Yield Locus Research Articles

User-friendly anisotropic hardening function with non-associated flow rule under the proportional loadings for BCC and FCC metals

This paper presents a user-friendly function under non-associate flow rule to capture anisotropic hardening (or evolution of yield surface) and the curvature of yield loci for body-centred cubic (BCC) and face-centred cubic (FCC) under the proportional loading conditions. The calibrated model for BCC and FCC metals is applied to illustrate the hardening behavior of uniaxial tension along different loading directions and of the equibiaxial tension for several steels and aluminum alloys (718AT, QP980, AA5182-O and AA 6xxx-T81) to evaluate its accuracy. The prediction is compared with the experimental results and modeling results of several popular plasticity models. The comparison demonstrates that the proposed model is pressure-insensitive, relatively simple for numerical implementation, capable of illustrating the different yielding behaviors between BCC and FCC metals and, capable of accurate modeling of flow curves under directional loading directions and equibiaxial tension both under the plane stress and triaxiality loading conditions. The proposed model is also flexible on adjusting the curvature of the yield surface between uniaxial tension and equibiaxial tension. Accordingly, the proposed anisotropic hardening model is useful to illustrate anisotropic hardening behaviors under uniaxial and equibiaxial tension.

Experimental characterization and theoretical modeling of size-dependent distortional hardening behavior of ultrathin metal sheets under multi-axial loading

The research on distortional hardening (DH) behavior of ultrathin metal sheets under multi-axial proportional loading affected by size effect is still missing and existing constitutive models cannot accurately predict the size-dependent distortional hardening (SDDH) behavior of ultrathin metal sheets. To solve the problem and fill the gap, experimental investigation and theoretical modeling of SDDH behavior of ultrathin SUS304 sheets under biaxial proportional loading is conducted in this paper. Firstly, the experimental yield loci of the ultrathin SUS304 sheets with different thicknesses and grain sizes are characterized through biaxial and uniaxial tensile tests. Experimental results show that DH behavior exists during the micro-scale plastic deformation of the ultrathin SUS304 sheets and that the degree of DH is amplified with the increase of the grain size. Subsequently, a mixed hardening constitutive model considering SDDH is proposed based on the Yld2000-2d yield criterion and its effectiveness to predict the evolving yield behavior of ultrathin SUS304 sheets with different grain sizes is verified. In addition, the capability of the developed model to predict the plastic flow behavior of the ultrathin SUS304 sheets under biaxial loading cases is explored. It is found that the prediction accuracy by the developed model is higher than that by the Yld2000-2d yield criterion. The developed constitutive model has a stronger capability than the Yld2000-2d yield criterion combined with isotropic hardening, extensively implemented in commercial software, to enhance the prediction precision of plastic deformation behavior of the ultrathin sheets in micro forming.

Effects of biaxial tensile mechanical properties and non-integer exponent on description accuracy of anisotropic yield behavior

It is necessary to identify differences among biaxial tensile mechanical properties to describe the plastic anisotropy and potential adjustment ability of yield criteria with the non-integer exponent for the yield surface. Therefore, in this study, uniaxial and cruciform biaxial tensile tests were performed under 17 different loading paths: uniaxial tension in seven different directions, cruciform biaxial tension in rolling/transverse and 45°/135° sampling directions with seven and three different stress ratios, respectively. Based on the BBC2008 yield criterion, the uniaxial yield stresses, rθ-values, yield loci on the normal plane, and shear yield loci on the diagonal plane, predicted using six parameter identification strategies, were quantitatively evaluated for MP980, DP490, 6016-T4, and 5182-O. Results show the constraining and regulation ability of the equi-biaxial tensile data for yield loci to be better than that of near-plane strain state data. The parameter identification strategy considering the non-integer exponent was observed to significantly improved the ability of the yield criterion to describe the anisotropic yield behavior. For a simplified evaluation system that considers only the prediction accuracy of the yield locus under the principal stress state, neglecting the prediction accuracy for the shear yield locus may lead to incorrect judgments regarding the best identification strategy.

