Plane Stress Yield Function Research Articles

Description of anisotropic properties of sheet metal based on spline curves and hole expansion test simulation of high-strength steel

A plane stress yield function with a yield surface described by Bézier curves comprising arbitrary numbers of segments is presented. This yield function can consider an arbitrary number of multiaxial stress states and their plastic strain rate directions as Bézier curve reference data. The planar anisotropy of the yield surface is described by the basis function of the cubic spline. By using the proposed model, the anisotropic properties of sheet metal are captured flexibly and precisely. To demonstrate the applicability of the proposed yield function, the accuracy of the captured anisotropic properties of high-strength steel was investigated. The proposed model was implemented in a commercial finite element analysis code, and hole expansion simulations of the high-strength steel were conducted to show the accuracy of the prediction.

A compact plane stress yield function formulation

The construction of anisotropic yield function by introducing linear transformations is a common practice with many phenomenological yield criteria. Based on the K1 − K2 representation method of principle stresses (represented by Barlat89 yield function), a compact 13-parameter plane stress function is proposed. The new yield function consists of five different linear transformations, and is designed to have a certain symmetry in both the overall structure and the distribution of linear transformation matrix components. The new 13-parameter yield function is applied to predict the yield surfaces (loci), plastic strain rate directions, flow stresses and r-values for both fcc and bcc materials. Compared with the yield functions of Yld2004-18p and CPB06ex2 (both reduce to 14 parameters in symmetric plane stress condition), the new yield function can achieve similar prediction accuracy with fewer parameters.

The formability of twinning—Induced plasticity steels predicted on the base of Marciniak-Kuczynski theory

Abstract The purpose of this work is to predict and analyze the formability of twinning – induced plasticity steels through the Marciniak-Kuczynski (MK) theory with emphasis on the solutions for improving the prediction results. The selected constitutive equations involve the Yld00-2d of Barlat et al. (2003) plane stress yield function, the Swift strain–hardening power law and the dislocation density based constitutive model proposed by Kim et al. (2013), taking into account the dislocation glide, twinning and dynamic strain aging. Three types of high manganese content TWIP steels sheet were selected. To understand the formability of the TWIP steel and the factors influencing it, a sensitive study on the effect of the mechanical properties of the TWIP steel on the MK theory concept and the predicted forming limits is performed. Using the dislocation density based microstructural model, the deformation twinning effect and the contribution of dynamic strain aging to the FLDs of TWIP steel is analyzed. The relevant influence of the sharpness of the yield surface in the biaxial stretching region in the prediction of FLDs of TWIP steels is highlighted. The extended MK model can be adapted to predict the forming limits of the TWIP steels by using an unusual high initial geometrical defect imposed by their high strain hardening. In this way it was showed that the MK theory cannot be applied for predicting the forming limits of TWIP steels unless by applying imperfection factors that are not physically reasonable. Therefore, new failure models are required for TWIP steel.

Anisotropic plasticity characterization of 6000- and 7000-series aluminum sheet alloys at various strain rates

This study aims to characterize the high strain rate constitutive behavior and the anisotropic plasticity of three high strength aluminum sheet alloys, AA6013-T6, AA7075-T6 and a developmental AA7xxx-T76 alloy. A comprehensive series of mechanical characterization tests were performed under uniaxial tension, simple shear and through-thickness compression (representing equal-biaxial tension) at room temperature and various strain rates ranging from 10−3 to 103 s−1. It was observed that all three alloys exhibited appreciable plastic anisotropy in terms of the measured r-values along various sheet orientations, although they exhibited minimal anisotropy in the stress response. For all three alloys, the flow stress slightly increases with increasing strain rate, which indicates mild positive strain rate sensitivity. However, the flow stress ratios and r-values do not show any explicit relation with respect to strain rate suggesting a negligible effect of strain rate on in-plane plastic anisotropy. Measurement of the constitutive response to large strains (beyond onset of diffuse necking) was achieved using shear specimens that are not subject to onset of necking. To model the flow stress dependence on equivalent plastic strain and strain rate, an extended Hockett–Sherby constitutive model is proposed and shown to fit accurately the experimental data over the studied range of strain and strain rates. The yield behavior of the sheet alloys was predicted using the Barlat YLD2000-2D plane stress yield function (Barlat et al., 2003) assuming both associative and non-associative (NA) flow rules, as well as for general stress states employing two 3D anisotropic models, Barlat YLD2004 (Aretz and Barlat, 2004) and modified–Yoshida (Lou and Yoon, 2018). An associative flow rule was assumed for both of the higher order 3D yield functions considered herein. The measured flow stress ratios and r-values from the uniaxial tension and equal-biaxial tension experiments, along with the stress ratios from the shear tests, were used to determine the coefficients of the yield functions. The calibrated yield functions exhibited good agreement with the experimental data for most loading conditions considered. The higher order Barlat YLD2004 and modified–Yoshida yield functions provided the better predictive accuracy. The non-associative Barlat YLD2000 yield function also performed reasonably well, while the associative Barlat YLD2000 yield function had the lowest accuracy. The constitutive models were implemented into the FE simulation, in order to evaluate the accuracy of the model parameters calibrated in this work.

