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In this study, a novel analytical method for predicting bending and springback behaviours of hexagonal close-packed (HCP) sheet metals is presented. The proposed analytical approach is developed by using the Cazacu-Barlat 2004 asymmetric yield function and isotropic plastic hardening rule. This model can be used to determine bending moment-curvature relationships and springback of HCP metals under uniaxial and plane strain loading conditions. Furthermore, to capture the nonlinearity in unloading and to improve springback prediction, the variable elastic modulus approach is implemented in the proposed model. The proposed new model reveals that reverse effects of the back force on springback behaviours cannot be found under the plane strain condition, which could not be found by using any existing models. Moreover, the analytical model is implemented into Abaqus via UMAT subroutine for its application in complex cases, and a numerical model is then developed as a showcase. The proposed methods are validated by using those experimental results available in literature. The results show considerable improvements by considering the plane strain condition and nonlinear unloading.

This study uses the Marciniak and Kuczynski (M-K) method to present an analytical forming limit diagram (FLD) for sheet metals. The procedure for the analytical FLD prediction is described in detail and step-wise manner, and an algorithm is written using MATLAB. First, an appropriate algorithm is determined to establish the theoretical analyses, and various anisotropic yield functions, such as Hill’s 48, Barlat 89, and Hosford, are considered. The predicted FLDs are compared with experiments involving a typical AA6016-T4 aluminum alloy. Second, the Gurson model that considers damage growth is implemented when Hosford is the yield function, as Hosford criterion predicts the best comparable analytical FLD with experiments among the yield functions. Third, a parametric study is performed to investigate the effects of parameters on the FLD prediction. Results indicate that an extremely low value for the initial void volume fraction in the safe and groove zones has minimal effects on the FLD prediction. Lastly, the values of void volume fractions are calculated assuming no geometrical imperfections and the imperfection is because of higher void volume fraction in groove zone than that in safe zone.

The forming limit curve (FLC) of zirconium alloys plays a significant role in the fabrications of spacer grid used for nuclear fuel assembly. The theoretical prediction of FLC has provided a convenient method to calculate the strain limit during the sheet forming process, but the prediction accuracy of zirconium alloy is still unsatisfactory due to its close-packed-hexagonal (HCP) structure. In this paper, to find a suitable yield surface exponent of zirconium alloys for improving the prediction accuracy, the theoretical FLCs of SZA6 zirconium alloy were calculated by self-programmed numerical codes with different exponent values and the results were compared with the experiments. To explain the mechanism of how the yield surface exponent affects the plastic behavior of SZA6 zirconium alloy, the strain distributions and element strain paths of Nakajima test were analyzed by finite element methods. At last, a suggested value of the yield surface exponent suitable for the FLC prediction of this zirconium alloy is acquired.

In the last decade, several phenomenological yield criteria for anisotropic material has been proposed to improve the modeling predictions about sheet metal-forming processes. In regard to this engineering application, two proprieties of models have been used. If the yield function and the plastic potential are not same (not equal), the normality rule is non associative flow rule (NAFR), otherwise, when the stresses yield has been completely coupled to the anisotropic strain rate ratio (plastic potential), is called the associated flow rule (AFR). The non-associated flow rule is largely adopted to predict a plastic behavior for metal forming, accurately about à strong mechanical anisotropy presents in sheet metal forming processes. However, various studies described the limits of the AFR concept in dealing with highly anisotropic materials. In this study, the quadratic Hill1948 yield criteria is considered to predict mechanical behavior under AFR and NAFR approach. Experiment and modeling predictions behaviour of normalized anisotropic coefficient r (θ) and σ (θ) evolved with θ in sheet plane. and the equibiaxial yield stress σb was assumed σb=1 but the rb-values was computed from Yld96 [15].

The rutting resistance design of asphalt pavement is full of compromise between technology and economy because it is difficult to predict rutting evolution and distribution. In order to investigate...

