Analytical Approach to Investigate the Effects of Through-Thickness Stress on Springback in Bending of Isotropic Sheet Metal

Within contemporary sheet metal forming processes like electromagnetic forming, the sheet experiences pronounced out-of-plane compression stress. This study focuses on the predicting the springback in the bending of an isotropic sheet metal in the presence of through-thickness compressive normal stress. In the analytical method, the longitudinal stress distribution across sheet thickness was calculated via equilibrium equations and the associated �ow rule in incremental plasticity based on the power law hardening model, obtaining the reverse bending moment. During the unloading phase, the external load was lifted, and the springback angle was estimated by assuming linear elastic behavior and neglecting the Bauschinger effect, employing the superposition method. A case study was conducted on an aluminum alloy sheet for different compressive stresses and bend curvatures. Subsequently, the reverse bending moments and springback angles obtained for each case were compared to the numerical outcomes from �nite element (FE) modeling. Analytical results indicate that elevating the compressive normal stress on the sheet surface to 75% and 100% of the yield stress will lead to a reduction in springback by 17.4% and 32%, respectively. At the compressive normal stress of 75% of the yield stress, the numerical model exhibited a 4.4% difference from the analytical model. For the validation of the analytical model, a four-point bending test was conducted with varying initial bend curvatures and angles. This involved comparing the experimental springback angle and curvature with both numerical and analytical predictions. In comparison, both the numerical and analytical models demonstrated strong agreement with the experimental results.


Introduction
Springback substantially in uences the quality and precision of components fabricated via sheet metal forming.It refers to the elastic deformation of a sheet after unloading and its relaxation from forming loads [1,2].The springback control and compensation approaches include sheet tension enhancement, tool angle compensation, and sheet hammering [3].As the tension of the sheet rises, longitudinal stress gradient in sheet thickness reduces, and, thus, the total bending moment and springback decrease [4].To compensate for the tool angle, it is required to accurately estimate springback.In traditional forming methods, the sheet is hammered in the bending angle and ironed in clearance between the die components to apply additional compressive stress on the sheet surface so that the plastic region of the bend section develops.As a result, springback decreases [5].
In the forming of sheet metals, the tool-sheet interaction often induces normal stresses in the interface.The stress state could be altered by contact stresses arising from the interaction between the tool and the sheet.This effect can be evaluated in two aspects.As a major consequence, normal stresses reduce the axial yield stress in tension.Thus, a smaller tensile stress would be required to induce a plastic deformation.For example, in the bending of a sheet using compressive stress on the surface, a lower bending moment is required to create a curvature.As another consequence, compressive stress on the surface may increase formability [6].In high-velocity forming, including electromagnetic forming and explosive forming, the tool-sheet interaction has signi cant effects [7].Iyama and Itoh [8] performed the bulge test in the underwater explosive forming of an annealed 5052 aluminum alloy sheet at a velocity of 6970 m/s and a pressure of 15.9 GPa.Schafer et al. [9] obtained a normal stress of 2.7-6.5 GPa in electromagnetic forming due to the tool-sheet interaction.Flexible-die forming methods, e.g., hydroforming and rubber pad forming, apply not only in-plane stresses but also normal stresses to the sheet surface [10].Brooke [11] showed that the normal stress on the sheet surface in the hydroforming of automotive industry components is nearly 50 MPa.When forming sheet metals, diminishing the radius/thickness ratio at sharp corners would nullify the assumption of plane-stress conditions [12].Experimental research has shown that the normal stress on the sheet surface in the plastic deformation processes of materials due to the pressure of the interface or die contact with the sheet has major effects on variables of the formed product.
Many studies have been conducted on springback in the forming of sheet metals in recent years.Zhang et al. [13] analytically investigated springback for the bulge process using the hydroforming of an aviation aluminum alloy sheet.They analytically obtained springback in bulge forming and validated the analytical model using both experimental tests and numerical simulations.Chu et al. [14] proposed an analytical model for springback in low-pressure tube hydroforming (LPTH) based on classical elasticity theory.The classical theory of elasticity assumes a direct correlation between springback and the bending moment in the forming of tubes.During this process, both a bending moment and a point force were applied on tube, simultaneously.While the internal pressure rises, the point force changes gradually from compression mode to tensile mode and the related bending moment enhances at rst and nally drops.The force state and bending distribution were analyzed, formulating the tensile force and bending moment of the tube.The tensile force and bending moment are directly related to the internal tube pressure.Furthermore, the corner radius and tube thickness are other effective parameters on the bending distribution and, therefore, springback.Analytical and experimental results demonstrated that the internal pressure was the most important parameter in reducing and even eliminating springback.
Research on the effects of through-thickness stress in sheet metal forming has mostly been focused on sheet formability.Hashemi et al. [15] studied the effects of normal stresses on the sheet surface on the forming limit diagram and formability limit stress diagram to estimate the onset of necking in T-shaped tube hydroforming.Imbert et al. [16] experimentally and numerically studied the effects of the tool-sheet interaction (i.e., sheet contact with the die) to evaluate electromagnetic forming damage.They showed that the tool-sheet interaction (substantial through-thickness normal stress) controlled damage (i.e., necking and fracture) and enhanced formability in electromagnetic forming.
Despite the signi cant effects of normal stress on springback, the effects of springback have not been analytically studied in the literature.This study proposes a simpli ed analytical model to evaluate springback in the bending of homogenous, isotropic sheet metals under the presence of normal stress.Incremental plasticity equations are used to nd the effects of normal stress on the longitudinal stress distribution, reverse bending moment, and springback angle.Then, the results are compared to numerical simulations for different normal stresses.In addition, the four-point bending test is performed at various initial bend curvatures and angles, comparing the experimental springback angle to the numerical model.

