Non-linear Strain Paths Research Articles

Plastic behaviour and forming limit during biaxial compressions of magnesium alloy AZ31 at room temperature

The plastic behaviour and compressive fracture of AZ31 magnesium alloy were examined at room temperature. The uniaxial compression tests and biaxial compression tests along linear and nonlinear strain paths were carried out until fracture using a unique biaxial compression testing machine developed by author. The results revealed the evolution of plastic anisotropy in biaxial compressions, that is the strain path dependency of work hardening. The adaptability of several forming limit criteria to the compressive fracture of AZ31 alloy was discussed. It was found that the uniform estimation of compressive forming limits by uniaxial and biaxial compressions was difficult by Freudenthal criterion and Oyane criterion. The result of Tresca energy criterion suggested that the criterion based on shear stress has possibility to predict the compressive forming limits at room temperature of AZ31 alloy.

Probabilistic modeling of stress-based FLD in tube hydroforming process

Recently a stress-based FLD due to its path-independency has become a new mainstream technology for prediction of forming limit in tube hydroforming processes, which usually have non-linear strain path. However, the random characteristics owing to experimental errors, inherent measuring equipment errors, dimensional tolerance of as-received blank, inconsistent lubrication condition and so on lead to uncertainty with regard to the position of the forming limit curve on the stress-based FLD. In this work, to take into account the uncertainty on the stress-based FLD, a probabilistic modeling on the stress-based FLD with confidence level is firstly attempted. With the assumption of all experimentally measured data as normally distributed random variables, stochastic evaluation of the reliability of the stress-based FLD is carried out. Moreover the statistical correlation between measured data during a bulge test and the material parameters is investigated by using the first-order approximated mean value and variance. The reliability is analytically calculated based on first-order reliability method (FORM) and verified with Monte-Carlo simulation (MCS). Finally, at a given deviation of measured data, the band width of forming limit curve on the stress-based FLD to be able to satisfy the required confidence level is determined.

Intrinsic diagram of sheet metal forming limits for arbitrary strain paths

Localized necking in sheets under biaxial tension is analysed by an Marciniak—Kuczynski localization approach (MK-analysis) along with a new plane-stress criterion. Analysis is developed for a rigid viscoplastic behaviour based on flow-theory of plasticity. The model is introduced in numerical calculations to determine forming limits to ductility under linear and non-linear strain paths. However, the results are presented in a new diagram that represent the effective strain as a function of the current strain-rate ratio. A comparison with classical forming limit diagrams shows the intrinsic character of the new diagram.

Experimental assessment of the response of sands under shear–volume coupled deformation

An experimental study of the behaviour of an alluvial sand under different strain increment paths representing shear–volume coupled deformation is presented. Both pore pressure and pore volume change simultaneously in these tests. Linear strain paths with different levels of limiting volumetric strain and nonlinear strain paths that simulate different pore pressure boundary conditions were applied to the soil specimen in the laboratory. The strain paths imposed included both expansive and contractive volumetric deformation. Nonuniform excess pore pressures generated during earthquakes (on account of the heterogeneity in natural soils) often lead to such deformation in situ following the end of strong shaking. The shear strength of the soil could decrease significantly when the pore pressure boundary conditions result in volume inflow that leads to a considerable reduction of the effective confining stress. The rate of volume inflow plays a significant role on the resulting stress–strain and pore pressure responses. Both the peak and the minimum shear strength mobilized during the test were significantly dependent on the strain path. The effective stress ratio at the instant of peak pore pressure is independent of the strain path followed, and it is equal to the effective stress ratio noted at the instant of phase transformation in undrained tests.

Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening

The main purpose of this work is to present two semi-analytical solutions for the von Mises elastoplasticity model governed by combined linear isotropic-kinematic hardening. The first solution (SOL ε) corresponds to strain-driven problems with constant strain rate assumption, whereas the second one (SOL σ) is proposed for stress-driven problems using constant stress rate assumption. The formulas are derived within the small strain theory Besides the new analytical solutions, a new discretized integration scheme (AM ε) based on the time-continuous SOL ε is also presented and the corresponding algorithmically consistent tangent tensor is provided. A main advantage of the discretized stress updating algorithm is its accuracy; it renders the exact solution if constant strain rate is assumed during the strain increment, which is a commonly adopted assumption in the standard finite element calculations. The improved accuracy of the new method (AM ε) compared with the well-known radial return method (RRM) is demonstrated by evaluating two simple examples characterized by generic nonlinear strain paths.