Motivated by experimental findings on sheet-metal forming, this article concerns the modeling of evolving anisotropies in finite plasticity. A covariant formulation of plasticity is employed in conjunction with evolution equations for the structural tensors that characterize the symmetry group of the yield function. A specific model is implemented into a finite element code to simulate tension and torsion tests. The results are then compared to experiments.

This paper presents a new Modified Cam Clay (MCC) type yield function, that is designed for robust and efficient use with implicit stress integration algorithms. The proposed yield function attains non-elliptical (e.g. tear and bullet) shapes, as well as the typical elliptical shape of the MCC model. Like that of MCC, and unlike most other yield functions with non-elliptical shapes available in literature, it is non-singular and unique throughout stress space. The experimental yielding stresses of a wide range of geomaterials have been accurately simulated using the yield surface. The yield function can be used in constitutive models based on classical elasto-plasticity theory.

Compacted bentonite is often considered as a possible buffer/backfill material for deep geological disposal of high-level radioactive waste. During the long-term lifespan, the bentonite barriers will undergo coupled thermo-hydro-mechanical (THM) progresses, which induce complex volume changes. In this work, the volume change behavior of compacted bentonites undergoing THM interactions was analyzed based on the framework of Barcelona Expansive Model, in which the thermal volume changes were taken into account by adopting the approach of Tang and Cui (Geotechnique 59:185–195, 2009). After analyzing the published experimental results of wetting/drying and heating/cooling cycles, new fI and fD functions were put forward for describing the interaction between micro- and macro-structures, while a thermal loading yield function curve was established for describing the TM behavior under unsaturated conditions. Some cyclic wetting/drying and thermal loading tests reported and modelled in literature were simulated to verify the relevance of the improvements undertaken. Comparisons among the original model, the modified model as well as the test results clearly showed a satisfactory performance of the model in describing the volume changes of compacted swelling clays upon THM processes.

Experiments are designed to experimentally characterize the plastic behaviour and fracture limits of an QP1180 steel sheet under complex loading conditions. The plastic behaviour is modelled by the Drucker yield function, while the fracture behaviour is illustrated the DF2016 [1] fracture criterion. The pressure-coupled Drucker yield function is calibrated by inverse engineering approach. The predicted load-stroke curves are compared with experimental results, which shows that the prediction matches with the experimental results with good agreement. Then the fracture data are then extracted from experiments and numerical simulations to calibrated two fracture criteria. The predicted fracture loci are compared with experimental results. The comparison demonstrates that both fracture criterion matches with the experimental results very well. Therefore, the Drucker function together with the Swift-Voce hardening law is recommended to model plastic deformation up to large plastic strain for various loading conditions. The fracture behaviour from shear to plane strain tension is recommended to be modelled by the DF2016 criterion and the pressure-coupled Drucker functions of sheet metal forming processes.

Directional anisotropic hardening under non-proportional loading is modeled. A recently proposed coupled quadratic-nonquadratic (CQN) yield function (Lee et al., 2017b) successfully captured the directional hardening behavior under the proportional loadings. This paper shows a constitutive equation to capture both directional hardening response and Bauschinger effect through an extended model of the CQN yield function with the homogeneous anisotropic hardening (HAH) model (Barlat et al., 2011). In the present study, the role of the CQN yield model is to capture the directional hardening behavior, and the HAH model is to follow non-proportional loading including the Bauschinger effect. The validation, with some material data, shows that the present model can follow the directional hardening under non-proportional loading in the stress-strain data. It is also shown that the present model can describe the yield surface distortion with Bauschinger and anisotropic hardening response. Finally, it is compared with other pre-existing models in order to analyze the utility of the proposed model.

