Study on the mechanism of unidirectional loading rubber bulging of bionic egg-shaped shell

Aiming at the problems of complex processing technology, uneven deformation, and poor forming accuracy of multi-curvature positive Gaussian slewing egg-shaped shells, it is proposed to use unidirectional loading rubber flexible medium bulging technology to improve the tube deformation distribution and enhance the processing accuracy of plastic forming of bionic egg-shaped shells. By analyzing the structure and manufacturing method of egg-shaped shells, this paper explores the optimization problems in the rubber bulging technology of non-homogeneous deformation egg-shaped shells. The simulation analysis and experimental verification methods are used to study the different technological parameters and forming accuracy of rubber bulging of egg-shaped shells under unidirectional loading and to master the deformation mechanism of rubber bulging of egg-shaped shell structural components under unidirectional loading. The study shows that optimizing technological parameters in unidirectional loading mode can effectively improve the uniformity of strain distribution and thickness distribution during the forming of egg-shaped shells, thus improving the processing accuracy of plastic deformation of bionic egg-shaped thin-walled parts. Furthermore, the simulation of the rubber bulging technology of the egg-shaped shell after the optimization of technological parameters is in good agreement with the experiment.


Introduction
In recent years, with the continuous promotion of the national strategy of strengthening the country's oceans and the increasing requirements for deep-sea exploration, the performance requirements of submersibles have become increasingly high. As an essential part of the submersible, the shell plays a role in ensuring the regular operation of the internal equipment and the health and safety of the personnel during the dive, and its weight accounts for about 1/2 to 1/4 of the total weight of the submersible [1]. The shell of a deep-sea submersible is generally a spherical shell structure with iso-curvature, which has excellent stability, low buoyancy coefficient, and high material utilization rate. However, its disadvantages are low space utilization, poor hydrodynamic characteristics, and difficult manufacturing. Whereas with the emergence of bionic structure, the typical egg-shaped structure shell has the advantages of excellent streamlining, weight-to-intensity ratio [2], which can obtain sufficient strength and stability with minimum material and exhibit superb pressure resistance and stability [3].
At present, there is a complex manufacturing technology for bionic shell structural components. The special-shaped pressure-resistant structural components are mainly manufactured by the combination of welding. However, the primary defect of this manufacturing method is uneven material distribution, which seriously affects the stability of the pressure-resistant structural components during service. Among the existing manufacturing technologies, the technology that can form the structural components with the curved surface integrally is flexible medium internal high-pressure bulging, including hydraulic bulging, granular medium bulging, and rubber bulging. Among them, hydraulic bulging and granular media bulging refer to the use of discrete media pressure instead of a rigid convex or concave die for plastic processing of the shell [4]. However, since liquids and particles are flowing media, the pressure load needs to be guaranteed, so the sealing requirements are stringent. Therefore, the equipment requirements and the manufacturing costs for hydraulic bulging and granular media bulging are high [5,6]. Rubber medium is a continuous medium. According to the mechanical characteristics, the bulging process of rubber medium can be divided into three stages: free forming stage, unstable forming stage, and stable forming stage. Since rubber has the characteristic of liquid isotropic flow under a high-pressure environment, its unique advantages are as follows: (1) Excellent versatility, simple forming equipment [7], simple operation process, and no sealing device is required [8], which will not cause leakage and pollution problems due to insufficient sealing.
(2) It can effectively shorten the forming process, shorten the production preparation cycle, reduce springback, and improve surface quality [9][10][11]. Since the bionic egg-shaped shell is a multi-curvature positive Gaussian rotary shell, which demands a low thinning rate and high processing accuracy in service, it is proposed to use rubber flexible medium bulging technology for plastic processing of structural parts of the bionic egg-shaped shell.
In the process of rubber bulging, the technological parameters have significant influence on forming accuracy. Presently, domestic and foreign scholars have conducted extensive research on rubber bulging. SUN et al. [12] investigated the effect of different friction coefficients on strain distribution during rubber bulging of Ti-15-3 alloy sheets. Moreover, they established a simulation analysis model to determine the friction coefficient for rubber bulging of Ti-15-3 alloy sheet at room temperature by comparing data from experimental and simulation. Liang et al. [13] researched the fabrication method of cylindrical collision absorbers with the discontinuous bulge. They proposed the fabrication method using continuous partial rubber local bulging and verified the formability by numerical simulation. Belhassen et al. [14] investigated the impact of different hardness rubbers on sheet forming properties in flexible rubber bulging of AA1050-H14 aluminum metal plate. It was shown that polyurethane rubber with a Shore A hardness of 70 gave better sheet formability after bulging compared to silicone rubber and natural rubber. Koubaa et al. [15] performed rubber bulging of aluminum alloy sheets. They analyzed the effects of different friction coefficients and rubber of different hardness from three aspects: thickness, damage and formability. The results showed that the thickness thinning areas differed with different friction coefficients. Furthermore, rubber with higher hardness can significantly reduce the thinning of the sheet, thus reducing the damage and improving the formability. Nosrat et al. [16] investigated the feasibility of using the rubber bulging method to manufacture cam-shaped 304 stainless steel tubes and compared it with the hydroforming method. The results showed that the rubber bulging method could compete with hydroforming under the proper setting of technological variables. Ramezani et al. [17] used polyurethane rubber to bulge OFHC copper sheets. Moreover, they proposed a new dynamic bulging test technique to evaluate the formability of sheet metal. Xu et al. [18] studied the forming behavior of AZ31 sheets in rubber thermoforming. They proposed the method of coupling temperature and rubber flexible medium for forming. The results showed that the formability of the AZ31 sheet was significantly improved by the coupling of temperature and rubber medium. Xiang et al. [19] established various deformation patterns using rubber with different structures to study the effect of different elastic deformation states of rubber on the formability of metal sheets. Their study showed that the forming load, bulging height, and strain could be adjusted using rubber with different structures. Although scholars at domestic and foreign level are deep enough in studying rubber flexible media bulging, they have studied rubber bulging primarily for metal plates, and few have studied rubber bulging methods for multi-curvature shells.
This paper establishes an egg-shaped shell rubber bulging model from the egg-shaped shell structure and manufacturing method. In order to optimize the rubber bulging technology of egg-shaped shells under unidirectional loading, the effects of tube blank size, mold structure, lubrication method, and hydraulic loading speed on the forming accuracy of egg-shaped shells from three aspects, namely, strain, thickness, and forming limit are analyzed. Moreover, this paper verifies the accuracy of the rubber bulging model of the egg-shaped shell after optimization of technological parameters by using the method of simulation analysis and experimental verification.

