Abstract
Accurately predicting the amount of springback has always been a prior focus in metal forming industry, particularly for creep age forming (CAF), for its significant effect on tool cost and forming accuracy. In this study, a closedform solution for CAF springback prediction covering deformation from elastic to plastic loadings was developed by combining the beam theory and Winkler’s theory, based on which an efficient springback compensation method for CAF was proposed. This developed solution extends the application area beyond the traditional beam theorybased springback prediction methods, maintaining its validity with large loading deflection in plastic range. Finite element (FE) simulation and fourpoint bending CAF tests adopting a 3rd generation AlLi alloy were conducted in both elastic and plastic forming regions and the results showed close agreement with the closedform springback predictions. For the proposed compensation method, an adjustment factor was introduced for complex flexible tool CAF to consider its deviation from the uniform stress loading and can be obtained using the closedform solution. The flexible tool CAF tests using the AlLi alloy demonstrated the applicability of the proposed compensation method to obtain the target shape within reasonable iterations, which can be further reduced by combining FE simulation.
1 Introduction
Creep age forming (CAF) is a metal forming method specially invented for fabricating largescale aluminium alloys’ panel components. Normally CAF process is comprised of three stages, i.e. loading, creepageing, and unloading. Deformation in CAF resulted by creep strain evolution can be induced below yield strength of the material. When compared with other traditional metal forming methods for panel fabrication, CAF has its advantages to obtain the final deformed panel [1]. However, springback is still unavoidable. Therefore, accurately predicting the springback becomes the key of applying CAF to deform panel with target shape/curvature. In the recent decade, advanced constitutive models have been proposed to more accurately predict creep deformation [2,3,4], and improved adjustable forming tool was developed to facilitate obtaining formed panel with target curvature in CAF [5]. Nevertheless, large springback occurring in unloading stage due to the limited portion of creep strain accumulated in creepageing stage presents a strong challenge for tool shape design, which significantly affects the tooling cost and the efficiency of fabricating CAF panels with the required curvature [6].
Springback happens in the unloading stage of metal forming as a result of recovery of elastic strain induced in loading stage, which has long been a focus in the whole metal forming field due to its substantial association with forming accuracy and fabrication efficiency. In the past decades, as reviewed by Wagoner et al. [7], analytical solutions and methods utilising finite element (FE) analysis were proposed to predict springback regarding to different metal forming conditions such as pure bending [8], bending combined with tension [9, 10], and draw bending [11, 12]. Additionally, in recent years, investigation of springback has extended to the nonlinear plastic bending behaviour with large deformation [13], bending of curved beam and tube [14, 15], and multiple reciprocating bending [16]. In the area of CAF, early investigation of springback analysis was conducted based on Euler–Bernoulli beam theory for predicting the formed sheet shape in singlecurvature case [17]. Latterly by independently considering relations of moment and bending curvature derived from the beam theory along length and width directions, prediction of springback for doublecurvature bending was fulfilled [18]. With the development of constitutive model for creepageing, by adopting the mechanismbased model to reflect stresscreep strain relationship in CAF, Jeunechamps et al. [19] further proposed an analytical springback prediction method which can be applied to doublecurvature bending case of CAF sheet. Li et al. [20] further extended the application of springback prediction surpassing elastic region, by utilising the beam theory and FE analysis.
In practical fabrication, the idea of springback control by compensating tool shape to achieve the target shape/curvature has been widely adopted. Displacement adjustment (DA) method, initially proposed by Gan et al. [21], is a direct geometric iterative control method by continually adjusting tool shape to let shape/curvature of the formed sheet gradually approaching the target value. Figure 1 shows schematically the DA method. The iterative algorithm of the DA method can be expressed as:
where \(C^i\) and \(S^i\) are the tool shape and the deformed shape after springback in \(i^{th}\) iteration, and \(D\) is the target shape to be formed. Hence the term (S^{i} – D) can be viewed as the geometric error between the deformed and target shape in \(i^{th}\) iteration as shown in Fig. 1b. C^{i + 1} is the tool shape taken in the (i + 1)^{th} fabrication iteration, as demonstrated in Fig. 1c, which is directly obtained by compensating the tool shape C^{i} in previous i^{th} iteration with the geometric error (S^{i} – D).
Based on the proposed DA method and utilising FE simulation, satisfactory convergence was reported to have been achieved between the shape of the sheet and its target surface formed by compensated flexible dies [22]. By considering compensation direction, Yang and Ruan [23] applied the DA method for the case of stamping of high strength steel, in which large displacement and rotation can occur during springback. In their study, errors between experimental and FE simulation results were observed after completion of one iteration. When the tool shape \(C\) and deformed sheet shape S were obtained from FE results to replace experimental ones, the validity of the DA method becomes highly dependent on the accuracy of simulated springback prediction results [7]. Hence the simulation error can become an issue for adopting FE analysis as an alternative tool, especially for fabrication of components requiring high precision. On the other hand, when the values of \(C\) and S are determined from experimental results without proper analytical method, it can be highly timeconsuming and costly on approaching the target value by iterations. It is of significance to find the balance between prediction accuracy of springback and cost of practical experiments to ensure the validity and effectiveness of the DA method.
Efforts have been made to seek progresses in DA method, aiming at reducing iterations and speeding up the convergence to deform the sheet to target shape [24, 25]. The variation of the DA method can be generalised as:
By introducing a compensation factor \(\alpha\) in the equation, the geometric error(S^{j}– D) in j^{th} iteration is revised to speed up the convergence rate and to broaden its application in complex cases such as deforming with varied curvature. The value of the compensation factor \(\alpha\) is highly dependent on the material, forming process and geometry of the forming tool [26]. As a result, \(\alpha\) can be different in each iteration and unpredictable, especially in the case of nonlinear forming with varying curvature [20]. To approach a proper value of α, a secant method was proposed by Li et al. [20] in recent study.
