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Detection of strain localization in numerical simulation of sheet metal forming
Abstract
This paper presents an investigation on the detection of strain localization in numerical simulation of sheet metal forming. Two methods to determine the onset of localized necking have been compared. The first criterion, newly implemented in this work, is based on the analysis of the through-thickness thinning (through-thickness strain) and its first time derivative in the most strained zone. The limit strain in the second method, studied in the authors’ earlier works, is determined by the maximum of the strain acceleration. The limit strains have been determined for different specimens undergoing deformation at different strain paths covering the whole range of the strain paths typical for sheet forming processes. This has allowed to construct numerical forming limit curves (FLCs). The numerical FLCs have been compared with the experimental one. Mesh sensitivity analysis for these criteria has been performed for the selected specimens. It has been shown that the numerical FLC obtained with the new criterion predicts formability limits close to the experimental results so this method can be used as a potential alternative tool to determine formability in standard finite element simulations of sheet forming processes.
... Another rate-dependent model can be found for evaluation of the non-linear strain paths. 23 Alternately, Lumelskyj et al. 24 proposed a finite element consideration of the sheet metal forming to detect strain localization by including the overall amounts of the strain paths. Another simple theoretical work for the localized thinning has been done for adhesively bonded steels based on a thickness gradient-dependent necking which incorporates a typical criterion in terms of the effective strain rate. ...
... We can use equation (25) for the single layer of the neo-Hooken. So, by combining differentiation of relations (24) and (25) leads to find the stress field of the neo-Hookean reinforce layers aŝ ...
In this paper, the influence of functional elastomeric substrate-supported layers for enhancing potential resistance capability against localized plastic failure of advanced high strength steels is considered based on a localized necking model of vertex theory. Application of this structure leads to postponing the plastic instability of the metallic part. By defining diffuse and localized modes of deformation in a general framework, the theoretical models are developed to predict necking limits at several stress states. In addition, the results of the Hookean and neo-Hookean elastomers are compared in terms of strain hardening with the anisotropy parameter of Hill’s yield criteria. Since necking band angle (NBA) is a principal factor for the necking prediction, its effect on bifurcation events is evaluated specifically for different ratios of stress rate, and quadratic and non-quadratic yield criteria. This analysis is performed by proposing a supported and yield-dependent necking bound angle (YD-NBA). All considerations are done by providing equilibrium conditions governed over the NBA. Finally, obtained results indicate good agreements between several theoretical considerations and experimental data.
... These methods were adapted to the finite element simulations and were thoroughly investigated in [8]. Here, the determination of a complete FLC by finite element simulations will be presented. ...
... Simulated strain paths and the numerical FLC compared with the experimental FLC[8]. ...
This paper presents an investigation on the determination of forming limit curves (FLCs) by finite element simulations. The numerical FLCs are determined applying the criteria of strain localization in simulations of the Nakazima formability tests. Two methods to determine the onset of localized necking have been compared. The first criterion is based on the analysis of the through-thickness thinning (through-thickness strain) and its first time derivative in the most strained zone. The onset of necking is assumed to occur at the point corresponding to a sudden change of the slope of the strain rate vs. time curve. The limit strain in the second method is determined by the maximum of the strain acceleration, which corresponds to the inflection point of the strain velocity vs. time curve. The limit strains have been determined for different specimens undergoing deformation at different strain paths covering the whole range of the strain paths typical for sheet forming processes. This has made it possible to construct numerical forming limit curves (FLCs). The numerical FLCs have been compared with the experimental one, showing quite a good agreement, especially in the case of the first criterion. This shows that finite element simulations can be used as a potential alternative tool to determine formability limits for sheet forming processes.
... To predict numerically the critical strains in the necking zone, Volk and Hora [31] presented an algorithm for the fully automatic and time-dependent determination of the onset of plastic instability based on physical effects, using the analysis of the first derivative of the strain in the necking area, the necking is assumed to start at the [18,32] developed a method to analyze the time evolution of the principal strains and their first and second time derivatives (called strain rate and strain acceleration, respectively) at a point in the fracture region. Lumelskyy [33] has used and validated the criterion of localization of the strain introduced by Volk and Hora. Yuan et al. [34] determined the forming limit curve by the FE analysis method, where the major strain increment of the necked element was compared with that of its adjacent element. ...