Characteristics and processing method of bionic egg-shaped shell
The contour curve of the bionic egg-shell is commonly described by the N-R egg-shaped function. The geometry Eq. (1) of the egg-shell curve is established according to the N-R egg-shaped function and the structural characteristics of the egg-shell. Furthermore, the egg-shaped structure model is made according to Eq. (1) egg-shell curve, and the coordinate diagram of bionic egg-shaped shell is obtained, as shown in Fig. 1. At present, the manufacturing of closed shells with curved surfaces (pressure-resistant shells, high-pressure vessels) is mainly prepared by dividing the closed shells into units and then assembling and welding them after completing the forming of the units. The way of dividing the unit uses the warp and weft line to divide the small unit, as shown in Fig. 2a, b. Figure 2a shows the use of warp lines to divide the units, and Fig. 2b shows the use of weft lines to divide the units. The contour of the egg-shaped shell is a positive Gaussian surface. The method of dividing the units increases the difficulty of machining, and the initial geometric defects are relatively significant. Meanwhile, this method will produce more welds, which will increase the risk of weld cracking and seriously threatens the gas tightness of the closed shell. Figure 2c shows the integrative bulging manufacturing method of egg-shaped shell structural members. This method divides the egg-shaped shell structural components into three parts and completes the preparation of the egg-shaped shell in the form of end caps at both ends and profiled tubes in the middle, which effectively reduces the number of units. This paper studies the technique by using scaled-down parts.
The advantages and disadvantages of the three different egg-shaped shell unitized welding fabrication methods are compared according to three aspects: the area of the unit, the number of welds, and the curvature of the unit, as shown in Table 1. Among the three types of collocation welding methods, weft unit collocation welding has the smallest unit area and the most welding seams. The curvature of the unit varies greatly and the processing procedure is tedious. The unit area, number of welds, and curvature variation of warp unit collocation welding are better than that of weft unit collocation welding, and the processing procedure is slightly less. The bulging collocation welding unit consisting of three parts has the largest area, the smallest number of welds, and the minor change in curvature of the bulging unit. Based on the comparison of the three values, the manufacturing technology of the bulging collocation welding unit can effectively reduce the processing procedure, reduce the number of welds and increase the forming accuracy of the egg-shaped shell.

