Analytical springback model for lightweight hexagonal close-packed sheet metal

Experiments have shown that magnesium alloy sheet a common hexagonal close-packed metal, exhibits mechanical behavior unlike that of sheets made of cubic metals (X.Y. Lou et al., 2007, Int. J. Plasticity, 24, 44). The unique stress–strain response includes a strong asymmetry in the initial yield and subsequent plastic hardening. In other words, the stress–strain curves in tension and compression are significantly different. A proper representation of the constitutive relationships is crucial for the accurate evaluation of springback, which occurs due to the residual moment distribution through the sheet thickness after bending. In this paper, we propose an analytical model for asymmetric elasto-plastic bending under tension followed by elastic unloading in order to evaluate the bending moment, which is equivalent to the springback amount. To simplify the calculations, the experimentally measured stress–strain curve of the magnesium alloy sheet was approximated with discrete linear hardening in each deformation region, and the material properties were characterized according to several simplifying assumptions. The bending moment was calculated analytically using the approximate asymmetric stress–strain relationship up to the prescribed curvature corresponding to the radius of the tool in sheet metal forming operations. A numerical example showed an unusual springback increase, even with an increase in the applied force; this is an unexpected result for conventional symmetric materials. We also compared the calculated springback amounts with the results of physical measurements. This showed that the proposed model predicts the main trends of the springback in magnesium alloy sheets reasonably well considering the simplicity of the analytical approach.     

Simulation of springback: Through-thickness integration

The number of through-thickness integration points (N_IP) required for accurate springback analysis following sheet forming simulation using shell elements is a subject of confusion and controversy. Li and Wagoner recommended, in 1999, based on a finite element analysis (FEA) of draw-bending springback, the use of 25 integration points (IP), with up to 51 IP required to ensure accuracies of 1%. Several researchers have since reported that N_IP between 5 and 11 are adequate, or even that 7 or 9 IP are optimal, with reduced accuracy for more IP. These apparent contradictions are addressed with an analytical model of elasto-plastic bending under tension, followed by elastic springback. The fractional error in the evaluated bending moment, which is equal to the fractional error in springback, was determined by comparing three numerical integration schemes, with various N_IP, to the closed-form result. The results illustrate the oscillatory nature of numerical integration error with small parametric changes, such that fortuitous agreement can be obtained in isolated simulations where the number of integration points is inadequate. The concept of an assured error limit is introduced as well as a maximum error limit (for a range of generally unknown sheet tensions). The assured error limit varies with the integration scheme, N_IP, bending ratio (R/t), and sheet tension. Guidelines for the number of integration points required for given error tolerances are reported to allow practitioners to choose numerical parameters appropriately.     

Creep and anelasticity in the springback of aluminum

Draw-bend tests, devised to measure springback in previous work, revealed that the specimen shapes for aluminum alloys can continue to change for long periods following forming and unloading. Steels tested under identical conditions showed no such time-dependent springback. In order to quantify the effect and infer its basis, four aluminum alloys, 2008-T4, 5182-O, 6022-T4 and 6111-T4, were draw-bend tested under conditions promoting the time-dependent response (small tool radius and low sheet tension). Detailed measurements were made over 15 months following forming, after which the shape changes were difficult to separate from experimental scatter. Earlier tests were re-measured up to 7 years following forming. The shape changes are generally proportional to log(time) up to a few months, after which the kinetics become slower. In order to understand the basis of the phenomenon, two models were considered: residual stress-driven creep, and anelastic deformation. In the first case, creep properties of 6022-T4 were measured and used to simulate creep-based time-dependent springback. Qualitative agreement was obtained using a crude finite element model. For the second possibility, novel anelasticity tests following reverse-path loading were performed for 6022-T4 and drawing-quality silicon-killed (DQSK) steel. Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys.     

中频感应局部加热弯管回弹的研究

对中频感应局部加热弯管在不同钢管材料及规格，不同弯曲半径及弯曲角度等工艺参数时的加载和卸载的力能参数，应力应变进行了分析和实验研究，提出了中频感应局部加热弯制大直径钢管的回弹理论，计算结果与有限元模拟结果以及实际弯制的２０，１０ＣｒＭｏ９１０和１２Ｃｒ１ＭｏＶ等钢管弯头实测的回弹量吻合良好。     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.     

Springback analysis for sheet forming processes by explicit finite element method in conjunction with the orthogonal regression analysis

A new method to evaluate the amount of springback in sheet forming processes based on the explicit finite element method and the orthogonal regression analysis is presented in this paper. To calculate springback accurately, a simple but effective contact searching algorithm is described and Lagrangian multiplier method was used to evaluate the contact force. The loading and unloading process could be simulated within one code. The numerical results by the present method were compared with the results by the commercial dynamic explicit code LS-DYNA3D, also with the experimental results and very good agreement was drawn. In order to obtain the springback conveniently for the purpose of practical use, the orthogonal regression analysis was implemented to establish the explicit relationship between the springback and some design parameters. The present method has been applied to the analysis of some actual sheet forming processes and very good agreement between the numerical results and the experimental results in the final geometry was obtained.     