Analytical model
Loading was modeled as free bending for simultaneous bending and coupling in the presence of a compressive normal stress on the sheet surface.The material is assumed to be homogeneous.In the loading phase, elastic behavior was assumed to be negligible compared to plastic behavior.In unloading, however, the sheet was assumed to have linear elastic and isotropic behavior.The strain-hardening behavior of the sheet was modeled using the power law hardening model (i.e., Hollomon's strain hardening law).Furthermore, the plastic behavior of the sheet was assumed to be isotropic and modeled using the von Mises yield criterion.The Bauschinger effect was neglected.The principal axes of the stress and strain tensors were placed at the longitudinal, transverse, and thickness axis.The normal stress distribution on the sheet surface was also assumed to be uniform.The sheet sections were assumed to remain planar with no distortions during the bending process.
A sheet with a thickness of t and an initial element length of l 0 was modeled, as shown in Fig. 1.Directions 1, 2, and 3 represent the longitudinal, transverse, and normal directions, respectively.The neutral axis is on the center of the sheet section, i.e., C 0 D 0 .AB is assumed to be at a distance of y from the neutral axis.
As shown in Fig. 1, l 0 is the length of AB before deformation, l s is the length of the neutral axis CD after deformation, and l is the length of AB after deformation.Here, l s = ρ 0 θ 0 and l = l s (1 + y ρ 0 ), where ρ 0 is the neutral axis radius (i.e., bend radius).For tension-free bending and ρ 0 ≫ t, the longitudinal strain is written as [12,17]:

Governing equations of plastic bending
The longitudinal, transverse, and thickness directions were assumed to be located on the principal axes of the plastic stress and strain tensors, the effective stress can be obtained from the principal stresses using the von Mises criterion [18]:

