Search

Dual phase (DP) steel sheets are increasingly being used for automotive applications due to relatively high strength and formability. In the current work, Marciniak formability tests were carried out to determine the forming limit curve (FLC) of DP600 steel sheet, and an extended version of Rousselier’s ductile damage model, which accounts for void nucleation, growth and coalescence was used to simulate the tests and predict strain localization and failure for three different strain paths: uniaxial tension, plane strain and biaxial tension. In addition, a combination of flat rolling and uniaxial tension tests were used to generate the extended flow curve of the material. Damage evolution in terms of Rousselier scalar damage variable and void volume fraction was assessed for each simulation condition. The FLC as well as neck and fracture morphologies and geometries were obtained from finite element simulations of the Marciniak tests and compared to experimental results. The sensitivity and dependency of the predicted necking limits, damage distribution and geometry predicted by the Rousselier damage model to the type of hardening model, strain path, void nucleation function and void coalescence criterion are discussed. The modified Rousselier model was shown to successfully predict the FLC, damage distribution and the final damage geometry of DP600 sheets.

There is a steady increase in the implementation of dual phase steels in stamped automotive components. Therefore, steel suppliers who develop dual phase steels are interested in predicting the microstructure-properties relationship for optimization of microstructural design. This goal is achievable by micromechanical modeling. The representative volume element (RVE) method has been a popular technique for micromechanical modeling of dual phase steels. It is generally considered that 2D modeling underestimates the flow curves and that 3D modeling predicts the experimental stress-strain curves more accurately. However, much of the research has focused on 2D modeling. This paper develops 3D micromechanical modeling of DP500 and bainite-aided DP600 steels by including statistical quantitative metallography data in the models. More than 3000 grains were analyzed in each steel. Hence, both volume fraction and morphology of martensite were statistically determined. This model predicted the ultimate tensile strength of these two dual phase steels with less than 0.5% error.

In this study, the tensile flow behaviour of AA5182-O sheet was experimentally obtained in different material directions (RD, DD, and TD) at strain rates ranging from 0.001 to 1000 s−1 and predicted by means of both phenomenological models and neural networks (NNs). Constants in Johnson–Cook (JC), Khan–Huang–Liang (KHL), and modified Voce were calculated using genetic algorithm (GA) and linear regression analysis and used to simulate the uniaxial tension tests. Two types of feed-forward back-propagation neural networks were also trained and validated to predict the rheological behaviour of the alloy without the limitations of a mathematical function. The weights and bias values of each network were then used to simulate uniaxial tensile deformation. Subsequently, the results were compared with experimental flow curves and accuracy parameters were calculated. It was found that the modified Voce constitutive equation was able to predict the flow behaviour of AA5182-O with better accuracy than JC and KHL models. Also, the NN was found to be the most accurate method of predicting the anisotropic rate-dependant behaviour of AA5182-O. Keywords: Phenomenological constitutive model, Genetic algorithm, Artificial neural network, 5000 series aluminium alloys, Moderate strain rates

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. Keywords: Forming limit curve, Formability, Yield criterion, MK analysis, Sheet metal, Loading path

The stress–strain response of some materials, such as advanced high strength steels, during unloading is nonlinear after the material has been loaded into the plastic deformation region. Upon reloading, the response shows a nonlinear elastic response that is different from that in unloading. Therefore, unloading–reloading of these materials forms a hysteresis loop in the elastic region. The Quasi-plastic–elastic model (Sun and Wagoner, 2011) was modified and combined with both isotropic-nonlinear kinematic hardening and two-surface plasticity models to simultaneously describe the nonlinear unloading response and complex cyclic response of sheet metals in the plastic region. The model was implemented as user-defined material subroutines, i.e. UMAT and VUMAT, for ABAQUS/Standard and ABAQUS/Explicit finite element codes, respectively. Uniaxial loading-unloading tests were performed on three common grades of automotive sheet steel: DP600, DP980 and TRIP780 steel. The model was verified by comparing the predicted material response with the corresponding experimental response. Finally, the model was used to predict the springback of a U-shape channel section formed in a plane-strain channel draw process. The results showed that the model was able to considerably improve springback predictions compared to the usual assumption of linear elastic unloading. Keywords: Plasticity, Constitutive model, Nonlinear elastic unloading, Springback

