Anisotropic Yield Criterion Research Articles

Control strategy of twist springback for aluminium alloy hybrid thin-walled tube under mandrel-rotary draw bending

The focus of this work is to investigate the control strategy of twist springback of an industrial aluminium alloy extruded tube after mandrel-rotary draw bending, which is a complex nonlinear physical process with coupling multi-factor interactive effects. A particularly pronounced twist springback characteristic of an AA6060-T4 hybrid thin-walled bent tube was observed. It leads to difficulty at the assembly stage due to the inaccurate geometry of product. For an accurate modelling of material behaviours, a non-quadratic anisotropic yield criterion integrated with combined isotropic and kinematic hardening model was adopted to describe the strain-stress behaviours including anisotropy, Bauschinger effects and permanent softening. The corresponding mechanical tests, e.g., enhanced forward-reverse simple shear tests, were performed to identify the material parameters. By using the improved and validated finite element model, two process control strategies related to the mandrel nose placement and the axial push assistant loading are proposed for the assessment of the control efficiency. The simulation results indicate that the mandrel nose placement mainly affects the longitudinal angular springback but not twist deformation in the circumferential sections. The angular springback angle firstly increases and then decreases after several tests of forward mandrel nose placement. The other control strategy, additional axial push assistant loading with high boost, is able to reduce twist deformation, but the compensation of angular springback is not attained and even a slight increase is observed. It also leads to the rise of the of wrinkle tendency in the intrados sidewall. Finally, the wipe die rake is put forward to reduce the wrinkle tendency due to the high boost applied for twist control.

A thermal forming limit prediction method considering material damage for 22MnB5 sheet

Hot stamping of high-strength steel is an innovative technology to achieve automobile lightweight and guarantee security simultaneously. However, the formability of high-strength steel is still limited at elevated temperatures by the evolution of void damage inside materials. Thus, the establishment of an efficient forming limit prediction method is urgently demanded. In the present work, the Gurson-Tvergaard-Needleman (GTN) model is extended by containing Hosford anisotropic yield criterion which can characterize void damage in normal anisotropic materials. The flow behavior of matrix material 22MnB5 at different forming temperatures is simulated by the modified Norton-Hoff hardening law. And the damage-related parameters are calibrated through FEM inverse analysis. Furthermore, the prediction method of forming limit combined with void damage is realized within the framework of the Marciniak and Kuczynski (M-K) model. Meanwhile, the high-temperature Nakazima test is conducted to obtain experimental thermal forming limit diagram (TFLD). The comparison of results by theoretical prediction and experimental test shows good consistency. Based on the proposed method, the effects of temperature, void damage, initial thickness imperfection, and anisotropy on the forming limit are analyzed. The formability improved with the increasing of deformation temperature but worse for other factors.

Numerical Investigation on Stress based Forming Limit Diagram for Ti-6Al-4V Alloy

Forming limit diagram (FLD) is an important tool in designing sheet metal forming processes. FLD is used to determine if a sheet metal forming process may induce necking in the material. However, these are only reliable for operations with linear strain paths. Stoughton [1] developed the stress based forming limit diagrams (FLSD) which are independent of strain path present in the material. This paper focuses on converting experimental FLD of Ti-6Al-4V at 400°Cpublished by Kotkunde et al. [2] to FLSDs. Various anisotropic yield criterion (Hill 48, Hosford 79, Barlat 89) and hardening rules (Hollomon Power Law, Swift Law and Modified Voce Law) are used.In addition, experimental FLDs and theoretical FLD curves obtained from Marciniak-Kuczynski (MK) analysis are converted to FLSDs and compared.

Experimental and Finite Element Studies of Stretch Forming Process for Ti-6Al-4V Alloy at Elevated Temperature

In this study, experimental and Finite Element (FE) analysis has been studied of stretch forming process for Ti-6Al-4V alloy at 400°C. Initially, uniaxial tensile tests experiments have been conducted to study the material properties at 400°C. Furthermore, stretch forming experiments have been performed using Nakazima test. The various quantitative parameters such as Forming Limit Diagram (FLD), Limiting Dome height (LDH) and thickness distribution are evaluated at 400°C. Additionally, Finite Element (FE) analysis of stretch forming process has been done using commercial available DYNAFORM software with LSDYNA server. Two different anisotropic yield criteria namely; Hill 1948 and Barlat 1989 with Hollomon Power Law (HPL) are consider for FE simulations. The results of FE simulations are compared with the experimental results. It has been observed that Barlat 1989 yield criterion prediction is more accurate than Hill 1948.

