Plasticity Improvement of Ball-Spun Magnesium Alloy Tube Based on Stress Triaxiality

The effects of thickness reduction, feed ratio, and ball diameter, and their coupling effects, on the average relative stress triaxiality during spinning are discussed via simulation results. The relationships among the parameters and the average value of relative stress triaxiality (AVRST) are fitted with multiple nonlinear functions to calculate the optimal process parameters. According to the trend of stress triaxiality, the corresponding process parameters are calculated for the minimum average value of relative stress triaxiality (AVRST). Room temperature experiments performed on an AZ31 magnesium alloy thin-walled tube with the optimal parameters reveal an improvement of cracking of the tube surface. The study reveals changes in the minimum AVRST and aids in selecting the process parameters to improve plastic performance.


Introduction
e ball-spinning process ( Figure 1) employs a support ring, conical ring, screw tube, and numerous balls that collectively constitute the ball-spinning mold. e ball-spinning mold is present on the outer wall of the workpiece. e mold and the workpiece rotate relative to each other, and the mold moves along the axis of the workpiece to produce the axial feed. en, the workpiece placed outside the mandrel comes into contact with the balls, and the workpiece is compressed to produce plastic deformation. e main parameters for the ball-spinning process are shown in Figure 2, where R is the ball radius, Δt is the thickness reduction, f is the feed ratio, and α is the spinning angle.
Rotarescu [1] performed a theoretical derivation and finite-element simulation to establish the relationship between the parameters for ball spinning. Abd-Eltwab et al. [2] studied the effects of processing variables pertaining to ball spinning on the forming load and the quality of the formed sleeves and determined the optimum values of these variables. Li et al. [3] obtained a formula for calculating the ballspinning pressure under the assumption of a plane strain state. Zhang et al. [4] analyzed the folding defects formed by ball spinning at the bottom of the inner grooves of copper tubes according to the results of finite-element analysis. Jiang et al. [5,6] simulated the ball spinning of a nickeltitanium shape memory alloy tube by the rigid-viscoplastic finite-element method and investigated the interface compatibility of the composite tube of copper and aluminum during ball spinning. In [7], the finite-element method was used to simulate the thin-walled tube ball spinning, and the reasonable process parameters were obtained. Kuss and Buchmayr [8,9] carried out a finite-element simulation and an experiment on the surface cracking phenomenon, which affects the spinning of the workpiece. Jiang et al. [10,11] simulated multipass backward ball spinning and carried out a study on the in uence of the ball size on deformability of thin-walled tubular part with longitudinal inner ribs.
As mentioned above, previous research on the ballspinning process parameters mostly considered the in uence of single-process parameters on the spinning tube, without taking into account the coupling e ects of various parameters. As a result, when a process parameter changes, the remaining process parameters cannot be correspondingly adjusted.

