Principally on sheet metal forming defects as described in the eleventh biennial congress of the International Deep Drawing Research Group (IDDRG)

The major defects encountered in sheet metal forming operations are listed and some appropriate references given. The most common defects that arise in press-shop situations as described in the recent congress of the IDDRG are briefly reviewed.Defect—“Want or absence of something necessary for completeness or perfection”.Failure—“Omission to perform or want of success”.From Webster's Dictionary of English.     

Damage evolution in creep bulging of thin sheet metal

The creeping motion of thin sheet metal, damaged by artificial cavities is observed in bulging tests and simulated ‘semi’-analytically. The sheet metal satisfies Norton’s Law for secondary creep and is subjected to a bi-directional stretch. The stretch is produced by creep bulging through elliptical dies with the virtue of sustaining nearly uniform background stress ratio for each aspect ratio of the die axes. In order to reach large deformations with significant shape evolution of the cavities, the tests were conducted at superplastic conditions. The sheet is double layered (only one layer is cavitated) made of Tin–Lead (50–50 Pb–Sn). The measured damage growth is compared to an approximate simulation. The simulation of the damage evolution, throughout its time history, makes repeated use of a so-called “Green-function solution” for the motion of a single isolated cavity in an infinite viscoplastic continuum. The solution is modified from Muskhelishvili’s elastic solution by replacing the elastic shear modulus by a “viscous-like” variable (“plastic shear modulus”) which depends (non-linearly) on the evolved average strain-rate. Similarly, the stresses in the ligaments between cavities were averaged to approximate the local stress concentrations. Due emphasis is given to the rotation of each elliptical cavity, beside its expansion (contraction) and elongation.     

金属板料冲压成形的数值模拟

本文采用有限元动力显式算法模拟金属板料冲压成形的加工过程。四结点蜕化壳单元和刚体壳单元分别用来建立权和模具的有限元模型;更新Ｌａｇｒａｎｇｅ法和速率型本构关系被用来处理板料变形中的大应变和大转动;材料模型采用塑性各向异性屈服与等向强化模型;通过主从面模型定义板料和模具的接触,接触算法采用运动约束法,摩擦力用库仓定律计算;并利用动力松弛法对回弹过程进行了计算。模拟结果和实际零件比较,证明模型合理,算法稳定,结果可靠,具有良好的应用价值。     

Analysis of cross and vertical buckling in sheet metal rolling

In sheet metal rolling, shape defects called “cross buckling” or “vertical buckling” sometimes appear, which are wrinkles like washboards. The direction of the crest line of the cross buckling inclines at a certain degree against the rolling direction, while that of the vertical buckling is parallel to the rolling direction. In this study, analysis of the cross and vertical buckling is performed using the elementary theory of buckling. First, we calculate the stress distribution in three-dimensional sheet metal rolling near the exit cross section inside the roll gap. Next, we calculate the residual stress distribution near the exit cross section outside the roll gap. Furthermore, sheet metal buckling is analysed using the residual stress distribution. Type of buckling, distance between neighboring wrinkles, inclination of the crest line of the wrinkles against the width direction and the region where the wrinkles appear are obtained. We compare analytical results with published experimental results, and find that the former agree well with the latter. Hence, we conclude that this method of analysis is valid, and that the cause of the cross and vertical buckling is the residual stress distribution near the exit cross section outside the roll gap.     

Forming severity concept for predicting sheet necking under complex loading histories

A new method of predicting neck formation in sheets under non-proportional loading is proposed, based on the concept of “cumulative forming severity”. This concept is borrowed from a macroscopic model of ductile fracture where the crack initiation is governed by the accumulated equivalent plastic strain modified by the stress triaxiality and the Lode angle parameter. Such an approach necessitates a representation of the forming limit diagram (FLD) in the space of the equivalent strain to neck and the Lode angle parameter.Another new factor is the assumption of the non-linear accumulation of forming severity for non-proportional and complex loading histories. A class of non-linear weighting function is proposed with only one free parameter. A starting point in the derivation is the known FLD corresponding to proportional loading. This can be determined from Hill's and Stören and Rice analytical solutions, from numerical simulation, or else taken directly from experiments. In the case of proportional loading, necking depends on the final state of stress or strain, so it does not matter if necking severity index is accumulated in a linear or non-linear way. For non-proportional loading, the unknown free parameter of the non-linear accumulation rule must be determined from a test.Experimental data on FLDs under complex strain paths for two types of material, aluminum alloy 6111-T4 [Graf A, Hosford W. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences 1994;36(10):897–910.] and aluminum-killed sheet steel [Muschenborn W, Sonne HM. Influence of the strain path on the forming limits of sheet metal. Archiv fur das Eisenhuttenwesen 1975;46:597–602], found in the literature are revisited by the proposed model. Calibrated from only one test with non-proportional loading condition, the model is able to predict the remaining tests of complex loading paths with good accuracy.     