A material model for more thorough analysis of plastic deformation of sheet materials is presented in this paper. This model considers the following aspects of plastic deformation behavior of sheet materials: (1) the anisotropy in yield stresses and in work hardening by using Hill’s 1948 quadratic yield function and non-constant stress ratios which leads to different flow stress hardening in different directions, (2) the anisotropy in plastic strains by using a quadratic plastic potential function and non-associated flow rule, also based on Hill’s 1948 model and r -values, and (3) the cyclic hardening phenomena such as the Bauschinger effect, permanent softening and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening model. Plasticity fundamentals of the model were derived in a general framework and the model calibration procedure was presented for the plasticity formulations. Also, a generic numerical stress integration procedure was developed based on backward-Euler method, so-called multi-stage return mapping algorithm. The model was implemented in the framework of the finite element method to evaluate the simulation results of sheet metal forming processes. Different aspects of the model were verified for two sheet metals, namely DP600 steel and AA6022 aluminum alloy. Results show that the new model is able to accurately predict the sheet material behavior for both anisotropic hardening and cyclic hardening conditions. The drawing of channel sections and the subsequent springback were also simulated with this model for different drawbead configurations. Simulation results show that the current non-associated anisotropic hardening model is able to accurately predict the sidewall curl in the drawn channel sections.

Electrohydraulic forming (EHF) is a high energy rate forming process in which the strain rate in the sheet metal can vary from 5 × 10 2 to 10 5 s −1 depending on various factors. Several mechanisms have been reported to cause an improvement in formability in EHF such as material deformation mechanisms, inertial effects and the dynamic impact of the sheet against the die. EHF is a complex high speed forming process and experimental work alone is not sufficient to properly understand this process. To understand the variation of some influential variables in EHF, electrohydraulic die-forming (EHDF) and free-forming (EHFF) of DP590 dual phase steel were simulated in ABAQUS/Explicit by considering the fluid/structure interactions. Three-dimensional finite element simulations were conducted by modelling the water with Eulerian elements with a view to investigating the effect of released energy on the sheet deformation profile history, strain distribution, loading path and damage accumulation type. The Johnson–Cook constitutive material model was used to predict the sheet behaviour and the parameters in this model were calibrated based on experimental test results available for DP590 at various strain rates. The Johnson–Cook phenomenological damage model was also used to predict the ductile failure (damage accumulation) in both EHDF and EHFF. Predicted final strain values and damage accumulation type showed good agreement with the experimental observations.

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.

Simple shear, uniaxial tension–compression and bending tests were used to determine the cyclic behaviour of two sheet metals: DP600 and AKDQ. The Yoshida–Uemori two-surface model along with Hill’s quadratic yield function was used to simulate the behaviour of these two materials in each test. For each test, a set of material constants was identified such that the error between the simulated and experimental responses is minimized. Using the material constants obtained from one test, the other tests were simulated to see whether the set of constants obtained from this test is able to describe the material response in the other tests. The results show that depending on the material, the set of constants obtained from one test may or may not be able to reproduce the material response in the other tests. Finally, each set of constants was used to simulate the springback of a U-shaped part formed in a channel draw process. The predicted springback profiles obtained from each set of constants were compared with the experimental profile. It was found that all three tests are suitable to characterize the behaviour of DP600 sheets in view of predicting the springback of channel sections. For AKDQ, however, the error between the predicted and experimental springback profiles was significant regardless of the type of characterization test performed. But for this channel draw process, simulations based on material data obtained from the reverse bending test provided the best prediction of springback.