In the automotive engineering community, the modified maximum force criterion proposed by Hora and co-workers is considered acceptable for theoretically estimating the forming limit curve (FLC) of sheet metals. Based on this criterion, a graphical method is proposed to simplify evaluation of the FLC. This paper investigates the proposed graphical method in a special case by using a power-hardening law and the Mises yield function, which leads to an explicit expression of the critical strains according the strain paths. The FLC of a DP590 sheet estimated using the proposed method is compared with that estimated using existing analytical methods. The proposed method provides the best prediction of the FLC of the tested material. For verification, the calculated FLC is then adopted into the finite-element method to predict the punch stroke at fracture of several notched specimens subjected to Nakazima tests. The good agreement between the predicted and experimentally determined values of the punch strokes verifies the ability and potential of the proposed method in industrial engineering.

Extended linear Drucker–Prager (DP) model has been a proper choice to capture the stress-strain of cylindrical concrete columns confined by fiber reinforced polymers (FRPs). However, most existing DP models calibrated by uniform active/passive tests are unsuitable for concrete under non-uniform passive confinement. This paper aims to calibrate the key parameters i.e., friction angle, plastic dilation angle and hardening needed for the DP model based on tri-axial test results on concrete cube subjected to non-uniform confinement. The effect of friction angle parameter is isolated from the hardening parameter within the yield function while the friction angle is defined as a function of concrete grade and hardening as a function of lateral confinement stiffness ratio. Here, the plastic lateral strains of non-uniform confined concrete, which could be obtained in cyclic tests but were usually left aside, are used to define a function for plastic dilation angle which works compatibly with the yield parameters. The resultant extended DP model could capture the stress-strain behaviour of concrete under both uniform or non-uniform passive pressure as well as the variable flow rules and different load paths occurring across the section of FRP-confined non-circular concrete.

Sheet metal forming forms in numerous industries like vehicle depend on the yielding of the sheet metals when strained. Yielding is portrayed by plastic flow of the materials when strained. The yield point if there should be an occurrence of uniaxial tension can be effectively decided from the pressure strain diagram, yet if there should arise an occurrence of multi axial Stresses it gets complicated. A connection between the principal stresses is required determining the conditions under which plastic flow occurs. This intricacy is tended to by the anisotropic yield capacities. Likewise, the tests used to acquire yield loci might be costly and time taking in such case these yield capacities end up being exceptionally viable. The yield criteria additionally help in deciding planar distribution of yield stresses and anisotropic coefficients, which gives a decent gauge of these mechanical parameters without having to through the pain of trial assurance. This project aims at using Hill 1948 criterion to obtain the Yield surface Diagrams for SS304 in annealed and original state and subsequently obtain the planar distribution of the uniaxial yield stress and anisotropic coefficient. Also, the performance evaluation of both the distributions will be done using accuracy index.

In the present work, the constitutive description of a 0.1-mm-thick ferritic stainless steel sheet was optimized for application to springback simulations after bending. Springback simulations require Bauschinger effects to be characterized carefully, which is a challenging task with an ultrathin sheet. The coefficients of the homogeneous anisotropic hardening (HAH) distortional plasticity model were calibrated with Zang's novel approach based on three-point bending conducted on pre-strained sheets (Zang et al., 2014). The Hockett–Sherby isotropic hardening law and the Yld2000-2d non-quadratic yield function were considered in this work to complete the HAH approach. Moreover, the degradation of elastic modulus was also accounted for in the finite-element simulations. The coefficients of the HAH model were calibrated using an inverse method by minimizing the difference between experimental and predicted specimen profiles after three-point bending and springback of pre-strained sheets. To validate the coefficients determined with this three-point bending test, U-draw bending tests were conducted and finite-element simulations were carried out. The springback predictions were found to agree well with the experimental results for all three pre-strains investigated, namely, 0% (as-received material) and 7.5% and 12.5% pre-strained ferritic stainless steel sheets.

