1 Introduction
2 Hill48 Yield Function (Using the Associated Flow Rule)
3 Determination of the Parameters of Hill48 Yield Function
3.1 General Formula for Yield Stress and Anisotropy along Different Directions
3.2 Solution of Regional Parameters
3.2.1 Yield Stress and Anisotropy Index Along Different Directions
3.2.2 Calculation of the Parameters of Hill48 Yield Function for Each Region
4 Performance of Different Yield Functions in Predicting Earing
4.1 Solution for Stress and Strain
Yield stress ratio | ||||
---|---|---|---|---|
\( {{\sigma_{0} } \mathord{\left/ {\vphantom {{\sigma_{0} } {\sigma_{0} }}} \right. \kern-0pt} {\sigma_{0} }} \)
|
\( {{\sigma_{22.5} } \mathord{\left/ {\vphantom {{\sigma_{22.5} } {\sigma_{0} }}} \right. \kern-0pt} {\sigma_{0} }} \)
|
\( {{\sigma_{45} } \mathord{\left/ {\vphantom {{\sigma_{45} } {\sigma_{0} }}} \right. \kern-0pt} {\sigma_{0} }} \)
|
\( {{\sigma_{67.5} } \mathord{\left/ {\vphantom {{\sigma_{67.5} } {\sigma_{0} }}} \right. \kern-0pt} {\sigma_{0} }} \)
|
\( {{\sigma_{90} } \mathord{\left/ {\vphantom {{\sigma_{90} } {\sigma_{0} }}} \right. \kern-0pt} {\sigma_{0} }} \)
|
1 | 0.9775 | 0.9550 | 0.9595 | 0.964 |
Anisotropy index | ||||
---|---|---|---|---|
r
0
|
r
22.5
|
r
45
|
r
67.5
|
r
90
|
0.7031 | 0.6089 | 0.6947 | 0.7207 | 0.8466 |
Material constants | Initial yield stress |
q
|
b
|
---|---|---|---|
Value | 99.11 | 180.79 | 14.45 |
4.2 Solution for Anisotropic Parameters
Region | Anisotropy parameters | |||
---|---|---|---|---|
F
|
G
|
H
|
N
| |
0°‒22.5° | 1.0195 | 0.5872 | 0.4128 | 1.5620 |
22.5°‒45° | 0.6421 | 0.6519 | 0.3427 | 1.5459 |
45°‒67.5° | 0.6247 | 0.6693 | 0.4604 | 1.5459 |
67.5°‒90° | 0.5827 | 0.9140 | 0.4933 | 1.5815 |
Solving area |
\( R_{11} \)
|
\( R_{22} \)
|
\( R_{33} \)
|
\( R_{12} \)
|
---|---|---|---|---|
0°‒22.5° | 1 | 0.8356 | 0.7889 | 0.9800 |
22.5°‒45° | 1.0027 | 1.0077 | 0.8791 | 0.9850 |
45°‒67.5° | 0.9408 | 0.9600 | 0.8791 | 0.9850 |
67.5°‒90° | 0.8430 | 0.9640 | 0.8174 | 0.9739 |
4.3 FE Simulation of Deep Drawing Test
Blank diameter (Db) | Blank thickness (t) | Die diameter (Dd) | Die radius (rd) | Punch diameter (Dp) | Punch radius (rp) |
---|---|---|---|---|---|
90 | 1.0 | 53.64 | 13.0 | 50 | 5.0 |