A rigid-plastic finite-element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications

General three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys Hill's quadratic yield function is considered in the present analysis. The explicit expressions of the yield criterion and the related equations are given in the surface curvilinear co-ordinate system. A convective co-ordinate system is used in order to take into account the effect of geometric change during one step in the incremental analysis. The expression of the effective strain increment during one step is obtained in closed form by introducing assumptions on the deformation path and by proper consideration of the rotation of the axes of anisotropy during deformation. Considering the equilibrium at the deformed state, a variational formulation is derived to determine the deformation during a step. The corresponding finite-element equations are found in order to analyze the general deformation of planar anisotropic sheet metals. Two computational examples are chosen and computed by using the developed FEM program in order to verify the present formulation. The flange deformation in deep drawing of a circular diaphragm of planar anisotropy is analyzed and compared with the existing solution. The effects of planar anisotropy in the circular diaphragm bulge test are also investigated.     

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.     

Finite element method for sheet forming based on an anisotropic strain-rate potential and the convected coordinate system

A variational formulation and the associated finite element (FE) equations have been derived for general three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys the strain-rate potential proposed by [11.]. By using the natural convected coordinate system, the effect of geometric change and the rotation of planar anisotropic axes were efficiently considered. In order to check the validity of the present formulation, a cylindrical cup deep drawing test was modeled for a 2008-T4 aluminum alloy sheet sample. Eating simulations were performed and planar anisotropic material properties were experimentally determined. Even though quantitative agreement was not fully achieved, reasonably good agreement was found between the FE simulation and the experiment in thickness strain distribution and caring. No numerical difficulty due to planar anisotropy was encountered, and the computational procedure was found to be very stable, requiring only moderate computational time. The results have shown that the present formulation for planar anisotropic deformation can provide a good basis for the analysis of sheet metal forming processes for planar anisotropic materials, especially for aluminum alloy sheets.     

Effect of sheet anisotropy on the wear in deep-drawing process of a cylindrical cup

In the present investigation, the deep-drawing process of a cylindrical cup is simulated using the finite element method, with the aim of investigating the effect of sheet anisotropy on wear depth in the die which is exposed to the most severe tribological condition in this process. Both the blank and the die are assumed to be deformable. For the blank, the elasto-plastic behavior is considered, while the die material is assumed to be elastic. The sheet is anisotropic, so Hill quadratic yield criterion is employed. Two kinds of anisotropy, i.e. planar anisotropy and normal anisotropy are considered in this study. To study the effects of sheet anisotropy on the wear depth on the die shoulder qualitatively a parameter called relative wear depth (RWD) is introduced. Different distributions of Lankford coefficients are chosen and several conclusive results are presented.     

板材的纯剪试验研究

在非对称几何形状的板成形过程中,金属在不同方向上的流动引起了平面剪切变形,而平面剪切变形是影响变形性质的一个重要因素。本文首先阐述了一种新的用于平面内简单剪切试验的夹具,其次介绍了从这个试验取得的三种铝板材（LY12M、LF21M、LC4M）的剪应力应力应变曲线。首次用试验方法取得了Hill新屈服准则中的幂指数m值和不同方向上的Bauschinger效应曲线。     

An exact solution for the elastic/plastic bending of anisotropic sheet metal under conditions of plane strain

A theoretical analysis for the elastic/plastic bending of sheet metal exhibiting a state of normal anisotropy is considered in this paper, assuming a plane strain condition to exist in the deformation process. The material is supposed to yield according to Hill's quadratic yield criterion and its associated normality rule of plastic flow. The relationship between the bending couple and the curvature of the bent sheet is presented in a graphical form that reveals the influence of anisotropy and strain-hardening on the bending characteristic of the sheet metal. The results indicate that the elementary bending theory significantly overestimates the magnitude of the bending couple to produce a given elastic/plastic curvature of the bent sheet.     

Analysis of crease-wrinkle interaction for thin sheets

In this paper, geometrically non-linear post-buckling analyses were performed to study the effect of sheet thickness, deployment angle, and load ratio on the crease-wrinkle interaction. A square sheet configuration with a single transverse crease was modeled using thin shell elements. The analysis proceeded by initially providing a realistic deployed state of a creased membrane sheet. Then an uneven corner loading was applied to introduce wrinkling. The effects of the induced anisotropy from the crease on the fine-scale detail of the wrinkle evolution, as a function of sheet thickness, loading, and crease deployment angle were systematically investigated. Significant differences were found in sheet compliance and crease-wrinkle interaction as these parameters were varied.     

