/MAT/LAW44 (COWPER)

Format

Definitions

Example (Metal)

#RADIOSS STARTER
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/UNIT/1
unit for mat
                  g                  mm                  ms
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#-  2. MATERIALS:
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/MAT/COWPER/1/1
metal
#              RHO_I
               .0078
#                  E                  NU
               20500                  .3
#                  a                   b                   n              C_hard          SIGMA_max0
                  50                 100                  .5                   1                  90
#                  c                   p       ICC  F_smooth               F_cut
                 100                   5         1         0                   0
#            EPS_max              EPS_t1              EPS_t2
                   0                   0                   0
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#ENDDATA
/END
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Comments

1. The difference between the Cowper-Symonds law and the standard Johnson-Cook model lies only in the strain rate dependent formulation:(1)
   $$\sigma =\left(a+b{\epsilon }_p{}^n\right)\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$$
   Where,

   ${\epsilon }_p$

   Plastic strain

   $\dot{\epsilon }$

   Strain rate

2. The law is only defined for solids and shells. The global plasticity option for shells is not available in the actual version.
3. Yield stress should be strictly positive.
4. The hardening exponent n must be less than 1.

   
   
   Figure 1.

5. The strain rate filtering is used to smooth strain rates.
6. ICC is a flag of the strain rate effect on material maximum stress ${\sigma }_{\max }$ :

   
   
   

   $\sigma ={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{1/p}\right)$	$\sigma ={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{1/p}\right)$
   ${\sigma }_{\max }={\sigma }_{\max 0}\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{1/p}\right)$	${\sigma }_{\max }={\sigma }_{\max 0}$

   Figure 2.

7. Strain rate filtering input (F_cut) is only available for shell and solid elements.
8. When ${\epsilon }_p$ reaches ${\epsilon }_p^{max}$ in one integration point, then based on the element type:
   • Shell elements: The corresponding shell element is deleted.
   • Solid elements: The deviatoric stress of the corresponding integral point is permanently set to 0, however, the solid element is not deleted
9. If ${\epsilon }_1>{\epsilon }_{t1}$ ( ${\epsilon }_1$ is the largest principal strain), the stress is reduced as:(2)
   $${\sigma }_{n+1}={\sigma }_n\left(\frac{{\epsilon }_{t2}-{\epsilon }_1}{{\epsilon }_{t2}-{\epsilon }_{t1}}\right)$$
10. If ${\epsilon }_1>{\epsilon }_{t2}$ , the stress is reduced to 0 (but the element is not deleted).