Anisotropic Yield Function Research Articles

A natural vector/matrix notation applied in an efficient and robust return-mapping algorithm for advanced yield functions

A fast and robust stress-integration algorithm is the key to full exploitation of advanced anisotropic yield functions in computational mechanics. Poor global convergence of a direct application of the Newton-Raphson scheme has been rectified by applying line search strategies during the Newton iterations. In this work the line-search approach is further improved by a better first guess. The new algorithm is implemented into a user-defined material subroutine (UMAT) in a finite-element (FE) software and tested. The implementation is made easier and more efficient by a new advantageous vector/matrix notation for symmetric second- and fourth-order tensors, which is the second result of this work. Benefits of this notation are discussed with respect to formulation of continuum-plasticity models as well as their implementations. FE simulations were run to demonstrate the performance of the new implementation, which is available as open-source software via GitLab repository (see Appendix). The new return-mapping algorithm implementation runs equally fast and robust as the simple von Mises and Hill standard implementations in the Abaqus/Standard software. This enables full exploitation of advanced yield functions as the new standard in industrial FE applications.

Applications of the Hill 1948 Yield Criterion Development, Performance Evaluation and Comparison for Describing the Behaviour of Different CRCA Grades

The sheet metal forming processes in several industries like automobile and aerospace suppose the yielding of the sheet metals once strained. Yielding is categorized by the plastic flow of the materials once strained. The yield purpose just in case of uniaxial tension may be simply determined from the stress strain graph, however just in case of multi axial stresses it gets complicated. A relationship among the principal stresses is required requiring the circumstances underneath that plastic flow happens. This complexity is addressed by the anisotropic yield functions. Also, the investigation to get yield loci could also be expensive and time taking. In such case these yield functions prove to be very effective. The yield criteria also facilitate in decisive planar distribution of yield stresses and anisotropic coefficients which gives a decent estimate of these mechanical parameters while not having to through the pain of experimental determination. This study aims at using Hill 1948 criterion to get the Yield Surface Diagrams for three different grades of CRCA Sheets such as ordinary (o), Deep Drawing (DD) and Extra Deep Drawing (EDD) to get the planar distribution of the uniaxial yield stress and anisotropic coefficient. Also, the performance analysis of different grades the distributions are done using accuracy index.

A virtual materials testing approach to calibrate anisotropic yield functions for the simulation of earing during deep drawing of aluminium alloy sheet

We present a pragmatic and computationally efficient scheme to combine polycrystal-plasticity models based on crystallographic texture with phenomenological yield functions into a multilevel modelling framework for consideration of the anisotropic materials response during finite-element (FE) simulations of forming operations. Materials tests are conducted virtually using the visco-plastic self-consistent (VPSC) polycrystal-plasticity model to provide the stresses and strain ratios in uniaxial tension, plane-strain tension and under balanced biaxial conditions. These anisotropic materials data are then utilized to calibrate an advanced phenomenological yield function, viz. Yld2011–27p due to Aretz and Barlat, to simulate the anisotropic materials behaviour during subsequent sheet metal forming operations, here, deep-drawing. Since the simulated materials data can be used just as experimental values, no special fitting procedure for the yield function is required and validation of the simulation results is straightforward. Moreover, the followed modelling scheme enables to combine materials data obtained from experimental and virtual tests in a so-called hybrid approach. The modelling scheme was validated by FE-simulations of the earing behaviour of various Al sheet alloys, including AA 1200-O, AA 3104-H19, AA 5182-O and AA 5050A-H44, with distinctively different textures. Since earing is almost entirely controlled by crystallographic texture, analysis of the earing behaviour provides for a meaningful but demanding proof-of-principle for the current hierarchical modelling scheme.

Local Strain Measurement in Tensile Test for an Optimized Characterization of Packaging Steel for Finite Element Analysis

The continuous development of packaging steels for thickness reduction processes requires an advanced process design. This process is increasingly supported by finite element analysis to simplify tool construction and material selection purposes. Therefore, the fundamental basis is always the precise material characterization of packaging steel commonly based on tensile tests to determine flow curve and Lankford coefficients. However, due to strong temper rolling and the occurrence of slip bands, most packaging steels just show little elongation in tensile test. Therefore, a method of Paul et al. to determine the flow curve with digital image correlation (DIC) methods in the necking zone was applied in this work to meet the requirements of packaging steel. For the use of anisotropic yield functions, it is necessary to determine Lankford coefficients. Thus, a new method is proposed to measure Lankford coefficients locally with a DIC system in tensile test, also in case that no homogenous forming condition is reached. With the presented approaches the packaging steel TH415 was characterized. In order to validate the developed methods, a demonstrator was simulated with anisotropic yield function Yld2000-2d . The comparison between simulation and experiment showed clear improvements in simulation accuracy when using the newly presented methods for packaging steel.

