TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007
Trang 49
THE SOFTENING IN PLASTIC DEFORMATION OF METAL
Truong Tich Thien
University of Technology, VNU-HCM
(Manuscript received on December14
th
, 2006; Manuscript received on June 28
h
, 2007)
ABSTRACT: In the plastic deformation stage of metal, work hardening always goes
together with softening. The nucleation, growth of small internal voids or cavities according to
plastic increasing is the microscopic mechanism of softening. The voids are nucleated at the
particle-matrix interface due to the agglutinate loss or the particle crack when the strain reaches
critical value. The growth of voids will then occur in company with the increase of plastic
deformation. The influence of void growth on material forming behaviour will be considered by
softening parameter of porous material.
Key words: hardening, constitutive softening, nucleation, growth, porous material.
1. INTRODUCTION
The nucleation, growth and coalescence of small internal voids or cavities are the
microscopic mechanism of ductile fracture in cold forming processes of metal. The microvoids
are nucleated under the tensile loading state at the impurities and hard particles in the ductile
metal. After nucleated microvoids, they will be grown due to plastic deformation and coalesced
together in order to create the microscopic ductile fracture when the critical state is reached. The
nucleation and growth of small internal voids or cavities are interpreted as the reason of the
strain softening of material. So, the strain hardening and the strain softening of material are two
phenomena occurring simultaneously during the plastic deformation of materials. At first the
strength of material increases since the strain hardening, but the material will be degraded due to
the growth of microvoids. This induces the strength degradation and the stress−strain
relationship will be shown by the curve with the negative slope.
The initial shapes of micro-voids are multiform and complex. On the other hand, their

()()
ab
eab
0eM
31 n
3
R
ln sinh
R21n 2 2
⎡⎤
−σ+σ
⎛⎞
εε+ε
=+
⎢⎥
⎜⎟
−σ
⎢⎥
⎝⎠
⎣⎦
(1)
For a material cell with several series of cylindrical voids (fig.4), interaction of neighbouring
void must be introduced in the model of void growth. At a uniform void distribution (initial void
distances), the growth parameters in radial direction a and b are defined according to 0
a
ca
0a

e
e2
2
.
.
T
T
h
h
e
ec
c
o
o
n
n
c
c
e
e
n
n
t
t
t
t
w
w
o
o
p
p
o
o
l
l
e
e
s
so
o
f
fh
h
a
a

u
u
r
r
e
e3
3
.
.T
T
y
y
p
p
e
e
s
so
o
f
f

.
.F
F
i
i
g
g
u
u
r
r
e
e1
1
.
.
T
T
y
y
p
p
e
e

s
s
u
u
l
l
p
p
h
h
i
i
d
d
e
ep
p
a
a
r
r
t
t
i
i
c
c

n
n
l
l
a
a
r
r
g
g
e
e4
4
1
1
5
5E
E
n
n
l
l
a
a

dlnF sinh d
n
⎧
⎡⎤
σ+σ
−
⎫
σ−σ
⎪
=+ε
⎢⎥
⎨⎬
−σσ
⎢⎥
⎭
⎪
⎣⎦
⎩
31
33
21 2 4
(3a)

()
()
()
()
ab
ab
cb e

Thus, an improvement of Mc.Clintock is necessary. Nguyen Luong Dung modified the
original Mc.Clintock model by adding a second model, fig.4c, to the original model, fig.4b, with

ab ab
aabb
and
∗∗
σ+σ σ+σ
⎛⎞ ⎛⎞
σ=− −σ σ=− −σ
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
22
(4)
The growth parameters in radial direction a and b of modified model are now defined according
to
()
*
a
ca a a
0
R
Fexp2
R
=ε−ε
(5a)

()
*
b

⎢⎥
⎪⎪
⎣⎦
⎩⎭
(6a)

()
()
()()
ab
ba
cb ecbe
eM eM
31 n
33
dlnF sinh d fd
21 n 2 4
⎧⎫
⎡⎤
−σ+σ
σ−σ
⎪⎪
=+ε=ε
⎢⎥
⎨⎬
−σσ
⎢⎥
⎪⎪
⎣⎦
⎩⎭

f
f
i
i
e
e
d
dm
m
o
o
d
d
e
e
l
l
.
.2
2
b
b
r
b
b
)
)
c
a
b
R
0

*
b
σ
*
a
σ
c
c
)
)= +
Science & Technology Development, Vol 10, No.06 - 2007

Trang 52
The other expressions for d(lnF
ab
) and d(lnF

abc bc
eM eM
31 n 2
2
33
d lnF sinh
21 n 2 4
31 n 2
2
33
sin h d
21 n 2 4
31 n 31 n
3
sinh cosh
1n 4 4
⎡
⎧⎫
⎡⎤
−σ+σ
σ−σ
⎪⎪
⎢
=++
⎢⎥
⎨⎬
−σσ
⎢
⎢⎥
⎪⎪

−σ σ
⎢
⎢⎥⎢⎥
⎪⎪
⎣⎦⎣⎦
⎩⎭
⎣
+
abc
eae
eM
3
dfd
4
⎤
σ−σ−σ
ε= ε
⎥
σ
⎦
(7a)
()
()
() ()
ijk jk
i
eM eM
ijk
eie
eM

Axis c
F
F
i
i
g
g
u
u
r
r
e
e6
6
.
.G
G
r
r
o
o
w
w
t


v
v
o
o
i
i
d
d
.
.

