A non-singular theory of dislocations in anisotropic crystals

Highlights

• We develop a non-singular theory of three-dimensional dislocation loops in anisotropic crystals. All key equations of anisotropic dislocation theory are derived as line integrals.

• The spreading function of the Burgers vector density is anisotropic, and it depends on up to six characteristic length scales.

• Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants.

• Without any fitting procedure, the non-singular stress field is compared to molecular statics calculations of virial stress in both cubic and hexagonal crystals.

• Anisotropic non-singular and atomistic results are in good agreement and, in contrast to the classical fields, their sign remain consistent in the dislocation core region.

Abstract

We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.

Introduction

Accounting for the elastic anisotropy of the medium is essential in studying several physical phenomena related to the mechanics of crystal dislocations. Elastic anisotropy has strong effects on dislocation behavior in irradiated materials, the microscopic deformation of crystals, the diffusion characteristics of atomic species, and on coupled elasto-magneto-electric phenomena. For example, elastic calculations in neutron-irradiated iron and copper (Seif and Ghoniem, 2013) have shown that anisotropy plays a significant role in the calculation of bias factors, a measure of the preferential absorption by dislocations of self interstitial atoms rather than vacancies, which ultimately results in swelling and material deterioration (Greenwood et al., 1959, Wolfer, 2007). Similarly, elastic anisotropy is known to alter stress-driven diffusion pathways in crystals (Larcht'e and Cahn, 1982), and in particular solute diffusion near dislocations (Garnier et al., 2013). This process finds practical applications in defect gettering in semiconductors (Perevostchikov and Skoupov, 2005), and may be relevant to improving the efficiency of Si-based solar cells for the photovoltaic industry (Ziebarth et al., 2015). Anisotropy is also responsible for particular features observed in the high temperature deformation of α-Fe, such as the presence of sharp corners in the shape of prismatic and glide loops (Fitzgerald and Yao, 2009, Fitzgerald and Aubry, 2010, Aubry et al., 2011), and the dependence of the critical stress necessary to bow out Frank-Read sources on its glide plane and orientation (Fitzgerald et al., 2012). The coupled elastic, electric and magnetic fields produced by an arbitrary three-dimensional dislocation loop in general anisotropic magneto-electro-elastic materials was developed by Han and Pan (2013). As a result of the coupling between the elastic, electric and magnetic fields, it was found that a dislocation, an electric potential discontinuity, or a magnetic potential discontinuity can induce simultaneous elastic, electric and magnetic fields. The coupling effect was also found to be very strong near the dislocation core. The elastic interaction between Self-Interstitial (SIA) Clusters, and between clusters and dislocations was included in kinetic Monte Carlo (KMC) simulations of damage evolution in irradiated bcc metals (Wen et al., 2005). The results indicate that near the dislocation core, SIA clusters, which normally migrate by 1D glide, rotate due to their elastic interactions. This rotation was found to be necessary to explain experimentally-observed dislocation decoration and raft formation in neutron irradiated pure iron. The elastic field of a general dislocation loop in anisotropic crystals was determined by incorporating numerically evaluated derivatives of Green's functions in the fast sum method (Han et al., 2003). The study showed that elastic anisotropy plays an important role in many dislocation mechanisms, such as local Peach-Koehler and self-forces, the operation of Frank-Read sources, dipole formation and break-up, and finally on dislocation junction strength. Compared to isotropic calculations, anisotropy was also found to yield image forces with opposite sign for dislocation models of bi-metal interface shearing (Chu et al., 2013), and therefore it may be important in predicting dislocation interactions with grain boundaries (e.g. Zbib et al., 2011, Wang et al., 2014). The elastic fields of three dimensional dislocation loops across anisotropic multilayer materials were studied by Han and Ghoniem (2005) and Ghoniem and Han (2005), while Zhang et al. (2016) considered the effects of core-spreading dislocation in anisotropic bi-materials.