A unified model for clays and sands, referred to as CSUH model, is proposed. The CSUH model develops the following aspects based on the unified hardening (UH) model: (a) a curved normal compression line in the e-lnp plane is proposed by introducing a parameter to describe the isotropic compression of sands; (b) a drop-shaped yield function, involving a critical state parameter, is adopted to get a better prediction of the shear yield and the critical state of sands; (c) a dilatancy parameter is introduced into the dilatancy equation to consider the density-dependent dilatancy of sands; and (d) different CSLs of sands with different lode’s angles in the e-p plane are considered. The CSUH model can degrade into the UH model to describe the behaviors of overconsolidated clays and can further degrade into the modified Cam-clay model to simulate the behaviors of normally consolidated clays. Finally, comparisons with the test data indicate that, the CSUH model cannot only predict the stress-strain relations of both clays and sands in a wide range of overconsolidation ratios or densities and stresses, but also is capable of describing triaxial compression, triaxial extension and three-dimensional behavior.

In this study, the mechanical properties of a high-strength laminated vibration damping steel sheet (LVDSS) with stretch-dominated deformation behavior were characterized and successfully applied to formability prediction. The high-strength LVDSS was fabricated with dual-phase 590 steel sheets as outer skins and polymeric adhesive layer as a core via roll bonding. The uniaxial and biaxial mechanical behaviors were evaluated by performing uniaxial tension and in-plane biaxial tension tests, respectively, while the forming limit was investigated by conducting punched dome tests. Then, based on the experimental data, the mechanical properties of the LVDSS were characterized, viz., the hardening curve, yield function, and forming limit curve. As an example of LVDSS application, a square cup drawing test was conducted and a dashboard prototype was fabricated using an LVDSS, and the deformation and failure behaviors were successfully predicted via a finite element simulation performed by inputting the characterized mechanical properties.

In this paper, a new double hardening constitutive model for frozen mixed soils is proposed, in which a new yield function is formulated based on the strength data of frozen soils. The associated f...

In elastoplastic analyses, a yield function needs convexities and C1 continuities, to meet the requirement of return mapping algorithms. However, many classic criteria in geotechnical engi...

The plastic anisotropy behavior of ferritic stainless steel (FSS) sheets was analyzed and modeled under associated and nonassociated flow rule approaches. Three orthotropic flow functions, known as quadratic Hill48 and nonquadratic (Yld2000-2d and BBC2005), were developed and employed under an associated and nonassociated flow rule hypothesis. For the NAFR based on the initial anisotropy, the mechanical behavior was described by the nonexponential model functions of Yld2000-2d and BBC2005 to predict the directional dependence of mechanical parameters. It provided a considerable advantage in terms of flexibility and good agreement with the experiment. According to the results, the polynomial fit functions of the transverse versus longitudinal true plastic strain curve were used to describe the designated properties corresponding to a selected level of strain. To describe the evolution of anisotropic hardening and potential plastic hardening, seven different loading conditions were considered. The proposed evolutionary non-AFR Yld2000-2d and BBC2005 criteria showed good accuracy in predicting the evolution of hardening yield and Lankford coefficients depending on the plastic deformation.

In this work, the plastic anisotropy of AA6022-T4 in hole-expansion is investigated focusing on prestraining effect, and its numerical solution is predicted by Homogeneous Anisotropic Hardening (HAH) model combined with an anisotropic yield function. For the prestraining, the material is subjected to a prestrain of 8% in uniaxial tension in the RD. Then, hole-expansion is conducted in a fully instrumented hydraulic press with a flat-headed punch. In both experiments, Digital Image Correlation (DIC) is used, to confirm the prevalence of uniaxial loading in the prestraining step and to measure thickness strain in the hole-expansion. The results show that the plastic anisotropy is manifested as a non-axisymmetric strain distribution around the hole. The prestraining, leading to non-proportional loading, changes the location of failure from ~45° to RD. The simulation shows the current version of HAH model is limited in capturing the thickness contours and the failure location.

