1. Chapter 5 Deep Drawing

2. Chapter 5 Deep Drawing
5.1 Deformation process and mechanical analysis in deep drawing
5.2 Deep drawing process of cylindrical workpiece
5.3 Calculation of the dimensions in the working portion of punch and die
5.4 Technological design of the drawn part
5.5 Typical structures of drawing die

3. The process to produce an opened hollow part with punch and die is called deep drawing.
Cylindrical, rectangular, conical, hemispherical, trapezoidal part, and panel part with complex shape can be made through deep drawing process. The deep drawn parts have a large size range and wide application.

4. The shapes of deep drawn workpieces are various, the characteristics and deformation rules for workpiece with different shapes are also different.
According to the equipment used, the deep drawing process can be classified into drawing with single-action, double-action and triple-action.

5. Fig.5.1 shows the diagrammatic sketch of the deep drawn parts.
According to whether the blank thickness changes or not during the deformation, the deep drawing process can be classified into drawing with or without thinning.
According to the shape of the deformed workpiece, the deep drawing process can be classified into cylindrical part, curved parts, rectangular box and complex part drawing.
Fig.5.1 shows the diagrammatic sketch of the deep drawn parts.

6. Fig. 5.1 Diagrammatic sketch of deep drawn parts
symmetrical rotational part (b) symmetrical rectangular part
(c) asymmetrical complex part
Fig. 5.1 Diagrammatic sketch of deep drawn parts

7. 5.1 Deformation process and mechanical analysis in deep drawing

8. 5.1.1 Deep drawing process of cylindrical part
The cylindrical part with diameter d and height h can be made from a circular blank with diameter D and thickness t through deep drawing (see Fig. 5.2).
During deep drawing process, the blank can be divided into five zones according to the different stress and strain states (see Fig. 5.3), where σ1, ε1 are the stress and strain in radial direction; σ2, ε2 are the stress and strain in thickness direction; and σ3, ε3 are the stress and strain in tangential direction.

9. Fig. 5.2 Deep drawing process

11. a1＞a2＞……＞a F1=F2

13. Fig. 5.3 Stress and strain of the blank during deep drawing

14. 1. Flange zone This is the main deformation zone.
In this zone, the material undergoes tensile stress σ1 in radial direction and compressive stress σ3 in tangential direction. If a blank holder is used, the compressive stress σ2 would exist in this zone due to the action of the blank holder.
The strain state in this zone is triaxial with tensile strain in two directions and compressive strain in one direction.
As the drawing proceeds, the absolute values of the stress and strain change continuously, resulting in the non-homogeneous distribution of the thickness and hardness of the workpiece.

15. 2. Die fillet zone This is the transitional zone.
As same as in the flange zone, the material in this zone under-goes tensile stress σ1 in radial direction and compressive stress σ3 in tangential direction. Furthermore, the material in this zone undergoes compressive stress σ2 caused by pressing and bending of the die fillet.
The material in this zone is stretched and thinned due to bending and sliding when passing over the die fillet zone. There is a little bit compressive deformation in the tangential direction also.

16. 3. Straight-wall zone
The drawing force is transferred to the flange through this zone. Because the stress σ2 in the thickness direction is equal to zero, this zone is in the plane strain state. Therefore the tangential stress σ3 (the intermediate stress) is equal to half of the axial stress, that is σ3=σ1/2.

17. 4. Punch fillet zone
This is also a transitional zone. The material in this zone undergoes radial tensile stress σ1 as well as tangential tensile stress σ3. At the same time, there is compressive stress σ2 in the thickness direction due to the pressing and bending by the punch.

18. In this zone, the material a little bit upward to the fillet (point a in Fig. 5.4), is situated between the punch and the die during the initial stage of drawing. There is only a little material to be transferred in this zone. The percentage deformation is small, the percentage work hardening is low, and there is no beneficial friction effect. As a result, the point a often becomes the weakest place of the whole drawn workpiece. Usually the cross-section through the point a is called critical section.
If the percentage deformation is very large, fracture or severe thinning phenomenon may occur at this place.

19. Fig.5.4 Variation of hardness and thickness of the drawn workpiece along the height direction

20. 5. Bottom zone
The material in this zone is in the plane tensile state. Because of the friction confinement at the punch fillet zone, both the stress and strain of the material in the bottom zone are small. The thickness variation before and after drawing is usually about 1~3% and can be neglected.

21. 5.1.2 Mechanical analysis of cylinder drawing process
The flange deformation zone of the blank undergoes tangential compressive stress as well as radial tensile stress during cylinder drawing (see Fig. 5.3). If the stress in the thickness direction is neglected, the solution of the tangential and radial stresses can be obtained by the equilibrium differential equation and also the plastic equation, which reflects the internal characteristics of the material.