√
The effective plastic strain increment is written as [18]: are the plastic strain increments in the principal directions.For a simple description of the deformation of an element and the correlation between strains in the principal directions, stress ratio α = σ 2 / σ 1 and strain ratio β = dϵ P 2 / dϵ P 1 were de ned and effective stress was rewritten as the following relation: Considering incompressibility in plastic deformation, the sum of the principal plastic strain increments is zero (dϵ P 1 + dϵ P 2 + dϵ P 3 = 0).Therefore, the plastic strain increment in the thickness direction is given by: The effective plastic strain increment is written based on the strain ratio as: For the plastic deformation, the associated ow rule can be written as [19]: where dϵ P ij is the plastic strain increment, f σ ij is the yield function, and dλ is the plastic multiplier.For the von Mises criterion, the strain ratio can be rewritten based on the stress components and stress ratio as: The power law correlates the effective stress and effective plastic strain during strain hardening stage as: where K is the hardening coe cient, and n is the strain-hardening exponent.For σ 3 = − P in Eq. ( 8) and putting its result in Eq. ( 6), and using Eqs.( 1) and ( 4), Eq. ( 9) can be rewritten as: Equation ( 10) is a nonlinear implicit equation σ 1 (y, P) | K , n , ρ 0 , α , where σ 1 represents the longitudinal stress (as a dependent parameter), K and n are material parameters, ρ 0 and α are process parameters, and P and y are independent parameters.To calculate the longitudinal stress distribution in the sheet section, it is required to solve Eq. ( 10) for σ 1 at different values of y.
When bending thin sheets, it is reasonable to assume α = 0, leading to the calculation of the strain ratio as: The longitudinal stress for α = 0 is obtained using Eq. ( 10) as: For σ 1 = 0 in Eq. ( 12), the neutral axis locus y * (P) | K , n , ρ 0 is obtained as: Once the bending-induced longitudinal stress distribution across the sheet section has been obtained, the bending moment can be calculated as:

Governing equations of elastic unloading
In the unloading phase, the reverse bending moment is applied in the opposite direction so that the resultant of the loading and unloading moments is zero ( As shown in Fig. 2, the neutral axis length l s remains unchanged after unloading.The change in bend angle can be expressed as: [12]: The unloading process was assumed to exhibit linear elastic behavior.Hence, the change in longitudinal stress during unloading is determined using Hooke's law.[12]: The maximum longitudinal stresses occur on the top and bottom surfaces of the sheet (i.e., y = t / 2).
The reverse bending moment per unit width is given by [12]: 17 Once E / Δρ has been derived from Eqs. ( 16) and ( 17), the following equality per unit width is obtained: The calculation of Δ 1 ρ using Eq. ( 18) and its insertion into Eq.( 15) gives the springback angle after elastic unloading: 3 Experimental setup Tensile testing was performed under ASTM E8 to obtain the mechanical properties of the AA6061 sheet.
Figure 3 shows the uniaxial tensile test using a 60-ton SANTAM universal testing machine.Table 1 lists the mechanical properties of the AA6061 sheet along with the normal anisotropy ratio − R and planar anisotropy ratio ΔR.To validate the analytical model, four-point bending test was carried out under the analytical model parameters as shown in Fig. 4.This test was used to induce pure bending between the two top supports during the loading phase.
( ) Five specimens with a length of 250 mm and a width of 20 mm from 0.8-thick AA6061 sheet were fabricated for the four-point bending test.Figure 5 illustrates the setup for the four-point bending test conducted on the 60-ton SANTAM universal testing machine.The bending of the sheet and, therefore, initial bend angle increase as the vertical displacement of the top supports (y c ) rises in the four-point bending test.
The specimens were photographed before and after loading, calculating the initial and nal bend angle and radius (θ 0 , ρ 0 and θ 1 , ρ 1 ) using image processing in GetData Graph Digitizer.Table 2 reports the geometric parameters of the sheet and bend, including the sheet thickness t, bend radius ρ 0 , curvature k 0 = 1 / ρ 0 , bent ratio ρ 0 / t, and bend angle θ 0 .

Numerical model
To analyze the springback phenomenon in a metallic blank, a Finite Element (FE) model was implemented.The initial stage of the bending process was executed using ABAQUS/Explicit software.Subsequently, the output eld variables obtained from this stage were imported into the unloading step.
In this phase, a quasi-static simulation was conducted employing the ABAQUS/Standard solver, with a speci c focus on calculating the springback angle and curvature radius.The sheet was modeled as a deformable solid part possessing dimensions of 100 mm in length, 20 mm in width, and 0.8 mm in thickness, as illustrated in Fig. 6.Since the system was axisymmetric, one-quarter of the part was simulated (i.e., symmetry about the x-axis on the y-z plane and about the z-axis on the y-x plane).To model a curvature radius of 34 mm, an analytical rigid-surface die was created.A reference point was assigned at the die's corner to accurately control its position.The isotropic elastic-plastic material model was then applied to the AA6061 blank based on the material properties de ned in Table 1.
The bending moment around the z-axis was prescribed by utilizing a coupling constraint on the reference point.A through-thickness compressive normal stress of 204 MPa (P/Y = 0.75) was imposed on both the top and bottom surfaces of the sheet.To expedite the computational process in the dynamic explicit analysis during the forming stage, the Mass scaling technique was implemented.The sheet was meshed using eight-node linear solid elements with reduced integration approach (i.e., C3D8R).The sensitivity of the model to the element size was analyzed, nding the optimal mesh to have 100 elements in the xdirection, 30 elements in the y-direction, and 8 elements in the z-direction (a total of 24000 elements).