A general analytic solution for plane strain bending under tension for strain-hardening material at large strains

A new analytic solution for plane strain bending under tension of a sheet is proposed for elastic-plastic, isotropic, incompressible, strain-hardening material at large strains. Numerical treatment is only necessary to calculate ordinary integrals and solve transcendental equations. No restriction is imposed on the hardening law. All governing equations and boundary conditions are exactly satisfied. The only exception is that the actual stress distribution over the ends of the sheet is replaced with a concentrated force and a concentrated bending moment. The through-thickness distribution of residual stresses and a measure of springback are also found. The range of validity of the solution is determined. An illustrative example is provided for Swift’s hardening law.     

Springback simulation and analysis of strong anisotropic sheet metals in U-channel bending process

The springback phenomenon of strong anisotropic sheet metals with U-channel bending as well as deep-drawing is numerically studied in detail by using Updating Lagrange FEM based on virtual work-rate principle, Kirchhoff shell element models and the Barlat-Lian planar anisotropic yield function. Simulation results are compared with a benchamark test. Very good agreement is obtained between numerical and test results. The focus of the present study is on the numerical simulation of the springback characteristics of the strong anisotropic sheet metals after unloading. The effects of the planar anisotropy coefficients and yield function exponent in the B-L yield function on the springback characteristics are discussed in detail. Some conclusions are given. The project supported by the National Natural Science Foundation of China (No. 19832020) and Provincial Natural Science Foundation of Jilin China (No.20000519)     

Effects of boundary conditions and anisotropy on elastically bent silicon

In four-point bending, the rollers that are used for load application impose additional constraints on the specimen that affect the anticlastic specimen curvature and cause the specimen displacement and stress profiles to deviate from the pure beam bending case. In this study, x-ray microdiffraction is used to map both the principal and anticlastic curvatures of elastically bent, rectangular (100)-type Si strips possessing width:thickness ratios of 40:1. We quantify the amount of roller constraint and show that the region over which the anticlastic specimen curvature is affected away from the roller is approximately five times the roller diameter. Consequently, for bending tests used to determine Poisson's ratio, if a region on the sample that is free from roller effects is not chosen, measurement errors as high as 46% can occur. Furthermore, we show that, due to the anisotropy of single crystal Si, this roller-constraining effect depends on crystallographic orientation and is more pronounced when the principal bending axis lies along the <100> direction as compared with the <100> direction.     

The bending moment and springback in pure bending of anisotropic sheets

Finite deformation rigid plastic and elastic–plastic analyses of plane strain pure bending of a plastically anisotropic sheet is presented. An efficient method for finding the exact solution is proposed by extending the previously developed method to the stage of unloading. Using this method the solutions are obtained in closed form or reduced to a numerical treatment of ordinary integrals, or an ordinary differential equation, or transcendental equations. An effect of plastic anisotropy and elastic properties on the bending moment is analyzed. The distribution of residual stresses is illustrated and an effect of material and process parameters on springback is investigated.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

Bending of a thin strip by a circular tool

We present a model of transverse bending of a wide thin elastic, elastoplastic, or rigid-plastic strip by a circular tool in the case of large displacements. The bending of the initial rectilinear strip is modeled by small increments in the load on the tool with analysis of the forces, moments, curvature, and the midline equation. The final form of the strip and the residual stresses are determined after elastic unloading. The model describes technological operations of sheet slab bending in dies and on roller-bending machines.     

Localization in NiTi tubes under bending

Nearly equiatomic NiTi can exhibit pseudoelastic behavior due to reversible solid-to-solid stress induced phase transformation at room level temperatures. In tension, the transformation leads to localized deformation of several percent that tends to spread at nearly constant stress. The deformation is recovered upon unloading while again localized deformation is exhibited. Under compression, while still pseudoelastic, the transformation strains are smaller, the stress is higher, the response is monotonic, and the deformation is essentially homogeneous. This study examines how this texture-driven, complex material asymmetry affects a simple structure: the bending of a tube. To this end, NiTi tubes are bent in a custom four-point bending facility under rotation control and isothermal conditions. The phase transformations lead to a closed moment-rotation hysteresis comprised of loading and unloading moment plateaus. During loading, localized nucleation of martensite results in a high curvature for the transformed sections of the tube and low curvature for the untransformed. Martensite, which corresponds to the higher curvature regime, spreads gradually while the moment remains nearly constant. The nucleation of martensite is in the form of bands inclined to the axis of the tube that organize themselves into diamond shaped deformation patterns on the tensioned side of the structure. The patterns are similar to those observed in bending of steel tubes with Lüders bands, however, for NiTi they develop only on the tensioned side due to the material asymmetry. A lower moment plateau is traced upon unloading with similar localized bending and the erasure of the diamond deformation patterns. This complex behavior was found to repeat for a number of temperatures in the pseudoelastic regime of NiTi with the moment-rotation hysteresis moving to higher or lower moment levels depending on the temperature.     