Stress analysis for the orthotropic pressurized structure of the cylindrical shell and spherical head

According to anisotropic shell theory, the pressurized structure of cylindrical shell and spherical head is analyzed, and the shear force, bending moment and stress solution at the juncture are obtained. The influence of anisotropic mechanical properties and geometric parameters on shear force, bending moment and stress distribution along the cylindrical shell is discussed, and the results are compared with the isotropic results. Additionally, Hill48 anisotropic yield criterion is also used to derive the equivalent stress intensity expression of the plane stress state, and the effect of thickness ratio and elastic modulus ratio on Hill48 equivalent stress at the juncture is discussed. Finally, finite element method is applied to verify the results of stress intensity calculation, which shows that the finite element solution and the analytical solution are in good agreement.

Prediction of failure in bending of an aluminium sheet alloy

This work is dedicated to numerical prediction of the bending of thin aluminium alloy sheets, with a focus on the material parameter identification and the prediction of rupture with or without pre-strains in tension prior to bending. The experimental database consists of i) mechanical tests at room temperature, such as tension and simple shear, performed at several orientations to the rolling direction and biaxial tension ii) air bending tests of rectangular samples after (or not) pre-straining in tension. The mechanical model is composed of the Yld2004-18p anisotropic yield criterion (Barlat et al. [3]) associated with a mixed hardening rule. The material parameters (altogether 21) are optimized with an inverse approach, in order to minimize the gap between experimental data and model predictions. Then, the Hosford-Coulomb rupture criterion is used in an uncoupled way, and the parameters are determined from tensile tests, both uniaxial and biaxial, with data up to rupture. In a second step, numerical simulations of the bending tests are performed, either on material in its original state or after pre-straining in tension, with pre-strain magnitudes increasing from 0.19 up to 0.3. The comparisons are performed on different outputs: load evolution, strain field and prediction of the rupture. A very good correlation is obtained over all the tests, in the identification step as well as in the validation one. Moreover, the fracture criterion proves to be successful whatever the amount of pre-strain may be. A convincing representation of the mechanical behavior at room temperature for an aluminium alloy is thus obtained.

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

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

Calibration of Gurson-type models for porous sheet metals with anisotropic non-quadratic plasticity

The growth and coalescence of voids in sheet metals are not only the main active mechanisms in the final stages of fracture in a necking band, but they also contribute to the forming limits via changes in the normal directions to the yield surface. A widely accepted method to include void effects is the development of a Gurson-type model for the appropriate yield criterion, based on an approximate limit analysis of a unit cell containing a single spherical, spheroidal or ellipsoidal void. We have recently [2] obtained dissipation functions and Gurson-type models for porous sheet metals with ellipsoidal voids and anisotropic non-quadratic plasticity, including yield criteria based on linear transformations (Yld91 and Yld2004-18p) and a pure plane stress yield criteria (BBC2005). These Gurson-type models contain several parameters that depend on the void and cell geometries and on the selected yield criterion. Best results are obtained when these key parameters are calibrated via numerical simulations using the same unit cell and a few representative loading conditions. The single most important such loading condition corresponds to a pure hydrostatic macroscopic stress (pure pressure) and the corresponding velocity field found during the solution of the limit analysis problem describes the expansion of the cavity.However, for the case of sheet metals, the condition of plane stress precludes macroscopic stresses with large triaxiality or ratio of mean stress to equivalent stress, including the pure hydrostatic case. Also, pure plane stress yield criteria like BBC2005 must first be extended to 3D stresses before attempting to develop a Gurson-type model and such extensions are purely phenomenological with no due account for the out- of-plane anisotropic properties of the sheet. Therefore, we propose a new calibration method for Gurson- type models that uses only boundary conditions compatible with the plane stress requirement. For each such boundary condition we use a spectral method to solve the limit analysis problem, using as spectral basis a newly developed Mie decomposition of incompressible velocity fields for ellipsoidal cells with confocal ellipsoidal voids, extending the well-known Lee and Mear family corresponding to the spheroidal axisymmetric case [1]. We thus obtain a series of points located on the Gurson-type yield surface and their corresponding normal directions. These points are used as input to some enhanced parameter identification method developed for anisotropic yield criteria [1] to determine the calibration of the Gurson-type models. We finally use these newly calibrated Gurson-type models to evaluate forming limits for porous sheet metals.

