/MAT/LAW92

A stress vs 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.

Format

Definitions

Example (Rubber with Parameter Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
Generic RUBBER
#              RHO_I
            1.000E-9 
#                 mu                   D                 LAM           
          2.8000E+01           1.4000E-1               1000.                 
#    IType    fct_ID                  NU              Fscale

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example (Rubber with Function Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
rubber
#              RHO_I
            1.000E-9 
#                 mu                   D                 LAM           
                 
#    IType    fct_ID                  NU              Fscale
         1         2               0.495
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
LAW92  e.strain vs  e.stress from uniaxial test(IType=1) 
#                  X                   Y
                   0                   0                                                            
                 .03                 .30                                                           
                 .06                 .55                                                            
                 .10                 .80
                 .20                 1.4
                 .30                 2.0
                 .50                 2.7
                 .70                 3.4
                 1.0                 4.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

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

   $$c_1=\frac{1}{2},c_2=\frac{1}{20},c_3=\frac{11}{1050},c_4=\frac{19}{7000},c_5=\frac{519}{673750}$$
   and (3)

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

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

   The Cauchy stress is computed as:(4)

   $${\sigma }_i=\frac{{\lambda }_i}{\text{J}}\frac{\partial W}{\partial {\lambda }_i}$$
2. If the stress strain curve fct_ID then the material parameters in line 3, $\mu$ , D and ${\lambda }_m$ must be defined and the line 4 input is not used. Poisson’s ratio is then calculated from the input as:(5)
   $$\nu =\frac{3K-2\mu }{6K+2\mu }$$
   The bulk modulus is calculated as:(6)

   $$K=\frac{2}{D}$$

   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 vs 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.(7)

   $$E={{\displaystyle \sum _{k=1}^{ndata}\left(\frac{N_k^{test}-N_k^{th}}{N_k^{test}}\right)}}^2$$
   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.(8)

   $${\lambda }_1={\lambda }_2=\lambda \text{and}{\lambda }_3={\lambda }^{-2}\text{with}\lambda =1+\epsilon$$
   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.(9)

   $$N_k^{th}=\frac{\partial W}{\partial {\lambda }_k}$$
   The nominal stress is computed for each mode assuming the full incompressibility:(10)

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

   • Uniaxial Mode:
     (11)

     $${\lambda }_1=\lambda \text{and}{\lambda }_2={\lambda }_3={\lambda }^{-\frac{1}{2}}\text{with}\lambda =1+\epsilon$$
     So(12)

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-2}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1={\lambda }^2+\frac{2}{\lambda }$$
   • Equibiaxial Mode:
     (13)

     $${\lambda }_1={\lambda }_2=\lambda \text{and}{\lambda }_3={\lambda }^{-2}\text{with}\lambda =1+\epsilon$$
     So(14)

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-5}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1=2{\lambda }^2+\frac{1}{{\lambda }^4}$$
   • Planar (Shear Mode):
     (15)

     $${\lambda }_1=\lambda ,{\lambda }_3=1\text{and}{\lambda }_2={\lambda }^{-1}\text{}\text{with}\lambda =1+\epsilon$$
     So (16)

     $$N^{\mathit{th}}=\frac{\partial W}{\partial \lambda }=2\mu (\lambda -{\lambda }^{-3}){\displaystyle \sum _{i=1}^5\frac{{\mathit{ic}}_i}{{({\lambda }_m)}^{2i-2}}}{\overline{I}}_1^{i-1}\text{with}{\overline{I}}_1={\lambda }^2+1+{\lambda }^{-2}$$
4. /VISC/PRONY must be used with LAW92 to include viscous effects.