Forming of aluminum alloys at elevated temperatures – Part 1: Material characterization

A temperature-dependent anisotropic material model for use in a coupled thermo-mechanical finite element analysis of the forming of aluminum sheets was developed. The anisotropic properties of the aluminum alloy sheet AA3003-H111 were characterized for a range of temperatures 25–260 °C (77–500 °F) and for different strain rates. Material hardening parameters (flow rule) and plastic anisotropy parameters (R₀, R₄₅ and R₉₀) were calculated using standard ASTM uniaxial tensile tests. From this experimental data, the anisotropy coefficients for the Barlat YLD96 yield function [Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997a. Yield function development for aluminum alloy sheets. J. Mech. Phys. Solids 45 (11/12), 1727–1763] in the plane stress condition were calculated for several elevated temperatures. Curve fitting was used to calculate the anisotropy coefficients of Barlat’s YLD96 model and the hardening parameters as a function of temperature. An analytical study of the accuracy and usability of this curve fitting technique is presented through the calculation of plastic anisotropy R-parameters and yield function plots at different temperatures.     

Plane stress yield function for aluminum alloy sheets—part 1: theory

A new plane stress yield function that well describes the anisotropic behavior of sheet metals, in particular, aluminum alloy sheets, was proposed. The anisotropy of the function was introduced in the formulation using two linear transformations on the Cauchy stress tensor. It was shown that the accuracy of this new function was similar to that of other recently proposed non-quadratic yield functions. Moreover, it was proved that the function is convex in stress space. A new experiment was proposed to obtain one of the anisotropy coefficients. This new formulation is expected to be particularly suitable for finite element (FE) modeling simulations of sheet forming processes for aluminum alloy sheets.     

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.     

Unit cell debonding analyses for arbitrary orientation of plastic anisotropy

Debonding of rigid inclusions embedded in the elastic–plastic aluminum alloy Al 2090-T3 is analyzed numerically using a unit cell model taking full account of finite strains. The cell is subjected to overall biaxial plane strain tension and periodical boundary conditions are applied to represent arbitrary orientations of plastic anisotropy. Plastic anisotropy is considered using two phenomenological anisotropic yield criteria, namely Hill [Proceedings of the Royal Society of London A 193 (1948) 281] and Barlat et al. [International Journal of Plasticity 7 (1991) 693]. For this material plastic anisotropy delays debonding compared to plastic isotropy except for the case of Hill’s yield function when the tensile directions coincided with the principal axes of anisotropy. For some inclinations of the principal axes of anisotropy relative to the tensile directions, the stress strain responses are identical but the deformation modes are mirror images of each other.     

Inflation and burst of aluminum tubes. Part II: An advanced yield function including deformation-induced anisotropy

In a recent study [Korkolis, Y.P., Kyriakides, S., 2008. Inflation and burst of anisotropic aluminum tubes for hydroforming applications. Int’l. J. Plasticity 24, 509–543], the formability of aluminum tubes was investigated using a combination of experimental and numerical approaches. The tubes were loaded to failure under combined internal pressure and axial load along radial paths in the engineering stress space. The experiments were then simulated using appropriate FE models and two established anisotropic yield functions. It was found that for some loading paths the computed deformations did not agree with the experimental ones, whereas rupture was generally overpredicted. In the current study the problem is tackled using a more advanced 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 function for aluminum alloy sheets – part I: theory. Int’l. J. Plasticity 19, 1297–1319]. Three different calibration schemes of this function are employed, in two of which the experimentally observed deformation-induced anisotropy is taken into account. It is demonstrated that both deformation and failure can ultimately be predicted successfully, albeit arduously, using a hybrid procedure detailed herein.     

Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals

In this paper, yield functions describing the anisotropic behavior of textured metals are proposed. These yield functions are extensions to orthotropy of the isotropic yield function proposed by Cazacu et al. (Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic yield criterion for hexagonal close packed metals. Int. J. Plasticity 22, 1171–1194). Anisotropy is introduced using linear transformations of the stress deviator. It is shown that the proposed anisotropic yield functions represent with great accuracy both the tensile and compressive anisotropy in yield stresses and r-values of materials with hcp crystal structure and of metal sheets with cubic crystal structure. Furthermore, it is demonstrated that the proposed formulations can describe very accurately the anisotropic behavior of metal sheets whose tensile and compressive stresses are equal.     

Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function

In this work, the recently proposed anisotropic yield function, Yld2004-18p [Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation based anisotropic yield function, Int. J. Plasticity 21, 1009], is implemented in a finite element (FE) code for application to the cup drawing simulation of a circular blank sheet. A short review of the Yld2004-18p relevant features is provided and the stress integration scheme for its implementation in FE codes is described. The simulation of the drawing process is conducted for an aluminum alloy sheet sample (AA2090-T3). The predicted and experimental cup height profiles (earing profiles) with six ears are shown to be in excellent agreement. Additional simulations on a ficticious material are performed in order to show that the yield function Yld2004-18p can lead to the prediction of cups with eight ears. In order to achieve these results, a sufficient number of input data are required to calculate the yield function coefficients. Finally, a simplified analytical approach that relates the earing profile to the r-value directionality is also presented in this paper. It is shown that this approach can be very useful as a first approximation of the earing profile of drawn cups.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

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.     

