Hill's Yield Function Research Articles

Experimentation and FE analysis of low carbon steel-based laser welded tailored blanks in case of single point incremental forming

This study explores the formability of laser-welded tailored blanks in the context of single-point incremental forming (SPIF), a flexible and dieless manufacturing process. Two approaches, variation in materials and thicknesses, were used to fabricate tailor-welded blanks (TWB) with CRCA and DD steel sheets. The quality of the weld was evaluated by metallographic studies and tensile tests. The ductility of the welded joint was reduced by 81%, and the hardness was increased by 133% compared to the base metal. SPIF was employed using a hemispherical tool to form a ‘D’ shape geometry on fabricated TWBs. Fracture was observed only at the weld line, irrespective of thickness and material combination in the TWB. The forming height achieved from the SPIF of TWBs was reduced by 24% and 34% for both differences in thickness and material when compared to their corresponding base metals. Finite Element simulations were performed in Abaqus software using a dynamic explicit approach employing von Mises and Hill48 yield functions. Further, the FE model was validated by comparing the thickness and surface major/minor strains at the same forming height as observed from the experiments. Reduced formability and no weld line movement were the main observations made in this study, and the same was manifested by the FE simulation.

A generalized, computationally versatile plasticity model framework - Part II: Theory and verification focusing on shear anisotropy

Shear-dominated deformation (SHDD) is pivotal in sheet metal forming; however, comprehensive modeling of plastic anisotropy in SHDD, specifically shear anisotropy considering both yield stress and plastic flow, has been inadequately addressed in existing literature. In this work, a generalized constitutive framework is introduced on the basis of stress triaxiality-dependent state variable to accurately emulate plastic anisotropy and the physics-based shear constraint in SHDD. The framework is capable to seamlessly integrate with existing yield criteria, preserving computational efficiency and versatility. Notably, the yield function, anisotropic parameters, and their optimization or analytical determination for the non-shear deformation state remain unaltered. When integrated with the Hill48 yield function, featuring either one or two anisotropic parameters within the generalized constitutive framework, precise characterization of yield strength and plastic flow in SHDD is achieved. The applicability of the framework extends to various anisotropic yield functions such as the widely employed Yld2k-2d and the sixth-order polynomial (Poly6) function as a class of associated flow rule-based yield functions, and one non-quadratic yield function for non-associated flow rule scenarios. Experimental validation with two engineering sheet metals, high-strength dual-phase steel DP980 and high-strength aluminum alloy AA7075-T6, was conducted. Comparative analyses with several recently proposed yield criteria, especially Poly6–18p, highlighted the efficiency of the proposed constitutive framework. Furthermore, this study explores intrinsic shear constraints, particularly the absence of through-thickness strains under in-plane shear stress. Additionally, it offers an enhanced description of plastic anisotropy in shear yield stress within the general framework, providing valuable insights into the complexities of SHDD.

Characterization and modeling of biaxial plastic anisotropy in metallic sheets

The anisotropic behavior of cold-rolled sheet metals has been extensively studied, typically characterized by uniaxial loading tests in different directions to determine yield strengths and plastic flow (Lankford r-values). However, biaxial principal stress states often focus solely on yield loci in the RD/TD (rolling/transverse) directions, with limited studies on other loading directions. This study analyzes the relationship between the principal stress yield loci and the material principal anisotropic stress directions based on the Hill48 yield function. Biaxial tensile tests are conducted on DP590 high-strength steel using cruciform specimens cut in various directions different from the sheet RD/TD directions. The evolution of the anisotropic yield locus and plastic flow of DP590 under biaxial tensile directions beyond the traditional RD/TD anisotropic directions is revealed. Furthermore, a modified Hill48 model that considers any principal stress direction of loading is proposed, accurately describing the continuous evolution of equi-biaxial tension stress, near-plane strain stress, and plastic flow direction for different principal stress directions of loading. The issues of failing to predict plastic flow under biaxial loadings in the original Hill48 model, and the degraded representation of existing constitutive models when accounting for out-of-plane anisotropy are addressed by decoupling the relationship between anisotropic parameters and stress state. In addition, the proposed calibration strategy of anisotropic parameters, using experimental data from various principal stress directions of loading, effectively captures the anisotropic evolution caused by changes in principal stress directions. Comprehensive evaluation and verification of the plasticity model should consider yield locus for any principal stress directions of loading, and also the normal and shear planes.

