Analysis of the wall thickness distortion in tube hydro-forging

The tensile deformation is the most common forming principle in the sheet forming process, but there is a new tube forming process—tube hydro-forging has been developed based on the principle of compression deformation. In addition to the instability problem, a new surface defect—thickness distortion—was found in the compression deformation. In this paper, the wall thickness distortion is analyzed from the point of view of mechanics and deformation. It is revealed that in the low pressure tube hydroforming process, a bending moment exists and leads to some small wrinkles in the straight wall. In the tube hydro-forging, the small wrinkles would develop to the thickness distortion with the increase of compressive deformation. The wall thickness and compressive strain are two major effect factors. It is verified by experiments and simulation. This study helps to recognize the new surface phenomenon in compressive deformation.


Introduction
With the development of industry and the increase of demand, hollow structural components tend to be lightweight and integrated, such as rocket fuel bunker, aircraft inlet, and other components. The stricter control of defects proposes put forward higher requirements to the plate and shell forming process. At present, tube hydroforming [1] is the most widely used process in tube forming. However, the wall thinning and bursting problem in forming parts with curve axis and small corner restricted its wide application.
Many other tube forming processes have been developed in response to different forming conditions, such as low pressure-sequence hydroforming [2,3] and low pressure tube hydroforming (LPTH) [4,5]. The improvement for tube hydroforming is at the level of the optimization of process steps, and these processes are based on the principle of expansion deformation.
The main defect in the expansion deformation process is fracture and bursting, which is caused by local excessive tensile stress, and the bursting problem in tube hydroforming has been well predicted [6,7]. Surface roughening is another common defect in tensile deformation [8], and it will worsen the surface quality and service performance of forming components [9]. It is known that many factors are related to the occurrence of surface roughening in sheet metal forming, such as strain accumulation [10], deformation conditions, grain size, grain orientation, and lattice structure [11]. Choi [12] pointed out that the deformation constraint at the free surface is relatively loose, and the surface has additional deformation freedom, resulting in surface roughening defects.
In recent years, a new forming process-tube hydroforging (THFG)-has been developed with the depth understanding of the forming principle [13]. As a new tube forming process, the principle of THFG is completely different from the tube hydroforming. The traditional plate and sheet forming processes are based on the tensile deformation, while the THFG is based on the compression deformation [14].
Buckle or wrinkle is the most significant defects in the compression deformation. For example, excessive axial feeding will result in excessive axial compressive stress in high pressure tube hydroforming, and lead to wrinkling [15]. Energy method [16] and bifurcation theory [17] were used to study the critical conditions. Cao and Boyce [18,19] proposed an analytical model for plate wrinkling based on the energy conservation and established a new criterion to predict the critical pressure of wrinkling in rectangle plate under biaxial compression and lateral tension. Based on the successful application of energy method in sheet wrinkling prediction, in THFG process, it was predicted that the wrinkle occurred in the compression straight wall and the required internal pressure was calculated [13].
In the THFG experiment, when the compression strain is large and the internal pressure is sufficient, the wrinkle can be suppressed, but a new phenomenon appeared in the tube wall-the thickness distortion. Similar to the surface roughening, the thickness distortion is a surface defect and the wall thickness distribution reveals a significant wave trend. Furthermore, the fatigue property of tube component is an important service performance, and the uneven wall thickness has a certain impact on the fatigue performance [20].
In this study, it is proved that the imperfect surface state exists in the preformed tube wall, and the effect of the imperfection on the thickness of formed part was studied, through mechanical analysis combined with simulation and experiment. In this paper, the thickness distortion was discussed only from the perspective of force and deformation, and provides a new method to research the defects of compression deformation. Meanwhile, the thickness distortion from the micro level also needs further research.

Thickness distortion analysis
The principle of THFG process is shown in Fig. 1. Firstly, the tube blank with diameter ϕ was flattened and put into the lower die. Under the support of internal pressure p 1 , the tube was pressed to a rectangular section and the circumference does not change; this process is called LPTH. Then, under the support of p 2 , the upper die continues to go down to reduce the circumference and increase the wall thickness, until it reaches the desired shape. The process from (c) to (d) is called THFG.
It can be seen that, the LPTH can be regarded as the preforming process of the THFG. From the point of view of deformation, the tube is mainly subject to the bending deformation in LPTH process, while the compression deformation in the THFG process. There is no analysis about the influence of LPTH on the THFG. Therefore, we begin the analysis with the LPTH [14].