Bursting failure prediction in tube hydroforming using FLSD

In tube hydroforming, circular components are hydrobulged or hydroformed from tubular blanks with internal pressure and simultaneous axial loading. Thus the tube can be fed into the deformation zone during the bulge operation allowing more expansion and less thinning without any defects such as wrinkling, buckling, and bursting. By contrast with the buckling and the wrinkling, the bursting is generally classified as an irrecoverable failure mode. Hence in order to obtain the sound hydroformed products, it is necessary to predict the bursting behavior and to analyze the effects of process parameters on this failure condition in hydroforming processes. In this study, a forming limit stress diagram (FLSD) is constructed by plotting the calculated principal stresses based on the local necking criterion. Using the theoretical FLSD, we carry out the numerical prediction of bursting failure in a hydroforming process, which usually has non-linear strain path. Finite element analyses are carried out to find out the state of stresses during simple hydroforming operation, in which the FLSD is utilized as the forming limit criterion for assessment of the initiation of necking, and influences of the material parameters on the formability are investigated. In addition, the numerical results obtained from the FEM combined with the FLSD are confirmed with a series of bulge tests in view of bursting pressure and show a good agreement. Consequently, it is shown that the theoretical and numerical approach to bursting failure prediction proposed in this paper will provide a feasible method to satisfy the increasing practical demands for assessment of the forming severity in hydroforming processes.

Plastic instability in complex strain paths and finite element simulation for localised necking prediction in sheet metal forming technology

Formability of sheet metals is a measure of its ability to deform plastically, being mainly limited by the occurrence of flow localisation or instability. Plastic instability numerical models have been used to predict such behaviour and recent and more accurate constitutive plasticity models have been applied in these calculations. Combination of Forming Limit Diagram (FLD) analysis with Finite Element (FE) simulations often fail to give the right answer, if complex strain paths are not included. This paper presents a Plastic Instability Model developed to predict localised necking under complex strain paths, which may be used as FLD prediction code or as a FE post-processing tool for necking prediction. It is shown that considering the non-Linear Strain Paths in the analysis, more accurate failure predictions are achieved. An experimental component with necking occurrence is studied. The numerical simulation of this component by FEM is performed and necking is predicted by developed code.

An Analysis of Deformation Microstructure and Subsequent Recrystallisation in Hot Deformed Aluminium Alloy AA5052 Using Forward and Reverse Torsion

The effects of strain path reversal on the microstructure in AA5052 have been studied using high resolution EBSD. Deformation was carried out using two equal steps of forward/forward (F/F) or forward/reverse (F/R) torsion at a temperature of 300°C and strain rate of 1s-1 to a total strain of 0.5. In both cases the deformation microstructure in the majority of grains analysed consisted of microband arrays clustering at specific angles to the macroscopic deformation axes. For the F/F condition microbands clustered around -20° and +45° to the maximum principle stress direction, whilst for the F/R condition significantly more spread in microband angle was observed. This suggests that the microbands formed in the forward deformation have or are dissolving and any new microbands formed are related to the deformation conditions of the final strain path. This leads to the conclusion that instantaneous deformation mode determines the orientation of new microbands formed whilst a non-linear strain path history influences the range of misorientation angle in the material through the dissociation of previously formed microbands and the formation of new microbands at the new straining condition, leading to a lower level of misorientation angle. Analysis of material subjected to static annealing at 400°C for 1 hour appears to correspond with these observations as the F/F material was completely recrystallised with a fine grain structure whilst the F/R material had no major signs of recrystallisation.