Strain rate dependent plasticity of lithium-ion pouch cells: Experiments and simulations

The safety of electric vehicles under crash event depends on the mechanical behavior of Lithium-ion cells under large deformation at high strain rates. Here, an extensive experimental campaign is performed on large-format pouch cells under out-of-plane indentation with two indenter shapes and speeds ranging from a few millimeters per minute up to ten meters per seconds, spanning six decades of strain rates. It reveals that the displacement at the onset of short circuit decreases with increasing strain rates, while a non-monotonic relationship is observed between the strain rate and the maximum force as well as the macroscopic cell tangent stiffness. Based on the experimental results, a phenomenological constitutive model is proposed making use of a Deshpande-Fleck yield locus and strain-rate dependent hardening. An internal variable is introduced to capture the softening at intermediate strain rates. The calibrated model is able to accurately reproduce the experimental data and is further validated on a hemispherical indentation performed at two meters per seconds.

Capacity of double-plate vertically loaded anchor in saturated marine fine sand

Experimental and finite element analyses were conducted to formulate the relationship between drag force and drag length as well as the ultimate loading capacity for the double-plate vertically loaded anchor (DPVLA) model in saturated marine fine sand. Results show that compared to the conventional VLA model, the DPVLA model can significantly enhance the peak drag force value and ultimate loading capacity. In the drag penetration test, the peak drag force value for the DPVLA model increases with the initial orientation and included angle. When the drag force is at its peak, the drag length of the DPVLA model is approximately six times the width of the model’s upper fluke; this drag length is smaller than that of the VLA model. In addition, increasing the included angle significantly improves the ultimate loading capacity of the DPVLA model in the ultimate loading test when the included angle is 0–30°. However, the DPVLA model cannot facilely penetrate the soil layer in the drag penetration test when the included angle exceeds 30°; hence, this limit must be observed. Increasing the bottom fluke length of the DPVLA model can also increase the ultimate loading capacity. This increases the anchor model weight, enabling the anchor to rapidly sink to the seabed. Based on numerical simulation results, a yield locus equation for the ultimate loading capacity of the DPVLA model subjected to combined loading in saturated fine sand was proposed. The penetration trajectory for the DPVLA model was also obtained from the yield locus equation of ultimate loading capacity.

Investigation on Plastic Flow Behaviors of FCC Polycrystalline Aluminum under Pre-Cyclic Tension-Compression Loading: Experiments and Crystal Plasticity Modeling.

The plastic flow behaviors of FCC polycrystalline aluminum after pre-cyclic tension-compression deformation are mainly investigated in tension–torsion stress space by the physically based crystal plasticity model introducing a back-stress. A global finite element model (GFEM) constructed of sufficient grains was established to simulate the same-size thin-walled tube specimen constrained and loaded as the experiments of yield surfaces. The computational results showed that the shape of subsequent yield surfaces and the plastic flow directions directly depended on the given offset strain levels and the applied re-loading paths under different pre-cyclic deformations. The angle deviation between the plastic flow direction and the theoretical orthogonal direction further indicated that there was a large difference between them in the inverse pre-straining direction, but the difference was negligible in the pre-straining direction. From the influence of the anisotropic evolution of the subsequent yield surfaces on plastic flow, we found that the plastic normality rule followed the smooth yield locus; conversely, the significant non-associated flow was attributed to the distorted yield locus. Furthermore, it was also demonstrated that the anisotropic evolution and the plastic flow trend of the subsequent yield surfaces obtained by experiments can be better reproduced by the crystal plasticity model.

Effect of shear localisation on yield surface for porous metals containing ellipsoidal voids

Diffused and localised plasticity in a narrow band between the pre-existing micro-voids are the dominant modes of deformation in a ductile material. In this paper, a yield function is presented for porous metallic materials accounting for these two modes of deformation. The model has been proposed by modifying Madou and Leblond’s model taking into account the effect of shear localisation between the inter voids ligaments on evolution of yield surface. The proposed model gives a smooth yield locus and captures the competition between defused plasticity and shear localisation as a function of void shape. The results of proposed model have been extensively validated with numerically evaluated yield locus using the finite element based limit analysis for a wide range of ellipsoidal void geometries. It has been shown that the model accurately predicts the effect of shear localisation on the yield surface.