Egg-shaped shell rubber bulging technology
Egg-shaped shell rubber bulging technology equipment including mold, steel column, rubber and tube, mold is divided into front and rear mold, rubber bulging technology schematic diagram is shown in Fig. 3. The mold and the steel column are made of CrWMn high carbon alloy tool steel. The rubber is made of cylindrical solid natural rubber with polyisoprene as the main component. The material of the tube blank is SUS304 stainless steel. The rubber bulging technology for egg-shaped shells is divided into three stages, namely the initial stage, the processing stage, and the snuggling closely to the mold stage.
In the initial stage of rubber bulging, the tube is put in with snuggling close to the rear mold, then a 250-mm-long cylindrical natural rubber with a steel column of the same diameter is put in. Finally, the front and rear molds are attached to make the mold closed, as shown in Fig. 3a. In the rubber bulging processing stage, the rubber is subjected to unidirectional pressure by unidirectional loading on the steel column. The rubber undergoes axial compression and expansion, resulting in axial feeding and plastic deformation of the tube, as shown in Fig. 3b. In the snuggling closely to the mold stage of rubber bulging, the tube blank is deformed under force in the thickness direction generated by the compression and expansion of the rubber until it fits the mold cavity completely, eventually completing the rubber bulging of the egg-shaped shell, as shown in Fig. 3c. The overall assembly of the egg-shaped shell rubber bulging technology is shown in Fig. 3d.

Rubber material constitutive model
The rubber material is a typical hyperelastic material. There are two main methods for describing the mechanical properties of hyperelastic materials. One is the phenomenological theory based on the mechanics of continuous media, and the common models are Mooney-Rivlin, Yeoh, Ogden, etc. The other is the statistical theory based on the change of molecular structure and conformational entropy, and the typical models are Arruda-Boyce, etc. The material behavior of isotropic hyperelastic rubber materials is often described by the strain energy density function of the phenomenological theory [20,21]. The rubber is an incompressible material in the phenomenological theory, and the Mooney-Rivlin model under this assumption is widely used to simulate the nonlinear deformation behavior of rubber materials [22]. The strain energy density function of the Mooney-Rivlin model is as follows: where W is the strain potential energy, C 10 and C 01 are the mechanical property constants to be determined, D is the coefficient describing the compressibility of the material, I 1 and I 2 are the first and second strain invariants, and J is the volume ratio. I 1 , I 2 is related to the stretching ratio i : Since rubber is assumed to be an incompressible material, J = 1 2 3 = 1 , obtaining the two-parameter Mooney-Rivlin model commonly used in engineering: According to the relationship between Kirchoff stress tensor and Green strain tensor, the relationship between the principal stress 1 and the principal stretch ratio 1 of incompressible rubber material under the uniaxial tensile test can be obtained: where 1 = 1 + 1 , 1 is the strain in the tensile direction, bringing Eq. (4) into (5) to obtain: The stress values 1 at different stretching ratios 1 were measured by the unidirectional tensile test, and the experimental data were fitted to a line with 1 1 as the abscissa and as the ordinate, with C 10 as the intercept and C 01 as the slope. Thus, the rubber material parameters of the twoparameter Mooney-Rivlin model were determined.

Analysis of the stress state of rubber under unidirectional compression
To investigate the mechanism of rubber bulging technology for egg-shaped shells under unidirectional loading, this paper analyzes the stress state of rubber under unidirectional loading by simulating the bulging process of cylindrical rubber in unidirectional compression. The stress state when the rubber expands is shown in Fig. 4. The rubber expansion region is in a three-dimensional stress state, subjected to compressive stress, circumferential tensile stress, and stress in the thickness direction. Under unidirectional loading force, the top of the rubber is subjected to uniform pressure, and the pressure is transmitted downward along the axial direction. The bottom part is subjected to the same size of reaction force and transmitted upward along the axial direction. The stresses in two different directions cause the rubber to expand near the middle part, generating force in the thickness direction. The tube blank is plastically deformed under the action of force in the thickness direction until snuggling closely to the mold, and finally, the egg-shaped shell structural part is obtained.

