11. Plastic Anisotropy e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail:...
-
Upload
geraldine-henderson -
Category
Documents
-
view
221 -
download
0
Transcript of 11. Plastic Anisotropy e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail:...
11. Plastic Anisotropy11. Plastic Anisotropy
Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: e-mail: [email protected]@gmail.com
Mechanical Engineering DepartmentGebze Technical University
ME 612ME 612 Metal Forming and Theory of Plasticity Metal Forming and Theory of Plasticity
If a material at different directions of the coordinate system attached to a point shows different properties then the material is said to be anisotropic.
The reason for the anisotropy is the mechanic or thermal operations applied to the metal. Especially anisotropy is seen in rolling operation, in the rolling direction.
There are a lot of studies performed on anisotropic material’s yield criteria. Hill determined the below yield criteria function:
Here F, G, H, L, M and N are the characteristic parameters determining the anisotropy. In case the material is isotropic
and when these are placed into Hill’s equation the new equation turns to be the same form as von-Mises equation which is valid for isotropic materials.
Dr. Ahmet Zafer Şenalp ME 612
2Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
1N2M2L2HGFf2 2222yx
2xz
2zyij
xyzxyz (11.1)
H3G3F3NML (11.2)
If tensile strengths in principal anisotropic directions are defined as X,Y,Z :
Figure 11.1. Principal anisotropy directions
If these equalities are solved for
Dr. Ahmet Zafer Şenalp ME 612
3Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.3)
(11.4)
HGX
12
FHY
12
GFZ
12
(11.5)
is obtained and X, Y, Z values determined by experiment are placed in the above equations to solve for F, G and H parameters. Unfortunately, it is not easy to measure Z value for sheet materials.L, M and N values are obtained from shear experiments.
Dr. Ahmet Zafer Şenalp ME 612
4Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.6)
(11.7)
(11.8)
222 X
1
Z
1
Y
1F2
222 Z
1
Y
1
X
1H2
222 Y
1
X
1
Z
1G2
Dr. Ahmet Zafer Şenalp ME 612
5Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
Figure 11.2. Equivalent stress-equivalent strain curves obtained in X, Y and Z directions
Material yield rule can be derived by using:
Here yield function. According to this yield criteria by taking the derivative of Eq (11.1) the following
relations for anisotropic materials are obtained:
6Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.9)
(11.11)
(11.10)
ij
ijij
fdd
ijf
zxyxx GHdd
xyzyy HFdd
yzxzz FGdd
yzyz Ldd
zxzx Mdd
xyxy Ndd
(11.12)
(11.13)
(11.14)
(11.15)
Dr. Ahmet Zafer Şenalp ME 612
Like Levi-Mises equations these equations are used by taking ratios. There is an r-value that is used to determine material’s anisotropy state which is defined as:
Generally; For steels r >1For aluminum r < 1For copper nearly 0.99
High r-value shows that yield strength in thickness direction is high.
7Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.16) 0
0
/ln
/ln
strain thickness
strainwidth alue-r
tt
wwv
t
w
Dr. Ahmet Zafer Şenalp ME 612
A test specimen cut in X direction and related to the experiment performed with this test specimen:
is valid.If these values are placed into the stress-strain relations valid for anisotropic materials:
is obtained. r-vaue is:
8Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.17)
x 0zy
xxxx )HG(d0G0Hdd
xxy Hd0H00Fdd
xxz Gd00F0Gdd
G:H:HGd:d:d zyx
(11.18)
(11.19)
(11.20)
G
H
G
H
d
dv
z
y
t
w
alue-r
(11.21)
Dr. Ahmet Zafer Şenalp ME 612
The r-value obtained from test in X direction is called rx or r0.Here 0 index is the angle of test specimen that makes with x axis.
For a test specimen cut in Y direction and the experiment with this specimen:
is valid.
If these values are placed into the stress-strain relations valid for anisotropic materials:
is obtained.
9Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.22)
(11.25)
(11.23)
(11.24)
G
Hrr 0x
y 0zx
yyx Hd00G0Hdd
yyyy )HF(d0H0Fdd
yyz Fd0F00Gdd F:HF:Hd:d:d zyx (11.26)
Dr. Ahmet Zafer Şenalp ME 612
r-value is:
The r-value obtained with the result of test in Y is called as ry or r90.
Here 90 index is the angle that specimen makes with X axis.
Sheet metal rolling direction is generally anisotropy direction and x axis is handled as rolling direction.
10Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.27)
(11.28)
F
H
F
H
d
dv
z
x
t
w
alue-r
F
Hr 90yr
Dr. Ahmet Zafer Şenalp ME 612
Hill proposed that the equivalent stress should be defined as:
in terms of principal stresses:
Equivalent strain in terms of principal strains can be written as:
Dr. Ahmet Zafer Şenalp ME 612
11Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
21
222222 222
2
3
HGF
NMLHGF yxxzzy xyzxyz
(11.29)
(11.30) 2
12
232
312
12
2
3
HGF
HGF
2
12
23
2
31
2
1221
3
2
HFGHFG
GdFdH
HFGHFG
FdHdG
HFGHFG
HdGdFHGFd
(11.31)
For a sheet material subjected to plane stress, with rotational symmetry about z-axis, so that:
Than the equivalent stress:
Equivalent strain in terms of principal strains can be written as:
Dr. Ahmet Zafer Şenalp ME 612
12Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
F
H
G
Hr (11.32)
(11.33) 2
1222
22
3
r
r yxyx
21
222
2)21(
2
3
2
yxzxzy ddrrddrddr
rd (11.34)
If a test specimen is cut in X-Y plane making an angle of with x axis the r-value for this case can be written and calculated as:
and before calculation of the below terms should be placed in stress-strain relations:
Here is applied stress.
13Mechanical Engineering Department, GTU
11. Plastic Anisotropy11. Plastic Anisotropy
(11.35)
(11.36)
z
2/
d
dr
zd 2/d
2x cos
2y sin
cossinxyτ(11.37)
(11.38)
Dr. Ahmet Zafer Şenalp ME 612