Anisotropic Yield Criterion Research Articles

Strain evolution in the single point incremental forming process: digital image correlation measurement and finite element prediction

Incremental Sheet Forming (ISF) is a relatively new class of sheet forming processes that allow the manufacture of complex geometries based on computer-controlled forming tools in replacement (at least partially) of dedicated tooling. This paper studies the straining behaviour in the Single Point Incremental Forming (SPIF) variant (in which no dedicated tooling at all is required), both on experimental basis using Digital Image Correlation (DIC) and on numerical basis by the Finite Element (FE) method. The aim of the paper is to increase understanding of the deformation mechanisms inherent to SPIF, which is an important issue for the understanding of the high formability observed in this process and also for future strategies to improve the geometrical accuracy. Two distinct large-strain FE formulations, based on shell and first-order reduced integration brick elements, are used to model the sheet during the SPIF processing into the form of a truncated cone. The prediction of the surface strains on the outer surface of the cone is compared to experimentally obtained strains using the DIC technique. It is emphasised that the strain history as calculated from the DIC displacement field depends on the scale of the strain definition. On the modelling side, it is shown that the mesh density in the FE models plays a similar role on the surface strain predictions. A good qualitative agreement has been obtained for the surface strain components. One significant exception has however been found, which concerns the circumferential strain evolution directly under the forming tool. The qualitative discrepancy is explained through a mechanism of through-thickness shear in the experiment, which is not fully captured by the present FE modelling since it shows a bending-dominant accommodation mechanism. The effect of different material constitutive behaviours on strain prediction has also been investigated, the parameters of which were determined by inverse modelling using a specially designed sheet forming test. Isotropic and anisotropic yield criteria are considered, combined with either isotropic or kinematic hardening. The adopted constitutive law has only a limited influence on the surface strains. Finally, the experimental surface strain evolution is compared between two cones with different forming parameters. It is concluded that the way the plastic zone under the forming tool accommodates the moving tool (i.e. by through-thickness shear or rather by bending) depends on the process parameters. The identification of the most determining forming parameter that controls the relative importance of either mechanism is an interesting topic for future research.

Forming of 2A12 Aluminum Alloy Double Curvature Thin-Wall Part Based on the Hill (1948) Anisotropic Yield Criterion

In order to solve the existing problems such as the less-plump corners and wrinkle during the forming process of a double curvature thin-wall part. Material parameter formulas are calculated based on Hill 1948 anisotropic yield criterion and flow rule. The material mechanical property parameters of 2A12 are obtained using uniaxial tensile experiment. Single factor test method is used and LS-DYNA is applied to conduct finite element analysis. So the influence law of holding force, blank shape and blank size imposed on the finite element model are calculated in sequence. On the basis of these laws, that the blanking force, blank dimension and blank shape are decided. Stamping die for the experiment model is designed and made, deformation measurement technology of coordinate grid circle is used to carry out strain analysis. From the comparison result between experiment and simulated data, it can be seen that Hill 1948 yield criterion and flow law can better solve the formability of 2A12 sheet.

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 ( Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.

Evaluation of yield criteria for forming simulations based on residual stress measurement

Process induced anisotropy in sheet metal is accounted for in analytical modeling by anisotropic yield criteria. The suitability of a yield criterion for predicting sheet metal forming process is generally validated by way of its ability to predict surface strains. However, the sensitivity of surface strains to yield criteria is dependent upon strain modes, with plane strain mode exhibiting higher sensitivity. To eliminate dependency on strain modes, stresses are used to evaluate yield criteria, since forming stresses are less sensitive to strain modes. In the study, the residual stresses remaining in a hemispherical cup formed in plain strain mode is predicted using Hill48 and Barlat89 criteria. The residual stresses are experimentally characterized by using X-Ray diffraction method. Suitable yield criterion for forming simulation is validated based on the correlation of theoretical predictions with experimental residual stress values.

