/MAT/LAW92

A stress versus strain curve can be defined as an input function in order to determine the material parameters by fitting this curve. This law is only compatible with solid elements.

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1. The Arruda-Boyce energy density.
   $$W=\mu {\displaystyle \sum _{i=1}^5\frac{c_i}{{({\lambda }_m)}^{2i-2}}}\left({\overline{I}}_1^i-3^i\right)+\frac{1}{D}\left(\frac{J^2-1}{2}-\ln (J)\right)$$

   With

   $$c_1=\frac{1}{2},c_2=\frac{1}{20},c_3=\frac{11}{1050},c_4=\frac{19}{7000},c_5=\frac{519}{673750}$$

   and

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

   with ${\overline{\lambda }}_k=J^{-1/3}\lambda J={\lambda }_1{\lambda }_2{\lambda }_3$

   The Cauchy stress is computed as:

   $${\sigma }_i=\frac{{\lambda }_i}{\text{J}}\frac{\partial W}{\partial {\lambda }_i}$$

2. If the stress strain curve, fct_ID is not defined, the material parameters in line 3, $\mu$ , $D$ and ${\lambda }_m$ must be defined and the line 4 input is not used.

   Ground shear modulus is computed as:

   $${\mu }_0=\mu \left(1+\frac{3}{5}\left(\frac{1}{{\lambda }_m^2}\right)+\frac{99}{175}\left(\frac{1}{{\lambda }_m^4}\right)+\frac{513}{875}\left(\frac{1}{{\lambda }_m^6}\right)+\frac{42039}{67375}\left(\frac{1}{{\lambda }_m^8}\right)\right)$$

   If $D$ is not defined, the bulk modulus is calculated as:

   $$K=\frac{2(1+\upsilon ){\mu }_0}{3\left(1-2\upsilon \right)}$$

   Where, $\upsilon =0.495$ and $D=\frac{2}{K}$ .

   If $D$ is defined, the formula should be $K=\frac{2}{D}$ , and the Poisson ratio is updated with $\upsilon =\frac{3K-2{\mu }_0}{3}$ .

   Note: For positive values of shear modulus, $\mu$ , and Limit of stretch, ${\lambda }_m$ , this model is always stable.
3. If the stress strain curve, fct_ID, is defined then the line 3 input parameters $\mu$ , $D$ and ${\lambda }_m$ are ignored and are automatically identified by fitting of the provided stress versus strain curve.

   A nonlinear least squares algorithm is used to fit the Arruda-Boyce parameters. The model is fully incompressible in fitting the Arruda-Boyce constants to the test data, except in the volumetric test.

   $$E={{\displaystyle \sum _{k=1}^{ndata}\left(\frac{N_k^{test}-N_k^{th}}{N_k^{test}}\right)}}^2$$

   Where E is relative error. The material constants are obtained using a least-squares-fit procedure to minimize the relative error between the theoretical nominal stress and given experimental data.

   $${\lambda }_1={\lambda }_2=\lambda \text{and}{\lambda }_3={\lambda }^{-2}\text{with}\lambda =1+\epsilon$$

   Where, $N_k^{\mathit{test}}$ is a stress value from the test data and $N_i^{\mathit{th}}$ is the theoretical nominal stress given by for each engineer strain i.

   $$N_k^{th}=\frac{\partial W}{\partial {\lambda }_k}$$
   The nominal stress is computed for each mode assuming the full incompressibility:

   $$J={\lambda }_1{\lambda }_2{\lambda }_3=1$$

   • Uniaxial Mode:
     $${\lambda }_1=\lambda \text{and}{\lambda }_2={\lambda }_3={\lambda }^{-\frac{1}{2}}\text{with}\lambda =1+\epsilon$$

     So

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-2}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1={\lambda }^2+\frac{2}{\lambda }$$

   • Equibiaxial Mode:
     $${\lambda }_1={\lambda }_2=\lambda \text{and}{\lambda }_3={\lambda }^{-2}\text{with}\lambda =1+\epsilon$$

     So

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-5}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1=2{\lambda }^2+\frac{1}{{\lambda }^4}$$

   • Planar (Shear Mode):
     $${\lambda }_1=\lambda ,{\lambda }_3=1\text{and}{\lambda }_2={\lambda }^{-1}\text{}\text{with}\lambda =1+\epsilon$$

     So

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-3}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1={\lambda }^2+1+{\lambda }^{-2}$$

4. /VISC/PRONY must be used with LAW92 to include viscous effects.