板材成形回弹数值分析的静力隐式方法

给出了板材成形加载过程，卸载过程数值模拟的静力隐式有限元方法，通过平衡迭代求解有限元方程，并采用隐式算法进行应力积分及接触压力计算，这种方法应力计算准确，因而回弹计算结果具有较高的精度，数值计算表明，纯弹性板大变形卸载后能够完全恢复到原始形状，说明该方法对几何非线性问题的分析是非常准确的，最后对板材柱面、球面成形的回弹进行了分析。     

Influence of end-conditions during tube hydroforming of aluminum extrusions

Tube hydroforming experiments were conducted to develop the forming limit diagram of AA6082-T4 by utilizing three types of end-conditions: (i) “free-end”, (ii) “pinched-end” or “fixed-end” and (iii) “forced-end”. It was found that “free-end” hydroforming gives the lowest forming limits followed by “pinched-end” and “forced-end” hydroforming. It was noticed that the tube failure occurs within 5° to the extrusion weld in the “free-end” experiments, within 7° in the “pinched-end” condition and extended up to 10° in the “forced-end” hydroforming experiments. Finite element simulations were carried out to capture the effects of the weld geometry, weld mechanical properties and the end-conditions of the extruded tube on the maximum induced stress and location of the maximum von Mises stress. It was found that the anisotropy of the weld material and the end-condition used during hydroforming experiments have the largest influence on the failure location with respect to the weld center.     

Buckling phenomena related to rolling and levelling of sheet metal

The paper deals with analytical and numerical considerations of buckling phenomena in thin plates or strips under in-plane loads which typically appear during rolling and levelling, i.e. straightening by stretching, of sheet metal. Buckling due to self-equilibrating residual stresses, caused by the rolling process, in eventual conjunction with global tensile stresses (denoted as “rolling buckling”) as well as buckling during the levelling process (denoted as “stretching buckling” or “towel buckling”) are considered. Analytical estimates are derived and compared against results of numerical simulations and field observations. Mode jumping by varying the global strip tension is explained on the basis of the derived analytical solutions. It is shown how from the waves, i.e. height and length, observed on the strip sliding over or lying on a rigid plane one can provide information about the distribution of the differences in the plastic strains over the width of the strip which lead to the buckled configuration. And, vice versa, knowledge of the plastic strain distribution can be used for estimating the expected wave heights representing a measure for the geometrical quality of the rolled product. The influence of the dead weight of the strip on the post-buckling pattern is also discussed on the basis of non-linear analyses.     

On the analysis of sheet metal wrinkling

An analysis for the prediction of wrinkling in curved sheets during metal forming is presented. Using a local approach, similar to that employed for conventional forming limit diagram representations, we construct “wrinkling limit curves” (WLCs) which represent the combinations of the critical principal stresses for wrinkling in curved sheet elements. Wrinkling limit curves are first determined using a bifurcation analysis for plastic buckling in short-wavelength shallow modes. A study of the effects of material properties and sheet geometry on the critical conditions for wrinkling is carried out. We then analyse the effects of geometric imperfections on wrinkling. This analysis is based on the implementation of a finite element scheme. The influence of nonproportional loading is also investigated. In our analysis the material is assumed to be isotropic, elastic-plastic with the plastic part modelled using both J₂ deformation theory and J₂ flow theory of plasticity.     

