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

mailto:[email protected]

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


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




(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.




(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




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:



(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)


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.



(11.16) 0

0

/ln

/ln

strain thickness

strainwidth alue-r

tt

wwv

t

w


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:



(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)


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.



(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)


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.



(11.27)

(11.28)

F

H

F

H

d

dv

z

x

t

w

alue-r

F

Hr 90yr


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:




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:




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.



(11.35)

(11.36)

z

2/

d

dr

zd 2/d

2x cos

2y sin

cossinxyτ(11.37)

(11.38)


11. plastic anisotropy e-mail: assoc.prof.dr. ahmet zafer Şenalp e-mail:...

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