Extended anisotropy yield criteria applied to Ti6Al4V at a high range of temperatures and considerations on asymmetric behavior

Heat assisted-forming processes are essential for Ti6Al4V sheet metal forming due to the poor formability of this alloy at room temperature. Although there exist works on the stress-strain response at high temperature, there is a lack of data in the literature on the yield criteria. The present paper presents an efficient experimental methodology to determine significant stress states at high temperature. Both plane strain data and compressive anisotropy values at high temperature allowed obtaining a more complete definition of the mechanical behavior of the Ti6Al4V, being the novel contribution of this research. Yield loci up to 900 °C were built according to different yield criteria that are fitted to the experimental data. Thus, Barlat1989 and CPB2006 were selected as the most suitable yield criteria to predict Ti6Al4V behavior. These criteria were extended to contemplate the anisotropy evolution under a wide range of working conditions. The formulated extended criteria for the entire range of conditions were close to the equivalent traditional ones applied to only a particular condition, which demonstrates the advanced made in the comprehensive modeling of Ti6Al4V alloy for all conditions applied in forming industries.

Effective yield criterion for a porous single crystal accounting for void size effects

The growth of sub-grain voids in crystalline materials is affected by plastic anisotropy as well as void size effects. In this paper, an effective yield criterion is derived for a porous single crystal using homogenization theory and limit analysis. A two dimensional planar model of a single crystal containing a random distribution of cylindrical voids is assumed. The effective yield criterion is derived using plastic limit load analysis of a hollow cylindrical representative volume element, containing a concentric cylindrical void in a single crystalline matrix. A conventional strain gradient plasticity model with an embedded material length scale is assumed for the matrix, in order to account for the void size dependence of yielding. The yield loci and void growth rates predicted by the model under plane strain conditions are validated by comparison with numerical results obtained using finite elements, as well as prior analytical results for the special case of size independent matrix behavior. The model predictions for the size dependence of the yield stress as well as void growth under proportional loading conditions are discussed with reference of known results from experiments and lower scale dislocation dynamics simulations.

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.

Approaches to analysing scatter in forming simulations: from fundamental to pragmatic

Including scatter in material properties has received attention in the last decades but, it is not fully mature yet. Industry has cooperated with academia to progress the topic, but at the same time devised simpler ways to start analysis anyway. Robustness analysis is a pre-requisite for process optimisation. Tata Steel has been in material modelling for Sheet Metal Forming for many years and has developed its own yield locus model to enable better customer support. The accuracy of this Vegter model comes at a price, however: more parameters to be measured. Wiebenga devised a robustness analysis but it requires more input than is usually available. At the same time leading software vendors introduced options to evaluate statistical behaviour. Obviously, process optimisation is only useful when accurate models can be employed, so in this paper we will focus on how the scatter in material properties translates in scatter in yield locus for the BBC2005 model vs. Vegter 2017. Automated statistical analysis was not available for Vegter 2017. So, finally, to enable estimation of scatter in forming processes in the short term with Vegter individual simulations were done on a selected number of input datasets. This is shown to give markedly different results to BBC2005.

Influence of the strain dependent material behaviour under plane strain on the yield locus modelling

Modern lightweight materials mostly exhibit a low formability especially in the plane strain state. Having in mind that over 80% of the occurred failures in the press shop in the automotive sector can be related to this strain state, the characterization under plane strain load should be further intensified. The evaluation of the plane strain flow curve is commonly done by notched tensile tests or biaxial tensile tests with optimized strain path. Both tests have some disadvantages in the flow curve evaluation especially at a high degree of deformation. With an elliptic hydraulic bulge test, the material behaviour under near plane strain condition can be evaluated up to fracture. The aim in this investigation is the analysis of the resulting material behaviour under plane strain of commonly used alloys AA5182, DC06 and DP600 with respect to the material anisotropy. Therefore, hydraulic bulge tests with a plane strain load path are performed in rolling direction and transversal direction. The resulting flow curves are used to analyse the material hardening with respect to the anisotropy. The results show differing hardening mechanisms for the investigated materials which will lead to different strategies for the material modelling.