22. 1. Drawing without blank holder
For an isotropic material, the principal stresses for the drawing without blank holder are the radial tensile stress σρ as well as the circumferential compressive stress σθ. The tensile stress would be regarded as positive.
The stresses acting on an element in the flange zone with radius ρ are shown in Fig The radial equilibrium equation is:

23. Fig.5.5 Stress analysis during cylinder drawing

24. （5－1） That is, （5－2） The radial equilibrium equation is:
For infinitesimal dθ, 。Simplifying the radial equilibrium equation and neglecting the higher-order terms, we obtain:
（5－1）
That is, （5－2）

25. Neglecting the work hardening of the material, the plastic condition of the maximum shearing stress becomes:
（5－3）
Substituting Equation 5.3 into Equation 5.2, we obtain:
After integration, we obtain:
（5－4）

26. Utilizing the boundary condition: when ρ=R, σρ=0, we obtain the integral constant C:
Substituting C into Equation 5.4, we obtain the radial tensile stress in the flange deformation zone σρ:
（5－5）
Furthermore, substituting Equation 5.4 into Equation 5.3, we obtain the tangential stress in the flange deformation zone σθ:
（5－6）

27. According to Equations 5. 5 and 5
According to Equations 5.5 and 5.6, the stress distribution in the flange deformation zone can be obtained, as shown schematically in Fig.5.5.
According to the distributions of the radial tensile stress σρ as well as the tangential compressive stress σθ, it is known that the absolute value of the tangential compressive stress σθ is larger than that of the radial tensile stress σρ, that is, the main deformation in this zone is compressive deformation. This shows that the cylinder drawing is a typical compression forming.
The tangential compressive stress σθ reaches its maximum value at the outmost edge of the deformation zone, i.e.,
（5－7）

28. The radial tensile stress σρ reaches its maximum value at the innermost edge of the deformation zone:
（5－8）
Fig. 5.5 shows that when the absolute value of the tangential compressive stress σθ equals to that of the radial tensile stress σρ, that is, , the position of the equal stress circle can be obtained as
（5－9）
for , ;
for ,

29. According to above analysis,
 it is known that in the cylinder drawing process, the principal strain of the blank near the die fillet zone is a radial tensile strain, and thinning occurs;
 whereas the maximum strain near the outer edge of the blank is a compressive strain, and a little bit thickening occurs.

30. 2. Drawing with blank holder
When drawing with blank holder, besides the stress σρmax which is necessary for the flange deformation, there are other tensile stresses existed at the straight-wall of the force transferring zone, such as the friction resistance σμ caused by the blank holding force acting on the surface of the flange deformation zone, the stress σWZ caused by the bending and flattening when the blank sliding over the die fillet and the corresponding friction resistance.

31. P Q

32. P=weμα

33. After calculating above stress items individually, we obtain the total stress at the straight-wall in the force transferring zone of the cylinder σ:

34. where α is the wrap angle between the blank and the die fillet
where α is the wrap angle between the blank and the die fillet. When α=π/2, ρ=r and σ reaches the maximum:
（5－10）
Because,
Therefore,
（5－11)

35. The theoretical calculation formula of the drawing force is:
（5－12）
Substituting Equation 5.11 into Equation 5.12, we obtain:
（5－13）
where, d is the cylinder diameter in mm; t is the blank thickness in mm; μ is the friction coefficient; rd is the fillet radius of die in mm.

36. The deep drawing force is mainly related to the material properties, the dimensions of the workpiece and blank, the fillet radius of die and the lubricant conditions.
Above theoretical derivation on the deep drawing force provides a good method and basis for the forming analysis and the process calculation, but it is not convenient for practical application. In practice, the following empirical formula is often used to calculate the deep drawing force:

37. （5－14） （5－15） For first-pass, the deep drawing force is:
For second and subsequent passes, the deep drawing forces are:
（5－15）
where, dl is the workpiece diameter after the first deep drawing pass;
di is the workpiece diameter after the i th deep drawing pass;
Pi is the drawing force in the i th deep drawing pass;
σb is the ultimate strength of the material;
K1 and K2 are the coefficients which can be obtained from Tables 5.1 and 5.2.

38. Table 5.1 Value of K1 Drawing coefficient
Relative thickness of the blank t/D0×100
Drawing coefficient
0.45
0.48
0.50
0.52
0.55
0.60
0.65
0.70
0.75
0.80 5
0.95
0.85
0.43
0.35
0.28
0.20 2
1.1
1.0
0.90
0.42
0.25
1.2
0.68
0.56
0.47
0.37
0.30
0.8
0.40
0.33
0.5
0.82
0.67
0.36
0.2
0.1
0.9

39. Table 5.2 Value of K2 Drawing coefficient
Relative thickness of the blank t/D0×100
Drawing coefficient
0.70
0.72
0.75
0.78
0.80
0.82
0.85
0.88
0.90
0.92 5
0.60
0.50
0.42
0.32
0.28
0.20
0.15
0.12 2
1.1
0.52
0.25
0.14
1.2
0.62
0.30
0.27
0.16
0.8
1.0
0.57
0.46
0.35
0.18
0.5
0.63
0.40
0.2
0.56
0.44
0.33
0.23
0.1
0.68
0.55

40. 5.2 Deep drawing process of cylindrical workpiece

41. 5.2.1 Quality of cylindrical workpiece
1. Wrinkle and fracture
The material properties, the blank dimensions and the stress states in deformation zone have great influence on the drawing process and product quality.
The main quality problems are wrinkle and fracture, which occur easily during deep drawing (see Fig. 5.6).
(a) wrinkle (b) fracture
Fig. 5.6 Failure of the drawn workpiece

42. As for the material property, the smaller value of the ratio of yield to tensile strength σs /σb is beneficial to the deep drawing process. This is because the smaller yield strength σs of material is favorable to the material flowing, and the larger tensile strength σb is favorable to prevent fracture forming.
Generally, when σs /σb ≤ 0.65 and the percentage elongationδ≥28%, the material owns good drawing- ability.

43. The relative thickness t/D mainly reflects the anti-instability capability of the blank.
The larger the t/D, the higher would be the anti-instability capability of the blank. When t/D reaches a certain value, the blank holding force can be removed or reduced, and the drawing force also reduces.
With t/D decreased to a smaller value, the tangential compressive stress at the outer edge of the deformation zone is very large, which often results in buckling or wrinkle.

