3.1 Wall thickness distribution

Under different forming conditions, the cylindrical parts with the drawing ratios are K1 = 2.125 and K2 = 2.25 respectively are shown in Fig. 5. After the experiment is completed, as shown in Fig. 6. measuring the wall thickness distribution of the two kinds of drawing ratios of cylindrical parts, and calculating the variation of the wall thickness. According to the distribution position of the measuring points, the cylindrical part can be divided into three deformation areas along the central axial section, are at the bottom deformation area I (points 1–4), the punch radius area II (points 5, 6) and the straight wall area Ⅲ (Points 7–13, 14) respectively.

Figure 6a)、b) show the wall thickness distribution of cylindrical parts with different drawing ratios under different forming conditions. We can find that the wall thickness at the bottom of the cylinder and the measuring points near the punch corner area are thinned after deep drawing, where the wall thickness of the punch corner area thining seriously and the straight wall area (such as measuring points 10–14) thickening. As shown in Fig. 6, compared with the wall thickness distribution of conventional deep-drawn parts, the uniformity of the wall thickness distribution of the parts have improved after using the MR fluid flexible medium for forming, and the wall thickness thining trend near the bottom area and thickening trend near the punch corner area of the cylinder parts are both decreaseed. In the inverse bulge and deep drawing composite forming process, the MR fluid acts as a force transmission medium to change the stress state of the sheet metal and suppress the occurrence of wrinkling defects effectively. With the increase of the inverse bulge height, the thinning of wall thickness decreases at the bottom and the punch corner area of the cylindrical parts. The thickening trend of the side wall of the cylindrical part is slowed down, and the distribution of the wall thickness is more uniform. However, when the pre-forming height is too high, the thickness of the bottom of the cylinder part becomes thinner because of the inverse bulge forming, where the material is enriched. The cylindrical wall was thicken and the wall thickness of the straight wall near the mouth of the cylindrical part is the most serious.

Calculate the chang rate of wall thickness at each measuring point of the cylindrical part with different drawing ratios, as shown in Fig. 6c)、d). It can be found that with the conventional drawing process conditions, the wall thickness of the punch fillet area of the formed part is thinned seriously and the maximum thinning rate are 10.1% and 11.3% with the drawing ratio of K1 = 2.125 and K2 = 2.25 respectively, the wall thickness of the straight wall area is thicken robviously, and the maximum thickening rate are 24.1% and 21.2% respectively. Compared with conventional deep drawing conditions, the maximum wall thickness thining rate at the punch corner area of the cylindrical part formed with the MR fluid soft medium has been reduced by 8.9% and 22.12% respectively, and the maximum wall thickness thickening rate of the cylinderical wall area is reduced by 11.6% and 12.7% respectively. The maximum wall thickness thining rate near the bottom center area (point 1) decreased by 36.4% and 27.7% respectively. The inverse bulge pre-forming process has a significant influence on the wall thickness distribution of the formed part. With the increase of the inverse bulge height, the wall thickness increase rate and thinning rate at different measurement points are reduced, and the overall forming performance is improved. The inverse bulge pre-forming process has a significant influence on the wall thickness distribution of the parts. With the increase of the inverse bulge height, the thickening and thinning rate of the wall thickness at different measurement points are reduced, and the overall forming performance of the cylinderical part has been improved. For cylindrical parts with drawing ratio K1 = 2.125, when the inverse bulge height is 9mm, the maximum thining rate of the bottom wall thickness of the cylindrical part is 3.7%, and the maximum thining rate of the wall thickness at the punch corner area is 5.2%, and the maximum thickening rate at the straight wall area is 6.1%. For cylindrical parts with drawing ratio K2 = 2.25, when the inverse bulge height is 5mm, the maximum wall thickness thining rate at the bottom area and the punch corner area of the cylinderical part is 2.9% and 9.1% respectively, and the maximum thickening rate at the straight wall area is 6%. Reasonable height of inverse bulge can suppress the phenomenon of wall thickness thining and thickening of the cylindrical part effectively. When the inverse bulge height is 11mm, the wall thickness of the bottom area of the cylindrical part with different drawing ratios are thinned extremely, extreme point appears near the punch corner area (point 7), where the thinning rates are 13.1% and 14.4% respectively. When the height of the inverse bulge pre-forming is lower, which is beneficial to the forming of the drawn part and the better the performance of the formed part.