Springback prediction of the vee bending process for high-strength steel sheets

The precise prediction of springback is a key to assessing the accuracy of part geometry in sheet bending. A simplified approach is developed by considering the thickness ratio, normal anisotropy, and the strain-hardening exponent to estimate the springback of vee bending based on elementary bending theory. Accordingly, a series of experiments is performed to verify the numerical simulation. The calculation of the springback angle agrees well with the experiment, which reflects the reliability of the proposed model. The effects of process parameters such as punch radius, material strength, and sheet thickness on the springback angle are experimentally tested to determine the dominant parameters for reducing the springback angle in the sheet bending process for high-strength steel sheets. Moreover, the effects of the thickness ratio, normal anisotropy, and the strain-hardening exponent on the springback angle in the vee bending process for high-strength steel sheets are theoretically studied. Therefore, improving understanding on and control of the springback reduction of the vee bending process in practical applications is possible.     

A reappraisal of the wedge-drawing test

The wedge-drawing test is re-evaluated in the light of improved methods of lubrication. The test is used to study the radial drawing of anisotropic sheet materials and the experimental results correlate well with theoretical predictions based on the orthotropic theory of plasticity. In the theoretical analysis planar anisotropy is considered as well as normal anisotropy. The materials tested were soft and half-hard aluminium, soft 70/30 brass, killed steel and titanium. It is clearly demonstrated that the wedge-drawing test does not simulate the instability conditions existing in deep-drawing. While in deep-drawing increasing R-values can lead to large increases in limiting drawing ratio, this is not the case in wedge-drawing.     

On the prediction of the forming-limit diagram of sheet metals

Models for predicting the formability of sheet metals which take into account a single mode of strain localization have not yet led to accurate predictions. In the present work an attempt is made to include some important experimentally determined parameters in the model, i.e. surface roughness, void growth, sheet thickness, grain size, strain-hardening exponent, strain-rate sensitivity index, and normal anisotropy.In this model, which is based on the Marciniak-Kuczynski (M-K) hypothesis, the initial size of the imperfection is determined by the surface roughness. Two possible concurrent modes of flow localization, viz. necking and shear-band formation, are assumed to contribute to the final failure. The solutions of equations resulting from the M-K theory and the Hutchinson-Tvergaard analysis for plane strain are combined in two ways to obtain the prediction equations for the formability of orthotropic, isotropically hardening materials. A reasonably good correlation with the experimental results is demonstrated.     

参数求解方法对屈服准则的各向异性行为描述能力的影响

详细分析基于应力各向异性和变形各向异性两种求解Hill48屈服准则参数的方法。在给出两种各向异性参数求解表达式的基础上,具体分析Hill48屈服准则本身的局限性。以5754O铝合金板为研究对象,进行不同方向的单向拉伸试验。采用两种各向异性参数求解方法,基于Hill48屈服准则推导不同方向拉伸过程中的理论应力-应变曲线和拉伸过程中的变形规律。通过对比理论与试验结果具体分析参数求解方法对屈服准则精度的影响。基于两种参数求解方法,进行5754O铝合金板拉深试验的有限元模拟,讨论不同求解方法对凸耳现象的描述精度。得出结论：当对应力各向异性为主的问题进行分析时,应采用应力各向异性法求解;当对变形各向异性为主的问题进行分析时,则应采用变形各向异性法求解。研究结果对屈服准则在板料成形方面的合理应用具有重要的参考价值。     

Blank optimization for sheet metal forming using multi-step finite element simulations

The present study aims to determine the optimum blank shape design for deep drawing of arbitrary shaped cups with a uniform trim allowance at the flange, i.e., cups without ears. The earing, or non-uniform flange, is caused by non-uniform material flow and planar anisotropy in the sheet. In this research, a new method for optimum blank shape design using finite element analysis is proposed. The deformation process is first divided into multiple steps. A shape error metric is defined to measure the amount of earing and to compare the deformed shape and target shape set for each stage of the analysis. This error metric is then used to decide whether the blank needs to be modified. The blank geometry change is based on material flow. The cycle is repeated until the converged results are achieved. This iterative design process leads to optimal blank shape. To test the proposed method, three examples of cup drawing are presented. In every case converged results are achieved after a few iterations. The proposed systematic method for optimal blank design is found to be very effective in the deep drawing process and can be further applied to other sheet metal forming applications such as stamping processes.     

三维板料成形过程的有限元分析

基于有限变形理论建立了三维金属板料成形过程的弹塑性有限元数学模型。数学模型采用物质坐标系中的Total Lagrange描述、J 2型本构方程和等向强化假设,考虑了板料的厚向异性,对于金属板料与模具的摩擦定律。为简化计算采用薄膜单元。为了改善节点接触状态变化时计算的收敛性,提出了“弹性边界层”方法。采用根据此模型编制的程序模拟了机油收集器基本件的成形过程,并与试验结果进行了比较。     

Finite-element simulation of hole-flanging process of circular sheets of anisotropic materials