44. During the drawing process, both the relative thickness of the blank and the maximum tangential compressive stress of the deformation zone are changing continually, the maximum tangential compressive stress keeps increasing to enhance the trend of instability.
With the proceeding of the drawing process, the relative thickness of the blank increases. Therefore the material stability in the flange deformation zone is strengthened. These two factors counteract mutually.

45. Experiments show that the moment for buckling and wrinkle to occur is the moment when the maximum radial tensile stress appears.
Rt=（0.7～0.9）R0

46. The slight wrinkle would affect the workpiece surface quality; the severe wrinkle would result in an exceeding tensile stress and fracture because it is impossible for the wrinkled section of the blank to pass through the clearance between the punch and die during drawing process.
The fracture of the material often occurs at the lower end of the cup wall near its junction with the punch fillet zone. Thickness thinning is happened there due to the bending deformation occurs in the punch fillet zone, and makes this district becomes the critical section.

47. The common method to prevent wrinkle in practice is to use a blank holder (see Fig. 5.7), which applies the proper blank holding force to the deformation zone.
If the blank holding force is too small, it cannot play an effective role in preventing wrinkle.
If the blank holding force is too large, it would result in an increasing of the radial tensile stress, and induce cracking.

48. Fig. 5.7 Deep drawing die with blank holder

49. Therefore, it is important to determine the proper blank holding force in the drawing process design.
Usually, the blank holder force Q is a bit larger than the minimum value that is necessary to prevent wrinkle. It can be calculated as follows:
（5－16）
where, A is the actual contact area between the blank holder and the blank at the initial stage of drawing; q is the blank holding force per unit area and can be selected from Table 5.3.

50. Table 5.3 Blank holding force per unit area q

51. To realize the holding function, two kinds of holding devices are used in practice.
One is the elastic blank holding device equipped with rubber, polyurethane rubber, spring, air or oil cylinder (see Fig. 5.8);
another one is the rigid blank holding device with a fixed gap (see Fig. 5.9).

52. Fig. 5.8 Elastic blank holding device
(a) Holding device using air cushion (b) Holding devices using spring, rubber or polyurethane rubber
Fig. 5.8 Elastic blank holding device

53. 1-inner slide 2-outer slide 3-drawing punch 4-blanking punch and blank holder 5-blanking die 6-drawing die
Fig. 5.9 Rigid holding device for double action punch press

54. (1) Elastic blank holding device
The relationship between the blank holding force Q and the drawing stroke of the elastic holding device is shown in Fig
It can be seen that the blank holding force Q of the elastic holding device using rubber, polyurethane rubber and spring increase oppositely with the increasing of the drawing depth; especially in the case of rubber and polyurethane rubber, the force Q increases more severely. So these kinds of blank holding devices can only be used in shallow drawing.
The holding device using air cushion is effective and can be used in deep drawing, but it also has its drawbacks.

55. Fig. 5.10 Relationship between the elastic
blank holding force Q and the drawing depth

56. (2) Rigid blank holding device
When such holding device is used, the blank holding force does not change with the press stroke. The die structure is simple and the drawing effect is quite good. This device is applicable for deep drawing and mostly suitable for the double action press (see Fig. 5.9).

57. For some thin-walled symmetrical deep drawn workpiece of complex shape, such as conical, hemispherical and parabolic curved part, the un-holding section of the blank section between the punch and die is large. Therefore the wrinkle prevention effect of the blank holder is not good, especially in the multi-pass drawing.
When the direct-reverse drawing methods is used (see Fig.5.11), the radial tensile stress σ1 increases and the tangential compressive stress σ3 decreases according to the plastic equation σ1+σ3=βσs. As a result, the wrinkle preventing effect of the blank holder is good.

58. Fig. 5.11 Re-drawing methods
(a) direct drawing (b) reverse drawing
Fig Re-drawing methods

59. The experiments show that when using reverse drawing method the drawing coefficient decreases by 10~15% as compared with direct drawing method. In reverse drawing, the minimum cylinder diameter d2 should be larger than or equal to (30~60) t, the roundness radius of punch rP should be greater than (2~6) t, and the die wall thickness is determined by the following equation:
（5－17）
where, td is the wall thickness of the die in mm;
d1 is the workpiece diameter after the first drawing in mm;
D2 is the workpiece diameter after the reverse drawing in mm;
m2 is the coefficient of the reverse drawing.

60. Table 5.4 Conditions whether using blank holder or not
Considering the strength of the die wall, td shouldn’t be too small. Therefore the application range of the reverse drawing is limited.
In the case of large relative thickness of material t/D and drawing coefficient, the blank holder may not be used. Whether using the blank holder or not can be determined according to the conditions shown in Table 5.4.
Table 5.4 Conditions whether using blank holder or not
Using blank holder
First drawing
Subsequent drawing
(t/D) ×100 m1
t/dn-1) ×100 mn
Necessary
＜1.5
＜0.6
＜0.8
Acceptable
1.5～2.0
0.6
1.5
0.8
Unnecessary
＞2.0
＞0.6
＞1.5
＞0.8

61. Usually, wrinkle is not the key problem in the drawing process, since it can be eliminated by using the blank holder, the drawing rib, or the methods of direct and reverse drawing. Instead, crack failure is the principal problem in the drawing process.
The fillet radii of the punch and die, the clearance between the punch and die also greatly influence the drawing process.

