Influence of critical resolved shear stress ratios on the response of a commercially pure titanium oligocrystal: crystal plasticity simulations and experiment
Introduction
In recent years, the use of commercially pure titanium (CP-Ti) has steadily expanded to a wide range of industrial applications. Among them, the biomedical, automotive, and aerospace industries particularly benefit from the high specific strength, corrosion resistance, and good biological compatibility of this metal (Brunette et al., 2012; Peters et al., 2003). CP-Ti sheets also exhibit stronger in-plane anisotropy under deformation than that in structural components of other metals, such as steel or aluminum. In particular, the work hardening behavior and Lankford value evolution of CP-Ti sheets are strongly dependent on the sample orientation with respect to the rolling direction. Stress–strain curves are also typically asymmetric between tension and compression. These properties are attributed to the combined influence of the low degree of symmetry of the hexagonal close packed (HCP) structure and the occurrence of twinning.
These mechanisms take place at the grain scale, and thus, the crystal plasticity finite-element method (CPFEM) (Peirce et al., 1982; Asaro, 1983) has constituted the natural framework to study the macroscopic deformation of CP-Ti sheets. The latest iterations of CPFEM models accurately reproduce the stress–strain curve asymmetry, Lankford value evolution, and texture evolution under various strain paths (Wu et al., 2007; Hama and Takuda, 2011; Gurao et al., 2011; Zambaldi et al., 2012; Warwick et al., 2012; Benmhenni et al., 2013; Kowalczyk-Gajewska et al., 2015; Sinha et al., 2016; Amouzou et al., 2016; Marchenko et al., 2016; Hama et al., 2017; Baral et al., 2018). However, the accurate identification of the numerous constitutive parameters involved in these models remains challenging. For instance, and contrary to body-centered cubic (BCC) and face-centered cubic (FCC) metals, the critical resolved shear stresses (CRSSs) vary from one slip system family to another. The CRSS values reported in the literature for prismatic slip in CP-Ti range from 30 MPa to 181 MPa, according to a recent survey by Wang and co-workers (Wang et al., 2017). These values also depend on grain size and chemical composition (Poty et al., 2011).
Moreover, CRSS magnitude ratios between slip families also fluctuate significantly (Warwick et al., 2012; Philippe et al., 1995), and this problem is emphasized for basal slip. As surveyed in our previous study (Hama et al., 2017), it may be reasonable to assume that prismatic slip has the smallest CRSS, followed in order by pyramidal <a> slip, twinning, twinning, and pyramidal <a+c> slip. However, the rank of the CRSS of basal slip is still undetermined because it is significantly different depending on the literature references considered (Wu et al., 2007; Philippe et al., 1995; Conrad, 1981). Therefore, in our previous study (Hama et al., 2017), the rank of the CRSS of basal slip was determined in an empirical manner. Although the parameters determined in the previous study were in good agreement with experimental results, it was difficult to justify the choice of this set of parameters thoroughly. The difficulty lies in identifying precise and unique CRSS values based solely on the comparison of the stress–strain curves and texture evolutions in simulations and experiments. To overcome this issue, researchers have been striving to gather additional experimental data directly at the grain scale. For pure Ti, Gong and Wilkinson (2009) have carried out bending tests on single crystal micro cantilever beams, and identified prismatic, pyramidal, and basal plane CRSSs from comparisons with CPFEM computations. The use of single crystals is definitely effective to identify the parameters for each system. However, because the effect of grain boundaries on the parameters cannot be evaluated, it may be difficult to use the determined parameters for polycrystalline materials directly. Moreover, it may also be difficult to find appropriate conditions to activate the arbitrary slip or twinning systems; thus, it is difficult to identify the parameters for all possible systems. Wang et al. (2017) proposed a direct evaluation of CRSSs utilizing X-ray microscopy to monitor the crystalline orientations and stress states evolution for a large number of grains in a tensile specimen. They estimated that the ratio of CRSS between basal slip and prismatic slip would be between 1.7 and 2.1 for the CP-Ti they tested. This approach is reliable to estimate this ratio, but some slip and twinning systems remain undetermined.