Experimental validation of numerical model for asymmetric deep drawing of DP780 steel sheet using digital image correlation

A validated numerical model for in-plane stress/strain prediction is essential in understanding the deformation behaviour of sheet metal forming process, in particular, asymmetric deep drawing of advanced high strength steel sheet. In this work, the Yld2000-2d anisotropic yield criterion integrated with the initial homogeneous anisotropic hardening model was employed to describe the complex material behaviours of DP780 steel as well as the adoption of Yoshida chord model for elastic modulus degradation. Digital image correlation technique was utilized to measure the in-plane strain and shape deviation of the developed P- channel. The validity of the numerical model was assessed by comparing the predicted strain distribution and twist springback with the measured results. It indicates that the developed numerical model based on the selected constitutive models is acceptable for the deformation analysis, although the predicted discrepancies still exist.

Material parameters identification procedure for BBC2003 yield criterion and earing prediction in deep drawing

Formation of wavy boundaries at the open end of a drawn cup is referred to as earing. Plastic anisotropy is the main parameter contributing to formation of ears. Therefore, in order to eliminate the undesirable ears in sheet metal parts, it is essential to predict the deformation behavior of sheet metals using advanced anisotropic yield criteria. Accordingly, the objective of this article is that to use BBC2003 yield criterion, to predict earing in deep drawing of cylindrical cups from AA3105 aluminum alloy. First, a new and simple approach is introduced to calculate the eight constants of this yield function. In this approach, only the uniaxial and plane strain tension tests results, were used. Nonlinear governing equations of these constants were derived and solved by means of well-known steepest descend method. In order to investigate performance and accuracy of this method, the calculated constants were utilized to predict normalized uniaxial yield stress, anisotropy parameters and plane stress yield surface. Then, based on this yield criterion and the associated flow rule, a VUMAT subroutine was developed and employed in the commercial finite element software ABAQUS/Explicit dynamics. The simulation of drawing process was conducted for anisotropic AA3105 aluminum alloy sheet. Cup height (earing profile) and thickness distribution were obtained from experiments and corresponding numerical simulations. Comparison of these results demonstrates great potential of BBC2003 yield criterion in accurate prediction of earing.

Damage and Failure Analysis of AZ31 Alloy Sheet in Warm Stamping Processes

In this study, a combined experimental-numerical investigation on the failure of AZ31 Mg alloy sheet in the warm stamping process was carried out based on modified GTN damage model which integrated Yld2000 anisotropic yield criterion. The constitutive equations of material were implemented into a VUMAT subroutine for solver ABAQUS/Explicit and applied to the formability analysis of mobile phone shell. The morphology near the crack area was observed using SEM, and the anisotropic damage evolution at various temperatures was simulated. The distributions of plastic strain, damage evolution, thickness, and fracture initiation obtained from FE simulation were analyzed. The corresponding forming limit diagrams were worked out, and the comparison with the experimental data showed a good agreement.

Influence of material models on theoretical forming limit diagram prediction for Ti–6Al–4V alloy under warm condition

Forming limit diagram (FLD) is an important performance index to describe the maximum limit of principal strains that can be sustained by sheet metals till to the onset of localized necking. It offers a convenient and useful tool to predict the forming limit in the sheet metal forming processes. In the present study, FLD has been determined experimentally for Ti–6Al–4V alloy at 400 °C by conducting a Nakazima test with specimens of different widths. Additionally, for theoretical FLD prediction, various anisotropic yield criteria (Barlat 1989, Barlat 1996, Hill 1993) and different hardening models viz., Hollomon power law (HPL), Johnson–Cook (JC), modified Zerilli–Armstrong (m-ZA), modified Arrhenius (m-Arr) models have been developed. Theoretical FLDs have been determined using Marciniak and Kuczynski (M–K) theory incorporating the developed yield criteria and constitutive models. It has been observed that the effect of yield model is more pronounced than the effect of constitutive model for theoretical FLDs prediction. However, the value of thickness imperfection factor (f0) is solely dependent on hardening model. Hill (1993) yield criterion is best suited for FLD prediction in the right hand side region. Moreover, Barlat (1989) yield criterion is best suited for FLD prediction in left hand side region. Therefore, the proposed hybrid FLD in combination with Barlat (1989) and Hill (1993) yield models with m-Arr hardening model is in the best agreement with experimental FLD.

