Influence of mandrel-cores filling on size effect of cross-section distortion of bimetallic thin-walled composite bending tube

Yingxi ZHU, Miomio WAN, Wei CHEN, Yun WANG, Wenin TU,Fn XU

a School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China

b School of Mechanical Engineering, Guilin University of Aerospace Technology, Guilin 541004, China

KEYWORDS Bending;Bimetallic composite tube;Cross-section distortion;Size factor;Size effect;Mandrel

Abstract Two new size factors of cross-section hollow coefficient and bending degree are introduced to reveal the size effect of bending forming of bimetallic composite tube.Hollow coefficient and bending degree can limit the commonly used bent tube to the size description range of(0,2.00).The evolution laws of the cross-section distortion forms in the hollow coefficient-bending degree interval are revealed as well as the action of the mandrel-cores on the size effect.Results show the mandrel-cores filling can expand the forming limit of the bent tube, but also bring two other forming defects of wrinkle and rupture.The identification factor(hollow coefficient multiply bending degree)provides a method for querying the cross-section distortion forms of all composite bending tubes.In the identification factor interval (0, 1.00), the distribution area of bending forming defects of the composite tube is continuous.The thin-walled composite bending tube collapses when identification factor in (0, 0.39), wrinkles when identification factor in [0.39, 0.50), and ruptures when identification factor in[0.50,1.00).The mathematical model of size effect is derived,by which the average cross-section distortion rate is found to distribute like a radial leaf in the hollow coefficient-bending degree qualified forming space.The best forming zone is hollow coefficient 0.46–0.68, and bending degree 0.25–0.47.

1.Introduction

Bimetallic composite tube is one kind of special tube composed of inner and outer double layers, which can synthesize the performance advantages of dual materials and meet the special performance requirements of tube in aviation,aerospace and other fields.1–4The copper-titanium thin-walled bimetallic composite tube uses T2 copper tube as the‘‘base tube”and the TA2 titanium tube as the‘‘covered tube”.Therefore, it has both good corrosion resistance and ductility of copper material,5,6and good vibration resistance and antioxidant of titanium material.7–9This remarkable material performance advantage makes the bending parts of copper-titanium bimetallic composite tubes gradually become irreplaceable parts in the aeroengine, marine pipeline transportation system and other fields.Even, they are gradually replacing the bending parts of traditional singlelayer tubes for more challenging and complex environments.10–12Thus, the plastic bending mechanism of coppertitanium bimetallic composite tube is the research object.

As we all know,the small-radius bending tube plays a huge role in changing the flow direction of liquids in the pipeline system, reducing pipeline assembly space and pipeline transportation pressure.The small radius bending forming of tubes has always been an important direction to overcome on the road of advanced manufacturing technology, whose processes mainly include push bending,13–17free bending18–22and rotary-draw bending,23–28etc.A large number of comparative studies had been carried out on different bending processes for single-layer metal tubes.It has been found that the Computer Numerical Control Rotary-Draw Bending (CNC RDB) process of the single-layer metal tubes is a bending process with high efficiency, reliable, high forming precision and small formable radius.23Therefore, the CNC RDB is introduced into the bending forming processes of copper-titanium composite tube, which has a groundbreaking and instructive role for the study of bimetallic composite bending tube.

At present, the CNC RDB forming technology of singlelayer metal tube has become mature.Numerous research results show that:(A) In the process of Rotary-Draw Bending(RDB)forming,it is inevitable that there will be some defects,such as cross-section collapse, inner wrinkle, outer bulge and rupture,which can be collectively referred to as cross-section distortion and have important correlations with the mandrel die.29–31(B)The correlations depend to a large extent, on the bending geometric parameters of the tube, i.e., the‘‘size factors”.The size factors include tube diameter D, wall thickness t, bending radius R, relative thickness D/t and relative bending radius R/D.Based on the above two conclusions, there is a significant‘‘size effect”on the cross-section forming defects of singlelayer metal tube, whose concept was first proposed by Ref.32.