3.2 Wall thickness change rate

As shown in Fig. 7, Comparing and analyzing the maximum change rate of wall thickness and the variance of the wall thickness of cylindrical parts with different drawing ratios. Figure 7a) shows the maximum thinning rate and thickening rate of the wall thickness of cylindrical parts under different forming conditions. The comparison of the wall thickness change rates of cylindrical parts with drawing ratios of K1 = 2.125 and K2 = 2.25 respectively, When the MR fluid assisted deep drawing, the maximum wall thickness thining rates are 9.2% and 10.6% respectively. Compared with the conventional drawing conditions, which are reduced by 8.9% and 6.19% respectively. The maximum wall thickness thining values appear at measurement points 5 and 7 respectively. The maximum wall thicknesst thickening rates are 21.3% and 18.5% respectively, which are reduced by 11.6% and 12.7%, and the maximum wall thickness thickening values appear at the measurement points 13 and 14 respectively.It can be found that a reasonable amount of inverse bulge pre-forming will increase the maximum wall thickness thining rate of the cylindrical parts, and with the increase of the inverse bulge height, the maximum wall thickness thining rate will decrease. However, when the inverse bulge height is 11mm, the maximum wall thickness thining rate increases obvisouly. When the drawing ratio K1 = 2.125, the maximum wall thickness thickening rate decreases with the increase of the inverse bulge height, and it will decreases to 6.1% as the inverse bulge height is 9mm. The maximum wall thickness thickening rate is the smallest, which is reduced to 6%, when the inverse bulge height is 5mm for the drawing ratio of the parts is K2 = 2.25, and as the inverse bulge height increases, the maximum wall thickness thickening rate increases.

Figure 7b) shows the mean value and variance of the wall thickness of the cylindrical parts with different drawing ratios under different forming conditions. The MR fluid medium is used for deep drawing. After the MR fluid is used for assisting deep drawing, the wall thickness variance of the cylindrical parts is compared with the conventional deep drawing, which are reduced by 40.8% and 37.2% respectively, and the uniformity of the wall thickness is improved. When the drawing ratio is 2.125, the mean value and variance value are continuously reduced with the increase of the inverse bulge height. When the inverse bulge height is 9mm, the variance value decreases by 86.31%, reduced to 0.0023. When the drawing ratio is 2.25, the wall thickness distribution is more uniform under the smaller inverse bulge pre-forming height, and the variance value is about 0.0025 as the inverse bulge height is 5mm, which is reduced by 82.8%. With the height of inverse bulge pre-forming increaseing, the variance value keeps increasing. When the inverse bulge height increases by 11mm, the wall thickness variance of the two kinds of drawing ratios of the cylindrical parts increases sharply and the uniformity of the wall thickness distribution is lower, that is the excessive inverse bulge height is not favorable to the deep drawing. When the drawing ratio is 2.125, the best pre-forming height is about 9mm. The best pre-forming height is constantly decreasing as the drawing ratio increase, this is because the large-diameter blank has a larger contact area with the flange area, which will increase the frictional force. And the inverse bulge deformation will reduce the thickness of the blank, resulting in the material is more difficult to flow. For cylindrical parts with a drawing ratio of 2.25, the best inverse bulge height is about 5mm.