Sheet Metal Forming Limit Predictions Based on Advanced Constitutive Equations

The present paper aims at analysing the strain and stress–based forming limits predicted by an advanced sheet metal forming limit model. The onset of localized necking is simulated through the Marciniak-Kuczinsky (MK ) analysis [1]., which connects the physically-based hardening model accounting for the evolution of the anisotropic work-hardening induced by the microstructural evolution at large strains of Teodosiu and Hu [2] with the phenomenological anisotropic yield criterion Yld2000-2d (Barlat et al., 2003) [3] Linear and complex strain paths are taken into account. The selected material is a DC06 steel sheet. The validity of the model is assessed by comparing the predicted and experimental forming limits. The remarkable accuracy of the developed software on predicting the forming limits is obviously due to the performance of the advanced constitutive equations applied on M-K theory, able to describe with great detail the material behaviour. The effect of the strain-induced anisotropy predicted by the microstructural hardening model on the evolution of formability under strain path changes is particularly aimed it.

Advanced High Strength Steel Sheet Forming and Springback Simulation

With the increasing use of finite element analysis method in sheet forming simulations, springback predictions of advanced high strength steel (AHSS) sheet are still far from satisfactory precision. The main purpose of this paper was to provide a method for accurate springback prediction of AHSS sheet. Material model with Hill’48 anisotropic yield criterion and nonlinear isotropic/kinematic hardening rule were applied to take account the anisotropic yield behavior and the Bauschinger effect during forming processes. U-channel forming and springback simulation was performed using ABAQUS software. High strength DP600 sheet was investigated in this work. The simulation results obtained with the proposed material model agree well with the experimental results, which show a remarkable improvement of springback prediction compared with the commonly used isotropic hardening model.

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.

Prediction of flange wrinkling in deep drawing process using bifurcation criterion

Surface distortions in the form of wrinkles are often observed in sheet metals during stamping and other forming operations. Because of the trend in recent years towards thinner, higher-strength sheet metals, wrinkling is increasingly becoming a more common and troublesome mode of failure in sheet metal forming. The prediction and prevention of wrinkling during a sheet forming process are important issues for the design of part geometry and processing parameters. This paper treats the phenomenon of flange wrinkling as a bifurcated solution of the equations governing the deep drawing problem when the flat position of the flange becomes unstable. Hill’s bifurcation criterion is used to predict the onset of flange wrinkling in circular and square cup drawing. In particular, the maximum cup height that can be drawn without the onset of flange wrinkling is predicted for the given set of process parameters. A parametric study of the maximum cup height is also carried out with respect to various geometric, material and process parameters. Finite element formulation, based on the updated Lagrangian approach, is employed for the analysis. The incremental logarithmic strain measure, which allows the use of a large incremental deformation, is used. The stresses are updated in a material frame. The material is assumed to be elastic–plastic, strain hardening, yielding according to an anisotropic yield criterion of Barlat et al. (2005) [23] (named as Yld2004-18p). Isotropic power law hardening is assumed. Inertia forces are neglected due to small accelerations. Modified Newton–Raphson iterative technique is used to solve the nonlinear incremental equations.

A new model for FLD prediction based on advanced constitutive equations

An advanced sheet metal forming limit model is developed. Using the Marciniak-Kuczinsky analysis, this approach intends to join advantages of physics-based aspects of plasticity with advantages of phenomenological material description. It aims thus at connecting the most advanced physically-based hardening model accounting for the evolution of the anisotropic work hardening induced by the microstructural evolution at large strains of Teodosiu and Hu (1995) with the advanced phenomenological anisotropic yield criterion Yld2000-2d (Barlat et al. Int J Plast 19:1297–1319, 2003). Two deep-drawing quality sheet metals are selected: a bake-hardening steel (BH) and AA6016-T4 aluminium alloy. Linear and complex strain paths are taken into account. By comparing the simulated and experimental results the model is validated.

Modeling of Tension-Compression Asymmetry in Off-axis Nonlinear Rate-dependent Behavior of Unidirectional Carbon/Epoxy Composites

