Isotropic Yield Function Research Articles

Earing Prediction of AA5754-H111 (Al-Alloy) with Linear Transformation-Based Anisotropic Drucker Yield Function under Non-Associated Flow Rule (Non-AFR).

The yield behavior of aluminum alloy 5754-H111 under different stress conditions for three kinds of plastic work is studied using an anisotropic Drucker model. It is found that when the plastic work is 30 MPa, the anisotropic Drucker model has the most accurate prediction. Comparing the Hill48 and Yld91 models with the Drucker model, the results show that both the anisotropic Drucker and Yld91 models can accurately predict the yield behavior of the alloy. Cylinder drawing finite element analysis is performed under the AFR, but it is not possible to accurately predict the position and height of earing appearance. The anisotropic Drucker model is used to predict the earing behavior under the non-AFR, which can accurately predict the earing phenomenon. Numerical simulation is conducted using three different combinations of yield functions: the anisotropic yield function and the anisotropic plastic potential function (AYAPP), the anisotropic yield function and the isotropic plastic potential function (AYIPP), and the isotropic yield function and the anisotropic plastic potential function (IYAPP). It is concluded that the influence of the plastic potential function on predicting earing behavior is more critical than that of the yield function.

Particle-matrix debonding with Strength-Differential effects

In this work the onset of failure caused by particle-matrix debonding in aluminum alloy 2090-T3 and magnesium alloy AZ31 under either overall uniaxial plane strain tension or compression load is studied numerically using a unit cell model. Especially for magnesium AZ31, experiments show a significant difference in the tensile-compressive yield stresses, the so-called Strength-Differential-effect (SD-effect). This SD-effect cannot be captured by any isotropic yield function and only by recent anisotropic yield functions. To accurately model AA 2090-T3 and magnesium AZ31 the general yield function by Yoon et al. (2014) is adopted in addition to the isotropic von Mises plasticity yield function and the anisotropic yield function of Hill (1948). The numerically predicted strain at which void nucleation initiates is generally found to be larger for magnesium AZ31 than for AA 2090-T3. However, depending on the orientation of the principal axes of plastic anisotropy this void nucleation strain varies significantly. This holds true for both materials, but the void nucleation strain of AA 2090-T3 in compression is numerically predicted to be practically identical whether or not isotropic or anisotropic plasticity is considered. In tension the void nucleation strain of AA 2090-T3 based on anisotropic plasticity is found to be more than double the strain obtained by von Mises plasticity. The numerically predicted void nucleation strain of magnesium AZ31 is significantly affected by the yield function adopted. For the von Mises yield function the value of the void nucleation strain is in between the predictions based on the anisotropic yield functions. For both tension and compression, the yield function of Yoon et al. (2014) results in the largest void nucleation strain, whereas the yield function of Hill results in the smallest strain value. This holds true for all five orientations of the principal axes of plastic anisotropy investigated.

Two novel alternative integration schemes for multi-invariants dependent isotropic finite deformation plasticity

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.

Bendability assessment of high strength aluminum alloy sheets via V-die air bending method

The bendability of high-strength aluminum alloy (AA) sheets for two different thicknesses was experimentally evaluated by the V-die air bending method and the numerical simulation procedure was attempted to comprehend the characteristics of the V-die air bending method. Despite similar bendability along the material direction for a given punch radius in the experiment, the distinguished fracture strains were predicted for each direction based on the punch force and displacement information. For the numerical simulation, V-die air bending was modeled with the full 3D non-quadratic anisotropic yield criterion and compared with the results of the isotropic yield function estimating that bendability along the rolling direction is higher than those along the transverse direction. The thinner material showed higher anisotropy in bendability and the characteristics of V-die air bending were also discussed suggesting a maximum value as an upper bound of the bendability. As a comparison, an anisotropic analytical formulation was also newly developed combined with the kinematic assumption and the limitation was also covered in the study. The analytical fracture strain was discussed as the equivalence to the maximum value suggested in the research.