Considering the recent and widespread use of half-hard rolled aluminum sheets in various industrial sectors, this study aims to characterize, determine and evaluate the formability of these sheets. The first phase is to experimentally determine the forming limit curve of laminated half-hard aluminum sheet AA1050-H24. The FLC is determined experimentally from the limiting strain values measured at the fracture location of the specimen using the Nakazima test. Different geometries of the laminated sheet were produced to obtain different deformation paths in the plane of the main deformations. However, experimentally determining a forming limit curve is very time intensive and requires dedicated and costly equipment. The second phase is to develop an alternative method to replace the experimental protocol. Indeed, we will propose a hybrid approach between the finite element method and necking criterion for determining the onset of localized necking in order to numerically predict this curve. In order to make a numerical prediction, Abaqus/Explicit was used to perform finite element modeling of the Nakazima test. The necking criterion based on the first component of the limit strain was used the time of appearance of necking and to plot the forming limit curve. A comparison of the experimental and numerical results is carried out to determine the effectiveness of the necking criterion in the numerical evaluation of the formability of aluminum sheet AA1050-H24. The necking criterion can numerically evaluate the formability of the AA1050-H24 sheet.
... Friction is a crucial consideration for deformation mode testing of metals [26,[53][54][55], and the present simulations suggest that friction can also influence the strain path in orthotropic composites. Increased friction shifts the strain ratio for g1-g4 specimens towards the right (Fig. 14a). ...
Woven thermoplastic composites are attractive to manufacturers due to their thermo-formability, but limitations in predicting complex failure behaviour still hinder widespread use. This work uses new specimen geometries and an associated apparatus to induce a selection of deformation modes and allows composite failure to be investigated under them. These specimens are tested with three different woven composites with varying matrix (polypropylene-PP, polycarbonate-PC), reinforcement (glass, PP), and weave (twill, satin). Strains captured with a Digital Image Correlation (DIC) system integrated with the hemispherical testing device show the strain evolution and deformation mode. Finite Element Analysis (FEA) demonstrates that friction, clamping, and bending have a minor effect on composite failure location and strain state. Experiments show significant differences in macroscale and mesoscale deformation and failure response due to composite constituents and weave architectures. A maximum fibre strain criterion is shown to be effective for fibre-reinforced composites across a range of deformation modes.
... Since finite element analyses have become widely used, both academic and industrial researchers carried out these methods to not only simulate the formability but also predict FLCs of the sheets. Lumelskyj and colleagues [45] constructed FLCs numerically and developed a through-thickness thinning method to predict formability limits. Continuum damage mechanics, coupled with the FEM, is a useful and practical tool to reduce the number of experiments in sheet metal forming. ...
The research mainly focused on the forming limit curves (FLC) of the 6061 Al alloy subjected to heat treatments with various (1 mm, 1.6 mm, 2 mm, and 2.5 mm) sheet metal thicknesses. Five peculiar heat treatments, including natural aging temper (T4) and highest-strength (T6) temper, were devised to study the influence of heat treatments on the formability of 6061 Al alloy sheets. Tensile tests were executed to determine the plastic behavior of the alloy under specific heat treatments. Laser marked FLC specimens were deep-drawn with the proposed spring-attached Nakajima deep drawing test setup. A finite element model (FEM) of the deep drawing experiment was constructed to investigate the nucleation and growth mechanism of the voids. The developed model was compared and verified with the experimental FLC, fracture locations, and dome height. The FEM results were differed from the experimental FLC by less than 5% considering all heat treatments and sheet thicknesses. The results declared that sheet metal thickness has a positive effect on the effective strain up to failure. On the other hand, the duration of artificial aging time reduced effective strain to failure controlled by nucleation and growth of voids.
An experimental metallographic method for determining strain distribution in a cold formed workpiece is presented in this paper. The method, based on the dependence of recrystallized grain size on prior deformation, was applied to a bulk formed element made of low carbon steel. Strain distribution was obtained by the calibration curve which gives the relation between recrystallized grain size and prior deformation. To determine strain values, the average grain size was measured in the longitudinal cross-section of the formed element. To accurately differentiate deformation zones, additional observation of carbides and nonmetallic inclusions morphology was utilized, also. The grain size and strain noticeably differ through the cross-section. It is observed that in the middle part, material flows freely through the die opening without significant deformation. From this zone, strain increases gradually, reaching the highest value at the contact surface of the die slope, while the top part flows radially having intermediate values of strain.