A phenomenological anisotropic theory of viscoplasticity that takes account of the difference between the nonlinear rate-dependent behaviors of unidirectional carbon fiber-reinforced composites under off-axis tensile and compressive loading conditions is presented. The plane-stress representation of the effective stress associated with the pressure-modified Hill’s anisotropic yield criterion is modified into a new form that allows consideration of the difference between the shear flow stress levels under transverse tension and compression, i.e., a shear flow differential effect, as well as of the difference between the transverse tensile and compressive flow stress levels, i.e., a transverse flow differential effect. The shear flow differential effect is modeled in two ways by assuming the sensitivity and insensitivity to transverse stress, respectively. The rate-dependent plastic deformation of unidirectional composites is modeled on the basis of the concept of overstress and the irreversible thermodynamics with internal variables. The viscoplasticity model proposed in the present study can be reduced to the form developed in an earlier study, which facilitates identification of material constants and comparison with other existing theories. Validity of the proposed viscoplasticity model is evaluated by comparing with the off-axis tension and compression test results on a unidirectional carbon/epoxy composite at different strain rates. It is demonstrated that consideration of both the transverse and shear flow differential effects is crucial for accurate prediction of the different nonlinear rate-dependent behaviors of unidirectional composites under off-axis tensile and compressive loading conditions and those two kinds of flow differential effects have successfully been modeled in the proposed theory of viscoplasticity.

Prediction of yield and long‐term failure of oriented polypropylene: Kinetics and anisotropy

AbstractThe time‐dependent yield and failure behavior of off‐axis loaded uniaxially oriented polypropylene tape is investigated. The yield and failure behavior is described with an anisotropic viscoplastic model. A viscoplastic flow rule is used with an equivalent stress, based on Hill's anisotropic yield criterion, and the Eyring flow theory combined with a critical equivalent strain definition. This model is based on factorization of the rate and draw ratio dependence and is capable of quantitatively predicting the rate, angle and draw ratio dependence of the yield stress as well as time‐to‐failure in various off‐axis tensile loading conditions characterized solely from the transverse direction. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2026–2035, 2009

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.

Femoral neck fracture prediction by anisotropic yield criteria

The fracture risk due to osteoporosis, is undertaken with Dual-Energy X-ray Absorptiometry (DEXA) which is an average of bone mineral density measurement, without taking into account the bone structure. The objective of this study was an experimental test to solicit the human proximal femurs by a physiological configuration (one leg stance phase of walking). For this, transversely isotropic finite element models were developed from CT scan acquisition. The failure load assessment was insured by anisotropic yield behaviour criteria based on distortion energy criterion (Hill’s criterion) and taking into account the difference between tension and compression yield properties (Tsai–Wu’s criterion). The results found in this study showed the significance part of anisotropic yield behaviour of bone on proximal femur.

Simulation of the anisotropic behavior of titanium alloys during sheet metal forming

This paper introduces a constitutive elastoplastic model based on anisotropic yield criterion. An evolving anisotropy during simulation is considered. The anisotropy axes are updated with the deformation gradient. An anisotropic yield criterion, an isotropic hardening and a kinematic hardening model the plastic behavior of titanium alloys. Different plastic yield criteria are compared to show the accuracy of each plasticity model for the simulation of forming processes.

Influence of advanced yield criteria on predictions of plastic anisotropy for aluminium alloy sheets

Extruded aluminium alloy sheets of AA7003-T6 are characterised by performing an experimental programme and numerical simulations. The anisotropic behaviour of the aluminium alloy is investigated using two linear transformation based anisotropic yield criteria Yld2000-2d (Barlat et al. [1]) and Yld2004-18p (Barlat et al. [2]). The yield criteria are calibrated using previously reported measured values of flow stress ratios and strain ratios obtained from uniaxial tensile tests and disc compression tests. The results from the model calibrations show that Yld2004-18p gives improved prediction. The evaluation of the selected yield functions is extended by performing numerical simulations of in-plane shear tests and plane-strain tension tests using the general-purpose non-linear FE code LS-DYNA. The results from the simulations show that the constitutive model based on Yld2000-2d predicts too low strength for the determined model parameters and indicates that the shape of the yield surface is inadequate. Predictions with the constitutive model based on Yld2004-18p gave better agreement with the experiments.

Residual stresses in press-braked stainless steel sections, I: Coiling and uncoiling of sheets

The manufacturing process of cold-formed thin-walled steel members induces cold work which can be characterized by the co-existent residual stresses and equivalent plastic strains and has a significant effect on their structural behaviour and strength. The present paper and the companion paper are concerned with the prediction of residual stresses and co-existent equivalent plastic strains in stainless steel sections formed by the press-braking method. This manufacturing process consists of the following two distinct stages: (i) coiling and uncoiling of the sheets, and (ii) press-braking operations. This paper presents an analytical solution for the residual stresses and co-existent equivalent plastic strains that arise from the first stage. In the analytical solution, the coiling–uncoiling stage is modelled as an inelastic plane strain pure bending problem; the stainless steel sheets are assumed to obey Hill’s anisotropic yield criterion with isotropic hardening to account for the effects of material anisotropy and nonlinear stress–strain behaviour. The accuracy of the solution is demonstrated by comparing its predictions with those obtained from a finite element analysis. The present analytical solution and the corresponding analytical solution for press-braking operations presented in the companion paper form an integrated analytical model for predicting residual stresses and equivalent plastic strains in press-braked stainless steel sections.