Effect of the loading-rate and stress state on the constitutive modelling and fracture of 2205 Duplex stainless steel

Shear cutting is a high-speed forming process where material separation is involved. The nature of the process leads to high strain rates and temperatures on the sheared zone of the material subjected to the operation. In this work, the mechanical behavior in terms of flow stress, plastic deformation and fracture of a 2205 Duplex stainless steel sheet is determined. Uniaxial tension, uniaxial compression, stack compression, plane strain tension and simple shear tests at different angles with respect to the rolling direction of the metal sheet are performed to determine the anisotropic behavior of the material. Strain rate and temperature dependence of the flow stress and the plastic behavior of the material is determined through notched tensile tests performed under various temperatures (20°C, 100°C, 300°C and 500°C) and strain rates (0.001 s-1, 10 s-1 and >100 s-1). Finally, fracture behavior under various stress states is determined through notched tensile, central hole, in-plane shear and plane strain tests. The temperature and strain rate dependence of the fracture behavior is determined through notched tensile, in-plane shear and plane strain tests performed under various temperatures (20°C, 100°C, 300°C and 500°C) and strain rates (0.001 s-1, 10 s-1 and >100 s-1). The anisotropic behavior and work hardening of the material are modelled by the Hill48 yield function and Swift-Hockett-Sherby hardening law, respectively. Among the two ductile fracture models tested, Hosford-Coulomb model showed the best agreement and is therefore used to account for the temperature and strain rate effects on the fracture behavior. By combining these three models, an accurate reproduction of the mechanical behavior of the 2205 Duplex stainless steel sheet is achieved.

Combined anisotropic and cyclic constitutive model for laser powder bed fusion fabricated aluminum alloy

This study presents new methods to effectively model the anisotropic yielding and hardening behavior of laser powder bed fusion fabricated aluminum alloy under both monotonic and cyclic loading conditions. The proposed model combines the yield surface-interpolation method to accurately describe the anisotropic hardening rates in various directions, with the Chaboche kinematic hardening rule to precisely reflect the cyclic characteristics. For numerical implementation of the combined anisotropic and cyclic constitutive model, a fully implicit stress integration algorithm based on return mapping method is provided. Moreover, the multiple parameters associated with the model are categorized and identified in an uncoupled manner. The isotropic and cyclic hardening parameters are determined by an inverse method, and the stability of the optimization outcomes is validated by applying different starting points for the parameters. Particularly, the back-stress effect on the identification of anisotropic parameters associated with the stress invariant-based Hill48 yield function is considered for the first time. This consideration leads to an improved prediction accuracy compared to the identification of anisotropic parameters without considering back-stress effect. The combined anisotropic and cyclic constitutive model, along with the calibrated parameters, are proven capable of accurately reproducing the intricate deformation behavior of laser powder bed fusion fabricated AlSi10Mg.

Experimental study and modelling of anisotropic behaviour of aluminium-lithium alloys in creep age forming

The anisotropic deformation behaviour of the 3rd generation aluminium-lithium (Al-Li) alloys can significantly affect the accuracy of prediction and fabrication, especially for fabricating complex shaped components. To address this challenge and shed light on improving the prediction accuracy with the existence of material anisotropy, this study systemically investigated the anisotropic behaviour during deformation and creep-ageing, and proposed a comprehensive anisotropic material model and its implementation method in the numerical simulation, which was verified with four-point bending creep age forming (CAF) tests. The anisotropic behaviours of the material including yielding, hardening, and creep deformation were investigated with uniaxial tensile and creep-ageing tests adopting specimens cutting along 0°, 15°, 30°, 45°, 70° and 90° to the rolling direction. The yield strengths varied in the range of 378 to 456 MPa for different directions, with the highest and lowest values appearing at 0° and 70°. The creep deformation also showed strong anisotropic behaviour and the accumulated creep strain for 90° (0.52%) was four times larger than the value for 0° (0.11%) under 415 MPa at 143 °C for 5 h An anisotropic material model was established based on the modified non-associated quadratic Hill48 yield function and can adequately capture the observed anisotropic behaviour. An implementation strategy using the bisection method with the map returning scheme was proposed for incorporating the proposed model in numerical simulation. A good agreement was achieved between the predicted and experimental results for four-point bending CAF tests, demonstrating the validity of the proposed anisotropic material model and the implementation method.