Results and Discussion
In analytical section, to nd the longitudinal stress distribution σ 1 in the sheet section and obtain the bending moment, Eq. ( 12) for σ 1 should be solved at different values of y.Eq. ( 12) is nonlinear, and the nonlinear least squares (NLS) method was used to solve it.The normalized compressive stress was de ned using the yield stress as σ 3n = P / Y. Five loading scenarios were assumed, including P / Y = 0, P / Y = 0.25, P / Y = 0.5, P / Y = 0.75, and P / Y = 1.The normalized distance from the neutral axis was de ned as y n = y / t. Figure 7 illustrates the non-dimensionalized longitudinal stress σ 1n = σ 1 / Y against the non-dimensionalized distance from the neutral axis, obtained by numerically solving Eq. ( 12).As illustrated in Fig. 7, an elevation in pressure signi cantly reduced the stress above the neutral axis, whereas the longitudinal stress below the neutral axis intensi ed in compressive nature, with the compressive stress magnitude increasing at a more gradual rate.Furthermore, the neutral axis shifts within the thickness, resulting in a larger portion of the sheet thickness being subjected to compressive stresses.The neutral axis displacement y * can be calculated using Eq. ( 13).
One can de ne the ratio of the thickness of the tension area t t to the thickness of the compressed area t c represented asζ = t t / t c .In the absence of compressive normal stress on the sheet surface ( P / Y = 0), ζ = 1 since the neutral axis located in the middle of section i.e. (y * = 0).As the compressive normal stress on the sheet increases, a larger portion of the sheet surface undergoes compressive longitudinal stresses.Here, ζ = 0 when the entire sheet section is under longitudinal compressive stresses.To evaluate the effects of compressive stress on the bending moment and springback, the characteristic curve of bending process M − k 0 can be plotted once the bending moment has been calculated via the stress distributions at different curvatures.This study according to Fig. 9 nondimensionalized the curvature using the critical elastic curvature (k e = 1 / ρ e ) in the absence of compressive stress ( k n = k 0 / k e ).The critical elastic curvature k e is the curvature at which the sheet is on the verge of entering the plastic region.Moreover, the bending moment M was nondimensionalized using the limiting elastic bending moment in the absence of compressive stress M e ( M n = \raisebox1ex$M$ \raisebox− 1ex$M e $).The limiting elastic bending moment refers to the moment at which the sheet is about to enter the plastic region.Marciniak et al. [12] calculated the limiting elastic bending moment and critical elastic curvature in the absence of normal stress on the sheet surface using Equations ( 20) and (21).