Finite element simulation of earing defect in deep drawing

Deep drawing is an extensively used press working process since it eliminates expensive machining and welding operations and enables the production of components at a very high rate. The workpiece material used in a deep drawing process is anisotropic in nature, due to a prior thermomechanical treatment. Earing is one of the major defects observed in a deep drawing process due to the anisotropic nature of the sheet material. Knowledge about the ear formation in deep drawing allows a prior modification of the process, which can result in a defect-free final product with financial savings. In this paper, a recently proposed anisotropic yield criteria by Barlat et al. for rolled sheets is used to model the anisotropy for simulating the earing defect in square and circular cup drawing processes. The effect of the tooling geometry and process parameters on the ear formation is studied. It is shown that, in the square cup, the uneven metal flow rate, rather than the material anisotropy, is mainly responsible for the flange earing. Finite element formulation, based on the updated Lagrangian approach, is employed for the analysis. The stresses are updated in a material frame and the logarithmic strain measure is used, which allows the use of a large increment size. Isotropic hardening is assumed, and it follows a power law. Inertia forces are neglected due to small accelerations. The modified Newton–Raphson iterative technique is used to solve the nonlinear incremental equations.

Femoral neck fracture prediction by anisotropic yield criteria

The fracture risk due to osteoporosis, is undertaken with Dual-Energy X-ray Absorptiometry (DEXA) which is an average of bone mineral density measurement, without taking into account the bone structure. The objective of this study was an experimental test to solicit the human proximal femurs by a physiological configuration (one leg stance phase of walking). For this, transversely isotropic finite element models were developed from CT scan acquisition. The failure load assessment was insured by anisotropic yield behaviour criteria based on distortion energy criterion (Hill’s criterion) and taking into account the difference between tension and compression yield properties (Tsai–Wu’s criterion). The results found in this study showed the significance part of anisotropic yield behaviour of bone on proximal femur.

An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations

The initial yield surface of 2D lattice materials is investigated under biaxial loading using finite element analyses as well as by analytical means. The sensitivity of initial yield surface to the dominant deformation mode is explored by using both low- and high-connectivity topologies whose dominant deformation mode is either local bending or strut stretching, respectively. The effect of microstructural irregularity on the initial yield surface is also examined for both topologies. A pressure-dependent anisotropic yield criterion, which is based on total elastic strain energy density, is proposed for 2D lattice structures, which can be easily extended for application to 3D cellular solids. Proposed criterion uses elastic constants and yield strengths under uniaxial loading, and does not rely on any arbitrary parameter. The analytical framework developed allows the introduction of new scalar measures of characteristic stresses and strains that are capable of representing the elastic response of anisotropic materials with a single elastic master line under multiaxial loading.

Applications of planar anisotropic yield criteria to porous sheet metal forming simulations

Planar anisotropic yield functions, with rounded vertexes especially near the equal-biaxial direction of the corresponding yield loci, appropriate for some structure metals are employed for the matrix surrounding voids in the present study. The widely adopted Hill anisotropic yield functions are also implemented into the matrix for comparisons. Mechanisms of the void growth, void nucleation, and void coalescence are simultaneously considered here. Effects of the yield function of the corresponding matrix on the sheet metal under two typical sheet forming operations, a hemispherical punch stretching operation and a cup drawing operation, are investigated via a finite element analysis. Thickness strains in various orientations of the sheet are then evaluated. Simulation results show that the yield function of the corresponding matrix plays important roles on the strain distribution and the strain localization as well. Early localization would be found for the sheet with relatively small initial void volume fraction in two operations. Yield functions of the matrix rather influence the earing phenomenon under the cup drawing procedure even similar displacement profiles of the outer boundary could be observed.