Multiaxial Plastic Deformation of Zircaloy-4 Nuclear Fuel Cladding Tubes

The following work is motivated by the desire to devise an internal pressure test that can mimic a displacement-controlled loading scenario and demonstrate how to apply the multiaxial stress and strain data from the test to develop an elastic/plastic constitutive model for a thin-walled tubular component. This is achieved by conducting simultaneous measurements of tangential and axial strain during the pressure test and integrating these strain measures into a feedback loop with the pressure controller. It is shown how data from such a test can be used to develop a large mechanical property data set relevant to biaxial loading conditions. The data obtained have high confidence evidenced by their low variability and alignment with other literature studies. Additionally, data from these internal pressure tests combined with full-tube axial tensile tests allow for the derivation of the Hill anisotropic yield function. The developed Hill yield function is validated by comparing the plastic strain ratios from the full-tube tension tests and by comparing the predicted yield stress in the tangential direction with measured values from ring tension tests in a previous study.

Measurement and modelling of strain-path dependent anisotropic hardening behaviors of high strength steels subjected to pre-strains

Anisotropic yielding and hardening behaviors are important parts of a constitutive model for material formability and strength assessment. In this work, the work hardening characteristics of a high strength steel sheet after it was subjected to 2% and 5% plastic strain in rolling and transverse directions were investigated by both the experimental measurement and mathematical modelling. It was observed that the material with the mild initial anisotropy showed a strong anisotropic work hardening effect after strain path was changed. A material model was proposed by Ma and Aung in order to reproduce the observed hardening phenomena under various conditions of strain path change. The proposed formulation has a simple form mathematically derived from a modified Swift law which can be easily implemented into existing FEM software as a user material model. This anisotropic hardening model for plane-stress condition was combined with Hill48 yield function and associative flow rule for numerical simulations. The model was validated through comparing the simulation results with experimentally measured stress-strain curves.

Limitation analysis of the Hill48 yield model and establishment of its modified model for planar plastic anisotropy

In this paper, the limitations of the original Hill48 yield model in predicting uniaxial tensile yield stresses and r-values were analyzed in detail, and then the necessary and sufficient conditions for judging the applicability of the original Hill48 yield model were proposed. On this basis, the approximate satisfactions of the steel sheet DC04, DP980 and the aluminum alloy AA5754-O to the necessary and sufficient conditions were investigated. The results show that the necessary and sufficient conditions can be used to qualitatively analyze the predicted error of the original Hill48 model. Based on the inherent relationship of Hill48 yield function, the direction of principal stress axis was introduced into the anisotropic parameters, and a modified Hill48 model suitable for plane stress state was developed. According to the variation of the predicted error of the original Hill48 yield model in biaxial tensile stress, an exponential interpolation method including material parameter was proposed to correct the asymmetric biaxial stress between uniaxial stress state and symmetric biaxial stress state. The theoretical predicted results show that the modified model can accurately capture the directional yield stresses, r-values and yield locus. The main advantage of the modified model is that it still maintains a simple quadratic form in the stress tensor, and can significantly improve the predicted accuracy of anisotropic behavior of aluminum alloy and other materials from the use of associated flow rule.

Anisotropic plasticity‐damage material model for sheet metal — Regularised single surface formulation

AbstractSheet metal forming as well as mechanical joining demand increasingly accurate and efficient material modelling to capture large deformations, the inherent sheet orthotropy and even process‐induced damage, which is expected to be influential. To account for large strains the additive logarithmic strain space is utilised that enables a straightforward incorporation of plastic anisotropy, herein modelled by a Hill48 yield function. A gradient‐enhancement is used to equip the ductile damage model with an internal length scale curing the damage‐induced localisation. An affine combination of the local and non‐local softening variable is derived enabling a more efficient single surface formulation for the regularised plasticity‐damage material model.