Egg-shaped shell rubber bulging model
The egg-shaped shell rubber bulging process is a quasistatic problem that includes large deformation of the rubber material, nonlinear material behavior, and complex frictional contact conditions. An explicit dynamic algorithm is used to simulate and analyze the problem. Explicit methods determine the solution by advancing the state of motion from one time increment to the next, without iteration. Therefore, for complex problems such as dynamic events, nonlinear material behavior, and complex intercontact interactions, explicit methods provide more accurate computational results for this problem. In the simulation analysis model, the mold uses a discrete rigid body, which can effectively reduce the calculation volume without affecting the calculation results. The tube blank is described by reduced-integral S4R Lagrangian cells, and the shell cells are used to ensure the accuracy of the calculation. The Mooney-Rivlin model is used for the rubber material with material parameters C10 of 3.2955, C01 of 1.1534 and D1 of 0.0004495, and the element model for the rubber is C3D8I. Hard contact is used between the tube and the mold, and the rubber, respectively. The coefficient of friction between the tube and the mold is 0.1 under the penalty friction formula, and the coefficient of friction between the tube and the rubber is 0.15. Unidirectional loading of the rubber in the axial direction deforms the rubber, thus inducing plastic deformation of the tube. SUS304 stainless steel is widely used in marine equipment manufacturing due to its abrasion resistance, low-temperature resistance, thermal expansion properties, and good thermal insulation performance. The material property parameters of SUS304 stainless steel obtained by the unidirectional tensile test are shown in Table 2. The stress-strain curve and tensile specimens are shown in  strain-hardening index of 0.31, a plasticity-strain ratio of 0.9, and an elongation of 55%.

Process of egg-shaped shell rubber bulging
The egg-shaped shell rubber bulging process is accomplished under a 100-t hydraulic press. Due to the unique nature of the rubber material, controlling the displacement of the indenter, i.e., controlling the compression of the rubber, can effectively control the processing accuracy of the egg-shaped shell. Therefore, the rubber bulging of the tube blank is carried out by controlling the lowering displacement of the hydraulic press head. The relationship between displacement and pressure is shown in Fig. 6. Before the displacement reaches 60 mm, the increased amplitude of pressure is more excellent. After the displacement reaches 60 mm, the increased amplitude of pressure is flatter. The egg-shaped shell rubber bulging process is shown in Fig. 7. As the displacement increases, the degree of rubber compression increases. In the initial bulging stage, the displacement reaches 50 mm, and the tube blank has already undergone significant plastic deformation. The stress distribution in the middle region of the tube blank is more uniform, with a size of about 400 MPa, and the ends are subjected to the least stress, at about 190 MPa. When the displacement reaches 70 mm, the stress in the middle region of the tube blank is between 700 and 800 MPa, and the deformation in the middle region is uniform. When the displacement reaches 100 mm, the tube blank has been snuggled closely to the mold, and the stress in the middle region of the egg-shaped shell is the largest, reaching about 1000 MPa. The stress is symmetrically distributed along the middle region of the egg-shaped shell, and the stress at the top and bottom of the egg-shaped shell is approximately 400 MPa.

Optimization of the tube blank size
In the rubber bulging technology of egg-shaped shell components, the size of the tube blank's initial diameter significantly influences the structure and forming accuracy of the egg-shaped shell when the length of the tube blank is 150 mm, and the thickness of 1 mm is always constant. Therefore, the size of the tube blank in the egg-shaped shell rubber bulging technology is optimized. In order to obtain an egg-shaped shell with excellent forming accuracy, optimization was conducted on the tube blank diameter between 60 and 80 mm. Compare the egg-shaped shell structures after bulging the tube blanks with diameters of 60 mm, 65 mm, 70 mm, 75 mm, and 80 mm, as shown in Fig. 8. Each egg-shaped structure has a different position in the whole egg-shaped shell structure and a different surface area. The surface area of each egg-shaped structure is shown in Fig. 9. When the diameter of the tube blank is 60 mm, the surface area of the egg-shaped shell after bulging is the largest, reaching 360.07 cm 2 ; when the diameter of the tube blank is 80 mm, the surface area of the egg-shaped shell after bulging is the smallest, reaching 282.99 cm 2 . It can be seen that the larger the diameter of the tube blank, the smaller the surface area of the egg-shaped shell after bulging, and the smaller the bulging region will be. To obtain the optimal values of the five tube blanks diameters, the maximum strain and maximum thinning rate of different egg-shaped shells after bulging were compared by simulating the bulging of the tube blanks with different diameters to judge the forming, as shown in Fig. 10. The strain indicates the degree of deformation of the tube blank. The thinning rate indicates the thickness change of the tube blank. Therefore, when the maximum strain of the tube blank is relatively large, and the maximum thinning is relatively small, the plastic deformation of the tube blank is sufficient, and the formability is better. It can be seen from the figure that although the maximum thinning rate of the egg-shaped shell at the tube blank diameter of 80 mm is the smallest, reaching 21.7%, the maximum strain is also the smallest, and the plastic deformation of the egg-shaped shell is not sufficient. Whereas the maximum strain of the egg-shaped shell at the tube blank diameter of 70 mm is relatively large, reaching 0.368, while the maximum thinning rate is 26.3%, so the plastic deformation of the egg-shaped shell is sufficient. Thus, the best formability of the egg-shaped shell is achieved for the tube blank diameter of 70 mm. So the final determination of the tube blank is 150 mm long, 70 mm diameter, and 1 mm thickness of SUS304 stainless steel tube.