Theoretical Basis and Related Hypotheses
Because of the close-packed hexagonal structure of the metal atom, the magnesium alloy shows poor plasticity and can be easily broken during spinning. erefore, it is important to select appropriate process parameters to improve the plasticforming ability and thus ensure surface quality.
Internal factors such as deformation temperature, deformation speed, and deformation methods as well as other external factors a ect the deformation behavior of magnesium alloys. At present, a large number of studies on the mechanical properties of magnesium alloys are gradually transferred from normal temperature and quasi-static conditions to di erent temperatures and di erent strain rates, including fracture strength and fracture ductility [12].
Rod parameter, soft coe cient, and stress triaxiality are the commonly used stress state parameters for studying the deformation and fracture of a metal. From multidirectional tension to multidirectional compression, the stress triaxiality and di erent stress states show a signi cant monotonic change; hence, it is imperative to describe the stress state of the material. e research results show that ductile fracture caused by plastic deformation is a ected by parameters such as strain rate and temperature as well as the stress triaxiality [13,14]. With an increase in stress triaxiality, the equivalent elastic modulus and equivalent yield stress of a magnesium alloy increase, but its fracture strain gradually decreases [15]. At present, a single stress or strain fracture criterion cannot explain the failure fracture behavior under the complex stress state of a magnesium alloy material. Considering the relationship between the stress triaxiality and the fracture strain as the core of the fracture criterion can help explain the magnesium alloy failure behavior in di erent stress states.
e stress triaxiality σ * force is given by where σ m is the spherical stress; σ 1 , σ 2 , and σ 3 are maximum, intermediate, and minimum principal stresses, respectively; and σ is the von Mises equivalent stress. Generally, the smaller the σ * value, the larger is the plastic deformation limit of the material and the better is the plastic-forming ability. El-Magd and Abouridouane [16] studied magnesium alloys and found that, under dynamic loading conditions (_ ε > 10 −3 ), there was an increase in deformation when the strain rate increased.
From the aspect of cracking of the material surface, the fracture failure of the metal is related to the strain rate and temperature in addition to the stress triaxiality. e most widely accepted and used fracture failure criterion is the Johnson-Cook fracture failure model, which is expressed as follows [17]: where ε f is the fracture strain; σ * is the stress triaxiality; σ e is the Mises equivalent stress; D 1 , D 2 , D 3 , D 4 , and D 5 are the material constants; _ ε is the strain rate; and T * is a temperature parameter.
According to the literature [17], in formula (2), stress triaxiality is the most important factor a ecting the fracture strain; when the hydrostatic pressure increases, the fracture strain decreases rapidly. e fracture strain mainly depends on the hydrostatic pressure state and is less dependent on the strain rate and temperature. us, stress triaxiality is the decisive factor for the fracture strain of a given material at medium and low strain rates. Although stress triaxiality and equivalent fracture strain can be calculated based on tested data, the material failure strain is not the same as the equivalent fracture strain. Hence, the actual relationship between equivalent strain and stress triaxiality cannot be determined experimentally. For this reason, a numerical simulation must be performed to obtain the accurate stress triaxiality of the specimen.
is study analyzes the change rule for the average value of relative strain triaxiality in the deformation in uence zone during the ball spinning of an AZ31 magnesium alloy thinwalled tube. A method for selecting the process parameters based on the stress triaxiality is presented.
Ball spinning is a complex stress-strain process, and the material stress-strain curve changes with the stress state; hence, calculation of the real stress triaxiality is very di cult. Based on the above analysis, the nite-element calculation in this paper has been carried out with the following conservative processing: the strain rate is in the medium-low range and has little e ect on the fracture strain; the simulation and experiment are carried out at room temperature, so the e ect of temperature on the fracture strain is neglected; a bilinear model of the stress-strain relationship of the material is used in the nite-element model. us, the stress triaxiality value at each point is not the true stress triaxiality but a relative representation of the stress triaxiality. e main purpose is to explore the change in stress triaxiality with di erent parameters and to provide a qualitative reference for the selection of process parameters toward a small stress triaxiality.

Model Establishment.
In this study, the commercial nite-element software ABAQUS is used to simulate the spinning process. e model is simpli ed accordingly. e support ring, screw tube, and conical ring are ignored, and ball movement is directly de ned. e ball, thrust ring, and mandrel are de ned as analytical rigid bodies, and only the tube is de ned as the elastoplastic body. e eight-node linear hexahedral element C3D8R is used, and the plastic deformation region is remeshed. As the local deformation is large, an enhanced hourglass control is set up. e niteelement model is shown in Figure 3.
To compare the e ects of di erent process parameters on the stress state of the workpiece (a thin-walled tube), multiple simulations must be conducted. Based on the above discussion, the elastic modulus and yield stress of the workpiece-magnesium alloy tube are given in a simple bilinear model [18] in Table 1. e material properties and process parameters of the tube are shown in Table 1.

Boundary Condition Settings.
In order to maximize the t of the actual spinning conditions, the boundary conditions for the simulation process are set as follows: (1) During spinning, the ball rotates in a three-dimensional manner. Hence, the simulation limits its three directions of translational freedom to retain the rotation freedom.
(2) e tube is in frictional contact with the mandrel and thrust ring at a friction coe cient of 0.08. e contact between the ball and the magnesium alloy material with lubrication corresponds to a friction coe cient of 0.1.
(3) e mandrel is fed axially with the workpiece, and the remaining directions of freedom are restricted.

Data Extraction from Simulation Results.
In ball spinning, besides the metal extrusion by the ball just below the ball, the nearby area is also a ected. us, this study considers the contact area between the ball and tube and the surrounding vicinity as a single ball-deformation-a ected area ( Figure 4). e average value of relative stress triaxiality (AVRST) in the a ected zone is taken as the basis for the selection of process parameters, which is mainly in the following considerations: First, the ball and the workpiece are theoretically in the point contact state, so the actual deformation-a ected area is very small. e location of the extreme value of stress triaxiality is usually not the position of the maximum position of the stress, and the AVRST can weaken the in uence of uctuations in the extreme value of stress triaxiality of an isolated unit.
Second, the balls are circumferentially distributed along the circumference of the workpiece, and the contact and noncontact states of the ball are continuously repeated at the same point on the workpiece. is repeated state is contained in the deformation-a ected zone. erefore, it is more reasonable to use the change in the AVRST in the deformation-a ected zone to investigate the plastic-forming ability of the deformation zone of the workpiece.