Chip formation in machining of anisotropic plastic materials—a finite element modeling strategy applied to wood

This paper presents an FEA modeling strategy for predicting the cutting forces generated during linear wood machining. The objective is to determine both the forces and paths of cracks propagating in elastoplastic and anisotropic materials. The model combines a bilinear representation of the material strain-stress relation and the Hill yield function. The proposed procedure also integrates the displacement extrapolation method to evaluate the stress intensity factors. It establishes the cutting forces from the contact pressures between the tool and the chip. These pressures are determined using the penalty method. The validation phase compares the model predictions with average experimental forces and shows correspondence levels higher than 91% and 92% for low (0.085e10−3 m/s) and high (6.8 m/s) wood-feeding speeds, respectively. The developed model maintains a high precision degree over a large range of feeding-velocity. This study demonstrates that the resistive force between the chip and the tool surface is a function of both the rake angle φ and the coefficient of friction (COF). The friction force prompts a self-energizing effect, which increases the resistive force. On the other hand, larger φ amplitudes reduce this effect. Furthermore, the rake angle φ defines the crack propagation mode. Larger φ amplitudes favor the opening mode, whereas smaller values promote the shear mode. The COF amplitude also influences the surface quality, with larger COFs producing more profound cutting dimples. Thus, reducing the COF should result not only in lower cutting forces but also in a better surface quality of machined parts.

Study on Limit Load of Orthotropic Cylindrical Pipe Under Different Combined Load Conditions

Abstract In engineering, many pressure pipes are made of steels with good plasticity, which are subject to internal pressure, axial force, shear force, bending moment, torsion moment or their combined loads. The plastic limit load is an important indicator of the load capacity of pressure pipe. According to Hill yield function, the theoretical solutions of limit load of orthotropic cylindrical pipe under various combined loads under internal pressure, axial force, shear force, torsion moment, and bending moment have been derived on the basis of elastic perfectly plastic constitutive model. The effects of radial stress on different combined limit loads of cylindrical pipe are explored and these results show that the radial stress should be considered about the limit load calculation especially for thick-walled cylindrical pipe. The interactions of various load combination are analyzed in detail and drawn with the interaction curves. For isotropic cylindrical pipe, the limit load increases with the yield strength. For the orthotropic cylindrical pipe, the limit loads of cylindrical pipe under axial force, bending moment, shear force, and torsion moment without internal pressure are only related to the axial yield strength. The limit bending moment is mainly dependent on the axial yield strength when internal pressure is lower, while the impact of the circumferential yield strength of orthotropic cylindrical pipe is obvious when internal pressure is some higher. When the axial yield strength of orthotropic cylindrical pipe is the same, the circumferential yield strength can enhance the limit axial load, limit torsion moment, and limit shear load. Under the different load conditions including internal pressure, bending moment, axial force, shear force, and torsion moment or their combined loads, the relation of limit bending moment with yield strength ratio is diverse, which is decide by the load combination, the circumferential yield strength, and the axial yield strength.

Four Earing’s Prediction in Deep Drawing of AISI 1008 Steel Sheet Conical Product

Earing is a phenomenon that appears in deep drawing of parts produced using this process because of the anisotropic material properties. Most of, recent theories didn’t fully employ the mechanical materials properties, and ears number achieved depends on the FE simulation approaches. To anticipate the ears form issue in conical parts of AISI 1008 steel sheet deep drawing, in this work a new method is used to predict the earing formation during deep drawing. This method proposed combines the yielding limits and anisotropy r-values of the material to determine Hill48 yield criterion variables using several conical angles with several punch velocities. Equation for the value of yielding and anisotropy index for different orientation is used. AISI 1008 steel sheet is used as a case in this work, the zone of deformation or blank is partitioned radially to equal parts according to the anisotropic behavior. Hill48 yield function constants solved based on the material yielding point and anisotropy index together for the parallel deformed regions. A simulation using FE software for the process on the base of this method is executed to compare it results of lab work. and it noticed that Increasing speed may contribute slightly in decreasing the severity of anisotropy effect and this can be explained the process propagate the material need time to flow and if the speed increase this lead to increasing in strain hardening of the material[m]. Increasing die angle shows that the different between the highest point to lowest point in the cup edge may increase. This work produces a conductive and dependable way to anticipate the forms of earing during cup drawing, which will have positive effective in manufacturing implementations and numerical outcomes and shows high agreement with lab work.

