Atomistic and mean-field estimates of effective stiffness tensor of nanocrystalline copper
Introduction
Metal polycrystals are examples of heterogeneous materials in which heterogeneity results mainly from different orientations of local anisotropy axes in each grain. When the size of grains is sufficiently large the continuum mechanics framework is applicable and usually the boundaries between the grains are treated as perfect interfaces that do not require special treatment.
In order to estimate effective properties of a coarse-grained polycrystalline material a micromechanical methodology is used. Besides the classical Voigt and Reuss estimates, the self-consistent model due to Kröner (1958) and Hill (1965) is most often employed. This methodology was successfully extended to the (visco)plastic crystals and large strain regime (Lebensohn, Tomé, 1993, Molinari, Canova, Ahzi, 1987). Verification of such micromechanical mean-field estimates is usually performed by means of full-field simulations employing either the finite element method (FEM) (Fan, Xie, Sze, 2010, Kamaya, 2009) or the fast Fourier transform (FFT) technique (Lebensohn, Kanjarla, Eisenlohr, 2012, Liu, Raabe, Roters, Eisenlohr, Lebensohn, 2010, Prakash, Lebensohn, 2009).
Nanocrystalline metals are polycrystalline materials with a grain size smaller than 100 nm (Gao, Wang, Ogata, 2013, Gleiter, 2000). In such materials two phases can be distinguished: grain cores and grain boundaries. The effect of the grain boundaries on the overall properties of a bulk material increases with decreasing grain size (Gao, Wang, Ogata, 2013, Sanders, Eastman, Weertman, 1997). At the macroscopic level a nanograined material is still described in the framework of continuum theories. Since there is a need for quick and simple estimates of effective properties, which use only basic knowledge of the material behavior: single grain properties, orientation distribution (texture) within the material volume and the averaged grain size, micromechanical mean-field estimates are also utilized in this case. Most of the mean-field schemes proposed for nanocrystalline materials describe them as composite media made of two or more phases. One phase is the grain core and the other phase (or phases) represents the behavior of the grain boundary. In the early mixture-based model (Carsley, Ning, Milligan, Hackney, & Aifantis, 1995) a nanophase metal was described as a mixture of a bulk intergranular region and a grain boundary built of amorphous metal. The overall strength of the material was obtained as a simple volume average of the strengths of the two phases. In Kim and Bush (1999) a refined mixture model was developed in which the intercrystalline phase was composed of three sub-phases: grain boundary, triple line junctions and quadratic nodes. Additionally pores were considered. All phases were modeled as isotropic and in order to obtain overall elastic moduli the implicit self-consistent relation was used. The concept of grain boundary subdivision into sub-phases was followed by Benson, Fu, and Meyers (2001); Qing and Xingming (2006), although using simple volume averages to obtain the overall Young’s modulus and strength, while in the approach presented by Zhou, Li, Zhu, and Zhang (2007) the overall properties were assessed by using subsequently isostrain and isostress assumptions for finding the average response of the grain core and the two grain boundary sub-phases. In Sharma and Ganti (2003) a nano-grained material was described as a crystalline matrix with embedded ellipsoidal flat disks representing the grain boundary. To estimate the overall strength that accounts for grain boundary sliding the disks’ stiffness moduli were anisotropic with one modulus vanishing, and the Mori–Tanaka (MT) scale-transition scheme was applied. In Jiang and Weng (2004) the idea of the generalized self-consistent (GSC) model was used to find the strength and stiffness of a nanograined polycrystal: the coating of the spherical grain was assumed to be made of an isotropic material representing the grain boundaries. The grain cores were assumed to be elastically isotropic, while retaining their plastic anisotropy. This idea was followed by Capolungo, Cherkaoui, and Qu (2007) and Ramtani, Bui, and Dirras (2009). In the former model the implicit formulae of the GSC scheme (Christensen & Lo, 1979) were replaced by the explicit relations of the Mori–Tanaka model under the assumption that the grains are embedded in the matrix of the grain boundary phase. Similarly, in Mercier, Molinari, and Estrin (2007) a two-phase elastic-viscoplastic model combined with the Taylor–Lin homogenization theory was used to find the overall yield stress of nanocrystalline copper. The core-shell modeling framework was also used in the case of nanowires (Chen, Shi, Zhang, Zhu, & Yan, 2006). The main difficulty of all these models is related to a valid description of the grain boundary phase(s) and quantification of its volume fraction.
