/MAT/LAW92

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.