With continued advancements in new generation aluminium alloys possessing superior mechanical properties, these high strength aluminium alloys have been widely adopted in practical fabrication recently [27]. Among these high strength alloys, the latest 3rd generation AlLi alloy has drawn much attention and is being viewed as a competitive substitution of its predecessors for its balanced synergy of strength, weight reduction, toughness, and corrosion resistance [28]. However, with the prominent strength promotion for this AlLi alloy, the creep resistance is enhanced simultaneously which in turn reduces the creep deformation in CAF: limited creep strain was observed in creepageing for this alloy [29]. Consequently, plastic strain can be needed and be introduced through large deformation in loading stage as supplement for CAF of the material. Hence, due to the required high stress in creepageing for the AlLi alloy and large deformation in loading stage, the applicability of the beam theory for springback prediction may be limited. When large deformation is exerted which induces plastic region in outer bending layers of the forming sheet, the Euler–Bernoulli beam theory, established within elastic region with assumption of small deflection, becomes inappropriate to describe deflection and induced stress: the relation of bending strain \(\varepsilon\) and curvature \(k\) can lose its geometry validity in the case of large deformation. Hence a new analytical method to properly reflect the stress states of material in large deformation for CAF with high strength AlLi alloys is urgently required, based on which the prediction accuracy of springback and the validity of corresponding developed compensation method can be ensured.
To fill the gap of analytical springback prediction in CAF with large loading displacement, a closedform prediction solution covering constitutive relations from elastic to plastic in CAF is developed in this study by combining the beam theory and Winkler’s theory. The solution is verified with fourpoint bending CAF tests adopting the latest 3rd generation AlLi alloy. To improve the efficiency in experimental iterations and solve the accuracy issue raised by prediction error from FE analysis, an efficient springback compensation method, based on the developed springback prediction solution, is proposed by introducing an adjustment factor for CAF with complex flexible tool. The effectiveness and adaptivity of the proposed method are verified through CAF of targeted singlecurvature sheets by using flexible tool and the same AlLi alloy with different thicknesses.
2 Closedform solution of CAF springback prediction
2.1 Geometric relation in CAF of single curvature sheet
In practical CAF fabrication, metal sheets are formed in forming tools with designed curvature, which can be continuous upper and lower dies, lower supporting dies/ribs in vacuum autoclave, or flexible tool with multiple loading and supporting points. Figure 2a schematically shows the sheet shape at various CAF stages. (i) Initially, the sheet is placed and supported on the forming tool, shown as the red initial shape. (ii) Then the downward loads/displacement (F_{0}, F_{1}, etc.) are applied and the sheet is deformed to the designed loaded shape in blue. \(O_a\) is the centre point of the loaded shape with curvature radius \(R_a\), and \(\theta_a\) is the centre angle for the loaded curvature between the two supporting points on the edges. (iii) After deforming to the loaded shape, the sheet is kept in this shape and creepaged at elevated temperature for designed ageing time. (iv) Finally, the sheet is unloaded after completion of creepageing and the final CAFed sheet is obtained after springback. As shown in Fig. 2a, the CAFed sheet has a curvature radius of R_{b} with O_{b} as the centre point and θ_{b} as the centre angle for the final CAFed shape. Δd_{f} is the deflection at midpoint of the final CAFed sheet from its initial undeformed state.
Figure 2b illustrates schematically the loading states with different forming tools. For loading with continuous die or autoclave vacuum, the load is applied in the form of pressure and can be treated as a line load q_{Load} when considering unit width. When adopting multipoint flexible tool, the sheet is deformed with multiple point loads F_{x}. When deforming with a flexible tool, elastic pads/splines are normally added between loading points and forming sheet to achieve more uniformly distributed stress and smoother forming curvature, thus minimising surface damages [20, 22, 30]. Therefore, as shown in Fig. 2b, the loading state for flexible tool with splines can be treated as combination of point loads \({F}_{x}^{\prime}\) and line load \({q}_{load}^{^{\prime} }\). Furthermore, when CAF fabrication is conducted in complex forming tool, the loaded structure is in hyperstatic state which significantly raises the difficulty of stress analysis for the deforming sheet. To simplify the analysis procedure, as demonstrated in Fig. 2c and d, fundamental loading models, i.e. fourpoint bending with two supporting and loading points and uniformly distributed stress bending, are adopted for the deforming process of the sheet. To approach the stress state of the sheet in complex forming tools, the sheet is deformed to the same shape using the loading models shown in Fig. 2c and d. Because the loaded shapes in Fig. 2c and d are the same as in Fig. 2a, the same centre point O_{α} and curvature radius R_{α} are used for the two loading models. In Fig. 2c, L is the distance between the two supporting points. L_{1} is the distance between the supporting point and the loading point with exerted loads of \({F}_{1}^{^{\prime\prime} }\). d_{ll} is the loading displacement of the loading points to reach designed loaded shape. In Fig. 2d, \({q}_{load}^{^{\prime\prime} }\) is the line load and d_{lm} is the displacement of the midpoint. The geometrical relation between loading point displacement d_{ll} and R_{α} can be obtained as:
For centre point displacement \({d}_{lm}\) and \({R}_{a}\), the geometric relation can be expressed as:
And the relation between \({\theta }_{a}\) and \({R}_{a}\) can be derived as:
Hence, \(R_a\) and \(\theta_a\) can both be determined with displacement d_{ll} or d_{lm}.
2.2 Strain and stress determination in loading stage
In loading stage, depending on the displacement exerted to achieve the target shape, the process can range from elastic to plastic loading. By dividing the initial undeformed sheet into equivalent \(m\) segments, Fig. 3a shows one of the sheet elements, defined as \(a_0b_0c_0d_0\). Within elastic loading, the sheet element \(a_0b_0c_0d_0\) is deformed to the shape \(a_1b_1c_1d_1\). k_{1} is defined as the deflection curvature and \(O_1\) is the centre point of k_{1}. A polar coordinate system (r, θ) with the origin at the centre point O_{1} is used to express the location of the sheet layers along thickness direction. As shown in Fig. 3a, θ_{1} is the centre angle of \(a_1b_1c_1d_1\). R_{n1} is the distance of the neutral plane to the centre point O_{1} in elastic loading state, which can also be defined as the curvature radius in this state. Additionally, a local Cartesian coordinate system (\(x,y,z\)) is established to depict the plane perpendicular to the neutral plane of the sheet, in which y and zaxes form the plane perpendicular to the deflection curvature and the neutral plane, and \(\mathrm x\) axis locates at centroid of yz plane and is parallel to the neutral plane, as shown in the right half of Fig. 3a. Based on the assumption of Euler–Bernoulli beam theory, the neutral axis and neutral plane pass the centroid of \(yz\) plane as the mid layer of the sheet and no strain is generated at neutral plane. δb is the unit width of the sheet segment and is defined as 1 mm in this study. t is the sheet thickness.