However, the preliminary research work of this paper shows that severe section collapse occurs when the dimensional parameters of the composite tube tend to be thin-walled,highsection hollow and large-bending.And with the increment of the size parameter, the cross-section distortion increases with a regular exponential function.Therefore, just as the singlelayer metal tube,the bimetallic composite tube also has the size effect of cross-section distortion,as shown in Fig.1.Mandrels should be used to suppress cross-section distortion, especially when the size factors tend to be unfavorable for the stable forming of bending tubes.

Fig.1 Size effect phenomenon of bimetallic composite tube.

It is thought that the size effect should be studied to reveal the mechanism of section distortion, and the best geometric forming parameters should be selected to avoid section distortion as much as possible.Therefore, many researchers have studied the size effect of RDB of single-layer metal tube under the condition of mandrel filling, or the effect of changing size factor values on the forming quality of bent tubes.It is found that the cross-section distortion of the bending tubes decreases with the increase of D and t when the mandrel is adopted in the bending process,32but the tendency to wrinkle also increases in tandem.33,34The filling of mandrel has a good effect on suppressing the cross-section distortion,while the risk of tube rupture increases.35,36Then,Ref.32 further discussed the effect of size factor D/t on cross-section distortion,and gave the values of D/t when the section distortion are the smallest.However,the above studies still do not clarify the influence of the mandrel on the size effect of cross-section distortion of the bending tubes.Moreover, they do not consider that the size factor R also has an important effect on the cross-section distortion,and this effect is usually played by R and D synergistically.

A majority of research took the size factors R and R/D as variables and studied their influence on the cross-section distortion of single-layer bending tubes.Huang et al.37studied the cross-section distortion of TA18 high-strength titanium tube,and found the distortion decreases as the bending radius R increases.Fang et al.38concluded that the cross-section distortion of high-strength TA18 bending tube decreases with the increment of R/D only when R/D ≥2.0.Yang et al.39studied the section distortion of Commercial Pure Titanium (CP-Ti)bending tube, and came to the same conclusion.Kajikawa et al.40found the C1220 single-layer tube can be bent well within the range of R/D = 2–3.Simonetto et al.41found the AISI 304L tube can be accurately bent when R/D=1.9.While research shows that the forming limit of AISI 304L bent tube is R/D = 1.6 if a suitable mandrel is applied.42,43

The above studies are based on the existence of reasonable mandrel filling conditions.The results show that the larger the R/D,the better the filling effect.When R/D ≤1.0,the effect of the mandrel on the bending accuracy of the single-layer metal tube will become complicated and even counterproductive.Li et al.44studied the effect of size factor R/D on the bending forming quality of 1Cr18Ni9Ti stainless steel tube.It is found that when R/D = 1.0, the cross-section of the tube is excessively deformed,and the section part filled by cores is prone to wrinkle.Safdarian45found that when R/D ≤1.0,the excessive distortion occurs for BS 3059 single-layer tube,which finally results in the rupture in the section part filled by mandrel.Then, the elastic mandrels are introduced in the R/D ≤1 CNC RDB forming process of the single-layer metal tubes,46but are difficult to remove after bending.In summary,the rigid mandrels are still the most commonly used supporting components for the tube bending.When the size factor R/D ≤1.0, its bending effect on the tube becomes complicated,and it is necessary to further find its mechanism of action in the size effect.