On the modelling of complex kinematic hardening and nonquadratic anisotropic yield criteria at finite strains: application to sheet metal forming

In the present paper, a finite strain model for complex combined isotropic-kinematic hardening is presented. It accounts for finite elastic and finite plastic strains and is suitable for any anisotropic yield criterion. In order to model complex cyclic hardening phenomena, the kinematic hardening is described by several back stress components. To that end, a new procedure is proposed in which several multiplicative decompositions of the plastic part of the deformation gradient are considered. The formulation incorporates a completely general format of the yield function, which means that any yield function can by employed by following a procedure that ensures the principle of material frame indifference. The constitutive equations are derived in a thermodynamically consistent way and numerically integrated by means of a backward-Euler algorithm based on the exponential map. The performance of the constitutive model is assessed via numerical simulations of industry-relevant sheet metal forming processes (U-channel forming and draw/re-draw of a panel benchmarks), the results of which are compared to experimental data. The comparison between numerical and experimental results shows that the use of multiple back stress components is very advantageous in the description of springback. This holds in particular if one carries out a comparison with the results of using only one component. Moreover, the numerically obtained results are in excellent agreement with the experimental data.

Mechanical design of a heterogeneous test for material parameters identification

The main goal of this work is to design virtually a heterogeneous test, with appropriate specimen shape and boundary conditions, which leads to an inhomogeneous strain field promoting the mechanical behavior characterization of thin metallic sheets under several strain paths and strain amplitudes. Finite element simulations were carried out with a virtual material, described by an anisotropic yield criterion (Yld2004-18p) associated to a mixed hardening law. The material parameters were derived from a large experimental database of quasi-homogeneous classical tests. A shape and boundary conditions optimization process was developed based on a quantitative indicator rating the strain field information and used as a cost function for guiding the test design. A heterogeneous test showing a butterfly shape was obtained, with strain states ranging from simple shear to plane strain tension. In addition, the designed heterogeneous test was used to determine the material parameters of the aforementioned constitutive model. The reliability of this identified material parameters set was then assessed and compared with the one coming from the experimental database composed by the quasi-homogeneous classical tests.

Analytical prediction of springback in bending of tailor-welded blanks incorporating effect of anisotropy and weld zone properties

An analytical model for the prediction of springback in bending of longitudinally welded tailor-welded blanks of different thickness is presented in this paper. The effect of strain hardening, anisotropy and weld zone has been incorporated in the analytical model. Power law of strain hardening and Hill’s anisotropic yield criterion have been used in the development of the analytical model for prediction of springback in tailor-welded blanks. The predicted values of springback are validated with experiments on V-bending of laser-welded blanks of Extra Deep Drawing quality steel sheets. Longitudinally welded specimens of three different thickness combinations with weld line oriented at 0°, 45° and 90° to the rolling direction are tested to investigate the effect of anisotropy. The springback values predicted by the analytical model incorporating the weld properties are found to be in good agreement with the experimental results in all of the cases. The springback has been found to be maximum when the weld line is oriented at 45° to the rolling direction.

Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets

This article is a review regarding recently developed inverse strategies coupled with finite element simulations for the identification of the parameters of constitutive laws that describe the plastic behaviour of metal sheets. It highlights that the identification procedure is dictated by the loading conditions, the geometry of the sample, the type of experimental results selected for the analysis, the cost function, and optimization algorithm used. Also, the type of constitutive law (isotropic and/or kinematic hardening laws and/or anisotropic yield criterion), whose parameters are intended to be identified, affects the whole identification procedure.