Potentials for material card validation using an innovative tool

For the numerical description of the material behavior of sheet metals, there is a large diversity of models. These models vary strongly in terms of complexity and calibration effort. Before using a material model for numerical analyses, it is essential to validate it. The paper presents an innovative tool that allows generating a well-founded database for a comprehensive validation of the yield locus description with few and simple tests: the MUC-Test (Material Under Control). The special geometry of the punch and die generates complex strain distributions in sheet metal samples during forming. The experiment covers the strain range between uniaxial and equibiaxial tension. The contribution shows the potential of this innovative tool by using the test results of three different material classes. Using numerical analyses, sensitivities of different material model parameters were investigated. Thereby, the focus lies on the curvature of the yield locus in the area between uniaxial and equibiaxial stress state.

Potential use of machine learning to determine yield locus parameters

The determination and validation of material parameters required for the finite element simulation of sheet metal forming processes can be realized by a full-field optical measurement of the deformation of a test specimen in combination with a simulation-based inverse approach. Development of such an inverse approach can be quite time consuming and requires programming skills and also expertise in FEM analysis and optimization. Emerging machine learning techniques offer a practical alternative to optimization and inverse approaches provided that the ground truth is completely known and generalized by the machine learning model. To be more precise, a machine learning model can directly compute the material parameters from the experimental measurements if the hypothetical mapping function between material parameters and deformation behavior is learned as ground truth. This paper presents such a machine learning based approach for the determination of validated yield locus parameters.

Shear Strength Characteristics of a Thermally Cured Sand-Bentonite Mixture

An experimental program was conducted to investigate the effects of curing time and curing temperature on shear behavior of a sand-bentonite mixture. The specimens were cured at temperatures of 40°C, 60°C and 80°C for 1, 3 and 5 days under 100kPa, 500kPa and 1000kPa confinements. The results of consolidated undrained triaxial shear tests showed that an increase in temperature from 40°C to 80°C at 1, 3 and 5 days of curing increased the shear strength by 25%, 24% and 23%, respectively. Also, the increase in curing time from 1 to 3 and from 1 to 5 days at 80°C increased the shear strength of samples 12% and 24%, respectively. The failure of pre-cured samples occurred in lower strains as a result of more induced brittleness. Moreover, the secant modulus as well as the size of yield loci and critical state line’s slope increased by pre-curing. The application of thermal cycles resulted in increasing shear strength and experiencing a negative pore water pressure which shows a transition towards the quasi-structured behavior. The results of scanning electron microscopy (SEM) studies confirmed the increase in void ratio during thermal curing.

The evolution of yield loci with finite plastic deformation along proportional and non-proportional loading paths in an annealed extruded AZ31 magnesium alloy

This investigation reports for the first time, the subsequent yield loci after finite plastic deformation based on 10με offset definition yield of an annealed extruded AZ31 magnesium alloy under proportional and non-proportional loading paths. The proportional loading paths comprise uniaxial tension, free end torsion, and combined tension-torsion radial paths. The tension loading followed by torsion, and torsion followed by tension are the non-proportional loading paths. Subsequent yield loci under proportional loading paths show a prominent nose in the loading direction and are almost flat in the unloading direction with evolution of significant isotropic hardening under tension and visible kinematic hardening under torsion loading path. Strong path-dependency is observed in subsequent yield loci under non-proportional loading directions along with hardening effects. All the yield loci are investigated in tension-torsion (σ₁₁-√3σ₁₂) stress space. The results are significantly different than observed for other metals, especially for aluminum alloys by Khan et al. (2009) and Iftikhar et al. (2021). Elastic moduli within each subsequent yield loci are also measured during these experiments with the increase in plastic strain. The observed small decreases in the elastic moduli with increasing plastic strains are very similar to the ones reported earlier (Iftikhar et al., 2021; Khan et al., 2009).