The above-mentioned studies have good reference, including simulation and analysis methods.But there are still the following problems.(A) Since the structure of bimetallic composite tube is more complicated than that of single-layer tube,so the coupling effect between the base tube and covered tube is inevitable.47Therefore, their conclusions of the singlelayer tubes are not completely applicable to the bimetallic composite tube.(B) In majority studies, the size factor is D/t or R/D, which is usually studied as independent variables.In fact, the research conclusion obtained in this way is onesided, because the dimensional structure of the bending tube includes three parts: D, t and R at the same time.(C) Their‘‘size effect”is expressed by the acquisition of influence laws or extreme forming conditions, but lacks a description of mathematical relationships.(D) The role of the mandrel in the cross-section distortion forms is not fully revealed.This leads readers to a lack of in-depth understanding of the macroscopic‘‘size effect”.Thus, the above studies have not made a precise revealing on the association between the distortion forms and structural characteristics of the tube.

Therefore,onthe basisofpreviousresearch work,48thispaper introduces a new method to study the size effect of RDB of thinwalled bimetallic composite tube.Two dimensionless size factors cross-section hollow coefficient λ and bending degree w are introduced,which solves the research difficulty caused by the coupling or superposition of many size factors.λ and w can limit the commonly used bent tube to the size description range of(0,2),which facilitates mathematical modeling and description of size effect.The mandrel-cores are used to suppress cross-section distortion,whose action on the size effect is revealed by studying the evolution laws of the cross-section distortion forms in the λ-w interval.The mathematical model of size effect is derived, by which the average cross-section distortion rate and the best forming zone can be predicted in the λ-w qualified forming space.Through this research method,hope to provide theoretical basis and process guidance for the production practice of multi-size and multispecification composite bending tubes.

2.Establishment of FE model of rotary-draw bending of coppertitanium thin-walled composite tube

To study the size effect of RDB of copper-titanium thin-walled composite tube,the Finite Element(FE)model has been established and verified based on the ABAQUS/explicit.T2 is selected as the base tube material, and TA2 is selected as the covered tube material.The material constitutive model adopts isotropic yield criterion and isotropic hardening model.Then the yield condition f is shown as

where s is the deviatoric stress; ε-p is the equivalent plastic strain; K and n are the material constants obtained by tensile test,49as shown in Table 1.49

The modeling processes have been discussed in detail in Ref.49, so it is not repeated here.The mandrel-cores die is composed of two parts,the mandrel and the rigid cores,whose connections are‘‘Join+Rotation”,as shown in Fig.2(a).The wire lines should be set in the order from the end core to the mandrel.This phenomenon follows the movement transmission sequence of the filling mold during the bending process,that is, the end core first enters the bending state, as shown in Fig.2(b).Therefore,the bending zone of the composite tube can be divided into three major zones,i.e.,the Mandrel Filling Zone (MFZ), the Cores Filling Zone (CFZ) and the Non-Filling Zone (NFZ).The FE model of the RDB of copper-titanium thin-walled composite tube is finally established for the prediction of distortion form, as shown in Fig.2(b).

Table 1 Material parameters of composite tube.49

Fig.2 FE model of RDB of copper-titanium thin-walled composite tube is established.

The comparison of experimental and simulation conditions is shown in Table 2.49–51The filling conditions are divided into two types: without filling and with mandrel-cores filling.The FE model of RDB of copper-titanium thin-walled composite tube without mandrel filling has been verified in Ref.49, so it is not repeated here.While the FE model with mandrel filling has been discussed in Fig.3.

It can be seen from Fig.3(a), the variation trends of the base tube and the covered tube of cross-section distortion rate obtained by the simulation are completely consistent with the experimental results.But the experimental data is generally larger than the simulated data.This phenomenon can be explained by the conclusion of Ref.52, that is, the diameter measurement error for titanium tube after stretching and bending is somewhat large with a vernier caliper.The error of average cross-section distortion rate of the base tube and the covered tube between simulation and experiment are 6.66%and 12.43% respectively, which are within the reasonable range.Fig.3(b) is the half section shapes of 90° bending tube,which show that the simulation and experimental results are consistent.Therefore, the establishment FE model of the copper-titanium thin-walled composite tube with mandrelcores filling can reliably predict the cross-section distortion.

