Nonlinear Loading Paths Research Articles

Investigation and Compensation of Biaxial Pre-strain During the Standard Nakajima- and Marciniak-test Using Generalized Forming Limit Concept

One of the most important tasks in sheet metal forming is the prediction of formability. The common criterion for ductile fracture in industrial application is the evaluation of the so-called Forming Limit Curve (FLC). The determination of FLCs is standardized in the international standard 12004-2 by two different experimental methods, the Marciniak- and the Nakajima-test. Several effects, for example bending or contact pressure, are leading to biaxial pre-forming and to a slightly non-linear load path. This results in inaccuracies of the FLC. The so-called Generalized Forming Limit Concept (GFLC) is a phenomenological approach, which was introduced to predict the local necking of non-linear load paths. The accuracy and robustness of the GFLC has already been shown in several papers. In this investigation the GFLC was used to compensate the biaxial pre-forming and to calculate a linear FLC. Input parameters are the Marciniak- and Nakajima-FLCs of the material HC340LA.

A Combined Isotropic, Kinematic and Cross Hardening Model for Magnesium AZ31B-H24 under Non-linear Strain Loading Path

A fully modularized framework was established to combine isotropic, kinematic, and cross hardening behaviors under non-monotonic loading conditions. Three sets of state variables were defined and applied to consider the effects of, a) loading history, b) twinning and de-twinning and c) different pre-strain. Experiments under two types of non-proportional loading conditions were conducted along different orientations, 1) uniaxial compression-tension reversal loading with different amounts of compressive strains, and 2) two-step uniaxial tension, known as cross-loading conditions, with different pre-strains. No apparent cross-hardening effect was observed for this material. The calibrated new hardening model, with an anisotropic CPB06ex2 yield criterion and an eMMC anisotropic fracture model, has been implemented into Abaqus/ Explicit as a user material subroutine (VUMAT). Good correlation was observed between experimental and simulation results.

Prediction of forming limit curves for nonlinear loading paths using quadratic and non-quadratic yield criteria and variable imperfection factor

Industrial sheet metal forming processes often involve complex deformation modes and it is necessary to consider nonlinear loading path effects when predicting forming limit curves. Moreover, the yield criterion plays a critical role in the accuracy of predicted forming limits.In this work the MK analysis was modified to relate the initial imperfection factor to a physical property such as the surface roughness, and the orientation of the imperfection was also allowed to vary. This model was used to predict the strain-based and stress-based forming limit curves (FLC and SFLC) of sheet materials that are subject to either linear or non-linear strain paths.Two different yield criteria were employed in this study, Hill's 1948 quadratic yield criterion and Hosford's 1979 non-quadratic yield criterion. The theoretical model was validated by comparing predicted FLC and experimental FLC curves obtained from the literature. FLCs and SFLCs predicted with these two yield criteria were compared for both linear and nonlinear loading paths.Results showed that both the quadratic and non-quadratic yield criteria predict the FLC with acceptable accuracy however on the whole the non-quadratic yield criterion generally provides a slightly better correlation with experimental data, especially on the right side of the FLC.

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.

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).

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.

Prediction of sheet forming limits with Marciniak and Kuczynski analysis using combined isotropic–nonlinear kinematic hardening

The forming limit curve (FLC), a plot of the limiting principal surface strains that can be sustained by sheet metals prior to the onset of localized necking, is useful for characterizing the formability of sheet metal and assessing the forming severity of a drawing or stamping process. Both experimental and theoretical work reported in the literature has shown that the FLC is significantly strain-path dependent. In this paper, a modified Marciniak and Kuczynski (MK) approach was used to compute the FLC in conjunction with two different work-hardening models: an isotropic hardening model and a mixed isotropic–nonlinear kinematic hardening model, which is capable of describing the Bauschinger effect. Predictions of the FLC using the MK analysis have been shown to be dependent on the shape of the initial yield locus and on its evolution during work hardening; therefore the hardening model has an influence on the predicted FLC. In this investigation, published experimental FLCs of AISI-1012 low carbon steel and 2008-T4 aluminum alloy sheets that were subjected to various nonlinear loading paths were compared to predictions using both hardening models. The predicted FLCs were found to correlate quite well with experimental data and the effects of strain path changes and of the hardening model on predicted FLCs are discussed.