Measurement of Ductile Forming Limit in Non-linear Strain Paths and Anisotropic Yield Conditions for 11% Cr Steel Sheets

The ductile forming limit in nonlinear strain paths is investigated for a 11% Cr steel which displays a high r-value and is used practically in difficult forming applications. Experimental measurements were performed to examine the forming limit in strain paths which involve plane strain loading in first-stage loading and loading at various strain ratios in the second stage. Because the forming limit diagram (FLD) is not applicable to nonlinear loading paths, the forming limit stress diagram (FLSD) theory is investigated. In order to evaluate the stress at the forming limit from measured strain, the anisotropic yield criterion of the material is investigated by uniaxial tension testing in three directions and biaxial stretching using hydraulic bulging. Hill quadratic yield criterion gives the best approximation for biaxial tension, whereas Hosford criterion is more appropriate for uniaxial tension. Yld2000-2d is the ideal criterion for both stress states. Considering the possible effect of dependency of r-values on measured strain, the evolution of r-values is measured. Under sufficiently large strain, the r-value becomes virtually constant. Based on the preliminary investigation outlined above, it was found that the evaluated forming limit stresses in nonlinear strain paths lie on one consistent line in the principal stress field. The limit stress line is not affected by changes in the amount of strain applied in the first loading stage, demonstrating that the FLSD theory of ductile fracture is adequate for this particular material.

Ductile Forming Limit Stress in Sheet Forming Processes with Nonlinear Strain Path Involving Out-of-plane Forming

The ductile forming limit in nonlinear strain paths which include out-of-plane forming such as rolling was investigated for a 11 % Cr steel sheet. For strain paths with only a plane stress state, e.g., in-plane strain, the forming limit stress diagram (FLSD) has been proven to be effective, even if the paths are nonlinear. In this research, a method of extending the FLSD theory to strain paths which include out-of-plane strain is investigated experimentally and theoretically. An experimental procedure is used to measure directly the stresses at the onset of diffused and localized necking and at the onset of separation with rolling-stretching loading paths. Theoretically, an assumption is introduced to consider out-of-plane strain in the M-K theory. The experimental and theoretical results of the forming limit stress show good agreement. When large rolling strain is induced in first-stage loading paths, this research demonstrated that there are cases where the FLSD theory is not effective for predicting the ductile forming limit. Those cases are characterized by a stress state of the re-yielding point in the final loading stage. If the point is outside the FLSD curve, the forming limit stress is not on the FLSD curve but located near the subsequent yield surface. Thus, a modified FLSD curve can be defined by the lines connecting the outer line of the conventional FLSD curve and the subsequent yield surface of the loading path under consideration. With this extension, it is possible to demonstrate the applicability of the FLSD method to processes which include out-of-plane loading paths such as rolling.

Evaluation of Ductile Forming Limit Stress in Sheet Forming Processes and Measurement of Actual Stress at Fracture

A stress-based ductile forming limit theory applicable to nonlinear strain paths is investigated with a 11 % Cr steel which displays a highly anisotropic behavior in plastic deformation. The forming limit stress is evaluated by a theoretical method based on the M-K theory, and is also measured directly in some cases with nonlinear loading paths. The evaluated and measured stresses are compared with an experimentally-obtained forming limit stress diagram, where the stresses are converted from strains by Stoughton's method. The theoretically-evaluated limit stress corresponds not to the measured stress at the onset of fracture but to the stress at the onset of localized necking. The evaluated forming limit stress and experimentally-obtained localized necking stress demonstrate no dependency on strains paths, thus verifying the applicability of the theory to this particular material. The experimentally-obtained forming limit stress diagram was corrected using the above results, which has improved the discrepancies between the experimentally- and numerically-obtained forming limit stress.

A comprehensive failure model for crashworthiness simulation of aluminium extrusions

A correct representation of the plastic deformation and failure of individual component parts is essential to obtaining accurate crashworthiness simulation results. The aim of this paper is to present a comprehensive approach for predicting failure in a component based on macroscopic strains and stresses. This approach requires the use of a number of different failure mechanism representations, such as necking (due to local instabilities), as well as ductile and shear fracture. All failure criteria have been developed in a way to include the influence of non-linear strain paths. The effectiveness of this approach in predicting failure is then discussed by comparing numerical results with test data by three point bending and axial compression tests of double chamber extrusion components. All studies presented in this paper were carried out on extrusions made from aluminium alloy EN AW-7108 T6.

