# /MAT/LAW44 (COWPER)

Block Format Keyword The Cowper-Symonds law models an elasto-plastic material. The basic principle is the same as the standard Johnson-Cook model; the only difference between the two laws lies in the expression for strain rate effect on flow stress.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW44/mat_ID/unit_ID or /MAT/COWPER/mat_ID/unit_ID
mat_title
${\rho }_{i}$
E $\nu$
a b n Chard ${\sigma }_{\mathrm{max}\text{​}0}$
c p ICC Fsmooth Fcut   VP
${\epsilon }_{p}^{max}$ ${\epsilon }_{t1}$ ${\epsilon }_{t2}$
Optional line for defining a yield stress function
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
fct_IDy   Fscaley

## Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Unit Identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

${\rho }_{i}$ Initial density.

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
E Young's modulus.

(Real)

$\left[\text{Pa}\right]$
$\nu$ Poisson's ratio.

(Real)

a Plasticity yield stress.

(Real)

$\left[\text{Pa}\right]$
b Plasticity hardening parameter.

(Real)

$\left[\text{Pa}\right]$
n Plasticity hardening exponent.

Default = 1.0 (Real)

Chard Plasticity Iso-kinematic hardening factor.
= 0
Hardening is full isotropic model.
= 1
Hardening uses the kinematic Prager-Ziegler model.
= between 0 and 1
Hardening is interpolated between the two models.

Default = 0.0 (Real)

${\sigma }_{\mathrm{max}\text{​}0}$ Plasticity maximum stress.

Default = 1020 (Real)

$\left[\text{Pa}\right]$
c Strain rate coefficient.
= 0 (Default)
No strain rate effect.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
p Strain rate exponent.

Default = 1.0 (Real)

ICC Strain rate computation flag. 6
= 0 (Default)
Set to 1.
= 1
Strain rate effect on ${\sigma }_{\mathrm{max}}$ .
= 2
No strain rate effect on ${\sigma }_{\mathrm{max}}$ .

(Integer)

Fsmooth Smooth strain rate option flag.
= 0 (Default)
No strain rate smoothing.
= 1
Strain rate smoothing active.

(Integer)

Fcut Cutoff frequency for strain rate filtering.

Default = 1030 (Real)

$\text{[Hz]}$
VP Formulation for rate effects.
= 0
Set to 2.
= 1
Plastic strain rate.
= 2 (Default)
Total strain rate.
= 3
Deviatoric strain rate.

(Integer)

${\epsilon }_{p}^{max}$ Failure plastic strain.

Default = 1020 (Real)

${\epsilon }_{t1}$ Tensile failure strain 1.

Default = 1020 (Real)

${\epsilon }_{t2}$ Tensile failure strain 2.

Default = 2x1020 (Real)

fct_IDy Yield stress function identifier.

(Integer)

Fscaley Scale factor for ordinate (stress) in fct_IDy

Default = 1.0 (Real)

$\left[\text{Pa}\right]$

## Example (Metal)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. The yield stress can be defined by the three stress coefficients (a, $b$ , and $n$ ), a function fct_IDy, or a combination of both. The stress is then scaled by the Cowper-Symonds strain rate coefficient.
• If fct_IDy is defined (> 0), a=0 and VP=1:(1)
$\sigma =fct_I{D}_{y}*Fscal{e}_{y}+\left(a+b{\epsilon }_{p}{}^{n}\right){\left(\frac{\stackrel{˙}{\epsilon }}{c}\right)}^{\frac{1}{p}}$
• If fct_IDy is defined (> 0) and a > 0:(2)
$\sigma =fct_I{D}_{y}*Fscal{e}_{y}*\left(1+{\left(\frac{\stackrel{˙}{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$
• If fct_IDy is not defined (= 0):(3)
$\sigma =\left(a+b{\epsilon }_{p}{}^{n}\right)\left(1+{\left(\frac{\stackrel{˙}{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$
Where,
${\epsilon }_{p}$
Plastic strain.
$\stackrel{˙}{\epsilon }$
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.
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. Fcut and Fsmooth are ignored.
• If VP = 2 or 3, and:
• Fsmooth = 0 + Fcut = 0.0, the strain-rate filtering is turned off;
• Fsmooth = 1 + Fcut = 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);
• Fcut0, Fsmooth 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 ${\sigma }_{\mathrm{max}}$ :
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).