Hill's Quadratic Yield Function Research Articles

Forming-limit strains of perforated steel sheets under biaxial stress state

The forming-limit strains of cold-rolled steel sheets perforated with regularly arranged round-, square-, and cross-shaped holes were experimentally and theoretically estimated. Two types of hole arrangements, i.e. square and triangular patterns, were considered for each hole shape and the experimental forming-limit strains were determined using the hemispherical dome test (also known as Nakazima test). The theoretical forming-limit curves were computed using the finite element method with Hill's quadratic yield function and a plastic instability criterion determined from the external force power. By comparing the experimental results and theoretical calculations under proportional loading, the formability prediction performance of the theoretical approach was evaluated. It was found that although the approach is not applicable for the triangular array of cross-shaped holes, it can achieve partially acceptable prediction for the other cases.

A quadratic yield function with multi-involved-yield surfaces describing anisotropic behaviors of sheet metals under tension/compression

A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed. Five mechanical properties are adopted to determine the coefficients of each part of the yield function. For particular cases, the proposed yield function can be simplified to Mises or Hill's quadratic yield function. The anisotropic mechanical properties are expressed by defining an angle between the current normalized principal stress space and the reference direction with the assumption of orthotropic anisotropy. The accuracy of the proposed yield function in describing the anisotropy under tension and compression is demonstrated.

A generalisation of the Hill's quadratic yield function for planar plastic anisotropy to consider loading direction

In this work, a new generalised quadratic yield function for plane stress analysis that is able to describe the plastic anisotropy of metals and also the asymmetric behaviour in tension-compression typical of the Hexagonal Closed-Pack (HCP) materials, is developed. The new yield function has a quadratic form in the stress tensor and it simultaneously predicts the r-values and directional flow stresses, which is shown to agree very well with experimental results. It also accurately describes the biaxial symmetric stress state which is fundamental for the accurate modelling of aluminium alloys. The new quadratic yield function represents the non-symmetric biaxial stress state by performing a linear interpolation from pure uniaxial loading to a biaxial symmetric stress state. The main advantages of this new yield function is that it can be used for the modelling of metals with any crystallographic structure (BCC, FCC or HCP), it only has five anisotropic coefficients and also that it is a simple quadratic yield criterion that is able to accurately reproduce the plastic anisotropy of metals whilst using an associated flow rule.

Determination of all elastic and plastic parameters for sheets of cubic metals only by uniaxial tension tests

Most of sheet metals belong to orthorhombic aggregates of cubic crystallites. Since both the elasticity tensor and the yield function of sheet metals depend on the crystalline orientation distribution, some correlations between elasticity and plasticity must exist. Employing the correlations, herein we derive the computational expressions of determining all elastic and plastic parameters in the elasticity tensor and the general Hosford yield function only by the data of uniaxial tension tests. By the elastic FEM numerical simulations and the plastic Taylor-Bishop-Hill model's numerical simulations, we verify the correlations and the computational expressions, and compare the general Hosford yield function with the Hill quadratic yield function and the Hill 1990 yield function. The main goal of this paper is to give a method of determining all the elastic and plastic parameters in the elasticity tensor and the general Hosford yield function only by uniaxial tension tests.

General Hosford yield functions of orthorhombic materials

Although the Hosford yield function is more suitable for describing both the yielding and the plastic deformation of orthorhombic materials than the Hill quadratic yield function, the Hosford yield function suffers from the restriction that the loading has to be coaxial with the orthotropy of the materials. To relax this restriction, herein we present a new general Hosford yield function for the orthorhombic materials. The new general Hosford yield function is suitable to any stress state of the orthorhombic materials. When η = 2, the new general Hosford yield function becomes the Hill quadratic yield function. The new general Hosford yield function is more general than the general Hosford yield function of Huang and Man (Int J Plast 41:97–123, 2013), which covers only weakly-textured sheets of cubic metals. Two examples show that the new general Horsford yield function with suitable η value gives much better fits than those of the Hill quadratic yield function (η = 2).

