Non-linear Strain Paths Research Articles

Characterization of the Shear Stress State Under Non-Proportional Strain Paths Realized by Biaxial Stretching in the Marciniak Test

Complex parts with sharp corners are on the rise to satisfy increasing customer demands on the design. To realize these parts in single or multistage forming processes the information of the material behavior under linear and non-linear strain paths is essential. As Bauschinger already showed for the material behavior under cyclic loading, a reduction of the subsequent beginning of plastic yielding can be observed. Different research works studied the Bauschinger effect under cyclic proportional loading or non-proportional strain paths mostly with uniaxial pre-straining of the material. In this paper the material behavior under shearing after previous biaxial stretching in the Marciniak test is analyzed for the aluminum alloy AA5182 and the advanced high strength steel DP600. The Marciniak test is suited for biaxial pre-straining due to an almost almost equibiaxial strain state and a homogeneous strain distribution in-plane on the bottom of the specimen up to higher strains than in a biaxial tensile test. The subsequent shear test is realized with a modified shear test specimen according to the ASTM standard. Finally, the influence of biaxal pre-straining on the beginning of plastic yielding under shear condition, the hardening behavior and the maximum reachable shear strain is investigated for a better understanding of the material behavior under non-proportional loading paths.

Effect of through-thickness normal stress on forming limits under Yld2003 yield criterion and M-K model

Forming limit curve considering through-thickness normal stress is theoretically obtained based on Yld2003 yield criterion and M-K model with the assumption that hydrostatic stress has no effect on plastic deformation. The suitability of Yld2003 yield criterion to describe the yield surface and forming limit under plane-stress condition for aluminum alloy AA6111-T43 is verified by the experimental data obtained from the literature. Then the effect of through-thickness normal stress on forming limit is analyzed. Different forms of forming limit presentation including traditional strain-based forming limit curve (FLC), forming limit effective strain curve (eFLC), stress-based forming limit curve (FLSC) and extended stress-based forming limit curve (XSFLC) are investigated in detail. It is found that forming limit curves in strain-space enhance with increased through-thickness normal stress under both linear and nonlinear strain paths with different pre-strains. However, the variation of forming limit curves in stress-space does not show single trend but depends on the variables of the coordinate plane. A new form of forming limit presentation named correcting FLSC is proposed. The correcting major stress enhances with through-thickness normal stress increase, and the floating range of the correcting FLSC due to through-thickness normal stress is much less than that of the traditional FLSC. The correcting FLSC can be taken as a steadier criterion for sheet metal forming limit prediction under three-dimensional stress state.

Experimental Researches on Nonlinear Strain Paths Forming for Dual Phase Steel

Strain paths during sheet metal forming are always complex and nonlinear. Forming limit diagram (FLD) is a common method to determine failure in the past decades. However, it is only suitable for linear strain path condition. Regarding dual phase steel DP780, a special experiment was designed and carried out on Zwick Cupping equipment to get nonlinear strain paths. And the strain status was analyzed in FLD. It was found that FLD cannot predict failure precisely in this case. A new approach proposed by Stoughton and Yoon which based on polar effective plastic strain was introduced to analyze this nonlinear strain paths condition, the result is in good agreement with experiment, which indicated that Polar Effective Plastic Strain Diagram was an effective and precise tool to determine failure especially for complex nonlinear strain paths forming.

Prediction of fracture in a dual-phase steel subjected to non-linear straining

In this work, selected fracture criteria are applied to predict the fracture of dual-phase steel subjected to non-linear strain paths. Furthermore, the effects of manufacturing history are studied. Four fracture criteria were calibrated in three tests using standard specimens. The fracture criteria were first validated in the circular Nakajima test. A second validation test case was included in order to validate fracture prediction for non-linear strain paths. In this test a sheet metal component was manufactured and subsequently stretched until it fractured. All fracture criteria included in this study predict fracture during the Nakajima test with reasonable accuracy. In the second validation test however, the different fracture criteria show considerable diversity in accumulated damage during manufacturing which caused substantial scatter of the fracture prediction in the subsequent stretching. This shows that manufacturing history influences the prediction of fracture.

