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.     

A Theoretical Study on Nonlinear Bending of Wires

The nonlinear bending of thin wires is a challenging topic in several applications where the final geometry of the wire after bending and springback has to be known. Typical examples are tyre manufacturing, helical spring design, spectacles frames. In order to develop analytical models able to set bending parameters for a required final shape of the wire, both account material behaviour (during the loading and unloading phases with springback effect) and geometrical nonlinearity have to be considered. In the case of plates bending, many analytical and numerical models are available in the literature, offering an accurate solution to this problem. However, the bending of thin wires could still be the subject of discussion and research. In this paper a new analytical model was developed, starting from the models available in the literature, in order to provide the designer with a simple model to predict the final shape of a wire by using mathematical codes. The model allows to predict with a higher level of accuracy the final shape of wires having different cross-sections after nonlinear bending. Since Bernoulli’s hypothesis is assumed, the model can be used in all the applications where the material behaviour of the wire guarantees that plane cross sections of the wire will remain plane after rotation due to bending, with negligible errors from the engineering point of view.     

一种弧形高强钢车身零件冲压回弹控制方法

高强度钢板材料冲压性能对冲压成形后零件精度的影响较大,主要表现在成形性能不稳定和回弹控制复杂.弧形零件具有典型的回弹特征,且回弹量较大,难以获取回弹的具体运动规律,给回弹控制和补偿带来了困难.本文在分析弧形高强钢零件回弹特征规律的基础上,开发典型截面回弹运动规律分析程序,定量分析回弹截面整体和局部的平动与转动量,采用反...     

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.     

Springback in cold metal-forming of AOO---H steel: nonhomogeneous deformation

The incremental theory of plasticity is applied in conjunction with finite elements to obtain the state of affairs in metal-forming. The allowable overforming for a AOO---H steel sheet bent around a circular die is investigated numerically. For a 90° bend, the springback angle is found to be approximately 25%. That is, an overform of 25% is required if the sheet is to remain permanently deformed at 72°. The residual energy is found by comparing the energy states before and after spring back. The amount depends on the rate of forming.A modified theory of plasticity is also employed to find the nonhomogeneous character of the deformation field in the forming of metal sheets. Changes in the local strain rates and strain rate history are accounted for by deriving individual constitutive relations for each of the finite elements. Parameters assigned to describe material properties no longer remained constant but changed according to the loading rates.The strain rates at different locations were found to change by more than three orders of magnitude that could not have been adequately described by a single constitutive equation. Inhomogeneous deformation could play a significant role in metal-forming when local strain rates vary over a wide range, as demonstrated in this work.     

Elastic springback in plates and beams formed by bending

A plane-strain solution is presented to predict the springback of plates formed by bending. The solution assumes that the strains are small, that the material is an isotropichardening von Mises material, and either that the stress-strain diagrams for tension and compression are identical or that an average diagram can be employed. The plastic portion of each stress-strain diagram is approximated by a finite number of straight lines. Both analytical solutions and experimental evidence indicate that the much simpler plane-stress solution can be used to predict springback for both beams and plates. Experimental data confirm that the type of die has a pronounced effect on the springback for beams and plates if the material exhibits a yield point.     

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.     

Inhomogeneous inverse problem in finite elasticity

For a class of materials for which the principal strain directions always coincide with the principal stress directions one can determine the stress field in an inhomogeneously deformed body from given boundary conditions and a known strain field without knowing the constitutive equations. Each point of the inhomogeneously deformed body contains information such as that derived from an individual homogeneous identification test. The practically important two-dimensional case leads to a problem of solving a linear hyperbolic system where two differential equations describe the principal stress components. The problem can be reduced to that of integration along characteristics.Under a certain globality condition the existence, uniqueness, and correctness of the solution are guaranteed in the whole test piece. It is shown that the globality condition is closely related to whether or not the test piece is isotropic and elastic. The influence of experimental error on the correctness of problem formulation is discussed.