A series of samples from four automotive materials - AKDQ, HSLA50, DP600, and AA6022-T43 - were deformed in a channel draw processes with different levels of draw bead penetration. As a result, varying magnitudes of deformations in plane strain mode and residual stresses were obtained. Through-thickness stress profiles were obtained non-destructively using a novel, high resolution X-ray diffraction technique.

Hydropiercing is an efficient way of piercing holes in mass produced hydroformed parts with complex geometries. By driving piercing punches radially into a hydroformed and fully pressurized tube, holes will be pierced and extruded into the tube-wall. Recent experimental studies have shown that the formability of advanced high strength steel (AHSS) tubes can be increased with the application of internal pressure. In this study, three-dimensional finite element simulations of a tube hydropiercing process of a dual phase steel (DP600) were performed in LS-DYNA, using phenomenological, micromechanical and combined damage criteria. Damage was included in the numerical analysis by applying constant equivalent plastic strain (CEPS), the Gurson-Tvergaard-Needleman (GTN), and the Extended GTN (GTN+JC) model. In order to calibrate the parameters in each model, a specialized hole-piercing fixture was designed and piercing tests were carried out on non-pressurized tube specimens. Of the various ductile fracture criteria, the results predicted with the GTN+JC model, such as the punch load-displacement, the roll-over depth, and the quality of the clearance zone correlated the best with the experimental data.

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).

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.

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948) . The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003) . The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989) . Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.

Pre-bending is a critical process required prior to hydroforming. The bending has an effect on the tube thickness and strain which will use up a portion of the formability of the as-received tube. To compensate for this loss of formability, a multi-objective optimization method was applied to improve the hydroforming process after pre-bending. A multi-objective genetic algorithm (MOGA) and Kriging surrogate model were used to optimize the loading path. The Kriging model was used to replace the finite element simulation in constraint handling. The optimal loading parameters in the hydroforming process were obtained for a tube that was previously bent 90°, and showed an improvement in reducing the corner radii of the part at the extrados and intrados of the bend (8.73 mm and 11.24 mm for the extrados and intrados of the bend, respectively). The corresponding corner fill expansion (CFE) was improved by 16.7% (or 1.79 mm) compared to the maximum expansion of 10.73 mm obtained experimentally.

The production of a tubular hydroformed part often requires a combination of internal pressure and axial force at the tube ends to fully form the tube to its specified geometry. A successful hydroforming process requires not only achieving a part that conforms to the design specifications, but also ensures that the part has a reasonably uniform thickness distribution and is free of defects, such as wrinkles, severe thinning, or fractures. The load path design (pressure vs. end feed history) largely determines the robustness of the process and the quality of the finished parts. In this paper, a hybrid constrained optimization method was proposed to solve this type of multi-objective problem by coupling a multi-objective genetic algorithm and a local search. The load path design procedure was developed by considering five objectives: four formability objectives (i.e., to minimize the risk of wrinkling, global and local thinning, and fracture) and a geometric objective (to minimize the corner radius). A Kriging predictor was used to accelerate the computation of genetic operations and generate new feasible solutions. Finite element simulations of the hydroforming process were also used after each generation to accurately evaluate the objectives of the offspring, and solutions with rank 1 were retained throughout all generations. Once the Pareto solutions were obtained by multi-objective genetic algorithm, a local search was carried out in the regions of interest with the assistance of visualization. This optimization method was applied to the hydroforming of a straight tube to create a part with an expanded region with a square cross section; the optimum load path produced a very safe part with a corner radius of only 9.115 mm and a maximum thinning of only 23.9%.

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ3), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (\( \beta = {{{{\sigma_3}}} \left/ {{{\sigma_1}}} \right.} \)). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain (\( \bar{\varepsilon } \geqslant 0.35 \) for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.