The effect of normal stress on the formability of sheet metals under non-proportional loading

Theoretical and experimental studies have shown that when a compressive stress is applied normal to the sheet surface its formability can be significantly improved. In forming processes such as hydroforming, the normal stress imposed at the surface of the sheet or tube can be very significant in specific locations and it is therefore necessary to account for this stress state when assessing the forming severity of industrial parts. It is also well known that the forming limit curve (FLC) can vary significantly in strain space as a result of non-linear strain paths. In this work, a numerical code based on the Marciniak and Kuczynski (MK) analysis (1967) was developed to predict the FLC of sheet metals by simultaneously accounting for the effects of strain path non-linearity and the normal stress. The FLC predicted with this code can be used to assess the forming severity of parts that are deformed under complex loading conditions in industrial metal forming processes. The predictive FLC model was validated using experimental data in which linear and bi-linear strain paths were applied to different steel and aluminum sheets or tubes under plane stress and three-dimensional stress conditions. This research work has two specific purposes: first, to investigate the influence of the magnitude of the prestrain in bi-linear loading on the sensitivity of the FLC to the normal stress, and second, to study the effect of the normal stress on the path dependence of the FLC. This work showed that, with increasing magnitudes of prestrain the influence of the normal stress on the formability of sheet metal will decrease, regardless of the loading path. Moreover, it was observed that even a significant normal stress will practically not affect the path dependence of the FLC. • Forming Limit Curves (FLC) were predicted in 3D stress states and non-linear loading paths. • The effect of prestrain magnitude on the sensitivity of the FLC to the through-thickness compressive stress was investigated. • The effect of normal stress on the path-dependence of FLC was studied. • It was shown that higher prestrain magnitudes will reduces the sensitivity of FLC to the normal stresses. • In most cases, the normal pressure on the sheet metal surface will not affect the path-dependence of FLC.

Measurement of Local Strain Path and Identification of Ductile Damage Limit based on Simple Tensile Test

To estimate the strength of automotive bodies under impact loading and to predict the crack occurrence of high strength steel sheets, the local fracture strain and the ductile damage limit must be measured by experiments to provide data for metal forming simulation by using FEM. In the present study, the local strains at the cracking position and its nonlinear strain path during a simple uniaxial tensile test were measured using a digital image grid method developed by authors. Then, a new identification method for the ductile damage limit of steel sheets was proposed with the aid of the measured nonlinear local strain historical path and local fracture strain. By changing the grid pitch length of the digital image grid method, the effect of grid pitch length on the measured local fracture strain and ductile damage limit were investigated. This experimental investigation is very useful to predict the fracture occurrence conditions if various sizes of finite element mesh were employed in the simulation. The ductile damage limit identified by the newly proposed method using a simple uniaxial tensile test agreed very well compared with the conventional press test.

Effect of nonlinear strain paths on forming limits under isotropic and anisotropic hardening

The path-dependence of the conventional Forming Limit Diagram (FLD) is an important issue for its applications in industry. Great efforts have been made to understand the nature of the path-dependence with both experimental and theoretical approaches, many of them attempting to find a path-independent way for the application of forming limits. In this paper, we focus on the nonlinear strain path effect on forming limit predictions using both isotropic and anisotropic hardening models. The Forming Limit Diagram (FLD), Forming Limit Stress Diagram (FLSD) and Forming Limit Effective Strain Diagram (epFLD) of sheet metals subject to linear and nonlinear strain paths are analyzed and compared using the Marciniak–Kuczynski approach. An anisotropic hardening model based on Yoshida and Uemori development is adopted in this study, and it is coupled with the traditional Hill’48 yield surface. This model is capable of describing the complex Bauschinger phenomenon after the material undergoes the reverse loading process such as the early re-yielding, work-hardening stagnation and permanent softening. Two different scenarios for the change of strain paths are also investigated. In the first scenario, the sheet material is initially loaded with a fixed strain increment ratio, unloaded to the free stress state, and then reloaded with a different strain increment ratio until the forming limit is reached. In the second scenario, the material does not undergo elastic unloading. Instead, the strain path is abruptly changed to a different strain increment ratio and the material undergoes continuous loading until the forming limit is reached. It is found that the work-hardening behavior after the pre-straining and the loading scenario plays an important role in the path dependent behavior of forming limits. Detailed analysis reveals that the M–K approach may have contributed to the significance of path-dependence observed in this study, especially at high pre-strain levels.