The present authors consider that oligocrystals (samples containing up to a few dozen grains) would be quite useful to further improve the parameters because of the following reasons. (1) Not only the macroscopic stress–strain curves but also the strain fields within each grain can be evaluated; thus, the effect of the parameters, including the effect of grain boundaries, can be examined at the grain scale. (2) Because active slip and/or twinning systems can be directly evaluated, it is expected that new insight into mesoscopic deformation and the effect of parameters at the grain scale can be examined in more detail (Poty et al., 2011; Chattopadhyay et al., 2012; Wang et al., 2015).
Approaches based on a systematic comparison of crystal plasticity analysis and full-field measurements for displacement and strain carried out on oligocrystals have shown promise to investigate the model accuracy and improve the constitutive parameter identification (Delaire et al., 2000; Zhao et al., 2008; Lim et al., 2014). Studying the deformation of oligocrystals rather than that of single crystals is also relevant because the grain boundaries and grain to grain misorientations effects (Lim et al., 2011; Klusemann et al., 2013), as well as grain size effects (Wang et al., 2012; Guery et al., 2016), can then be examined. To reduce the effect of the unknown underlying microstructure on the surface response, these works focused on small tensile samples containing columnar grains. Electron backscatter diffraction (EBSD) measurements were carried out on undeformed samples to accurately reproduce grain geometries and orientations in numerical models. The surface deformation was then recorded under mechanical loading, using in-situ (Delaire et al., 2000; Lim et al., 2014) or ex-situ (Zhao et al., 2008) digital image correlation (DIC) (Sutton et al., 1983, 1986; Chu et al., 1985) providing the surface strain fields. Performing additional EBSD measurements on the deformed sample allowed comparing the texture evolution within individual grains with the simulation results (Delaire et al., 2000; Zhao et al., 2008; Klusemann et al., 2012, 2013; Turner et al., 2012). Local slip activity at the grain scale can also be scrutinized by comparing the slip lines observed on the deformed sample surface with the model prediction, considering the most active slip systems in the simulation (Havliček et al., 1990; Yao and Wagoner, 1993; Ziegenbein et al., 1998; Delaire et al., 2000; Guery et al., 2016; Guan et al., 2017). Finally, this framework provides additional comparison points such as deformed sample shapes (Klusemann et al., 2012, 2013; Zhao et al., 2008; Lim et al., 2014) and surface roughening (Zhao et al., 2008; Lim et al., 2014).
However, most previous works studied either FCC (Cheong and Busso, 2006; Delaire et al., 2000; Zhao et al., 2008; Turner et al., 2012; Yao and Wagoner, 1993; Ziegenbein et al., 1998; Guan et al., 2017) or BCC (Lim et al., 2011, 2014; Klusemann et al., 2012, 2013; Havliček et al., 1990) metals. In the case of HCP metals, few studies dedicated to surface strain measurement of oligocrystals can be found in the literature. Utilizing a multi-scale experimental setup, Efstathiou and co-workers were able to examine in detail the macroscopic and mesoscopic strain distributions in a CP-Ti polycrystal (Efstathiou et al., 2010). With a similar experimental setup, Aydiner and colleagues (Aydiner and Telemez, 2014) detailed the impact of twinning on the mesoscopic and macrospic strain fields of a magnesium alloy. A more local response was recorded by Barkia et al. (2015b), who carried out an in-situ tensile test of CP-Ti instrumented with the grid method to record the strain distribution within a small number of grains and highlighted the influence of grain boundaries on strain localization. The previous studies provide detailed results at the grain scale. However, the experimental findings were not directly confronted with CPFEM simulations. Recently, Zhang et al. (2018) examined the micro slip distribution in a Ti-6Al-4V polycrystal using crystal plasticity analysis, and found a fairly good agreement with the experimental results obtained with high-resolution DIC (HR-DIC) and high-resolution EBSD (HR-EBSD). Direct comparisons between crystal plasticity predictions and DIC results remain, however, scarce for HCP metals, thus providing the motivation for the present study.