Experimental and numerical axial cutting of AA6061-T6 circular extrusions under both dynamic and quasi-static loading conditions were completed using single- and dual-cutter configurations to investigate load/displacement and collapse behaviour of the extrusions. Circular specimens with various wall thicknesses were considered for impact and quasi-static testing in this research. A steel cutter (AISI 4140) with four blades, having blade tip widths of 1.0 mm or 0.75 mm and blade lengths of 7 mm or 26.1 mm were used to cut through the extrusions. Straight and curved deflector profiles were used to flare the cut petalled sidewalls and facilitate the cutting system. Further quasi-static cutting tests using dual cutters were completed with or without the presence of a spacer to examine the load/displacement response as an adaptive energy absorption system. Results from the experimental impact tests illustrated that a higher peak cutting force, with a magnitude of approximately 1.09–1.98 times that of the force necessary under quasi-static testing conditions, was needed to initiate the cutting deformation mode. After this initial high force, the load/displacement responses were observed to be similar to those from the quasi-static tests with the exception of minor variations which resulted from material fracture that occurred on the petalled sidewalls during dynamic testing. Larger lengths of cutter blades and the curved deflector eased the flaring of the petalled sidewalls and reduced the occurrence of material fracture. The blade tip width had minor effects on the initial peak cutting force and mean cutting forces for extrusions under impact loading. The mean cutting force from the dynamic tests was determined to be 0.82–1.2 times that from the quasi-static experimental tests. Finally, quasi-static axial crushing of extrusions was completed to compare crashworthiness measures with the adaptive energy absorption system under the cutting deformation mode. A finite element model incorporating an Eulerian formulation was selected for the numerical model to simulate the cutting process. Simulation results generally agreed well with the experimental tests with a maximum over prediction of approximately 33% and 18% for the cutting force under impact and quasi-static loading, respectively.

A semi-implicit integration scheme was used to implement the Yoshida–Uemori two-surface model into the finite element method. Hill’s quadratic yield function was employed to account for the orthotropic behaviour of the metal sheet. The model was used to predict the cyclic simple shear response of DP-600. In order to evaluate the capability of the model for sheet metal forming, the model was used to simulate both the forming stage of a channel draw process in the presence of drawbeads and the subsequent springback stage. The results show that the model predicts the springback profile very well, especially at deeper drawbead penetrations.

An experimental investigation to determine the strain distribution and collapse behaviour for AA6061-T6 square cross-sectional extrusions with and without circular discontinuities under quasi-static axial compressive loading was completed. Three-dimensional digital image correlation (DIC) was utilized for strain assessment. In order to validate the results of the optical strain measurement system, tensile tests were first conducted employing both the DIC technique and a traditional extensometer. Strain observations from both methods were found to be very consistent prior to strain localization in the test specimen. Quasi-static axial crushing tests were then conducted. Extrusions considered for the present research had a nominal side width, wall thickness and length of 38.1 mm, 3.15 mm, and 200 mm, respectively. A centrally-located circular hole with diameter of either 14.29 mm, 10.72 mm or 7.14 mm was incorporated into the extrusion. Square tubes without any discontinuities were also considered in the experimental testing program. Testing results showed that the collapse mode of the extrusion altered from global bending to a cutting and splitting deformation mode with the presence of the circular discontinuity. Strain localization occurred near the vicinity of the holes for all specimens. For discontinuities sized 14.29 mm and 10.72 mm the location of strain localization and the initiation of material fracture was at the edge of the discontinuity while the location for extrusions with a 7.14 mm hole was found to occur at the intersection of the extrusion side walls. Maximum values of the effective strain were found to vary from approximately 60% to 100%. The region of strain localization was consistent with the location where material fracture initiated.