62. As drawing proceeds, the blank of the flange zone is drawn progressively into the die fillet zone under the radial tensile stress and causing the bending deformation. The deformed material slides continually along the die fillet zone until it comes out of this zone, and straights up through reverse bending.
When the edge of the blank enters the die fillet zone, the radial tensile stress is very small. As a result, this part of material cannot be bended close to the die. When this part of material leaves the die fillet zone, it cannot be straight up completely. Therefore malformation usually occurs at the workpiece opening, which affects the workpiece quality.

63. At the same time, the fillet radii of the punch and die have a decisive effect on the drawing process to determine whether the process success or not. Supposing that the roundness radii of the punch and die are zero, the die is not a drawing die anymore and becomes a blanking die, and the drawing process becomes a blanking process.
So in the designing of the die, how to select the proper fillet radius of the die and the clearance between the punch and die is the critical points to avoid the product defects and failures.

64. 2. Earring and residual stress
In the drawing process, besides wrinkle and fracture, the earring and residual stress are also the quality problems usually appeared.
The punching sheet is produced through rolling. The mechanical properties of the sheet metal along the rolling direction are different from other directions. Therefore in the drawing process, the thickness variation and radial deformation of the material in different directions are also different. This results in local bulging at the workpiece opening and forming uneven convexes, so-called earring see Fig The earring at the workpiece opening can be eliminated through trimming process.

66. During cylinder drawing, the residual stress on the outer surface of the workpiece is tensile, and the stress on the inner surface is compressive. The residual stress at the workpiece opening reaches the maximum value when the blank slides over the die fillet zone due to bending and straightening. In extreme case, crack may occur on the cylinder wall due to stress corrosion.
The method to overcome this problem is to adopt the process of drawing with thinning to have all of the material at the straight wall zone entering the yield state. In this way, the influence of the residual stress can be reduced greatly.

67. 5. 2. 2 Drawing coefficient and drawing number
The ratio of the diameter after drawing d to the diameter before drawing D is called drawing coefficient m. It is usually expressed by following equation:
（5－18）
where, d is the cylinder diameter after drawing in mm; D is the blank diameter in mm.
The drawing coefficient can be used to express the percentage deformation. The above equation shows that the smaller the drawing coefficient, the greater would be the percentage deformation.

68. In practice, the reciprocal value of the drawing coefficient, which is the drawing ratio K, is also used to express the percentage deformation:
（5－19）
For each kind of material, there exists a limit for the percentage deformation. Therefore the drawing coefficient of each kind of material has a minimum value, which is called the limit drawing coefficient.
When the drawing coefficient of the part is less than the limit drawing coefficient, multi-pass drawing is needed.

69. The number of drawing can be determinated by flowing way:
In the case of multi-pass drawing, the drawing coefficient of each drawing pass is calculated as follows.
For the first drawing pass,
For subsequent drawing passes,

70. ．
where, D is the blank diameter in mm;
Dn is the product diameter in mm;
d1, d2, d3…dn-1 is the diameter of the semi-product in subsequent drawing pass in mm (see Fig. 5.12).

71. Fig.5.12 Diagrammatic representation of drawing process

72. （5－20） The drawing coefficient of the workpiece is:
So, the relationship between the drawing coefficient of the workpiece and that of the subsequent drawing pass is:
（5－20）
If the selected drawing coefficient is too small, wrinkle, crack, or severe thinning would occur in the drawn workpiece.
The limit drawing coefficient is the smallest drawing coefficient to avoid wrinkle and crack in the workpiece during drawing.

73. Theoretically, the limit drawing coefficient m1 of the material may reach 0.37 without considering the influence of the friction, the fillet radius of die and the work hardening.
In practice, the limit drawing coefficients for various materials are shown in Tables 5.5 and 5.6 after considering all kinds of actual conditions. The number of drawing and the dimension of the intermediate drawn workpiece can also be determinated according to Equation 5.20.

74. Relative thickness of the blank (t/D)×100
Table 5.5 Limit drawing coefficient of the cylindrical part with blank holder
Drawing coefficient
Relative thickness of the blank (t/D)×100
2.0～1.5
1.5～1.0
1.0～0.6
0.8～0.3
0.3～0.15
0.15～0.08 m1 m2 m3 m4 m5
0.48～0.50
0.73～0.75
0.76～0.78
0.78～0.80
0.80～0.82
0.50～0.53
0.75～0.76
0.78～0.79
0.80～0.81
0.82～0.84
0.53～0.55
0.75～0.78
0.79～0.80
0.81～0.82
0.84～0.85
0.55～0.58
0.82～0.83
0.85～0.88
0.58～0.60
0.83～0.85
0.88～0.87
0.60～0.63
0.85～0.86
0.87～0.88
Note: (1) The drawing coefficients in the table are suitable for mild steels, such as 08, 10, 15Mn, and the softened brass H62. For the materials with poor drawing property, such as 20, 25 and Q215, Q235 steel, and hard aluminum, the values in this table should be increased by 1.5~2.0%. For the materials with good plastic property, such as 05 steel and 08, 10 deep punching steel and annealed aluminum, the values can be decreased by 1.5~2.0%.
(2) The values in the table are suitable for drawings without intermediate annealing operation. If an intermediate annealing operation is used, the values in the table can be decreased by 2~3%.
(3) The smaller values in the table are suitable for the large fillet radius of die [rd =(8~15) t]; and the larger values in the table are suitable for the small fillet radius of die [rd = (4~8)t].

