Optimum blank design for sheet metal forming based on the interaction of high- and low-fidelity FE models

This paper deals with an optimization methodology for the design of the sheet-metal-forming process. In this study, a new design optimization system is developed which employs an iterative optimization technique and numerical simulation of a sheet-metal-forming process. The main feature of this new optimization method is that it is based on the interaction of high- and low-fidelity simulation models in order to reduce overall computing time. In the iterative optimization procedure, only the corrected low-fidelity model is used. The high-fidelity model, which requires much longer computing time, is used only for the correction of the low-fidelity analysis and validation of the final solution. To demonstrate the developed optimization method on a practical application, it is applied to the optimum blank design for deep-drawing process of a rectangular box. In deep-drawing, the flange of the drawn product is usually trimmed off to obtain the desired product geometry, and the trimmed material is wasted. Therefore, the formulation of the optimization problem is to determine the optimum initial blank geometry which minimizes the amount of the trimmed material, that is, the waste of material. It is confirmed that the blank design was optimized successfully in remarkably short computing time by the developed optimization method.     

A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming

Spitzig and Richmond [Acta Metall. 32 (1984) 457] proposed that plastic yielding of both polycrystalline and single crystals of steel and aluminum alloys shows a significant sensitivity to hydrostatic pressure. They further showed that under the associated flow rule, this pressure sensitivity leads to a plastic dilatancy, i.e. permanent volume change, that is at least an order of magnitude larger than observed. Indeed, the plastic dilatancy for most materials is on the order of the measurement error and must be zero in the absence of phase change and significant void nucleation during plastic deformation. A non-associated flow rule based on a pressure sensitive yield criterion with isotropic hardening is proposed in this paper that is consistent with the Spitzig and Richmond data and analysis. The significance of this work is that the model distorts the shape of the yield function in tension and compression, fully accounting for the strength differential effect (SDE). This capability is important because the SDE is sometimes described through kinematic hardening models using only pressure insensitive yield criteria.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Path-dependence of the forming limit stresses in a sheet metal

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczyński model and a phenomenological plasticity model with non-normality effects [Kuroda, M., Tvergaard, V., 2001. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids 49, 1239–1263]. Forming limits are simulated for linear stress paths and two types of combined loading: a combined loading consisting of two linear stress paths in which unloading is included between the first and second loadings (combined loading A), and combined loading in which the strain path is abruptly changed without unloading (combined loading B). The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. The strain-path dependence of the forming limit stress is discussed in detail by observing the strain localization process.     

A new method for circular grid analysis in the sheet metal forming test

Computer vision systems are employed to determine the major and minor lengths of deformed elliptic grids while determining a sheet metal's workability. The existing method identifies the ellipse using the least squares analysis. It suffers two drawbacks: assumptions in direct conflict with the observed real-world processes and an undesirable property of orientation dependence. For the remedy, this paper presents a new method that, in addition to achieving the desired property of orientation invariance, discards assumptions that conflict with real-world processes. The proposed method is implemented and tested using simulated and real-world data. Results are reported and compared with those obtained by the existing method.     

The effect of plastic anisotropy on compressive instability in sheet metal forming

The wrinkling behavior of a thin sheet with perfect geometry is associated with compressive instability. The compressive instability is influenced by many factors such as stress state, mechanical properties of the sheet material, geometry of the body, contact conditions and plastic anisotropy. The analysis of compressive instability in a plastically deforming body is difficult considering all the factors because the effects of the factors are very complex and the instability behavior may show a wide variation for a small deviation of the factors. In this study, the bifurcation theory is introduced for the finite element analysis of puckering initiation and growth of a thin sheet with perfect geometry. All the above mentioned factors are conveniently considered by the finite-element method. The instability limit is found by the incremental analysis and the post-bifurcation behavior is analyzed by introducing the branching scheme proposed by Riks. The finite-element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite-element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. In order to investigate the effect of plastic anisotropy on the compressive instability, a square plate that is subjected to compression in one direction and tension in the other direction is analyzed by the above-mentioned finite-element analysis. The critical stress ratios above which buckling does not take place are found for various plastic anisotropic modeling methods and discussed. Finally, the effect of plastic anisotropy on the puckering behavior in the spherical cup deep drawing process is investigated. From the results of the finite-element analysis, it is shown that puckering behavior of sheet metal is largely affected by plastic anisotropy.     