Prediction of Strain Path Changing Effect on Forming Limits of AA 6111-T4 Based on a Shear Ductile Fracture Criterion

Strain path changing is a phenomenon in the stamping of complex panels or multiple-step stamping processes. In this study, the influence of the strain path changing effect was investigated and assessed for an aluminum alloy of 6111-T4 with a shear ductile fracture criterion. Plastic deformation of the alloy was modeled by an anisotropic Drucker yield function with the assumption of normal anisotropy. Then the shear ductile fracture criterion was calibrated by the fracture strains at uniaxial tension, plane strain tension and equibiaxial tension under proportional loading conditions. The calibrated fracture criterion was utilized to predict forming limit curves (FLCs) of the alloy stretched under bilinear strain paths. The analyzed bilinear strain paths included biaxial tension after uniaxial tension, plane strain tension and equibiaxial tension. The predicted FLCs of bilinear strain paths were compared with experimental results. The comparison showed that the shear ductile fracture criterion could reasonably describe the effect of strain path changing on FLCs, but its accuracy was poor for some bilinear paths, such as uniaxial tension followed by equibiaxial tension and equibiaxial tension followed by plane strain tension. Kinematic hardening is suggested to substitute the isotropic hardening assumption for better prediction of FLCs with strain path changing effect.

Hole-Expansion: Sensitivity of Failure Prediction on Plastic Anisotropy Modeling

The influence of yield function parameters on hole-expansion (HE) predictions are investigated for an anisotropic AA6022-T4 aluminum sheet. The HE experiment is performed in a fully-instrumented double-action hydraulic press with a flat-headed punch. Full strain fields are measured by a stereo-type digital image correlation (DIC) system. The stress state gradually changes from uniaxial to plane-strain tension to biaxial tension in the radial direction. Besides HE, the plastic anisotropy of AA6022-T4 is characterized by uniaxial tension and plane-strain tension experiments. Uniaxial tension is considered as the most important, since it is the stress state along the hoop direction in the hole. For the finite element (FE) simulation, the Yld2000-2d non-quadratic anisotropic yield function is used with two different parameter sets, calibrated by: (1) uniaxial tension only (termed Calib1) and, (2) both uniaxial and plane-strain tension (Calib2). The strain field predictions show a good agreement with the experiments only for Calib2, which takes into account plane-strain as well uniaxial tension. This indicates the importance of biaxial modes, and in particular plane-strain tension, for the adopted yield function to produce accurate HE simulations.

Extrapolation and Ce‐based implicit integration of anisotropic constitutive behavior

AbstractFor finite strain plasticity with both anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner‐Lee decomposition of the deformation gradient to obtain a differential‐algebraic system (DAE) in the semi‐implicit form and solve it by an implicit Richardson‐extrapolated method based on intermediate substeps. The source is here the right Cauchy‐Green tensor and the consistent Jacobian of the second Piola‐Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first‐order differential equation and a non‐smooth algebraic equation. The development of a Richardson‐extrapolated implicit integrator for any hyperelastic case and any yield function is the goal of this work. The integration makes use of a backward‐Euler method for the flow law complemented by the solution of a yield constraint. The resulting system is solved by the Newton‐Raphson method to obtain the plastic multiplier and the elastic right Cauchy‐Green tensor . To ensure power consistency, we make use of the elastic Mandel stress construction. Iso‐error maps for three yield functions and three numerical examples are presented.

Study of forming limit diagram (FLD) prediction of anisotropic sheet metals using Gurson model in M-K method

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.

Nonlinear computation of cables with high order solid elements using an anisotropic material model

AbstractDue to their complex structure cables exhibit an anisotropic behaviour and undergo large deformations in various applications. The large deformations are simulated using blended high order solid elements that can represent large deformations efficiently. To reduce the computational complexity the cross section consisting of many single wires and different layers of material is homogenized and represented by a transversely isotropic material model. Frictional effects and possible reordering of the parts are modelled through elastoplastic material behaviour with an anisotropic yield function. The simulations show that for a simple tension test and a free torsion test the material parameters can be satisfactorily identified.