75. Relative thickness of the blank (t/D)×100
Table 5.6 Limit drawing coefficient of the cylindrical part without blank holder
Drawing coefficient
Relative thickness of the blank (t/D)×100
1.5
2.0
2.5
3.0
＞3.0 m1 m2 m3 m4 m5 m6
0.65
0.80
0.84
0.87
0.90
----
0.60
0.75
0.55
0.53
0.50
0.70
0.78
0.82
0.85
Note: This table is suitable for mild steels, such as 08, 10 and 15Mn. The notes for Table 5.4 are also applicable here.

76. The drawing coefficient is an important parameter in drawing process, which can be used to express the percentage deformation during drawing.
The smaller the drawing coefficient, the larger would be the percentage deformation.
There are many factors affecting the allowable limit drawing coefficient. In the designing of the drawing process and the practical stamping production, to make full use of the beneficial factors, to adopt the useful measures to increase the strength and the load capacity in the force transferring zone of the blank, to decrease the forming force of the deformation zone, to make the deformation zone easier to be deformed, are the key links to reduce the limit drawing coefficient of each pass, to decrease the number of drawing and to achieve success in drawing forming.

77. 5.2.3 Determination of blank dimension
The cylindrical drawn workpiece is drawn from a circular blank. To simplify calculation of the blank dimension, the variation of the material thickness can be ignored.
According to the constant volume condition in plastic deformation, the dimension of the drawn blank can be calculated directly by the equivalent surface area of the blank before drawing and that of the workpiece after drawing.

78. According to above principle, the drawn workpiece is divided into several simple geometric shapes first (see Fig. 5.13). The area of the individual geometric shape is then summed up and the blank dimension is calculated as:
That is (5-21)

79. Fig. 5.13 Calculation of blank dimension for cylindrical workpiece

80. Calculate the areas of the individual parts in Fig. 5.13 respectively:
Substituting them into Equation 5.21, we obtain:
（5－22）

81. Trimming compensation Δh
where, d is the outer diameter of the cylinder in mm;
R is the inner fillet radius of the cylinder bottom in mm;
H is the height of straight-wall of the cylinder in mm. In practical calculation, a trimming compensation Δh should be added, which can be obtained from Table 5.7.
Table 5.7 Trimming compensation Δh for the cylindrical drawn workpiece (mm)
Workpiece height H
Trimming compensation Δh
10～50
50～100
100～200
200～300
1～4
2～6
3～10
5～12
Note: Trimming compensation is unnecessary in shallow drawing.

82. The above calculation is of close approximation
The above calculation is of close approximation. In the case of small relative height H/d, the trimming process can be omitted and the compensation is unnecessary. In practice, the calculation results should be amended according to the specific condition.

83. 5.3 Calculation of the dimensions in the working portion of punch and die

84. 5. 3.1 Fillet radii of punch and die
1. Fillet radius of die rd
The fillet radius of die has an important effect on the drawing process. It affects the deep drawing force, the thinning in the straight-wall zone, the drawing coefficient, the numbers of drawing, the die life and whether the wrinkles occur or not.
The larger the fillet radius of die, the smaller would be the drawing force, which obviously advantageous in decreasing the drawing coefficient, the drawing number, and the thinning of material in the straight-wall zone, and to increase the die life.

85. But excessively large fillet radius of die would cause the blank to depart from the blank holder too early and wrinkles may occur.
The principle for determining the fillet radius of die is to choose the fillet radius as large as possible under the condition of wrinkle-free.
Fig and Fig show the experimental results of the effect of the fillet radius of die rd on the drawing force and the coefficient.
It can be seen that the effect of rd on the drawing force or the limit drawing coefficient is significant, especially the effect of the relative fillet radius of die rd/t. But when rd/t >10, the effect is relatively small.

86. Fig. 5.14 Effect of the fillet radius of die rd on the drawing force P (material: 08F)
Fig Effect of the relative fillet radius of die rd on the limit drawing coefficient
(material: Brass)

87. The fillet radius of die rd is related to the blank properties, material thickness t, percentage deformation (m), drawing speed, numbers of drawing, drawing method and the height of drawn workpiece. In practice, it can be determined referring to Table 5.8.
Note: Upper limit is for thin blank, lower limit for thick blank.

88. For the subsequent processes, rd can be calculated by the following equation (Note that it should be equal to or greater than twice the thickness):
When drawing workpiece with wide flange, the effect of the blank holder would not be abated. The rd can be 0.5 to 1.0 times larger than that shown in Table 5.8.

89. 2. Fillet radius of punch rP
The effect of the fillet radius of punch on the drawing force is very small.
But if the fillet radius of punch is too small, it would severely affect the material thinning, increase the bending stress at the “dangerous region” and weaken the material, resulting in the increase of the limit drawing coefficient. Moreover, local thinning mark would remain on the sidewall of the semi-product in the subsequent drawing process, thus affecting the product quality.

90. In practice, the fillet radius of punch rP can be decided by the flowing equations:
in the first drawing pass of the multi-pass drawing: rP=(0.7~1.0)rd
in the subsequent drawing passes of the multi-pass drawing:
The fillet radius of punch is equal to the fillet radius of product in single-pass drawing or the last drawing in multi-pass drawing and should be larger than (2~3) t.
If the fillet radius of product is less than (2~3) t, the fillet radius of punch should still equal to (2~3) t, except in shallow drawing. The desired fillet radius of product can be obtained through an additional sizing process.

91. 5.3.2 Structure of punch and die
Whether the structure of punch and die is reasonable, not only relates to the product quality but also affects the drawing percentage deformation (drawing coefficient) directly.

