20
4.5 Constitutive Modeling
It is well-known that the dynamic response of material largely depends upon the deformation rate, temperature, and plastic
strain. In order to describe the flow behavior of a material, different constitutive models are proposed by various researchers
incorporating the effect of work hardening, temperature, and strain rate. The most famous and extensively used constitutive
model to estimate the behavior of elastoplastic structure is Johnson-Cook material model (Eq. 4.3). This model includes the
effect of isotropic strain hardening, strain rate hardening, and thermal softening. The JC model expressed as
s
e
e
e
=
+
é ë
ù û * +
æ
è
ç
ö
ø
÷
æ
è
ç ç
ö
ø
÷ ÷
* -
-
-
A B
C
T T
T T
n
pl
pl
pl
ref
M
r ef
1
1
0
ln


æ æ
è
ç
ö
ø
÷
é
ë
ê
ê
ù
û
ú
ú
m
(4.3)
where σ is the flow stress,ε pl is the equivalent plastic strain, 
e
pl is the current plastic strain rate, and 
e 0
pl is the reference
plastic strain rate, T is the current temperature, T ref (25 °C) is the reference temperature, and T M (625 °C) is the melting temperature. The parameter of the JC model A, B, n, C, and m are evaluated from the experimental results. A is the yield stress
-200
0
2 00
400
600
800 1000 1200 1400
1.00
1.02
1.04
1.06
1.08
1.10
1.12
Ud / Us
Yd / Ys
Strain Rate (S
-1
)
Yd /
Ys
1.00
1.05
1.10
1.15
1.20
1.25
1.30
Ud /
Us
s
s
s
s s
s
s
s
Fig. 4.5 Variation of
(σ YT /σ YRT ) and (σ UT /σ URT ) With
temperature σ YRT , σ URT are
yield and ultimate strength at
room temperature. σ YT , σ UT are
yield and ultimate strength at
T temperature
-200
0
200
400
600
800
1000 1200 1400
10
12
14
16
18
20
f (%)
X (%)
Strain Rate (S
-1
)
f
)
%
(
30
32
34
36
38
40
X (%)
e
e
Fig. 4.6 Variation of ε f (%)
and area reduction X (%) with
temperature
P. Chakraborty et al.