A Plane Stress Yield Function Described by Multi-Segment Third Order Bézier Curves

A plane stress yield function which is described by multi-segment spline curve is proposed. This model is able to consider an arbitrary number of multi-axial stress states and its normal directions on the yield surface. Moreover by using interpolation based on cubic spline function, planar anisotropy is also described precisely. To show the applicability of the model, some evaluation of material properties and simulation results of hole expansion test are discussed.

A plane stress yield function described by multi-segment spline curves and its application

A plane stress yield function which is described by multi-segment spline curve is proposed. This model is able to consider an arbitrary number of multi-axial stress points and its normal directions on the yield surface. To show the applicability of the model, some evaluation examples are shown and its accuracy is discussed.

A Yield Function for Anisotropic Sheet Material Described by 3<sup>rd</sup>-Degree Spline Curve and its Applications

A new plane stress yield function using the 3rd-degree spline curve is proposed for the anisotropic behavior of sheet metals. This yield function considers the evolution of anisotropy in terms of both r values and stresses. In order to demonstrate the applicability of the proposed yield function, hole expanding tests with mild steel and 6000 series aluminum alloy sheets were simulated.

Plane stress yield function described by 3rd-degree spline curve and its application

In this study, a plane stress yield function which is described by 3rd-degree spline curve is proposed. This yield function can predict a material anisotropy with flexibility and consider evolution of anisotropy in terms of both r values and stresses. As an application, hole expanding simulation results are shown to discuss accuracy of the proposed yield function.

3次スプライン降伏関数およびその異方硬化モデルの提案

3次スプライン降伏関数およびその異方硬化モデルの提案

A Theoretical Study of the Effect of the Double Strain Path Change on the Forming Limits of Metal Sheet

The present paper aims at a theoretical study of the forming limits of a sheet metal subjected to double strain path changes by using as reference material the AA6016-T4 aluminum alloy sheet. The simulation of plastic instability is carried out through the Marciniak-Kuczynski analysis. The initial shape of the yield locus is given by the Yld2000-2d plane stress yield function. The strain hardening of the material is described by the Voce type saturation law. Linear and several complex strain paths involving single and double strain path changes are taken into account. The validity of the model is assessed by comparing the predicted and experimental forming limits under linear and selected one strain path change. A good accuracy of the developed software on predicting the forming limits is found. A sensitive analysis of the influence of the type and value of the double prestain in the occurrence of the plastic flow localization is performed. A remarkable effect of the double strain path change on the sheet metal forming limits is observed.

