Modeling bending of α-titanium with embedded polycrystal plasticity in implicit finite elements
Introduction
Titanium and titanium alloys constitute an important class of metals with widespread applications, ranging from aerospace, to medical applications, to consumer products, due to their outstanding properties such as high specific strength, good formability, good corrosion resistance and biocompatibility. Pure titanium (α-Ti) has a hexagonal close-packed (hcp) crystal structure. Hcp single crystals are known to exhibit strong mechanical anisotropy. The mechanical response of polycrystalline α-Ti aggregates is also highly anisotropic with marked tension–compression asymmetry, mainly due to pronounced texture (non-random distribution of crystal orientations), which is a consequence of thermo-mechanical processing. The complete sets of elasto-plastic anisotropic properties associated with all possible textures for α-Ti have theoretically been identified [1], [2]. Grain morphology was also found to play a major role in anisotropy of α-Ti [3]. The behavior of α-Ti is further complicated by a wide variety of plastic deformation mechanisms with different activation stresses, which evolve differently with deformation, giving rise to anisotropic macroscopic hardening. A large number of studies have been performed to identify the deformation mechanisms that are active under quasi-static deformation of pure α-Ti at room temperature. The experimental data reported concern the mechanical response and microstructure evolution under various loadings, including plane-strain compression and simple shear [4], uniaxial and biaxial tension [5], uniaxial compression [4], [6], equal-channel angular extrusion [7], and rolling [8]. It was established that prismatic <a> slip is the easiest glide mode in α-Ti. This was attributed to its c/a ratio of 1.587, which is lower than the ideal c/a ratio of 1.633. basal <a> slip also operates in α-Ti but require higher activation stresses than the prismatic systems, or elevated temperatures [9]. pyramidal <c+a> slip systems are also harder than prismatic slip, but offer the additional degrees of freedom necessary to accommodate arbitrary plastic strains.
In addition to slip, these studies also reported occurrence of two main deformation twinning modes in α-Ti. The tensile twinning and the compressive twinning modes result in tensile and compressive strains along the parent grain’s c-axis, respectively, and re-reorient the latter by 84.8° about the direction and by 64.6° about the direction, respectively [10]. Due to these microstructural changes, twinning has a profound influence on texture evolution and strain hardening of the material during plastic deformation [4], [11]. In addition, primary twins can favorably reorient a crystal for secondary and, subsequently, even for ternary twinning. The occurrence of these second- and third-generation twins in high-purity α-Ti and their influence on the mechanical behavior, especially under dynamic loadings was recently reported [6].
Three competing effects are generally accepted to explain the significant contribution of deformation twinning to strain hardening: (i) the texture hardening due to crystal reorientation, from softer to harder [12], (ii) the Hall–Petch effect due to grain subdivision [13], and (iii) the Basinski hardening mechanism due to transmutations of dislocations from glissile to sessile [14]. Thus, a realistic description of the mechanical behavior and microstructure evolution of α-Ti requires accurate twinning models. Four different approaches were proposed to address this outstanding problem and incorporated in crystal plasticity based constitutive laws: (i) the predominant twin reorientation (PTR) method [15], (ii) the volume fraction transfer (VFT) scheme [16], (iii) the total Lagrangian approach [17], and (iv) the composite-grain method [18]. All these methods were implemented within crystal plasticity models to account for microstructure evolution. Models such as Taylor-type [19], [20], [21] and self-consistent-type [22], [23] have been rigorously formulated and widely used to understand and predict the mechanical response and microstructure evolution in metals subject to finite strains. The incorporation of crystal plasticity models into finite-element (FE) codes has also received substantial attention. For a detailed overview, the reader is referred to [24]. In particular, the strategy for embedding the viscoplastic self-consistent (VPSC) model in an implicit FE framework in the commercial FE code ABAQUS was presented in [25]. However, FE calculations where a polycrystalline aggregate is associated with each FE integration point are computationally extremely expensive, thus limiting its applicability to problems that do not require a fine spatial resolution. In view of applications to detailed analyses of complex forming processes for which the FE meshes are routinely of the order of thousands of elements, a robust macroscopic level elastic/plastic model are useful. The predictive capabilities of such a model can be assessed not only by comparison with experimental data but also through comparison with simulations using advanced polycrystalline models.