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Along the circumferential direction of the workpiece shell, the tension zone between two balls appears at intervals, immediately below the ball; eight units are taken from each side in the ball feeding direction to constitute the deformationaffected zone.
e stress triaxiality value of each element in the set is extracted, as shown in Table 2.
As mentioned above, the stress triaxial values are relative, but its change can be derived from multiple sets of process parameters; this can qualitatively guide the selection of the process parameters in favor of plasticity improvement.

Results of Finite-Element Calculation
e three main process parameters-ball diameter, thickness reduction, and feed ratio-affect the stress state of the workpiece during spinning, and the coupling effects between these parameters are also significant. erefore, the relationships between one of these parameters and the other two parameters are studied. e AVRST in the deformation-affected zone under different parameter configurations for each group in Table 3 is plotted as a graph. Cloud diagrams of relative stress triaxiality by the finite-element method, corresponding to each group of process parameters, are extracted. e areas in which the relative stress triaxiality is greater than zero are set in white color for significant distinction, as shown in Figures 5-10, for each graph and cloud diagram.

Discussion
According to the calculated data, the relative stress triaxiality for different ball diameters, amounts of thinning, and feed ratios is analyzed and discussed as follows. Figure 5, as the ball diameter gradually increases, the AVRST in the deformationaffected zone decreases first and then increases. is observation indicates that excessively small or excessively large ball diameters are not suitable for the plastic deformation capacity.

Effect of Ball Diameter. As seen in
As can be seen from curves 1 and 3 in Figure 5, the minimum AVRST in the deformation-affected zone appears at R � 3 mm, while the spinning angle is Curve 2 shows the minimum value when R � 4.5 mm, and the corresponding spinning angle is is angle is consistent with the best spinning angle obtained by the production practice mentioned in the literature [19].
From the contrasting trend for curves 1 and 3 in Figure 6, it is seen that with an increase in the ball diameter, the difference in AVRST increases. e corresponding AVRST plotted on curves 1 and 3 increases rapidly, but curve 2 is relatively flat. is indicates that when a larger ball diameter is used, a smaller feed ratio and larger thickness reduction should be adopted.
To analyze the distribution of stress triaxiality in Figure  6, a nodal flow vector diagram of the section of the contact area between the ball and the workpiece is extracted, as shown in Figure 11.
Notably, the contact area of the ball is squeezed during spinning. In this case, the relative stress triaxiality is small. During the movement of the ball along the circumference of the workpiece, the material flow velocity is lower on the adjacent front and rear areas of the ball than in the ball contact area. us, the frontal pressure and rear tensile stress states are formed.
Moreover, a band-like tensile stress region is generated on the workpiece surface in the direction of about 45°because of the large shearing stress.
When the ball diameter is small, the deformation area is also small. In this case, the relative stress triaxiality in most areas is small and negative. With an increase in ball diameter, the area of plastic deformation and the area in which the relative stress triaxiality is positive increase, but the relative stress triaxiality pole value decreases from 5.16 to 4.71.
Moreover, when the ball diameter is R � 4 mm, the minimum value of relative stress triaxiality is larger than that at R � 3 mm, and this minimum value generally appears immediately below the ball. is indicates that as the ball diameter  increases, the plastic limit of the material decreases, and particularly, the extent of the thickness reduction is diminished. Moreover, when the ball radius increases, the extremum of relative stress triaxiality in the tension region increases, so excessively small ball diameters are highly undesirable. Figure 7, the AVRST decreases rst and then increases with increasing thickness reduction. is observation indicates that excessively high or low thickness reductions are not conducive for ductileforming ability. From the three curves in Figure 7, when the ball diameter is R 3 mm, the thickness reduction corresponding to the minimum AVRST is 0.2. When the ball diameter is R 4.5 mm, the thickness reduction corresponding to the minimum AVRST is 0.3. ese two values satisfy the following relation:       Advances in Materials Science and Engineering is correspondence implies that the optimum spinning angle is always about 21°, which is consistent with the analysis results in Section 5.1.

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In addition, with an increase in the thickness reduction, the area of the unspun section of the workpiece in which the relative stress triaxiality is greater than 0 shows a decreasing trend. is is because as the thickness reduction increases, the radial spinning force component increases faster than the axial force and tangential force component [20]; therefore, a larger thickness reduction is advantageous for reducing circumferential torsional failure and axial pressure buckling.