92. 1. Drawing without blank holder
（1）Shallow drawing The workpiece is deformed in single-pass drawing process. The structures of punch and die shown in Fig can be adopted, Fig (a) is suitable for large workpiece while Figures 5.16 (b) and (c) are suitable for small one.
（2）Deep drawing The workpiece is deformed in two or more drawing passes. The structures of punch and die shown in Fig can be adopted.

93. Fig. 5.16 Die structures without blank holder
(a) arc shape (b) conical shape (c) involute shape
Fig Die structures without blank holder

94. Fig. 5.17 Multi-drawing without blank holder

95. 2. Drawing with blank holder (see Fig. 5.18)
(a) suitable for d＞ (b) suitable for d≤100
Fig Multi-passes drawing with blank holder

96. During conical die drawing, the blank is deformed into a curved shape in the initial stage (see Fig. 5.19), which has the stronger anti-instability capability to prevent wrinkle. Moreover, the conical die is beneficial to the flowing of the deformed material. It decreases the friction resistance and the bending deformation, thus decreasing the drawing force. As a result, smaller drawing coefficient can be adopted. The sidewall quality of product is furthermore improved due to the decreasing of the repeated bending.
No matter what kind of die structural is used, it should pay attention to the relationships between the fillet radii of the punch and die, and also the relationship between the fillet radii of the blank holder in the successive drawing passes (see Fig and Fig. 5.17).

97. Fig. 5.19 The sketch of conical die drawing

98. 5.3.3 Clearance between punch and die
The clearance between the punch and die in drawing is a very important parameter, it affects:
1. Drawing force
The smaller the clearance, the larger would be the drawing force.
2. Product quality
If the clearance between the punch and die is too large, the wrinkle occurs easily. The thickening in the workpiece opening zone can’t be eliminated, and coning occurs in the straight wall zone of the product. However if the clearance between the punch and die is too small, fracture or severe thinning may occur.
3. Die life
The smaller the clearance between the punch and die, the more serious would be the wearing of the punch and die.

99. Therefore, the principle to determine the clearance between the punch and die is to consider not only the thickness and tolerance of the blank, but also the thickening in the workpiece opening zone.
The clearance Z (single-sided) should be a bit larger than the blank thickness and can be calculated by the following equation:
Z = tmax+c t
where, tmax is the maximum blank thickness in mm, tmax=t+Δ;
Δ is the positive deviation of the blank dimension in mm; c is the coefficient considering the material thickening, it can be obtained in the relevant manuals.

100. Intermediate drawing pass
Generally, the clearance during cylindrical workpiece drawing can be selected according to Table 5.9. For stainless steels and high temperature alloys, Z can be selected as (1.20~1.25) t.
Table 5.9 Clearance Z between the punch and die for the cylindrical workpiece
Material
Clearance
First drawing pass
Intermediate drawing pass
Last drawing pass
Mild steel
（1.3～1.5）t
（1.2～1.3）t
1.1t
Brass, aluminum
（1.3～1.4）t
（1.15～1.2）t
Note: If the workpiece has high tolerance demand, the clearance of the last drawing pass Z can be adopted as: Z=1.05t

101. 5.3.4 Dimension and manufacturing tolerance in the working portion of punch and die
The dimensional accuracy of the drawn workpiece is determined by the dimension and tolerance in the working portion of punch and die during the last drawing pass directly. So the latter should be determined according to the demand of the drawn workpiece.

102. When the dimension tolerance is marked on the outer shape of the workpiece, as shown in Fig. 5.20a,
the dimension in the working portion of die is:
(5－23)
and the dimension in the working portion of punch is:
(5－24)
When the dimension tolerance is marked on the
inner shape of the workpiece, as shown in Fig.5.20b
the dimension in the working portion of die is:
(5－23)
(5－25)
and the dimension in the working portion of punch is:
(5－26)

103. For the first and the intermediate passes of multi-pass drawing, it is unnecessary to give strict demands on the dimensional tolerance of workpiece. If taking the die as reference,
the dimension in the working portion of die is:
(5－27)
and the dimension in the working portion of punch is:
(5－28)

104. (a) the tolerance is marked on the (b) the tolerance is marked on the
outer shape of the workpiece inner shape of the workpiece
Fig Dimensions of the workpiece and die

105. where the manufacturing tolerance grade of the punch δp and die δd can be took as IT6~IT9 or selected from Table 5.10.
Table 5.10 Manufacturing tolerance of punch δp and die δd (mm)
Material thickness (mm)
Diameter of the drawn workpiece
≤20
20～100
＞100 δd δp
≤0.5
0.02
0.01
0.03 ——
＞0.5～1.5
0.04
0.05
0.08
＞1.5
0.06
0.10
Notes: If necessary, the manufacturing tolerance grade of the punch can be increased to IT6~IT7.

