A two-pronged approach for springback variability assessment using sparse polynomial chaos expansion and multi-level simulations

In this study, we show that stochastic analysis of metal forming process requires both a high precision and low cost numerical models in order to take into account very small perturbations on inputs (physical as well as process parameters) and to allow for numerous repeated analysis in a reasonable time. To this end, an original semi-analytical model dedicated to plain strain deep drawing based on a Bending-Under-Tension numerical model (B-U-T model) is used to accurately predict the influence of small random perturbations around a nominal solution estimated with a full scale Finite Element Model (FEM). We introduce a custom sparse variant of the Polynomial Chaos Expansion (PCE) to model the propagation of uncertainties through this model at low computational cost. Next, we apply this methodology to the deep drawing process of U-shaped metal sheet considering up to 8 random variables.     

A new anisotropic elasto-plastic model with degradation of elastic modulus for accurate springback simulations

Springback, a phenomenon that is governed by elastic strain recovery after the removal of forming loads, is of great concern in sheet metal forming. There is no doubt that in this regard, physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial for controlling this problem. Unfortunately, by currently available approaches, springback still cannot be adequately predicted in general. In this paper, a new constitutive model is proposed which considers simultaneously sheet anisotropy, damage evolution and strain path-dependent stiffness degradation during sheet metal forming. For parameter identification of the built constitutive model, a particular experimental procedure is developed and an optimization procedure is employed to solve the inverse problem that arises. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel sheet. The simulation results, which prove to be in good agreement with the experimental ones, lead to the conclusion that accurate modelling only of anisotropic yielding is not enough to accurately predict the springback phenomenon; the constitutive model should also include the strain path-dependent change of the elastic moduli.     

Plane strain transversely anisotropic analysis in sheet metal forming simulation using 6-component Barlat yield function

In most FEM codes, the isotropic-elastic and transversely anisotropic-elastoplastic model using Hill??s yield function has been widely adopted in 3D shell elements (modified to meet the plane stress condition) and 3D solid elements. However, when the 4-node quadrilateral plane strain or axisymmetric element is used for 2D sheet metal forming simulation, the above transversely anisotropic Hill model is not available in some FEM code like Ls-Dyna. A novel approach for explicit analysis of transversely anisotropic 2D sheet metal forming using 6-component Barlat yield function is elaborated in detail in this paper, the related formula between the material anisotropic coefficients in Barlat yield function and the Lankford parameters are derived directly. Numerical 2D results obtained from the novel approach fit well with the 3D solution.     

大幅面板材渐进折弯成形的本构建模及模拟

为了提高大幅面板材成形的模拟精度,在板材折弯平面应变假设条件下,推导出基于Hill各向异性屈服准则的弹塑性本构方程.借助ABAQUS有限元软件本构模块用户子程序接口,通过编程将上述推导的应力-应变本构关系显示表达式嵌入ABAQUS分析平台.以超长大开口半椭圆形工件成形为例,建立了大幅面钢板渐进折弯的三维弹塑性有限元模型,并数值模拟了多道次渐进折弯成形及回弹全过程.模拟效果和工程应用结果表明,与传统的基于平面应力假设的本构关系模型相比,采用平面应变假设的本构关系模型的模拟结果更接近实验值.     

Influence of out-of-plane compression stress on limit strains in sheet metals

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ ₃), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.     

Evaluation of yield criteria for forming simulations based on residual stress measurement

Process induced anisotropy in sheet metal is accounted for in analytical modeling by anisotropic yield criteria. The suitability of a yield criterion for predicting sheet metal forming process is generally validated by way of its ability to predict surface strains. However, the sensitivity of surface strains to yield criteria is dependent upon strain modes, with plane strain mode exhibiting higher sensitivity. To eliminate dependency on strain modes, stresses are used to evaluate yield criteria, since forming stresses are less sensitive to strain modes. In the study, the residual stresses remaining in a hemispherical cup formed in plain strain mode is predicted using Hill48 and Barlat89 criteria. The residual stresses are experimentally characterized by using X-Ray diffraction method. Suitable yield criterion for forming simulation is validated based on the correlation of theoretical predictions with experimental residual stress values.     

Full-field measurement technique and its application to the analysis of materials behaviour under plane strain mode

The purpose of the paper is to provide a comprehensive experimental and numerical analysis of one of the encountered and critical state modes in sheet metal forming processes. The study is carried out with the help of the full-field measurement techniques. In order to confer some generality to the proposed work, several materials and different specimen shapes are considered that exhibit more or less homogeneous strain field. The proposed experimental study of the plane strain test is completed by a preliminary identification of the material parameters for non-linear behaviour at finite strains, using heterogeneous strain field.     