Tube hydroforming is a manufacturing process used to produce structural components in cars and trucks, and the success of this process largely depends on the careful control of parameters such as internal pressure and end-feed force. The objective of this work was to establish a methodology, and demonstrate its effectiveness, to determine the optimal process parameters for a tube hydroformed in a die with a square cross section. The Taguchi method was used to establish a design of virtual hydroforming experiments, and numerical simulations were carried out with the finite element code LS-DYNA®. A sensitivity analysis was also carried out with analysis of variance. Multi-objective functions that consider necking/fracture, wrinkling, and thinning were formulated, and the response surface methodology was used with the most sensitive factors to obtain a defect-free part. An objective function, based on the final corner radius in the part, was also included in the optimization model. The forming severity of virtual hydroformed parts was evaluated using the forming limit stress diagram and the forming limit (strain) diagram. Finally, the normal-boundary intersection method and the L2 norm were used to obtain the Pareto-optimal solution set and the optimal solution within this set, respectively. The hydroforming process for this part was also optimized using the commercial optimization software LS-OPT®, with two different single-objective algorithms. However, the optimum load path predicted with the proposed methodology was shown to achieve a smaller corner radius. The proposed optimization technique helped to define a process window that leads to a robust manufacturing process and improved part quality.

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.

The objective of this work is to predict the springback of Numisheet’05 Benchmark#3 with different material models using the commercial finite element code ABAQUS. This Benchmark consisted of drawing straight channel sections using different sheet materials and four different drawbead penetrations. Numerical simulations were performed using Hill's 1948 anisotropic yield function and two types of hardening models: isotropic hardening (IH) and combined isotropic-nonlinear kinematic hardening (NKH). A user-defined material subroutine was developed based on Hill's quadratic yield function and mixed isotropic-nonlinear kinematic hardening models for both ABAQUS-Explicit (VUMAT) and ABAQUS-Standard (UMAT). The work hardening behavior of the AA6022-T43 aluminum alloy was described with the Voce model and that of the DP600, HSLA and AKDQ steels with Hollomon's power law. Kinematic hardening was modeled using the Armstrong–Frederick nonlinear kinematic hardening model with the purpose of accounting for cyclic deformation phenomena such as the Bauschinger effect and yield stress saturation which are important for springback prediction. The effect of drawbead penetration or restraining force on the springback has also been studied. Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior. Comparisons between simulation results and experimental data showed that the IH model generally overestimated the predicted amount of springback due to higher stresses derived by this model. On the other hand, the NKH model was able to predict the springback significantly more accurately than the IH model.

A biaxial testing apparatus was used to investigate the elastic-plastic behaviour of an 1145 aluminum sheet alloy. Flat cruciform specimens were deformed up to effective strains of approximately 0.15 in biaxial stretching, along seven different proportional strain paths. A finite element analysis of each test was carried out using four different phenomenological models of anisotropic plasticity. An iterative procedure was coupled with the numerical analyses in order to determine the anisotropic parameters in the various yield functions and thus obtain best fits to these functions. This method of numerically analyzing cruciform specimens leads to fairly accurate biaxial flow curves, as well as plastic work contours. The results are also compared with crystal plasticity simulations using a Taylor-type model.

A material model for more effective analysis of plastic deformation of sheet materials is presented in this paper. The model is capable of considering the following aspects of plastic deformation behavior of sheet materials: the anisotropy in yielding stresses in different directions by using a quadratic yield function (based on Hill’s 1948 model and stress ratios), the anisotropy in work hardening by introducing non-constant flow stress hardening in different directions, the anisotropy in plastic strains in different directions by using a quadratic plastic potential function and non-associated flow rule (based on Hill’s 1948 model and plastic strain ratios, r-values), and finally some of the cyclic hardening phenomena such as Bauschinger’s effect and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening (so-called Armstrong-Frederick-Chaboche model). Basic fundamentals of the plasticity of the model are presented in a general framework. Then, the model adjustment procedure is derived for the plasticity formulations. Also, a generic numerical stress integration procedure is developed based on backward-Euler method (so-called multi-stage return mapping algorithm). Different aspects of the model are verified for DP600 steel sheet. Results show that the new model is able to predict the sheet material behavior in both anisotropic hardening and cyclic hardening regimes more accurately. By featuring the above-mentioned facts in the presented constitutive model, it is expected that more accurate results can be obtained by implementing this model in computational simulations of sheet material forming processes. For instance, more precise results of springback prediction of the parts formed from highly anisotropic hardened materials or that of determining the forming limit diagrams is highly expected by using the developed material model.