Numerical simulation of springback using an extended return mapping algorithm considering strain dependency of elastic modulus

Several studies have reported that the elastic modulus decreases during elastoplastic deformation. However, the return mapping algorithms commonly used to implement the elastoplastic constitutive equations into finite element programs assume that the elastic modulus remains constant during plastic deformation. In order to consider the decrease of elastic modulus in elastoplastic constitutive models, the return mapping algorithms need to be generalized. In this study, a numerical procedure was developed for implementation of elastoplastic constitutive laws assuming that the elastic modulus was defined as a function of effective plastic strain. Using the developed numerical procedure, the Hill's quadratic yield function and the combined isotropic-nonlinear kinematic hardening model was implemented as a user subroutine for ABAQUS commercial finite element package. Different numerical tests were carried out to evaluate the performance of the developed algorithm. The results showed that the algorithm was robust and as accurate as the other return mapping algorithms. Finally, the model was used to simulate both the forming stage and the subsequent springback of a U-shaped part made of DP600 steel. The results showed that the accuracy of springback simulation considerably improved when the decrease of elastic modulus was described in the model.

Finite element simulation of springback for a channel draw process with drawbead using different hardening models

The objective of this work is to predict the springback of Numisheet’05 Benchmark#3 with different material models using the commercial finite element code ABAQUS. This Benchmark consisted of drawing straight channel sections using different sheet materials and four different drawbead penetrations. Numerical simulations were performed using Hill's 1948 anisotropic yield function and two types of hardening models: isotropic hardening (IH) and combined isotropic-nonlinear kinematic hardening (NKH). A user-defined material subroutine was developed based on Hill's quadratic yield function and mixed isotropic-nonlinear kinematic hardening models for both ABAQUS-Explicit (VUMAT) and ABAQUS-Standard (UMAT). The work hardening behavior of the AA6022-T43 aluminum alloy was described with the Voce model and that of the DP600, HSLA and AKDQ steels with Hollomon's power law. Kinematic hardening was modeled using the Armstrong–Frederick nonlinear kinematic hardening model with the purpose of accounting for cyclic deformation phenomena such as the Bauschinger effect and yield stress saturation which are important for springback prediction. The effect of drawbead penetration or restraining force on the springback has also been studied. Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior. Comparisons between simulation results and experimental data showed that the IH model generally overestimated the predicted amount of springback due to higher stresses derived by this model. On the other hand, the NKH model was able to predict the springback significantly more accurately than the IH model.

Large deformation of anisotropic austenitic stainless steel sheets at room temperature: Multi-axial experiments and phenomenological modeling

A newly developed multi-axial testing technique for sheet materials is employed to investigate the inelastic response of a temper-rolled stainless steel 301LN under isothermal quasi-static loading conditions at room temperature. The experimental technique consists of a flat sheet specimen, which is subject to combinations of shear and normal loading using a custom-made dual-actuator system. The large deformation behavior under monotonic loading is determined along more than 20 distinct radial paths in the stress space. The experimental results indicate that Hill's quadratic yield function along with an associated flow rule provides a good approximation of the initial yield behavior of this anisotropic two-phase FCC/BCC sheet material. Based on the experimental data for radial monotonic loading, it is concluded that conventional isotropic-kinematic hardening models cannot successfully describe the strain hardening of this austenitic steel. Instead, a non-associated anisotropic hardening model is proposed that relates the increase in yield strength to an isothermal martensitic transformation kinetics law. The comparison of the model predictions with the experimental results shows very good agreement for all biaxial and uniaxial experiments.

Sheet metal forming analyses with an emphasis on the springback deformation

Abstract An accurate modeling of the sheet metal deformations including the springback is one of the key factors in the efficient utilization of FE process simulation in the industrial setting. In this paper, a rate-independent anisotropic plasticity model accounting the Bauschinger effect is presented and applied in the FE forming and springback analyses. The proposed model uses the Hill's quadratic yield function in the description of the anisotropic yield loci of planar and transversely anisotropic sheets. The material strain-hardening behavior is simulated by an additive backstress form of the nonlinear kinematic hardening rule and the model parameters are computed explicitly based on the stress–strain curve in the sheet rolling direction. The proposed model is employed in the FE analysis of Numisheet’93 U-channel benchmark, and a performance comparison in terms of the predicted springback indicated an enhanced correlation with the average of measurements. In addition, the stamping analyses of an automotive part are conducted, and comparisons of the FE results using both the isotropic hardening plasticity model and the proposed model are presented in terms of the calculated strain, thickness, residual stress and bending moment distributions. It is observed that both models produce similar strain and thickness predictions; however, there appeared to be significant differences in computed residual stress and bending moments. Furthermore, the springback deformations with both plasticity models are compared with CMM measurements of the manufactured parts. The final part geometry and overall springback distortion pattern produced by the proposed model is mostly in agreement with the measurements and more accurate.

