1 Introduction

Creep age forming (CAF) is a metal forming method specially invented for fabricating large-scale aluminium alloys’ panel components. Normally CAF process is comprised of three stages, i.e. loading, creep-ageing, 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 creep-ageing 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 non-linear 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 single-curvature 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 double-curvature bending was fulfilled [18]. With the development of constitutive model for creep-ageing, by adopting the mechanism-based model to reflect stress-creep strain relationship in CAF, Jeunechamps et al. [19] further proposed an analytical springback prediction method which can be applied to double-curvature 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:

Fig. 1
figure 1

Schematic of DA method for springback compensation


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 (Si – D) can be viewed as the geometric error between the deformed and target shape in \(i^{th}\) iteration as shown in Fig. 1bCi + 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 Ci in previous ith iteration with the geometric error (SiD).

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 time-consuming 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(SjD) in jth 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 Al-Li 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 Al-Li alloy, the creep resistance is enhanced simultaneously which in turn reduces the creep deformation in CAF: limited creep strain was observed in creep-ageing 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 creep-ageing for the Al-Li 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 Al-Li 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 closed-form 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 four-point bending CAF tests adopting the latest 3rd generation Al-Li 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 single-curvature sheets by using flexible tool and the same Al-Li alloy with different thicknesses.

2 Closed-form 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 (F0, F1, 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 creep-aged at elevated temperature for designed ageing time. (iv) Finally, the sheet is unloaded after completion of creep-ageing and the final CAFed sheet is obtained after springback. As shown in Fig. 2a, the CAFed sheet has a curvature radius of Rb with Ob as the centre point and θb as the centre angle for the final CAFed shape. Δdf is the deflection at midpoint of the final CAFed sheet from its initial undeformed state.

Fig. 2
figure 2

Schematic of forming in CAF fabrication and two proposed loading models

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 qLoad when considering unit width. When adopting multipoint flexible tool, the sheet is deformed with multiple point loads Fx. 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. four-point 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 Oα and curvature radius Rα are used for the two loading models. In Fig. 2c, L is the distance between the two supporting points. L1 is the distance between the supporting point and the loading point with exerted loads of \({F}_{1}^{^{\prime\prime} }\). dll 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 dlm is the displacement of the midpoint. The geometrical relation between loading point displacement dll and Rα 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 dll or dlm.

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 \(a_1b_1c_1d_1\). k1 is defined as the deflection curvature and \(O_1\) is the centre point of k1. A polar coordinate system (r, θ) with the origin at the centre point O1 is used to express the location of the sheet layers along thickness direction. As shown in Fig. 3a, θ1 is the centre angle of \(a_1b_1c_1d_1\). Rn1 is the distance of the neutral plane to the centre point O1 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 z-axes 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.

Fig. 3
figure 3

Schematic of deforming process in elastic loading and stress distribution through thickness. The labels in bracket in b are for the condition at the end of elastic loading

When loading begins as exerting downward displacements on the sheet, elastic strain and stress are generated along x-axis direction in yz plane. The relation of deflection and induced moment depends on different loading model. For four-point 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 z-axis. 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:

$$\begin{array}{cc}\sigma=\frac{M_1}I\left(r-R_{n1}\right),&R_{n1}-\frac t2\leq r\leq R_{n1}+\frac t2\end{array}$$

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 θ2 to θ3, and the centre point and neutral plane respectively change from position O2 to O3 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 \(a_3b_3c_3d_3\).

Fig. 4
figure 4

Schematic of deforming process from elastic to plastic loading state in CAF and stress distribution through the thickness of the sheet

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:

$${\varepsilon }_{y}={\varepsilon }_{2}^{^{\prime}}+\Delta {\varepsilon }^{^{\prime}}$$

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:

$$\left\{\begin{array}{c}{\varepsilon }_{2}^{^{\prime}}={\varepsilon }_{y}(\frac{r-{R}_{n2}}{t/2})\\ \Delta {\varepsilon }^{^{\prime}}=\frac{\Delta \theta (r-{R}_{n3})}{r{\theta }_{2}}\\ {\varepsilon }_{y}={\varepsilon }_{2}^{^{\prime}}+\Delta {\varepsilon }^{^{\prime}}\end{array}\right., r={r}_{i} \;\;\text{or}\;\; r={r}_{0}$$