双相钢冲压成形应变路径影响规律研究

设计特征盒形件,使其包含双拉、拉-压、平面应变以及双线性应变路径,反映了覆盖件成形中常见的应变路径状态。通过正交试验方法,分析了工艺参数（压边力、摩擦系数、板厚）对其成形过程中特征区域应变路径影响的显著性,并得出了显著因素对应变路径的影响趋势。初步取得了控制DP600高强钢成形过程中特征区域应变路径的方法,为控制DP高强钢成形质量提供了有力的指导。     

2D adaptive mesh methodology for the simulation of metal forming processes with damage

In this work, a fully adaptive 2D numerical methodology is proposed in order to simulate with accuracy various metal forming processes. The methodology is based on fully coupled advanced finite strain constitutive equations accounting for the main physical phenomena such as large plastic deformation, non-linear isotropic and kinematic hardening, ductile isotropic damage and contact with friction. The adaptivity concerns the space discretization using FEM as well as the applied loading sequences. Mesh size distribution is based on various error indicators making use of the hessian of the plastic strain rate combined with a specific damage error function and a specific local curvature error function evaluated at contact boundaries. 2D mesh size can be refined or coarsened when necessary according to these error indicators. Particularely, the smallest size is found to be inside the zones where the damage is highly active. The applied loading paths are also adaptively decomposed into various sequences depending on both number and size of the fully damaged elements. The adaptive procedure is validated through various sheet and bulk metal forming examples. In this paper, a plane stress tensile test, an axisymmetric blanking process of two materials with different ductilities and a cold extrusion process are presented.     

Plane strain finite element simulation of shear band formation during metal forming

A plane strain finite element formulation and solution procedure for shear band failure during the plane strain metal forming process are developed and presented. The large strain elastic-plastic formulation includes a 5-node 10-degree-of-freedom (d.o.f.) ‘crossed-triangle’ element, a 4-node 8-d.o.f. element with selective reduced integration, an 8-node 16-d.o.f. element and a 4-node 8-d.o.f. element with an embedded shear band. The formulation includes an elastic-plastic material model with a modified Gurson yield function and combined isotropic-kinematic hardening. The solution procedure is based on a Newton–Raphson incremental-iterative method with an orthogonal projection of zero or negative eigen-modes when required. Two different examples of plane strain tension test are studied with results compared with available numerical solutions to evaluate the present formulation and solution procedure of the four different elements. The results demonstrate that both types of the 4-node quadrilaterals are comparable to the 5-node crossed-triangle element as well as the 8-jiode element. To further validate and to demonstrate the predictive capability and practical applicability of the present development, two plane strain metal forming examples are investigated. The first application is a numerical simulation of a sheet-stretching test with results compared with experimental data for a commercially pure aluminium–magnesium 5182-O sheet. The load vs. extension history and the through-thickness strain are compared. The good agreement suggests that it is possible to numerically determine the parameters needed for the modified Gurson yield function. The second application is a numerical simulation of the formation of dead metal zones in the extrusion process. A plane strain extrusion of a short aluminium billet through straight-sided dies is presented and characteristic features of the formation of dead metal zone are observed.     

Modelling studies of sheet metal formability of friction stir processed Mg AZ31B alloy under stretch forming

Magnesium alloys have poor formability at room temperature. The formability can be improved through hot forming at the cost of deterioration in strength and other mechanical properties. Improvements in texture and grain refinement are the alternate ways for formability improvement. The economically viable process for such applications is alloying or grain refinement technologies like equal channel angular pressing (ECAP), friction stir processing (FSP), and accumulative roll bonding (ARB), etc. Friction stir processing is an emerging solid state microstructure modification technique that can produce homogeneous microstructure with fine-grains in a single pass. The desirable characteristics for sheet formability are the maximum limiting dome height under plane and biaxial strain deformation conditions and the major fracture strain limits through forming limit diagrams (FLDs). Equiaxed homogeneous microstructure with fine grains through FSP results in the enhancement of formability of the material. The objective of the present work is to establish the methodology for viable sheet metal forming practices by altering the process conditions. This needs a clear understanding of the friction stir processed Mg alloy under different strain conditions to get optimized process parameters.     