3.3 Equivalent stress distribution after inverse bulge pre-forming

Under different forming processes, the equivalent stress distribution of the cylindrical parts with two drawing ratios after inverse bulge pre-forming is shown in Fig. 8. It is defined that the side of the sheet material close to the blank holder is the outside, and the side close to the die is the inside. During the inverse bulge pre-forming process, the outside of the blank at the fillet of the blank holder is stretched, and the inner side of the bulging center area of the sheet is compressed, and the outer side is stretched. As shown in Fig. 8a) and Fig. 8b), when the inverse bulge height is 7mm, the material in the bulging deformation area becames yield almost at the same time. Under the combined action of the MR fluid and the blank holder, plastic deformation has occurred around the fillet area of the blank holder. The maximum equivalent stress value exists on the inside of the sheet, and the minimum value appears on the inside of the transition area between the fillet area and the free bulging forming area. The equivalent stress has a stress gradient along the thickness direction in the deformation area. As the height of the inverse bulge increases, the plastic deformation area of the rounded corner area expands continuously and the stress gradient along the thickness direction increases, the deformation begins to move to the center of the bulging forming area. When the inverse bulge height is 11mm, the maximum equivalent stress exists on the outer surface of the highest point of the bulging area. The circumferential stress and radial stress of the bulging center area are basically equal, which is obvious two-way tensile stress state, and the wall thickness is severely thinned. As shown in Fig. 8c), the maximum equivalent stress of cylindrical parts with different drawing ratios in the pre-forming stage increases with the height of the inverse bulge increase, and the difference value of the maximum equivalent stress decreases firstly and then increases. When the height of the inverse bulge pre-forming is the same, the cylindrical parts with a drawing ratio of 2.125 enters the deformation state earlier than the drawing ratio of 2.25, the plasic deformation area is larger and the inverse bulge effect is more obvious.

3.4 Stress analysis of deep drawing

The cylindrical part during deep drawing forming is shown in Fig. 9. We divide the cylindrical part into five parts: flange area, die corner area, straight wall area, bottom corner area and bottom area, and the side close to the punch is the inner side of the cylinderical wall, and the side close to the die is the outer side of the cylinderical wall. Under the action of an external magnetic field, the aluminum alloy after inverse bulge pre-forming is deep-drawn forming with soft die using MR fluid medium, and analyzed the stress and strain state of cylindrical parts under different forming processes.

Since the inverse bulge pre-forming process has a significant effect on the drawing performance and stress state of the cylindrical part, we take measuring points along the outer area of the wall in order to further study the influence of the inverse bulge pre-forming effect on the deep drawing process. In the early stage of the deep drawing forming, the inverse bulge area appears obvious and the sheet at the corner of the die forms a soft drawbead. As the drawing displacement increases, the inverse bulge pre-forming area is reduced gradually. When the drawing displacement is about 12mm, the inverse bulge deformation area basically disappears. And the sheet is drawn into the die with the increase of the drawing displacement.

Figure 10a) shows the maximum radial stress of the cylindrical parts under different forming conditions at the early stage of deep drawing (the displacement of the punch is 12mm). During the deep drawing process, the deformation of the plate at the flange fillet is the most severe, and stress concentration occurs in the inner area of the sheet at the transition between the die corner and the flange area, where the maximum radial tensile stress occurs. Meanwhile, the wrinkling defect here is the most difficult to overcome, and the tendency of wrinkling at the fillet can be suppressed by improving the fluidity of the material in the flange area. In the case of using the MR fluid as the soft mold medium for forming, the maximum tensile stress values of the cylindrical parts with drawing ratios of 2.125 and 2.25 are smaller than that under conventional drawing conditions. At the early stage of deep drawing forming, the radial tensile stress value of the sheet in the flange decreases and the absolute value of the radial compressive stress increases under the action of back pressure, which is beneficial for the material to flow into the cavity of the die. Therefore, the tendency of wrinkling defects and lug phenomenon is suppressed effectively, and the surface quality of the side wall is better. When the inverse bulge height is 5mm, the amount of material deformation in the fillet area is small and the effect of the soft drawbead is weaker, the maximum radial tensile stress value decreases and the compressive stress value increases. The soft drawbead has a significant effect after pre-forming with the increase of the deformation height. and the friction drag and resistance to deformation increaseing accordingly after the material at the die corner area has strain strengthening after the bending and anti-bending deformation processes.

When the drawing displacement is the same, the pre-forming effect has a similar influence on the material deformation of the flange area of the cylindrical part with different drawing ratios. Therefore, the stress state of the sheet material in the flange area of the cylindrical part with a drawing ratio of K1 = 2.125 is analyzed. In the entire inverse bulge and deep drawing forming process, the deformation of the materials in each area of the cylindrical part affects each other. The fluidity of the sheet matel in the flange area is very critical to the quality of the final formed part. The flow resistance of the material can be adjusted to control its uniformity of deformation.

