J. Cent. South Univ. (2016) 23: 2845-2857

DOI: 10.1007/s11771-016-3348-x

Stress-strain analysis on AA7075 cylindrical parts during hot granule medium pressure forming

DONG Guo-jiang(董国疆)¹, YANG Zhuo-yun(杨卓云)², ZHAO Jian-pei(赵建培)²,

ZHAO Chang-cai(赵长财)², CAO Miao-yan(曹秒艳)³

1. College of Vehicles and Energy, Yanshan University, Qinhuangdao 066004, China;

2. Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education of China

(Yanshan University), Qinhuangdao 066004, China;

3. National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University,

Qinhuangdao 066004, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: Hot granule medium pressure forming (HGMF) is a technology in which heat-resistant granules are used to replace liquids or gases in existing flexible-die forming technology as pressure-transfer medium. Considering the characteristic of granule medium that seals and loads easily, the technology provides a new method to realize the hot deep-drawing forming on high strength aluminum alloy sheet. Based on the pressure-transfer performance test of granule medium and the material performance test of AA7075-T6 sheet, plastic mechanics analysis is conducted for the areas, such as the flange area, force-transfer area and free deforming area, of cylindrical parts deep-drawn by HGMF technology, and the function relation of forming pressure is obtained under the condition of nonuniform distribution of internal pressure. The comparison between theoretical result and experimental data shows that larger deviation occurs in the middle and later period of forming process, and the maximum theoretical forming force is less than the experimental value by 24.6%. The variation tendency of the theoretical thickness curve is close to the practical situation, and the theoretical value basically agrees well with experimental value in the flange area and the top area of spherical cap which is in the free deforming area.

Key words: granule medium; aluminum alloy sheet; deep-drawing; hot forming; limit drawing ratio

1 Introduction

Flexible-die hot forming technology on sheet metal is a kind of advanced flexible forming technique to realize one-time forming of thin-walled and complex parts on light alloy sheet. The existing flexible-die hot forming techniques on sheet metal include warm hydro-mechanical deep drawing (WHDD), quick plastic forming (QPF) with gas as medium, warm viscous pressure forming (WVPF) and hot granule medium pressure forming (HGMF).

WHDD [1-4] technology is applied to manufacturing the parts on sheet metal with characteristics, such as complex shape, varied size, high-quality appearance and small-batch production, which can make the process of manufacturing complex-shape parts on sheet metal more simple and flexible as well as reduce the cost of mould. There are many studies on this technology. POURBOGHRAT et al [1], PALUMBO and PICCININNI [2] and LANG et al [3] respectively studied the constitutive relation, yield performance and forming limit of AA5754 sheet, AA6061 sheet and AA7075 sheet at elevated temperatures. On the basic of these studies, the characteristics and forming mechanism of warm hydro- mechanical deep drawing technology were also explored. However, the technology needs some additional equipments such as a big supercharger, some valves and pipelines which can bear high temperature and pressure, and the heat-resistant property of these equipments limits the temperature range of this technology in the practical application. However, QPF [5-7] technology with advantages, such as optimizing product performance and high production efficiency, can realize quick forming (strain rate is at the level of 10^-2 s^-1) in a larger temperature range, and it overcomes the defects of low production efficiency and expensive material in the process of super plastic forming (SPF). The exploitation of high strain rate material is the key point for the application of QPF technology because the material of product needs to get large deformation at a high strain rate. WVPF [8-10] technology can control the flow of viscous medium and reasonably adjust the magnitude and direction of tangential adhesion force, so the forming quality of parts can be improved. HGMF [11-14] technology is a new technology, in which granule medium replaces the medium in existing flexible-die hot forming process, such as liquids, gases or viscous medium. Hot forming of thin-walled parts on light alloy sheet and tube can be realized by general pressure equipment based on the properties of granule medium, such as withstanding high temperature and pressure, filling well, sealing and loading easily. Taking advantage of the nonuniform pressure-transfer of granule medium and the friction between granules and sheet metal, the forming property of sheet metal can be improved effectively [15-17].

