/MAT/LAW44 (COWPER)
Block Format Keyword The CowperSymonds law models an elastoplastic material. The basic principle is the same as the standard JohnsonCook 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  C_{hard}_{}  ${\sigma}_{\mathrm{max}\text{}0}$  
c  p  ICC  F_{smooth}  F_{cut}  VP  
${\epsilon}_{p}^{max}$  ${\epsilon}_{t1}$  ${\epsilon}_{t2}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

fct_ID_{y}  Fscale_{y} 
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) 

C_{hard}  Plasticity
Isokinematic hardening factor.
Default = 0.0 (Real) 

${\sigma}_{\mathrm{max}\text{}0}$  Plasticity maximum
stress. Default = 10^{20} (Real) 
$\left[\text{Pa}\right]$ 
c  Strain rate coefficient.
(Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
p  Strain rate
exponent. Default = 1.0 (Real) 

ICC  Strain rate
computation flag. 6
(Integer) 

F_{smooth}  Smooth strain rate
option flag.
(Integer) 

F_{cut}  Cutoff frequency
for strain rate filtering. Default = 10^{30} (Real) 
$\text{[Hz]}$ 
VP  Formulation for
rate effects.
(Integer) 

${\epsilon}_{p}^{max}$  Failure plastic
strain. Default = 10^{20} (Real) 

${\epsilon}_{t1}$  Tensile failure
strain 1. Default = 10^{20} (Real) 

${\epsilon}_{t2}$  Tensile failure
strain 2. Default = 2x10^{20} (Real) 

fct_ID_{y}  Yield stress
function identifier. (Integer) 

Fscale_{y}  Scale factor for
ordinate (stress) in
fct_ID_{y} Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Example (Metal)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/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
#12345678910
#ENDDATA
/END
#12345678910
Comments
 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 CowperSymonds 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.
 If fct_ID_{y} is
defined (> 0), a=0 and
VP=1:
 The law is compatible with truss, beam, shell, and solid elements.
 Yield stress should be strictly positive.
 The hardening exponent n must be less than 1.
 The strain rate filtering
is used to smooth strain rates, with the following:
 If VP = 1, the strainrate 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 strainrate filtering is turned off;
 F_{smooth} = 1 + F_{cut} = 0.0, the strainrate 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 strainrate filtering uses the cutoff frequency provided by the user.
 ICC is a flag of the strain rate effect on material maximum stress
${\sigma}_{\mathrm{max}}$
:
$\sigma ={\sigma}_{y}\left(1+{\left(\frac{\dot{\epsilon}}{c}\right)}^{1/p}\right)$ $\sigma ={\sigma}_{y}\left(1+{\left(\frac{\dot{\epsilon}}{c}\right)}^{1/p}\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}0}\left(1+{\left(\frac{\dot{\epsilon}}{c}\right)}^{1/p}\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}0}$  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
 If
${\epsilon}_{1}>{\epsilon}_{t1}$
(
${\epsilon}_{1}$
is the largest principal strain), the stress is reduced
as:
(4) $${\sigma}_{n+1}={\sigma}_{n}\left(\frac{{\epsilon}_{t2}{\epsilon}_{1}}{{\epsilon}_{t2}{\epsilon}_{t1}}\right)$$  If ${\epsilon}_{1}>{\epsilon}_{t2}$ , the stress is reduced to 0 (but the element is not deleted).