Limit Strain Curves Research Articles

An improved Marciniak-Kuczynski approach for predicting sheet metal formability

The Marciniak-Kuczynski (M-K) approach is the most renowned numerical method for forming limit strain curve (FLC) determination. In the commonly used M-K model, the thickness imperfection orientation (ψ) in the sheet plane is variable and the analysis is looped for different initial imperfection orientations (ψ0) for each strain route (ρ), which causes enormous computational waste. In addition, the predicted necking angle may obviously deviate from the theoretical one. This paper investigates the initial imperfection orientation and the necking angles (ψ*) of the AA5182-O sheet by both Nakazima experiments and M-K analysis. The effects of anisotropy on the necking angles and forming limit strains are investigated numerically. The results indicate that the necking orientation at equibiaxial stretching corresponds to the direction of the maximum r-value. The reason of necking angle deviation is well explained. Furthermore, the theoretical equation of necking angle is modified by an empirical formula, which is related to the ratio of r0 to rmax near the equibiaxial expansion. In light of the constant orientation of the imperfection deformation band during stretching, an improved M-K method is proposed based on the pre-assigned and non-evolving groove orientation. Compared with the modified M-K approach, this improved method achieves a higher calculation efficiency with a 97.7% reduction in computing time while maintaining the same level of accuracy.

Development of bending limit strain and stress curves of bendability prediction for assembling automotive body panels

Development of bending limit strain and stress curves of bendability prediction for assembling automotive body panels

On the local evaluation of the hydrogen susceptibility of cold-formed and heat treated advanced high strength steel (AHSS) sheets

Hydrogen embrittlement (HE) is a crucial problem for many advanced high strength steels (AHSS) used in the automotive industry. To contribute to the microstructural understanding on the resistance against hydrogen embrittlement a complex phase (CP) steel and dual phase (DP) steel were investigated systematically in the present work. Both steels have the same tensile strength, but different microstructures. Thermal desorption spectroscopy (TDS) and electrochemical permeation (EP) were used to study the effect of plastic pre-deformation on the chemical diffusivity and solubility. The results were used to parametrize a fully coupled diffusion-mechanical finite element (FE) model. Constant load tests (CLTs) were applied on electrochemical pre-charged notched samples to evaluate the role of hydrogen charging on the mechanical properties. A local evaluation of the CLTs with the fully coupled FE model revealed local limit strain curves (LSC). Comparing these curves yields a very strong effect of the microstructure on the resistivity on hydrogen embrittlement.

Formability prediction of Ti6Al4V titanium alloy sheet deformed at room temperature and 600 °C

Formability predictions of Ti6Al4V titanium alloy sheets deformed at room temperature and 600 °C, using D-Bressan’s shear stress rupture criterion and the critical strain gradient macroscopic modelling are presented and discussed in relation to limit strain results obtained experimentally. Ti6Al4V Forming Limit Strain Curves were predicted and compared with experimental curves at 25 °C and 600 °C and strain rate 0.1 s−1. The analytical models were calibrated by means of tensile tests performed on samples cut at 0°, 45° and 90° to the rolling direction at different temperatures and strain rates to obtain the Lankford coefficients and material strain and strain rate hardening behavior. The applied critical shear stress rupture criterion showed to give a fairly good fit with experimental limit strains outcomes, proving that the shear stress rupture nature of specimens deformed at room and elevated temperatures were well reproduced, despite a not-null strain rate sensitivity coefficient at 600 °C. Fracture occurrence by fast shear stress mechanism through thickness direction was corroborated by experimental fractograph observations close to fracture surfaces. It was shown that predicted limiting major true strain of fracture by shear stress curve is governed by normal anisotropy, strain hardening exponent, pre-strain and normalized critical shear stress parameter, which depends on temperature and strain rate whereas the strain hardening exponent depends largely on temperature. In contrast, the critical strain gradient modelling for onset of localized necking showed poor correlation with the experimental limit strains. Best fit of the critical shear stress criterion with experimental limit strain curves was given by specimens deformed near plane strain or FLCo. Hence, a common feature of various strain-rate independent metals, annealed or cold rolled, is that the main fracture mechanism of thin sheet metals can be considered as fast shear stress through sheet thickness without a visible localized necking in biaxial stretching.