Investigating these phenomena in the context of discrete dislocation dynamics (DDD) simulations requires the use of the anisotropic elastic theory of dislocations (e.g Mura, 1987). Compared to the isotropic case, the numerical implementation of the anisotropic elastic theory of dislocations suffers two main drawbacks, both of which are ultimately related to the Green's tensor of the anisotropic Navier differential operator (Lifshitz and Rosenzweig, 1947, Synge, 1957). The first drawback consists in the increased computational cost of the Green's tensor, which in classical anisotropic elasticity requires an additional integration on the equatorial circle of the unit sphere in Fourier space. In order to mitigate the heavy computational costs of Green's tensor, Aubry and Arsenlis (2013) have recently proposed an expansion in spherical harmonics, while Bertin et al. (2015) have proposed a fast Fourier Transform (FFT) method applicable to periodic dislocation systems. The second issue is due to the fact that the classical elastic fields of dislocations inherit the singularity of the Green's tensor at the origin, and therefore their numerical implementation requires some form of regularization. Although regularized theories of three-dimensional dislocation loops exist for isotropic materials (Cai et al., 2006, Banerjee et al., 2007, Lazar, 2012, Lazar, 2013, Po et al., 2014), their anisotropic extension is non-trivial. In fact, their common regularization technique consists in removing the singularity from the derivatives of the distance function R appearing in the Green's tensor by convolution with isotropic regularization functions. However, not only the anisotropic Green's tensor does not involve the derivatives of R explicitly, but the very assumption that in anisotropic materials the regularization function remains isotropic is questionable.

In this paper we develop a non-singular theory of three-dimensional dislocation loops in anisotropic media. The theory is derived within a simplified version of Mindlin's anisotropic strain-gradient elasticity with up to six length scale parameters, a framework already introduced by Lazar and Po (2015b) and called Mindlin's anisotropic gradient elasticity with separable weak non-locality (see also Lazar and Po, 2015a, Seif et al., 2015). In recent years, the use of non-locality as a mean to describe the elastic fields of defects cores has received renewed attention (e.g. Taupin et al., 2014, Taupin et al., 2017). In the proposed framework, the non-local parameters appear as the coefficients of an anisotropic Helmholtz differential operator. The Green's function of such operator is taken as the spreading function for the Burgers vector density. This choice is based on the ansatz that the volume affected by the plastic distortion is determined by the characteristic length scale parameters of the elastic continuum. This choice is justified by several considerations. First, the Green's function possesses unit integral over infinite space, and therefore it does not change the total Burgers vector density, a necessary condition for the non-singular fields to converge to their classical counterparts a few characteristic lengths away from the dislocation core. Second, because atomistic calculations show that such length scale parameters are in the order of nearest-neighbor interatomic distances, the plastically distorted volume remains localized between interatomic planes. Third, because the anisotropic Helmholtz operator is defined by a symmetric tensor of rank two, which must be invariant under material symmetry operations, the theory entails spreading functions which possess a different number of length scale parameters for different classes of material symmetry. In particular the spreading function possesses six independent length scales in triclinic, four in monoclinic, three in orthorhombic, two in tetragonal, hexagonal, and trigonal, and one for cubic crystals. For low symmetry crystals, therefore, the characteristic length scales of the “core regularization” are different along different crystallographic directions. Fourth, regardless of the class of material symmetry, the proposed non-singular dislocation theory regularizes the classical one maintaining formally identical dislocation key formulas. As a matter of fact, all the non-singular anisotropic dislocation fields can formally be obtained from their classical counterparts by replacing the singular Green's tensor with its regularized version given in Lazar and Po (2015b). The anisotropic non-singular versions of all classical dislocation equations are derived, and in particular: both Volterra and Burgers equations for displacement (Volterra, 1907, Burgers, 1939a, Burgers, 1939b), the Mura-Willis distortion equation (Mura, 1969, Willis, 1967), the Peach-Koehler stress and force equations (Peach and Koehler, 1950), and Blin's formula for the interaction energy between two dislocation loops (Blin, 1955). Apart from the generalized solid angle appearing in the displacement field, all equations are expressed in terms of non-singular line integrals, and therefore they are well-suited for applications in DDD simulations. An interesting technical remark is that the interaction energy equation is derived without the use of a stress function, a result which, to the best of our knowledge, was never obtained before.

The paper is organized as follows. In section 2 we develop the anisotropic non-singular theory of three dimensional dislocation loops, as a systematic generalization of the classical anisotropic theory. In section 3 we provide a physical interpretation of the simplified Mindlin framework, and we propose methods to determine the tensor of gradient length scale parameters for several crystal structures. In section 4 molecular statics calculations of the stress field in both cubic and hexagonal crystals are compared to the proposed anisotropic non-singular dislocation theory. Elastic calculations are performed using the DDD method, that is by numerical line integration of the proposed elastic kernels along the dislocation lines. Moreover, because the length-scale parameters are found by a deterministic method, the comparison is made without any fitting procedure. In section 5 we discuss the topic of self energy of dislocation loops, and we provide several results useful to compare non-singular anisotropic, non-singular isotropic, and classical theories. Finally, discussion and conclusions are presented in section 6.