Prediction of forming limit curves using an anisotropic yield function with prestrain induced backstress

The Forming Limit Diagram (FLD), a plot of the maximum major principal strains that can be sustained by sheet materials prior to the onset of localized necking, is a useful concept for characterizing the formability of sheet metal. Both experimental and numerical results in the literature have shown that the level of the FLD is strongly strain path dependent and the prediction of FLD depends on the shape of the initial yield function and its evolution. In this work, to improve the accuracy of FLD prediction under nonlinear strain paths for a given material, the evolution of the yield function is proposed in terms of the changes of its center and its curvature. The center of the subsequent yield surface after preloading and unloading will be determined via a backstress tensor, and the curvature change will be reflected by changing the exponent in the yield function. Both parameters are functions of the effective plastic strain and will be determined using the forming limit strains obtained from two nonlinear tests. Using this approach, a combination of Marciniak–Kuczynski (M–K) analysis (Marciniak, Z., Kuczynski, K. 1967. Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609.) and a general anisotropic yield criterion developed by Karafillis and Boyce (Karafillis, A.P., Boyce, M.C. 1993. A general anisotropic yield criterion using bounds and transformation weighting tensor, J. Mech. Phys. Solids, 41, 1859) is used to predict nonlinear FLDs of both Al2008-T4 and Al6111-T4. Excellent agreements were obtained between computed FLDs with the experimental data of Graf and Hosford (Graf, A., Hosford, W.F. 1993a. Calculations of forming limit diagrams for changing strain paths. Metall. Trans. A. 24, 2497; Graf, A., Hosford, W.F. 1993b. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. A. 24, 2503; Graf, A., Hosford, W.F. 1994. The influence of strain path changes on forming limit diagrams of Al 6111-T4. Int. J. Mech. Sci. 36, 897). This prediction capability provides a powerful tool in the design and optimization process of 3D sheet metal forming where strain path changes are inevitable.

Experimental and analytical necking studies of anisotropic sheet metals

Two approaches are detailed for the determination of forming limit diagrams (FLDs). The first one is experimental: a correlation technique has been developed for the displacement field measurement on a sheet. The second approach is theoretical: a necking criterion based on the load-instability and plane strain localisation assumption is proposed. It includes the possibility to introduce the modified Gurson–Tvergaard damage model taking account the Hill’s or others anisotropic yield surfaces and the influence of triaxial stress state in the neck. The criterion is written in an intrinsic form and is as well applicable to linear strain paths as well as to non-linear strain paths. Finally, the FLDs for an aluminium alloy, a mild steel and a titanium alloy obtained by experimental and theoretical ways are compared.

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.

Use of two behaviour laws for the determination of the forming-limit stress diagram of a thin steel sheet: results and comparisons

In this paper two behaviour laws used for the determination of the forming-limit stress curves of orthotropic steel sheets are compared. The first is the classical quadratic Hill’s criterion and the second is the 3G theory proposed by CRM of Liege, which assumes that the straining mechanism is described by sliding in the three planes showing maximum shear stresses, i.e. the planes at 45° from the principal stress directions in the isotropic case. Both of the theories enable the determination of the forming-limit stress states by means of a step-by-step plastic calculation along the strain paths determined in an experimental way on laboratory-drawn parts. For the two cases it is shown that the forming-limit stress curves are independent of the shape of the strain path. The interest of the forming-limit stress curve is to be a very good test stop enabling the prediction of the onset of necking in the numerical simulation of industrial drawing operations showing non-linear strain paths.