106. (一) 网格变化
1.横向变形前Δl1=Δl2=Δl3 变形后Δl1＞Δl2＞Δl3.
纵向变形前 h1= h2= h3 Δh3＞Δh2＞Δh1靠近底部变形小，
靠近口部变形大，直壁中间部分变形最小（接近弯曲变形）
3.圆角部分变形程度比圆筒小，即网格线不与底面垂直（或斜线）。
4.直边部分变形最小靠近圆角的拉深变形最大。

107. (二)应力分布特点:
1.σρ分布不均，圆角处最大，直边部分最小，盒形件平均应力小于圆筒件平均应力。所以就危险断面的载荷来说，盒形件应力小。所以拉深深度更深些。取决于r/B
2.σθ在角度大，直边部小。与角部相应的圆筒件相比，材料的稳定性加强了。σθ↓起皱可能性↓。
3.直边和圆角互相影响的大小，随着盒形件形状的不同而不同。如r/B、t/B、h/r，在毛坯计算和工序计算的方法上有很大不同。由h/r代表变形程度。h/r相对高度受材料r/B、t/B影响。

108. 5.4 Technological design of the drawn part

109. 5.4.1 Technological property analysis of the drawn part
The technological property of the drawn part determines the quality, cost and production time of the product, and whether it can be produced through deep drawing or not. A drawn part with good technological property can meet not only the application demand of the product but also can be produced by the simplest, quickest and most economic way.

110. where h is the height of the drawn workpiece in mm;
The demands on the technological properties of the drawn part are as follows:
1. Demand on the outer dimension of the drawn part
When designing a drawn part, its height and flange width should be reduced as much as possible in order to reduce the drawing number. For different shapes of drawn workpiece, the conditions for adopting the single-pass drawing are as follows:
(1) The condition to produce a cylindrical workpiece through single-pass drawing is:
h ≤ (0.5~0.7)d
where h is the height of the drawn workpiece in mm;
d is the diameter of the drawn workpiece in mm counted by the middle centerline of the material thickness.

111. Relative drawing height (h/d)
The allowable limit heights of the single-pass drawing for different materials are shown in Table 5.11.
Table 5.11 Limit heights of single-pass drawing
Material
Aluminum
Hard Aluminum
Brass
Mild steel
Relative drawing height (h/d)
0.73～0.75
0.60～0.65
0.75～0.80
0.68～0.72

112. (2) The condition to produce a rectangular workpiece through single-pass drawing is: when the fillet radius of rectangle r = (0.05~0.20) B (where B is the width of the short side of rectangle), the height of the drawn workpiece h ≤ (0.3~0.8) B.
(3) The condition to produce a flanged workpiece through single-pass drawing is:
the ratio of the cylinder diameter d to the blank diameter D satisfies d/D ≥0.4.

113. 2. Demands on the shape of the drawn part
(1) When designing a drawn part, it should be noted clearly either the outer or the inner shape of the drawn workpiece should be guaranteed. The inside and outside dimension of the workpiece cannot be noted at the same time.
(2) The hole should be situated at the flat plane of the drawn workpiece. It is better not to put the hole in the straight-wall zone or the fillet zone of the workpiece, otherwise the hole would difficult to be deformed.
(3) It is better not to adopt complex and asymmetrical drawn workpiece.

114. The shapes of the drawn workpiece can be divided into three types: symmetrical, asymmetrical and that with 3D space curved surface (such as the automobile panel).
The symmetrical workpiece is the easiest for drawing, and the drawing of the 3D space curved surface workpiece is the most difficult.
If possible, the shape of the drawn workpiece should be simplified as much as possible to benefit the drawing forming. The semi-open or the asymmetrical hollow workpiece could be deformed by combining two or more workpieces together first and then separated it by cutting (see Fig. 5.21).

115. Fig. 5.21 Drawing two parts together

116. (4) When drawing a hollow workpiece with complex shape, the intermediate locating datum for the blank should be taken into account.
(5) In the case of drawing workpiece with concave on the flange (see Fig. 5.22), if the axial line of the concave aligns with the drawing direction, it can be drawn out; if the axial line is perpendicular to the drawing direction, it can only be sized after drawing.

117. Fig. 5.22 Drawn part with the concave on the flange

118. 3. Demands on the fillet radius of the drawn part
The fillet radius at the bottom and the flange of the drawn workpiece should not be too small.
If the fillet radius is too small, a sizing process should be added.
The smallest fillet radius without sizing process is shown in Fig

119. Fig. 5.23 Minimum roundness radius

120. 4. Demands on the dimension tolerance grade and surface quality of the drawn part
(1) The tolerance grade of the cross-section dimension of the drawn part is often lower than IT11. If the demand is higher, it can be done through adding a sizing process.
(2) The thickness variation of the drawn part: the thickness of the straight wall along the drawn workpiece could be about 1.2t to 0.6t, and the thicknesses at the four corners of the rectangular workpiece are also increased.
(3) The indentation produced in multi-pass drawing on the outer-wall or the flange surface of the workpiece is allowable.

121. 5.4.2 Drafting of drawing process specification
Take the product shown in Fig as an example to briefly illustrate the drafting of the process specification and its procedure.

122. Fig. 5.24 Diagram of the outer-shell of the temperature-saving device
material: H62M; thickness: t=0.8mm; annual output: pieces
Fig Diagram of the outer-shell of the temperature-saving device

123. 1. Process analysis of the product
(1) Analysis in view of the product material
The plasticity of H62M is good for drawing and forming. The thickness is 0.8 mm. It can be deformed without difficult.
(2) Demand on the product shape and tolerance grade
The section of Φ60 mm at the dimension is a cylinder with a flange, and can be deformed through drawing. The straight-wall section of the dimension Φ34mm can be deformed through drawing or flanging. The roundness radius R=1.5mm, equivalent to R=2t, can be drawn directly. The smaller roundness radius R0.5 can be obtained through sizing.

124. The demands on the tolerance grade of the outer dimension are as follows:
the tolerance grade of the dimension Φ is equivalent to IT12 or IT13. Through drawing process, the tolerance grade of IT12 can be obtained. The tolerance grade of the outer diameter Φ can be obtained through trimming process (trimming by punch). The tolerance grade of the height
, 15.2±0.2 can be
obtained through sizing process.
The tolerance grade of the
height can be realized
through ordinary machining.

