Multi-scale modeling and experimental study of twin inception and propagation in hexagonal close-packed materials using a crystal plasticity finite element approach; part II: Local behavior
Introduction
Characterization of the mechanical response of materials under varied loading conditions can provide insight into the operating micro-deformation mechanisms. Various experimental and numerical methods have been implemented to answer critical questions about deformation mechanisms with the goal of e.g., improvement of in-service life-time of materials. While in-situ deformation studies, such as neutron or synchrotron diffraction are very powerful, and can provide extensive data as to the operating micro-mechanisms (Hutchings et al., 2005) they can be difficult or expensive to perform. Hence laboratory based techniques, such as digital image correlation (DIC), are also an important characterization tool. DIC is a versatile technique that has been used to understand deformation at various length scales from micro to macro. The basic concept behind image correlation is to find features in the images taken from areas of interest at different loading stage and try to find one-to-one correspondence between positions of these features (Lee et al., 1987, Fonseca et al., 2005, Tan et al., 2005, Amy, 2006, Peters and Ranson, 1982). Once the relations between images are resolved, strains can be extracted and deformation evolution can be studied. As the images are usually captured from the surface, strains at the surface can be measured and bulk information, in contrast to penetrating diffraction methods, is not monitored. Crystal plasticity in the finite element framework has been used in parallel with image correlation techniques to answer experimentally-irresolvable questions. Study of materials at this scale usually consist of mapping of grain shapes into a finite element solver and investigating the effects of grain–grain interaction on the average and local behavior of materials (Nakamachi et al., 2007, He′ripre′ et al., 2007, Ge′rard et al., 2009, Demir et al., 2010, Merzouki et al., 2010).
Because of their low absorption cross sections towards neutrons, zirconium and its alloys have been extensively used in nuclear reactors. Alpha-zirconium has Hexagonal Close-Packed (HCP) crystal structure with high tendency to twin under tensile loading along the c-axis of the crystal. It is known that twins form readily at the stress concentrations of crack tips in zirconium alloys (Kerr et al., 2010). Hence, a precise understanding of the interaction of local neighborhood on twinning has potentially significant practical interest as well as providing insight into the physical mechanism of twinning. In this paper, we report a study of twin formation, including both inception and propagation, in Zircaloy-2 samples.
Deformation twinning at the grain scale in a CPFE framework has been studied to some extent. Texture evolution as well as deformation and activity of twin variants of each grain were studied by Prakash et al. (2009) and Choi et al. (2010). To reduce numerical instability, an implicit-dynamic formulation was implemented by Barton et al. (2009) to model twin formation in a 125 grain polycrystal. A good picture of twin formation is presented in Barton et al.'s work, but the lack of CPFE results being compared with experiment, leaves questions as to the assumptions made. Formation of twins in single crystal zinc has been studied by Forest and Parisot (2000) where twin nucleation was virtually controlled by introducing a geometrical defect and twin growth was the result of the motion of the localization front on one or both sides of the twin, in the spirit of Maugin (1998). In the current study, local twin formation in Zircaloy-2 polycrystal samples is studied experimentally as well as numerically. Grain orientations and geometries at the surface of the samples are measured for different samples before and after uniaxial straining. In-situ tensile tests are carried out in an SEM chamber where grains' strains are measured by the digital image correlation technique. Grain maps are imported into the ABAQUS FE solver to study inception and propagation of twins using different assumptions within a crystal plasticity formulation. The CPFE results are compared against experimental results both locally and statistically.
Section snippets
Experimental procedure
Set one of dog bone samples (S1) with 14 mm gauge length, 3 mm width, and 2 mm thickness were prepared from a previously well characterized Zircaloy-2 slab (Xu et al., 2008b, Xu et al., 2009, Mareau and Daymond, 2010). The initial texture of the slab is shown in Fig. 1a. Due to having a relatively small grain size (∼13 μm) and an inability to readily resolve grain boundaries in S1, cubes from the original slab were cold-rolled to 10% (compression along previous rolling direction with 10% thickness
Input models
The pre-deformation orientation maps of grains, either directly determined (S2) or extrapolated (S1) were used to generate input files for the FE solver. For this purpose, the average orientation of each grain was calculated (Cho et al., 2005, Pantleon et al., 2008) and used as the initial input orientation of each grain. Hence, orientation variations within grains before deformation are neglected. Grain maps were subsequently imported into the FE solver and meshed with C3D8 cubic elements with
Twin inception and propagation
Results of the simulations for the two cases shown in Fig. 2a and b are shown in Fig. 4, Fig. 5, respectively. The M1 scheme is used for both cases with the implementation of plane stress boundary conditions (see Section 4.3). Random colors are assigned to each grain in Fig. 4a and b; however, black dots in Fig. 4b represent the reoriented (twinned) elements. The size of these black dots is proportional to the number of reoriented IPs in each element. In order to compare CPFE results with the
Discussion
First we would like to summarize our CPFE observations and discussions. A CPFE code has been used to study twin inception and propagation in different measured microstructures of Zircaloy-2 samples. In the current approach, the stress state at each IP is the only parameter controlling twin inception and propagation; it is shown that twins tend to initiate at grain boundaries in both “soft” and “hard” grains due to stress concentration originating from mismatch between elastic and plastic
Conclusion
Various different approaches are tested in a CPFE framework to investigate the ability of CPFE to explain twin formation in an HCP material. Two dimensional measured grain maps are implemented as the input files for FE simulations. It is assumed that twin inception and propagation are exclusively controlled by stress state at each IP and the predominant twin variant is responsible for twin formation. CPFE simulations revealed that twins tend to form at grain boundaries especially at the
Acknowledgments
This work was supported by a Discovery Grant from the Canadian Natural Sciences and Engineering Research Council.
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