A viscous shell formulation for the analysis of thin sheet metal forming

A finite element method to analyse large plastic deformations of thin sheets of metal is presented. The formulation is based on an extension of the general viscoplastic flow theory for continuum problems to deal with thin shells. Axisymmetric situations are considered first and here the simple two noded reduced integration element is used. Numerical results for the stretch forming and deep drawing of circular sheets are presented and comparison with experimental results is made. The second part of the paper deals with the deformation of sheets of arbitrary shape. The general viscous shell element is derived from the standard reduced integration, “thick shell element. Numerical results for simple 3-D sheet forming problems are given.     

板材冲压成形的晶体塑性有限元模拟

将率相关晶体塑性本构理论引入Mindlin曲壳单元模型与动力显式有限元法,采用切线系数法计算剪切应变率,根据晶体取向正态分布规律在单元积分点处分配晶体取向,按各晶体取向体积分数的加权平均计算多晶体应力,开发晶体塑性动力显式有限元程序,实现板材冲压成形过程模拟和晶体取向演化预示.以主要初始织构为铜织构和S织构的轧制铝板为对象,对方盒件冲压成形过程及织构演化进行数值模拟,计算结果与试验结果呈现出较好的一致性.通过晶体弹塑性有限元法不仅可以预示板材宏观成形构形变化,而且能够预测板材织构的演化情况.模拟结果显示在方盒件冲压成形过程中,铜织构和S织构为不稳定取向,变形后逐渐转到其他取向.     

Accurate determination of biaxial stress-strain relationships from hydraulic bulging tests of sheet metals

The meridional strain gradient in hydraulic bulging of sheet metal through a circular aperture is severe. The resulting thickness gradation affects the bulge shape and complicates the evaluation of representative strain and stress at the pole of the bulge. An earlier solution for calculation of polar strain is extended to provide accurate explicit formulations of the thickness variation and the effective radius of curvature near the pole of the bulge. The representative stress—strain relationship determined for the biaxial stress state at the pole is shown to be significantly affected by these corrections.     

Integration of RP and explicit dynamic FEM for the visualization of the sheet metal forming process

This paper develops a FORTRAN program to convert the explicit dynamic finite element method (FEM)-simulated deformed sheet to the stereolithography (STL) format used in the rapid prototyping (RP) apparatus. Such integration of the RP/FEM can rapidly produce a visualized 3D physical part of the sheet deformation state. Three cases – cylindrical drawing, bore expanding and square cup drawing processes, simulated by explicit dynamic FEM – were investigated to verify the integration system. The wrinkled flange in the cylindrical drawing process, the circle hole expansion in the bore expanding process, and the square cup in the square cup drawing were successfully predicted by explicit dynamic FEM, and the rapid prototyping 3D physical parts also showed good visualization of the deformed sheet for the above three cases. It proves that the integration system of RP/FEM will be able to supply a useful method for the visualization of the 3D physical part in the sheet metal forming process.     

面向设计的板料成形快速仿真系统FASTAMP

FASTAMP是基于改进的有限元逆算法和动力显式算法的板料成形快速仿真软件。改进的逆算法求解器采用了考虑弯曲效应的DKQ四边形单元及方程组快速求解算法,真实考虑了摩擦、压边力、背压力和曲压料面等实际工艺条件,在模拟精度和速度上均有较大的提高。系统结合了两种算法的优势,将产品设计、选材和工艺设计三个独立的过程紧密结合起来,可快速分析产品设计中的潜在缺陷,为工艺设计人员提供有效的工艺设计参考和强有力的设计辅助分析工具。     

Ideal stretch forming for minimum weight axisymmetric shell structures

A companion paper “Minimum Weight Design of Axisymmetric Shell Structures” by Richmond and Azarkhin describes the design of axisymmetric thin-walled structures and parts that can support specified loads with minimum material. The result is an optimum shape and thickness distribution in the final part when the strength of the material is assumed to be uniform. Here, we demonstrate the application of ideal forming theory to design sheet stretching processes that can produce the optimum shapes and thickness distributions from flat sheets of uniform thickness. Specific designs are achieved for producing minimum weight shell structures that will support a specified uniform pressure assuming both the Mises and the Tresca yield criteria along with the rigid-perfectly plastic flow condition. In the case of the Tresca yield condition, the optimum structure is a spherical shell segment with uniform thickness, and an associated ideal stretching process is hydraulic bulging. Because the effects of strain hardening have been neglected in the structural optimization theory, it has been possible here to design the minimum weight structure and its forming process sequentially. In subsequent work, we plan to include the effects of strain hardening on the shell strength, which will then require coupled design of the structure and its forming process. Also extension of these methods to three-dimensional geometries will be considered.     