A hole-flanging operation on a flat circular sheet with a hole in the center is simulated by an incremental elasto-plastic finite-element method, which incorporates strain-hardening and anisotropy in the direction normal to the sheet, with care taken to describe the boundary conditions of penetration, separation and the alternation of the sliding—sticking state of friction. The simulation clearly demonstrates the processes of generation of deformation shape until unloading. The calculated sheet geometries and the relationship of punch load to punch stroke are in good agreement with the experimental data.The stress at the hole periphery in the flange is assumed to a state of circumferential uniaxial tension, in order to simplify the fracture mode as a simple tension test. By making use of the instability of uniaxial tension, an approximate relationship to determine the onset of necking of the hole periphery in the hole-flanging process is derived and it is found to be influenced by the process geometry and the plastic properties of the material, such as the stress-concentration factor K, strain-hardening n and normal anisotropy R, and the estimated value, being obtained by the derived equation, agrees well with the experimental data.It is noted that the derived relationship for estimating the instability of the hole-flanging process can be combined into the developed finite-element model to simulate the critical condition of the limiting deformation of the hole-flanging process. This combined method could possibly be applied towards improving both the manufacturing process and the design of tools for the hole-flanging operation.     

Characterization of magnetic field rotation of steel sheet under uniaxial stress

The rotation of magnetic field strength on a steel plate caused by uniaxial stress has been known for years. However, previous study investigated this phenomenon rather experimentally using hall plates and searching coils. This paper presents a fundamental study of magnetic anisotropy based on an analytic model using magnetic scalar potential. The magnetic field strength of the steel sheet can be solved mathematically by performing a special coordinate transformation. The calculated angular displacement due to stress induced magnetic anisotropy was later found to be highly consistent with FEM simulation results. Finally, the quantization of magnetic field rotation of an electrical sheet (ES) under variable tension stress was performed. Using 50CS470 ES sheet as the testing sample, it was discovered the rotation as much as 25° occurred when 50 MPa tension load was applied, corresponding to a permeability ratio of 2.8.     

A method for visualization of surface texture anisotropy in different scales of observation

Anisotropy of functional surfaces can in many practical cases significantly influence the surface function. Tribological contacts in sheet forming and engine applications are good examples. This article introduces and exemplifies a method for visualization of anisotropy. In a single graph, surface texture properties related to the anisotropy as a function of scale are plotted. The anisotropy graph can be used to explain anisotropy properties of a studied surface such as texture direction and texture strength at different scales of observation. Examples of milled steel surfaces and a textured steel sheet surface are presented to support the proposed methodology. Different aspects of the studied surfaces could clearly be seen at different scales. Future steps to improve filtering techniques and an introduction of length-scale analysis are discussed.     

An analytical and experimental study of stretch flanging

A stretch flanging operation on a flat circular sheet with a hole in the center has been analyzed by means of a total strain membrane theory of rigid-plasticity which incorporates strain hardening and anisotropy in the direction normal to the sheet. Numerical procedures are used to calculate the stress and strain distributions in the sheet. Numerical results so obtained indicate that the state of stress in the flange is dominantly uniaxial. By making use of this observation, an approximate theory for the stretch flanging has been developed, and a closed-form expression for the maximum strain in the flange has been obtained in terms of the geometrical variables of the flange. Experimental data, using aluminum-killed steel sheets, have been obtained and compared with the corresponding theoretical results. It was found that, for strains less than 100 per cent (which is the region of practical interest), the calculated results are in good agreement with the experimental data.     

Optimal Sequencing Rules for Some Large-Scale Flexible Manufacturing Problems Under the Manhattan and Chebychev Metrics

The purpose of this paper is to develop optimal tool partitioning policies and strip sequencing strategies for a class of flexible manufacturing problems. The problems under consideration involve a large number of operations to be performed by a series of tools on a two-dimensional object. For example, these operations could consist of drilling holes in a metallic sheet. Tools are arranged in a carousel or along a toolbar according to a predetermined sequence. Operations are performed by repeatedly moving the sheet to bring the hole locations under the tool. During each pass, as all operations involving a series of consecutive tools are executed, two main problems are to be solved: (1) how to move the sheet during each pass, (2) how to partition the tools into blocks of consecutive tools. A strip strategy is used to move the sheet. Given this policy, optimal strip widths and tool partitioning policies are determined jointly. Analytical solutions are derived under two metrics corresponding to different operating modes. A numerical example is provided.     

Estimation of the measurement uncertainty in the anisotropy test

The evaluation of the sheet metal drawability in mechanical shaping processes depends on a large number of analysis, among which the anisotropy evaluation. Nowadays, there is no Brazilian test laboratory accredited by the Metrology, Quality and Technology National Institute (INMETRO) to perform this analysis. So, the object of the present work is to establish a procedure for the estimation of the measurement uncertainty in the plastic anisotropy ratio of sheet metals, in accordance to the Guide to the Expression of Uncertainty in Measurement (GUM), aiming at the accreditation of this test in the Physical Metallurgy Laboratory. As results, we present the calculations performed and the uncertainty form proposed, and an analysis of which sources contributed the most for the uncertainty in the execution of a test. Finally, we propose improvement actions aiming at the reduction of the calculated uncertainty and the adequacy of the Measurement System for the desired application.