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

AbstractA micro‐mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group‐theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises ‐ Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro‐mechanically motivated generalization of the von Mises ‐ Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension‐compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface.

Full Three-Dimensional Cavitation Instabilities Using a Non-Quadratic Anisotropic Yield Function

Abstract Full three-dimensional cell models containing a small cavity are used to study the effect of plastic anisotropy on cavitation instabilities. Predictions for the Barlat-91 model (Barlat et al., 1991, “A Six-Component Yield Function for Anisotropic Materials,” Int. J. Plast. 7, 693–712), with a non-quadratic anisotropic yield function, are compared with previous results for the classical anisotropic Hill-48 quadratic yield function (Hill, 1948, “A Theory of the Yielding and Plastic Flow of a Anisotropic Metals,” Proc. R. Soc. Lond. A193, 281–297). The critical stress, at which the stored elastic energy will drive the cavity growth, is strongly affected by the anisotropy as compared with isotropic plasticity, but does not show much difference between the two models of anisotropy. While a cavity tends to remain nearly spherical during a cavitation instability in isotropic plasticity, the cavity shapes in an anisotropic material develop toward near-spheroidal elongated shapes, which differ for different values of the coefficients defining the anisotropy. The shapes found for the Barlat-91 model, with a non-quadratic anisotropic yield function, differ noticeably from the shapes found for the quadratic Hill-48 yield function. Computations are included for a high value of the exponent in the Barlat-91 model, where this model represents a Tresca-like yield surface with rounded corners.

Prediction of Eight Earings in Deep Drawing of 5754O Aluminum Alloy Sheet

Earings appear easily during deep drawing of cylindrical parts owing to the anisotropic properties of materials. However, current methods cannot fully utilize the mechanical properties of material, and the number of earings obtained differ with the simulation methods. In order to predict the eight-earing problem in the cylindrical deep drawing of 5754O aluminum alloy sheet, a new method of combining the yield stress and anisotropy index (r-value) to solve the parameters of the Hill48 yield function is proposed. The general formula for the yield stress and r-value in any direction is presented. Taking a 5754O aluminum alloy sheet as an example in this study, the deformation area in deep drawing is divided into several equal sectorial regions based on the anisotropy. The parameters of the Hill48 yield function are solved based on the yield stress and r-value simultaneously for the corresponding deformation area. Finite element simulations of deep drawing based on new and existing methods are carried out for comparison with experimental results. This study provides a convenient and reliable way to predict the formation of eight earings in the deep drawing process, which is expected to be useful in industrial applications. The results of this study lay the foundation for the optimization of the cylindrical deep drawing process, including the optimization of the blank shape to eliminate earing defects on the final product, which is of great importance in the actual production process.

Experimental and numerical study on the deformation mechanism of straight flanging by incremental sheet forming

The geometric accuracy is a key quality indicator for the application of flanging by incremental sheet forming (ISF), namely incremental flanging (IF). As a local loading forming process, the deformation process in IF is different from that in conventional flanging formed by press-working operation in which considerable geometric error can be introduced. Thereby, the deformation process of straight flanging by ISF is investigated through both experimental and numerical simulation approaches. First, the effects of different yield functions (Mises and Hill48) and hardening models (A–F, isotropic and kinematic) on the prediction accuracy are investigated. In particular, the Hill48 yield function is not only solved by r-values, but also calibrated by material properties of strain states corresponding to the actual deformation condition of straight flanging by ISF. Among different yield functions, calibrated methods and hardening models, a A–F hardening model with the calibrated Hill48 yield function is found to have the best performance in terms of prediction accuracy of both forming force and cross-section profile. Then, based on this model, the deformation process of the straight flanging by ISF is analyzed. The results show that the strain states of straight flanging by ISF are plane and uniaxial tension strain states. In the forming process of straight flanging by ISF, the whole blank experiences bending-unbending deformation in which the material contacting with the die undergoes reverse loading. In addition, the main deformation modes of straight flanging are bending and stretch deformation through internal energy analysis. Finally, the deformation process of the straight flanging by ISF is theoretical analyzed. It is found that the elastic moment in the reverse loading mode is the main source for geometric error of the straight wall.