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:

$$\left\{\begin{array}{cc}\begin{array}{c}{\varepsilon }_2^{''}={\varepsilon }_y\left(\frac{r-R_{n2}}{t/2}\right)\\\triangle{\varepsilon }^{''}=\frac{\triangle\theta\left(r-R_{n3}\right)}{r\theta_2}\\\begin{array}{c}{\varepsilon }_e={\varepsilon }_2^{''}+\triangle{{\varepsilon }^{''}}\\\sigma_e=E{{\varepsilon }_e}\end{array}\end{array},&r_i\leq r\leq r_0\end{array}\right.$$

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 \(dA=\delta bdy=dy\) = \(d\left(r-R_{n2}\right)=dr\), the equation can be written as:

$$\int_{R_{n2}-t/2}^{R_{n2}+t/2}\sigma dr=\int_{r_i}^{r_0}\sigma_edr+\int_{r_i}^{R_{n2}-t/2}\sigma_pdr+\int_{r_0}^{R_{n2}+t/2}\sigma_pdr=0$$

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:

$${M}_{3}={M}_{2}+\Delta M=\int \sigma (r-{R}_{n2}) dA$$

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 creep-ageing

In creep-ageing 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 creep-ageing 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 creep-ageing, by applying proper constitutive model reflecting the relation of stress and creep strain, \(\varepsilon_c\left(t_a\right)\) can be obtained as:


fdenotes the constitutive model being used. The creep strain accumulated after completion of creep-ageing at time of \(t_{end}\) is denoted as εc(tend).

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 creep-ageing, 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 creep-ageing in elastic/plastic loading state is demonstrated in Fig. 5.

Fig. 5
figure 5

Schematic of strain (x direction) distribution along thickness of the sheet (y direction) after completion of creep-ageing. \(\varepsilon\) is the bending strain in x direction. The dotted lines stand for neutral plane of the sheet

As shown in Fig. 5, after completion of creep-ageing, 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 creep-ageing/start of unloading stage, can be expressed as:

$$\left\{\begin{array}{c}\varepsilon_e\left(d\right)=\varepsilon\left(d\right)-\varepsilon_p\left(d\right)-\varepsilon_c\left(t_{end}\right)\\M_{end}=\int E\varepsilon_e\left(d\right)y_ddA\end{array}\right.$$

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 four-point bending cases, by applying Eq. (6), the recovered deflection at loading point (\(d_{rl}\)) during springback can be obtained as \(d_{rl}=\frac{M_{end}}{6EI}(3L_1L-4L_1^2)\). For uniform stress bending cases, by applying Eq. (7), the recovered deflection at midpoint of the sheet (drm) can be derived as \(d_{rm}=\frac{40M_{end}L^2}{384EI}\). By utilising Eqs. (3) and (4) respectively for four-point bending and uniform stress bending cases, the curvature \(1/R_b\), as the final curvature of CAFed sheet, can be derived as \(R_b^2={(L/2-L_1)}^2+\left(\sqrt{R_b^2-\left(L/2\right)^2}+\left(d_{ll}-d_{rl}\right)\right)^2\) 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:

$$\Delta d_f=R_b-\sqrt{R_b^2-\left(L/2\right)^2}$$

3 Springback compensation method based on closed-form solution

The above closed-form solution of springback prediction for CAF is based on two simplified loading models, i.e. four-point 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 state-of-the-art 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:

$${d}_{lm}^{^{\prime}}=\alpha ^{\prime}\frac{40{M}_{1}{L}^{2}}{384EI}$$

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 closed-form 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 solution-based 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. (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 ith iteration CAF can be derived based on the modified closed-form springback prediction solution (Eqs. (8) to (24). For the first iteration, the initial adjustment factor \({\alpha }_{i}^{^{\prime}}\) is set to be 1.

  2. (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. (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 closed-form 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 closed-form solution-based 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).

Fig. 6
figure 6

Flowchart of closed-form solution-based springback compensation method

4 Experimental procedure

4.1 Material

The material used was a 3rd generation Al-Li 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 as-received 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 Al-Li alloys [27]. The material constants for this Al-Li alloy are listed in Table 1, where h and n are the constants used in Eq. (14). The adopted creep-ageing constitutive model \(f_c\) is presented in Appendix 2.

Table 1 Material constants for the Al-Li alloy

4.2 Four-point bending CAF experiments

Four-point bending tests were conducted to verify the established closed-form 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 four-point 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 four-point 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 creep-ageing, 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.