Optimization of the mold structure
In the egg-shaped shell rubber bulging technology, the structure of the mold also influences the processing accuracy of the egg-shaped shell. Especially the edge line at the top of the egg-shaped cavity in the mold, it will not only make the flow of the tube blank axial feeding poor but also cause the stress concentration in the tube blank at the location of this edge line, eventually leading to the cracking of the tube blank. Therefore, in order to eliminate the internal stress at the edge line and increase the flowability of the tube blank axial feeding, the rounded corner should be rounded at this edge line position. However, the size of the rounded corners also affects the formability of the egg-shaped shell. Therefore, to obtain egg-shaped shells with better forming accuracy and machining precision, the rounded corner sizes are optimized. According to the mold size, the radius optimization range of the rounded corners at the top edge line of the egg-shaped cavity inside the mold was determined to be 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm, and the mold structure with different rounded corners as shown in Fig. 11.
The rubber bulging of five different egg-shaped shells with different rounded corner sizes was simulated separately to obtain the equivalent stress distribution at the top edge of different egg-shaped shells after bulging. The path of the top edge of the egg-shaped shell is shown in Fig. 12. When the rounded corner radius is 1 mm, the equivalent stress at the top edge of the eggshaped shell after bulging fluctuates wildly and is not uniformly distributed. When the rounded corner radius is 2 mm, compared with other sizes of rounded corner radius, the equivalent stress at the top edge of the eggshaped shell after bulging has the slightest fluctuation   Fig. 13. The mean and standard deviation of the equivalent stress at the top edge of the egg-shaped shell after bulging with different rounded corners sizes are shown in Fig. 14. The smaller the average value of the equivalent stress, the smaller the internal stress at the edge. The standard deviation indicates the uniformity of the equivalent stress distribution at the edge, and the smaller the standard deviation, the more uniform the equivalent stress distribution. When the rounded corner radius is 1 mm, the average equivalent stress at the top edge of the egg-shaped shell after bulging is the largest among the five rounded corner sizes, and it is still affected by the internal stress. When the rounded corner radius is 3 mm, the average value of the equivalent stress at the top edge of the egg-shaped shell after bulging is the smallest, and the internal stress is also the smallest among the five rounded corner sizes. Combining the average value of the equivalent stress and the uniformity of the equivalent stress distribution, a rounded corner with a radius of 3 mm has the best effect of reducing the internal stress, and the equivalent stress is also uniformly distributed at the edge line. Therefore, the radius of the rounded corner at the top of the edge line of the egg-shaped cavity in the mold is determined to be 3 mm.