Predictions of Forming Limit Curves of AA6014 Aluminium Alloy at Room Temperature

The aim of present work is to examine D-Bressan mathematical model to accurately predict the surface limit strains by the critical fast shear stress criterion of conventional AA6014 aluminium alloy sheets, after loading with non-linear strain paths at room temperature. Two kinds of limit strain curves can be plotted in the Map of Principal Surface Limit Strains (MPLS): the surface local necking limit, FLC-N, and the fast shear stress fracture limit curve FLC-S. D-Bressan´s critical shear stress criterion model combined with Barlat´s Yld 2000-2D yield criterion is proposed for theoretical prediction of FLC-S curve in sheet metal forming operations and were compared with experimental outcomes of AA6014. In order to calibrate the Bressan-Barlat macroscopic model, tensile tests at two different strain rates on specimens cut at 0o, 45o and 90o to the rolling direction and bulge test were carried out at room temperature to obtain the material coefficients of plastic anisotropy, strain and strain rate hardening law. Experimental bi-linear strain paths were carried out in specimens with 1.04 mm thickness by pre-stretching in uniaxial stress direction up to two strain levels before performing Nakajima testing experiments for obtaining the forming limit strain curves. D-Bressan critical fast shear stress modelling, combined with Barlat´s Yld 2000-2d yield function, predicted quite well both formability of AA6014 sheet as received and the reduced FLC-S after 10% and 20% uniaxial pre-stretching and bi-linear strain path. Comparisons of experimental FLC-S and predictions with Barlat´s Yld 2000-2D and Hill 79 yield criteria were analyzed and discussed. D-Bressan approach with Barlat´s Yld 2000-2d yield function, for exponent m=6, provided better fitting to the experimental FLC-S curves than D-Bressan model combined with Hill 79 yield function.

Influence of Pre-Stretching Levels on the Forming Limit Strain and Stress Curves of High Strength Steel Sheet

In this work, Forming Limit Curves (FLCs) of the conventional and pre-stretched High Strength Steel (HSS) sheet grade 440 (SCGA440-45) were investigated. The conventional forming limit curve was experimentally determined by using the Nakajima stretching test. Subsequently, the non-linear strain path FLCs were precisely developed through the Nakajima stretching test after the specimens were pre-stretched in biaxial direction up to several levels on the Marciniak In-plane stretching test. The gained non-linear strain path FLCs were compared with the conventional FLC.Additionally, the experimental Forming Limit Stress Curve (FLSCs) were calculated using the experimental FLC and non-linear strain path FLCs data from investigated steel sheet. The yield criterion Hill’48 was employed in combination with the Swift strain hardening law to describe anisotropic deformation and plastic flow behavior of the HSS sheet, respectively. Hereby, the influence of pre-stretching levels on the experimentally determined the FLCs and FLSCs were examined. The results prove a significant influence of the pre-stretching levels on the both FLCs and FLSCs of the investigated HSS sheet. For a low pre-stretching in biaxial loading the FLCs demonstrated a reduced formability and the FLSCs exhibited the limited stress levels depending on the experimental FLC data.

Reverse engineering as a way to optimize and design parts produced by elastic-medium drawing

On the example of an aeronautical part of the classical form for a box-type part, optimization of the cavity elastic forming process using a cover with guaranteed clearance is considered, as well as classical optimization of a part based on modeling and optimization based on the technology of “technological reconstruction”. Schemes of classical cavity elastic forming process and cavity elastic forming process with guaranteed clearance are presented. The main aspects of using NURBS technology for restoring the shape of a part from a finite element mesh are described. The problems arising in the case of classical optimization using simulation and the ways of their solution using “technological reconstruction” are shown. The results of calculation in the form of gradients are presented: plastic deformations; thickness and quality zones based on the Keller-Goodwin limit strain curve.