When loading begins as exerting downward displacements on the sheet, elastic strain and stress are generated along xaxis direction in yz plane. The relation of deflection and induced moment depends on different loading model. For fourpoint loading model demonstrated in Fig. 2c, by utilising the Euler–Bernoulli beam theory, the relation between the loading point displacement \(d_{ll}\) and the generated moment \(M_{\mathit1}\) in yz plane (Fig. 3b) in elastic loading can be expressed as [31]:
where E is the Young’s modulus and I is the moment inertia of yz plane for zaxis. For uniform stress loading case demonstrated in Fig. 2d, the relation between the midpoint loading displacement \(d_{lm}\) and the moment \(M_1\) in elastic loading is derived as [32]:
Within linear elastic state, the distribution of stress in x direction along the thickness in yz plane is shown in Fig. 3b. In the figure, r is the distance of a random layer to the centre point \(O_1\), \({\mathit\mathit\sigma}_{\mathit m}\) and \(\sigma_{\mathit m}\) are respectively the maximum compressive and tensile stresses achieved at the top and bottom surface of the sheet. The stress distribution can be expressed as:
As demonstrated in Fig. 3b, in elastic loading, the stress follows linear distribution along thickness (y direction).
With continued increasing of deflection, the strain and stress in x direction increases. When the absolute value of \({\sigma}_m\) and \(\sigma_m\) reaches the yield stress \(\sigma_y\), the current stress state is defined as end of elastic loading state. As illustrated in Fig. 3b, the deformed sheet segment, curvature, centre point, centre angle, and moment at the end of elastic loading state are respectively defined as \(a_2b_2c_2d_2\), \(k_2\), \(O_2\), \({\mathrm\theta}_2\), and \(M_2\), and the distance of neutral plane to the centre point \(O_2\) at the end of the state is defined as \(R_{n2}\).
Further increase in deflection will lead to the stress surpassing the elastic region, i.e. reaching plastic region. Figure 4 shows the case after entering the plastic loading state. In the plastic loading state, the elastically loaded shape of sheet element \(a_2b_2c_2d_2\) further deforms to \(a_3b_3c_3d_3\), with deflection curvature increasing from \(k_2\) to \(k_3\) as shown in Fig. 4a. Simultaneously, the centre angle increases from θ_{2} to θ_{3}, and the centre point and neutral plane respectively change from position O_{2} to O_{3} and \(R_{n2}\) to \(R_{n3}\) (\(R_{n3}\) is the distance of neutral plane to the centre point \(O_{\mathit2}\) in plastic loading). In this situation, the linear elastic stress–strain relation is broken down in the outer bending layers, and plastic strain is generated in the plastic regions near the top and bottom surfaces of the sheet as shown in the red regions in Fig. 4b. The location of the transition boundary from elastic to plastic region near the top surface is defined using \(r_i\) as the distance to the centre point \(O_2\), while \(r_o\) is used to mark the distance for the position of transition boundary near bottom. The layers above \(r_i\) and underneath \(r_o\) are the plastic regions. With the increase of deflection, the upper elastic–plastic boundary will move downwards, and the lower boundary will move upwards, i.e. \(r_i\) increases and \(r_o\) decreases. \(\triangle\sigma\) and \(\triangle M\) in Fig. 4b are respectively the increment of stress and corresponding moment in deformation from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\).
By applying the Winkler’s theory, the strain generated at layer at position r when the element deforms from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\) can be expressed as:
where \(\Delta\varepsilon\) is the strain accumulated at layer of position r in the transformation process from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\), and \(\triangle\theta\) is the change of centre angle as \(\triangle\theta=\theta_3\theta_2\) as illustrated in Fig. 4a. From Eq. (9), zero strain increment was derived in the layer of neutral plane \(R_{n3}\). When layer r locates underneath \(R_{n3}\), i.e. \(r>R_{n3}\), positive strain is generated and the layer in that position is in tension state. In contrast, when located above the neutral plane (\(r<R_{n3}\)), negative strain is induced as in compression state. When r is equivalent to \(r_i\) or \(r_o\), the layer is at the transition boundary from elastic region to plastic region. According to the definition, the strain at elastic–plastic boundary at \(r_i\) and \(r_0\) equals to:
where \(\varepsilon_y\) is the strain at yield stress \(\sigma_y\) at transition boundary \(r_i\) and \(r_0\). Additionally, under the assumption that transverse shear strain is neglected in Winkler’s theory, \(\varepsilon_y\) can be derived as comprised of two parts as:
where \({\varepsilon }_{2}^{^{\prime}}\) is the strain generated at the same layer at \({r}_{i}\) or \({r}_{0}\) when the sheet is loaded to the end of elastic loading state. \(\Delta {\varepsilon }^{^{\prime} }\) is the accumulated strain at \({r}_{i}\) or \({r}_{0}\) during deformation from the end of elastic loading \({a}_{2}{b}_{2}{c}_{2}{d}_{2}\) to \({a}_{3}{b}_{3}{c}_{3}{d}_{3}\) in plastic loading state. By utilising the linear stress–strain relation shown in Fig. 3b and applying Eqs. (10) and (11), \({\varepsilon }_{y}\) can be further expressed as:
Due to the linear relation for strain and stress within elastic region, Eq. (12) can be further extended to describe the elastic strain \({\varepsilon }_{e}\) and stress \({\sigma }_{e}\) at random position \(r\) within elastic region (\({r}_{i} \le r\le {r}_{0}\)) when deforms to \({a}_{3}{b}_{3}{c}_{3}{d}_{3}\) as:
where \({\varepsilon }_{2}^{^{\prime\prime} }\) is the strain generated at position r in elastic region (\({r}_{i} \le r\le {r}_{0}\)) when deforms to \(a_2b_2c_2d_2\) as the end of elastic loading. Δε’’ is the accumulated strain at r in plastic loading state from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\). \(\varepsilon_e\) and \(\sigma_e\) are respectively the strain and stress at position r in linear elastic region of \(a_3b_3c_3d_3\).
As shown in Fig. 4b, plastic strain is induced in the layers located above \(r_i\) and below \(r_0\). To reflect the relation between plastic strain and flow stress beyond linear elastic relation in the plastic region, the classical power law function was used to express work hardening behaviour of the material as:
where \(\sigma_p\) is defined as the flow stress in the plastic region. \(\sigma_y\) is the yield stress of material and \({\mathrm\varepsilon}_{\mathrm p}\) is the plastic strain. h and n are material constants. By combining Eqs. (9) and (14), the flow stress at specific layer r in plastic region can be expressed as:
In the yz plane of the sheet, the force equilibrium function can be derived as:
where A is the area of the yz plane of the sheet. With further transformation of \(dA=\delta bdy=dy\) = \(d\left(rR_{n2}\right)=dr\), the equation can be written as:
By inserting Eqs. (13) and (15) into Eq. (17), Eq. (17) can be expressed in a function which \(R_{n3}\) is the only parameter as:
Detailed function derivation process is given in Appendix 1. With the determination of \({R}_{n3}\), by adopting Eqs. (13) and (15), the strain and stress in both elastic and plastic regions can be determined. Hence the moment in \(yz\) plane around \(z\) axis for \({a}_{3}{b}_{3}{c}_{3}{d}_{3}\), defined as \({M}_{3}\), can be derived as:
where \(M_2\) is the moment at \(a_2b_2c_2d_2\) and \(\triangle M\) is the moment increased during the deformation from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\) as respectively shown in Figs. 3b and 4b.
2.3 Springback prediction after creepageing
In creepageing stage, stress relaxation will be induced due to accumulation of the creep strain. For layer at location r in the deformed element with elastic loading \(a_1b_1c_1d_1\) or plastic loading \(a_3b_3c_3d_3\), the stress variation in creepageing stage in respect of ageing time \(t_{\mathit a}\) can be expressed as:
where \(\sigma\left(t_a\right)\) is the stress at ageing time \(t_a\) of the layer, \(\mathrm\sigma\) is the stress generated at the end of loading stage. \(\sigma_r\left(t_a\right)\) is the extent of stress relaxation at ageing time \(t_a\) due to generation of creep strain, which can be calculated as \(\sigma_r\left(t_a\right)=E\varepsilon_c\left(t_a\right)\) where \(\varepsilon_c\left(t_a\right)\) is the creep strain accumulated at layer r at ageing time \(t_a\). It is also known that the evolution of creep strain is determined by the stress in creepageing, by applying proper constitutive model reflecting the relation of stress and creep strain, \(\varepsilon_c\left(t_a\right)\) can be obtained as:
f_{c }denotes the constitutive model being used. The creep strain accumulated after completion of creepageing at time of \(t_{end}\) is denoted as ε_{c}(t_{end}).
In unloading stage, with the upward movement of the loading points and decrease of \(d_l\) and \(d_{lm}\), driven by the recovery of remaining elastic strain after creepageing, springback of the deformed sheet is induced. The springback of the sheet will stop when recovery of elastic strain is completed. Hence, the strain and stress in springback follows linear elastic relationship and the unloading process can be viewed as a reversed process of elastic loading. The strain distribution along the thickness of the sheet at the end of creepageing in elastic/plastic loading state is demonstrated in Fig. 5.
As shown in Fig. 5, after completion of creepageing, the remaining elastic strain \(\varepsilon_e\left(d\right)\) at specific layer d of the sheet, along with the induced moment \(M_{end}\) for yz plane around z axis as the recovery moment at the end of creepageing/start of unloading stage, can be expressed as:
where \(\varepsilon\left(d\right)\), \(\varepsilon_p\left(d\right)\), and \(\varepsilon_c\left(t_{end}\right)\) are respectively the strain, plastic strain, and creep strain in x direction at layer d of the sheet, and \(y_d\) is the distance of the layer to the neutral plane. For fourpoint bending cases, by applying Eq. (6), the recovered deflection at loading point (\(d_{rl}\)) during springback can be obtained as \(d_{rl}=\frac{M_{end}}{6EI}(3L_1L4L_1^2)\). For uniform stress bending cases, by applying Eq. (7), the recovered deflection at midpoint of the sheet (d_{rm}) can be derived as \(d_{rm}=\frac{40M_{end}L^2}{384EI}\). By utilising Eqs. (3) and (4) respectively for fourpoint bending and uniform stress bending cases, the curvature \(1/R_b\), as the final curvature of CAFed sheet, can be derived as \(R_b^2={(L/2L_1)}^2+\left(\sqrt{R_b^2\left(L/2\right)^2}+\left(d_{ll}d_{rl}\right)\right)^2\) for the former case and \(R_b=\sqrt{R_b^2\left(L/2\right)^2}+\left(d_{lm}d_{rm}\right)\) for the latter. By applying geometric transformation derived in Eq. (4), the final deflection of the sheet at the midpoint can be expressed as:
3 Springback compensation method based on closedform solution
The above closedform solution of springback prediction for CAF is based on two simplified loading models, i.e. fourpoint bending and uniform stress bending. In practical CAF fabrication, however, specialised forming tool may be used, in which a higher complexity of loading state for the forming sheet is involved. Multipoint flexible tool, as one of the stateoftheart forming tools, is characterised by its flexibility in setting required loading shape and has been adopted to not only CAF but also other sheet forming processes [5, 22, 33]. When deforming with flexible tool, depending on the designed loaded shape, the loading process can span from linear elastic to nonlinear plastic loading. For elastic loading with a multipoint flexible tool, the loading state is varying with the increase of loading displacement, which can be difficult to analytically derive the relation between loading deflection and stress states of the forming sheet. Furthermore, the derived solution will lose its validity once the locations of loading points are modified. By applying the simplified uniform stress bending model and Eq. (7), when the sheet is deformed in elastic loading with flexible tool, an equation with higher convenience and robustness to express the relation between loading displacement \({d}_{lm}^{^{\prime}}\) and induced moment \({M}_{1}\) is proposed as:
where \({d}_{lm}^{^{\prime}}\) is the displacement at midpoint of the sheet adopting flexible tool, and a’ is the adjustment factor to approach real relation between generated moment and \({d}_{lm}^{^{\prime}}\) within elastic loading. When \({d}_{lm}^{^{\prime}}\) continues to increase and reach the end of elastic loading state, Eq. (24) can be further written as \({d}_{lm}^{^{\prime}}={\alpha }^{^{\prime}}40{M}_{2}{L}^{2}/384EI\), where \(M_2\) is the moment induced at the end of elastic loading as illustrated in Fig. 3b.
When the stress surpasses elastic region during loading, as shown in Eq. (9), the strain increment \(\triangle\varepsilon\) in plastic loading state is derived based on geometrical change with the increase in deflection from the end of elastic loading. Consequently, the proposed solution for stress analysis and springback prediction in plastic loading state (from Eqs. (9) to (23) remains its validity in the case of multipoint flexible forming tool. Hence, by replacing Eqs. (7) with (24), a modified closedform springback prediction solution (Eqs. (8) to (24) is obtained for multipoint flexible tool. As can be noticed, the accuracy of this modified springback prediction solution for CAF with flexible tool is highly depending on the determination of adjustment factor a’. This adjustment factor not only helps to approach true relation of deflection and the generated moment within elastic loading, but also defines the transformation boundary from elastic to plastic loading. The solutionbased springback compensation method, aiming at forming sheet with target curvature by approaching appropriate adjustment factor \({\alpha }^{^{\prime}}\) through iterations, can be generalised as follows:

(1)
With a determined target curvature as \({1/R}_c\) and adjustment factor \({\alpha }_{i}^{^{\prime}}\), the loaded shape with curvature as 1/\({R}_{a}^{i}\) for i^{th} iteration CAF can be derived based on the modified closedform springback prediction solution (Eqs. (8) to (24). For the first iteration, the initial adjustment factor \({\alpha }_{i}^{^{\prime}}\) is set to be 1.

(2)
The \({i}^{th}\) iteration CAF is conducted with flexible forming tool which is adjusted to the derived loaded curvature \(1/{R}_{a}^{i}\). After completion of CAF, curvature of the CAFed sheet in that iteration is obtained and denoted as \(1/{R}_{b}^{i}\). \({R}_{b}^{i}\) is then compared with the target curvature radius . \(\Delta e\) is the shape error at midpoint between the CAFed and target shape and can be obtained by using Eq. (23) as \(\Delta e=\left({R}_{c}\sqrt{{{R}_{c}}^{2}{\left(L/2\right)}^{2}}\right)\left({R}_{b}^{i}\sqrt{{{R}_{b}^{i}}^{2}{\left(L/2\right)}^{2}}\right)\).

(3)
If the absolute value of shape error \(\left\Delta e\right\) exceeds the tolerance range \(Tol_1\), with the obtained CAFed curvature \(1/R_b^i\) in \(i^{th}\) iteration, the adjustment factor \({\alpha }_{i+1}^{^{\prime}}\) for next \({(i+1)}^{th}\) iteration is updated based on the modified closedform springback prediction solution (Eqs. (8) to (24). With the updated adjustment factor, the next iteration is started from step (1).
A flowchart of the proposed closedform solutionbased springback compensation method is demonstrated in Fig. 6, where i stands for the \(i^{th}\) iteration with the method and function g stands for the modified springback prediction solution (Eqs. (8) to (24).
4 Experimental procedure
4.1 Material
The material used was a 3rd generation AlLi alloy with major compositions of 0.6–0.9 wt% Li, 3.4–4.5 wt% Cu, and 0.6–1.1 wt% Mg. The asreceived material was in the form of sheets with 2, 4, and 6 mm thickness in T8 temper which is the commonly adopted finish temper for the 3rd generation AlLi alloys [27]. The material constants for this AlLi alloy are listed in Table 1, where h and n are the constants used in Eq. (14). The adopted creepageing constitutive model \(f_c\) is presented in Appendix 2.
4.2 Fourpoint bending CAF experiments
Fourpoint bending tests were conducted to verify the established closedform springback prediction solution. Strip shape specimens with dimension of 220 mm × 20 mm and thickness of 2, 4, and 6 mm were adopted in the testes. The fourpoint bending tool is composed of two upper loading cylinders and two supporting cylinders as shown in Fig. 7. The specimen was placed on the two supporting cylinders of the fourpoint bending tool with dimension as illustrated in Fig. 7. The specimens were deformed with designed loading displacement \(d_{ll}\) covering both elastic and plastic loading by the loading cylinders which are then fixed during creep age forming at 143 °C for 5 h. After completion of the creepageing, the specimen was cooled down in the air to room temperature with the bending tool and unloaded to allow happening of springback to obtain the final deformed curvature. The test groups for CAF under constant maximum surface stress of 430 MPa in elastic loading and with constant target curvature are listed in Tables 2 and 3 respectively. In Table 2, the specimens of different thicknesses were loaded with different displacements to reach the same maximum surface stress of 430 MPa within elastic region of the material. In Table 3, different loading displacements were adopted to creep age forming specimens with the same target curvature of 1/2000 mm^{−1}.
4.3 Flexible tool CAF experiments
Specimens of the same material were used in the tests to verify the proposed springback compensation method for CAF. The specimen dimensions were 335 mm × 165 mm for 2 mm sheets, and 320 mm × 165 mm for 4 mm sheets. A flexible tool as shown in Fig. 8a was employed. The setup, dimension and pin arrangements are illustrated in Fig. 8b and c. x’, y’ and z’ are the axes parallel to the specimen’s length, thickness, and width directions, and the origin of the coordinate system x’y’z’ locates at the centre point of the specimen bottom surface as shown in Fig. 8c. A 3 (rows) × 6 (columns) pin matrix was adopted for lower pins and a 3 (rows) × 4 (columns) matrix for the upper pins. To achieve more evenly distributed pressure applied on the specimen, for each row of pins, two steel splines with a dimension of 400 mm × 30 mm and 1.2 mm thickness were placed on the top and bottom of the specimen, aligned with the pin row. Rubber sheets were used between the upper and lower splines and the specimen to eliminate/minimise any damage to the specimen, as shown in Fig. 8b. The centre line interval for each twoneighbouring upper/lower pins in all rows (x’ direction) and columns (y’ direction) is 60 mm. The heights of the pins in each column were adjusted to be consistent with the designed loading curvature of the specimen. The same target curvature of 1/2000 mm^{−1} was set for CAF fabrication with flexible tool as shown in Table 4. In each iteration, the specimen was deformed to the loaded curvature \(1/R_a^1\) and then fixed for creepageing at 143 °C for 5 h. After completion of the creepageing, the specimen was cooled down in the air with the flexible tool to room temperature and unloaded to obtain the final deformed curvature after springback. For verification purpose, the error tolerance Tol_{1} of absolute shape error ∆e was set to be 1/10 of the target Δd_{f} as 0.56 mm.
5 Results and discussions
5.1 Fourpoint bending CAF experiments
Figure 9 shows the springback results for all thicknesses, including the prediction curves of the midpoint loading deflection d_{lm} with the final deflection Δd_{f} at the same midpoint of the CAFed specimen, and the moment release curves in creepageing for the moment at the end of creepageing stage M_{end} as a function of the moment at the end of loading stage M_{load} The evolutions of maximum absolute values of creep strain ε_{c}(t_{end}) and plastic strain \(\varepsilon_p\) at the end of creepageing, which were obtained at upper surface layer of the sheet, are also demonstrated with the increase of M_{load} in the figure. These results (solid lines) were obtained using the closedform solution proposed in this study. For comparison, the dotted lines are the prediction results derived from the traditional beam theory with assumption of small deflection as [20]:
where \(k_b\) is the loaded curvature as \(1/R_b\), σ(d) is the corresponding induced stress at layer d. The experimental data and numerical simulation results are also presented using diamond and star shape symbols respectively. The FE model for obtaining the presented data is presented in Appendix 3. In addition, the comparisons of all obtained results of springback percentage (sp) in the two test groups with different thicknesses, calculated using \(sp=\left(1\Delta d_f/d_{lm}\right)\times100\%\), are demonstrated in Fig. 10.
As demonstrated in Figs. 9 and 10, for all thicknesses, satisfactory convergences were achieved among results of the solutionbased predication, fourpoint bending CAF experiments, and numerical simulation, indicating validity of the proposed closedform solution. The predicted springback results by the solution have a maximum difference of 0.49 mm from the experiments and 0.27 mm from the FE simulations, while the beam theorybased results have much larger deviations, especially in the plastic loading region, with a maximum prediction error of 1.6 mm for the 4 mm thickness sheet. In addition, by comparing the obtained prediction curves of the beam theorybased and the solutionbased solutions respectively in Fig. 9a, c, and e, a similar trend which can be identified for all thicknesses is that with the same loading displacement d_{Im}, the prediction result d_{fc} for the beam theorybased method is always lower than the solutionbased method, indicating that a lower stress is calculated by the former method than the latter.
In addition, based on the obtained prediction curves, different evolution trend can be observed in elastic and plastic loading conditions for all thicknesses as shown in Fig. 9. In elastic loading, the CAFed deflection Δd_{f} and moment release (M_{load}–M_{end}) gradually increased at the beginning and surged with higher rate when approaching the plastic loading. In plastic loading, a high increase rate of Δd_{f} and (M_{load}–M_{end}) were obtained for all thicknesses. On the other hand, as demonstrated in Fig. 9b, d, and f, the final creep strain surged when approaching plastic loading state and maintained at a level slightly above 0.2% with minor increase in plastic loading, while the plastic strain during loading increased sharply with increasing moment. The observed surge for deflection Δd_{f}, moment release (M_{load}–M_{end}), and maximum creep strain when approaching plastic loading can be attributed to the nonlinear increase of creep deformation of the alloy when the applied stress surpasses 400 MPa, which was observed in uniaxial creepageing tests at 143 °C for this material [29]. The high increase rate of Δd_{f} and (M_{load}–M_{end}) achieved in plastic loading is due to the effects of plastic strain and larger creep strain respectively induced in stages of loading and creepageing. Additionally, for this AlLi alloy, springback is high and can offset the majority of the loading displacement. As demonstrated in Fig. 10, the lowest springback percentage is 79.5% for 6 mm sheet for CAF of a target curvature of 1/2000 mm^{−1}. It can be noted that due to high strength and creep resistance of the alloy, even for fabricating panels with a small curvature of 1/2000 mm^{−1}, CAF need be conducted with plastic loading state.
5.2 Flexible tool CAF experiments
Figure 11 shows the photos of the CAFed sheets using the solutionbased springback compensation method proposed in this study. As demonstrated in Fig. 11, the final deflection increases with increasing iteration for both 2 and 4 mm sheets. Within 4 iterations, the final error in the shape/curvature falls in the range of error tolerance Tol_{1} of 0.56 mm. Figure 12 is the comparison of the curvatures for the target value with the loaded, CAFed, and FE simulated results in the final 4th iteration for 2 and 4 mm sheets. As shown in the figure, similar to the fourpoint bending tests, for CAF of 1/2000 mm^{−1} curvature sheets using flexible tool, large loading displacement and high springback were observed. For 2 and 4 mm sheets, the maximum loading displacement d_{lm} was respectively 12.9 and 7.4 times larger than the target deflection Δd_{f}, with a springback percentage of 92.9% and 87.4%.
Figure 13 presents the evolution of adjustment factor \({\alpha }^{^{\prime}}\) and shape error \(\left\triangle e\right\) with increasing iterations, together with the FE simulation results of adjustment factor (α'_{s}) and shape error (\(\left\triangle e_s\right\)) based on the loaded shape of 4th iteration. As shown in the figure, for 2 and 4 mm sheets, after 4 iterations, the CAFed sheets satisfied the tolerance range of 0.56 mm with shape error of 0.5 mm for 2 mm and 0.38 mm for 4 mm. The adjustment factors were derived with values greater than 1 for both thicknesses, indicating that lower stiffness of the structure was induced for both thicknesses when loading with flexible tool than the proposed uniform stress bending model. With increasing iteration, the adjustment factor increased for both thicknesses, but the increasing rate gradually decreased. This decreasing trend of the rate demonstrated that the factor was converging to an appropriate value to reflect the true relationship between the deflection and the generated moment with the increase of iterations.
In this study, the initial value of adjustment factor was set to 1 to investigate the proposed method. In practical fabrication, the initial value can be set based on FE simulation. As demonstrated in Fig. 13a and b, the FE simulation results are close to the experimental data. Therefore, by combining FE analysis as preliminary estimation, a more proper initial value greater than 1 can be adopted to decrease the required iterations for the target shape and to improve the cost efficiency when deforming with flexible tool.
6 Conclusions
In this study, a closedform solution for creep age forming springback prediction covering elastic to plastic behaviours of the material has been proposed. Based on the solution, a springback compensation method for CAF fabrication with complex flexible forming tool has been developed. Both the proposed solution and the compensation method have been verified with corresponding CAF tests adopting the latest 3rd generation AlLi alloy. Major conclusions can be summarised as:

(1)
Due to superior strength and creep resistance of the AlLi alloy, a large loading displacement is required and a high springback percentage has been observed in CAF fabrication of a singly curved sheet with curvature of 1/2000 mm^{−1}. When adopting flexible tool, the loading displacement was 12.9 and 7.4 times larger than the target deflection with springback percentage of 92.9% and 87.4% respectively for the 2 and 4 mm sheets, which noticeably raised the difficulty of springback prediction and compensation.

(2)
A closedform springback prediction solution is developed by combining the traditional beam theory and Winkler’s theory, for CAF covering deformation from elastic to plastic loadings. Furthermore, an efficient springback compensation method for CAF is proposed based on the developed solution method.

(3)
Good convergences have been achieved among the results from the developed prediction solution, FE simulation, and experiments for fourpoint bending CAF tests, conducted in both elastic and plastic loading ranges with different sheet thicknesses. The validity of the proposed closedform springback prediction solution has been well supported with a maximum error of 0.49 mm compared with the experimental results.

(4)
Successful CAF fabrication of target singlecurvature sheet using multipoint flexible tool has been achieved by utilising the proposed springback compensation method within four iterations and with acceptable tolerances. An adjustment factor greater than 1 has been obtained for both 2 and 4 mm sheets in CAF with flexible tool, indicating a lower stiffness structure has been induced when loading with flexible tool than the proposed uniform stress bending model.

(5)
The applicability of the developed springback prediction solution and the solutionbased compensation method in plastic loading, verified through corresponding CAF tests, has extended the application area beyond the traditional beam theorybased prediction and compensation methods to elastic–plastic CAF covering cases of large deflection in plastic range.
Abbreviations
 a, a’ :

Compensation factor for modified displacement adjustment method and adjustment factor for the solutionbased springback compensation method
 C ^{i}, \({C}^{i+1}\), \({C}^{j}\), \({C}^{j+1}\) :

Tool shape \(C\) in the \({i}^{th}\) and \({(i+1)}^{th}\), \({j}^{th}\) and \({(j+1)}^{th}\) iterations respectively for displacement adjustment and modified displacement adjustment methods
 D :

Target shape to be formed
 d _{ll}, \({d}_{lm}\), \({d}_{lm}^{^{\prime}}\) :

Loading displacements at the loading points (fourpoint bending), at the midpoint, and at the midpoint adopting the flexible tool
 d _{ rl, } \({d}_{rm}\) :

Springback recovered deflection at loading points (fourpoint bending) and at the midpoint
 E :

Young’s modulus of the material
 f _{ c } :

Adopted constitutive model for creepageing
 I :

Moment inertia of \(yz\) plane for \(z\) axis
 k _{1}, \({k}_{2},\) \({k}_{3}\) :

Curvatures of sheet element in elastic loading, at the end of elastic loading, and in plastic loading, respectively
 L, \({L}_{1}\) :

Distance between supporting points and distance between the supporting and loading points
 M _{1} , \({M}_{2}\) , \({M}_{3}\) :

Moments generated in local \(yz\) plane for \(z\) axis in elastic loading, at the end of elastic loading, and in plastic loading
 M _{load}, \({M}_{end}\) :

Moments in \(yz\) plane for \(z\) axis at the end of loading stage and at the end of creepageing stage after stress relaxation before unloading
 O _{a}, \({O}_{b}\) :

Centre points of loaded curvature and CAFed, i.e. final deformed curvature of the sheet
 O _{1}, \({O}_{2}\), \({O}_{3}\) :

Centre points of sheet segment with the polar coordinate in elastic loading, at the end of elastic loading, in plastic loading
 R _{a}, \({R}_{b}\) , \({R}_{c}\) :

Curvature radii of loaded shape, CAFed shape, and target shape
 R _{n} _{1}, \({R}_{n2}, {R}_{n3}\) :

Distances of the neutral plane to the origin point \({O}_{1}\) in elastic loading, and to the origin point \({O}_{2}\) at the end of elastic loading and in plastic loading, respectively
 r, \({r}_{i}\), \({r}_{o}\) :

Distance of random specific plane to the origin point. Distances of the top and bottom elastic–plastic boundary to the origin point \({O}_{2}\) in plastic loading state
 S ^{i}, \({S}^{j}\) :

Obtained formed shape \(S\) in \({i}^{th}\) and \({j}^{th}\) iterations
 t :

Sheet thickness
 t _{ a } :

A certain time point during creepageing
 Tol _{1} :

Error tolerances respectively for modified springback compensation method
 x, y, z :