Forming limit diagram (FLD) is the most intuitive method to evaluate and analyze the forming performance of sheet metal, which is widely used in production. To examine the formability of AZ31 magnesium alloy and 7050 aluminum alloy, the simplified bulging models based on the Nakazima experiment are established by ABAQUS finite element (FE) software, and the maximum punch force criterion is adopted as the instability criterion. The forming limit diagrams of 7050 high-strength aluminum alloy at room temperature and AZ31 magnesium alloy at warm working conditions are obtained by extracting the in-plane strain of the adjacent element of the maximum strain element at the moment of instability. Compared with experimental observation shows that the Nakazima virtual model established in this paper can accurately predict FLD. In addition, the influences of lubrication conditions and virtual punching speeds on the bulging process of AZ31 and AA7050 sheet metals are also investigated. The results show that the better the lubrication environment, or the lower the punching speed, the better the formability of the sheet, and reducing the punching speed has a more significant improvement effect on the formability of AZ31 sheets.
The NUMPRESS System has been developed in IPPT PAN as a result of a project financially supported by European Regional Development Fund (within the Innovative Economy Programme) and is dedicated to small and middle enterprises dealing with sheet metal forming. The program consists of (i) an analytical module for analysis of forming processes with the finite element method, (ii) an optimization module controlling execution of the analytical module and performing optimization with respect to selected process parameters, in both deterministic and robust formulation, (iii) a reliability analysis module controlling execution of the analytical module to assess how random distribution of design parameters affects forming results, and (iv) a graphical user interface enabling communication between modules and easy definition of design parameters and optimization criteria. The analytical module consists of two independent programs up to the user's choice: NUMPRESS-Flow, a faster and less accurate program for implicit quasi-static analysis of rigid-viscoplastic shells (based on the flow approach) and NUMPRESS-Explicit, a program for explicit dynamical analysis of elastic-plastic and elastic-viscoplastic shells. Both programs are interfaced to a well-known commercial graphical pre-and postprocessor GiD. Fundamentals of formulations employed in the system and numerical examples are presented in the paper.
This paper presents results of experimental studies of forming limit curves (FLC) for sheet forming under complex strain paths. The Nakazima-type formability tests have been performed for the as-received steel blank and for the blank pre-strained by13%. Prestraining leads to abrupt change of strain path in the blank deformation influencing the forming limit curve. The experimental FLC of the pre-strained blank has been compared with the FLC constructed by transformation of the as-received FLC. Quite a good agreement has been found out. The concept of strain-path independent FLCs in polar coordinates has been verified. Two types of the polar diagrams have been considered, the first one with the strain-path angle and effective plastic strain as the polar coordinates, and the second one originally proposed in this work in which the thickness strain has been used instead of the effective plastic strain as one of the polar coordinates. The second transformation based on our own concept has given a better agreement between the transformed FLCs, which allows us to propose this type of polar diagrams as a new strain-path in dependent criterion to predict sheet failure in forming processes.
TWBs) are tailor made for different complex component designs by welding multiple metal sheets with different thicknesses, shapes, coatings or strengths prior to forming. In this study the Hemispherical Die Stretching (HDS) test (out-of-plane stretching) of TWBs were simulated via ABAQUS/Explicit to obtain the Forming Limit Diagrams (FLDs) of Stainless steel (AISI 304) laser welded blanks with different thicknesses. Two criteria were used to detect the start of necking to determine the FLD for TWBs and parent sheet metals. These two criteria are the second derivatives of the major and thickness strains that are given from the strain history of simulation. In the other word, in these criteria necking starts when the second derivative of thickness or major strain reaches its maximum. With having the time of onset necking, one can measure the major and minor strains at the critical area and determine the forming limit curve.