Prediction of limit strains in hot forming of aluminium alloy sheets

Theoretical analysis for the prediction of the limit strains of AA5083 aluminium alloy sheets deformed at room and elevated temperatures are presented and discussed in relation to experimental limit strains. The AA5083 forming limit strain curves, FLC, were predicted by employing D-Bressan and Marciniak–Kuczynski macroscopic concept models and were compared with the experimental FLC from Nakazima tests performed at room and high temperatures. In order to calibrate the models, tensile tests on specimens cut at 0°, 45° and 90° with respect to the rolling direction were carried out in a wide rage of temperatures and strain rates in order to obtain the coefficients of plastic anisotropy as well as material strain and strain rate hardening behaviour at different temperatures. The applied critical shear stress rupture criterion and strain gradient evolution models showed to give a satisfactory agreement with the forming limit strain curves of AA5083 aluminium sheet, proving the suitability of a novel concept of shear stress rupture and local necking evolution in sheet metals deformed at room and elevated temperatures. The correlation of M–K predicted FLC-N curve with the experimental points for room and high temperatures was also fairly good.

Modelling of Forming Limit Strains of AA5083 Aluminium Sheets at Room and High Temperatures

Present work analyses mathematical modelling to predict the onset of localized necking and rupture by shear in industrial processes of sheet metal forming of aluminium alloy 5083 such as biaxial stretching and deep drawing. Whereas the AA5083 sheet formability at room temperature is moderate, it increases significantly at high temperature. The Forming Limit Curve, FLC, which is an essential material parameter necessary to numerical simulations by FEM, of AA 5083 sheet was assessed experimentally by tensile and Nakajima testing performed at room and 400°C temperatures. Tensile test specimens at 0o, 45o and 90o to the direction of rolling (RD) and Nakazima type specimens at 0o RD of aluminium AA5083 were fabricated. Simple tensile tests at room and 400°C temperatures were performed to obtain the coefficients of plastic anisotropy and material strain and strain rate hardening behavior at different temperatures. Nakazima biaxial tests at room and high temperature, employing spherical punch were carried out to plot the limit strains in the negative and positive quadrant of the Map of Principal Surface Limit Strains, MPLS, of aluminium AA5083 sheet. The “Forming Map of Principal Surface Limit Strains”, MPLS, shows the experimental FLC which is the plot of principal true strains in the sheet metal surface (ε1,ε2), occurring at critical points obtained in laboratory formability tests or in the fabrication process of parts. Two types of undesirable rupture mechanisms can occur in sheet metal forming products: localized necking and rupture by induced shear stress. Therefore, two kinds of limit strain curves can be plotted in the forming map: the local necking limit curve FLC-N and the shear stress rupture limit curve FLC-S. Localized necking is theoretically anticipated to occur by two mathematical models: Marciniak-Kuczynski modelling, hereafter M-K approach, and D-Bressan modeling. Prediction of limit strains are presented and compared with the experimental FLC. The shear stress rupture criterion modeling by Bressan and Williams and M-K models are employed to predict the forming limit strain curves of AA5083 aluminium sheet at room and 400°C temperatures. As a result of analysis, a new concept of ductile rupture by shear stress and local necking are proposed. M-K model has good agreement with both D-Bressan models.

Experimental and numerical investigations on limit strains in ductile fracture

Several incidents in the past showed the risk of a human and/or environmental caused accident which exceeds the design limit of a component. Therefore the quantification of safety margins becomes necessary. According to technical standards, the safety assessment is usually based on stress criteria. The deformation capability of the material is hardly taken into account with these criteria. Limit strain based safety assessment concepts can overcome this disadvantage.The main influence factors on limit strains are the stress triaxiality, the component size, the loading path and the strain rate. To quantify these factors, different experiments are performed with specimens made of the steel 20MnMoNi5-5, which is representative for German nuclear power-plants.In the range of high stress triaxiality values, different specimens are tested. All the specimens are simulated using Rousselier model to derive the crack initiation location and time. The experimental and numerical results can be used to derive a limit strain curve. The influence of loading paths on failure strains is shown at pre-loaded notched tensile specimens.Finally, an outlook on the planned experimental and numerical investigations in the range of small stress triaxiality values is given.