Consideration of strength differential effect in sheet metals with symmetric yield functions

A simple approach is proposed to extend symmetric yield functions for the consideration of strength differential (SD) effect in sheet metals. The SD effect is coupled with symmetric yield functions by simply adding a weighted pressure term for anisotropic materials based on the approaches of Spitzig et al. (Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metall 1995;23:885–893), Spitzig and Richmond (The effect of pressure on the flow stress of metals. Acta Metall 1984;32:457–463) and Stoughton and Yoon (A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming, Int J Plast 2004;20:705–731). This approach is applied to the symmetric Yld2000-2d yield function (Barlat et al. Plane stress yield function for aluminum ally sheets—Part I: Theory, Int J Plast. 2003;19:1297–1319. The Yld2000-2d yield function is then modified to describe the anisotropic and asymmetric yielding of two aluminum alloys with small SD effect: Al 2008-T4 and Al 2090-T3 and four hexagon-closed packed (HCP) metals with strong SD effect: pure textured magnesium, an Mg 0.5% Th Sheet, an Mg 4% Li sheet and a Ti–4Al–1/4O2 titanium sheet. The predicted yield surfaces are compared with experimental results for the evaluation of the modified yield function. The comparison reveals that both anisotropy and asymmetry are correctly modeled by the proposed approach.

Modeling of anisotropic plastic behavior of ferritic stainless steel sheet

Modeling of anisotropic plastic behavior of ferritic stainless steel sheet (Type 409) was investigated using the three yield functions of Hill [A theory of the yielding and plastic flow of anisotropic metals. Proceedings of Royal Society of London, Series A 1948;193:281–97.], Barlat and Lian [Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity 1989;5:51–66] and Barlat et al. [Plane stress yield function for aluminum alloy sheet. Part I: Theory. International Journal of Plasticity 2003;19:1297–319.] (referred to as Yld2000-2d) criteria. Mechanical behaviors were characterized based on uniaxial tension, balanced biaxial bulge, and disk compression tests. Directionalities of yield stresses and r values were predicted from the three criteria and compared with experimental results. In order to verify the modeling accuracy of the three functions under complex loading conditions, cylindrical cup drawing and limiting dome height tests were carried out numerically and experimentally. It has been demonstrated that the result from Yld2000-2d criterion exhibits good agreement with experimental data. The effects of anisotropic hardening on earing and necking were also investigated based on the different levels of plastic work.

Path-dependent failure of inflated aluminum tubes

Our recent investigation on the formability of Al alloy tubes under combined internal pressure and axial load is expanded by examining the effect of the loading path traced. A set of Al-6260-T4 tubes were loaded along orthogonal stress paths to failure and the results are compared to those of the corresponding radial paths. It is confirmed that failure strains are path-dependent, but also is demonstrated that failure stresses become path-dependent if the prestrain is significant. The experiments are simulated using the previously developed finite element models and the calibration of the Yld2000-2D [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets-part I: theory. Int. J. Plasticity 19, 1297--1319] anisotropic yield function shown in [Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of anisotropic aluminum tubes. Part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plasticity 24, 1625–1637] to yield accurate predictions of rupture for nine radial paths. The models are shown to reproduce the path dependence of the failure stresses and strains quite well. A group of additional radial and corner paths are subsequently examined numerically to enrich the existing data on path-dependence of failure. It is again shown that the amount of plastic prestraining in either of the two directions influences the difference of the failure stresses and strains between the radial and the corner stress paths.

Yield function calibration for orthotropic sheet metals based on uniaxial and plane strain tensile tests

Plastic anisotropy in numerical analysis of sheet metal forming operations can be described very efficiently by means of analytical yield functions. However, for an anisotropic sheet material acceptable calibration of an appropriate yield function requires numerous mechanical testing procedures involving different loading modes such as directional uniaxial tensile tests and an equibiaxial tensile or bulge test. Realization of the mentioned loading modes requires the usage of different testing devices that are not always available. Therefore, a calibration procedure for advanced yield functions involving only a single tensile test machine seems appealing. In the present article a calibration method involving three directional uniaxial tensile tests and two plane strain tensile tests is proposed. This calibration method is applied to the recently introduced yield function Yld2003 [H. Aretz, Applications of a new plane stress yield function to orthotropic steel and aluminium sheet alloys, Model. Simul. Mater. Sci. Eng. 12 (2004) 491–509; H. Aretz, A non-quadratic plane stress yield function for orthotropic sheet metals, J. Mater. Process. Technol. 168 (2005) 1–9] using different materials. The obtained results are compared to those obtained by using the conventional calibration based on uniaxial and equibiaxial test data. It is shown that for all considered materials the proposed calibration procedure gives satisfying results. Finally, for selected materials and with respect to the different calibration procedures sensitivity analyses concerning the predicted forming limit diagram (FLD) are carried out using the finite element based FLD simulation approach proposed by Lademo et al. [O.-G. Lademo, T. Berstad, O.S. Hopperstad, K.O. Pedersen, A numerical tool for forma-bility analysis of aluminium alloys. Part I. Theory, Steel Grips 2 (2004) 427-431; O.-G. Lademo, T. Berstad, O.S. Hopperstad, K.O. Pedersen, A numerical tool for formability analysis of aluminium alloys. Part II. Experimental validation, Steel Grips 2 (2004) 433–438].

