Influence of Bauschinger Effect and Anisotropy on Springback of Aluminum Alloy Sheets
2017 年 58 巻 6 号 p. 921-926
詳細
2017 年 58 巻 6 号 p. 921-926
For accurate springback calculations, the development of accurate constitutive equation “Yoshida-Uemori model” must always be taken into account. However, several springback calculations of other sheet metals by Yoshida-Uemori model have shown wrong agreements with the corresponding experimental data. The reason is why most of them have been calculated from the view point of the accuracy of Bauschinger effect without the strong anisotropy of sheet metal. In the present paper, we have investigated how the initial anisotropy affects the amount of springback for aluminum sheet metals with Bauschinger effect. Specifically, hat bending experiments in R.D. and T.D. were compared with the corresponding calculations. From the above mentioned comparisons, we found that the optimum combination of an anisotropic yield functions and Yoshida-Uemori model is very important for accurate springback analysis.
This Paper was Originally Published in Japanese in J. JILM 65 (2015) 582–587.
For the last decades, CAE systems, which are capable to reduce the cost and the number of steps in a manufacturing process dramatically, have been mainly introduced for the research and development divisions of press forming in the automobile industries. However, there are still plenty of problems with the press forming of aluminum alloy sheet metals. One of the problems is to predict precisely the extremely large springback of aluminum alloy sheet metals due to the low Young's modulus. Springback of the materials are also well-known as a serious failure reducing the work efficiency for press forming.
Taking account of the cyclic hardening and Bauschinger effect have been commonly recognized to predict springback precisely. Although the constitutive equation with the Bauschinger effect have been utilized in the press forming simulations, the springback still remains the difficult problem. One of the reasons is that the initial plastic anisotropy of the materials have been ignored. The plastic anisotropy of the aluminum alloy sheet material makes the mechanism of springback deformation more intricate. Since neither von Mises nor Hill' 481,2) can describe the plastic anisotropy of the materials very well, high order yield functions (Gotoh3,4) and Barlat et al5,6)) are needed for accurate simulations.
In the present research, the four kinds of anisotropic yield functions and the two types of hardening rules were introduced into the commercial finite element code to investigate how important it is to utilize an accurate kinematic hardening model incorporating with an appropriate choice of anisotropic yield function for the accurate springback calculations. Furthermore, we investigated how large springback deformations are predicted for aluminum alloy sheet metals in the U-bending and the hat bending processes.
The test piece utilized in the present research were A5052-O sheet metal with 1.2 mm thickness and AA6016-T4 sheet metal with 1.0 mm thickness. The mechanical properties (proof stress, equi bi-axial yield stress, and Lankford values) of these metals are listed in the Table 1.
| Material | Angle from rolling direction | ||||||
|---|---|---|---|---|---|---|---|
| 0° | 22.5° | 45° | 67.5° | 90° | biaxial | ||
| 0.2% proof stress/MPa |
A5052-O* | 166 | 169 | 166 | 163 | 163 | 166 |
| AA6016-T4# | 195 | 197 | 187 | 191 | 186 | 195 | |
| r -value | A5052-O¶ | 0.72 | 0.6 | 0.51 | 0.56 | 0.59 | — |
| AA6016-T4¶ | 0.76 | 0.4 | 0.26 | 0.33 | 0.61 | — | |
Superscripts *, # and ¶ indicate plastic strains of 0.04, 0.03 and 0.1, respectively, where stresses and/or r-values were obtained.
The schematic illustration of U-bending procedure is shown in the Fig. 1. The blank (200 mm × 50 mm) was hold on the counter punch by 100 kN in the whole deformation process. The punch speed and stroke are 2.0 mm/min and 80 mm, respectively. The clearance between punch and die were kept in 1.5 mm.
Schematic illustration of experimental apparatus for U-bending test.
The schematic illustration of U-bending procedure is shown in the Fig. 2. The blank (320 mm × 50 mm) was hold on the blank holder by 30 kN in the whole deformation process. The blank was also hold on the counter punch by 100 kN. The punch speed and stroke are 2.0 mm/min and 80 mm, respectively.
Schematic illustration of experimental apparatus for hat-bending test.