Study of Forming Parameters in Hydroforming of a Thin-Walled ASTM C11000 Copper Tube

Tube hydroforming technology is still considered a new technique growing fast in automotive and aircraft industries. Many researches on all aspects of this process are still required. Contact friction is one of the most effective parameters on tube wall thinning. To successfully fulfill the process without any common defects, it is very important to determine the proper internal pressure and axial feeding loading paths. In this paper, the effect of lubrication on tube wall thinning on ASTM C11000 copper alloy is discussed as well as the effect of internal pressure and axial feeding. An axisymmetric bulged tube is investigated using theoretical, numerical and experimental methods. Improved linear and non-linear pressure and feeding loading paths are applied and the predicted results are experimentally proved. It is observed that non-linear pressure application gives smoother results. Also proper lubrication plays an important role in success of the process.

Analytical solutions for inherently incremental, similar elastic contact problems with bulk stress

Analytical solutions are obtained for a class of inherently incremental, similar elastic half-plane frictional contact problems which involve simultaneous application of a changing, remote bulk-stress during oblique loading. The method is based on Abel Integral Equations. The spatial gradient of the slip function is obtained uniquely by the conditions determining the existence of a bounded solution to the shear traction Cauchy SIE and global shear equilibrium, provided there is stick at all points in the contact throughout the loading. Solutions are obtained for both the cylindrical indenter and the flat punch with rounded edges. The method also lends itself easily to generalization to non-linear load paths and a number of such problems are considered. Problems involving the development of slip zones within the contact may be solved in some special cases.

Solution to the Incorrect Benchmark Shell-Buckling Problem

A benchmark, geometrically nonlinear, shell-buckling problem is reviewed and the correct solution is identified and discussed vis-a-vis the incorrect solution from the open literature. The problem is that of a hinged, thin, isotropic cylindrical shell section, point-loaded transversely at the center, undergoing large deformations (5 x thickness). This benchmark problem and solution were introduced in 1972 and have been reproduced in at least 30 journal articles and books in the more than three decades since that time. The benchmark problem is of interest because it demonstrates many features possible in shell buckling, including a load-limit point (snap-through buckling) and a deflection-limit point (snap-down, or snap-back). The benchmark problem is typically used to demonstrate the capability of commercial/private finite element codes to traverse such complicated nonlinear load paths. The existing incorrect benchmark solution involves the nonlinear growth of only symmetric deformation modes. The correct solution to this benchmark problem involves bifurcation from the nonlinear load path into a simple asymmetric deformation mode. Experimental data do not exist for verification of the calculated benchmark response, and so a similar problem using a laminated composite shell is suggested as an alternative benchmark problem for shell stability analysis.

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.

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.

H- and d-adaptive FE methods for two-dimensional structural problems including post-buckling of shells

The paper first treats the locking problem of shear-elastic plates in bending in the context of multi-grid-methods. A new plate bending element with shape functions of the Allman-type is presented and used for linear and non-linear problems. An important issue is the presentation of a flexible mesh generator with different refinement strategies according to a priori criteria. Different a posteriori criteria for adaptive approximations are given, such as a gradient-based error with one and two indicators for geometrically non-linear load paths, the first being residuum based and the second controlling the geometrical nonlinearities. Branch switching and post-buckling deformation processes are investigated adaptively using these indicators and demonstrated with examples. Another important topic is the automatic dimensional adaptivity of beams and plates within disturbed zones as boundary layers, concentrated load areas and sharp changes of thicknesses This d-adaptivity is combined with h-adaptivity and illustrated by some examples.