For the flexible-die forming technology, the main theoretical study is about WHDD technology at present, and this technology generally assumes that the distribution of pressure is uniform and there is no friction between liquid and sheet metal. The main characteristics of HGMF technology studied in this work are the nonuniform pressure-transfer and strong friction between medium and sheet metal, which makes the theoretical mechanical model of HGMF technology different from that of existing technologies. In this work, the mechanical model is established to describe the forming process of AA7075 cylindrical part formed by HGMF technology, stress-strain analysis is conducted, and the forming force and thickness distribution curve are obtained as well as verified by test. All these works provide experimental basis and theoretical foundation for the application of this technology.

2 Material performance test

2.1 Material performance test of AA7075 sheet

The experimental material is AA7075 super strength aluminum alloy cold-rolled sheet made by Southwest Aluminum Holdings Ltd., China, and its constitutive equation is obtained by hot uniaxial tensile test (as shown in Fig. 1):

                (1)

where is equivalent stress; is equivalent strain; is equivalent strain rate; T is deformation temperature.

The test indicates that the yield strength σ_s of AA7075 sheet monotonically decreases with the rise of temperature. This relationship can be fitted by the linear equation as follows:

                           (2)

When the strain rate is 0.001 s^-1, the total elongation at maximum force of tensile samples are in 8.09%-12.53% at 25-300 °C (as shown in Fig. 2). And all the samples do not produce necking when the value of deformation degree is not more than 8%. Therefore, when the strain rate is 0.001 s^-1 and engineering strain is 7%, the strain value of tensile sample should be collected so that the Lankford coefficient r can be obtained. The weighted average of plastic strain ratio can be precisely fitted by the following linear equation:

                         (3)

Fig. 1 True stress-true strain curves of AA7075 sheet

Fig. 2 Total elongation at maximum force at different temperatures and strain rates

The r-value of AA7075 sheet is larger in the moderate temperature interval of 250-300 °C, and the value of elongation is the largest at 250 °C. As a result, the best forming temperature is about 250 °C.

2.2 Material performance test of granule medium

As a type of friction material [14-16], granule medium has different constitutive relation and flowing rule compared with liquid and gas. Basing on the requirement of technology in this work, 5^# NMG, a kind of non-metallic granule (NMG) whose diameter is in the range of 0.12-0.14 mm, is regarded as granule medium. Due to the fact that the constituent of NMG, the mechanical and chemical properties can keep stable at moderate temperature (under 400 °C), NMG keeps a certain hardness (HRC 48-55), and there is no cohesion among granules in the condition of high temperature and pressure, namely the cohesive force is zero.

The apparatus of pressure-transfer performance test of NMG is shown in Fig. 3. The curves of the radial pressure p_r along z axis and the axial pressure p_z along r axis can be obtained by the sensors in the wall and bottom of stock bin, and conversions are made for radial pressure p_r and coordinates of z axis as follows:

,                           (4)

where p is the load of punch; r_t is the diameter of stock bin. Consequently, quadratic function relation of α-β which reflects the radial pressure of 5^# NMG medium is obtained as follows:

               (5)

Fig. 3 Principium of solid granule medium pressure-transfer performance test

Similarly, conversions can also be made for axial pressure p_z and coordinates of r axis as follows:

,                           (6)

By the conversions, the quadratic function relation of χ-γ for 5^# NMG medium can be accurately fitted as follows:

                          (7)

3 Stress analysis on HGMF technology

In order to analyze the stress state of cylindrical parts in deep drawing process at moderate temperatures (200-300 °C), the plastic deformation model of sheet metal under non-uniform internal pressure is established, and assumptions are made as follows:

1) Sheet metal is under plane stress state, namely σ_t=0;

2) Sheet metal has the characteristics of planar isotropy and normal anisotropy, and plastic flow fits Hill yield criterion;

3) The thickness of sheet metal is constant in the forming process, namely dε_t=0;

4) The free deformation surface is spherical surface, and the spherical crown is tangent to the fillet of die at the beginning of deep drawing, so it is tangent to the straight wall of die after straight wall is deformed.