The yield function at the initial state, f0, has a general form:
| \[f_0 = \phi(\boldsymbol{\sigma}) - Y = 0,\] | (1) |
| \[f = \phi(\boldsymbol{\sigma}, \boldsymbol{\alpha}) - Y = 0,\] | (2) |
| \[\boldsymbol{D}^p = \frac{\partial f}{\partial \boldsymbol{\sigma}} \dot{p},\] | (3) |
In the present research, we utilized Y-U model7–10) for describing the kinematic motion of the yield surface. The model is constructed in the framework of the two surface modeling, wherein the yield surface moves kinematically within the bounding surface F. The bounding surface F is represented by the following equation
| \[F = \phi(\boldsymbol{\sigma}, \boldsymbol{\beta}) - (B + R) = 0,\] | (4) |
| \[\boldsymbol{\alpha}^* = \boldsymbol{\alpha} - \boldsymbol{\beta},\] | (5) |
| \[\mathring{\boldsymbol{\alpha}^*} = C \left[ \left( \frac{a}{Y} \right)(\boldsymbol{\sigma} - \boldsymbol{\alpha}) - \sqrt{\frac{a}{\bar{\alpha}^*}} \boldsymbol{\alpha}^* \right] \dot{p},\] | (6) |
| \[\dot{p} = \sqrt{\frac{2}{3} \boldsymbol{D}^p:\boldsymbol{D}^p},\quad \bar{\alpha}^* = \phi(\boldsymbol{\alpha}^*),\quad a = B + R - Y,\] | (7) |
| \[\dot{R} = m(R_{sat} - R) \dot{p},\] | (8) |
| \[\mathring{\boldsymbol{\beta}} = m \left[ \left( \frac{b}{Y} \right) (\boldsymbol{\sigma} - \boldsymbol{\alpha}) - \boldsymbol{\beta} \right] \dot{p},\] | (9) |
Besides the kinematic hardening model, the following model of plastic strain dependent Young's modulus was utilized in the present calculations:
| \[E = E_0 - (E_0 - E_a) [1 - \exp (-\xi p)],\] | (10) |
In addition to the above mentioned models, the following isotropic hardening model are utilized in the present research:
| \[\bar{\sigma} = k(\varepsilon_0 + p)^n\] | (11) |
The material parameters of these model for aluminum alloy sheet metals are shown in Table 2. These values are identified by the commercial software Matpara.
| Material | Yoshida-Uemori model | Swift model | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Y /MPa |
a0 /MPa |
C | b /MPa |
m | R sat /MPa |
h | F /MPa |
n | e0 | |
| A5052 | 60 | 40 | 1000 | 15 | 13 | 150 | 0.2 | 440 | 0.33 | 0.005 |
| AA6016 | 105 | 55 | 850 | 15 | 10 | 155 | 0.1 | 450 | 0.25 | 0.01 |
It is very easy to incorporate any of widely accepted yield functions of orthotropic anisotropy with Y-U model. In the present research, we introduced the following four types of yield functions (von Mises, Hill'48, Gotoh's bi-quadratic yield function, Barlat Yld2000-2d) into a commercial finite element code. The yield functions under plane strain condition are written as follows:
- Hill'48 quadratic yield function:
| \[ \begin{split} f & {} = \frac{1}{2} \left\{ F\sigma_{yy}^2 + G\sigma_{xx}^2 + H \left( \sigma_{xx} - \sigma_{yy} \right)^2 + 2N\tau_{xy}^2 \right\} - \frac{G + H}{2} \bar{\sigma}^2 \\ & {} = 0 \end{split} \] | (12) |
- Gotoh's bi-quadratic function:
| \[ \begin{split} f = {} & A_1\sigma_{xx}^4 + A_2\sigma_{xx}^3 \sigma_{yy} + A_3\sigma_{xx}^2\sigma_{yy}^2 + A_4\sigma_{xx}\sigma_{yy}^3 + A_5\sigma_{yy}^4 \\ & {} + (A_6\sigma_{xx}^2 + A_7\sigma_{xx}\sigma_{yy} + A_8\sigma_{yy}^2) \tau_{xy}^2 + A_9\tau_{xy}^4 - A_1\bar{\sigma}^4 = 0 \end{split} \] | (13) |
- Barlat Yld2000-2d function:
| \[\phi = \phi' + \phi'' = 2\bar{\sigma}^a\] | (14) |
| \[\phi' = \left| X'_1 - X'_2 \right|^a,\quad \phi'' = \left| 2X''_2 + X''_1 \right|^a + \left| 2X''_1 + X''_2 \right|^a\] | (15) |
To verify the validity of orthotropic yield functions and comparing the experimental results of yield loci, the bi-axial tensile tests for the materials are conducted11–13). From finite element simulation of bi-axial stretching deformations by using the isotropic hardening model with von-Mises yield function, it's confirmed that the errors of calculated stresses are guaranteed within ±3.1% for stress ratios of $0.5 \le \sigma_y/\sigma_x \le 2.0$11). The Fig. 3 shows the comparing the experimental initial yield surfaces with the corresponding yield surfaces calculated by four types of yield function. The shape of the yield surface for A5052-O shows symmetric due to its isotropy. By comparing the yield surfaces, it is found that the yield surface calculated by Gotoh's bi-quadratic yield function correspond reasonably well with the experimental observation. Although the calculated yield surface by von Mises can describe the responses in the R.D., there is a large discrepancy between the calculation and experiment in T.D. direction. The calculated yield surface by Barlat Yld2000-2d cannot describe the accurate responses in the R.D. In contrast to the above mentioned yield functions, Hill'48, which are well-known high accurate function for high tensile strength steel sheets, cannot describe the yield surface for aluminum alloy sheet metal.