In this paper a new anisotropic yield function that describes the elastoplastic behavior of sheet metals depends not only on the second stress invariant J2 for the yield, but also on the third stress invariant J3, both may affect the shape of the yield surface. The new anisotropic yield function is general and applicable to three-dimensional stress states. The proposed model is identified by calibration parameters from experimental data of material including uniaxial and biaxial tests. In this study, the predictive capabilities of the new anisotropic yield function are demonstrated by comparison between the predictions and experimental data for yield locus, normalized yield stress, and r-value. The result indicates that the new anisotropic model is reasonable to predict not only the symmetric shape but also the asymmetric shape of yield surface of the sheet metals.

In this paper, the proposed yield function is the function of stress, hardening parameter, and strain rate. To fully describe the stress state and strain rate condition of material during d...

In the present paper an orthotropic elastoplastic constitutive formulation for sheet metals is analyzed. The formulation is based on the non-associated flow rule, the isotropic hardening concept and utilizes orthotropic four parametric Karafillis-Boyce stress function as yield function/plastic potential. In defining the evolution equation for the equivalent plastic strain, as hardening parameter, two approaches are considered. By full approach, the equivalent plastic strain increment is proportional to the ratio of plastic potential and yield function where the plastic multiplier is proportionality factor. By simple approach, the equivalent plastic strain increment corresponds to the plastic multiplier. For considered constitutive descriptions, algorithmic formulations, based on the implicit return mapping, are developed and analyzed in predicting simple deformation tests and the cylindrical cup deep drawing process for AA5754-H22 sheet sample. Influence of the equivalent plastic strain evolution equation on the model predictions is assessed.

In this paper, a fully-involved anisotropic failure model including the effect of stress triaxiality and Lode angle proposed by Ganjiani [Ganjiani, M., 2020. A damage model for predicting ductile fracture with considering the dependency on stress triaxiality and Lode angle. European Journal of Mechanics-A/Solids, 104048] is extended to predict the fracture phenomena in the ma-terials. For considering the anisotropy effects, Hill’s 48 yield function is used. The proposed model is applied to construct the fracture loci and the corresponding Fracture Forming Limit Diagram (FFLD) to validate the performance of the model. The fully-involved anisotropic means that in constructing FFLD, the anisotropy is involved on both the stress triaxiality and the fracture strain. The predicted results are in good agreement with the experimental data over a wide range of stress triaxialities. Comparison between the current model and some fracture criteria is also provided, and the results indicate the significant potential of the model to predict ductile fracture as well as FFLD especially for anisotropic materials.

The Forbush decreases of the cosmic ray intensity observed on 24 December 2014 and on 8 September 2017 were chosen for cosmic ray spectral analysis. At first an analytical study of the solar and geomagnetic parameters of these events was carried out due to the fact that both are typical cosmic ray events. Hourly cosmic ray data of the neutron monitor stations obtained from the high-resolution neutron monitor database were used for calculating the cosmic ray density and anisotropy variations. Following the method of the coupling coefficients, the galactic cosmic ray spectral index was calculated using the technique of Wawrzynczak and Alania (Adv. Space Res. 45, 622, 2010). A newly presented yield function by Mishev, Usoskin, and Kovaltsov (J. Geophys. Res. 118, 2783, 2013) including a geometrical correction factor, already used in the spectral analysis of the cosmic ray ground level enhancements, was applied for the first time to the case of Forbush decreases. A comparison of these results during the events is performed by using two other coupling functions: the function presented in the work of Clem and Dorman (Space Sci. Rev. 93, 335, 2000) and the one in the work of Belov and Struminsky (Proc. 25th Int. Cosmic Ray Conf. 1, 201, 1997). The latter includes an extension for neutron monitor stations with rigidity $1~\mbox{GV} < R < 2.78~\mbox{GV}$ . The obtained spectral index and the calculated cosmic ray intensity in the heliosphere during the two Forbush decreases after the coupling by these three functions are presented and discussed.

A plane stress yield function which is described by multi-segment spline curve is proposed. This model is able to consider an arbitrary number of multi-axial stress states and its normal directions on the yield surface. Moreover by using interpolation based on cubic spline function, planar anisotropy is also described precisely. To show the applicability of the model, some evaluation of material properties and simulation results of hole expansion test are discussed.