薄板冲压成型中板料起皱的临界应力分析和预测

针对薄板冲压成型中起皱这一常见的材料失效形式，运用板料压缩失稳理论，提出虚拟刚度变量的概念和板料产生起皱的临界应力计算方法，并借助于计算机仿真技术和有限元计算方法，搜寻并计算得到反映板料各处发生起皱难易程度的临界因子，生成起皱云图来观察材料各部分的稳定状态，预测可能出现起皱的部位，为修改冲压工艺和修模提供依据，通过分析起皱云图得出的结论与实际情况非常接近。     

Initiation and growth of wrinkling due to nonuniform tension in sheet metal forming

An experimental investigation was conducted on the initiation and growth of wrinkling due to nonuniform tension using the Yoshida buckling test. The initiation of wrinkling was detected by strain gages mounted on both surfaces of the samples in the loading and transverse directions. The bifurcation of aluminum auto body sheets appeared to be smooth and much less abrupt than that observed in a steel sheet. A special fixture was designed to, perhaps for the first time, continuously measure the in situ growth of the buckle heights so that the rates of buckle growth were monitored as functions of strain and stress in the loading direction. In contrast to what is commonly believed, it was found that the buckle height is not predominantly determined by the material yield strength, and lower averager value does not increase the rate of buckle growth. Crystallographic texture components and pole figures of the test materials were also measured, and the relationship of plastic anisotropy with wrinkling behavior was investigated by experiments with specimens aligned in the rolling direction, the transverse direction and 45-deg to the rolling direction of the sheet materials.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

On the implementation of anisotropic yield functions into finite strain problems of sheet metal forming

In this article the implementation of anisotropic yield functions into finite element investigations of orthotropic sheets with planar anisotropy is discussed within a plane-stress context. Special attention is focused on the proper treatment of the orientation of the anisotropic axes during deformation into the finite-strain range. As an example problem the hydrostatic bulging of a membrane is considered in conjunction with a recently proposed anisotropic yield function. It is shown that the aspects of the plane-stress assumption, which do not come into consideration in isotropic analyses, can play an important role on the accuracy of the solution when the rotation of the orthotropic axes enters the computation directly due to the presence of material anisotropy. When the material anisotropy is considered and when the deformation of the workpiece is not limited to the plane of the undeformed sheet (such as cup drawing, hydrostatic bulging, etc.), the numerical experiments indicate that the only correct formulation is the one based on numerically imposing the requirement that for the plane-stress application, the in-plane material axes have to remain in the plane of the sheet during the deformation.     

板材多点成形过程的有限元分析

多点成形过程采用静力隐式格式进行数值模拟是比较合适的。本文建立了用于多点成形过程分析的静力隐式弹塑性大变形有限元方法 ,给出了对稳定迭代收敛过程效果较好的板壳有限单元模型、处理多点不连续接触边界的接触单元方法以及增量变形过程中应力及塑性应变计算的多步回映计算方法。基于这些方法编制了计算软件 ,应用该软件进行了矩形板的液压胀形过程及球形模具拉伸成形过程的有限元分析 ,数值计算结果与典型的实验结果及计算结果吻合很好。最后给出了球形、圆柱形目标形状的实际多点成形过程的数值模拟结果。     

Marciniak–Kuczynski type modelling of the effect of Through-Thickness Shear on the forming limits of sheet metal

The Marciniak–Kuczynski (MK) forming limit model is extended in order to predict localized necking in sheet metal forming operations in which Through-Thickness Shear (TTS), also known as out-of-plane shear, occurs. An example of such a forming operation is Single Point Incremental Forming. The Forming Limit Diagram (FLD) of a purely plastic, isotropic hardening material with von Mises yield locus is discussed, for monotonic deformation paths that include TTS. If TTS is present in the plane containing the critical groove direction in the MK model, it is seen that formability is increased for all in-plane strain modes, except equibiaxial stretching. The increase in formability due to TTS is explained through a detailed study of some selected deformation modes. The underlying mechanism is a change of the stress mode in the groove that results in a delay of the onset of localized necking.     