Forming limit diagram (FLD) is a measure of the formability of a sheet material. The major-minor strain pairs that are closest to the neck on multiple specimens of various strain paths are utilized to construct a boundary between safe and unsafe zones. The challenge to obtain the FLD is the determination of incipient necking. Three approaches to determine the limit strains have been investigated and compared in this research in order to establish the optimal one for implementation: (1) commonly used Bragard criterion (ε 1) Br periodic grids; (2) tracking the region of large local strains from strain history to locate the instance when critical major strain (ε 1) cr happens; (3) post-processing of strain history to locate the inflection in the major strain rate curve (ε̈ 1) max of localization. The last criterion of inflection in strain rate (ε̈ 1) max., carries both a numerical and a physical meaning towards developing an understanding of flow localization, formability and fracture.
This paper provides an elementary introduction into modeling of localized inelastic processes such as cracking or formation of shear bands. The basic concepts are explained using a simple one-dimensional example. It is shown that a stress-strain law with softening used in a standard continuum description leads to physically meaningless results and that the numerical solution suffers by a pathological sensitivity to the finite element discretization. The paper presents an overview of methods that provide an objective description of the localization problem. Regularization techniques are illustrated by examples dealing with the nonlocal damage model and with the softening plasticity model enhanced by the second gradient of the cumulative plastic strain.
This paper presents the investigation on detection of strain localization in experimental research and numerical simulation of sheet metal forming. Experimental tests and numerical simulations of the Nakazima test have been performed for the DC04 grade steel sheet. The onset of localized necking has been determined using the criterion based on analysis of the major principal strain and its first and second time derivatives in the most strained zone. The strain localization has been evaluated by the maximum of strain acceleration which corresponds to the inflection point of the strain velocity vs. time. The limit strains have been calculated numerically and experimentally for specimens undergoing deformation at different strain paths. It has been shown that the numerical model predicts formability limits close to the experimental results. Analyzed criterion can be used as a potential alternative tool to determine formability in standard finite element simulations of sheet forming processes.
This is a review of the profile method and the evolution of profiles with sheet thickness. The principles of a new method to determine the limiting strains at the onset of necking, and its application are described.
Sheet-metal fabricators are considering the use of new materials with higher strength-to-weight ratios than commonly-used low-carbon steel. The Keeler-Goodwin forming limit curve (FLC) developed primarily from production stampings of low-carbon steels cannot be used for these materials. A simple technique was developed to determine FLCs in the laboratory. Sheet specimens, securely clamped at the periphery, were stretched to failure over a 4in dia hemispherical steel punch. Failures were generated over a range of minor-to-major strain-ratios from minus one-half to 1 by increasing the lubrication on the positive minor strain side and by decreasing the specimen width on the negative minor strain side. To be consistent with press-shop use, the FLC was defined by the major and minor strains at the onset of visible, localized necking in a deformed circle of an original diameter of 0. 1in. The FLCs were drawn to fall below the strain in necked and fracture-affected zones and above the strains found just outside these zones.
Many practitioners of the metal forming community remain faithful to the idea that strain metrics are useful for formability assessment. However, it is only valid when deformation occurs along linear strain paths. Current simulations of multi-stage sheet forming processes for rigid-packaging and automotive components result in higher rejection rates due to the inaccuracy of forming and fracture limit models. In this work, we establish a new approach considering path-independency in forming limits based on the stress-based forming limit and polar EPS (Effective Plastic Strain) diagram which appear to be an effective solution for nonlinear effects. The related theory has been implemented into a user material model in commercial software.
In this paper, we present a new approach to obtain the asymptotic expansion and super convergence for the linear element on Union Jack mesh. First, we construct a generalized interpolation function and its discrete harmonic extension by using the energy embedding method and the method of separation of variables. Then, we present elaborate estimates for the generalized interpolation function and the harmonic extension. Finally, the asymptotic expansion, super convergence and extrapolation are obtained based on those estimates.
This paper introduces an advanced, forming-condition-based formability diagram for accurately assessing the formability of complex stampings. The forming-condition-based formability diagram consists of a formability index that rationally define two curves (marginal line and safe line) and two zones (marginal zone and safe zone). The formability index measures deformation severity, tracks deformation history and predicts deformation trend. The safety factor, defined as the width of the marginal zone by means of the formability index, takes into account the forming conditions including deformation zone size, forming mode, bending process model, deformation history, metal flow pattern and post-necking deformation capacity of sheet metal. It is demonstrated that, rather than the traditional formability limit diagram, this formability diagram can more accurately quantify the formability status and can be used to determine the metal flow adjustment ranges for solving split problems. Its application to an automotive body side outer panel shows that it is a robust tool for formability engineering and troubleshooting through using numerical simulations and/or circle grid analysis.