Electrohydraulic Forming (EHF) is a high velocity forming process, in which the strain-rate in the sheet metal can reach very high values depending on the prescribed input energy, the chamber geometry, the die geometry, instrumentation efficiency and the mechanical properties of the sheet material. In EHF, a high voltage discharge between electrodes that are submerged in a water-filled chamber generates a plasma channel that leads to propagation of a shockwave through the water that forms the sheet, with or without a die, in less than a millisecond. EHF generates a complex pressure pulse history that is extremely challenging to simulate. In this work, three-dimensional finite element simulations of DP590 sheet were completed in free-forming (EHFF) and die-forming (EHDF) conditions using ABAQUS/Explicit and a combination of Eulerian and Lagrangian elements. The Johnson-Cook constitutive plasticity model was selected and the parameters were calibrated based on uniaxial tensile test data at different strain-rates. A comprehensive numerical study was carried out with a view to understanding the differences between EHFF and EHDF in terms of the history of the deformation profile of the specimen, the strain-rate history, the loading path and through-thickness stresses. Higher strain-rates and more complex strain-paths were predicted in EHDF compared to EHFF due to dynamic sheet/die interaction. Good correlation between the experimental and numerical results demonstrated the ability of numerical models to accurately predict the history of the deformation profile in both EHDF and EHFF conditions.

Another factor that has a very significant effect on sheet formability is the loading path. Indeed, non-linear strain paths cause the forming limit curve (FLC) to translate in principal strain space and therefore the actual FLC for a given location on a part may be quite different from the as-received FLC. Consequently, the virtual design of a forming process will only be reliable if the forming limits used to assess the forming severity of the part were determined with due consideration of the (non-linear) strain paths in each critical location of the part. The FLC of sheet metals have often been predicted using the MK method which was proposed by Marciniak and Kuczynski (1967) and which models the inhomogeneity of the sheet material by a very shallow groove. Predictions of theoretical FLC using the MK method generally assume plane stress conditions, but in this work the MK analysis was modified to consider both the nonlinearity of the loading path and the presence of a through-thickness stress, simultaneously. Different magnitudes of prestrain were applied to a virtual sheet so as to simulate bi-linear strain paths and different values of the normal stress were applied for the prediction of the FLC. Published experimental FLC data obtained for different bilinear loading conditions were used to validate the theoretical work in both plane stress and general three-dimensional stress conditions. It was found that this modified MK model predicts FLC that correlate well with the limited experimental data.

Interferometrically measuring the index of refraction variation (index homogeneity) of glass blanks requires that the blanks be made transparent to the interferometer laser. One method for achieving this is to 'sandwich' a rough ground blank between two polished flats while adding an index matching liquid at each surface interface. This is better known as oil-on-flat (OOF) or oil-on-plate testing. Another method requires polishing both surfaces and is better known as polished homogeneity (PHOM) testing or the Schwider method. Corning Inc. historically has used OOF testing to measure the index homogeneity of disk-shaped, fused silica boules over multiple 18' diameter apertures. Recently a boule polishing and PHOM testing process was developed by Corning for measuring the homogeneity over 24' diameter apertures to support fused silica production for the National Ignition Facility (NIF). Consequently, the PHOM technique has been compared to the OOF process using a number of different methods including repeatability/reproducibility studies, data stitching, and vibration analysis. The analysis performed demonstrates PHOM's advantages over OOF testing.

Due to an increasing usage of dual phase (DP) steels in the automotive industry, it is necessary to fully investigate the mechanical response of DP steels during sheet forming processes or crash events. In this work, the mechanical response of DP600 obtained at different strain rates was used to calibrate the Yld2004-18P yield function at each level of strain rate, and the effect of strain rate on the flow surface was evaluated. Results showed that variation in the material anisotropy coefficients as a function of strain rate can alter the shape of the flow surface of DP600. The current modelling approach is shown to provide a frame work for the incorporation of strain rate-dependence of anisotropy within finite element simulations of sheet metal forming and crashworthiness.