An invariant-based elasto-visco-plastic model for unidirectional polymer composites at finite strains

An invariant-based constitutive model for unidirectional composites, including viscous effects in the elastic and plastic regimes at finite strains, is proposed. The model is based on the multiplicative decomposition of the deformation gradient, using an isoclinic configuration to avoid the non-uniqueness of intermediate configurations. The visco-elastic behaviour is represented by the generalised Maxwell model, combined with a transversely isotropic yield function and a non-associative plastic potential. Visco-plastic effects are introduced through the Perzyna overstress function. The theoretical basis of the material model and the computational treatment within the finite element solution procedure are presented in detail. Two algorithms are proposed for an implicit integration of the evolution equations: (i) a semi-implicit stress update, and (ii) a fixed-point iterative fully implicit stress update. A comparison of their performance reveals that the former becomes faster, even though the latter guarantees the quadratic convergence rate in the Newton–Raphson scheme. The predicted stress–strain response for different fibre orientation angles is in close agreement with experimental data obtained under quasi-static and dynamic loading. The model can also capture the important re-orientation of the fibres due to external loading.

Limit load instabilities of anisotropic tubes under combined tension and torsion

Under tensile loadings, thin-walled structures such as sheet metal and tubes develop load maxima, or limit loads, beyond which deformation localizes leading to rupture. For uniform stress states, Considère-type estimates of limit loads can serve as forming limits in applications. The paper explores the effect of anisotropy on such estimates for thin-walled Al-alloy tubes under combined tension and torsion. Anisotropy is modeled using Yld04-3D with an exponent of 8. Material hardening originates from a simple shear test using this yield function, taking into account material axes rotation caused by the shearing. A Considère formulation is developed for the problem, which also incorporates the effect of material frame rotation. The analysis is used to establish limit loads for a set of circumferentially constrained tension–torsion experiments tested under radial nominal tension-shear stress paths. The predictions reproduce the strains measured at the limit loads for the range of biaxiality ratios considered. By contrast, corresponding results produced using the isotropic yield functions of von Mises and Hosford(8) increasingly deviate from the measured results as the shear stress increases. Considère-type formulation is also developed for the same tension–torsion loadings for a uniform thickness tube. The results exhibit a similar trend but the limit strains for shear dominant paths are significantly lower, demonstrating the stabilizing effect of the circumferential constraint used in the experiments.

Modeling of thin sheet forming processes by combining solid-shell finite element with isotropic elastoviscoplastic model. Application to magnetic pulse forming processes

Sheet metal alloys are used in many industries to save material, reduce weight and improve the overall performance of products. For the last decades, many types of elements have been developed to resolve the locking problems encountered in the simulation of thin structures. Among these approaches, a family of assumed-strain solid-shell elements has proved to be very efficient and attractive in simulating thin 3D structures with various constitutive models. Furthermore, these elements are able to account for anisotropic behavior of thin structures since isotropic yield functions cannot capture the real physics of some forming processes. In this work, von Mises isotropic yield criterion with Johnson-cook hardening model are combined with a linear prismatic solid-shell element to simulate sheet metal forming processes. A new element assembly technique has been developed to permit the assembly of prismatic elements in a tetrahedral element-based software. This technique splits the prism into multiple tetrahedral elements in such a way that all the cross-terms are accounted for. Furthermore, a tetrahedral based partitioning code has been modified to account for the new prismatic element shape without changing the core structure of the code. More accurate results were obtained using low number of solid-shell elements compared to its counterpart tetrahedral element (MINI element). This reduction in the number of elements accelerated the simulation, especially in the coupled magnetic-structure simulation used for magnetic pulse forming process. The proposed element and criteria are implemented into FORGE (in-house code developed at CEMEF) for simulating magnetic pulse forming process.

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.

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.

Correlative Study on Plastic Response and Formability of Ti-6Al-4V Sheets under Hot Forming Conditions

Abstract In this study, the formability behavior of Ti-6Al-4V alloy sheets under uniaxial and multi-axial loading conditions at 923 K (650 °C), which has received less attention till now, has been extensively investigated by experiments and numerical modeling. Uniaxial tension tests at various strain rates and limit dome height (LDH) tests at the elevated temperature, covering the uniaxial stretch, plane strain, and biaxial stretch deformation modes, were conducted, and the corresponding constitutive models were constructed. Based on the experimentally characterized mechanical properties and associated constitutive laws, finite element simulations of the LDH tests were conducted, and the formability of the investigated material was analyzed using the Marciniak-Kuczyinski theory by employing the von-Mises isotropic yield function and Voce hardening law, including strain-rate sensitivity. At the elevated temperature, a strong positive rate sensitivity effect was observed. Finally, the predicted load-displacement responses of the LDH tests were in good agreement with the measured data. The corresponding formability analysis can contribute to the design of optimal hot forming processes for Ti-6Al-4V sheets.