An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming

Up to now, several computational methods have been proposed for crystal plasticity models. The main objective of these computational methods has been to overcome the problem with the non-uniqueness of active slip systems during the plastic deformation of a single crystal. Crystal plasticity models based on a single crystal yield function have been proposed as alternative algorithms to overcome this problem. But the problem with these models is that they use a highly non-linear yield function for the crystal, which makes them computationally expensive. In this paper, a computational method is proposed that would modify a single crystal yield function in order to make it computationally efficient. Also to better capture experimental data, a new parameter is introduced into the single crystal yield function to make it more flexible. For verification, this crystal plasticity model was directly applied for the simulation of hydroforming of an extruded aluminum tube under complex strain paths. It was found that the current model is considerably faster than the previous crystal plasticity model based on a power-law type single crystal yield surface. Due to its computational efficiency, the current crystal plasticity model can also be used to calculate the anisotropy coefficients of phenomenological yield functions.     

Earing predictions based on asymmetric nonquadratic yield function

A nonquadratic yield function (Yld96; Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J. Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997. Yield function development for aluminium alloy sheet. J. Mech. Phys. Solids, 45, 1727) which simultaneously accounts for the anisotropy of uniaxial yield stresses and r values was newly implemented in a finite element code. Yield surface shapes, yield stress and r-value directionalities of Yld96 were investigated and compared with those of the previous yield function, Yld91 (Barlat, F., Lege, D.J., Brem, J.C. 1991a. A six-component yield function for anistropic metals. Int. J. Plasticity, 7, 693). Complete formulations for Yld96 implementation and the calculation of coefficients were also discussed for the convenient use of Yld96. A 2090-T3 aluminum alloy sheet sample was modeled and earing formation during a cup drawing test was simulated using the FEM code. The results of earing and thickness strain profiles were compared with the results obtained with Yld91. Investigations were further carried out with a translated yield surface to account for the strength differential effect observed in this material. Computation results with the translated yield surface were in very good agreement with experimental results. It was shown that the yield surface shape and translation have a significant influence on the prediction of the cup height profile during the drawing of a circular blank.     

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.     

Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets

Finite element (FE) simulations of the simple shear test were conducted for 1050-O and 6022-T4 aluminum alloy sheet samples. Simulations were conducted with two different constitutive equations to account for plastic anisotropy: Either a recently proposed anisotropic yield function combined with an isotropic strain hardening law or a crystal plasticity model. The FE computed shear stress–shear strain curves were compared to the experimental curves measured for the two materials in previous works. Both phenomenological and polycrystal approaches led to results consistent with the experiments. These comparisons lead to a discussion concerning the assessment of anisotropic hardening in the simple shear test.     

A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming

Spitzig and Richmond [Acta Metall. 32 (1984) 457] proposed that plastic yielding of both polycrystalline and single crystals of steel and aluminum alloys shows a significant sensitivity to hydrostatic pressure. They further showed that under the associated flow rule, this pressure sensitivity leads to a plastic dilatancy, i.e. permanent volume change, that is at least an order of magnitude larger than observed. Indeed, the plastic dilatancy for most materials is on the order of the measurement error and must be zero in the absence of phase change and significant void nucleation during plastic deformation. A non-associated flow rule based on a pressure sensitive yield criterion with isotropic hardening is proposed in this paper that is consistent with the Spitzig and Richmond data and analysis. The significance of this work is that the model distorts the shape of the yield function in tension and compression, fully accounting for the strength differential effect (SDE). This capability is important because the SDE is sometimes described through kinematic hardening models using only pressure insensitive yield criteria.     

Anisotropic work hardening of steel sheets under plane stress

This work is a follow-up of the previous report by Kim and Yin [Kim, K.H., Yin, J.J., 1997. Evolution of anisotropy under plane stress. J. Mech. Phys. Solids 45, 841–851] regarding the anisotropic work hardening of cold rolled steel sheets. Tensile prestrain has been applied at angles to the rolling direction and then tensile uniaxial yield stress and R-value distributions are measured. As reported earlier, the orientations of local maxima and minima in the yield stress are altered when the prestrain axis is not in the rolling direction. This led Kim and Yin [Kim and Yin (1997)] to suggest that the orientations of orthotropy axes are altered by the tensile prestrain at angles to the rolling direction. However, R-value distribution is found to be hardly affected by the prestrain. The unchanging R-value distribution shows that the material remembers the rolling direction even after the prestrain. An attempt is made to approximate the observed yield and flow behavior based upon isotropic-kinematic hardening with the quadratic yield function (Hill, 1948). The degree of approximation raises the issues of yield point definition, flexibility of yield function, non-associated flow rules, distortional hardening and others.