The objective of this work is to study the appropriate CRSS ratios for a grade 2 CP-Ti by examining the deformation behavior of a grade 2 CP-Ti oligocrystal, especially focusing on the basal slip CRSS. To this end, a uniaxial tensile test was conducted using an oligocrystal that had 32 grains in its gauge section, and the deformation fields on the sample surface obtained by DIC were compared on a grain per grain basis with the crystal plasticity finite element simulations for several sets of CRSSs. The sample surface was scanned using EBSD before and after the tensile test to reproduce the sample microstructure in the numerical model accurately and to examine the texture evolution. The paper is organized as follows: Section 2 describes the material, sample preparation, and experimental setup. Section 3 details the crystal plasticity model and choice of reference constitutive parameters. Section 4 presents and compares the results of the simulation with experimental data, with respect to texture evolution, strain fields, and slip activity. The effect of CRSS ratios on the predictive accuracy of these results, especially focusing on basal slip, is then discussed in Section 5. Finally, the conclusions obtained in this work are summarized in Section 6.
Section snippets
Material
The investigated material was grade 2 CP-Ti with coarse grain size (5 mm on average). It was obtained from continuous casting conducted in a plasma oven. As a result of this process, the grains were typically elongated in the casting direction. To obtain a columnar microstructure, the tensile samples were prepared perpendicularly to this direction. The tensile dogbone sample geometry is presented in Fig. 1, along with a micrograph illustrating the typical microstructure aspect. The chemical
Crystal plasticity model
The crystal plasticity model employed in this study followed that developed in a previous study for CP-Ti sheets (Hama et al., 2017). The basic formulations of this model were first developed to study the deformation behavior of Mg alloy sheets (Hama and Takuda, 2011). Then the model was modified to take into account detwinning processes (Hama and Takuda, 2012a), which led to accurate predictions of the deformation behavior of Mg alloy sheets under a variety of loading paths, including
Strain fields
Simulations using the reference constitutive parameters were carried out by using the fine mesh model (Fig. 3b). The axial strain field () evolution obtained experimentally is shown in Fig. 4, along with the stress–strain curve recorded during the test. The averaged axial strain as estimated by DIC in the gauge section was used to produce the stress–strain curve.
The axial strain distribution are highlighted for several points of the stress–strain curve in Fig. 4. Point A corresponds to the
Choice of CRSSs ratios
The determination of CRSS values for CP-Ti at room temperature has been examined in previous studies (Philippe et al., 1995; Wu et al., 2007; Warwick et al., 2012; Wang et al., 2017). They concurred in reporting that the deformation mode with the smallest CRSS value is prismatic slip, followed in order by pyramidal <a> slip, twinning, twinning, and pyramidal <a+c> slip, but the rank of basal slip CRSS reported in these works differed as follows. In their literature review, Wang
Conclusion
In this study, the ability of crystal plasticity simulations to reproduce the deformation behavior of a grade 2 CP-Ti oligocrystal was examined in detail. The use of adapted parameters from a previous study on grade 1 CP-Ti led to reasonable simulation–experiment agreements with respect to texture evolution, strain field distributions, and slip line predictions. A method based on Radon transform of DIC residuals to automatically detect and measure slip line orientations to facilitate
Acknowledgments
This work has been funded by the Japan Society for the Promotion of Science under the short-term fellowship program (PE16708), and the authors gratefully acknowledge its support. The authors would also like to thank Mr. Sohei Uchida of TRI Osaka for his kind assistance in performing the EBSD measurements. TH acknowledges that this study was also supported by Japan Society for the Promotion of Science, JSPS, KAKENHI, grant numbers 17H03428 and 17K06858. Last but not least, the authors would like
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