Shear bands are common in dense quasi-static granular flows. They can appear in the interior of the flowing material or at confining boundaries and are typically of the order of ten particle diameters in thickness. Deformation tends to be localized in shear bands separating non-deforming or weakly deforming regions. Dilatancy and sharp velocity variation are typical in these shear layers. Much work has been reported in the literature concerning the development of non-local quasi-static rheological models to predict the flow behaviour in shear layers. In a recent article, Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) derive a non-local extension to a classical plasticity model by postulating that some local quantities appearing in the yield function, which stipulates the relationship between different components of the stress for the material to undergo sustained yielding, and the flow rule which provides information on the rate of deformation tensor to within an arbitrary multiplicative constant, should be replaced by their local averages. They then obtain an explicit non-local model which does not involve new microstructural variables and they show that the model captures velocity and volume fraction fields in simple shear flows, although some model parameters must be fitted to achieve quantitative agreement. This article discusses the work of Dsouza & Nott (2020) and comments on work ahead for further testing and developing the model.

Ground-based neutron counters are a standard tool for detecting atmospheric showers from GeV range primary cosmic rays of either solar or galactic origin. Bare neutron counters, a type of lead-free ...

In this study, yield surfaces of austenitic stainless steel produced by a cold-rolling process are measured using uniaxial and biaxial tensile tests. Using results obtained by electron backscatter diffraction, information on crystal orientation is introduced into a computational model for a multiscale crystal plasticity simulation. Finite element simulations for polycrystal of fine-grained austenitic stainless steel under biaxial tension are performed in order to predict yield surfaces of fine-grained austenitic stainless steel. Validity of predicted yield surfaces is evaluated by comparison between yield surfaces obtained by numerical simulations and experimental tensile tests.

Although significant advancement has been made over recent years with respect to three-dimensional upper bound calculations of tunnel facing, a considerable difference still exists between analytically and empirically based stability values. The current work suggests that the difference may well be the outcome of the traditional use of Tresca yield criterion for the upper bound calculations, which, by definition, does not distinguish among the shearing modes (compression, extension, plane strain). Consequently, this paper suggests and discusses a new yield function, which allows for asymmetric yielding. Such yielding is only beneficial in the case of three-dimensional and continuous velocity fields, and therefore a numerical procedure that generates relevant kinematically admissible fields for classical upper bound calculation is suggested. The procedure involves conversion from a load controlled boundary value problem to a velocity controlled problem at the limit state of collapse. The analysis results in significantly lower upper bound values than those presented earlier (for Tresca material), and the values are much closer to the stability curves of Kimura and Mair (1981), which are commonly used in design. Keywords: Tunnels, Plasticity, Stability, Upper bound

This study presents the development of a thermodynamics-based constitutive model for describing the macro-mechanical behaviour of porous rocks. The model formulation is developed within the well-established framework of generalised thermodynamics, or thermodynamics with internal variables (TIV), in order to guarantee the thermodynamics admissibility of the results. A new way for coupling different mechanisms of energy dissipation is proposed within the framework of TIV, so that the essential aspects of macro-mechanical behaviour of porous rocks ranging from brittle to ductile, dilation and compaction, as well as various modes of localised failure, under shearing at a wide range of confining pressures are correctly captured. The pressure sensitive behaviour of porous rocks is captured by introducing a shift stress in the formulation of the yield function, through the concept of ‘frozen elastic energy’ or non-dissipative part of the plastic work, which is stored within the material during loading. The proposed model allows the investigation of several important aspects of macro-mechanical behaviour of porous rocks at both material and specimen levels.