Table 2 Comparison of boundary conditions used in FE simulation and experiment.49–51

Fig.3 Comparisons of cross-section distortion rates between experimental and simulative results with mandrel-cores filling.

3.Definition of size effect and setting of simulation conditions

The size effect of thin-walled composite tube shows the relationship between size-structure feature and cross-section distortion of the tube.Thus, the mathematical description methods for size-structure feature, cross-section distortion and their relationship are described in detail as follows.

3.1.Two dimensionless size factors λ and w for size-structure feature description

The original inner diameter (thickness) of the base tube is D1(t1),the original outer diameter(thickness)of the covered tube is D2(t2), the bending radius to the neutral layer of the bent tube is R, and R/D2is the relative bending radius.

Two size factors λ and w are introduced to describe the sizestructure feature.48λ is the cross-section hollow coefficient,which is redefined as

where λ is dimensionless by introducing reference diameter D0,D0=50 mm.Then the hollowness of current commonly used tube sizes can be limited within the range of (0, 2).The larger the λ, the higher the degree of cross-section hollowness.

w is the bending degree, which is described in

where w is also dimensionless processing and reference radius R0, R0= 45 mm.Then the bending degree of current commonly used tube sizes can be limited within the range of (0,2).The smaller the w, the higher the bending curvature of the tube, and the greater the difficulty of forming.

3.2.Judgment of four types of cross-section distortion

In the bending process, the cross-section distortion includes cross-section collapse (Fig.4(a)), inner wrinkle (Fig.4(b),where ω is bending direction), outer bulge (Fig.4(c)) and rupture.Which distortion form the thin-walled composite tube exhibits during the bending process can be judged according to the cross-section distortion rate Δδij, the number of wrinkles N and the damage value Cl.The judgment method is as

Fig.4 Cross-section distortion forms of thin-walled composite tube after bending.

According to Eq.(5)and Fig.4(b),the cross-section distortion rate Δδ and the number of wrinkles N have the relationship, that is, when Δδi-1<Δδi>Δδi+1(where Δδi–1is the cross-section distortion rate of section i–1, Δδi+1is the crosssection distortion rate of section i + 1), an inner wrinkle appears on the inside of the composite tube, and it is marked as Nw= 1; when Δδi-1>Δδi<Δδi+1, an outer bulge appears on the outside of the composite tube, and it is marked as Nb= 1.So the number of wrinkles N = Nw+ Nb.

The damage value Clof Cockcroft-Latham damage criterion53is adopted to judge whether the composite tube is cracked, as

where C is the critical damage value obtained by experiment,C = 0.25 for the base tube, C = 0.32 for the covered tube;49σ-,σ*and ε- are the equivalent stress, the equivalent strain and the maximum tensile stress, respectively; the subscript l means the iterative step l.In the FE simulation, Clis iterated and accumulated in each iterative step l.When Cl<1,the rupture does not appear, otherwise, the rupture occurs.

3.3.Mathematical modeling of size effect

Size effect among cross-section distortion and λ-w is revealed by solving relation expressions F12(wc, λ), F22(λc, w) and scaling law Δs1(β(w)), Δs2(β(λ)),48where β is scaling factor corresponding to w and λ.

where F31((R/D2)c, R/D2) is the relation expression with specific value of R/D2.

3.4.Simulation conditions for size effect

According to the Eqs.(8)–(11),parameter selection ranges of λ and w could take D2,R and R/D2as reference objects,and then there are three kinds of Simulation conditions,I,II,III,for the size effect study in total, as shown in Tables 3–4.The value ranges of λ and w can also be seen in Tables 3–4.The thicknesses are 2 mm and 1 mm respectively for the base tube and the covered tube, which remain unchanged.