This paper presents the results of numerical simulations of the formability tests carried out for a pre-stretched 1 mm thick DC04 steel sheet. Simulation consisted of the subsequent stages as follows: uniaxial stretching of the sheet, unloading and stress relaxation, cutting specimens out of the pre-stretched sheet and bulging the blank with a hemispherical punch. Numerical modeling has been verified by comparison of the simulation results with the experimental ones. Good concordance of the results indicates correct performance of the numerical model and possibility to use it in further theoretical studies.
Forming Limit Curves (FLCs) are an important tool in steel sheet metal forming. Experimental measurements of FLCs are costly, and therefore, empirical prediction methods are of practical use. Difficulties in accurately defining FLCs for new steel grades, such as AHSS, have necessitated a review of the existing prediction methods. Four points were defined to characterise an FLC, and correlations between the coordinates of these points and the mechanical properties from tensile testing were found. The results show that the total elongation, Lankford coefficient and thickness are strongly related to the FLC values. Predictive equations were derived from the statistical relations between the measured FLC points and the mechanical properties. To verify the predictive equations, predicted FLCs for approximately fifty steel grades in various thickness ranges were compared with measured FLCs. It was found that the newly developed method accurately predicts the FLCs.
The objective of the paper is to define a new method for the experimental determination of the Forming Limit Curves (FLCs). The procedure is based on the hydraulic bulging of two specimens. The most important advantages of the method are the capability of investigating the whole strain range specific to the sheet metal forming processes, simplicity of the equipment, and reduction of the parasitic effects induced by the friction, as well as the occurrence of the necking in the polar region. The comparison between the FLCs determined using the new procedure and the Nakazima test shows minor differences.
A new methodology is proposed to obtain the forming limit diagram (FLD) of sheet materials by utilizing routinely obtained experimental punch load versus displacement traces from hemispherical punch stretching experiments and by analyzing strain history of the test samples from finite element simulations of the experiments. The simulations based characteristic points of diffuse and localized necking are utilized to obtain the limit strains. The proposed method for FLD determination considers out-of-plane displacement, punch–sheet contact and friction, and avoids using experimental strain measurement in the vicinity of the neck on the dome specimens and the rather arbitrary inhomogeneity factor to trigger localization such as in the commonly used Marciniak–Kuczynski method. A criterion of maximum in the ‘pseudo’ major strain acceleration for the onset of localized necking, proposed earlier by the present authors, is utilized to determine the limit strain in FE simulations as well as in FLD verification experiments. The proposed overall approach for obtaining FLD is rapid and accurate and could be implemented for routine FLD generation in a laboratory setting with significant reduction in cost, effort and subjectivity.
A theory is suggested which describes, on a macroscopic scale, the yielding and plastic flow of an anisotropic metal. The type of anisotropy considered is that resulting from preferred orientation. A yield criterion is postulated on general grounds which is similar in form to the Huber-Mises criterion for isotropic metals, but which contains six parameters specifying the state of anisotropy. By using von Mises' concept (1928) of a plastic potential, associated relations are then found between the stress and strain-increment tensors. The theory is applied to experiments of Korber & Hoff (1928) on the necking under uniaxial tension of thin strips cut from rolled sheet. It is shown, in full agreement with experimental data, that there are generally two, equally possible, necking directions whose orientation depends on the angle between the strip axis and the rolling direction. As a second example, pure torsion of a thin-walled cylinder is analyzed. With increasing twist anisotropy is developed. In accordance with recent observations by Swift (1947), the theory predicts changes in length of the cylinder. The theory is also applied to determine the earing positions in cups deep-drawn from rolled sheet.