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.

Advances in anisotropy of plastic behaviour and formability of sheet metals

This paper reviews the most recent models for description of the anisotropic plastic behavior and formability of sheet metals. After a brief review of classic isotropic yield functions, recent advanced anisotropic criteria for polycrystalline materials of various crystal structures and their applications to cup drawing are presented. Next, the discussion focuses on novel formulations of anisotropic hardening. A brief review of the experimental methods used for characterizing and modeling the anisotropic plastic behavior of metallic sheets and tubes under biaxial loading is presented. The experimental methods and theoretical models used for measuring and predicting the limit strains, development of new tests for determining the Forming Limit Curves (FLC), as well as on studying the influence of various material or process parameters on the limit strains are presented.

A non-quadratic pressure-sensitive constitutive model under non-associated flow rule with anisotropic hardening: Modeling and validation

A non-associated flow rule (non-AFR) constitutive model is proposed to describe the evolving yield stress and plastic potential surfaces of sheet metal under plane stress conditions. The yield stress function is a multiplication of two yield functions, within the first part includes an asymmetric pressure-sensitivity that considers anisotropic and asymmetric yielding of sheet metals. This yield stress function can describe anisotropic hardening behavior by directly employing the hardening functions under seven different loading conditions, i.e., uniaxial tension along 0°, 45°, 90° to the rolling direction (RD) of sheet metal, uniaxial compression along the same orientations and equi-biaxial tension without any interpolation or optimization procedure at a discrete level of equivalent plastic strain. The second part is a non-quadratic isotropic function, which is the sum of two isotropic yield functions with high flexibility in shaping the yield surface. A new plastic potential function is proposed to capture the differential Lankford coefficients (r-values) between uniaxial tension and compression obtained from mechanical testing. Eight parameters in the plastic potential function were calibrated from seven r-values under the same loading conditions and one yield stress as calibrating parameters for the yield stress function. The proposed constitutive model was validated by experimental data of three engineering sheet metals, i.e., one dual-phase steel DP980, one TRIP-assisted steel QP980 and one aluminum alloy AA5754-O and compared with several classical yield criteria. The evolving yield behavior was accurately described by the proposed non-associated flow rule (non-AFR) constitutive model.

Cyclic Plasticity of the As-Built EOS Maraging Steel: Preliminary Experimental and Computational Results

This short communication offers a preliminary view on ongoing research conducted on the as-built EOS maraging steel 300. The material’s cyclic elastoplastic characteristics under strain-controlled loading have been investigated experimentally. Specimens fabricated under two primary orientations, horizontally and vertically to the build plate, have been tested. The obtained stress–strain hysteresis loops exhibited symmetry, with the vertical specimen showing a higher plastic strain energy dissipation capability than the horizontal specimen. Modelling of the material’s elastoplastic behaviour was performed with a commonly used kinematic hardening rule, combined with both isotropic and anisotropic yield functions and elasticity moduli. The obtained simulations of the hysteresis loops, from the implementation of these two plasticity models, indicate the advantage of the anisotropic modelling approach over the isotropic approach. The anisotropic plasticity model describes in a more representative way the inherent elastic and plastic anisotropy of the as-built material. Further research is underway to explore the low cycle fatigue performance of this additively manufactured metal.