On linear transformations of stress tensors for the description of plastic anisotropy

The derivation of anisotropic yield functions based on the approach of linear transformations of a stress tensor is investigated for general and plane stress states. The number of coefficients available for the description of plastic anisotropy is discussed. A few specific yield functions are given to illustrate the concept. Among these examples, a plane stress formulation is described in more detail, namely, Yld2000-2d [Barlat, F., Brem, J.C., Yoon, J.W., Dick, R.E., Choi, S.H., Chung, K., Lege, D.J., 2000. Constitutive modeling for aluminum sheet forming simulations. In: Khan, A.S, Zhang, H., Yuan, Y. (Eds.), Plastic and Viscoplastic Response of Materials and Metal Forming, Proceedings of the 8th International Symposium on Plasticity and its Current Applications, Whistler, Canada, July 2000. Neat Press, Fulton, MD, pp. 591–593; Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets – Part I: theory. Int. J. Plasticity 19, 1297–1319; Yoon, J.W., Barlat, F., Dick, R.E., 2000. Sheet metal forming simulation for aluminum alloy sheets. In: Sheet Metal Forming Simulation: Sing-Tang 65th Anniversary Volume, SAE paper 2000-01-0774, Society of Automotive Engineer, SAE, pp. 67–72]. It is shown that other recently published anisotropic yield functions are, in fact, Yld2000-2d presented in different forms.

Forming of AA5182-O and AA5754-O at elevated temperatures using coupled thermo-mechanical finite element models

A temperature-dependent anisotropic material model was developed for two aluminum alloys AA5182-O and AA5754-O and their anisotropy parameters were established. A coupled thermo-mechanical finite element analysis of the forming process was then performed for the temperature range 25–260 °C (77–500 °F) at different strain rates. In the developed model, the anisotropy coefficients for Barlat’s YLD2000-2d anisotropic yield function [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets – Part 1: Theory. Int. J. Plasticity 19, 1297–1319] in the plane-stress condition and the parameters for the isotropic strain hardening were established as a function of temperature. The temperature-dependent anisotropic yield function was then implemented into the commercial FEM code LS-DYNA as a user material subroutine (UMAT) using the cutting-plane algorithm for the integration of a general class of elastoplastic constitutive models [Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006b. Forming of aluminum alloys at elevated temperatures – Part 2: Numerical modeling and experimental verification. Int. J. Plasticity 22 (2), 342–737]. The temperature-dependent material model was used to simulate the coupled thermo-mechanical finite element analysis of the stamping of an aluminum sheet using a hemispherical punch under the pure stretch boundary condition (no material draw-in was allowed). Simulation results were compared with experimental data at several elevated temperatures to evaluate the accuracy of the UMAT’s ability to predict both forming behavior and failure locations. Two failure criteria were used in the analysis; the M–K strain based forming limit diagrams ( ε-FLD), and the stress based forming limit diagrams ( σ-FLD). Both models were developed using Barlat’s YLD2000-2d anisotropic model for the two materials at several elevated temperatures. Also, as a design tool, the Genetic Algorithm optimization program HEEDS was linked with the developed thermo-mechanical models and used to numerically predict the “optimum” set of temperatures that would generate the maximum formability for the two materials in the pure stretch experiments. It was found that a higher temperature is not needed to form the part, but rather the punch should be maintained at the lowest temperature possible for maximum formability.