역공학과 해석적 방법을 이용한 관재벌지시험에서의 관재물성치 결정

In numerical analysis for hydroforming process, the stress calculation is effected by flow stress which is general obtained by stress-strain relationship from uni-axial tension test, so the result of the analysis, especially in tube hydroforming, has limitation of accuracy, tubes are made in roll-forming process and become work-hardened. Then roll forming process causes material properties between rolling direction and circumstantial direction of the tube to be different. So it is difficult to predict material behavior in the process condition of bi-axial stress state. In this study, the flow stress of the tube is determined by inverse engineering approach and bulge test that is widely used for formability test in the condition of bi-axial stress. And Hill's quadratic yield function and flow rule are used to consider the anisotropy of the tube in the roll forming process.

A Plastic Constitutive Model for Anisotropic Materials and Finite Element Analysis of Strain Localization Process

A plastic constitutive model for initially orthotropic but subsequently skewed anisotropic materials is proposed in this paper. The variation of anisotropic axes is discussed within the framework of objectivity, and the evolution of the axes is formulated. It is shown that the yield function proposed is identical to Hill's quadratic yield function when the anisotropic axes lie along the base vectors. Applying the conventional laws for both work-hardening and material flow to the yield function, an anisotropic constitutive model is completed. A finite element equation is obtained next, based on the updated Lagrangian formulation. A fully three-dimensional analysis is carried out to predict the strain localization behavior of anisotropic sheet metals. The direction of strain localization is strongly affected by the initial configuration of the anisotropic axes, and the simulated result shows fairly good agreement with the experimental one.

Rigid-Plastic Deformation of Metal with Elliptic Crystal Grain Deformed by Slip.

The rotation of the grain in polycrystalline metal is caused by crystal slips in each grain. In order to clarify the rotation of a grain surrounded by neighbouring grains, a plane model is introduced in which an anisotropic elliptic inclusion is embedded in an isotropic matrix. Hill's quadratic yield function allowing for orthotropic anisotropy is adopted to express the slip in the inclusion. The deformation and the rotation of the inclusion are analyzed with the rigid-plastic finite element method and discussed based on the calculated results. The inclusion deformed by slip rotates so that the slip direction coincides with the tensile direction. The rotation of the grain is restricted when the shape of grain deviates from the circular shape.

Evolution of anisotropy during twisting of cold drawn tubes

Cold drawn tubes of SAE 1020 steel have been twisted and then loaded by combinations of tension—torsion—internal pressure. After different amounts of twisting, tensile uniaxial yield stress has been measured at angles to the tube to confirm the orthotropic symmetry. Finite rotations of the orthotropy axes are observed. Measured plane stress yield locus show significant shape change during twisting. At each stage, experimental observations are very well represented by Hill's quadratic yield function. Discussions follow on the extension of Hill's theory to large deformation problems.

The application of an anisotropic yield function of the sixth degree to orthotropic material

By combining Drucker's yield function with Hill's quadratic yield function, an anisotropic yield function of the sixth degree is proposed. The effects of the third deviatoric stress invariant and initial anisotropy are also included. Experimental evaluation is made on 1050 aluminium tubes under multiaxial stress states. The tubes in the as-received condition are subjected to progressive reductions in the hot extruding and cold drawing processes. They are annealed by heating at 200 °C for 1 h. By applying proportionally combined loadings of axial load, internal pressure, and torsion to the specimens, a change of yield stress with a rotation of the principal stress axes and a difference between the directions of the principal stress and principal strain increment are examined. In the tension-internal pressure stress field, it is found that this aluminum tube exhibits orthotropic anisotropy of high strength in the tangential direction. The yield surface in the tension-torsion stress field lies outside von Mises’ yield surface. The torsional yield stress deviates considerably from von Mises criterion. Such behavioural characteristics can be expressed precisely by the proposed yield function. In addition, it is experimentally verified that the normality rule is obeyed in strain behaviour.