3.1 Stress analysis on flange deformation area

The micro-body is taken from flange area (as shown in Fig. 4), and the equilibrium equation in radial direction can be obtained without blank holder. If the angle dθ between the tangent planes of micro-body is small enough, then The equilibrium equation can be given as follows:

                           (8)

where σ_ρ is the radial stress of micro-body (MPa); σ_θ is circumferential stress of micro-body (MPa); dσ_ρ is the increment of radial stress (MPa); ρ represents radial position (mm); dρ represents the increment of radial position (mm).

Fig. 4 Mechanical model of flange deformation zone

The equivalent stress of the material with characteristic of planar isotropic and normal anisotropic can be expressed as

     (9)

In the flange deformation area, σ₁=σ_ρ and σ₃=σ_θ; if dε_t=0, then

                       (10)

By substituting Eq. (9) into Eq. (10), it can be obtained that

              (11)

Substituting Eq. (11) into Eq. (8), the following equation can be obtained:

                             (12)

where R_w is the radius of flange of formed part (mm).

When there is the function of blank holder in the forming process, the radial tensile stress in flange area will increase [18], and the increment value is

                                (13)

where σ_f is the additional radial stress produced by the function of blank holder (MPa); t is the thickness of sheet metal (mm); μ_w is the friction coefficient between sheet metal and die; F_B is the force on flange applied by blank holder (N).

                (14)

where R_b is the radius of die for forming cylindrical part; R_d is the fillet radius of die for forming cylindrical part; p is the smallest unit blank holder force.

In this work, p is given by empirical formula [18]:

       (15)

where R₀ is the original radius of forming sheet metal.

The sheet metal is curved and straightened after going through the fillet of die, the impact of which on radial tensile stress can be obtained as follows:

                          (16)

At the exit position of the fillet of die, the angle of lap is π/2 and ρ=R_b. Therefore, the maximum tensile stress of sheet metal under the common function of blank holder force and the process of curving and straightening can be given as follows:

  (17)

3.2 Stress analysis on free deforming area

The mechanical model of free deforming area is shown in Fig. 5. The assumptions given above set two states in the forming process of parts, namely the spherical crown is respectively tangent to the fillet and wall of die. Therefore, the following relationship can be obtained.

1) If the forming height of part H≤R_b+R_d, then

                  (18)

                        (19)

              (20)

2) If the forming height of part H>R_b+R_d, then

                                    (21)

                                   (22)

               (23)

The analysis on the forming part cut along Section A is implemented (as shown in Fig. 5(a)), and the equilibrium differential equation along axis z of free deforming area is obtained as follows:

μ_nrtanφdrdφ      (24)

where σ_z,b is the axial stress at Section B in free deforming area (MPa); φ and dφ are respectively the angle and the increment of angle between any meridional plane and axis z (rad); r and dr are respectively the radius of latitudinal circle and the increment of radius in any section (mm); μ_n is the friction coefficient between granule medium and sheet metal.

When H>R_b+R_d, the stress state of free deforming area is analyzed by cutting the deforming part along Section B (as shown in Fig. 5(b)), and the equilibrium differential equation as the same as Eq. (24) can also be obtained. At this moment, φ₀=π/2 and r₀=R_b.

Fig. 5 Mechanical model of free deforming area:

According to Eq. (7) which describes the distribution of axial pressure in the pressure-transfer process of 5^# NMG medium, it can be found that

   (25)

where p_d is the internal pressure of solid granule medium which is vertical to the free deforming area (MPa)；p_y is the pressure on the surface of punch (MPa)；h is the distance between the plane in which the radius of latitudinal circle is r and the lower surface of punch (mm). According to the geometrical relationship shown in Fig. 5, h can be expressed in the whole forming process as

                     (26)

where h_w is the distance between the lower surface of punch and the flange of part (mm).

