1. 11. Plastic Anisotropy e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail.comazsenalp@gmail.com Mechanical Engineering Department Gebze Technical University ME 612 Metal Forming and Theory of Plasticity

2. 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 Anisotropy (11.1) (11.2)

3. 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 Anisotropy (11.3) (11.4) (11.5)

4. 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 Anisotropy (11.6) (11.7) (11.8)

5. Dr. Ahmet Zafer Şenalp ME 612 5Mechanical Engineering Department, GTU 11. Plastic Anisotropy Figure 11.2. Equivalent stress-equivalent strain curves obtained in X, Y and Z directions

6. 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 Anisotropy (11.9) (11.11) (11.10) (11.12) (11.13) (11.14) (11.15) Dr. Ahmet Zafer Şenalp ME 612

7. 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 >1 For aluminum r < 1 For copper nearly 0.99 High r-value shows that yield strength in thickness direction is high. 7Mechanical Engineering Department, GTU 11. Plastic Anisotropy (11.16) Dr. Ahmet Zafer Şenalp ME 612

8. 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 Anisotropy (11.17) (11.18) (11.19) (11.20) (11.21) Dr. Ahmet Zafer Şenalp ME 612

9. The r-value obtained from test in X direction is called r x or r 0. 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 Anisotropy (11.22) (11.25) (11.23) (11.24) (11.26) Dr. Ahmet Zafer Şenalp ME 612

10. r-value is: The r-value obtained with the result of test in Y is called as r y or r 90. 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 Anisotropy (11.27) (11.28) Dr. Ahmet Zafer Şenalp ME 612

11. 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 Anisotropy (11.29) (11.30) (11.31)

12. 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 Anisotropy (11.32) (11.33) (11.34)

13. 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 Anisotropy (11.35) (11.36) (11.37) (11.38) Dr. Ahmet Zafer Şenalp ME 612