Evaluation of yield function including effects of third stress invariant and initial anisotropy

By means of combining Drucker's yield function with Hill's quadratic yield function, the anisotropic yield function of the sixth degree is proposed. It is able to include the effects of the third deviatoric stress invariant and initial anisotropy. The experimental evaluation is made on thin-walled cylindrical specimens of mild steel (in the fully annealed condition and the stress-relief annealed condition after a tensile pre-strain) and 2024 aluminum alloy (in the -0 and -T6 tempered conditions). By applying proportional combined loadings of axial load, internal pressure, and torsion to the specimens, a change of yield stress with a rotation of the principal stress axes and a difference between the directions of the principal stress and principal strain increment are examined Under tension–internal pressure and tension–torsion, the yield surfaces and strain behaviour are determined. The fully annealed steel is almost isotropic for yielding although it reveals the effect of the third deviatoric stress invariant. The stress-relief annealed steel and the 2024–0 and -T6 aluminum alloys exhibit axi-symmetric anisotropy. All of the yield surfaces can be expressed precisely by the proposed yield function. In addition, the normality rule is obeyed in strain behaviour.

Plane-stress crack-tip fields for perfectly plastic orthotropic materials

Plane stress mode I crack-tip fields for perfectly plastic orthotropic materials are studied. Plastic orthotropy is described by Hill's quadratic yield function. The construction of crack-tip fields is based on the general crack-tip field analysis for elastic perfectly plastic materials given by Rice (1) and guided by the correspond- ing low-hardening power-law solutions. Two very different types of plane-stress crack-tip fields emerge as plastic orthotropy is varied. The first one consists of a centered fan sector in front of the crack tip and two neighboring constant stress sectors. The second one consists of a constant stress sector in front of the crack tip, a constant stress sector bordering the crack face, and a centered fan sector between the two constant stress sectors. All the perfectly plastic crack-tip solutons are verified by the corresponding low-hardening power-law solutions. General trends of crack-tip stress solutions as functions of plastic orthotropy and implications of these solutions to the design of ductile composite materials are discussed.

Plane-stress crack-tip fields for power-law hardening orthotropic materials

Plane stress mode I near-tip fields in orthotropic materials are examined. Plastic orthotropy is described by Hill's quadratic yield function and the strain hardening behavior is given by an appropriate generalization of a uniaxial tensile power-law stress-strain relation. Pronounced changes in the pattern of the angular variations of crack-tip fields have been observed with the degree of plastic orthotropy and the amount of strain hardening. Possible shapes and sizes of plastic zones (as inferred from effective stress contours) are presented for high- and low-hardening materials and a wide range of plastic orthotropy. The shape of the plastic zone for a particular case of plastic orthotropy agreed remarkably well with the zone of intense straining induced by an appropriately orientated crack within a graphite/epoxy laminate.

Plane-strain crack-tip fields for power-law hardening orthotropic materials

Near-tip stress and strain fields for power-law hardening orthotropic materials under plane-strain conditions are presented. Plastic orthotropy is described by Hill's quadratic yield function. The angular variations of these HRR-type fields depend on a single parameter which specifies the state of plastic orthotropy. Near-tip fields for highly orthotropic materials differ substantially from the fields for isotropic materials. Mode I (symmetric) and mode II (anti-symmetric) solutions for different degrees of plastic orthotropy are given. The angular stress distributions for the low-hardening material agree remarkably well with the plane-strain slip-line fields. Based on the singularity fields, effective stress contours are constructed. The applicability of these fields in the context of a fiber-reinforced composite containing a macroscopic flaw is discussed.

A rigid-plastic finite-element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications

General three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys Hill's quadratic yield function is considered in the present analysis. The explicit expressions of the yield criterion and the related equations are given in the surface curvilinear co-ordinate system. A convective co-ordinate system is used in order to take into account the effect of geometric change during one step in the incremental analysis. The expression of the effective strain increment during one step is obtained in closed form by introducing assumptions on the deformation path and by proper consideration of the rotation of the axes of anisotropy during deformation. Considering the equilibrium at the deformed state, a variational formulation is derived to determine the deformation during a step. The corresponding finite-element equations are found in order to analyze the general deformation of planar anisotropic sheet metals. Two computational examples are chosen and computed by using the developed FEM program in order to verify the present formulation. The flange deformation in deep drawing of a circular diaphragm of planar anisotropy is analyzed and compared with the existing solution. The effects of planar anisotropy in the circular diaphragm bulge test are also investigated.