Experimental results of initial yield loci, compared with the predictions calculated by several yield functions.
For AA6016-T4 sheet metal, the shape of yield surface is unsymmetrical due to the strong anisotropy. The discrepancy between the calculation and experiment in R.D. is especially distinguished. Due to the strong plastic anisotropy, the calculated yield surface by the low order yield functions as Hill and von Mises cannot capture the corresponding experimental data. In contrast to the functions, Gotoh's bi-quadratic yield function and Yld2000-2d can describe the above mentioned strong anisotropy very well. As same as the A5052-O, Gotoh's bi-quadratic yield function have a good agreement with the experimental data very accurately.
In order to verify the accuracy of constitutive models, the above mentioned models were introduced into the commercial finite element code LS-DYNYA (ver.971). The work-hardening behaviors were represented by either the swift type of Isotropic hardening model or Yoshida-Uemori kinematic hardening model. The yield functions installed into the commercial finite element code are von Mises, Hill'48, Gotoh and Yld2000-2d. The shell element with seven full integration points in the thickness direction were utilized to model the blank. The elements size were set to 1.5 mm square. The friction coefficient between blank and tools were kept 0.1.
The definition of springback angle is shown in Fig. 4. The calculated springback angles in the R.D. by the yield functions are shown in Fig. 5. The springback angles calculated by Y-U model are larger than IH model, and Y-U model very well can capture the springback angles of both materials. Since bending deformation in the blank is occurred once during whole deformation for U-bending processes, the accuracy of stress strain response in uniaxial tension (or compression) deformation is very important to predict the springback. Figure 6 (a) shows schematic illustrations of stress-strain curve during bending deformation. There is no difference between two models during tensile deformation. The springback deformation is strongly affected by the un-loading curves. Figure 7 shows the difference in the calculated stress-strain curves between two models during un-loading for A5052-O aluminum alloy sheet metal. The linear response during un-loading is predicted by IH model. On the other hands, the small plastic deformation due to the non-linear response is calculated by Y-U model. That is the reason why the springback due to the plastic strain recovery of Y-U model is larger than that of IH model. The same reason can be applied to the springback prediction for AA6016-T4 material.
Definition of springback angle θ.
Springback angle θ, in U bending (R.D.), predicted by IH model and Y-U model combined with several yield functions: (a) A5052-O; and (b) AA6016-T4.
Schematic illustrations of stress-strain curve during bending: (a) hardening rule; and (b) yield function.
Strain-Stress relationship for A5052-O during springback process calculated by IH model and Y-U model.
Although the most accurate calculated result are obtained by von Mises, the difference between experiment and the other models (Gotoh's bi-quadratic yield function and Barlat Yld2000-2d) are very small. In order of the quantity of the amount of springback, von Mises, Gotoh's bi-quadratic yield function, Barlat Yld2000-2d, and Hill'48 are ranked. The calculated stress state by Gotoh's bi-quadratic yield function and von Mises can capture the experimental data very well (see in Fig. 3). In contrast to this, Hill'48 yield function is not able to capture the springback well. The main deformation of U-bending process is assumed to be conducted under plane strain condition, so the accuracy of yield function under plane strain condition is very important key for bending procedures. From this results, we can conclude that the difference of springback are caused by the stress state under plane strain condition. For further investigations, we investigate the amount of springback in the T.D. for AA6016-T4 aluminum alloy sheet metal, which has a very strong plastic anisotropy. The Y-U model is utilized for this material, because the Bauschinger effect during stress-reversal cannot be neglected. Figure 8 shows the springback angles for this material. Experimental springback in the T.D. shows smaller than that in the R.D. The reason is the stress state under plane strain condition. As already described in the section 3.2, the shape of experimental yield surface for AA6016-T4 aluminum alloy sheet metal is not symmetric to the bi-axial stretching line. The stress value in R.D. is little bit larger than that in T.D. Since the finite element calculation by von Mises yield function cannot describe the unsymmetrical yield surface, the difference between R.D. and T.D. never appear. On the other hands, the Gotoh's bi-quadratic yield function can capture the difference of stress level under plane strain condition. That is the main reason of the Gotoh's bi-quadratic yield function has good agreement with the experimental data.
