/MAT/LAW100 (MNF)

It consists of having a specific number of networks with an elastic component and an optional flow component. This law is only compatible with solid elements.

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Definition

Comments

1. This material is only compatible with solid elements with Lagrange type total strain. The strain formulation flag is automatically set to I_smstr =10 in /PROP/SOLID.
2. The response of the material can be represented using a set of parallel networks. Network 0 is the equilibrium network with a nonlinear hyperelastic component and optional creep component. In secondary networks, a nonlinear hyperelastic component is in series with a nonlinear viscoelastic flow element, and hence is the time-dependent network. All networks have the same hyperelastic behavior, scaled by a stiffness weight factor for the secondary networks.
   The sum of the stiffness weight factors must be equal to 1:(1)

   $${\displaystyle \sum _{i=0}^N{S}_i}=1$$

   
   
   Figure 1.

3. Same polynomial strain energy potential is used for the hyperelastic components in all networks. In secondary networks, this potential is scaled by a factor $S_i$ .
4. Flag_HE
   1. 1 = Polynomial form: The energy density is then written(2)
      $$W_0={\displaystyle \sum _{i+j=1}^3{C}_{ij}{\left({\overline{I}}_1-3\right)}^i{\left({\overline{I}}_2-3\right)}^j}+{\displaystyle \sum _{i=1}^3\frac{1}{D_i}{\left(J-1\right)}^{2i}}$$
   2. 2 = Arruda-Boyce: The energy density is then written:(3)
      $$W_0=\mu {\displaystyle \sum _{i=1}^5\frac{c_i}{{({\lambda }_m)}^{2i-2}}}\left({\overline{I}}_1^i-3^i\right)+\frac{1}{D}\left(\frac{J^2-1}{2}-\ln (J)\right)$$
   3. 3 = Neo-Hook: The energy density is then written:(4)
      $$W_0=C_{10}\left({\overline{I}}_1-3\right)+\frac{1}{D}{\left(J-1\right)}^2$$
   4. 4 = Mooney-Rivlin: The energy density is then written:(5)
      $$W_0=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+\frac{1}{D}{\left(J-1\right)}^2$$
   5. 5 = Yeoh: The energy density is then written:(6)
      $$W_0=C_{10}\left({\overline{I}}_1-3\right)+C_{20}{\left({\overline{I}}_1-3\right)}^2+C_{30}{\left({\overline{I}}_1-3\right)}^3+\frac{1}{D}{\left(J-1\right)}^2$$
   6. 13 = Neo-Hook with temperature: The energy density is then written:(7)
      $$W_0=\frac{\mu \left(T\right)}{2}\left({\overline{I}}_1-3\right)+\frac{K\left(T\right)}{2}{\left(J-1\right)}^2$$
   and the energy density for each secondary network: (8)

   $$W_i=S_i{W}_0$$
   Then, the total energy density for the secondary network is (9)

   $$W={\displaystyle \sum _{i=0}^N{W}_i}$$
   Note: (10)

   $${\displaystyle \sum _{i=0}^N{S}_i}=1$$

   (11)

   $${\overline{I}}_1={\overline{\lambda }}_1^2+{\overline{\lambda }}_2^2+{\overline{\lambda }}_3^2$$

   (12)

   $${\overline{I}}_2={\overline{\lambda }}_1^{-2}+{\overline{\lambda }}_2^{-2}+{\overline{\lambda }}_3^{-2}$$

   (13)

   $${\overline{\lambda }}_i=J^{-\frac{1}{3}}{\lambda }_i$$

   The Cauchy stress is computed as:(14)

   $${\sigma }_i=\frac{{\lambda }_i}{J}\frac{\partial W}{\partial {\lambda }_i}$$
5. The networkID must be left justified, and the name must use the form "NETWORK_i"

   Where, i is the networkID. Other names like "network1" or "NET1" are not allowed.

6. Polynomial form:
   1. The initial shear modulus and the bulk modulus are computed as:(15)
      $$G=2\left({\displaystyle \sum S_i}+1\right)\left(C_{10}+C_{01}\right)$$
      and (16)
      $$K=\frac{2}{D_1}\left(1+{\displaystyle \sum S_i}\right)$$
   2. If D₁ = 0, an incompressible material is considered.
7. The effective creep strain rate
   1. For Bergstrom Boyce viscous model, the expression is:(17)
      $${\dot{\epsilon }}_B^v=A_1{\left(\tilde{\lambda }-1+\xi \right)}^C{\left(\frac{{\overline{\sigma }}_B}{{\tau }_{ref}}\right)}^M$$
      Where, (18)
      $$\tilde{\lambda }=\sqrt{\frac{{\overline{I}}_1}{3}}$$
   2. For Hyperbolic sine viscous model, the expression is:(19)
      $${\dot{\epsilon }}_B^v=A_2{\left(\sinh B\overline{\sigma }\right)}^{n_2}$$
   3. For Power law viscous model, the expression is:
      (20)

      $${\dot{\epsilon }}_B^v=A_3{\left\{{\overline{\sigma }}^{n_3}{\left[\left(M_3+1\right){\epsilon }^v\right]}^{M_3}\right\}}^{\frac{1}{M_3+1}}$$
      Flow rule for equilibrium network:(21)

      $${\dot{\epsilon }}_{cr}=A_{pl}{\left(\frac{\overline{\sigma }}{\widehat{\sigma }}\right)}^{n_{pl}}$$

      (22)

      $$\widehat{\sigma }={\widehat{\sigma }}_0\left[f_f+\left(1-f_f\right)\exp \left(\frac{-{\epsilon }_{cr}}{\widehat{\epsilon }}\right)\right]$$