The influence of lubrication method on rubber bulging
In the egg-shaped shell rubber bulging process, the different lubrication conditions significantly impact the forming results of the egg-shaped shell. The main ways of lubrication are liquid lubrication and solid lubrication. Among them, liquid lubrication is mainly with lubricating oil, according to the type of oil is  On the other hand, solid lubrication mainly includes graphite powder, molybdenum disulfide powder, etc. Its advantages are light weight and small size, which can be used in special working conditions and harsh environments. But the disadvantage is that the coefficient of friction is large, usually 50 to 100 times that of liquid lubrication, which will produce friction chips, noise, etc. Therefore, to reduce the friction coefficient between technological equipment as much as possible and improve the forming accuracy of the egg-shaped shell, lubricating mechanical oil is chosen as the lubricating medium.
In the egg-shaped shell rubber bulging technological equipment, the tube blank and mold both belong to the steel material, the coefficient of friction is about 0.15 under the condition of no lubrication, and the coefficient of friction is about 0.1 under the condition of using lubricated mechanical oil. Due to the different materials of rubber and tube blank, the friction coefficient between them with and without lubrication has an enormous difference. The coefficient of friction between rubber and tube blank under non-lubricated condition is between 0.3 and 0.6 and decreases to between 0.15 and 0.2 when using lubricated mechanical oil. Since the difference in friction coefficient between the tube blank and the mold is small with or without lubrication, the influence of lubrication between the rubber and the tube blank on the formability of the egg-shaped shell is mainly investigated. The rubber bulging technology of egg-shaped shells under different lubrication conditions were simulated. The lubrication between the tube blank and the mold is always maintained, and the different lubrication conditions are indicated by changing the friction coefficient between the rubber and the tube blanks. Under the condition of no lubrication, the tube blank generates axial compressive stress in the process of axial feeding. As the axial loading of rubber increases, the axial compressive stress on the tube blank increases. When the axial compressive stress exceeds the critical value, the material given to the upper part of the tube blank in the axial direction cannot be replenished to the bulging area in time, which will result in the accumulation of material and the occurrence of circumferential wrinkling destabilization. The egg-shaped shell obtained with no lubrication in the manufacturing process is shown in Fig. 15a. Under the condition of using lubricating mechanical oil, although the tube blank will also produce axial compressive stress due to the reduction of friction, the axial compressive stress always does not exceed the critical value. Therefore, the upper part of the tube blank axially given to the material can also be timely replenished to the bulging area, and there will be no wrinkling destabilization. The egg-shaped shell obtained with lubricating oil in the manufacturing process is shown in Fig. 15b.
In order to investigate further the effect of lubrication conditions between rubber and tube blanks on the formability of egg-shaped shells, strain distribution, thickness distribution, and forming limit of egg-shaped shells were investigated. The strain distributions between the rubber and the tube blank under different lubrication conditions are shown in Fig. 16. Under unlubricated conditions, the maximum strain in the wrinkled region of the egg-shaped shell reaches 0.26. Compared to the egg-shaped shell with lubricant oil, the overall strain in the area from the top of the egg-shaped shell to the wrinkled area is higher. However, the strain tends to decrease in the wrinkled transition region, reaching a minimum of 0.114. This is because the ring bulge is generated above the transition region due to wrinkling, in which region the relative deformation is minor, and the strain is low. The strain starts to increase in the bulging region after the transition region. Since there is no axial material replenishment in the bulging region, the strain in the equatorial region of the egg-shaped shell, where plastic deformation is most significant, is significantly less under unlubricated conditions than under lubricated oil conditions. From the overall strain distribution, the strain distribution of the egg-shaped shell in the case of lubricant oil between the rubber and the tube blank is uniform, reaching the maximum strain of 0.383, and the forming accuracy is excellent. In contrast, the egg-shaped shell's maximum strain without lubricant oil reaches 0.374, and the overall strain distribution is not uniform, with poor processing accuracy and formability.
Under non-lubricated conditions, the wrinkled area of the egg-shaped shell increases in thickness due to the accumulation of materials, with the maximum thickness being about 1.04 mm, which is greater than the initial thickness of the tube blank. In the wrinkled transition region, the thickness decreases and reaches a minimum of 0.89 mm. The thickness starts to decrease further in the bulging region after the transition region. The thickness of the egg-shaped shell in the condition of using lubricating oil shows a trend of decreasing and then increasing due to the timely replenishment of the material. From the overall thickness distribution, the thickness distribution of the egg-shaped shell in the case of using lubricant oil between rubber and tube blank is uniform, with a minimum thickness of 0.737 mm and better forming accuracy.
In the case of no lubricating oil, the minimum thickness reaches 0.83 mm, but the overall thickness is greater than the egg-shaped shell in the case of using lubricating oil, and the thickness distribution is not uniform. The thickness distributions of the egg-shaped shell under different lubrication between rubber and tube blank are shown in Fig. 17.
Under the condition that there is no lubrication between rubber and tube blank, most areas of the tube blank can be deformed typically in the plastic deformation process. In contrast, a few areas have the tendency to wrinkle and destabilization. In addition, a small part of the area is entirely wrinkled, resulting in defects and poor overall forming accuracy of the egg-shaped shell. In the case of using lubricating oil between rubber and tube blank, the tube blank can be plastically deformed in anticipation until snuggling closely to the mold, and there is no tendency of wrinkling and wrinkling instability. Moreover, there is no rupture of the egg-shaped shell, and its overall forming accuracy is excellent. Forming limit curve (FLC) is an effective method to determine the formability of a material. The FLC was obtained by the rigid convex die bulging test method. The final plot was obtained by drawing the specimens of different widths of 304 stainless steels and measuring the major and minor strains in the necking or rupture zones. The forming limits of egg-shaped shells with different lubrication conditions between the rubber and tube blank are shown in Fig. 18.
From the strain distribution, thickness distribution, and forming limit of the egg-shaped shell under different lubrication conditions between the rubber and the tube blank, the forming accuracy of the egg-shaped shell is better under the lubrication conditions between the rubber and the tube blank. In contrast, the egg-shaped shell without lubrication has the defect of wrinkling and instability, and the forming accuracy is poor.