Approximate yield criteria for anisotropic porous ductile sheet metals

An approximate macroscopic yield criterion for anisotropic porous sheet metals is developed under plane stress conditions in this paper. The metal matrices are assumed to be rigid perfectly plastic and incompressible. The Hill quadratic and non-quadratic anisotropic yield criteria are used to describe the matrix normal anisotropy and planar isotropy. The voids in sheet metals are assumed in the form of through-thickness holes. Under axisymmetric loading, a closed-form upper bound macroscopic yield criterion is derived as 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 plane stress upper bound solutions for different in-plane strain ratios can be fitted well by the closed-form macroscopic yield criterion.     

On the identification of an effective cross section for a cruciform specimen

Biaxial tensile testing of sheet metals is becoming increasingly popular for sheet metal forming. Determining equivalent stresses in biaxial tensile specimens is more complicated than in conventional uniaxial tensile specimens. In the present study, we compare four different approaches to calculate effective stresses during biaxial tensile loading of a cruciform specimen: (a) partial unloading method, where stresses are determined based on force–strain curves; (b) identification with uniaxial tensile testing; (c) an analysis of equivalent biaxial tests; and (d) numerical simulations. Considering experimental results for an AA1050 aluminium alloy and for a low‐carbon steel DC06, we show that, for the cruciform sample studied here, two methods do not yield physically reasonable results: The uniaxial approach does not properly take into account the effect of transverse loading, and the equivalent biaxial approach exhibits uncertainties in strain measurement data. The most comprehensible approach is the numerical method, because it also yields detailed information about the local stress and strain states. The numerical results are in excellent agreement with the partial unloading method in terms of the initial flow stress and of effective stress–strain curves for strains up to 0.02, with both methods predicting a similar effective cross section of 18.0 mm² for the considered specimen.     

Experimental and numerical evaluation of forming limit diagram for Ti6Al4V titanium and Al6061-T6 aluminum alloys sheets

The forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metal. In this work, the formability, fracture mode and strain distribution during forming of Ti6Al4V titanium alloy and Al6061-T6 aluminum alloy sheets has been investigated experimentally using a special process of hydroforming deep drawing assisted by floating disc. The selected sheet material has been photo-girded for strain measurements. The effects of process parameters on FLD have been evaluated and simulated using ABAQUS/Standard. Hill-swift and NADDRG theoretical forming limit diagram models are used to specify fracture initiation in the finite element model (FEM) and it is shown that the Hill-swift model gives a better prediction. The simulated results are in good agreement with the experiment.     

Material model calibration for superplastic forming

Superplastic forming is a slow forming process. The forming time can be minimized by optimizing the pressure profile applied to the forming sheet. The optimization of the superplastic forming pressure is usually done such that a target strain rate at a high strain rate sensitivity is maintained. Careful consideration of the strain rate is required, since localized thinning can occur when the material is strained too quickly. This paper demonstrates that it is essential to explicitly include strain rate sensitivity data, obtained from strain rate jump tests, during the calibration of material model used for superplastic forming simulations. Conventional calibration methods only consider stress–strain data at different strain rates. Such an approach implicitly assumes that a material model that matches the stress–strain data at the different strain rates, will automatically match strain rate sensitivity data. However, by explicitly including the strain rate sensitivity data, the selected material model is more susceptible to localized thinning as the applied strain rate is increased. It is essential for the selected material model to exhibit this behaviour to prevent superplastic forming simulations at high strain rates from predicting stable deformation, when in fact localized thinning will occur.     

Thinning as a failure criterion during sheet metal forming

Thinning during forming is often considered a failure criterion in the metal forming industry. It is believed that a critical amount of thinning takes place in a sheet metal before failure. In this study, varying widths of low-carbon steel sheets were punch stretched under laboratory conditions. Thinning during punch stretching was measured at various locations along the steel sheets. These measurements demonstrated that thinning during forming is not constant, but that it is a function of the strain path followed by the sheet. Hence, thinning should not be used as a failure criterion during forming of sheet metals.     

Proposed life prediction model for laser-formed high-strength low-alloy curved components

Techniques employed for material processing using laser technology are progressing at a rapid pace. One such technique is that of forming sheet metal plates. This high‐intensity localized heating process allows for forming of metallic sheet materials without the need for expensive tools and dies or any mechanical assistance. The fundamental mechanisms related to this process are reasonably well understood and documented but there remain areas that require further research and development. One such area is the fatigue behaviour of sheet materials manufactured by this novel process. Hence, the proceeds of this paper deal with fatigue life prediction of sheet metal components laser‐formed to a radius with a curvature of approximately 120 mm. The approach to this proposed model considers the mean stress relationship as given by Gerber and a prediction model derived from combining the aspects of life prediction models according to Collins and Juvinall & Marshek.     