A Cruciform Shape to Study the Influence of Strain Paths on Forming Limit Curves

The optimization of sheet metal forming processes requires an accurate prediction of material behavior and forming abilities, especially for aluminum alloys which exhibit generally a low formability compared with typical mild steels. This study presents an original technique based on the use of a cruciform shape for experimental characterization and numerical prediction of forming limit curves (FLCs). By using a cruciform shape, the whole forming limit diagram is covered with a unique geometry thanks to the control of the displacements in the two main directions of the specimen. The test is frictionless and the influence of linear and non-linear strain paths can be easily studied since the strain path is controlled by the imposed displacements, independently on the specimen geometry. The influence of strain paths is first studied by introducing a linear prestrain (uniaxial, plane strain or equi-biaxial), in rolling direction. Afterwards non linear prestrain paths arealso tested.

Development of a new failure prediction criterion in sheet metal forming

In industrial try-out processes in sheet metal forming usually the forming limit curve is used as failure criterion in order to describe the onset of localized necking. Forming limits, however, are strain-path dependent. Up to today many different approaches how the strain-path dependent behavior of the forming limit curve can be avoided have been published. Best known are the approaches based on forming limit stress curves published by Arrieux, and the approach of Muschenborn published in 1975. An overview over existing failure criteria is given in this contribution. The failure criterion forming limit stress curve as well as several failure criteria based on Muschenborn’s approach will be evaluated with newly recorded experimental data on forming limit curves for non-proportional loading. A new approach, that is in contrast to the two mentioned approaches not based on assumptions but on experimental observations, is presented herein. The suggested approach is presented as failure surface where strains above the surface indicate the onset of localized necking. The failure surface is given as function of the loading mode and the level of effective pre-stretching. Different suggestions how to use and simplify the new approach are given in this paper. The prediction accuracy of the addressed approaches as well as of the newly suggested approach is evaluated by transforming the data of the approaches back into the conventional forming limit curve for several non-linear strain paths. The comparison of the described approaches shows that forming limits for non-proportional loading can be well-predicted with the suggested approach.

Strain hardening response and modeling of EDDQ and DP780 steel sheet under non-linear strain path

The anisotropic hardening behaviors of dual phase (DP780) and extra deep drawing quality (EDDQ) steel sheets under non-proportional strain paths were investigated. Two-step uniaxial tension tests, which consisted of the first loading in the rolling (RD) or transverse (TD) and the second loading in every 15° from the first loading axis, were conducted. For DP780 steel, a significant Bauschinger effect accompanied by a transient hardening behavior after reverse loading was the prominent phenomenon. In contrast, EDDQ exhibited stress overshooting followed by strain hardening stagnation with respect to the monotonic flow curve near cross-loading conditions. The extended HAH model combined with the Yld2000-2d yield function were used to reproduced the anisotropic hardening behavior of the two materials. For DP780, the extended HAH model could capture the Bauschinger effect and transient hardening behavior well for tension reloading at 0°, 15°, 75° and 90° from the RD or TD prestraining direction. However, the predictions at 30°, 45° and 60° were slightly different from the experiments. For EDDQ, this approach reproduced the strain hardening anisotropy well including flow stress overshooting followed by a stage of strain hardening stagnation.

Evolution of Yield Loci for Aluminum Alloy AA6016 and Deep Drawing Steel DC06 under the Influence of Non-Linear Strain Paths

The application of modern materials plays an important role directly under the aspect of lightweight potential. To exploit these options effectively a numerical accurate reproduction of the material behavior is indispensable. Especially in the case of large deformations a directional and strain rate dependent hardening behavior can be observed. By disregarding this effect significant failure in the computed stress state can arise, which can conduct to a corruption of the spring-back forecast. Within this contribution a new test method for analyzing the evolution of subsequent yield loci under strain path changes for the aluminum alloy AA6016 and the deep drawing steel DC06 is presented. In the first stage of the experimental investigations, yield loci with linear strain paths were considered to characterize the material behavior for the initial condition. On further experiments with several stress states the strain path dependent hardening behavior of the material is determined. The non-linear strain paths are realized through uniaxial prestrained primary specimens with following extraction of secondary samples for following stress states, e.g. a modified ASTM simple shear test specimen. Subsequent yield loci are investigated and compared to the yield surfaces Hill48 and Barlat 2000 (Yld2000-2d) with an isotropic hardening behavior. With this study the evolution of the yield locus for prestrained specimens is evaluated. The research of the subsequent yield loci for strain path changes serves as basis for further scientific investigations with a view to assess different approaches of isotropic-kinematic hardening models in consideration of the analyzed steel and aluminum sheet metals.

Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability

The optimization of sheet metal forming processes requires accurate evaluations of material forming abilities. This paper presents an original technique based on the use of a cruciform shape for experimental characterization and numerical prediction of forming limit curves. The whole forming limit diagram is covered with a unique geometry by controlling displacements in the two main directions of the cruciform shape. The test is frictionless and the influence of linear and non-linear strain paths can be easily studied. The modelling of the cruciform shape with the finite element method permits to plot forming limit curves without any calibration step, essential for the classical Marciniak–Kuczynski (M–K) model. Experimental and numerical results are presented for an aluminium alloy 5086. These results are respectively compared with the ones from classical techniques: Marciniak test and numerical M–K model.

Comparative analysis between stress- and strain-based forming limit diagrams for aluminum alloy sheet 1060

The forming limit diagram (FLD) and forming limit stress diagram (FLSD) of aluminium alloy 1060 under linear and nonlinear strain paths are investigated. The calculation of FLSD is based on experimental FLD using the method proposed by Stoughton. Different from the FLD that varies with the strain path, the FLSD is not sensitive to the strain path. Therefore, FLSD is convenient as a forming limit criterion for multi-stage sheet forming. The influences of the material's yield criteria on FLSD are also discussed by comparison of the Hill's 48, Hill's 79 and Hosford non-quadratic criterion. The impacts of material hardening laws (Voce and Swift models) on translation of FLD and FLSD are analyzed. The Voce hardening law and the Hosford yield criterion are appropriate for the FLSD calculation of the aluminium alloy 1060. The stress calculation program and display interfaces of FLD and FLSD are developed on MATLAB, where the strain data can be input from experiment measurement or FEM calculations.

Influence of Different Pre-Stretching Modes on the Forming Limit Diagram of AA6014

In order to evaluate the formability of sheet materials forming limit diagrams (FLD) are recorded which represent the values of major and minor strain when necking occurs. FLDs are recorded based on the assumption that exclusively linear strain paths occur. In real forming parts, however, particularly in those with complex shapes, predominantly non-linear strain paths occur which reduce the accuracy of the failure prediction according to a conventional FLD. For this reason forming limits after loading with non-linear strain paths have to be investigated. In this contribution a systematic analysis of the forming limits of a conventional AA6014 alloy after loading with non-linear strain paths is presented. This material is pre-stretched in uniaxial, plane strain and biaxial direction up to several levels before performing Nakajima experiments in order to determine FLDs. During the pre-stretching process as well as during the Nakajima experiment the strain distribution can be measured online very precisely with the optical deformation measurement systems GOM Aramis or VIALUX. The gained curves are compared to the FLD of the as-received material. The results prove a significant influence of the pre-stretching condition on the forming limits of the used aluminum alloy. For a low pre-stretching in uniaxial as well as in biaxial direction the FLDs show a slightly reduced formability while after higher pre-stretching levels the forming limit can be improved such as for biaxial loading after uniaxial pre-stretching. The formability after pre-stretching in plane strain direction was changed. Also, a shift of the FLD depending on the direction of pre-stretching can be observed.

Identification of Process-Based Limit Stress States Applying a Stretch-Bending-Test

The evaluation of forming simulation by using the forming limit curve has only limited validity if it is applied on car body components with non-linear strain paths. If modern high strength materials are used, the forming limit criteria can also provide invalid predictions. Especially high strength multiphase steels show a specific behaviour in forming, necking and crack initiation. If bending loads are applied to these materials, the onset of cracking occurs partially not within the range of the forming limit curve (FLC). The stress limit indicates the failure beginning more accurate. It is independent of the forming history and should be less sensitive to the behaviour of high strength steels. In the post processing of a simulation it could be used similar to the forming limit. A limit curve applied on the in-plane-stress-diagram of an analysed component defines areas that are more vulnerable for cracking. The required stress limit curve will be obtained in this research by applying a stretch-bending-test. It is selected in order to reach loads, which are comparable to the forming process in the components’ production. The forming state that is affecting the specimen is a combination of bending and stretching load. Different load conditions can be applied at the test by altering the stamp-radius and the specimen geometry. Since stresses cannot be measured directly in the experiment, the test is modelled in the simulation where the stresses can be calculated for a given material model. Finally the stress limit criterion was applied on the test parts’ stress distribution diagram. Occurring stresses above the stress limit curve are displayed on the simulation. Thereby it is possible to show a good correlation in critical areas between the failure prediction in the simulation and occurring rupture on the test component.