Efficient modelling of the elastoplastic anisotropy of additively manufactured Ti-6Al-4V

The expanded design freedom offered by Additive Manufacture (AM) technologies is stimulating innovation in new directions, but it is also introducing new challenges in material characterisation. The inherent anisotropic behaviour observed in AM materials, like Ti-6Al-4V alloy, is of critical importance to safe and reliable implementation in structurally significant roles. To fully leverage the design freedom offered by AM, an accurate modelling capability is required to characterise this material behaviour. Building on the limited research efforts to date, this paper proposes a model that combines the Hill48 anisotropic yield function with the nonlinear multicomponent Armstrong-Frederick kinematic hardening rule. This model permits the accurate simulation of the anisotropic behaviour of these materials both under monotonic tensile and cyclic elastic-plastic loading. The performance of the model was evaluated by comparing the results of simulation with previously published stepped symmetric strain-controlled experimental data for Ti-6Al-4V fabricated via SLM in the three primary build orientations. An efficient process to establishing the model parameters from limited tests is also presented. The results of the simulations are in good agreement with the experimental data, thus demonstrating the proposed model’s capacity to simulate efficiently the difference in cyclic stress-strain evolution between different build orientations.

Effect of non-associated flow rule on fracture prediction of metal sheets using a novel anisotropic ductile fracture criterion

The ductile fracture behavior of anisotropic materials was investigated and modeled by the uncoupled ductile fracture criterion for aluminum alloys 6016-AC200. The ductile fracture model takes into account micro-mechanisms of void nucleation, void growth, and evolution of void coalescence. The anisotropic yield function and non-associated flow rule were applied to increase the accuracy of the ductile fracture prediction. Series of uniaxial tensile tests, equi-biaxial tension test, and hydraulic bulge test were utilized to identify coefficients of the plasticity model. Various standard fracture tests covering a wide range of stress triaxiality from negative to moderate and high-stress triaxiality were used in order to calibrate the ductile fracture parameters for AA6016-AC200. The strain field on the surface of specimens was captured by the non-contact measurement DIC system for all tests. The fracture surface, fracture locus, and fracture forming limit diagram constructed by the proposed ductile fracture criterion were plotted. These basic fracture tests and a square cup drawing test were simulated by the ABAQUS/Explicit commercial software to verify the efficiency of the proposed fracture criterion and the advanced plastic model in predicting the onset of ductile fracture behavior. The results indicated that the proposed ductile fracture criterion can be utilized for predicting the onset of the anisotropic ductile fracture behavior in sheet metal forming.

Anisotropic plasticity model forms for extruded Al 7079: Part II, validation

This is the second part of a two-part contribution on modeling of the anisotropic elastic-plastic response of aluminum 7079 from an extruded tube. Part I focused on calibrating a suite of yield and hardening functions from tension test data; Part II concentrates on evaluating those calibrations. A rectangular validation specimen with a blind hole was designed to provide heterogeneous strain fields that exercise the material anisotropy, while at the same time avoiding strain concentrations near sample edges where Digital Image Correlation (DIC) measurements are difficult to make. Specimens were extracted from the tube in four different orientations and tested in tension with stereo-DIC measurements on both sides of the specimen. Corresponding Finite Element Analysis (FEA) with calibrated isotropic (von Mises) and anisotropic (Yld2004-18p) yield functions were also conducted, and both global force-extension curves as well as full-field strains were compared between the experiments and simulations. Specifically, quantitative full-field strain error maps were computed using the DIC-leveling approach proposed by Lava et al. The specimens experienced small deviations from ideal boundary conditions in the experiments, which had a first-order effect on the results. Therefore, the actual experimental boundary conditions had to be applied to the FEA in order to make valid comparisons. The predicted global force-extension curves agreed well with the measurements overall, but were sensitive to the boundary conditions in the nonlinear regime and could not differentiate between the two yield functions. Interrogation of the strain fields both qualitatively and quantitatively showed that the Yld2004-18p model was clearly able to better describe the strain fields on the surface of the specimen compared to the von Mises model. These results justify the increased complexity of the calibration process required for the Yld2004-18p model in applications where capturing the strain field evolution accurately is important, but not if only the global force-extension response of the elastic–plastic region is of interest.

Experimental Investigation and Plasticity Modeling of SS316 L Microtubes Under Varying Deformation Paths

Abstract In this paper, results for SS316 L microtube experiments under combined inflation and axial loading for single and multiloading segment deformation paths are presented along with a plasticity model to predict the associated stress and strain paths. The microtube inflation/tension machine, utilized for these experiments, creates biaxial stress states by applying axial tension or compression and internal pressure simultaneously. Two types of loading paths are considered in this paper, proportional (where a single loading path with a given axial:hoop stress ratio is followed) and corner (where an initial pure loading segment, i.e., axial or hoop, is followed by a secondary loading segment in the transverse direction, i.e., either hoop or axial, respectively). The experiments are designed to produce the same final strain state under different deformation paths, resulting in different final stress states. This difference in stress state can affect the material properties of the final part, which can be varied for the intended application, e.g., biomedical hardware, while maintaining the desired geometry. The experiments are replicated in a reasonable way by a material model that combines the Hill 1948 anisotropic yield function and the Hockett–Sherby hardening law. Discussion of the grain size effects during microforming impacting the ability to achieve consistent deformation path results is included.