4.Results and discussion

Based on the established FE model and Simulation conditions I and II, the evolution laws of the cross-section distortion forms with the size factors λ and w are discussed and revealed,as well as the characteristics of partition distribution of the cross-section distortion.Finally, mathematical model of size effect is established by solving F12(wc, λ), F22(λc, w) and Δs1(β(w)) with Simulation conditions I–III.The distribution of Δδ in λ-w space has been developed by using the mathematical model.

4.1.Size effect of cross-section distortion based on Simulation condition I (w = wc)

This section is discussed based on Simulation condition I of Table 3.The evolution laws of the cross-section distortion in the λ interval are discussed and revealed.The relationship of Δδ-λ is obtained, which is mathematically described by F12(wc,λ)in the qualified λ interval.The influence of the filling action of mandrel-cores on size effect is summarized by comparing the relationships of Δδ-λ with mandrel-cores filling and without filling.

Table 3 Simulation conditions I and II for size effect.

Table 4 Simulation condition III for size effect.

4.1.1.Evolution laws of cross-section distortion in λ interval

Fig.5 shows the distribution of the cross-section distortion rate Δδ with different λ along the bending direction.It can be seen that the distribution of Δδ in the interval [0.20, 0.46]are very stable and all positive values, which indicates that the composite tube has cross-section collapse.And Δδ decreases with the increment of λ in the interval [0.20, 0.46].This is because the larger the hollow structure, the larger the contact area between the mandrel die and the tube blank,and the better the mandrel supporting effect on the composite tube.When λ reaches 0.97, Δδ increases significantly and fluctuates violently in both positive and negative directions, indicating the section of the composite tube is wrinkled.The wrinkle height is increasing with the increment of λ, as shown in Fig.5(a).It also has been found that Δδ present two main characteristic areas, i.e., the mandrel-cores filling zone (CFZ and MFZ) and the NFZ.Δδ of composite tube in CFZ and MFZ fluctuates more obviously than that in NFZ (Fig.5(b)), so CFZ and MFZ are more prone to wrinkle, bulge and even rupture under extreme forming conditions.

Evolution laws of the cross-section distortion in the λ interval are further analyzed in Figs.6 and 7.It is found that the variation trends of the average cross-section distortion rate Δδ with λ are consistent for the base tube and the covered tube.Both of them will show three gradual transitional distortion forms as their hollowness increase, that is, cross-section collapse, wrinkle and rupture (as shown in Figs.6 and 7).And correspondingly, there are three forming areas on the Δδ-λ curve forming graph, namely the qualified area(λ-interval [0.11, 0.77), the wrinkle area λ-interval [0.77, 1.17)and the rupture area (λ-interval [1.17, 1.57]).The Δδ of the qualified area is significantly lower than those of the wrinkle and rupture areas.

4.1.2.Size effect relationships of Δδ-λ

In the qualified area (λ-interval [0.11, 0.77)), Cl< 1, N = 0,and Δδij≥0, thus the distortion form is cross-section collapse according to Eq.(4).Due to Δδ < 3%, the composite tubes with λ within this range [0.11, 0.77) all can be well formed.The mathematical model for size effect relationships of Δδ-λ is presented in Eq.(12),which means Δδ is reduced in the form of a logarithmic function (as shown in Fig.8(a)).In wrinkle and rupture area [0.77, 1.57],Δδ is increased in Boltzmann function.

where a1, b1, and z1are parameters, whose values have been obtained as shown in Table 5.

Fig.5 Distribution of Δδ with different λ along bending direction.

In the wrinkle and rupture area(λ-interval [0.77,1.57]),Δδ is increased in the form of a Boltzmann function, as shown in Fig.8(b).

In the wrinkle area (λ-interval [0.77, 1.17)), Cl< 1,N ≥2,and Δδij≥0 and Δδij<0 are coexist,thus the distortion forms are inner wrinkle and outer bulge according to Eq.(4).This phenomenon is also verified by Fig.7.In the first half of the λ-interval [0.77, 1.17), the wrinkle form is dominated by inner wrinkles, and in the second half of the interval, outer bulge become the main wrinkle feature.

