^{1}

^{2}

In this paper, the traditional empirical coefficient method and a novel loop material analytical method presented by the authors have been compared and discussed by analyzing several practical cylindrical workpieces’ deep drawing projects. Also, some conclusions about how to modify the traditional analyzing method of cylindrical workpieces’ deep drawing process could be concluded and the necessity of this modification could be proved.

Technically in traditional way, the deep drawing coefficient (table) is generally taken as an experimental parameter for the analysis of cylindrical workpieces’ stamping and deep drawing process. The drawing coefficient can be defined as the ratio of the outer diameter of slab to the diameter of tube after drawing [

In order to solve this problem, a novel loop material theory analytical method, which is established by using 3 loop material parameters (loop material width b, plate thickness t and inner diameter of drawing cylindrical workpieces d), has been proposed by Lin et al. in 2009 [

It is important that the novel loop material theory could improve the accuracy and reasonability of the traditional method of cylindrical workpieces’ deep drawing process in a practical way. Several process analysis examples will be taken to analyze and compare these two methods in this paper.

Practically, when rigid-plastic plate forms into cylindrical workpieces by stamping and deep drawing process, only the loop material outside the cylinder bottom will suffer plastic yield, which means the part of cylinder bottom remains unchanged. The result of cylindrical workpieces deep drawing process is usually determined by the extent of plastic deformation of the loop material, so how to obtain the extent of plastic yield is a key point to distinguish the experimental analysis and scientific analysis. However, rare researches have been done to solve this problem.

In the loop material theory, the extent of plastic deformation can be expressed by mathematical terms (b/d) and log(b/t), which are the basic mathematical characteristics of loop material plastic yield, and these two terms can also be called Loop Material Plastic Yield Genes (LPG).

According to reference [

As shown in

where D is outer diameter and d is inner diameter.

In addition, because of the influence of cold-work hardening and relative width of loop material (b/t), a relative width coefficient of loop material log(b/t) should be considered. Hence, the material digestible rate of loop material can be defined as:

Simply, the material digestible rate of loop material can be seen as the product of geometry digestible rate and relative width coefficient. Like the genes in biology, N can be seen as a basic parameter (or can be called as basic characteristic factor of drawing deformation) which can indicate the extent of plastic yield of loop material or can be regarded as a basic characteristic factor in other related process analysis project, so it can also be called as Loop Material Plastic Yield Genes.

In order to simplify the analytical model, N could be used. Moreover, the less magnitude of N means smaller deformation of the workpieces, which means it will be easier to drawing, and vice versa. With the help of this Loop Material Plastic Yield Gene, more process analysis problems such as multiple drawing, suitable drawing load or blank holder force, cylinder blank modification and suitable selection of die fillets can be solved efficiently.

In the traditional way, the blank relative thickness (T = 100 t/D) is generally used to divide the interval value as a criteria, and then the range of ultimate drawing coefficient can be obtained from the related table, for example in

Unfortunately, the only thing we can know is “when the die fillet radius R = 8 ~ 15 t, m_{1} should be the smaller value, and when R = 4 ~ 8 t, m_{1} should be the bigger one”. However, a judgement formula can be obtained in this novel loop material theory [

where (b/d) is the geometry digestible rate of loop material and (b/t) is the relative width of loop material. On the other side of the formula, A is a constant parameter related to the property of material, J is a coefficient parameter related to mould gap, blank holder force, lubrication condition, drawing speed, etc. _{d} is the radius of die fillets, x, which is usually equals 8, is a multiple of t and n is an exponent.

At the initial analysing step, the influence coefficient of die fillets can be assumed to be

Blank relative thickness (T = 100 t/D) | 2 - 1.5 | 1.5 - 1.0 | 1.0 - 0.6 | 0.6 - 0.3 | 0.3 - 0.15 | 0.15 - 0.08 |
---|---|---|---|---|---|---|

First ultimate drawing coefficient m_{1} | 0.48 - 0.50 | 0.50 - 0.53 | 0.53 - 0.55 | 0.55 - 0.58 | 0.58 - 0.60 | 0.60 - 0.63 |

experiments [

As can be seen from Equation (4), the material digestible rate of loop material N will not be larger than

By comparison, some conclusions can be concluded. Firstly, significant impacts of the plate thickness and cold-work hardening on deep drawing process are considered in both methods. In traditional way, the ultimate drawing coefficients are the same in the case of equal relative thickness, regardless of the thickness of plate. Similarly, the results will be the same with calculating by Equation (4).