Fig. 7
figure 7

Schematic of four-point bending tool (dimensions in mm) and photo with loaded specimen

Table 2 Four-point bending CAF experiments in elastic loading, with the same maximum loading surface stress of 430 MPa
Table 3 Four-point bending CAF experiments with the same targeted curvature

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 two-neighbouring 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 creep-ageing at 143 °C for 5 h. After completion of the creep-ageing, 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 Tol1 of absolute shape error |∆e| was set to be 1/10 of the target Δdf as 0.56 mm.

Fig. 8
figure 8

Multipoint flexible tool and its dimensions

Table 4 Experiment programme for CAF of target curvature with flexible tool

5 Results and discussions

5.1 Four-point bending CAF experiments

Figure 9 shows the springback results for all thicknesses, including the prediction curves of the midpoint loading deflection dlm with the final deflection Δdf at the same midpoint of the CAFed specimen, and the moment release curves in creep-ageing for the moment at the end of creep-ageing stage Mend as a function of the moment at the end of loading stage Mload The evolutions of maximum absolute values of creep strain εc(tend) and plastic strain \(\varepsilon_p\) at the end of creep-ageing, which were obtained at upper surface layer of the sheet, are also demonstrated with the increase of Mload in the figure. These results (solid lines) were obtained using the closed-form 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]:

Fig. 9
figure 9

Creep age forming behaviour for 2, 4, and 6 mm sheets, showing the predicted, FE simulated, and experimental final deflection Δdf vs loading displacement dIm; the moment Mend, strain εc, and plastic strain εp at the end of creep-ageing as a function of loading moment Mload

$$\left\{\begin{array}{c}\begin{array}{cc}\varepsilon(d)=k_by_d,&-t/2\leq y_d\leq t/2\\\sigma\left(d\right)=E\varepsilon\left(d\right),&-\sigma_y/(Ek_b)\leq y_d\leq\sigma_y/(Ek_b)\\\sigma\left(d\right)=\sigma_y+h{(\varepsilon\left(d\right)+\sigma_y/E)}^n,&\sigma_y/(Ek_b)\leq y_d\leq t/2\\\sigma\left(d\right)=\sigma_y+h{(\varepsilon\left(d\right)-\sigma_y/E)}^n,&-t/2\leq y_d\leq{-\sigma}_y/(Ek_b)\end{array}\end{array}\right.$$

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.

Fig. 10
figure 10

Comparison of the springback percentage results in two test groups with different thicknesses. The legends from left to right respectively stand for results of experiment, numerical simulation, closed-form solution, and the beam theory-based solution

As demonstrated in Figs. 9 and 10, for all thicknesses, satisfactory convergences were achieved among results of the solution-based predication, four-point bending CAF experiments, and numerical simulation, indicating validity of the proposed closed-form 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 theory-based 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 theory-based and the solution-based 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 dIm, the prediction result dfc for the beam theory-based method is always lower than the solution-based 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 Δdf and moment release (Mload–Mend) gradually increased at the beginning and surged with higher rate when approaching the plastic loading. In plastic loading, a high increase rate of Δdf and (Mload–Mend) 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 Δdf, moment release (Mload–Mend), 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 creep-ageing tests at 143 °C for this material [29]. The high increase rate of Δdf and (Mload–Mend) achieved in plastic loading is due to the effects of plastic strain and larger creep strain respectively induced in stages of loading and creep-ageing. Additionally, for this Al-Li 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 solution-based 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 Tol1 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 four-point 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 dlm was respectively 12.9 and 7.4 times larger than the target deflection Δdf, with a springback percentage of 92.9% and 87.4%.

Fig. 11
figure 11

Photos of CAF iteration results of sheets with different thickness

Fig. 12
figure 12

Comparison of the 4th iteration loaded curvature 1/\(R_a^4\), CAFed curvature 1/\(R_b^4\), FE simulated result 1/\(R_s^4\), and target curvature 1/\(R_c\), \(R_c\) = 2000 mm, for 2 mm and 4 mm sheets. The dot symbols are measured points at corresponding positions of x’ axis

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.

Fig. 13
figure 13

Evolution of adjustment factor and shape error with increased 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 closed-form 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 Al-Li alloy. Major conclusions can be summarised as:

  1. (1)

    Due to superior strength and creep resistance of the Al-Li 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. (2)

    A closed-form 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. (3)

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

  4. (4)

    Successful CAF fabrication of target single-curvature 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. (5)

    The applicability of the developed springback prediction solution and the solution-based compensation method in plastic loading, verified through corresponding CAF tests, has extended the application area beyond the traditional beam theory-based prediction and compensation methods to elastic–plastic CAF covering cases of large deflection in plastic range.