By substituting Eq. (25) and Eq. (26) into Eq. (24), the expression of σ_z,b can be obtained. According to the stress continuity condition, σ_ρ,a=σ_z,b at Section A shown in Fig. 5(a). By combining the above result and Eq. (17) and substituting the geometric parameters of forming part and mould, the pressure of punch p_y can be obtained when H≤R_b+R_d.

3.3 Stress analysis on force-transfer area

Due to the lateral pressure of granule medium on sheet metal, the contact pressure and friction exist between granules and the inner side of sheet metal, and the blank sticks to the straight wall of die tightly, producing a frictional resistance (as shown in Fig. 6).

The height of straight wall is h_z, and the equilibrium equation along z direction is given as follows:

 (27)

where σ_z,a and σ_z,b are radial stresses in the force-transfer area of cylinder wall (MPa); p_r is the internal pressure of granule medium which is perpendicular to the straight wall of cylindrical part (MPa); h_a is the vertical distance between the lower surface of punch and the plane at Section A (mm). According to Fig. 6, h_a=h_w+R_d. h_b is the vertical distance between the lower surface of punch and the plane at Section B (mm), and h_b=h_w+R_d+h_z, and h_z is the height of the straight wall of forming part.

According to Eq. (5) which describes the distribution of radial pressure of 5^# NMG medium, r=R_b in the force-transfer area of cylinder wall, and then p_r(r,h) can be transformed to p_r(h) which is the single value function of h, namely:

     (28)

Based on the continuity condition of stress, σ_ρ,a=σ_z,a at Section A, as shown in Fig. 6. By combining the analysis results of flange and free deforming area, it can be obtained that

              (29)

By combining Eq. (29) with the mechanics analysis expression, the pressure of punch p_y can be obtained when H≥R_b+R_d.

Fig. 6 Mechanical model of force-transfer area

According to the above stress analysis on the flange, free deforming area and force-transfer area in forming process of part, the forming forces under different conditions can be solved. The calculation process is shown in Fig. 7.

The forming force is calculated by MATLAB and the parameters of calculation example are given in Table 1. The theoretical calculation values are compared with the loading curves obtained by technology test, as shown in Fig. 8. Forming temperature is 250 °C, the radius of plate blank is 67.5 mm and the forming height is 45 mm. The loading curve obtained by theoretical calculation starts to deviate far from the experimental result in the middle forming process, and the maximum forming force obtained by theoretical calculation is lower than measured value by 16 kN. Firstly, in the theoretical derivation, the deformation strengthening effect is ignored, which leads to reducing the calculated value of forming force with the development of deformation. Secondly, in the theoretical derivation, it is assumed that the thickness of sheet metal is constant, so the outer edge of flange shrinks faster theoretically than that in actual working condition, leading to the decrease of deep drawing resistance in flange area, then reducing the forming force. These problems become more serious in the last time of forming process, which magnifies the deviation between theoretical result and experimental one. The diameter of outer edge of forming part is 99.52 mm that is obtained by theoretical calculation, while the measured result of technological test is 105 mm. In addition, the discreteness of granule medium makes the pressure-transfer process complicated, and it is difficult to be described by precise mechanical model, which is the main reason of making error. At present, we devote to combining discrete element method and finite element method to realize the coupling of granules and sheet metal. Based on the above reasons, even though there exists deviation between the loading curves obtained by theoretical calculation and the measured curves, the overall variation trend of curves is similar.

Fig. 7 Flowchart for calculating forming force

Table 1 Parameters for calculating forming force

Fig. 8 Comparison of loading curves

4 Strain analysis of HGMF technology

The thickness of sheet metal is considered constant in the process of stress analysis, but the thickness of sheet metal varies seriously in the actual forming process of sheet metal, while the through-thickness normal stress is relatively small. Therefore, the sheet metal is considered to be in plane stress state in the process of strain analysis. From the study on the material performance test of AA7075 sheet, it can be known that Lankford coefficient of aluminum alloy sheet is close to 1 at elevated temperatures (200-300 °C). Therefore, if aluminum alloy sheet is considered as isotropic material in hot forming process, Mises yield criterion can be adopted, and then the stress-strain relationship can be obtained as follows:

                  (30)

where ε_ρ, ε_t and ε_θ are respectively radial strain, through- thickness normal strain and zonal stress. According to the incompressibility condition for plane stress problem, impose α=σ_ρ/σ_θ, and Eq. (30) can be transformed to

                               (31)

Then:

                             (32)

where t₀ and t are respectively thicknesses before and after deformation (mm); R₀ is the initial radius of sheet metal before deformation (mm); R is the radial radius of any position on sheet metal after deformation (mm).