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Abstrakte Für die Klasse der Materialen, für welche die Grundspannungsrichtungen und die Grundverformungsrichtungen gleich sind, kann man das Spannungsfeld in einem inhomogen deformierten Körper für gegebene Randbedingungen und einem bekannten Verformungsfeld ohne Kenntnis der detailierten Form der Dehnungs-Spannungsgleichung bestimmen. Jeder Punkt des inhomogen deformierten Körpers liefert dieselbe Information wie ein individueller homogener Test. Der praktisch wichtige zweidimensionale Fall führt zu einem Problem der Lösung eines linear hyperbolischen Systems von zwei Differentialgleichungen für die Grundspannungskomponenten. Das Problem kann auf die Integration entlang der Charakteristik des Systems reduziert werden.Unter einer Bedingung der Globalität das die Existent, Eindeutigkeit und Richtigkeit der Lösung in der ganzen Testprobe gewährleistet sind. Es wird gezeigt, dass diese Globalitätsbedingung damit zusammenhängt eng, ob die Testprobe isotrop und elastisch ist. Der Einfluss der experimentellen Fehler auf die Richtigkeit der Formulation des Problems wird auch analysiert.

     

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。     

Prediction of elastic strain recovery of a formed steel sheet considering stiffness degradation

The aim of this work is to show first, how the springback of a steel sheet drawn part is affected by the stiffness degradation, as it results from the damage evolved during forming process, and second, to build a respective modeling approach to take this degradation into account. For the consideration of the orthotropic elastic properties degradation we develop an approach, based on the Mori-Tanaka theory, where damage is considered by inclusion of ellipsoidal cavities. The respective void shape evolution is proposed to be identified with the measurements of elastic modulus in two perpendicular directions during the uniaxial tensile test of a flat specimen at different loading stages. The proposed approach is coupled with the Gurson-Tvergaard-Needleman (GTN) plastic potential, though it could be substituted by almost any other continuum damage model. At the end the presented approach is experimentally validated by a simple springback test, developed by authors. A very good agreement between by calculation predicted and measured springback amount is observed.     

Minimal models for squealing of railway block brakes

Block brakes have been used to brake railway vehicles for approximately 200 years. During the last 40 years, disk brakes have replaced the block brakes at passenger cars but block brakes are still used for all freight wagons. One problem of block brakes is that they show an enormous tendency to squeal. Although the block brake is a very common and old technical component, there exists almost no scientific work on its noise behavior regarding squeal. On the other hand, a lot of work has been done on the problem of disk brake squeal especially concerning the modeling of the excitation mechanism. The goal of this paper is to investigate whether and how models from disk brake squeal can be modified to model block brake squeal. The starting point of these investigations is a minimal model for an automotive disk brake introduced by von Wagner et al. (J Sound Vibration, 51(1–2):223–237, 2007) that is adapted to the block brake problem. It can be shown that such a simple converted model does not show any instability at all. A deeper analysis suggests that the reasons for squeal in block brakes could originate from in-plane vibrations of the brake disk or specific geometrical properties of the railway wheel. The self-excited vibrations explaining the squeal occur at relatively low rotational speeds far below the first critical rotor speed which has rarely been observed in rotor dynamics.     

Anticlastic curvature in draw-bend springback

Draw-bend springback shows a sudden decline as the applied sheet tension approaches the force to yield the strip. This phenomenon coincides with the appearance of persistent anticlastic curvature, which develops during the forming operation and is maintained during unloading under certain test conditions. In order to understand the mechanics of persistent anticlastic curvature and its dependence on forming conditions, aluminum sheet strips of widths ranging from 12 to 50 mm were draw-bend tested with various sheet tensions and tool radii. Finite element simulations were also carried out, and the simulated and measured springback angle and anticlastic curvature were compared. Analytical methods based on large deformation bending theory for elastic plates were employed to understand the occurrence and persistence of the anticlastic curvature. The results showed that the final shape of a specimen cross-section is determined by a dimensionless parameter, which is a function of sheet width, thickness and radius of the primary curvature in the curled region of an unloaded sample. When the normalized sheet tension approaches 1, this parameter rapidly decreases, and significant anticlastic deflection is retained after unloading. The retained anticlastic curvature greatly increases the moment of inertia for bending, and thus reduces springback angle.     