An advanced analysis of axisymmetric plastic sheet bending including transverse shear deformation

Finite deformation theory for axisymmetric elastic-plastic sheet bending is developed. The theory incorporates transverse shear deformation by adopting the extended Kirchhoff-Love's hypothesis, i.e. during bending deformation, a line element normal to the undeformed mid-surface is allowed to change the angle between itself and the mid-surface while its straightness is retained. A new idea, called “equivalent curvature”, is proposed which plays a similar role to the curvature in conventional plate bending theory but incorporates the effects of transverse shear deformation. Numerical calculations based on this theory have been performed for two examples of sheet-forming processes, i.e. the hydrostatic bulging of a circular sheet and the U-type sheet-bending process. Results show that the proposed theory can predict more precise information concerning the forming processes for a moderately thick sheet than the conventional sheet bending theory which ignores transverse shear deformation.     

Limit strain predictions for strain-rate sensitive anisotropic sheets

The combined effects of material strain-rate sensitivity and anisotropy on necking or “limit” strain predictions are examined for thin sheets with transversely isotropic properties. Various rate dependent constitutive laws based on flow theory and deformation theory of plasticity are considered.The strong effect of material strain-rate sensitivity in increasing the amount of straining prior to localized necking is first emphasized. We then discuss the joint influence of rate dependence and anisotropy on the theoretical limit strains and forming limit curves. Both strain-rate sensitivity and the local shape of the anisotropic yield surface are shown to significantly affect the predicted limit strains.A necking-band bifurcation analysis is also carried out to reveal in an explicit manner the remarkable sensitivity of overall forming limit diagram shapes to the parameters in the anisotropic yield function.     

An experimental study of the in-plane stretching of sheet metal

Experimental results are presented of the limit strains for the in-plane stretching of steel, aluminium and brass sheets. The development of surface strains is studied in incremental tests and attention is drawn to “crazing” of the surface of sheet specimens and to “banding” in aluminium specimens. Despite the wide difference in the mechanical properties of the three materials the values for the limit strain are remarkably similar. For steel and aluminium the limit strains are greater than the predicted instability strains, but for brass the limit strains are considerably less than the predicted instability strains. The wide scatter in the experimental limit strains perhaps indicates that failure is largely influenced by random effects.     

A new model for springback prediction in which the Bauschinger effect is considered

Springback prediction is an important issue for the sheet metal forming industry. Most sheet metal elements undergo a complicated cyclical deformation history during the forming process. For an accurate prediction of springback, the Bauschinger effect must be considered to determine accurately the internal stress distribution within the sheet metal after deformation. Based on the foundations for isotropic hardening and kinematic hardening, Mroz multiple surface model, plane strain assumptions, and experimental observations, a new incremental method and hardening model is proposed in this paper. This new model compares well with the experimental results for aluminum sheet metal undergoing multiple-bending processes. As is well known, aluminum is one of the most difficult sheet metals to simulate. The new hardening model proposed in this paper is not only a generic model for springback prediction but also a hardening model for sheet metal forming process simulation.     

Analytical solution for interface stresses due to concentrated surface force

The two-dimensional analytical solution for interface stresses due to concentrated surface force has been deduced, by introducing infinite mirror points which are the images of the load point reflected by the interface and the free surface, and adopting the interchange law of differentiation. The analytical solution can be represented in terms of the summation of the “partial” Goursat's complex stress functions defined in the local coordinate systems with their origins placed at each of the mirror points. It is found that the “partial” stress functions corresponding to a higher order mirror point can be determined from those to the lower one. It is also found that the contribution of the “partial” stress functions to the stress field decreases with the increase of the corresponding mirror point order, therefore, only considering the stress functions corresponding to the first several order mirror points can give the accurate enough solution. Numerical examination by the use of boundary element method has also been carried out to verify the theoretical development.