The substitute presentations of the conventional forming limit diagram (FLD) are stress-based forming limit diagram (FLSD), extended stress forming limit diagram (XSFLD), and polar effective plastic strain forming limit diagram (PEPS-FLD). These diagrams have already been proposed as alternative criteria to the conventional FLD for predicting forming limits in processes that sheet does not experience proportional or in-plane loading conditions. The present study provides a complete comparison of different forms of forming limit diagrams. For this purpose, FLD and extended FLDs are predicted based on the modified Marciniak and Kuczynski (M-K) model with Yld2011 anisotropic yield function while various factors that cause a change in the FLD of a specific alloy including initial imperfection coefficient (f0), non-linear strain path, through-thickness normal and shear stresses are considered. The results indicate that all forms of extended diagrams are independent of strain path and the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to the strain path than PEPS-FLD. In this regard, the main weakness of XSFLD, especially on the right-hand side, is that the safety margin cannot be visualized easily. By increasing the initial imperfection coefficient, the level of all forms of diagram increases. However, the effect of through thickness stresses on different diagrams is not the same. Increase in normal and shear stresses results in a downward shift in FLSD and an upward shift in other types of forming limit diagrams.

Many anisotropic phenomenological yield functions have been proposed in the literature in which their predictive capabilities strongly depend on the experimental calibration data as well as the calibration procedure to identify the anisotropy parameters. In this paper, emphasis is placed upon the experimental and numerical calibration procedure of anisotropic yield functions in the region of shear loading (zero hydrostatic stress with equal and opposite in-plane principal strains and stresses). Conventional anisotropic calibration procedures are shown to introduce non-physical artefacts into constitutive models which manifest as a non-zero hydrostatic stress or through-thickness strains generated under in-plane shear stress that violate the definition of the shear loading condition. To overcome this issue, a new physically necessary constraint is applied on the plastic potential to enforce equal and opposite principal strains in the shear state and correct the shear region of anisotropic yield functions. Using this necessary constraint, the widely used Yld2000-2d anisotropic yield function was calibrated using an associated flow rule for aluminum alloy sheet using published data for AA2090-T3 to demonstrate how enforcing this constraint can be readily implemented to correct the shear region of the anisotropic yield surface. Furthermore, to investigate the influence of the shear constraint, an AA7075-T6 alloy was experimentally characterized in uniaxial tension, equal-biaxial tension and shear. It was revealed that with the additional shear constraints, non-physical artefacts of plane-stress anisotropic yield functions such as Yld2000-2d can be removed during the calibration procedure. However, due to the additional shear constraints, available anisotropic models may become over-constrained and alternate yield functions with more flexibility or non-associated flow rules may be required.

This paper evaluates the performance of anisotropic yield functions based on mathematical methods which consider the prediction accuracy of yield stresses and R-values. The anisotropic yield functions being evaluated include one quadratic yield function and six non-quadratic ones. These yield functions are applied to describe the anisotropy of steel sheets and aluminum alloy sheets to evaluate their predictability. The root-mean-square errors (RMSEs) are computed to quantitatively assess their performance. The computation results of RMSEs demonstrate that the Yld2004-18p yield function exhibits the highest accuracy but requires extensive tests to calculate anisotropic parameters. The Yld2000-2d and BBC2000 yield functions are the same and thus have the same prediction accuracy. The application to cylindrical cup drawing shows that, associated with the Yld2004-18p and Yld2000-2d yield functions, the finite element (FE) simulations of cup drawing process can predict cups with six or eight ears. Considering both efficiency and accuracy, the Yld2000-2d and BBC2000 yield functions, which have less anisotropic parameters to be calculated, are recommended for metals with intermediate anisotropy.

The compression, tension, shear and combined shear-compression tests with different loading rates were conducted to investigate the mechanical behaviour of PA66 specimens. The initial yield and post-yield behaviour were analysed to determine the parameters in theoretical models. Results showed that the yield criterion based on three stress invariants gives the best description of the initial yield behaviour of PA66. The post-compression/tensile behaviour of PA66 was described by a hyperbolic function. The subsequent yield function was proposed based on the combination of initial yield surface and mixed work hardening model. The ratio of compression and tensile strength was utilized to appropriately characterize the kinematic hardening. Additionally, the effects of loading rate on PA66 yield stress and post-yield behaviour can be described by Eyring theory. On the basis of the general observations and classical elastoplastic theory, a macroscopic phenomenological constitutive was constructed to describe the mechanical behaviour of PA66.