Firstly according to reference [_{1} = 0.45, T = 0.8, m_{1} = 0.52 and T = 0.08, m_{1} = 0.63 can be taken as examples, and N is 0.696, 0.682 and 0.694, respectively. Secondly, the relative thickness of the whole blank is chosen to be considered in traditional method, but in the proposed novel method, the relative thickness of loop material which has practical plastic deformation is used, and can be defined as b/t.

More importantly, in traditional method, the parameters of ultimate drawing coefficient are dependent on the relative thickness of the whole blank (defined as D/t), will be shown as independent variables in a particular equation. In details, this particular equation is a regression equation which can be obtained by the intermediate value from the deep drawing coefficient table (e.g. m_{1} = 0.53 in case of T = 1.0, m_{1} = 0.55 in case of T = 0.6), and can be written as:

Comparing to the results obtained from the deep drawing coefficient table, the curves of regression equation are close to the curves of intermediate value (errors are all less than 0.02), which means the regression equation is effective. As mentioned before, D or (D/t) using as independent variables are incorrect, because the fact that only loop material has the plastic deformation has been ignored [

Thirdly, it is hard to know which kind of parameters is better in the traditional method in the practical application. For instance, it is hard to decide which range (T < 1.5 ~ 1.0 or T < 1.0 ~ 0.6) should be referred to m_{1} when the relative thickness is 1 exactly. In this novel method, the influence of die fillets on ultimate deep drawing process can be reflected directly by using the influence coefficient of die fillet

Fourthly, the calculation of drawing times of multi-stage deep drawing can be obtained conveniently by this judgment formula, which can be defined as:

where N_{t} is the total material digestible rate of loop material (from the original blank to the final cylindrical workpiece).

According to the data of multi-stage deep drawing in existing traditional documents, some comparison and calculation between deep drawing process and the material digestible rate of loop material can be obtained. It can be found that the material digestible rate of loop material during the second-stage deep drawing process is even larger than the one of blank during the first-stage deep drawing process. Moreover, the one of the multi-stage deep drawing process will decrease with the decreasing of die radius after the second-stage deep drawing process, which means N ≤ AJ in case of

In traditional method, the nominal maximum drawing force P can be defined as:

Actually, it has been found that it is not reasonable in practical application. Hence, a parameter K_{1}, which is related to the extent of drawing deformation, can be used efficiently to refer as formula:

However, in loop material theory, it is much more efficient when the formula includes the judgment formula mentioned above. Specifically, at the critical state that drawing force equals to the ultimate tensile stress, the resistance coefficient of loop material of loop material is

Compared to the results which is calculating by using the table in page 258 of reference [_{1} can be substituted as the resistance coefficient of loop material

In the traditional method, the expansion blank size will be reduced proportionally according to the particular formula based on the average thinning coefficient or the area change coefficient, but sometimes the size of the blank with large diameter and low height will be reduced to even less than the drawing cylindrical workpieces. As shown in

where

by drawing deformation.

Specifically,

where

Similarly,

where b is the width of loop material, R is the die fillet radius, Z is the ratio of the die side clearance to the plate thickness, K is a comprehensive coefficient (generally equals 1) related to pressure condition, lubrication condition, drawing speed, etc. The precise dimension value of blank can be obtained by subtracting the correction quantity from the value of blank calculating by formula.

Equation (10) is suitable for cylindrical workpieces with various dimension, and the errors can be controlled within 0.3 mm [

For the narrow loop of the outside loop material which is formed by drawing without blank holder, the ratio of the diameter of the cylinder and the diameter of the blank is used as the drawing coefficient in the traditional way, but it is not reasonable because the bottom part, which is included in the parameter calculation and occupies a large part of the blank area, will keep unchanged during the deep drawing process. In addition, it is hard to know whether the deep drawing process is successful or not. It is not reasonable that when d/t > 75 (d/t is the relative cylinder diameter), the width of loop material which will suffer plastic yield remains at 12.5. Also, the influences of particular process conditions (e.g. die fillets, etc.) are not reflected in traditional method.

In the loop material theory, a standard that judge whether the deep drawing process without blank holder is successful or not is firstly proposed. The rim of formed cylinder becomes irregular can be regarded as failure and the one keeps smooth can be regarded as success.

Considering factors such as die fillets, the diameter of cylinder, etc., a shape coefficient W can be used, and another judgement formula can be defined as:

where

Equation (13) indicates that the ultimate width of loop material, drawing without blank holder, will decrease with the diameters of both cylinder and die fillets, based on basic width Ct.