Differently than in the case of coarse-grained materials, in the case of nano-grained polycrystals the use of full-field simulations based on continuum theory for the purpose of verification or calibration of the above-discussed models is questionable. Therefore atomistic simulations based on molecular dynamics are performed to assess the elastic moduli and strength of such materials (Chang, 2003, Choi, Park, Hyun, 2012, Fang, Huang, Chiang, 2016, Gao, Wang, Ogata, 2013, Mortazavi, Cuniberti, 2014, Schiøtz, Vegge, Di Tolla, Jacobsen, 1999). An alternative approach is to use refined constitutive models, for example those incorporating scale effects through gradient enhancements, within the framework of FEM calculations (Kim, Dolbow, & Fried, 2012).
In atomistic simulations of elastic properties, usually only the tensile Young’s modulus is determined, see e.g. Gao et al. (2013); Schiøtz et al. (1999). It is observed that this modulus decreases with a decreasing grain size. This tendency is qualitatively and quantitatively reproduced by two- or multi-phase micromechanical models after proper adjustment of the grain boundary stiffness and the volume fraction, e.g. (Gao, Wang, Ogata, 2013, Jiang, Weng, 2004, Ramtani, Bui, Dirras, 2009).
The goal of the present paper is to estimate the effective elastic properties of nanocrystalline copper taking the anisotropy of a single crystallite fully into account. To this end, a series of static molecular simulations are performed to find all 21 components of the stiffness tensor and an anisotropic mean-field core-shell model of a nanograined polycrystal is formulated within the continuum mechanics framework. The results of both approaches are compared.
The paper is constructed as follows. The next section describes the details of the performed atomistic simulations. Section 3 presents the proposed formulation of the two-phase model of a nano-grain polycrystal. Additionally, the isotropisation procedure is outlined for the anisotropic stiffness acquired in the simulations. Section 4 discusses the results of the atomistic simulations and compares their outcomes with the respective mean-field estimates. The paper is closed with conclusions.
Section snippets
Computational methods
All molecular simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) (Plimpton, 1995) and a scientific visualization and analysis software for atomistic simulation data OVITO was used to visualize and analyse simulation results (Stukowski, 2010). Due to interest in the static behavior of the material, the molecular statics (MS) approach at the temperature of 0K (Maździarz, Young, Dłuzewski, Wejrzanowski, Kurzydłowski, 2010, Maździarz, Young,
Continuum mechanics estimates of elastic moduli
Overall elastic properties of a coarse-grained polycrystalline material are usually found employing the micromechanics methodology set in a continuum mechanics framework. According to the micromechanical theories polycrystals are one-phase heterogeneous materials in which heterogeneity roots from the different orientation of crystallographic axes from grain-to-grain in the representative volume of polycrystalline continuum. Effective properties are assessed assuming that local elastic
Results of atomistic simulations
The computational samples analysed by atomistic approach are denoted as where NUC is a number of unit cells, Ng - a number of grain orientations and SYS denotes the system of grain distribution, i.e.: BCC, FCC or random, see Table 1. The finite set of Ng orientations has been considered, namely polycrystalline representative volumes composed of copper grains with 16, 54, 128 or 250 randomly selected orientations have been analysed. The orientation was defined in terms of three Euler
Conclusions
The paper discusses the applicability of continuum-mechanics mean-field one-phase and two-phase models to estimating effective elastic properties of bulk nanocrystalline FCC copper. The methods used to verify such micromechanical mean-field estimates in the case of coarse-grained materials, such as full-field simulations on representative aggregates using the FEM or FFT framework, when they are combined with the classical linearly elastic constitutive law, are not applicable in the present
Acknowledgments
The simulations were performed at the Interdisciplinary Centre for Mathematical and Computational Modeling of Warsaw University (ICM UW) and the computing cluster GRAFEN at Biocentrum Ochota. The research was partially supported by the project no. 2016/23/B/ST8/03418 of the National Science Centre, Poland.
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