Fracture behaviour of aluminium alloy sheet under roller hemming process

Roller hemming is a tendency for hemming flexibility. While for aluminium alloy sheets, fracture happens easily. According to comparison of flat surface straight line flanging and limit strain under necking theory, the failure mode of aluminium alloy sheet under roller hemming is ductile fracture, with the aid of analysis of cross-section of the fracture location. Then ductile fracture criterion critical parameter is determined by simulation and experiments of uniaxial tensile test. Consequently, forming limit strain curve at fracture could be calculated from the assumptions including stress–strain curve, plane-stress condition, linear strain path, Deformation Theory of Constitutive Equation, Cockcroft–Latham fracture criterion, etc. Finally, the feasibility of ductile fracture criterion and bending effect on roller hemming process are analysed.

Reliability analysis of forming limits of anisotropic metal sheets with uncertain material properties

This paper presents a probabilistic method for estimating the reliability of the forming limits of anisotropic metal sheets with uncertain material properties. This method is based on Marciniak–Kuczynski (M–K) analysis and a first-order reliability method. The M–K approach is applied to the computation of both the strain-based and stress-based forming limits, and the anisotropic behaviour of the metal sheets is described by the Yld2000-2d yield criterion. Based on the measured statistics of material properties, forming limit curves with various reliability levels are calculated. The effects of uncertain material properties on the calculated forming limit curves are discussed. In addition, the effects of correlation strength among the uncertain properties are also investigated. Numerical results show that (i) the variation of the calculated forming limit strain curves with an increase in the reliability level is different both qualitatively and quantitatively from that of the forming limit stress curves and (ii) correlations among uncertain material properties may greatly affect the failure probability of metal sheets.

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ee ≥ 0.37 for AISI-1012 steel and ee ≥ 0.25 for AA-2008-T4 aluminum).

A Shear Fracture Criterion to Predict Limit Strains in Sheet Metal Forming

Present work examines a mathematical model to predict the onset of shear fracture in the industrial processes of sheet metal forming such as biaxial stretching. Historically, sheet metal formability has been assessed by simple testing such as the Erichsen test. Lately, the concept of experimental Forming Limit Curve, FLC, was developed to evaluate formability. The Forming Limit Diagram shows the FLC which is the plot of principal strains in the sheet metal surface, e1 and e2, occurring at critical points obtained in the laboratory formability tests or in the fabrication process. Two types of undesirable failure mechanisms can occur in sheet metal forming operations: local necking and shear fracture. Therefore, two kinds of limit strain curves can be plotted: the local necking limit curve FLC-N and the shear fracture limit curve FLC-S. The D-Bressan shear instability criterion model proposed for the theoretical prediction of forming limit strain curve owing to the onset of local necking, FLC-N, in sheet metal forming is reformulated to predict the shear fracture strain limit, FLC-S, of sheet metal forming operations. Shear fracture is anticipated to initiate in the direction of pure shear when the shear stress attains some critical value. The Barlat Yld2000-2d anisotropic yield function proposed by Barlat et al. is employed and the material is assumed to display both planar and normal anisotropy. The new approach investigate the influence of mechanical plastic properties such us the plastic anisotropy parameters, pre-strain and work hardening coefficient in sheet metal formability. Some experimental and theoretical results of forming limit curve for aluminium obtained from present model for two values of the power coefficients a = 5 and 6 are shown. The relevant issues related with the FLC-S is presented and discussed.

Deformation and densification laws of powder cold forging

The deformation and densification laws of preform upsetting and closed-die forging were researched based on experimental results of cold forging of deoxidized Fe powder sintering porous material under different initial conditions such as friction factor, ratio between height and diameter and relative density. The fracture limit criterion for powder cold-forging upsetting and the limit strain curve were achieved. The effect of friction factor, ratio between height and diameter and relative density on fracture strain limitation was emphatically analyzed. The limit process parameter curves for the deformation of upsetting were also confirmed. Laws of deformation, densification and density distribution for closed-die forging of powder perform during cold-forging were further analyzed and discussed with the help of experimental phase analysis. As a result, this experiment established theoretical foundations for the design of preform and die as well as optimization of technological process parameters.