According to Equation (3), Equation (13) can be rewritten as:

where

Equation (14) indicates that the rigid digestible rate of loop material, after drawing without blank holder, must be not larger than

For the calculation of internal bore flanging force, the formula in traditional method can be defined as:

where б_{s} is the yield stress of material, d is the diameter of cylinder after flanging, and d_{0} is the diameter of prefabricate bore.

However, it is also not reasonable because no matter what is the size of cylinder, the flanging forces will be the same if the loop materials’ width is equal, and in practice, the fact that the flanging loop material is combined closely with the outer basis should not be ignored.

In addition, it is not correct that the yield stress of material б_{s} has been taken as the parameter because the actual average stress might be greater than б_{s} or even close to the ultimate tensile stress б_{b}. The analytical results with Equation (15) are likely to less than the practical one, and the errors will increase with the growing of diameters of cylinder, which will have a negative impact.

According to the loop material theory proposed in this paper, the internal bore flanging process must consider two kinds of forces: 1) the bending forces caused by the deformation generated by the connection between flanging loop material and blank matrix, 2) the stretching force generated by the flanging loop material. Hence, the analytical model of the internal bore flanging force, which can be suitable for various cases, is proposed [

where d is the diameter of material that forms to cylinder, б_{s} is the yield stress of material, b is the width of loop material that forms to the straight cylinder, A and B are two calculated distribution coefficients, and m and n are two exponents. (b/d) is the geometric area extension rate of the loop material that forms to the straight cylinder (it will increase if the flanging resistance grows), and _{s}. In addition, _{b} or much less than б_{s}.

In order to simplify the formula, ^{n}, and n = 1/4, A = 0.27, B = 8 can be assumed, then the car prefabricated bore flanging force caused by the cylinder head punch can be written as:

where F_{1} is the bending force on the cross section of cylinder and F_{2} is the stretching force on the loop material that forms the cylinder. The analytical results are almost the same as the experimental results, and the errors between them are all less than 4%, when AISI 1008 Steel has been taken as an example [

The deep drawing process of cylinders with notches or lugs is another problem which is different from the deep drawing process of a whole circle loop material or the curve flanging. Besides the target of improving the production efficiency, solving this problem can help to understand deeply about the internal formation during deep drawing process. As shown in

a formula, which can show the relationship between the dimension of the notch of the cylinder after deep drawing and the dimension of the notch of the original blank, has been proposed [

where _{1} is a coefficient parameter (m_{1} = 1 ± 0.3) related to blank holder condition, lubrication condition, mould gap, etc. In addition, n_{1} is the tiny notch extrusion coefficient, which can be assumed to be _{1} should be required in the case of_{0} is impact coefficient of both notch sides, and its value is at a range between 0 and 1 (k_{0} will decrease when the width of side area decreases).

Specifically,

where θ is the central angle of the notch of the blank.

where E and F are two constants and assumed to be (E/F) = 0.078, b_{1} is the radial width of loop material outside the blank notch (_{1} can be ignored), h_{1} is the height of the part which below the bottom of the formed cylinder notch (_{1} can be ignored), and

According to Equation (18) mentioned above, it can be noticed that the notch width of the formed cylindrical workpieces is the subtraction that between the notch width (or arc length) of the formed cylinder which is drawn from a whole loop material and the value of abnormal extrusion on each notch side.

Interestingly, it is a deep drawing process of a whole circle loop material problem in the case of b_{1} = 0, and it is a curve flanging problem in the case of h_{1} = 0, which means Equation (18) is a comprehensive formula that can reflect various kinds of situations. Compared to the experimental result, the errors between analytical results and the practical dimension are all less than 0.5 mm.

It can be concluded that the traditional method has many disadvantages, and the analytical method proposed in this paper can explain clearly and calculate precisely via these formed process examples. The analytical results can be validated by correspondent experiments, and it is analyzed by comparing to the traditional method which uses the deep drawing coefficient (table) in this paper [

It can be seen that this proposed loop material theory is more efficient and effective than the traditional method since it includes all the factors which should be considered, even though some of formulas seem to be complicated. Moreover, the analytical formula of extension of cylinder can also be deduced by using the relative width of loop material (b/d), which can show the connection between them.

According to the similarity between analytical results and experimental results we have obtained, it can be said that it is an overturning innovation in which the novel loop material theory replaces the traditional method. This new method will not only help to understand the principle inside the deep drawing process, but also help to perfect the computer codes which are used to do the simulation.

Yang, R.X. and Lin, Z.W. (2017) Modifying the Traditional Analyzing Method of Cylindrical Workpieces’ Deep Drawing Process by Using a Novel Loop Material Theory Method. Engineering, 9, 351-360. https://doi.org/10.4236/eng.2017.94019