Influence of Evolution in Anisotropy During Strain Path Change on Failure Limits of Sheet Metals

Effects of evolution in anisotropy during plastic deformation under strain path changes on the formability and failure were investigated in the present study. The evolution in anisotropic property of the extra deep drawing steel was considered by implementing the non-quadratic anisotropic yield function Yld2000-2d as a function of effective plastic strain, and the corresponding forming behaviour in two-step forming processes was analysed. For the strain path effect, pre-strain was applied under biaxial mode using Marciniak in-plane stretch forming set-up, followed by the secondary deformation using the out-of-plane stretch forming tool. For failure prediction of the proposed two-step forming, different failure limit approaches were investigated. First, a strain based forming limit diagram (FLD), proposed as the Marciniak–Kuczynski model was modified to include the evolution in anisotropic yield function. The dynamic shift in FLD was also determined by taking strain path change into consideration. In addition, the influence of evolution of yield function on the strain path independent failure limit criteria was also assessed in terms of stress based forming limit diagram. Finally, the prediction accuracy of the failure limit criteria was compared among different models in terms of failure location and limiting dome height (LDH). It was observed that the incorporation of evolution in anisotropic yield surface improved the prediction of formability in terms of the LDH and strain distribution for the investigated material.

Study on the influence of the yield surface shape in the hole expansion test

The hole expansion test has become popular since it allows to study the formability of metallic sheets, in particular the onset of necking in stretch flanging areas. The accurate prediction of strain localization and consequent fracture requires a proper description of the plastic behavior, particularly the anisotropic yield function. In the context of strain localization prediction, the yield criterion adopted plays an important role, particularly when using an associated flow rule, since the direction of the plastic strain tensor is modelled by the normal to the yield surface. In this work, the parameters of an advanced yield criterion are calibrated considering a wide set of experimental data, which includes results from uniaxial and biaxial tension tests. This enables establishing yield surfaces with similar shape in the plane defined by the stress components in the rolling and transverse directions and a null shear component in the same plane. However, their shape changes slightly when considering non-null values for that shear component. The numerical simulations of the hole expansion test demonstrate the impact of these slight differences on the thickness strain distribution and, consequently, in the instant and location of the necking.

Response, Localization, and Rupture of Anisotropic Tubes Under Combined Pressure and Tension

Abstract The inelastic response and failure of Al-6061-T6 tubes under combined internal pressure and tension is investigated as part of a broader study of ductile failure of Al-alloys. A custom experimental setup is used to load thin-walled tubes to failure under radial paths in the axial-hoop stress space. All loading paths achieve nominal stress maxima beyond which deformation localizes into a narrow band. 3D digital image correlation (DIC) was used to monitor the deformations in the test section and successfully captured the rapid growth of strain within the localization bands where they burst. The biaxial stress states generated are first used to calibrate the nonquadratic anisotropic Yld04-3D yield function (Barlat et al., 2005, “Linear Transformation-based Anisotropic Yield Functions,” Int. J. Plasticity, 21(5), pp. 1009–1039). The constitutive model is then incorporated through a UMAT into a finite element analysis and used to simulate numerically the experiments. The same calculations were performed using von Mises (VM) and an isotropic nonquadratic yield function. The material hardening responses adopted were extracted for each constitutive model from the necked zone of a tensile test using an inverse method. The use of solid elements captures the evolution of local deformation deep into the localizing part of the response, producing strain levels that are required in the application of failure criteria. The results demonstrate that the adoption of a nonquadratic yield function, together with a correct material hardening response are essential for large deformation predictions in localizing zones in Al-alloys. Including the anisotropy in such a constitutive model produces results that are closest to the experiments.