The analysis of LPTH
According to Nikhare's research [21], during the LPTH process, the die closing force produces a bending moment M A at the tangent point of the corner, so that the corner radius gradually decreases, and the tube surface gradually contacts to the die, and the length of the straight wall gradually increases. The force analysis and the bending moment M A distribution are shown in Fig. 2.
The corner is subjected the coupling effect of bending moment M A and tensile force T. Base on the research, the hoop force T under the internal pressure is [21]: According to the classic plastic bending theory, the M A and T are satisfied: where the T p is the fully yield tension, T p = σ s t. The plastic bending moment M p is: Substituting Eqs. (1) and (3) into Eq. (2) results in: The principle of THFG [14] When the M A = 0, the hoop force T is satisfied: Consequently, the bending moment M A disappears when the tube is subjected to the yield pressure p s , where the p s is: However, in practice, the applied pressure is less than p s . Nikhare proposed a formula to estimate the critical pressure p c of buckling in the vertical straight wall, that is [5]: where r is the corner radio and L is the length of straight wall AB, and the pressures p c , p s are varied with the geometry parameter r and L. In the LPTH process, the r is decreased and the L is increased, normally, the p c < p s , and the applied pressure is hardly equal to p s .
For the straight wall AB, the inner layer is subjected to uniformly distributed pressure p, while the outer layer is subjected to the contact stress p n . Due to the existence of bending moment M A , the p n < p at the section of length d, as shown in Fig. 2(c). With the increase of L, the section of length d will exist in the straight wall and cannot be completely contact with die; thus, a small wrinkle was existed.

The analysis of THFG
When the corner is completely contact with die surface, the bending moment M A disappears, but the small wrinkle still exists.
Next, under a sufficient pressure supporting, the upper die going down to make the vertical straight wall compressed. The following analysis would discuss the development of the small wrinkle. It is assumed that the small wrinkle is a trigonometric function, as shown in Fig. 3, and the curve equation is described as: where δ is the maximum deflection, and m is the frequency: The wrinkle length is assumed as l + ∆l, and based on the arc curve integration, the wrinkle length can be written as: ∆l can be simplified as: Fig. 2 Force and moment in LPTH [21] Assuming that the internal pressure can completely flatten the small wrinkle, the hoop strain generated by flattening (taking the absolute value) is: At the same time, the compressive strain ε c is produced in the THFG process, and then the hoop strain can be written: Figure 3 shows the stress state acting on the tube cross section. In the THFG process, the tube cross section was compressed to form the design shape, and the tube length was unchanged, so the axial strain ε z = 0 and the plane strain state condition and isotropic material model were assumed. The hoop strain is − ε θ , and the thickness strain ε t = ε θ based on the volume constancy.
The tube cross section was subjected to 3D compressive stress in the THFG process. The hoop stress is − σ θ , the axial stress is − σ z , the normal stress is − σ t , and the normal stress in the straight wall σ t = p.
When the shear stress is neglected in the plane strain state, the axial stress can be obtained as: The equivalent stress and strain were described as:   where K is the strength coefficient; n is the strain hardening exponent, and the factor c = 2/√3. The deformation force F is: Next, the force acting on the small wrinkle will be analyzed. According to the symmetry of wrinkle, a half model is taken as the analysis object, as shown in Fig. 4. The resultant force of internal pressure is P. The vertical component of reaction force Q at point C is equal to pl/2. Based on the force balance, the axial force T C at point C can be expressed as: where the θ is the angle between the line Q and the horizontal line. And it can be simplified as follows: The force T C is produced by the internal pressure applied on wrinkles. When T C > F, the wrinkle will be deformed and reached a balance shape: the geometric parameter δ b and l b , so the δ b and l b required the following condition to be satisfied: The wrinkle remains a stable state with the shape of δ b and l b . As shown in Fig. 5(a), the small wrinkle was compressed with the upper die going down, and the wall thickness is increased. In the compression process, the outer surface of small wrinkle will be filled up and straighten, and the inner surface will be more prominent. Multiple small wrinkles will develop to a wavy inner surface, as shown in Fig. 5(b). This phenomenon is called thickness distortion.

Parametric study
In the section, small wrinkles with wavelengths l b = 1, 2, and 3 mm were taken as the example. Figure 6 corresponds to the maximum deflection δ b changed with wall thickness at different wavelengths. It can be seen that the larger the wall thickness, the small the deflection δ b that can be flattened, and thickness distortion is less obvious. The upper right figure illustrates that the δ b /l b 2 is same at different wavelengths.  The effects of material parameters K value and n value on the maximum deflection δ b are given in Fig. 7. The internal pressure p is directly proportional to the K value [13], so the p/K is constant in this discussion. The greater the n value, the larger δ b , the distortion will be more serious.

Materials
The material used in this study is AA6063-O tubes with outer diameter 51 mm and initial thickness t 0 = 1.5 mm. The true strain-stress curve and mechanical properties are shown in Fig. 8

Experimental scheme
The THFG experiments were performed to form a stepped rectangle tube with corner r = 5 mm and width 32 mm, and the length of middle compression zone is 40 mm, and the punch stroke ∆ can be adjusted by replacing the die inserts. The experimental setup consists of hydraulic system and press machine as shown in Fig. 9. The hydraulic system can control the gas pressure to follow the loading path, the control precision is 0.5 MPa, and the displacement control precision is 0.1 mm.
The ContourGT-K 3D optical profiler of BRUKER Company is used to measure the 3D topography of the tube inner surface.