This study clarified the influence of the post-necking prediction of the hardening law on the theoretical forming limit curve (FLC) of aluminium sheets subjected to punch-stretching tests. A procedure was developed to identify the parameters of a recently developed hardening model (Kim–Tuan hardening model) based on the curve fitting method. Subsequently, this model was used to capture the post-necking behaviours of two aluminium alloy sheets (AL5052-O and AL6016-T4), which were compared with those of various other hardening models, including the Swift model, Voce model, Hockett–Sherby model, Ghosh model, and a linear combination of the Swift and Voce models. These hardening models, with their extrapolations for post-necking prediction, were employed to analytically calculate the FLCs of the tested materials based on the modified maximum force criterion (MMFC). Furthermore, a simple and effective method was found to estimate the level of limited strain in the plane strain mode, namely FLC0. The proposed method was used to clarify the influence of the hardening law and yield function on the level of the computed FLC. The results indicated that the flow curves predicted by the Kim–Tuan hardening model effectively matched the experimental data obtained from uniaxial tensile tests, while their extrapolations provided intermediate predictions of the post-necking behaviour between the flow curves of the Swift and Voce models. In addition, the proposed hardening model improved the accuracy of the computed FLC for the studied materials.

This article investigates the macroscopic behavior of uniaxially pre-strained large-scale DP780 and CHSP45R sheet specimens. A novel grip system was designed for the pre-straining of the large tensile specimens. After this first loading, the uniformly strained gauge-section of the large-scale specimen was machined to smaller standard specimens. After pre-strain, further uniaxial tension tests at 45° and 90° from the pre-tensile direction, compression test and in-plane biaxial tension tests at 1:1, 2:1 and 1:2 force ratios were conducted. The flow curves and the instantaneous r-values of the pre-strained steel in the aforementioned uniaxial loading directions were compared with their monotonic response. In addition, the pre-strained compression test was incorporated to the π-plane at several incremental offset strains. The yield function- and isotropic hardening-based homogeneous anisotropic hardening (HAH) model (Barlat et al., 2014) was selected to predict the material response after non-linear strain path change. The Swift law characterized the isotropic hardening, while the Yld2000-2d anisotropic yield function (Barlat et al., 2003a) represented the yield locus. All the coefficients of the distortional plasticity model were manually determined and validated with an optimization algorithm. The HAH-predictions of the pre-strained flow curves and yield loci at several offset strains were in reasonable agreement with the experimental data, while the instantaneous r-value evolution was qualitatively well captured.

Commercially pure titanium (CP Ti) has been actively used in the plate heat exchanger due to its light weight, high specific strength, and excellent corrosion resistance. However, researches for the plastic deformation characteristics and press formability of the CP Ti sheet are not much in comparison with automotive steels and aluminum alloys. The mechanical properties and hardening behavior evaluated in stress–strain relation of the CP Ti sheet are clarified in relation with press formability. The flow curve denoting true stress–true strain relation for CP Ti sheet is fitted well by the Kim–Tuan hardening equation rather than Voce and Swift models. The forming limit curve (FLC) of CP Ti sheet as a criterion for press formability was experimentally evaluated by punch stretching test and analytically predicted via Hora's modified maximum force criterion. The predicted FLC by adopting Kim–Tuan hardening model and appropriate yield function shows good correlation with the experimental results of punch stretching test.

This paper presents a general numerical scheme for the fractional plastic flow rule, dedicated to a wide class of materials manifesting the non-normality of plastic flow and induced plastic anisotropy. To determine the vector of the plastic flow, a special numerical procedure has been developed, which is applicable for any smooth and convex yield function. The obtained approximation is verified based on an analytical solution. The paper also presents a set of numerical results for the generalised Drucker–Prager model.