In the early stage of deep drawing process, the circumferential stress distribution of the sheet metal in the flange area under different forming conditions is shown in Fig. 10 (b). It can be seen that the material in the flange area is subjected to circumferential compressive stress, and the sheet material is easy to lose stability and wrinkle. In the initial stage of the conventional deep drawing forming, the circumferential compressive stress in the flange area is relatively large. With the assistance of the magnetic mediam, the back pressure can be transmitted through the magnetic particles, so that the sheet metal is close to the surface of the punch, and the friction resistance is increased. The circumferential compressive stress value near the corner of the die increases. At the same time, the MR fluid has a lubricating effect, which reduces the friction between the sheet material and the surface of the die, increases the fluidity of the sheet material, and effectively reduces the circumferential compressive stress in the flange area.

However, the circumferential stress distribution of the sheet metal in the flange area is extremely uneven, and wrinkling defects are obvious during these two forming processes. After inverse pre-forming, a soft drawing bead is formed near the corner of the die, which increases the flow resistance of the sheet in the flange area and the amount of circumferential deformation, and reduces the value of compressive stress. And with the increase of the inverse bulge height, the surface area of the sheet material used for the bottom forming of the cylindrical member gradually expands, and the suspended area increases accordingly, as shown in Fig. 11. The area of the suspended area under different forming processes is A, B, C and D, that is, A < B < C < D, the area of the soft draw bead is enlarged during the deep drawing process, and the plastic deformation ability of the sheet metal is improved. When the inverse bulge height is 5mm, the radial tensile stress near the corner of the die increases, and the circumferential compressive stress decreases. As the height of inverse height increases, the value of the circumferential compressive stress at the corner of the die increases, the distribution of compressive stress in the flange area is gradually uniform. When the inverse bulge height is about 9mm, the distribution of the compressive stress in the flange area is the most uniform, and the formation of the soft draw bead increases the area of the sheet metal against the punch and improves the forming performance of the cylindrical part. When the inverse bulge height is 11mm, the circumferential compressive stress value is too larget to cause wrinkling.

In the early stage of deep drawing, there is bending of the sheet at the corners of the die-reverse bending is caused by the effect of pre-forming deformation, and the inner and outer sides of the sheet are plastically deformed, and the stress has a stress gradient along the thickness. Analyzing the cylindrical parts with a drawing ratio of K1 = 2.125, as shown in Fig. 12, study the radial stress and equivalent stress distribution along the thickness direction at the fillet of the deep part die under different inverse bulge forming conditions. The inner side of the cylinder wall is the radial tensile stress and the outer side is the radial compressive stress, and there is a stress gradient along the thickness direction, the greater the radial tensile stress is, the more likely the fracture occurs. In the conventional deep drawing process, the maximum radial tensile stress at the fillet is 188Mpa, the maximum compressive stress is 48.2Mpa, and the stress gradient is 236.2Mpa, where the equivalent stress gradient is large, and the plastic deformation at the fillet is obvious. In the process of MR fluid soft die forming, the deformation at the corner of the die is small under the back pressure of the magnetic medium. At this time, the radial tensile stress and compressive stress are smaller, the radial stress gradient and equivalent strain gradient decrease. After the inverse bulge and deep drawing composite forming, due to the effect of soft drawbead in the early stage of deep drawing, the outer tensile stress and inner compressive stress decrease and the radial stress gradient decreases when the reverse bulging height is small. As the amount of pre-deformation increases, the tensile stress value at the outside of the sheet increases, and the compressive stress value at the inside decreases. When the inverse bulge height is about 9mm, the outside tensile stress is 187Mpa, and the inside compressive stress is 1.6Mpa, the stress gradient is 188.6Mpa. The stress neutral layer has moved, the radial compressive stress on the inner side turns into tensile stress. The equivalent stress gradient is the smallest and the material is easy to flow under tensile stress. When the inverse bulge height is 11mm, the tensile stress decreases along the thickness direction continuously, and the inner side of the sheet changes from compressive stress to tensile stress, and the stress gradient decreases. However, the bottom of the cylinder becomes in a cone after deep drawing due to the excessive pre-deformation and insufficient back pressure, and the forming effect is not good.