Fracture behavior of recrystallized and stress-relieved Zircaloy-4 cladding under biaxial stress conditions

ABSTRACT Pellet-cladding mechanical interaction (PCMI) under reactivity-initiated accident conditions may lead to failure of high-burnup fuel rods. The biaxial stress states of Zircaloy cladding tubes induced by PCMI loading presumably makes the tubes more susceptible to failure. To clarify the influence of the anisotropic mechanical properties of Zircaloy cladding tubes on their fracture behavior under biaxial stress conditions, biaxial tensile tests were performed, and the measured stresses and strains were converted to reduced parameters such as equivalent strain, equivalent stress, and stress triaxiality by using the anisotropic constants of the Hill yield function derived in our previous study. The minimum fracture strains of recrystallized (RX) and stress-relieved (SR) specimens were located where the stress ratio of axial to circumferential is 0.75. The equivalent plastic fracture strains tended to decrease monotonously with increasing stress triaxiality, which is a typical trend observed in ductile fracture, in the range of 0.65–0.78 for both specimens. In the case of SR specimens, however, the analysis with stress triaxiality did not reduce the fracture strains well to a single trend curve, showing that another influential factor is required such as an anisotropic fracture behavior or a plastic behavior deviated from that predicted by the yield function.

Extension of the DF2016 isotropic model into an anisotropic ductile fracture criterion

The paper extends the recently proposed ductile fracture criterion of DF2016 (Lou, Chen, Clausmeyer, Tekkaya and Yoon, 2017. Modeling of ductile fracture from shear to balanced biaxial tension for sheet metals. International Journal of Solids and Structures 112, 169-184) to consider the anisotropic ductile fracture behaviour of sheet metals. The DF2016 criterion is coupled with the Hill48 yield function to model the anisotropy in ductile fracture. For the verification purpose, experiments are conducted for AA6082-T6 sheet with a thickness of 1.0 mm in various loading conditions: in-plane shear, uniaxial tension, plane strain tension, and the balanced biaxial tension. Tests are carried out in the rolling direction, diagonal direction and transverse direction. Fracture strains are measured by digital image correlation along different loading directions. The anisotropic fracture of the sheet is modelled by the DF2016 criterion by incorporating an anisotropic yield function. The predicted anisotropic fracture loci are compared with experimental results in shear, uniaxial tension, plane strain tension and balanced biaxial tension. The comparison demonstrates that the anisotropic DF2016 criterion accurately models the anisotropic fracture behaviour of AA6082-T6 sheet in wide loading conditions from shear to balanced biaxial tension.

The Hosford yield function for weakly textured sheets of cubic crystal orthotropic metals in any stress state

Most rolled sheet metals are orthotropic aggregates of cubic crystallites. The texture coefficients, characterized by the preferred orientation of the crystallites, are important to set up the yield function. Although the Hosford yield function is more suitable than the Hill yield function for describing both the yielding and plastic deformation of orthotropic material, it suffers from the restriction that the three principal stresses must be coaxial with the orthotropy of materials. This paper proposes the new Hosford yield function for weakly textured sheets of cubic crystal orthotropic metals in any stress state by expanding the introduced orientation-dependent functions to its sixth-order Taylor series expansion. Also, the new yield function, which covers three material parameters and seven texture coefficients, is more general than the existing Hosford yield function. Finally, both the plastic anisotropy of the q-value and the yield stress obtained from the new yield function agree well with experimental results. This yield function can lay a theoretical foundation for analyzing the mechanical properties of metal materials.