Damage prediction of hot stamped boron steel 22MnB5 with a microscopic motivated ductile fracture criterion: Experiment and simulation

With the increasing demand of the reduction of energy consumption and enhancement of automobile safety, the steels with ultra-high strength are widely utilized in automobile industry. Because of the high achievable strength, hot stamping process can be used to supply promising automobile components. The increasing use of high-strength steels leads to great challenges to model the fracture evolution because of their high strength and low ductility. The paper is concerned with plasticity and ductile fracture modelling of hot stamped boron steel 22MnB5, where martensite phase with high yield strength, work hardening and ductility are present during the quenching process. The Lou's ductile fracture criteria which involve void nucleation, growth and coalescence in metallic materials are theoretically reviewed and extended and implanted using users’ subroutine VUMAT in ABAQUS/Explicit. Parameters of the selected fracture criterion are determined using calibration tests with various kinds of specimens. Dogbone-like uniaxial tensile specimens are tested to determine the deformation hardening parameters, and specimens with central hole, in-plane shear specimens and notched specimens, balanced biaxial tensile specimens are tested to identify the fracture behavior in various loading stresses. Digital Image Correlation (DIC) method is applied to monitor the surface pattern and record the strain evolution of the specimen surface during the experiment. Because of fully austenitized microstructure, the Mises isotropic yield function is adopted. The hybrid method is used to obtain the fracture strain in various stress directions and implemented to determine the material parameters of fracture model. The constructed ductile fracture surfaces calibrated with fracture strain using DIC calculation and hybrid method are compared in Lode parameter and triaxiality coordinate as well as the principal stress space, respectively. The comparison reveals that the fracture strain obtained by hybrid method is more reasonable. The determined ductile fracture criterion is applied to forecast the fracture onset for seven kinds of specimens and compared with the experiment data. The comparison indicates the simply extended Lou's criterion of ductile fracture can precisely describe the evolution of ductile fracture from uniaxial tensile to biaxial tensile stress state for hot stamped boron steel 22MnB5.

Effective behaviour of porous ductile solids with a non-quadratic isotropic matrix yield surface

In this study, we examine the macroscopic yielding of isotropic porous ductile solids having a matrix yield function dependent on the second and third deviatoric stress invariants. Numerical limit analyses using a three-dimensional finite element model of a hollow sphere with a Hershey-Hosford matrix yield function are conducted for different shapes of the matrix yield surface and porosity levels. These numerical results are then used to elucidate first-order effects of the third deviatoric stress invariant on the macroscopic yielding and further used as reference data to assess the performance of two porous plasticity models that incorporate effects of the third deviatoric stress invariant using the isotropic non-quadratic Hershey-Hosford yield function. The first model is derived from an upper-bound limit analysis of the hollow sphere representative volume element using the Gurson-Rice trial velocity field, but with a rather general isotropic matrix yield function. The second model is a simple, heuristic extension of the Gurson model incorporating the equivalent stress measure of the Hershey-Hosford yield function.From the numerical limit analyses, it is found that the contours of the macroscopic yield surface in the deviatoric plane transform from the hexagonal shape of the underlying matrix yield surface to a rounded triangular shape that converges to the circular shape of the Gurson model as the macroscopic stress triaxiality ratio increases. This shape transformation is dependent upon the porosity level. The upper-bound model was found to be in very good agreement with the numerical data for all stress states, shapes of the matrix yield surface, and porosity levels. The heuristic model provides good predictions for low and moderate levels of porosity pertinent to ductile fracture, but the predictions deteriorate when the stress triaxiality ratio and the porosity level increase.We also address the issue of how representative the spherical unit cell is for the description of real porous solids. To that end, we make comparisons between a space-filling representative volume element in the form of a cubic unit cell with a centric spherical void and the hollow sphere model. These results show that the hollow sphere model generally provides slightly higher values for the yield limits. The shape of the yield loci is similar for the two models in the case of non-quadratic matrix yield surfaces, while the cubic model gives a different shape of the yield loci for low and intermediate stress triaxiality ratios when the von Mises yield function is used for the matrix.