Effect of continuous strain path changes on forming limit strains of DP600

The strain path may change in actual sheet metal‐forming processes, so the determination of formability of sheet metal should consider the nonlinear strain path. For identifying the forming limit (FL) strains under nonlinear strain path, a conventional two‐step procedure with unloading is classically used to produce the strain path change, which results in no continuous measure of strain. The in‐plane biaxial tensile test with a cruciform specimen is an interesting alternative to overcome the drawbacks of conventional method. The strain path change can be made without unloading during a single test. In this work, the experimental FL strains of DP600 sheets under two types of nonlinear strain path are investigated and then compared with those under linear strain paths. The Oyane ductile fracture criterion is used in the finite element simulation to predict the experimental results.     

Modelling the correlation between the geometrical features and the forming limit strains of perforated Al 8011 sheets using artificial neural network

In the recent years, sheet metals are produced with perforations in various shapes and patterns to improve the appearance of sheet and to save weight of components. As in conventional metal sheets, it is important to form the perforated sheet metals also within their safe strain regions to avoid the forming failures like necking, fracture and wrinkling. The Forming Limit Diagram (FLD) is an appropriate tool to determine the forming limit strains. The limiting strains of perforated sheet metals mainly depend on the geometry of the perforations and forming variables. This leads to large increase in number of test to be conducted with various geometry and forming variables for determining the forming limit strain for perforated sheets. Aiming to reduce the number of experiments needed, in this work, an Artificial Neural Network (ANN) model has been developed for forming limit diagram of perforated Al 8011 sheets based on experimental results and correlated with the geometrical features of the perforated sheets. This model is a feed forward back propagation neural network (BPNN) with a set of geometrical variables as its inputs and the safe true strains as its output. This ANN model can be applied for prediction of FLD of perforated sheet having any geometry.     

Finite element modelling and experimental investigation of single point incremental forming process of aluminum sheets: influence of process parameters on punch force monitoring and on mechanical and geometrical quality of parts

New trends in sheet metal forming are rapidly developing and several new forming processes have been proposed to accomplish the goals of flexibility and cost reduction. Among them, Incremental CNC sheet forming operations (ISF) are a relatively new sheet metal forming processes for small batch production and prototyping. In single point incremental forming (SPIF), the final shape of the component is obtained by the CNC relative movements of a simple and small punch which deform a clamped blank into the desired shape and which appear quite promising. No other dies are required than the ones used in any conventional sheet metal forming processes. As it is well known, the design of a mechanical component requires some decisions about the mechanical resistance and geometrical quality of the parts and the product has to be manufactured with a careful definition of the process set up. The use of computers in manufacturing has enabled the development of several new sheet metal forming processes, which are based upon older technologies. Although standard sheet metal forming processes are strongly controlled, new processes like single point incremental sheet forming can be improved. The SPIF concept allows to increase flexibility and to reduce set up costs. Such a process has a negative effect on the shape accuracy by initiating undesired rigid movement and sheet thinning. In the paper, the applicability of the numerical technique and the experimental test program to incremental forming of sheet metal is examined. Concerning the numerical simulation, a static implicit finite element code ABAQUS/Standard is used. These two techniques emphasize the necessity to control some process parameters to improve the final product quality. The reported approaches were mainly focused on the influence of four process parameters on the punch force trends generated in this forming process, the thickness and the equivalent plastic deformation distribution within the whole volume of the workpiece: the initial sheet thickness, the wall angle, the workpiece geometry and the nature of tool path contours controlled through CNC programming. The tool forces required to deform plastically the sheet around the contact area are discussed. The effect of the blank thickness and the tool path on the punch load and the deformation behaviour is also examined with respect to several tool paths. Furthermore, the force acting on the traveling tool is also evaluated. Similar to the sheet thickness, the effect of wall angle and part geometry on the load evolution, the distribution of calculated equivalent plastic strain and the variation of sheet thickness strain are also discussed. Experimental and numerical results obtained allow having a better knowledge of mechanical and geometrical responses from different parts manufactured by SPIF with the aim to improve their accuracy. It is also concluded that the numerical simulation might be exploited for optimization of the incremental forming process of sheet metal.