E ect of Feed Ratio.
As seen in the three curves in Figure  9, the AVRST rst decreases and then increases with increasing feed ratio. is plot indicates that an excessively large or small feed ratio is not conducive for improving the plastic-forming ability of the tube, and all the feed ratios corresponding to the minimum AVRST is about 0.2. By comparing curve 1 and curve 2, it can be seen that, at a larger thickness reduction, we must use a smaller feed ratio to achieve better stress states. By comparing curve 2 and curve 3, it can be seen that when the ball diameter increases, the feed ratio used should also be high for a smaller AVRST. Figure 10 shows that when the feed ratio is small, the AVRST of the deformation-a ected zone is large. e area mainly distributed in the spinned region, where the relative stress triaxiality is greater than 0 is large, but the maximum relative stress triaxiality is 2.27, which is smaller than that for the other cases, indicating that it is di cult to break the material under these conditions.

Multivariate Nonlinear Function Fitting
From the above analysis, the trend of AVRST with the change of process parameters is obtained, so the nonlinear function is tted according to the existing calculation data in the following text. So that when a process parameter changes, it is easy to match the remaining process parameters.
According to the simulation results, the three-variable cubic polynomial is selected as tting function. During the tting analysis using the standard ternary cubic polynomial model, it is found that a signi cant collinearity relationship exists among the four items of RΔtf, RΔt, Rf, and Δtf in the polynomial. However, when these four items are applied into the tting function model, the model becomes distorted, and the tting results are not estimated. erefore, these four items on the standard ternary cubic polynomial model are eliminated, and the nal tting function model is attained consequently as follows:    Advances in Materials Science and Engineering e data in Table 2 are used, and the results are shown in Table 4. e tting degree of the tting function is also considered, and the determination parameters are shown in Table 5. e plot in Figure 12 compares the compatibility between the results of FEM and tting function.
In Figure 12, the compatibility between the measured value of the AVRST and the calculated value of the tting function is high with no point of complete deviation, so the tting function model given in this paper is reliable. At the given range of ball diameter of 2.5 mm ≤ R ≤ 4.5 mm, thickness reduction of 0.1 mm ≤ Δt ≤ 0.5 mm, and the feed ratio of 0.1 mm/r ≤ f ≤ 0.3 mm/r, the optimal process parameters that correspond to the minimum AVRST are obtained as follows: R 3.01, Δt 0.205, and f 0.208.

Experimental Verification
e material used in the experiment is a magnesium alloy AZ31B extruded tube. e horizontal spinning machine used in the experiment is shown in Figure 13, and it can achieve feed ratios of 0.1, 0.2, and 0.3 mm/r. However, the inner diameter of the conical ring is limited, so the ball diameter cannot be changed arbitrarily to adjust the range of thickness reductions. erefore, the experimental ball diameter is xed R 3.0 mm, and the experiment only explores the changes of thickness reduction and feed ratio. In line with the previous nite-element analysis, the number of balls used in the experiment is 9, and the spinning mold is lled with grease.
To clearly observe the tube surface after spinning for comparative analysis, the spinned tube surface is examined by an ultradepth microscope.
Spinning experiments are carried out for di erent thickness reductions and feed ratios. e experimental results are shown in Figures 14 and 15. Figure 14(a) shows that the pipe surface is smoother and shows minor cracks. In Figure 14 original scratches on the surface of the tube. e tube surface in Figure 14(c) is seriously damaged, and deep cracks are visible along the tube circumference.
In Figure 15(a), the pipe surface shows no obvious cracks and debris but displays a poor and dim nish. Figure 15(b) is the same as Figure 14(b). In Figure 15(c), the surface shows visible cracks and a rolled skin, and the micrographs reveal a stack of layers on the surface.
It can be seen from the experimental results that the quality of the spinned tube is closely related to the AVRST, and the failure of the tube after spinning is consistent with the simulation results. It is thus demonstrated that the method for using the AVRST to characterize the plasticforming ability of the material is feasible.

Conclusion
In this paper, the in uence of the process parameters on the stress state of the spinning deformation zone during ball spinning is described by nite-element simulation. e relationship among the three parameters-ball diameter, feed ratio, and thickness reduction-and the average stress triaxiality are discussed. Finally, spinning experiments are carried out, and the following conclusions are drawn.
e AVRST for the ball-spinning deformation rst decreases and then increases with changes in the three main process parameters. Excessively large or small values of the ball diameter, feed ratio, and thickness reduction are not conducive for improving the plastic-forming ability of the tube. When a large thickness reduction is used, a large ball diameter can improve the stress state. When the feed ratio is large, the ball diameter is reduced, and the stress state in the deformation-a ected zone is improved; increasing the ball diameter and reducing the feed ratio is bene cial for improving the plastic-forming capacity of the tube. e tting formula used in this paper can predict the AVRST of the deformation-a ected zone of the workpiece accurately within a certain range of process parameters.

Conflicts of Interest
e authors declare that there are no con icts of interest.