The effect of hydraulic loading speed on rubber bulging
Among the egg-shaped shell rubber bulging technological equipment, in addition to the lubrication conditions between the rubber and the tube blank have a significant impact on the forming accuracy of the egg-shaped shell, the hydraulic pressure head loading speed also influences the forming accuracy of the egg-shaped shell. Therefore, to investigate the effect of hydraulic loading speed on the formability of egg-shaped shells, the rubber bulging of egg-shaped shells at hydraulic loading speeds of 1000 mm/s, 1200 mm/s, 1400 mm/s, 1600 mm/s, 1800 mm/s, and 2000 mm/s were simulated. Then, the strain distribution, thickness distribution and forming limits of egg-shaped shells under different hydraulic loading speeds were analyzed. When the hydraulic loading speed is 1000 mm/s, 1200 mm/s, 1400 mm/s, and 1600 mm/s, respectively, the strain distribution of the egg-shaped shell is uniform, the overall strain disparity is slight, and the strain increases with the increase of the loading speed. When the hydraulic loading speed reaches 1800 mm/s, the overall distribution of the egg-shaped shell strain remains uniform, although the increased amplitude is large. When the hydraulic loading speed reaches 2000 mm/s, the rubber is not fully expanded due to the excessive loading speed, resulting in the egg-shaped shell not being fully formed. Moreover, the middle and lower part of the shell is not filled in time, thus producing wrinkling destabilization defects. At this time, the maximum strain is about 0.315, the strain is wavy, and the strain distribution is not uniform. The strain distributions of the egg-shaped shell under different hydraulic loading speeds are shown in Fig. 19.
When the hydraulic loading speed is 1000 mm/s, 1200 mm/s, 1400 mm/s, and 1600 mm/s, respectively, the thickness of the egg-shaped shell is evenly distributed. Meanwhile, the thickness thinning rate of the egg-shaped shell is low and the overall thickness disparity is slight. The thickness decreases with the increase of the loading speed at this time. When the hydraulic loading speed reaches 1800 mm/s, the thickness thinning rate of the egg-shaped shell is relatively large, and the overall thickness distribution is uniform. When the hydraulic loading speed reaches 2000 mm/s, the lower and middle part of the tube blank do not have sufficient plastic deformation, and the phenomenon of wrinkling and destabilization defects appears. At this time, the overall thickness of the egg-shaped shell is more significant, with a minimum When the hydraulic loading speed is 1000 mm/s, 1200 mm/s, 1400 mm/s, and 1600 mm/s, respectively, the tube blank can be fully plastically deformed in the egg-shaped shell rubber bulging technology, and the processing accuracy is excellent. When the hydraulic loading speed reaches 1800 mm/s, the egg-shaped shell does not appear to be wrinkled during the rubber bulging process, although there is a tendency to be wrinkled. When the hydraulic loading speed reaches 2000 mm/s, the tube blank appears wrinkled and destabilized defects. At this time, plastic deformation occurs in a small area of the egg-shaped shell, while most of the area has a tendency to be wrinkled and unstable. Another few area are wholly wrinkled and produce destabilization defects. From the forming limits of egg-shaped shells at different hydraulic loading speeds, it can be seen that as the hydraulic loading speed increases, the risk of wrinkling and instability of the egg-shaped shell increases. The forming limits of egg-shaped shells at different hydraulic loading speeds are shown in Fig. 21.
The strain distribution, thickness distribution, and forming limits of egg-shaped shells under different hydraulic loading speeds are compared. When the hydraulic loading speed is between 1000 and 1800 mm/s, the variation of the loading speed has less effect on the forming accuracy of the egg-shaped shell. When the hydraulic loading speed reaches 2000 mm/s, the egg-shaped shell shows wrinkled and destabilized defects. The faster the speed of hydraulic loading, the faster the speed of rubber bulging of the eggshaped shell. Therefore, considering the economic cost, the speed can be appropriately accelerated in the range of 1000 to 1800 mm/s.