Numerical analysis of diffuse and localized necking in orthotropic sheet metals

In the present paper the diffuse and localized necking models according to Swift [Swift, H.W., 1952. Plastic instability under plane stress, Journal of the Mechanics and Physics of Solids, 11–18], Hill [Hill, R., 1952. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 1, 19–30] and Marciniak and Kuczyński [Marciniak, Z., Kuczyński, K., 1967. Limit strains in the process of stretch-forming sheet metal. International Journal of Mechanical Sciences 9, 609–620], respectively, are considered. A theoretical framework for the mentioned models is proposed that covers rigid–plastic as well as elastic–plastic constitutive modelling using various advanced phenomenological yield functions that are able to account very accurately for plastic anisotropy. The mentioned necking models are applied to different orthotropic sheet metals in order to assess their predictive capabilities and to stress out some potential sources for discrepancies between simulations and experiments. In particular, the impact of the applied hardening curve and the equibiaxial r-value, which was recently introduced by Barlat [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Choi, S.-H., Pourboghrat, F., Chu, E., Lege, D.J., 2003. Plane stress yield function for aluminium alloy sheets – part 1: theory. International Journal of Plasticity 19, 297–1319], on formability prediction is investigated. Furthermore, the impact of the Portevin–LeChatelier effect on the formability of aluminum sheet metals is discussed.

Modeling of aluminum alloy sheets based on new anisotropic yield functions

New anisotropic models (Yld2000-2d, Barlat et al. [F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, S.H. Choi, F. Pourboghrat, E. Chu, D.J. Lege, Plane stress yield function for aluminum alloy sheets, Int. J. Plastic. 19 (2003) 1297–1319]; Yld2004-18p, Barlat et al. [F. Barlat, H. Aretz, J.W. Yoon, M.E. Karabin, J.C. Brem, R.E. Dick, Linear transformation-based anisotropic yield functions, Int. J. Plastic. 21 (2005) 1009–1039]) that accurately describe the anisotropic behavior of aluminum alloy sheets were implemented in a finite element code, LS-DYNA User MATerial subroutine (UMAT). The Yld2004-18p especially, requires sufficient experimental input taken every 15° from the rolling direction. As planar anisotropic behavior, earing simulation was carried out in order to show the capability of Yld2004-18p to predict more than four ears. A general automotive panel simulation which covers forming, trimming and springback was performed to show the generality and efficiency of the two new yield functions. The simulation results led to the excellent predictions.

A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states

An anisotropic plane stress yield function based on interpolation by second order Bézier curves is proposed. The parameters for the model are readily derived by four mechanical tests: a uniaxial, an equi-biaxial and a plane strain tensile test and a shear test. In case of planar anisotropy, this set of tests must be repeated for a number of directions. An arbitrary number of directions can be incorporated in the model. This flexibility can be used to describe, e.g., the yield stress and R-value as a function of the loading direction more accurately than with other common analytical yield functions. The relevance of an accurate description of the yield locus and the flexibility of the proposed yield function are demonstrated by the prediction of forming limit diagrams. It is demonstrated that the sensitivity of the FLD to small changes in the yield locus can also be used to determine some of the material parameters by inverse analysis.

A non-quadratic plane stress yield function for orthotropic sheet metals

In this article, a new plane stress yield function for orthotropic sheet metals called ‘Yld2003’ is proposed. The yield function contains eight anisotropy parameters which can be fitted to selected experimental input data. The main attractivity of the new yield function lies in its very simple mathematical form which makes it very effective and thus quite interesting for implementation in finite element codes. Special attention is paid to the flexibility of the new yield function to reproduce experimental input data. Some applications to orthotropic sheet materials and comparison to the recent yield function ‘Yld2000-2D’ proposed by Barlat et al. [F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, S.-H. Choi, F. Pourboghrat, E. Chu, D.J. Lege, Plane stress yield function for aluminium alloy sheets – part 1: theory, Int. J. Plast. 19 (2003) 1297–1319] confirm the capabilities of the proposed yield function. Furthermore, it is shown that the proposed yield function has nearly the same flexibility like Yld2000-2D.