125. The width 1. 2 mm of the small long hole equals to 1
The width 1.2 mm of the small long hole equals to 1.5t, can be punched directly. The tolerance grade of the height can also be obtained through punching. Although there are sharp angles in the small, long hole that is difficult to be deformed, this problem can be solved by improving the die structure (using a insert die for instance).
Based on above analysis, the product can be produced through stamping.
The main processes are blanking, drawing, punching, flanging, sizing, trimming and machining the small end of the product.

126. 2. Drafting of technology scheme
A product can often be produced with different technology schemes. It should be discussed and demonstrated according to the practical conditions of the factory. The selected technology scheme most be the reasonable one in economical and technological viewpoint.
At least, the following schemes can be used to produce the product as shown in Fig
Scheme 1: blanking, drawing (Φ60), drawing (Φ34), sizing, trimming, punching the side hole, punching the small, long hole, machining;
Scheme 2: blanking, drawing, punching, flanging, sizing, trimming, punching the side hole, punching the small, long hole, machining;

127. Scheme 3: compound process of blanking, drawing and punching, compound process of flanging and sizing, trimming, punching the side hole, punching the small, long hole, machining.
The difference between scheme 1 and scheme 2 is in making Φ34 straight-wall. It is made through drawing and trimming in scheme 1, which saves a set of dies but consumes more material. This scheme is not reasonable for large batch. More material can be saved by using scheme 2. Although one more process is needed, but compound process can be used to solve this problem. According to the given production output, scheme 2 is more reasonable than scheme 1.

128. The difference between scheme 2 and scheme 3 is that the compound processes are used in most of the forming processes in scheme 3, but only single process is used in scheme 2. For large batch production, adopting compound process may shorten production time, improve labour productivity and reduce product cost, so scheme 3 is more reasonable than scheme 2.
Therefore scheme 3 is used finally.

129. 3. Calculation of the technology procedure (simplified)
(1) Determination of the blank dimension and the sheet width.
(2) Determination of the numbers of drawing.
(3) Determination of the hole dimension to be punched before flanging.
4. Auxilary processes (simplified)
According to the workpiece material and process sequence, the processes such as intermediate heat treatment, lubrication and checking should be schemed. Moreover, the final heat treatment, surface treatment, checking and even the warehouse entry work should also be schemed according to the demands of the product.

130. 5.4.3 Quality analysis of the drawn workpiece
According to practical experience, the reasons causing unqualified workpiece or waste are as follows:
1. The product design can’t meet the drawing process demand;
2. The inappropriate or unqualified material is used;
3. The technological design is unreasonable;
4. The die design and manufacture can’t meet the process demand;
5. The die is incorrectly adjusted or there is an operation mistake.

131. The types and the reasons of the drawn waste, its prevention and solving methods in practice are given in Table 5.12.

136. 5.5 Typical structures of drawing die
According to different working conditions and the equipment used, the structures of the drawing die are also different. The first and subsequent drawing dies are usually classified as follows:

137. 1. Simple first-pass drawing die without blank holder
The typical structure of the simple first-pass drawing die without blank holder is shown in Fig The structure of the die is simple. The upper die (punch) usually is made as a whole. If the diameter of the punch is too small, a upper bolster is needed to increase the contact areas between the punch and slide to prevent the direct contact between the punch bottom section and the press slide, where the contact compressive stress may exceed the allowable stress of the slide material (≤80~90Mpa).

138. In order to strip the workpiece from the punch conveniently, a vent with diameter great than Φ3 mm should be made in the punch.
The guide between the upper and lower die is formed naturally by the material and clearance between the punch and die. It is generally unnecessary to use a pillar and bushing.
The above structure is usually suitable for the drawn workpieces with small drawing depth and the material thickness greater than 2 mm.

139. Fig. 5.25 Drawing die without blank holder
(a) the workpiece is dropped downwards (b) the workpiece is pushed upwards
Fig Drawing die without blank holder

140. 2. Drawing die with blank holder
The typical structure of the drawing die with blank holder is shown in Fig
Recently, the blank holding devices used in factory are mostly installed at the bottom of the press, thus the elastic parts (such as spring, rubber, polyurethane rubber) can be made with large height, or an air cushion can be used to meet the deep drawing demand.
The blank holding device installed in the punch or die can only be used in very shallow drawing.

141. (b) drawing die used in the double action press
First drawing pass
Second drawing pass
(b) drawing die used in the double action press
(a) drawing die used in the single action press
5.26 Drawing die with blank holding device

142. Fig. 5.27 Drawing die used in hydraulic press
The typical structure of the drawing die used in hydraulic press is shown in Fig
(a) first drawing pass (b) second drawing pass
Fig Drawing die used in hydraulic press

144. 落料凹模
成形凹模
成形凸模

145. 固定推料板
组合定位块

146. 上底板
下底板

147. Thank You ！

148. Homework: 1 最小弯曲半径: (中英文) 2 拉深系数： (中英文) 3 图3.2中件1-13的中英文说明(P61)
4 论述拉深的主要问题和解决措施.

149. (当法兰的切向压应力超过了板料临界压应力 P274) 采用适当的拉深系数和压边力.
凸缘主变形区的压力状态
当t/D 较小时，由于变形区外边缘切向压应力很大，易失稳起皱。
措施：压边圈
弹性压边圈和刚性压边圈