This paper presents some advances of finite element explicit formulation for simulation of metal forming processes. Because of their computational efficiency, finite element programs based on the explicit dynamic formulation proved to be a very attractive tool for the simulation of metal forming processes. The use of explicit programs in the sheet forming simulation is quite common, the possibilities of these codes in bulk forming simulation in our opinion are still not exploited sufficiently. In our paper, we will consider aspects of bulk forming simulation.We will present new formulations and algorithms developed for bulk metal forming within the explicit formulation. An extension of a finite element code for the thermomechanical coupled analysis is discussed. A new thermomechanical constitutive model developed by the authors and implemented in the program is presented.A new formulation based on the so-called split algorithm allows us to use linear triangular and tetrahedral elements in the analysis of large plastic deformations characteristic to forming processes. Linear triangles and tetrahedra have many advantages over quadrilateral and hexahedral elements. Linear triangles and tetrahedra based on the standard formulations exhibit volumetric locking and are not suitable for large plastic strain simulation. The new formulation allows to overcome this difficulty.New formulations and algorithms have been implemented in the finite element code Stampack developed at the International Centre for Numerical Methods in Engineering in Barcelona. Numerical examples illustrate some of the possibilities of the finite element code developed and validate new algorithms.
Out-of-plane (dome) formability test is used extensively to determine forming limit diagrams (FLDs) of sheet metals. Strain paths are generated to form the FLD using this test. In this research, a comparative study was performed between measured and predicted strain paths. Numerical predictions are well suited with the experimental results. The strain paths from finite element simulations are found fairly acceptable to represent both sides of the FLD. The effects of mesh size and lubrication conditions on the forming limit diagram were also investigated.
The failure prediction in sheet metal forming is typically realized by evaluating the so called forming limit curves (FLC).
The standard experimental method is the Nakajima test, where sometimes also the Marciniak test setup is used. Up to now, the FLC determination was performed with failed specimens and an one-directional intersection
line method or by manually analyzing the estimated strains before cracking. Both methods determine the failure by considering
the occurrence of cracking and do not consider the possibility of time continuous recording of the Nakajima test. Consequently
forming limit curves which have been evaluated in such way are often “laboratory dependent” and deviate for identical materials
significantly. This paper presents an algorithm for a fully automatic and time-dependent determination of the beginning plastic
instability based on physical effects. The algorithm is based on the evaluation of the strain distribution based on the displacement
field which is evaluated by optical measurement and treated as a mesh of a finite element calculation. The critical deformation
states are then defined by 2D-consideration of the strain distribution and their time derivates using a numerical evaluation
procedure for detecting the beginning of the localization. The effectiveness of the proposed algorithm will be presented for
different materials used for the Numisheet’08 Benchmark-1 with Nakajima test.
KeywordsFLC–Local necking–Optical detection of strain localizations–Nakajima-test
This paper examines the conditions for instability of plastic strain under plane stress for a material conforming to the Mises-Hencky yield condition and strain-hardening according to a unique relationship between root-mean-square values of shear stress (q) and incremental strain (δψ).If, under fixed loading conditions, the material undergoes a strain increment which is consistent with the applied stress system, the conditions are stable or unstable according as the increment in representative yield stress is greater or less than the increment in representative induced stress.The strain at which instability arises is found in terms of the biaxial stress ratio p2/p1 under different conditions of applied loading, and the effect is demonstrated of strain-hardening according to an empirical relation of the type q = c (a + ψ)n.The analysis is also applied to certain cases of non-uniform stress distribution. In the case of the hydrostatic bulge results are obtained showing a critical thinning ranging from 26 per cent for a non-hardening material to about 45 per cent for typical strain-hardening materials, values in general agreement with experimental data. Conditions over the punch head in the pressing of a cylindrical shell are discussed but computations are not attempted.
Permissible discontinuities of stress, velocity, and surface slope are investigated in a plastic-rigid sheet deformed in its plane. One such discontinuity of velocity is shown to be the mathematical idealization of localized necking; the necessary restrictions on the stress-state and rate of workhardening are obtained for any yield function and plastic potential. The results are illustrated by an examination of the modes of necking in notched tension strips. The constraint factors at the yield point are obtained for notches with wedge-shaped or circular roots.
Influence of friction on strain distribution in Nakazima formability test of circular specimen
- Lumelskyy
Effect of strain ratios on the deforming limit of steel sheet and its application to the actual press forming
- Kobayashi
Calculations of forming limit diagram for changing strain paths
- Graf
Nowa metoda wyznaczania utraty stateczności blachy oraz napręŜeń szczątkowych
- Zimniak
Numerical simulations of Nakazima formability tests with prediction of failure
- Lumelskyj