Comparison of springback angle θ between R.D. bending and T.D. bending.
The final shapes of hat bending procedure calculated by three types of constitutive equations as von Mises with IH model, Hill'48 with IH model and Gotoh's bi-quadratic yield function with Y-U model are shown in Fig. 9. The corresponding experimental data is also shown in the same figure. The R.D. of blanks are set to the longitudinal direction. From this figure, we can find that the difference of final shapes between IH model and Y-U model are very small. The both calculated final shapes are equal to the experimental data. Since the plastic dependent Young's modulus of A5052-O material is not observed11–13), the amount of springback of this material is determined by the accumulated stress before springback deformation. The amount of springback of IH models are larger than that of Y-U model, due to the accumulated stress values before springback. From the results, it is found that the influence of Bauschinger effect are not so important for A5052-O sheet metals11–13). On the other hand, the yield functions are very sensitive to the final shapes of hat bending. The amount of springback calculated by Hill'48 with IH model are larger than that of von Mises with IH model. From this result, we can find that the accurate description of yield function under plane strain condition is essential to the final shapes. Figure 10 shows the comparisons of experiment and the corresponding calculated results for the final shapes under two bending conditions (R.D. bending and T.D. bending). Gotoh's bi-quadratic yield function with Y-U model is utilized in the calculation. The amount of experimental springback in T.D. bending is much smaller than that of R.D. This result indicate that the strong plastic anisotropy affects the final shape of hat bending. The calculated final shape in R.D. bending show a good agreement with the experiment, while the final shape in T.D. cannot capture the corresponding experimental data. The difference are caused by the inaccuracy of the yield function under the plane strain condition in T.D. The calculation is little bit larger than the experiment. The comparisons between experiment and calculations are shown in the Fig. 11. The calculations were carried out by using two types of constitutive models (Gotoh's bi-quadratic yield function with IH model and von Mises yield function with Y-U model). Figure 11 (a) and Fig. 11 (b) show the R.D. bending and the T.D. bending, respectively. The calculated final shape in R.D. by von Mises yield function with Y-U model show a good agreement with the corresponding experimental observation, while the calculated shape in T.D. by the above mentioned model underestimate the experimental result. This is because von Mises yield function is not able to describe the plastic anisotropy of the material. Since the function can describe the either stress stat in the R.D. or T.D., the accurate springback deformation in both directions cannot be predicted. The calculated final shape in R.D. by the Gotoh's bi-quadratic yield function with IH model shows little bit smaller than the experiment, while the calculated shape in T.D. shows little bit larger than the experiment. From the results in Fig. 3, it is found that the accuracy of yield function under plane strain condition is essential to springback predictions. It is well-known fact that the springback angle of high tensile strength steel sheets strongly depend on the accurate stress strain models response during stress reversal. In addition to the accurate kinematic hardening model, both the plastic strain dependent Young's modulus and plastic anisotropic yield functions should be introduced for further investigations and improvement of springback calculations for aluminum alloy sheet metals.
Comparison of final shapes after springback between experimental and calculated results by several constitutive models for A5052-O.
Comparison of final shapes after springback between experimental and calculated results by constitutive models combined Y-U model and Gotoh's yield function for AA6016-T4.
Comparison of final shapes after springback between experimental and calculated results by several constitutive models for AA6016-T4: (a) R.D. bending; and (b) T.D. bending.
The plastic strain dependent Young's modulus, the Bauschinger effect and initial plastic anisotropy have great influences on the springback calculations. Especially for the sheet metals with strong plastic anisotropy, the springback deformation might depend on the bending directions. For the accurate springback calculations, it is essential to utilize an accurate kinematic hardening model incorporating with an appropriate choice of anisotropic yield function.