Analytical determination of anisotropic parameters for Poly6 yield function

The present work mainly focuses on the analytical determination of anisotropic parameters for Poly6 yield criterion, which is mathematically equivalent to Yoshida2013 yield function. The parameters of Poly6 yield criterion are expressed with the r-values and yield stresses without any optimization method, which has been successfully applied for highly anisotropic materials. The Poly6 yield criterion is analytically calibrated using two different methods under associated flow rule; i.e, 1) the shear flow stress and rb value and 2) the stresses under σxx = 2σyy and 2σxx =σyy. The first type of analytical calibration from shear stress and rb value is compared with the Yoshida2013 yield criterion whose parameters are obtained from an optimization process, which shows that they are equivalent for describing stress and strain distributions. The yield loci of FCC materials predicted from the second method is similar to Yld2000-2d. Both methods of calibrating the analytical Poly6 yield criterion are shown to describe the r-value and uniaxial yield stress directionalities. The accuracy and effectiveness of the proposed analytical methods are verified through the selected FCC and BCC materials. The ability to describe the evolutions of materials’ anisotropic behaviors with the proposed analytical Poly6 has been verified by applying it to AA5182-O.

The Use of Bore-Expansion Test for Determining The Influence of Anisotropic Yielding on The Formability of Metal Sheets.(Dept.M)

In sheet metal forming operations, formability is usually governed by a great number of parameters such as strain hardening, strain rate sensitivity, and the shape of anisotropic-yield loci. in the present work, the deformation and formability in bore-expanding were examined theoretically and experimentally from aspect of the shape of anisotropic yield loci employing Bassani yield criterion. According to this criterion, a numerical analysis was performed for axi-symmetrical bore-expanding using three investigated materials, AL-Killed steel, EG-LOW carbon steel, and Brass 70/30. The bore-expanding limit was also studied using various necking theories and results were compared with the experimentally determined values. It was concluded that Bassani criterion was useful to predict the deformation and formability in bore-expanding test only at a certain value of r*.

Determination of the Biaxial Anisotropy Coefficient Using a Single Layer Sheet Metal Compression Test

Numerical process design leads to cost and time savings in sheet metal forming processes. Therefore, a modeling of the material behavior is required to map the flow properties of sheet metal. For the identification of current yield criteria, the yield strength and the hardening behavior as well as the Lankford coefficients are taken into account. By considering the anisotropy as a function of rolling direction and stress state, the prediction quality of anisotropic materials is improved by a more accurate modeling of the yield locus curve. According to the current state of the art, the layer compression test is used to determine the corresponding Lankford coefficient for the biaxial tensile stress state. However, the test setup and the test procedure is quite challenging compared to other tests for the material characterization. Due to this, the test is only of limited suitability if only the Lankford coefficient has to be determined. In this contribution, a simplified test is presented. It is a reduction of the layer compression test to one single sheet layer. So the Lankford coefficient for the biaxial tensile stress state can be analyzed with a significantly lower test effort. The results prove the applicability of the proposed test for an easy and time efficient characterization of the biaxial Lankford coefficient.

Characterization on the thermal anisotropic behaviors of high strength AA7075 alloy with the Yld2004-18p yield function

Hot forming is an effective method to improve the formability of high strength AA7075 alloy. When formed at elevated temperature, AA7075 alloy still exhibits distinct anisotropic behaviors. The main purpose of this paper is to characterize the thermal anisotropic behaviors of AA7075 alloy. Hot uniaxial tension, plane strain tension and disk compression tests were firstly performed at 300 °C, 350 °C and 400 °C. The obtained thermal normalized stresses and r-values indicate that the thermal strain anisotropy is relatively strong compared with the thermal stress anisotropy. The advanced Yld2004-18p function was introduced to describe the thermal anisotropic behaviors of AA7075 alloy. The coefficients in Yld2004-18p function were calibrated on the basis of the experimental data at each test temperature. Subsequently, the Yld2004-18p function was employed to draw the thermal yield loci of AA7075 alloy, among which high similarity is observed. Furthermore, the Yld2004-18p function was implemented in the ABAQUS/Explicit to conduct hot cup drawing simulations. The numerically predicted earing profiles were compared with the experimental ones. It is found the spatial fluctuation of the earing profiles can be precisely predicted by implementing the Yld2004-18p function in drawing simulations. Four ears are observed near 45°, 135°, 225° and 315°at each temperature. As temperature increases from 300 °C to 400 °C, the earing profiles are slightly elevated and the valleys at 0° and 180° are slightly deepened.