Finite element model
The Abaqus 6.13-1 software with an explicit dynamic finite element formulation is used to establish the plane strain model for the numerical simulation. The half part is taken as the analysis object considering the symmetry of the formed parts, as shown in Fig. 10. The upper die, lower die, and flattened die are analytical rigid bodies. The lower die is fixed and the upper die moves down. Surface to surface contact element is used at the tube/ die interface and the friction coefficient is 0.1 [22]. The tube was assumed isotropic material model and discretized by elastic-plastic strain quadrilateral elements with the element type of CPE4R (a 4-node bilinear plane strain quadrilateral, reduced integration). The structured element shape is used and 8 elements in the thickness direction. The circumferential element size was about 0.2 mm.
As described in Fig. 1, first, the flattened die moves 10 mm to the right to flatten the tube and then get back. Next, the upper die moves down with no pressure until the tube reaches the Fig. 1(b) state. At last, the LPTH process and THFG process were carried on.

Results and discussion
In the LPTH process, as the upper die is going down, the length of straight wall is increased. If the shape of corner is assumed as quarter of circle, and the radio of upper corner and lower corner is the same, the L would increase from 29.8 to 42.4 mm and the corner radio r would decrease from 16 to 5 mm. Thus, the critical pressure in LPTH is p c = 1 ~ 0.7 MPa and the yield pressure p s = 7.2 MPa to 24 MPa.
However, in the actual simulation and experiment, due to the existence of friction, the upper and lower corners are elliptical and different, so it is difficult to calculate the yield pressure p s during the LPTH, and also hardly control the bending moment M A which is equal to zero. Thus, it is impractical to reduce the thickness distortion by adjusting the internal pressure.
In the THFG process, the required pressure to prevent wrinkle is about 24 MPa when the punch stroke ∆ = 11 mm.

The analysis of simulation results
First, the simulation of tube 1 was carried out, where the thickness t = 1.5 mm, the pressure p 1 = 3 MPa in the LPTH process, p 2 = 24 MPa, and ∆ = 11 mm in the THFG process. The surface state of the straight wall was discussed. The equivalent stress distribution of formed tube and the curves of outer surface and the inner surface were extracted, as shown in Fig. 11. It shows that there is an obvious thickness distortion phenomenon, the inner surface raised due to the increase of wall thickness, but the thickness distortion becomes more obvious with the increasing of punch stroke ∆. The outer surface has slight wrinkles at ∆ = 0 mm and the small wrinkle is disappeared with the punch stroke ∆ increase.
In order to achieve a better understanding of surface morphology, the volatility of the inner surface was measured. Firstly, the wall thickness distribution calculated by the theoretical model is subtracted from the simulated thickness to obtain the wave curves, as shown in Fig. 12. It shows that with the punch stroke ∆ increase, the fluctuation range of curves becomes greater.
The standard deviation σ was calculated for quantitative comparison of surface morphology, where the average value is assumed as zero. The variation of standard deviation σ with punch stroke ∆ is shown in Fig. 13. It manifests that the standard deviation σ first increases and then decreases slightly with the increase of punch stroke ∆. When the ∆ > 8 mm, the standard deviation σ increases rapidly. Furthermore, the punch stroke ∆ has a linear correlation with the compressive strain at point A according to previous research [14]. To summarize, the thickness distortion initially increased and then decreased slightly as the strain increased when the strain was less than 0.2. As the compressive strain was larger than 0.2, the thickness distortion would become serious rapidly.

The analysis of experiment results
The experiments of tube 1, tube 2, and tube 3 were carried out and the formed tubes are shown in Fig. 14, where the experiment parameters of tube 2 are t = 1.5 mm, p 1 = 3 MPa, p 2 = 24 MPa, and ∆ = 8 mm; tube 3 are t = 2.1 mm, p 1 = 3 MPa, p 2 = 24 MPa, and ∆ = 8 mm. Take a section of the vertical straight wall as sample to analyze the inner surface morphology, and three scanning areas are shown in Fig. 15. It can be seen from the figures that there is obvious roughness on the inner surface; furthermore, tube 1 has the most serious surface defect and tube 3 has slight defect.
In order to quantitatively analyze the degree of thickness distortion, the experimental data of path 1, path 2, and path 3 were extracted for analysis, and the actual data was smoothed with FFT filter method for reduced noise. Figure 16 shows the experimental data of path 1 and three smooth paths, where the horizontal plane for surface morphology analysis is used as datum plane of zero, and the change of wall thickness is not considered for the data acquisition principle.
It can be seen that the curves have obvious fluctuation trend. The standard deviation σ was calculated, and the σ 1 of path 1 is 18.43, the σ 2 = 12.7, and σ 3 = 8.54, where the average value is assumed as zero. Obviously, the smaller the wall thickness or the larger the punch stroke ∆, the thickness distortion phenomenon is more obvious. This also supports the above conclusions.

Conclusions
The tube hydro-forging, as a rare process based on the principle of compression deformation, appears a new surface defect, named as thickness distortion. There is less literature about the surface roughness in compressive deformation compared with surface roughness in tensile deformation. In this article, the thickness distortion was studied from the perspective of mechanics and deformation. It is proved that the bending moment was existed in the straight wall during LPTH process, and resulted in some small wrinkles at the straight wall.
In the THFG process, the small wrinkles would develop to the thickness distortion with the increasing of the