For the outer edge of flange (σ_ρ=0), the thickness is obtained as follows based on Eq. (32):

                            (33)

where R_w is the radius of outer edge of flange after deformation (mm).

In order to obtain the thickness distribution of part deformed by HGMF technology, some stipulations and assumptions are made as follows (as shown in Fig. 9).

1) The original sheet metal is divided into many concentric annuluses along the radius direction, and the concentric annuluses are numbered 1, 2, …, N from the outer edge of flange to the centre. The width of each annulus is △R before deformation, namely △R=R₀/N, and the thickness distribution of the annulus is uniform after deformation. The stress on the contact surface between two adjacent annuluses is continuous.

Fig. 9 Strain analysis in forming process:

2) It is assumed that the stress in the internal surface of the biggest annulus (namely, No. 1 annulus) is in accordance with the radial stress distribution law which is obtained under the condition of invariable thickness.

3) The thickness of annulus at arc OA (or OA′), namely, the fillet of die, is considered invariable. In addition, it is assumed that the annulus at Point A is in plane strain state, namely

4.1 Deformation analysis on flange area

1) Analysis on the element of No. 1 annulus in flange area.

According to Eq. (33), the thickness of the No. 1 annulus after deformation is given by

                          (34)

Based on the incompressible condition, it can be obtained that

                             (35)

where △R is the initial width of annulus (mm); is the length of generatrix of the No. 1 annulus in flange area (mm). Then, the expression of zonal strain of No. 1 annulus in flange area can be obtained:

                          (36)

According to the basic assumptions above and Eq. (12), the radial stress of the No. 1 annulus in flange area can be given by

                      (37)

where β is the correction factor of stress.

2) Analysis on the element of No. k annulus in flange area.

The radial stress of the element of No. k annulus in flange area is given by

                         (38)

where and are respectively the radial stress and radial stress increment of the element of No. k-1 annulus in flange area (MPa).

Based on Mises yield criterion the zonal stress of the element of No. k annulus in flange area can be obtained as follows:

                           (39)

By substituting Eq. (38) and Eq. (39) into Eq. (31) and setting the through-thickness normal strain of the element of No. k annulus can be given by

                  (40)

Then, the thickness of the element of No. k annulus in flange area can be obtained as follows:

                      (41)

According to the incompressible condition, the length of generatrix of the element of No. k annulus after deformation can be given by:

                   (42)

Therefore, the zonal strain of the element of No.k annulus can be expressed as

                       (43)

For the element of any annulus in flange area, the equilibrium equation of radial stress can be obtained as follows:

(44)

According to the above deformation analysis on flange area, the thickness distribution of sheet metal can be calculated by the iterative operation which stops when . Then, the deformation analysis on the force-transfer area or free deforming area is implemented according to the geometrical relationship of the objective forming part and mould.

4.2 Deformation analysis on force-transfer area

If and , then the height of A′B in force-transfer area is and the spherical crown in free deforming area at the bottom part is tangent to the straight wall of die at point B. At this moment, the element of No. k+1 annulus in flange deforming area begins to turn into force-transfer area, namely the Section A′ shown in Fig. 9(b). The element can be considered as the element of No. 1 annulus in force-transfer area.

1) By ignoring the thickness variation of the element at arc OA′ (the fillet of die), the boundary condition of the element of No. 1 annulus in force- transfer area is

                            (45)

                                   (46)

                                    (47)

                                (48)

                    (49)

where and are respectively the radial stress (MPa) and the circumferential strain on upper surface of the element of No. 1 annulus in force-transfer area; , and are respectively the thickness, the radius of internal surface and the length of generatrix of the element of No. 1 annulus in force-transfer area (mm).