Prediction of localized thinning in sheet metal using a general anisotropic yield criterion

The forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metal. The ability to accurately predict the FLD for a given material has been shown to depend on the shape of the selected yield function. In addition, both experimental and numerical results have shown that the level of the FLD is strongly strain path dependent. In this work, a combination of Marciniak–Kuczynski (M–K) analysis and a general anisotropic yield criterion developed by Karafillis and Boyce (Karafillis, A.P., Boyce, M.C., 1993. A general anisotropic yield criterion using bounds and transformation weighting tensor. J. Mech. Phys. Solids 41, 1859) is used to predict localized thinning of sheet metal alloys for linear and nonlinear strain paths. A new method for determining the constants in the yield criterion is proposed. The optimal values are obtained by fitting the initial yield stresses and calculated FLD under linear strain paths with the experimental measurement. Using this approach, accurate yield functions can be defined for both Al2008-T4 and Al6111-T4. Comparisons of computed FLDs with the experimental data of Graf and Hosford (Graf, A., Hosford, W.F., 1993b. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. A. 24, 2503; Graf, A., Hosford, W.F., 1994. The influence of strain path changes on forming limit diagrams of Al 6111-T4. Int. J. Mech. Sci. 36, 897) show good agreements.

Analytical and experimental studies of necking in sheet metal forming processes

Abstract Two approaches are detailed for the determination of forming limit diagrams (FLD). The first one is experimental: a correlation technique has been developed for the displacement field measurement on a sheet. The second approach is theoretical: a necking criterion based on the load-instability and plane strain localization assumptions is proposed. It includes the possibility to introduce the modified Gurson–Tvergaard damage model taking account Hill's or Barlat and Lian's anisotropy. The criterion is written in an intrinsic form and is applicable to linear strain paths as well as to non-linear strain paths. Finally, the FLD for three aluminum sheets and a mild-steel sheet obtained by experimental and theoretical ways are compared.

Nonlinear subincremental method for determination of elastic–plastic–creep behaviour

AbstractThe frequently used subincremental method has so far been based on a linear interpolation of the total strain path within each main step. This method has proven successful when elastic–plastic behaviour and secondary creep is involved. The present paper proposes a nonlinear subincremental method applicable to general elastic–plastic–creep behaviour including problems with a highly nonlinear total strain path caused by the occurrence of creep hardening. This nonlinear method degenerates to the linear‐approach for elastic–plastic behaviour and when secondary creep is present. It is also linear during step loadings and it becomes increasingly more nonlinear, the more creep hardening deformations dominate the behaviour. A wide range of structures are analysed and the results from both subincremental methods are compared; the nonlinear strategy increases the accuracy by a factor of typically 3–5 without affecting the computer time. Moreover, the implementation of the nonlinear method is extremely simple. The optimum number of substeps in each main step is found to be around 5. For such a choice, the advantage of using the subincremental method as compared to the more conventional solution technique, where only one type of time step is used, is a significant reduction in computer time without, in practice, affecting the accuracy.

Prediction of the forming limit diagrams of anisotropic sheets in linear and non-linear loading

The formability of sheet metals is often evaluated from strain analysis using the concept of forming limit diagrams (FLDs). Numerous experimental results show that the formability of the material strongly depends on the loading history and on the plastic anisotropy of the rolled sheet. The limiting strains at the onset of localized necking can be predicted from a plastic instability analysis introduced by Marciniak and coworkers. However, the hypothesis of proportional loading is no longer valid for complex industrial stampings. Multistage forming operations often involve non-linear strain paths, and abrupt changes in the strain ratio can be observed. In the present work the occurrence of plastic instabilities in the general case of a rate-sensitive anisotropic material with orthotropic symmetry under non-proportional loading is analysed. Theoretical FLDs are determined for proportional loading, for a sequence of two proportional loadings, for a sequence of two proportional loadings and for non-proportional loading (curved strain paths). The simulations are carried out for a wide range of loading conditions, from uniaxial tension in the rolling direction to uniaxial tension in the transverse direction through biaxial stretching. The influence of the anisotropic behaviour of the sheet is studied using Hill's theory of plastic anisotropy for both linear and non-linear strain paths in terms of the anisotropic coefficient measured during uniaxial tensile stretching in three different directions referred to the rolling direction (r o, r 45 and r 90). A good correlation is obtained between the theoretically and experimentally determined FLDs, and it is shown that important increases in formability may be achieved through careful specification of strain paths, blank-holder pressure and anisotropic blank orientation.