F
F
i
i
g
g
u
u
r
r
e
e5
5
.
.

r
r
i
i
a
a
l
l
.
.

b

a
2R
0

0
a
l
0
b
l
0
c
l
c
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007
Trang 53
where

The accumulated damage of the momentary semi-axis i due to void growth is given as

()
()
(
)
()
()
e
N
e
e
N
e
ijk
ii
eM
jk
ijk
eie
eM eM
31 n
3
AlnF sinh
1n 4
31 n
cosh d f d
4
ε
ε

⎪⎪⎪
⎩⎭
⎭
⎦
∫
∫
(8)
The influence of void growth on material forming behaviour was considered by softening
parameter of porous material σ
e
/σ
eM
. The general form of yield function for porous material is
given by TRUONG Tich Thien
()
2
ee
ij eM
eM eM
,,f BcoshA C0
⎛⎞ ⎛ ⎞
σσ
φσ σ = + − =
⎜⎟ ⎜ ⎟
σσ
⎝⎠ ⎝ ⎠
(9)
where σ
eM
is the equivalent stress of matrix material (no voids), the factors A, B, C depend

0
14
1
23
(11)
The process of micro ductile fracture prediction is shown in the flowchart of figure 7.

Science & Technology Development, Vol 10, No.06 - 2007

Trang 54

2.2. The finite element model
For the symmetry, the FEM analysis model only includes one-sixteenth of material cell
(fig.8).
F
F
i
i
g
g
u
u
r
r
e
e7
7

f
f
r
r
a
a
c
c
t
t
u
u
r
r
e
ep
p
r
r
e
e
d
d
i
i
c
c

i
i
o
o
n
n
.
.S
S
t
t
a
a
r
r
t
tN
o
Yes
I
I
n
n
p

e
r
r
i
i
a
a
l
lp
p
r
r
o
o
p
p
e
e
r
r
t
t
i
i
e
e
s

*L
L
o
o
a
a
d
ds
s
t
t
a
a
t
t
e
e
:
:S
S
1

a
i
i
n
ni
i
n
n
c
c
r
r
e
e
m
m
e
e
n
n
t
t
:
:Δ


q
q
c
c<
<l
l
n
n
F
F
m
m
a
a
x
xResult output:
•Text file
•Graphic file
Stop
°Define:

σf
f
=
=f
f
o
o
;
;q
q
c
c=
=0
0
;

a
x
x=
=β
β
l
l
n
n
3
0
14
23f
⎛⎞
π
⎜⎟
⎜⎟
⎝⎠TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 10, SỐ 06 - 2007
Trang 55
Figure 9a.Equivalent strain distribution o
f
model.
n = 0.2
o
f
0
= 0.1
Figure 9b. Equivalent stress distribution
of model.

n = 0.2
f
0
= 0.1
F
F
i
i
g
g
u
u
r
r
e
e8


M
M
a
a
t
t
e
e
r
r
i
i
a
a
l
lc
c
e
e
l
l
l
l1

e
l
l
l
lb
b
)
)F
F
E
E
M
Mm
m
e
e
s
s
h
h


e
Figure 11a. Material softening according to different yield functions in triaxial load.
0.80
0.85
0.90
0.95
1.00
0.0 0.1 0.1 0.2 0.2 0.3 0.3
Equi val e nt s tr ai n
Le mait re
Gu rs o n
Dung
Thien
Finite
ε
e
f
0
= 0.01
n = 0.2
σ
e
/ σ
eM
Figure 11b. Material softening according to different yield functions in high triaxial load.
0.2
0.4
0.6
0.8
1.0

Tri-
Axia-
lity
High
Tri-
Axia-
lity
Uni-
Axia-
lity
Tri-
Axia-
lity
High
Tri-
Axia-
lity
Uni-
Axia-
lity
Tri-
Axia-
lity
High
Tri-
Axia-
lity
Lemaitre 1.38 0.615 0.155 0.89 0.395 0.095 0.73 0.325 0.08
Gurson 1.39 0.65 0.405 0.89 0.46 0.35 0.73 0.4 0.33
Dung 1.39 0.66 0.42 0.91 0.49 0.425 0.75 0.45 0.425

hạt-mạng do mất đi sự dính kết hay do nứt hạt khi biến dạng dẻo đạt đến giá trị giới hạn. Tiếp
theo sự tăng trưởng của lỗ sẽ xảy ra trong điều kiện biền dạng dẻo gia tăng. Ảnh hưởng của sự
tăng trưởng lỗ hổng sẽ được khảo sát bởi thông số biền mềm của vậ
t liệu xốp.
Science & Technology Development, Vol 10, No.06 - 2007

Trang 58
REFERENCES
[1]. W. F. Chen, D. J. Han, Plasticity for Structure Engineers, Springer-Verlag- New York
- Berlin - Heidelberg - London - Paris – Tokyo, (1988).
[2]. B. Dodd and Y. Bai, Ductile Fracture and Ductility, Academic Press, London (1987).
[3]. E. Doege, H. M. Nolkemper and I. Saeed, Fliesskurvenatlas metallischer Werkstoffe,
Hanser Verlag, Muenchen (1986).
[4]. N. L. Dung, Fortschr Ber. VDI Reihe 2 Nr. 175, VDI-Verlag, Dusseldorf (1989).
[5]. T. T. Thien, The Model for Ductile Fracture Prediction in Metal Forming, Doctor
Thesis, Ho Chi Minh City University of Technology, (07/2001).
[6]. Truong Tich Thien, Vu Cong Hoa, A Process of Micro Ductile Fracture Prediction for
Metal, Proceedings of International Conference for Mechanical and Automotive
Technologies (ICMAT) 2005, Chonbuk National University, Korea, June 1 ~ 3, (2005).
[7]. M. J. Worswick and R. J. Pick, J. Mech. Phys. Solid 38, 601 (1990).