Local Cartesian coordinate system, in which \(y\) and \(z\)axes form the plane perpendicular to the neutral plane, and \(x\) axis locates at centroid of \(yz\) plane and is parallel to the neutral plane
 x’, y’, z’ :

Axes parallel to the length, thickness, and width directions of the specimen with the origin at the centre point of the specimen bottom surface
 y _{ d } :

Distance of layer \(d\) of the sheet to the neutral plane
 ∆d _{f} :

Deflection at midpoint of the CAFed sheet compared with its undeformed initial state
 ∆e :

The shape error at midpoint between the CAFed and target shape of the sheet
 ∆θ, \(\Delta \varepsilon\), \(\Delta M\) :

Increments of the centre angle, strain, and moment in the transformation process from \({a}_{2}{b}_{2}{c}_{2}{d}_{2}\) to \({a}_{3}{b}_{3}{c}_{3}{d}_{3}\)
 ε, ε_{2},ε_{y} :

Strain generated along \(x\) direction, accumulated strain at the end of elastic loading, and strain at 0.1% offset yielding point
 ε_{c}, ε_{e} , ε _{p} :

Accumulated creep strain, elastic strain, and plastic strain along \(x\) direction
 θ, θ _{ a } , θ _{ b } :

Centre angle of the sheet segment. Centre angles of loaded curvature and CAFed curvature of the sheet
 θ _{1} , θ _{2} , θ _{3} :

Centre angles of deformed segment in elastic loading, at the end of elastic loading, and in plastic loading
 σ, σ _{ e } , σ _{ p } :

Stress along \(x\) direction, stress at layer \(r\) along \(x\) direction in elastic region and plastic region in plastic loading, respectively
 σ _{ r } :

Relaxed stress along \(x\) direction in \(yz\) plane
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Acknowledgements
Financial support (grant number MESM_PSG110) from Commercial Aircraft Corporation of China (COMAC) is gratefully acknowledged.
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Xi Wang: investigation, writing—original draft, visualisation. Rong Qi: writing—review and editing. Zhusheng Shi: conceptualization, supervision, funding acquisition, project administration, writing—review and editing. Jianguo Lin: writing—review and editing.
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Appendices
Appendix 1
1.1 Derivation of Eq. (18)
Based on Eq. (12), r_{i} and r_{o} can be expressed as functions of R_{n3}, i.e. \(r^\ast=f_1(R_{n3})\):
where A’, B’, and C’ are derived as:
with determined ε_{y}, θ_{2} , Δθ, t, and R_{n2}.
By substituting Eqs. (13) and (15) into Eq. (17), as the following equation can be obtained:
where A’’ is derived as:
By adopting Eq. (15) and applying Taylor expansion, B” can be derived as:
Similarly, C” is expressed as:
where h and n are determined material constants; Eq. (17) or (28) can be expressed as a function of R_{n3} as Eq. (18), i.e. f(R_{n3})= 0.
However, the derived function cannot be solved analytically to obtain the solution of R_{n3}. Bisection method is applied as an iterative algorithm tool to approach the value of R_{n3}. The process of utilising bisection method to obtain the value of R_{n3} is manifested in Fig. 14. Considering the possible extent of deflection increase from \(a_2b_2c_2d_2\) to \(a_3b_3c_3d_3\) in CAF and corresponding potential neutral plane variation range in plastic loading state, in the first iteration, the initial values of R_{n3α}^{i} and R_{n3b}^{i}, as the upper and lower limits of R_{n3} for bisection method, are given with values as \(R_{n3a}^1=R_{n2}+t/5\) and \(R_{n3b}^1=R_{n2}t/5\). Tol_{2} is defined as the error tolerance comparing with obtained results of f(R_{n3}), which is set to be the value of 5E4 in this study.
Appendix 2
2.1 Adopted constitutive model
The constitutive equations adopted in this study are shown below [34]:
where \(\varepsilon_c\left(t_a\right)\) and \(\sigma\left(t_a\right)\) are respectively the creep strain and stress at layer r at creepageing time \(t_a\); is the threshold stress; H is a creep variable to reflect the hardening effect induced by evolution of normalised dislocation density \(\overline{\mathrm\rho}\) and normalised precipitate radius \(\overline r\) during creepageing. ε'_{p} and \({\overline\rho}_0\) are respectively the plastic strain and the normalised dislocation density accumulated in the loading stage. Other symbols used in the equations are material constants. The dot above a symbol indicates time derivative. The initial values and material constants used for the constitutive equations are presented respectively in Appendix Table 5 and Appendix Table 6.
Appendix 3
3.1 Fourpoint bending FE model
As shown in Fig. 15, a threedimensional ABAQUS FE model was established according to the setup and dimensions in Fig. 7a. The constitutive model was implemented in the simulation through CREEP subroutine. For the loading cylinders, as shown in Fig. 15, freedom for all directions apart from displacement along y’ (U_{y},) axis was fixed, allowing the loading cylinders only move parallelly along y' axis. The supporting cylinders were completely fixed in all directions. Surface contacts were set up between the cylinders and specimen which were modelled with penalty tangential behaviour with friction coefficient of 0.25. The specimen was composed of S4R shell element with mesh size of 2 × 2 mm and 11 integration points through the thickness direction. The cylinders were composed of discrete rigid elements with size of 2 × 2 mm. The general information of the model is summarised in Appendix Table 7. During simulation, the specimen was deformed by the loading cylinders with the designed loading displacement (given in Tables 2 and 3) and was kept at the obtained shape for creepageing with the required time. The final defection prediction result was obtained after the specimen was fully unloaded with the upward movement of the loading cylinders.
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Wang, X., Rong, Q., Shi, Z. et al. An efficient closedform solution for springback prediction and compensation in elastic–plastic creep age forming. Int J Adv Manuf Technol 125, 1115–1133 (2023). https://doi.org/10.1007/s00170022106073
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DOI: https://doi.org/10.1007/s00170022106073
Keywords
 Springback prediction
 Springback compensation
 Closedform solution
 Creep age forming
 AlLi alloy
 Multipoint flexible tool