The equilibrium equation of the element of No. 1 annulus in force-transfer area along direction z can be expressed as follows:

          (50)

where and are respectively the radial stress increment (MPa) and the positive pressure given by granule medium (MPa) of the element of No. 1 annulus in force-transfer area.

2) The element of any No. m annulus in force- transfer area.

Radial stress and zonal strain of the element of No. m annulus in force-transfer area can be given by

                         (51)

     (52)

where and are respectively the radial stress (MPa) and radial stress increment (MPa) of the element of No. m-1 annulus in force-transfer area; and are respectively the zonal strain and zonal strain increment of the element of No. m-1 annulus in force-transfer area.

According to Mises yield criterion, zonal stress of the element of No. m-1 annulus in force-transfer area can be given byImpose and the thickness of the element of No. m annulus in force-transfer area after deformation is

                      (53)

The radius of internal surface of the element of No. m annulus in force-transfer area can be given by . According to the incompressible condition, the generatrix length of the element of NO. m annulus after deformation can be given by

             (54)

Based on the equilibrium equation of the element of No. m annulus along direction z, it can be obtained that

  (55)

where is the pressure which is given by granule medium on the element of No. m annulus in free deforming area (MPa).

The thickness distribution of sheet metal can be calculated by the iterative operation which stops when . Then, the deformation analysis on free deforming area begins to be implemented.

4.3 Deformation analysis on free deforming area

If and , the force-transfer area has not formed at this moment, and the spherical crown in free deforming area at the bottom of part is tangent to the fillet of die. At this moment, the element of No. k+1 annulus in flange deforming area just enters into die, namely, the Section A shown in Fig. 9(a). For analyzing expediently, the element is considered as the element of No. 1 annulus in free deforming area.

1) The element of No. 1 annulus in free deforming area.

Geometric boundary condition is

                  (56)

                        (57)

                                (58)

where r₀ is the radius of spherical crown in free deforming area (mm); is the radius of latitudinal circle of the element of No. 1 annulus in free deforming area (mm); is the angle between the element of No. 1 annulus in free deforming area and z axis (rad).

By ignoring the thickness variation of the element at arc OA (the fillet of die), the boundary condition of the element of No. 1 annulus in free deforming area is

                            (59)

                                   (60)

                                    (61)

                    (62)

where , , and are respectively the radial stress (MPa), zonal strain, thickness (mm) and the length (mm) of generatrix of the element of No. 1 annulus in free deforming area; σ_w is the stress to straighten the bend when sheet metal goes through the fillet of die, which can be calculated based on Eq. (16).

The force equilibrium equation of the element of No. 1 annulus along direction z can be obtained as follows:

              (63)

where is the pressure which is given by granule medium on the element of No. 1 annulus in free deforming area (MPa).

(2) The element of any NO. j annulus in free deforming area.

The geometric boundary condition can be obtained from Fig. 9(a):

                            (64)

                               (65)

where is the angle between the element of No. j annulus in free deforming area and z axis (rad); is the radius of latitudinal circle in the element of No. j annulus in free deforming area (mm).

The radial stress and zonal strain of the element of NO. j annulus in free deforming area are respectively expressed as follows:

                          (66)

 (67)

where and are respectively the radial stress and radial stress increment of the element of No. j annulus in free deforming area (MPa); and are respectively zonal strain and zonal strain increment.

According to the Laplace equation in the non-torque thin shell theory, there is

                              (68)

where R_ρ and ρ_θ are respectively the radial and zonal curvature radii of any point in free deforming area (mm); p is the internal pressure on free deforming area (MPa)

When the free deforming area is the surface of spherical crown, . By substituting Eq. (66) into Eq. (68), the zonal stress of the element of No. j annulus can be obtained:

                        (69)

where is the pressure which is given by granule medium on the element of No. j annulus in free deforming area (MPa).