A finite strain constitutive model for non-quadratic yield criteria and nonlinear kinematic/isotropic hardening: application to sheet metal forming

In this paper, a finite strain material model for complex plastic anisotropy with nonlinear isotropic and kinematic hardening is consistently derived. The model is based on the classical multiplicative decomposition of the deformation gradient and derived in a thermodynamically consistent way. An important new aspect of the work is the straightforward implementation of general and anisotropic yield criteria into a constitutive model, which is entirely formulated in the reference configuration. Nevertheless, and for the sake of illustrating the potential of the model, in this work a Barlat-type (Yld2004-18p) yield criterion is employed. The kinematic hardening formulation follows the continuum mechanical extension of the classical rheological model of Armstrong–Frederick hardening. The numerical integration of the evolution equations is carried out using the exponential map approach, which is able to preserve plastic incompressibility. For numerical efficiency purposes, the exponential tensor functions are evaluated in a closed form using the spectral decomposition, and special attention is given to the preservation of the internal variables’ symmetry. The model is assessed by means of several numerical simulations for anisotropic materials at finite strains, including sheet metal forming processes with comparison to experimental data.

Twist springback characteristics of dual-phase steel sheet after non-axisymmetric deep drawing

This paper aims to investigate the twist springback characteristics of advanced high strength steel sheet subjected to deep drawing. A C-rail benchmark, which leads to a particularly pronounced twist springback characteristics, was developed. For an accurate numerical modeling of the process, a non-quadratic anisotropic yield criterion integrated with combined isotropic and kinematic hardening model was used to describe the strain-stress behavior including anisotropy and Bauschinger effects. The corresponding mechanical experiments, namely uniaxial tension and forward-reverse simple shear tests were performed to determine the material parameters. The digital image correlation technique was applied for component tests as well as the deformation and stress-strain analysis. The experimental validation of the elastic-plastic finite element model was assessed by comparing maximum in-plane strain, thickness reduction distribution and twist springback of the drawn rail. To explore the source of twist springback, the deformation associated with in-plane stress and bending moment was analyzed. The results indicate that the bending moment before springback caused by non-symmetric stress states play an important role in twist springback and control. Certain regions of the die radius were varied in a numerical analysis to control the bending moment for the minimization of twist springback as well as the preliminary results of the relationship between the ratio of variable die radius and twist springback.

A general anisotropic yield criterion for pressure-dependent materials

Many engineering materials exhibit strong anisotropy along with a pressure dependence on the plastic yield behavior. The ability to predict the effects of pressure dependence and plastic anisotropy of these materials on their respective yield surfaces is important for accurately analyzing their behavior during deformation. The current work develops a general anisotropic yield criterion that includes the capability of modeling pressure-dependent effects either additively or multiplicatively in the yield criterion at the onset and during the evolution of plastic deformation. The developed yield criterion can be used to represent both quadratic and non-quadratic yield surfaces and has no restrictions on the symmetricity of the plastic behavior, allowing the entire spectrum from isotropic to fully anisotropic. The pressure dependence of the yield criterion and the corresponding plastic rate of deformation equation are expressed as general functional forms to allow development and incorporation of new pressure dependence functions or to accept existing pressure dependence functions directly. Therefore, the developed yield criterion can be used as a framework for accurately modeling the plastic behavior of various material systems, largely reducing the complexity in yield surface description for different classes of materials and allowing a systematic approach for implementation in numerical analysis procedures such as the finite element method. Two existing pressure-dependent models are reformulated to show the applicability of the general yield criterion framework presented herein to developing new non-quadratic anisotropic models with a dependence on pressure. While the general yield criterion is developed as an extension of a specific non-quadratic anisotropic yield criterion, the same model development methodology can be applied to other novel or existing yield criteria that are defined in the deviatoric stress space.

Application of a Dung’s Model to Predict Ductile Fracture of Aluminum Alloy Sheets Subjected to Deep Drawing

In this paper, prediction of failed evolution of anisotropic voided ductile materials will be developed based on Dung’s microscopic damage model. An isotropic and anisotropic formulation of the Dung’s damage model that using von Mises yield criterion and Hill’s quadratic anisotropic yield criterion (1948) integrated with isotropic hardening rules of matrix material used to simulate the deep drawing process of aluminum alloy sheets. The model is implemented as a vectorized user-defined material subroutine (VUMAT) in the ABAQUS/Explicit commercial finite element code. The predictions of ductile crack behavior in the specimens based on void nucleation, growth and coelescence are compared with Gurson – Tvergaard – Needleman (GTN) model and experiment results from reference.