Analysis of hot limit strains of a superplastic 5083 aluminum alloy under biaxial tension

In this paper, limit strains of a commercial superplastic 5083 aluminium alloy sheet were investigated under biaxial tensions at an optimal temperature of 823 K . The sheet was deformed under stress ratios of 1, 0.8293, 0.6667, 0.5985 using a superplastic gas pressure forming tester with dies of aspect ratios of 1:1, 4:3, 2:1 and 8:3. An experimental limit strain curve was obtained and its shape was found to be different from that of conventional forming limit curve. An instability model based on the Marciniak–Kuczynski model and the damage mechanics approach was developed to predict the limit strains of the alloy. In the model, an initial geometrical defect and the inhomogeneity in cavity distribution were considered. The predicted limit strains were found to increase with a decrease in geometrical defect size and the degree of inhomogeneity in cavity distribution. It was demonstrated that the model was able to predict limit strains successfully without assuming any initial geometrical defects. The trend of the predicted limit strain curve was found to be in reasonable agreement with the experimental findings.

Forming evaluation of complex tube hydroformed components

Currently, the main feedstock of tubular blank material for tube hydroforming is derived from slit steel coil, which has been roll formed and electric resistance welded (ERW). From ERW tubular blank, a number of intermediate processing stages such as prebending and preforming may be necessary to achieve the final formed component. The end result is a component that may have experienced a complex strain path history. The traditional forming limit (strain) curve (FLC) is inherently affected by non-linear strain paths; hence, a suitable alternative forming limit criterion has been studied and is proposed for complex tube hydroforming processes. The criterion proposed is the forming limit stress curve (FLSC), which is theoretically independent of strain path and prestrain history. The methods used to generate the FLC and FLSC for tubular blanks are described, and prebending is used as an example to demonstrate how complex tubular component forming operations, and subsequent formability, may be evaluated through use of the FLSC.

The use of a shear instability criterion to predict local necking in sheet metal deformation

A shear instability criterion can provide a consistent approach to the onset of local necking in sheet metal forming under biaxial stretching. The neck is anticipated to initiate in the direction of pure shear when the shear stress attains some critical value. The yield function proposed by Hill is employed and the material is assumed to display only normal anisotropy. Empirically it is found that the additional material parameter required by this yield function is simply related to R, the coefficient of normal anisotropy, for a number of materials. This allows the limit strain to be predicted in terms of three well established plastic properties, viz. work hardening coefficient, coefficient of normal anisotropy and initial pre-strain. The influence of these on the limit strain curve is analysed and the coefficient of work hardening shown to play the most important role. Data available in the literature are employed in a comparison of the present theory with that due to Marciniak. In general the predicted limit strains are in reasonable agreement with the trend of experimental results for a wide range of materials. In the case of isotropic materials with work hardening coefficients in the range 0·2-0·6 predictions from the present theory are almost identical with those from that presented by Stören and Rice. The theory presented here exhibits a good correlation with experimental limit strains for materials with high work hardening coefficients, of approx. 0·4 or more. Generally, for low work hardening materials, with coefficients of 0·25 or less, the shear instability theory tends to an underestimate of limit strains and a Marciniak type of analysis may be more appropriate. However, bearing in mind the scatter of the experimental data the present theory constitutes a safe lower bound on limit strains and, in addition, has the advantage of simplicity in the mathematical calculation required.

Influence of the plastic properties of a material on the forming limit diagram for sheet metal in tension

A development of the theory of the formation of a groove, which appears before fracture in sheet metal subject to tension is presented, and is based on the assumption of initial non-homogeneity of the material in order to facilitate the determination of the limit strain of sheet metal subject to biaxial tension. Additional considerations in the present paper include strain-rate sensitivity, plane anisotropy and the difference in the values of the fracture strain which depends on the direction of the fracture with reference to that of rolling. The influence of the above properties of the material on the limit strain curve is analysed over the entire range of biaxial tension. Theoretical curves of limit strain are compared with experimental results for sheet metal in biaxial tension under conditions which eliminate the influence of friction and which ensure uniform strain distribution over the entire surface of the test piece, except in places of strain concentration. The discrepancies between the experimental and theoretical limit strain curves are analysed.