Impose and the thickness of the element of No. j annulus can be expressed as

                       (70)

According to the incompressible condition, the length of generatrix of the element of No. j annulus in free deforming area after deformation can be obtained as follows:

         (71)

The equilibrium equation of the element of No. j annulus along direction z can be expressed as follows:

             (72)

When the force-transfer area of part has formed, and the free deforming area is a spherical crown which is tangent to the straight wall in force- transfer area at Section B (as shown in Fig. 9(b)). Considering the deformation analysis of the flange area and the force-transfer area, the element at Section B in free deforming area is the No. k+m+1 element of sheet metal. If the No. k+m+1 element of sheet metal is considered as the No. 1 element in the free deforming area, according to the basic assumptions of deformation analysis above, the geometric boundary condition of the free deforming area can be transformed into

                                    (73)

                                   (74)

                                (75)

If the following expression for the generatrix length of the No.1 element of the spherical crown in free deforming area is given:

            (76)

The stress at Section B where the spherical crown in free deforming area is tangent to the straight wall in force-transfer area is continuous, and then,

                          (77)

The equilibrium equation of the element of No. 1 annulus along direction z can be expressed as follows:

              (78)

When the free deforming area is the spherical crown, the original deformation conditions are given by the expressions from Eq. (73) to Eq. (78). Then, iterative operation can be conducted by expressions from Eq. (64) to Eq. (72) that describe the deformation analysis of the element of any No. j annulus in free deforming area.

According to the above deformation analysis on the free deforming area, the iterative operation stops when , and the thickness of the element of each annulus can be calculated.

Based on the assumptions above, the wall thickness distribution of thin cylindrical part can be predicted by the deformation analysis of the flange area, the free deforming area and the force-transfer area in forming process. The flowchart is shown in Fig. 10.

Fig. 10 Flowchart for calculating forming thickness

According to the parameters in Table 1, the forming temperature T=250 °C, the initial radius of sheet metal R₀=67.5 mm and the height of deformed part H=30 mm, and then, the pressure of punch p_y and the radius of outer edge of flange R_w could be calculated by the calculation procedure of forming pressure. Then, the above calculated parameters were input into the calculation procedure of thickness to obtain the theoretical thickness distribution curve of forming part. The theoretical curve is compared with the measured curve of part under the same condition, as shown in Fig. 11.

Fig. 11 Comparison of thickness curves

It is known from Fig. 11 that there exist disparities in the fillet transition zone between the theoretic thickness distribution curve and the measured data in the test. The main reason is that the deformation of the element in flange area directly turns into the free deforming area by ignoring the deformation of the element in the fillet of die during the theoretical derivation of the thickness of part. But the variation tendency of the theoretical thickness curve is close to the measured data and they are basically consistent in the flange deformation area and the top area of spherical crown in free deforming area.

5 Conclusions

1) The plastic mechanics analysis on the flange area, force-transfer area and free deforming area is conducted in the forming process of cylindrical parts deformed by the HGMF technology. The function relation of forming pressure in the drawing process of cylindrical parts is obtained under the condition of the non-uniform distribution of internal pressure. The theoretical loading curve is compared with the measured data, which shows that the deviation occurs in the middle and later period of forming process and the maximum theoretical forming force is less than the measured data by 24.6%. However, the variation tendency of the loading curves is similar.

2) The deformation analysis on the flange area, force-transfer area and free deforming area is conducted in the forming process of cylindrical parts deformed by the HGMF technology. The variation tendency of the theoretical thickness curve is close to the measured data, and the theoretical thickness of flange deformation area and the top area of spherical crown in free deforming area are basically consistent with the experimental results.

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(Edited by YANG Bing)

Foundation item: Projects(51305386, 51305385) supported by the National Natural Science Foundation of China; Project(E2013203093) supported by the Natural Science Foundation of Hebei Province, China

Received date: 2015-09-07; Accepted date: 2016-11-18

Corresponding author: ZHAO Chang-cai, Professor, PhD; Tel/Fax: +86-335-8057031; E-mail: zhao1964@ysu.edu.cn