In the rupture area (λ-interval [1.17, 1.57]), the outer bulge becomes more and more excessive as λ increases, which eventually causes the material of the outer forming surface to rupture, as shown in Fig.7.

Fig.6Δδ-λ curves forming graph of base tube and covered tube.

4.1.3.Comparisons of relationships ofΔδ-λ with mandrel-cores filling and without filling

Based on the previous research results of the size effect of composite tube without filling,48Fig.9 shows the comparison analysis of the relationships of Δδ-λ with mandrel-cores filling and without filling.Three conclusions can be drawn as follows:

(1) Under the action of the mandrel-cores, the variation trends of Δδ-λ curves become more complicated.Instead of just increasing as Boltzmann functions,they first decrease logarithmically and then increase as Boltzmann functions, as shown in Fig.9(a).Thus,in the qualified area,there is no such phenomenon like the smaller the hollow coefficient, the better the forming quality.

(2)Under the filling action of the mandrel-cores,the λ qualified forming area is expanded from[0.11,0.58)to[0.11,0.77),while Δδ decreases from 10% to 3%.Therefore, the mandrelcores can well suppress the cross-section distortion, as shown in Fig.9(b).

(3) The filling of the mandrel-cores leads to the diversification of cross-section distortion forms.With the increment of λ,the composite tube bent without filling will only experience cross-section collapse, while the composite tube bent with mandrel-cores filling will experience cross-section collapse,wrinkle and rupture, respectively.Because the wrinkle and rupture seriously affect the shape accuracy of composite bent tube, the composite tubes in the λ-interval [0.77, 1.57] are all unqualified, and the other filling mandrels should be considered to effectively control the wrinkle and rupture.

Fig.7 Distortion forms of thin-walled composite tube.

Fig.8 Mathematical models ofΔδ-λ.

Table 5 Parameter values of F12(wc, λ), F22(λc, w) and Δs1(β(w)).

Fig.9 Comparison analysis of relationships of Δδ-λ with mandrel-cores filling and without filling.

4.2.Size effect of cross-section distortion based on Simulation condition II (λ = λc)

This section is discussed based on Simulation condition II of Table 3.The evolution laws of the cross-section distortion in the w interval are discussed and revealed.The relationship of Δδ-w is obtained, which is mathematically described by F22(λc,w)in the qualified w interval.The influence of the filling action of mandrel-cores on size effect is summarized by comparing the relationships of Δδ-w with mandrel-cores filling and without filling.

4.2.1.Evolution laws of cross-section distortion in w interval

Fig.10(a) shows the distribution of the Δδ with different w along the bending direction.It can be seen that the smaller the w, the better the forming quality of the composite tube.With the increase of w,the degree of bending of the composite tube increases, the uneven distortion of the material becomes more intense.So, three cross-section distortion forms of cross-section collapse, wrinkle and rupture will occur successively.It also can be seen that the inner wrinkle(Δδij≥0)distortion form gradually transformed into outer bulge distortion form (Δδij< 0).The same as the phenomena in Fig.5, Δδ of composite tube in CFZ and MFZ fluctuates more obviously than that in NFZ, as shown in Figs.10(a) and (b), so CFZ and MFZ are more prone to wrinkle, bulge and even rupture under limit conditions.

Fig.10 Distribution of Δδ with different w along bending direction.

Fig.11Δδ-w forming graph of base tube and covered tube.

where a2,b2,z2and d2are the parameters of polynomial function, whose values have been obtained as shown in Table 5.

(2) Under the filling action of the mandrel-cores, the w qualified forming area is expanded from [0.30, 0.60] to [0.30,0.75], while Δδ decreases from 12.78% to 2.13%.Therefore,the mandrel-cores can expand qualified area and ensure that more composite tubes are accurately formed.