Numerical simulation of the interface molten metal air in the shot sleeve chambre and mold cavity of a die casting machine

The objective of this study relates to the numerical simulation of the free surface during the two-dimensional flow and solidification of aluminum in the horizontal cylinder and mold cavity of the high pressure die casting HPDC machine with cold chamber. The flow is governed by the Navier–Stokes equations (the mass and the momentum conservations) and solved in the two phase’s liquid aluminum and air. The tracking of the free surface is ensured by the VOF method. The equivalent specific heat method is used to solve the phase change heat transfer problem in the solidification process. Considering the displacement of the plunger, the geometry of the problem is variable and the numerical resolution uses a dynamic grid. The study examines the influence of the plunger speed on the evolution of the interface aluminum liquid–air profile, the mass of air imprisoned and the stream function contours versus time. Filling of a mold is an essential part of HPDC process and affects significantly the heat transfer and solidification of the melt. For this reason, accurate prediction of the temperature field in the system can be achieved only by including simulation of filling in the analysis.     

多次透射公式飘移问题的控制方法

多次透射公式(multi-transmitting formula, MTF)是在近场波动数值模拟中一种广泛应用的人工边界条件, 具有形式简单、精度可控和通用性好的优点, 但高阶MTF与有限元方法相结合有时会出现飘移问题. 现有的几种MTF消飘方法往往会显著地影响边界精度, 为此本文提出一种新的消飘因子修正MTF格式, 能够在较高精度水平下实现对飘移问题的有效控制. 该方法保持MTF的一次透射项不变, 仅对各高次透射误差项进行修正, 从而大幅降低了因消飘因子造成的精度损失. 消飘因子设置格式确保在零频和零波数情形下能够满足GKS准则, 从理论上保证了消飘目标的实现. 进一步给出该方法的高阶统一形式, 并将传统的消飘因子修正MTF的方法归结为该统一形式的一个特例. 通过反射系数分析, 证明本文方法不仅具有精度优势, 而且消飘因子选取的适应性更强、取值范围更广. 数值算例表明, 本文的消飘因子对波动能量比较集中的法向和小角度透射波动的模拟精度影响很小, 在控制高阶MTF飘移问题和保持模拟精度方面, 均能够取得明显效果.      

Numerical simulation of polyhedral crystal growth based on a mathematical model arising from non–local thermomechanics

In this work we present some results of the numerical simulation of the growth of a crystal from its melt, taking into account faceting. The simulation is based on a numerical solution of a three–dimensional generalized Stefan problem. That problem arises from a non–local thermomechanical theory applied to a continuous system with an interface and embodies ideas from the dislocation theory of crystal growth. In the model, the crystal surface is an isotherm and the growth velocity of a crystal face depends on the velocities of the other faces and on the whole crystal configuration as well as on the temperature gradient. A front fixing formulation of the model is considered. This is a conservative form of the Isotherm Migration Method [6, 7, 8, 9, 10, 11] in spherical coordinates. The numerical solution is based on an explicit finite difference discretization of the resulting non–linear equations. We develop a theoretical analysis of the interface equations that drive the crystal face motion. Numerical results, showing evolution of complex crystals with configuration changing during the growth, are in accord with experimental results. Furthermore, numerical experiments offer useful information on the influence of certain parameters in the model on the growth process. Received: March 21, 1996     

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.     

Local nonsimilarity method for the two-phase boundary layer in mixed convection laminar film condensation

The two-phase boundary layer in laminar film condensation was solved by Koh for the free convection regime and forced convection regime using the similarity method. But the problem on mixed convection does not admit similarity solutions. The current work develops a local nonsimilarity method for the full spectrum of mixed convection, with a generic boundary layer formulation reduced to two specific cases mathematically identical to Koh’s formulations on the two limiting cases for either free or forced convection. Other solution methods for mixed convection in the literature are compared and critically evaluated to ensure a high level of accuracy in the current method. The current solution is used to extend Fujii’s correlation for mixed convection to the region where the energy convection effect is significant but has been hitherto neglected. The modified Fujii correlation provides a practical engineering tool for evaluating laminar film condensation with a mixed convection boundary layer.     

Viscoplastic dynamics of isotropic plates of variable thickness under explosive loading

A problem of viscoplastic dynamic bending of isotropic plates of variable thickness is formulated. A method for integrating the initial-boundary problem is developed. Numerical results are compared with a known analytical solution obtained within a rigid-plastic model; good agreement is demonstrated. The efficiency of the method developed is verified by numerical computations. It is shown that the final flexure of plates can be reduced severalfold by applying rational design. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 123–134, March–April, 2007.     

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)     

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.