Failure prediction for nonlinear strain paths in sheet metal forming

The Forming Limit Curve (FLC) is a conventional failure criterion to estimate sheet metal formability for proportional loading conditions in Finite Element Analysis. Previous studies found that a standard FLC is not suitable for predicting the influence of nonlinear strain paths. This paper introduces a new method for the description of failure behavior in two-step forming operations by using a metamodeling technique. The main objectives of this approach are the cost effectiveness of the required experimental calibration and its practical applicability. The predicted forming limits determined with the proposed method are presented and validated by experimental results.

A practical method to evaluate the forming severity of tubular hydroformed parts

In view of the prevalence of non-linear strain paths that develop in parts that are formed in multiple stages, such as bent and hydroformed structural components, the conventional forming limit curve (FLC) cannot be used to assess the forming severity of this manufacturing process. A path-independent stress-based forming limit criterion has been shown to be far more suitable to evaluate such parts, and this paper shows how this failure criterion can be effectively used to evaluate tubular hydroformed parts “on the shop floor”. Knowing the strain history in a given location of a part, a shifted FLC can be computed from the stress-FLC and used to determine the safety margin at this location. This methodology was used to evaluate the forming severity of an automotive instrument panel beam. It was found that this approach is user-friendly and provides a significant improvement in the ability to assess process robustness and product quality compared to the conventional method. The FLCs obtained using the proposed method were found to be in good agreement with those predicted with an MK-based calculation code. Finally, it is shown that a numerical simulation of the entire forming process is recommended to confirm the estimated strain path in critical locations and improve the accuracy of the method.

Dynamic forming limits and numerical optimization of combined quasi-static and impulse metal forming

Subject of this work is the incorporation of forming limits in the numerical optimization of technological forming processes for sheet metal. Forming processes with non-linear load paths and strongly varying strain-rate, such as, e.g., combinations of deep drawing and electromagnetic forming are of particular interest. While in the latter impulse forming process inertial forces play a significant role, the first one is of quasi-static nature such that inertial forces may be neglected. Although classical forming limit diagrams provide an easily accessible method for the prediction of forming limits, they cannot be applied in situations involving pulsed loading along non-linear strain paths. Hence, they are extended to forming limit surfaces here. The target function to be minimized is computed via finite-element simulation. To avoid a large number of simulations, an interior point method is employed as optimization method. In this algorithm, forming limits appear via a logarithmic barrier function, which has to be computed sufficiently fast. The optimization algorithm is exemplarily applied to an identification problem for a two-stage forming process.

Numerical Prediction of Damage Evolution in Sheet Metal Forming Processes with Nonlinear and Complex Strain Paths

Various thin-walled parts with fairly complex shapes are produced from sheet metals such as automotive panels and other structural parts. In these processes, damage and fracture may be observed on the work piece, and formability plays a fundamental role. Therefore, determination of forming limits and prediction of rupture modes in these operations is very important for process design engineers. In this paper, first, based on plane stress elasto-plasticity and finite strain theories a fully coupled elastic-plastic-damage model is used to predict damage evolution in one sheet metal forming process with nonlinear and complex strain paths. As the plane stress algorithm is valid for thin sheet metals and finite strain theory is recommended for large deformations or rotations, the model is able to quickly predict both deformation and damage behaviour of the parts with nonlinear and complex strain paths. The numerical simulations are compared with experimental tests. Comparison of the numerical and experimental results shows that the proposed damage model is accurate for various forming conditions. Hence, it is concluded that finite element method combined with continuum damage mechanics, can be used as a reliable and rapid tool to predict damage evolution in sheet metal forming processes with nonlinear and complex strain paths.

A study of high strength steels undergoing non-linear strain paths—Experiments and modelling

Abstract This paper presents an evaluation of the constitutive behaviour, including plastic anisotropy and mixed isotropic-kinematic hardening of two high strength steels, Docol 600DP and Docol 1200M, during strain path changes. A series of tensile and shear tests was performed on both virgin and pre-strained materials. The initial anisotropy and work hardening parameters were obtained from tensile tests, shear tests and a bulge test of the virgin material, whereas the kinematic hardening parameters were identified by comparing numerical predictions to experimental results related to the pre-strained materials. Numerical predictions using the obtained parameters agree well with the experimental results, both in the case of proportional, and under non-proportional strain paths.