(3) The distortion forms are completely different.With the increment of w,the composite tube without filling only experiences the cross-section collapse,while the composite tube with mandrel-cores filling experiences cross-section collapse, wrinkle and rupture.

(4)Without filling,there is obvious‘‘necking”in the experimental and simulation results, as shown in Fig.14(b).While mandrel-cores filling can avoid‘‘necking”phenomenon.It is necessary to adopt mandrels to improve bending accuracy of composite tube, especially for those applied in harsh environment of petroleum transportation.

4.3.Size effect of cross-section distortion based on Simulation condition III

Fig.12 Distortion forms of λc composite tube with different w.

Fig.13 Mathematical model ofΔδ-w.

Fig.14 Comparison analysis of relationships of Δδ-w with mandrel-cores filling and without filling.

Therefore, it is unreasonable to use size factor R/D2to study the evolution laws of the cross-section distortion.Thus,the identification factor λ × w is introduced.The Δδ distribution of the cross-section distortion form has been discussed on the λ×w interval(0,1.00),as shown in Fig.15(b).In the λ×w interval, the interval partitions where collapse, wrinkle and rupture occur respectively are quite distinct.The composite bending tube collapses when λ × w?(0, 0.39), wrinkles when λ × w ?[0.39, 0.50), and ruptures when λ × w ?[0.50,1.00).Therefore, the identification factor λ × w provides a method for querying the cross-section distortion forms of all thin-walled composite bending tubes.

Fig.15 Size effect relationships between the distortion forms and λ-w.

Fig.16 Δs1(β(w))-β(w) relationships of base tube and covered tube.

Fig.17 λ-w qualified forming space of composite tubes.

5.Conclusions

The evolution laws of the cross-section distortion forms in the λ-w interval are revealed as well as the action of the mandrelcores on the size effect of cross-section distortion.The mathematical model of size effect is derived.The research results are as follows.

(1) The use of mandrel-cores makes the forming defects of the composite tube after bending have obvious three-zone distribution characteristic, i.e., the Mandrel Filling Zone (MFZ),the Cores Filling Zone (CFZ) and the Non-Filling Zone(NFZ).Tubes in CFZ and MFZ are more prone to wrinkle,bulge and even rupture under extreme forming conditions.

(2) Cross-section collapse is the main forming defect of bending forming of bimetal composite tube.The use of mandrel-cores brings about two other cross-section distortion forms, namely wrinkle and rupture.As λ and w increase, the cross-section distortion of composite tube gradually changes from cross-section collapse, wrinkle to rupture.

(3)The mandrel-cores can expand qualified area and ensure more composite tubes are accurately formed.Under the filling action of the mandrel-cores, the λ qualified forming area is expanded from [0.11, 0.58) to [0.11, 0.77), while Δδ decreases from 10% to 3%.The w qualified forming area is expanded from [0.30, 0.60] to [0.30, 0.75], while Δδ decreases from 12.78% to 2.13%.

(4) The identification factor λ × w is introduced, providing a method for querying the cross-section distortion forms of all composite bending tubes.In the λ × w interval (0, 1.00), the distribution area of bending forming defects of the composite tube is continuous.The thin-walled composite bending tube collapses when λ × w?(0, 0.39), wrinkles when λ × w ?[0.39, 0.50), and ruptures when λ × w ?[0.50, 1.00).

(5) The mathematical model of size effect is established,which is the product of logarithmic function and exponential function in λ-w qualified forming space.The size effect model predicts that Δδ is distributed in a radial leaf shape in λ-w qualified forming space.The larger the λ and the smaller the w,the better the forming quality of the composite tube.The best forming zone is Zb={λ, w|0.46 ≤λ ≤0.68, 0.25 ≤w ≤0.47}.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(Nos.51601070 and 51875263),the Open Project of Guangdong Key Laboratory of Precision Equipment and Manufacturing Technology, China (No.PEMT202102), and the Natural Science Foundation of Jiangsu Province, China (No.BK20181447).