Prediction of ductile fracture on 6016-T4 aluminum alloy sheet metal forming considering anisotropic plasticity

This study is aimed to predict ductile fracture of sheet metal forming for 6016-T4 aluminum alloy (AA6016-T4). Six shapes of tension tests in three directions with respect to the rolling direction have been carried out up to fracture to measure ductile fracture for the sheet. Since the fracture-related parameters determined by finite element simulation are sensitive to plasticity model, the Yld2000-2d and anisotropic Drucker yield functions with a combined Swift-Voce hardening model were firstly identified for AA6016-T4 and then used to compare the prediction accuracy. These constitutive models are implemented into ABAQUS for plane stress condition and S4R shell elements are set for the finite element model. Comparison of the experimental and simulation results indicates that the anisotropic Drucker yield function is more suitable for AA6016-T4 by considering both the computational accuracy and efficiency. Based on these results, fracture loci for AA6016-T4 were represented by four common ductile fracture criteria (Cockcroft–Latham, Rice–Tracey, Clift and DF2012). Three kinds of specimen were employed to calibrate the fracture models by using a hybrid experimental–numerical method. The results show that the DF2012 criterion is more flexible and accurate for AA6016-T4 with the comparison of force–displacement curves. For further validation, these four ductile fracture criteria are also applied to predict the fracture of an automobile inner panel in drawing forming, and the results show that the DF2012 criterion provides sufficient prediction precision for AA6016-T4.

Identification of strain localization-induced failure in hot-rolled steel sheets: A hybrid numerical–experimental approach to the virtual forming limit test

A coupled numerical–experimental inverse calibration approach was implemented and systematically validated to evaluate the formability of high-strength, hot-rolled, hyper-burring (HB780) steel sheets and dual-phase (DP780) steel sheets. The proposed methodology featured the determination of hardening behavior beyond the uniform elongation limit at large strain and shapes of 3-D, non-quadratic yield functions: Hosford and Yoshida models. To accurately identify the 3-D yield functions and directly monitor the strain localizations of hemispherical domes for the virtual numerical forming limit analyses, stretching tests were recommended for the investigated hot-rolled sheets. The modeling procedure was evaluated by comparing the forming limits of the stretching tests’ various sample widths to those of the experiments, which confirmed the validity of the identified constitutive model parameters by predicting the localized failure strains and punch force–displacement curves under different deformation modes. For the purpose of validation, simulations and experiments of cylindrical circular cup drawing tests were performed. The proposed numerical procedure could predict the profiles of eared cups, locations of fracture, and drawing limit ratios of the two hot-rolled sheets without the input of the forming limit diagram when proper calibrations of the 3-D, non-quadratic, anisotropic yield function and strain hardening at large strains were guaranteed.

Prediction of Strain Limits via the Marciniak-Kuczynski Model and a Novel Semi-Empirical Forming Limit Diagram Model for Dual-Phase DP600 Advanced High Strength Steel

The prediction capability of a forming limiting diagram (FLD) depends on how the yield strength and anisotropy coefficients evolve during the plastic deformation of sheet metals. The FLD predictions are carried out via the Marciniak-Kuczynski (M-K) criterion with anisotropic yield functions for DP600 steel of various thicknesses. Then, a novel semi-empirical FLD criterion is proposed, and prediction capabilities of the criterion are tested with different yield criteria. The results show that the yield functions are very sensitive to anisotropic evolution. Thus, while the FLD curves from the M-K model and the proposed model are not the same for each thickness, the proposed model has better prediction than the M-K model.

On the experimental characterization of sheet metal formability and the consistent calibration of the MK model for biaxial stretching in plane stress

Analytical formability models based upon in-plane proportional plane stress loading such as the Marciniak–Kuczynski (MK) model are commonly evaluated against forming limit curves (FLCs) obtained using Nakazima tests with out-of-plane deformation, non-linear strain paths and tool contact. The present study has conducted a comprehensive experimental characterization of a DP1180 advanced high strength steel with relatively low formability to demonstrate that the use of Nakazima FLC data can lead to a physically inconsistent MK model. Although good agreement can be obtained using the MK model with Nakazima data, it is attributed to calibration bias in the hardening model and selection of the imperfection factor. Marciniak tests should be performed to characterize the FLC or process-corrections be applied to the Nakazima FLC to then calibrate the MK imperfection factor. An analysis of the MK framework reveals a false dependence of the imperfection factor on the hardening model due to artefacts in the hardening model calibration. When the Considère constraint for the uniform elongation is imposed during the hardening model calibration, the MK imperfection factor becomes nearly independent of the choice of hardening model. The accuracy of the FLC predicted by the MK model, when appropriately constrained and calibrated in a deterministic manner, was in a reasonable agreement with the experimental data only after accounting for material anisotropy. In contrast, the deterministic Modified Maximum Force Criterion (MMFC) model provided a very close estimate of the experimental forming limit strains of the mildly anisotropic DP1180 using both isotropic and anisotropic yield functions.