the flag Iform:
   • Iform = 1 (Default):
     $$W_A={\displaystyle \sum _{i+j=1}^3{C}_{ij}{\left({\overline{I}}_1-3\right)}^i\cdot {\left({\overline{I}}_2-3\right)}^j}+{\displaystyle \sum _{i=1}^3\frac{1}{D_i}{\left(J-1\right)}^{2i}}$$
   • Iform = 2:
     $$W_A={\displaystyle \sum _{i+j=1}^3{C}_{ij}{\left({\overline{I}}_1-3\right)}^i\cdot {\left({\overline{I}}_2-3\right)}^j}+K\left(J-1-\ln J\right)$$
     Where,
     • $K=2/D_1$
     • $D_2=D_3=0$
     • $W_B=S_b\cdot W_A$
     • ${\overline{I}}_1={\overline{\lambda }}_1^2+{\overline{\lambda }}_2^2+{\overline{\lambda }}_3^2$
     • ${\overline{I}}_2={\overline{\lambda }}_1^{-2}+{\overline{\lambda }}_2^{-2}+{\overline{\lambda }}_3^{-2}$
     • ${\overline{\lambda }}_i=J^{-\frac{1}{3}}{\lambda }_i$
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:
   $$\mu =2\left(S_b+1\right)\left(C_{10}+C_{01}\right)$$

   and

   $$K=\frac{2}{D_1}\left(1+S_b\right)$$

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:
   $${\dot{\epsilon }}_B^v=A{\left(\tilde{\lambda }-1+\xi \right)}^C{\left(\frac{{\overline{\sigma }}_B}{{\tau }_{ref}}\right)}^M$$
   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