21
The calculated limiting elastic bending moment and critical elastic curvature for the case-study sheet are 29 N.mm and 0.0098 mm ¹, respectively.
For the AA6061 sheet, the longitudinal stress distribution at different compressive stress was obtained using Eq. ( 12), as shown in Fig. 7.Then, the bending moment was calculated using the characteristic curve of bending process at a given curvature (Table 2) and different compressive stresses.The insertion of the bending moment into Eq.( 19) resulted in the springback at different compressive normal stresses.The springback was nondimensionalized using the initial bend angle (Δθ n = Δθ / θ 0 ).Table 3 provides the bending moment and bend angle change due to the springback at different compressive stresses for the AA6061 sheet.According to Table 3, an increase in the compressive normal stress on the sheet surface to the yield stress (P / Y = 1) reduced the bending moment and springback by 32%.
To evaluate the impact of through-thickness compressive normal stress, numerical analyses were conducted to obtain longitudinal stress distributions on the sheet surface, both in the absence and presence of compressive normal stress.The distribution of longitudinal stress in the absence of throughthickness stress for a bending moment of 46.4 N.mm (which was obtained from the analytical model for a bend radius of 34 mm) is achieved as illustrated in Fig. 10.
Moreover, for the extraction of the longitudinal stress component across the thickness (y-direction) a path was de ned on the y-z symmetry plane.Figure 11 illustrates the nondimensionalized longitudinal stress distributions for scenarios without (P/Y = 0) and with (P/Y = 0.75) compressive normal stress.
Consistent with the analytical ndings, the numerical results also con rm that applying a pressure of 0.75Y on the sheet causes a shift in the longitudinal stress towards the compressive zone, resulting in reduced amounts of unloading elastic moment.Consequently, in this scenario, the natural axis shifts to an upper region of the sheet, causing a larger portion of the cross-section to experience compression.
The unloading model had the same parameters as the loading one, except that the bending moment and compressive stress on the sheet surface were excluded.The deformations induced by loading were utilized as pre-de ned initial conditions for the sheet.The springback angle (Δθ) was computed by tracking the positions of two points located on the wall surface of the bent sheet after both the bending and unloading steps.Table 4 presents a comparison of the numerical springback angles without compressive normal stress (P/Y = 0) and under a compressive normal stress of 204 MPa (P/Y = 0.75).Indeed, the springback angle exhibits a reduction of 16.2% when the through-thickness stress increases to 0.75Y.In general, it can be deduced that the results closely align, and any minor discrepancies could be attributed to the analytical model's imprecision in estimating longitudinal stress at the sheet's center, where a small degree of strain is generated.
To validate the theoretical models, four-point bending tests were conducted as described in section 3.
Figure 14 shows the four-point bending specimens after bending through different initial bend angle and radii applied by the vertical displacement of the top tool.Furthermore, Table 5 reports the four-point test results at different initial bending angles and radii.It is clear that with the increase of the bending angle, the springback decreases and the radius of curvature increases.θ 0 = 180 ∘ and ρ 0 = 34mm for P / Y = 0 and P / Y = 0.75.According to Table 7, as the compressive normal stress on the sheet surface increased to 75% of the yield stress, the analytical and numerical springback angles reduced by 17.4% and 16.2%, respectively.The difference between the numerical and analytical results can be attributed to the underestimation of strains at the center of the sheet in the analytical model.6 Conclusion This study analytically evaluated the bending behavior of an isotropic sheet metal, considering the power-law strain-hardening model in the presence of through-thickness compressive stress on the sheet surface.The effects of the through-thickness stress on the longitudinal stress distribution, reverse bending moment, and springback angle were analyzed.In a case study involving AA061 sheet, an elevation in the compressive normal stress on the sheet surface to the yield stress level (i.e., P/Y = 1) resulted in a signi cant 32% reduction in springback.To validate the analytical model, the four-point bending test was performed at different initial bend angles and radii.The analytical model demonstrated an error of less than 10% in estimating the springback angle for various initial bend angles and radii.A numerical nite element model was implemented to estimate springback, comparing the numerical and analytical results.Both the analytical and numerical models indicated that an increase in the compressive normal stress on the sheet surface to 75% of the yield stress would result in a reduction of springback by 17.4% and 16.2%, respectively.Consequently, precision in sheet products can be enhanced by minimizing springback through controlled through-thickness compressive normal stresses on the sheet surface.

Declarations
Ethical Not applicable.

Consent to participate
All the authors listed have approved to participate the study.

Consent for publication
All the authors listed have approved the manuscript to publish.Four-point bending specimens after bending along with the initial and nal bend angles

Figure 8
Figure 8 plots the nondimensionalized neutral axis position y * n = y * / t versus the normalized compressive stress, and ζ versus the P / Y ratio for the AA6061 sheet.According to Fig. 8, the entire sheet section experienced longitudinal compressive stresses i.e. ζ = 0 and y * n = 0.5 when P / Y equals to 1.25 or P = 340 MPa.

Figure 5 Four
Figure 5

Table 5
Initial and nal bend angles and radii in the four-point bending test and their changes Table6presents comparisons between analytical and experimental springback angles at various initial bend curvatures and radii, speci cally in the absence of compressive normal stress.The results indicate a noteworthy agreement between the analytical model and experimental data for different bend curvatures and radii when there is no compressive normal stress on the surface.The error may arise from not accounting the anisotropic properties of sheet metal, both in terms of elastic and plastic behavior.