A hybrid membrane/shell method for calculating springback of anisotropic sheet metals undergoing axisymmetric loading

To reduce the computation cost of finite element analyses aiding die design for sheet metal stamping, a hybrid membrane/shell method was developed to determine the springback of anisotropic sheet metal undergoing axisymmetric loading. The hybrid membrane/shell method uses a membrane model to analyze the stamping operation. The bending/unbending strains and stresses varying through thickness are calculated analytically from the incremental shape determined by the membrane analysis. These bending strains and stresses and the final membrane shape are used with a shell finite element model to unload the sheet and calculate springback. The accuracy of the springback prediction with the hybrid method was verified against the springback of 2036-T4 aluminum and a DQAK steel sheet drawn into a cup. It was found that, in comparison with a full shell model, a minimum of 50% CPU time saving and a comparable accuracy was achieved when the hybrid method was used to predict springback.     

Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity

Sheet metal forming processes generally involve non-proportional strain paths including springback, leading to the Bauschinger effect, transient hardening, and permanent softening behavior, that can be possibly modeled by kinematic hardening laws. In this work, a stress integration procedure based on the backward-Euler method was newly derived for a nonlinear combined isotropic/kinematic hardening model based on the two-yield’s surfaces approach. The backward-Euler method can be combined with general non-quadratic anisotropic yield functions and thus it can predict accurately the behavior of aluminum alloy sheets for sheet metal forming processes. In order to characterize the material coefficients, including the Bauschinger ratio for the kinematic hardening model, one element tension–compression simulations were newly tried based on a polycrystal plasticity approach, which compensates extensive tension and compression experiments. The developed model was applied for a springback prediction of the NUMISHEET’93 2D draw bend benchmark example.     

Tension–compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior

In-plane tension and compression experiments on copper alloy sheets (phosphor bronze) and 6000 series aluminum alloy sheets (AA6016-T4) were conducted using a specially designed testing apparatus. The apparatus is equipped with comb-type dies so that stress–strain curves of a sheet specimen subjected to tension followed by compression, and vice versa, can be measured without buckling of the specimen, as well as those for monotonic tension and compression. A difference was observed in the flow stresses between tension and compression for the as-received copper alloy, but not for the aluminum alloy. Moreover, stress reversal tests, such as tension followed by compression and compression followed by tension, were carried out in order to measure the Bauschinger effect. In the second part of the experiment, bending moment–curvature diagrams were measured for the as-received and pre-stretched specimens. The bending moment–curvature diagrams were compared with those calculated using the stress–strain curves obtained from the tension–compression tests, and were in good agreement with those calculated with the tension–compression asymmetry and the Bauschinger effect correctly reproduced.     

A reliability study of springback on the sheet metal forming process under probabilistic variation of prestrain and blank holder force

This work deals with a reliability assessment of springback problem during the sheet metal forming process. The effects of operative parameters and material properties, blank holder force and plastic prestrain, on springback are investigated. A generic reliability approach was developed to control springback. Subsequently, the Monte Carlo simulation technique in conjunction with the Latin hypercube sampling method was adopted to study the probabilistic springback. Finite element method based on implicit/explicit algorithms was used to model the springback problem. The proposed constitutive law for sheet metal takes into account the adaptation of plastic parameters of the hardening law for each prestrain level considered. Rackwitz-Fiessler algorithm is used to find reliability properties from response surfaces of chosen springback geometrical parameters. The obtained results were analyzed using a multi-state limit reliability functions based on geometry compensations.     

Zigzag carbon nanotubes—Molecular/structural mechanics and the finite element method

This paper discusses in details the relation between the bond bending stiffness used in molecular mechanics and the bending stiffness used in structural mechanics for zigzag carbon nanotubes (CNTs).Recent publications assumed the structural bending stiffness EI/a to be a constant and set it equal to the molecular bond bending stiffness C. By developing a closed form expression for the deformation of zigzag CNTs under simple tension, we suggest that the relation between EI/a and C is more complex. It actually depends on the bond bending stiffness C, the torsional angle φ and the lattice translational index n. In the limit of an infinite tube radius, which represents a graphene sheet, EI/a tends to C/2. Numerical simulations are also presented that validate the results.     

中频感应局部加热弯管的回弹理论分析和试验研究

对中频感应局部加热弯管在不同钢管材料及规格、不同弯曲半径及弯曲角度等工艺参数时的加载和卸载的力能参数，应力应变进行了理论分析和实验研究，提出了中频感应避部加热弯制大直径钢管的回弹理论，并与实际弯制的２０、１０ＣｒＭｏ９１０和１２ＣｒＭｏＶ三种钢管弯头实测的回弹量进行了比较，理论计算和试验结果吻合良好。