Verification of optimization of rubber bulging process parameters for the egg-shaped shell
After optimizing the rubber bulging technology parameters of the egg-shaped shell, the radius of the mold cavity rounding was determined to be 3 mm, and the diameter of the tube blank was determined to be 70 mm.
To reduce friction, grease was used between the tube blank and the mold, and lubricated mechanical oil was used between tube blank and the rubber. To verify the accuracy of the simulation analysis, the analysis was carried out in terms of both the axial strain distribution and thickness distribution of the egg-shaped shell. Figure 22 compares the egg-shaped shell's simulated and experimental axial strain distributions. The strain distribution of the simulated analysis is slightly smaller than that obtained from the experiment. The experimentally obtained axial maximum strain is 0.383, and the simulated analytical axial maximum strain is about 0.363, with an error of only 5%. From the error of the overall axial strain distribution, the error does not exceed 10%, with an average error of 6%. The maximum error is located at the top of the egg-shaped shell, where the strain is slight. Figure 23 compares the egg-shaped shell's simulated analysis and experimental axial thickness distribution. The thickness distribution of the simulated analysis is slightly smaller than that obtained from the experiment. The maximum axial thickness obtained from the experiment is 0.739 mm, and the maximum axial thickness of the simulation analysis is about 0.707 mm, with an error of only 4%. From the overall view, except for a slight difference in thickness at the top of the egg-shaped shell, the thickness distributions of the remaining parts are basically the same, with an overall average error of 4.1%. Integrated comparison of the axial strain and thickness distribution of the egg-shaped shell, the simulation analysis agrees well with the experiment, so the simulation analysis results can be considered accurate.

Conclusions
In this paper, the deformation mechanism of rubber bulging of the bionic egg-shaped shell under unidirectional loading is studied. The rubber bulging technology of the egg-shaped shell is optimized. The influence of different technological parameters on the forming accuracy of the rubber bulging of the egg-shaped shell under unidirectional loading is analyzed to guide the manufacturing and forming of a high-precision egg-shaped shell. The conclusions from the simulation analysis and experimental verification are as follows: 1. In the rubber bulging technology of the egg-shaped shell, the tube blank size and mold structure have a significant influence on the processing accuracy of the egg-shaped shell. Optimizing the tube blank size and mold structure can effectively improve the uniformity of strain and thickness distribution of the egg-shaped shell, reduce stress concentration and improve the forming accuracy of the egg-shaped shell. 2. The lubrication condition between the rubber and the tube blank affects the forming accuracy of the eggshaped shell. When there is no lubrication between the rubber and the tube blank, wrinkling defects occur in the egg-shaped shell, and the forming accuracy is poor. When there is lubrication between the rubber and the Minor true strain(mm/mm) Major true strain(mm/mm) tube blank, the strain and thickness of the egg-shaped shell are uniformly distributed, and the forming accuracy is excellent. 3. The faster the hydraulic loading speed, the greater the strain on the egg-shaped shell, the greater the wall thickness thinning rate, and the smaller the egg-shaped shell thickness. When the hydraulic loading speed is between 1000 and 1800 mm/s, the change in loading speed has less influence on the forming accuracy of the egg-shaped shell, and the speed can be increased appropriately in this range. When the hydraulic loading speed reaches 2000 mm/s, the egg-shaped shell has wrinkling and destabilization defects, and the forming accuracy is poor.
Funding All the experiments were financed and supported by the National Natural Science Foundation of China (52205372), the pre-research funds project in Zhangjiagang city (ZKCXY2131), the foundation provided by Jiangsu University of Science and Technology (BY2021286), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX22_1924).

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Conflict of interest
The authors declare no competing interests.