Effects of solution and quenching treatment on the residual stress in extruded ZK60 magnesium alloy

The quench induced residual stress in extruded ZK60 magnesium alloy is investigated. X-ray diffraction and layer removal technique are employed to determine the residual stress in extrusion and long transverse direction through the thickness of the extruded plate after solution and quenching treatment. FEM simulation is employed to clarify the origin of the stress state. The residual stress is not uniform through thickness, it is compressive on the surface and first decreases and then increases in absolute magnitude with increasing depth and becomes tensile in the center. Anisotropic residual stress can be detected, with larger stress in long transverse direction and smaller stress in extrusion direction. This is rationalized by the anisotropic flow stress caused by the tilted basal texture. Anisotropic residual stress can be replicated in simulation using an isotropic yield function at high temperature and an anisotropic yield function at low temperature during quenching, the smaller flow stress in extrusion direction facilitates plastic flow at low temperature and results in smaller compressive residual stress in extrusion direction. The fact that no twins can be found after quenching demonstrates that the asymmetric yield strength associated with twinning does not affect the residual stress.

Limit analysis and homogenization of nanoporous materials with a general isotropic plastic matrix

In this paper, a closed-form expression of a macroscopic strength criterion for ductile nanoporous materials is established, by considering the local plastic behavior as dependent on all the three isotropic stress invariants and by referring to the case of axisymmetric strain-rate boundary conditions. The proposed criterion also predicts void-size effects on macroscopic strength properties. A homogenization procedure based on a kinematic limit-analysis is performed by addressing a hollow-sphere model comprising a rigid-ideal-plastic solid matrix. Void-size effects are accounted for by introducing an imperfect-coherent interface at the cavity boundary. Both the interface and the solid matrix are assumed to obey to a general isotropic yield function, whose parametric form allows for a significant flexibility in describing effects induced by both stress triaxiality and stress Lode angle. Taking advantage of analytical expressions recently provided by Brach et al. [Int J Plasticity 2017; 89: 1–28] for the corresponding support function and for the exact velocity field under isotropic loadings, a parametric closed-form relationship for the macroscopic strength criterion is obtained as the solution of an inequality-constrained minimization problem, the latter being faced via the Lagrangian method combined with Karush-Kuhn-Tucker conditions. Finally, several comparative illustrations are provided, showing the influence of local-yield-function parameters on the established criterion, as well as the model capability to describe the macroscopic strengthening, typical of nanoporous materials, induced by a void-size reduction for a fixed porosity level.

Fracture-based forming limit criteria for anisotropic materials in sheet metal forming

This paper is concerned with modeling of fracture-based forming limit criteria for anisotropic materials in sheet metal forming to predict the sudden fracture in complicated forming processes. The Lou–Huh ductile fracture criterion is modified using the Hill's 48 anisotropic yield function instead of the von Mises isotropic yield function to take account of the influence of anisotropy on the equivalent plastic strain at the onset of fracture. For the derivation of an anisotropic ductile fracture criterion, the principal stresses (σ1,σ2,σ3) are expressed in terms of the stress triaxiality, the Lode parameter, and the equivalent stress (ηH,LP,σ¯H) based on the Hill's 48 yield function. Three different kinds of fracture-based forming limit criteria are suggested and investigated with an assumption that the stress state is under the plane stress condition with proportional loading. To determine the parameters of the model proposed, the two-dimensional digital image correlation (2D-DIC) method is utilized to measure the strain histories on the surface of three different types of specimens during deformation and the measurement results are investigated to identify the anisotropy effect on the equivalent plastic strain at the onset of fracture. This paper also discusses about a scaling method for a strain-based fracture forming limit criterion in order to capture the onset of fracture using a single forming limit curve for an anisotropic material. From the comparison of various forming limit criteria suggested, it is concluded that a polar effective plastic strain-based (PEPS) fracture forming limit diagram (FFLD) is suitable for prediction of the sudden fracture in AHSS sheets in complicated sheet metal forming processes on the basis of its path independence and simplicity of measuring strains in real forming processes.

On the modeling of asymmetric yield functions

A first degree homogeneous yield function is completely determined by its restriction to the unit sphere of the stress space; if, in addition, the function is isotropic and pressure independent, its restriction to the octahedric unit circle, the π -circle, is periodic and determines uniquely the function. Thus any homogeneous, isotropic and pressure independent yield function can be represented by the Fourier series of its π -circle restriction. Combinations of isotropic functions and linear transformations can then be used to extend the theory to anisotropic convex functions. The capabilities of this simple, yet quite general methodology are illustrated in the modeling of the yielding properties of AZ31B magnesium alloy.