In the present paper, we demonstrate that using the VPSC framework the main features of the mechanical response of high-purity α-Ti are captured with very good accuracy. The plastic deformation mechanisms during monotonic tension and compression tests along different orientations are identified based on comparison with a comprehensive set of characterization data collected using electron backscattered diffraction (EBSD) and neutron diffraction. Due to the lack of reliable single-crystal data, the VPSC material parameters are calibrated based on stress–strain macroscopic response in monotonic tension and compression.
We first show that the VPSC model is able to reproduce stress–strain response and microstructure evolution for all orientations. Validation of the VPSC model is provided by applying it to testing conditions that were not used for the identification of the material parameters. To this end, simulations of the three-dimensional deformation of α-Ti beams subjected to four-point bending along different directions with respect to the hard-to-deform crystallographic <c> axis predominant orientation of the material are presented.
Furthermore, we compare the simulation results in bending obtained by FE-VPSC to the results obtained using the analytic macroscopic yield criterion proposed in [23] coupled with a new distortional hardening model. It is shown that the accuracy of the results obtained with the macroscopic model is comparable to those obtained with the polycrystal model. We show that both models predict with great accuracy the evolution of tension-compression asymmetry of the material and the rigidity of the response along the crystallographic <c> direction. Quantitative agreement with experimental data is presented. Specifically, the shifts of the neutral axis of the beams and changes in the cross-section geometry during deformation are well captured. Such a comparison is important in view of structural and process modeling applications.
Section snippets
Material and experiments
The starting material used in the present work is high-purity (99.999%), single-phase α-Ti. The material was supplied in the form of a 15.87 mm thick cross-rolled disk of 254 mm. The aggregate exhibits equiaxed twin-free grain structure with an average grain size of about 20 μm [12]. Fig. 1 shows the initial macro-texture measured by neutron diffraction at Los Alamos Neutron Science Center (LANSCE), and the initial micro-texture measured by electron back scattered diffraction (EBSD). The pole
Polycrystalline modeling
In order to model the mechanical behavior of polycrystalline α-Ti, we used a self-consistent model. This class of models is based on the solution of the problem of an ellipsoidal inclusion embedded in a homogenous effective medium. The inclusion is taken to be an individual grain while the homogenous medium represents the polycrystalline aggregate. A detailed description of the VPSC model used in this study can be found elsewhere [e.g. 22]. Here we only report the equations necessary to follow
Embedding of a self-consistent crystal plasticity model in implicit finite elements
The strategy of embedding the VPSC model at meso-scale level in an implicit FE framework was discussed in [25]. A finite element integration point is considered as a polycrystalline material point, whose meso-scale mechanical response is obtained interrogating the VPSC model. As a result, we introduce another level of homogenization from the meso-scale to the macro-scale. In the implementation of VPSC as a User Material Subroutine (UMAT) in ABAQUS-Standard, the total strain increment Δε is
Orthotropic elasto-plastic constitutive model
The above direct implementation of a polycrystal model into an advanced commercial FE code has the advantage that it follows the evolution of anisotropy and tension-compression asymmetry due to evolving texture. As mentioned earlier, these kinds of FE calculations are computationally very intensive. In view of applications to detailed analyses of complex forming processes, a robust macroscopic level elastic/plastic model is useful.
In the following, we present such a macroscopic formulation and
Bending of α-titanium
We simulate a set of four-point beam-bending experiments on the beams of α-Ti using the VPSC and J2-J3 models embedded in ABAQUS-Standard. Bending conditions were selected for cross-validation of the models capabilities to predict the effects of the directionality of twinning. Depending on the loading direction with respect to the <c> axis of the grains, we expect to find qualitative differences between the response of the upper (top/compressive) and lower (bottom/tensile) fibers of the beam
Conclusions
The deformation behavior of polycrystalline α-titanium was examined using experiments and crystal plasticity modeling. The mechanical response, characterized in tension and compression along the main plate directions, was highly anisotropic and exhibited strength-differential effects. Twinning statistics and texture evolution were determined based on EBSD and neutron diffraction data. The twin volume fraction in a sample deformed along RD to a strain of 0.3 was measured to be over 60%. We
Acknowledgments
M. Knezevic gratefully acknowledges the Seaborg Institute for the Post-Doctoral Fellowship through the LANL/LDRD Program with the U.S. Department of Energy. R.A. Lebensohn acknowledges support from LANL’s Joint DoD/DOE Munitions Technology Program. The authors also acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy & Microanalysis Research Facility at the Australian Centre for Microscopy and Microanalysis, The University of Sydney, especially
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