/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   Fsmooth               F_cut
                 100                   5         1         0                   0
#            EPS_max              EPS_t1              EPS_t2
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
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Comments

1. The yield stress can be defined by the three stress coefficients (a, $b$ , and $n$ ), a function fct_ID_y, or a combination of both. The stress is then scaled by the Cowper-Symonds strain rate coefficient.
   • If fct_ID_y is defined (> 0), a=0 and VP=1:(1)
     $$\sigma =fct\_I{D}_y*Fscal{e}_y+\left(a+b{\epsilon }_p{}^n\right){\left(\frac{\dot{\epsilon }}{c}\right)}^{\frac{1}{p}}$$
   • If fct_ID_y is defined (> 0) and a > 0:(2)
     $$\sigma =fct\_I{D}_y*Fscal{e}_y*\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$$
   • If fct_ID_y is not defined (= 0):(3)
     $$\sigma =\left(a+b{\epsilon }_p{}^n\right)\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$$
   Where,

   ${\epsilon }_p$

   Plastic strain.

   $\dot{\epsilon }$

   Plastic strain rate for VP =1.

   Total strain rate for VP =2.

   Deviatoric strain rate for VP =3.

2. The law is compatible with truss, beam, shell, and solid elements.
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, with the following:
   • If VP = 1, the strain-rate filtering is set by default and the cutoff frequency is automatically computed by Radioss according to time step value. F_cut and F_smooth are ignored.
   • If VP = 2 or 3, and:
     • F_smooth = 0 + F_cut = 0.0, the strain-rate filtering is turned off;
     • F_smooth = 1 + F_cut = 0.0, the strain-rate filtering uses a cutoff frequency which is automatically computed by Radioss according to time step value (as for VP = 1);
     • F_cut ≠ 0, F_smooth is automatically set to 1 and the strain-rate filtering uses the cutoff frequency provided by the user.
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. When ${\epsilon }_p$ reaches ${\epsilon }_p^{max}$ in one integration point, then based on the element type:
   • Truss and Beam elements: The element is deleted
   • 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
8. If ${\epsilon }_1>{\epsilon }_{t1}$ ( ${\epsilon }_1$ is the largest principal strain), the stress is reduced as:(4)
   $${\sigma }_{n+1}={\sigma }_n\left(\frac{{\epsilon }_{t2}-{\epsilon }_1}{{\epsilon }_{t2}-{\epsilon }_{t1}}\right)$$
9. If ${\epsilon }_1>{\epsilon }_{t2}$ , the stress is reduced to 0 (but the element is not deleted).