Numerical study for influences of material parameters on hydrostatic bulging of metal sheet

In this paper, the influences of various material parameters, the hardening exponent (n), the rate sensitivity (m). the thickness anisotropy parameter (R) and the index M in the Hosford and Hill yield function, on the hydrostatic bulging of a circular clamped sheet of ductile metal materials are analysed by introducting a rigid-viscoplastic finite element method. By numerical studies, an empirical relationship within the average limit thickness strain – ₃ ^* and the material parameters (n andm) is obtained. Besides, it has been found that the influences of surface shapes of the yield function on the average limit thickness strain can be reflected by the Barlat'sP value which represents the effects ofR andM values.This work is supported by the National Natural Science Foundation and Natural Science Foundation of the Youth of China.     

非概率可靠指标和可靠度定义的比较研究

首先对几种主要的非概率可靠指标、可靠度进行了整理介绍;然后针对区间和椭球两个基本模型,对非概率可靠指标和可靠度的定义、几何意义以及计算公式进行了系统的比较分析。结果表明:线性情况下,区间可靠指标对结构功能函数具有不唯一性,椭球可靠指标与结构功能函数一一对应。因此,基于均值和离差之比定义的非概率可靠指标不适用于区间模型用来衡量结构的可靠程度。本文还将区间可靠指标与可靠度、椭球可靠指标与可靠度进行了比较,并将非概率可靠度与服从均匀分布、正态分布的概率可靠度以及服从三角形分布的非精确概率可靠度进行了比较,结果表明:区间模型相对于其他模型较为保守,椭球可靠度与概率可靠度和三角形分布下的非精确概率可靠度相接近。     

An experimental study of the effects of prestrain on the propagation of small disturbances in neoprene filaments

A small disturbance was caused to propagate along a long, slender, prestrained Neoprene filament. The particle velocity of the pulse was measured at two stations along the length of the filament by means of electromagnetic transducers which operate on the Faraday principle. Particle velocity vs. time data were obtained from oscilloscope photographs of the transducer outputs for each level of prestrain from 0 percent up to 400 percent engineering strain. The two particle-velocity records for each level of prestrain were subjected to linear viscoelastic analysis which employed the use of numerical Fourier transforms of the particle-velocity records. Computer programs were written which allowed computation of the numerical transforms and from them the computation of the phase velocity and attenuation coefficients of the material over the narrow frequency bandwidth of the Fourier spectra of the particlevelocity pulses. Data analysis revealed that, at a given frequency, the phase velocity increases significantly and that the attenuation coefficient decreases significantly with an increase in prestrain level over the range of prestrains of the tests. These material properties, that of a decreasing attenuation and an increasing phase velocity with increasing prestrain, are suggestive of the open possibility of the ability of the material to develop and support a shock wave for a large-amplitude disturbance.     

Practical limitations to the influence of through-thickness normal stress on sheet metal formability

According to a recent (original) model, when hardening properties and the ratio of through-thickness normal stress to the first principal stress (γ≡σ₃/σ₁) are held constant, sheet metal formability can be increased dramatically through the introduction of a compressive through-thickness normal stress, σ₃. In practice, however, both the hardening properties and γ evolve with the progression of deformation. To manage most efficiently the evolution of the hardening properties and γ, the original model is cast into a more compact form and presented as a proposed alternative form (proposed model). When the evolution of the hardening properties and γ is considered, the proposed model is shown to be in very good agreement with observed data; the influence of through-thickness normal stress on sheet metal formability is quite small for all practical purposes. Because the structure of the original model is similar to that of the proposed model, the original model is also validated. Ultimately, it is verified that although the theory of the original and proposed model may be acceptable, the implications of such theories are less profound than those first proposed when practical limitations are considered. This work serves as a useful basis for: (1) further understanding the limitations of the influence of compressive through-thickness normal stress on sheet metal formability and (2) exploring opportunities for improving sheet metal formability.