Journal of Materials Science: Materials Theory
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Nonsingular straight dislocations in anisotropic crystals
Journal of Materials Science: Materials Theory volume 8, Article number: 5 (2024)
Abstract
A nonsingular dislocation theory of straight dislocations in anisotropic crystals is derived using simplified anisotropic incompatible first strain gradient elasticity theory. Based on the nonsingular theory of dislocations for anisotropic crystals, all dislocation keyformulas of straight dislocations are derived in generalized plane strain, for the first time. In this model, the singularity of the dislocation fields at the dislocation core is regularized owing to the nonlocal nature of strain gradient elasticity. The nonsingular dislocation fields of straight dislocations are obtained in terms of twodimensional anisotropic Green functions of simplified anisotropic strain gradient elasticity. All necessary Green functions, including the twodimensional Green tensor of the twofold anisotropic HelmholtzNavier operator and the twodimensional \(\varvec{F}\)tensor of generalized plane strain, are derived as sum of the classical part and a gradient part in terms of Meijer Gfunctions. Among others, we calculate the regularization of the Barnett solution for the elastic distortion of straight dislocations in an anisotropic crystal. In the framework of simplified anisotropic first strain gradient elasticity, the necessary material parameters are computed for cubic materials including aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using a second nearestneighbour modified embeddedatommethod interatomic potential. The elastic distortion and stress fields of screw and edge dislocations of \(\frac{1}{2} \langle 111\rangle\) Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of \(\frac{1}{2} \langle 110\rangle\) Burgers vector in fcc copper and fcc aluminum have been computed and presented in contour plots.
Introduction
This paper is dedicated to Professor Nasr Ghoniem, and it celebrates his illustrious and exemplary career in the field of the mechanics and physics of defects in crystals. His holistic research style, often involving experiments, theory, and numerical modeling, has created many valuable opportunities to connect researchers in the field. Our collaboration started about ten years ago, and it stemmed from an attempt to include characteristic length scales in the elastic theory of dislocations. Strengthened by several mutual visits between UCLA and TU Darmstadt, such collaboration led to several manuscripts (Po et al. 2014; Lazar and Po 2014, 2015a, b; Seif et al. 2015; Po et al. 2018; Lazar and Po 2018a, b; Po et al. 2019; Cui et al. 2019; Lazar et al. 2020). The present work builds on our simplified strain gradient elasticity theory of dislocations in anisotropic crystals, and it derives specialized results for straight dislocations.
Classical continuum theories like the theory of linear elasticity are intrinsically size independent. For the study of dislocations in anisotropic crystals, classical anisotropic elasticity theory is often used (e.g., Bacon et al. (1980); Ting (1996); Hirth and Lothe (1982); Steeds (1973)). In twodimensional (2D) anisotropic elasticity, the displacement fields of straight dislocations were derived by Stroh (1958, 1962) using the socalled Stroh formalism (see also Ting (1996)) and by Asaro et al. (1973) using the socalled integral formalism (see also Bacon et al. (1980); Balluffi (2012)). The integral formalism was originally derived from the Stroh formalism by Barnett and Lothe (1973). In two dimensions (2D), the elastic distortion and the strain energy of infinitely long straight dislocation lines with Burgers vector \(\varvec{b}\) in an anisotropic medium were given by Barnett and Swanger (1971). Using the twodimensional anisotropic Green tensor of generalized plane strain, a Burgerslike formula for straight dislocations has been given by Lazar (2021) leading to a new derivation of the integral formalism (see also Lazar and Kirchner (2021)). It is wellknown that classical anisotropic elasticity is not valid at small scales leading to singularities in the dislocation fields at the dislocation core. However, the nearfield behaviour of the dislocation fields is of high importance for applications and for the understanding of physics within the dislocation core.
Dislocations are lattice defects of great significance, since they cause plasticity and hardening in crystals. A dislocation is a line defect in a crystal breaking locally the translational symmetry of the perfect crystal and leading in this way to a lower symmetry at the defect region of the imperfect crystal, namely at the dislocation core. In fact, the dislocation core is just an arrangement of atoms without any crystal symmetry. From the crystallographic point of view, the translational symmetry of crystals is disturbed by the lattice defect (dislocation) so that the symmetry of the point group of the dislocation core region is lower than the symmetry of the original point group of the perfect crystal. The broken symmetry in the dislocation core is important for many physical phenomena like plastic deformation, superalloys at high temperature, and birefringence (see, e.g., Kosevich (1979)). However, in some cases, it can be useful to look at the imperfect crystal from the point of view of approximate symmetry. Moreover, crystals have a discrete structure. The range of interaction can never be less than the discrete length, which is a finite length proportional to the lattice constant. Discreteness itself gives rise to nonlocality.
Therefore, a generalized continuum field theory, which possesses nonlocality and avoids singularities at small scales, is needed for an improved modelling of dislocations in crystals. Generalized continuum theories such as strain gradient elasticity and nonlocal elasticity are theories valid down to the Ångströmscale (see, e.g., Eringen (2002); Lazar (2017); Lazar et al. (2020, 2022)). Mindlin (1964) (see also Mindlin (1968)) derived the theory of compatible first strain gradient elasticity. Compatible first strain gradient elasticity incorporates the first gradient of the elastic strain tensor in the elastic energy in addition to the elastic strain tensor. For the isotropic case, this framework is characterized by the two Lamé constants and five strain gradient parameters leading to two characteristic lengths. In the early days of strain gradient elasticity, several trials (e.g., Lardner (1971); Rogula (1973)) to find nonsingular fields produced by dislocations were not successful, leading only to additional singularities in the dislocation fields. More than three decades later, Altan and Aifantis (1997) derived a simplified version of Mindlin’s first strain gradient elasticity. Using such a simplified first strain gradient elasticity theory with only one characteristic length scale parameter, Gutkin and Aifantis (1996, 1997) found, for the first time, nonsingular elastic strain fields of screw and edge dislocations in the framework of gradient elasticity. Lazar and Maugin (2005) (see also Lazar et al. (2005); Lazar (2017)) have shown how nonsingular stress and strain fields of screw and edge dislocations can be computed in simplified first strain gradient elasticity including eigenstrain fields called simplified incompatible strain gradient elasticity. Such simplified first strain gradient elasticity is a particular version of Mindlin’s first strain gradient elasticity where the double stress tensor can be expressed in terms of the gradient of the Cauchy stress tensor (see, e.g., Lazar and Maugin (2005); Lazar (2016)). Simplified incompatible strain gradient elasticity (gradient elasticity of Helmholtz type) provides robust nonsingular solutions including one length scale parameter for the elastic distortion, plastic distortion, stress and displacement fields of screw and edge dislocations. An important mathematical property of simplified strain gradient elasticity is that it provides a straightforward regularization based on partial differential equations (PDEs) of higher order where the characteristic length scale parameter plays the role of a regularization parameter. The nonsingular expressions of all dislocation key equations were given by Lazar (2012, 2013, 2014) for dislocation loops using simplified strain gradient elasticity. For dislocations, the incompatible version of simplified strain gradient elasticity including plastic distortion and dislocation density tensors is used leading to an incompatible strain gradient elasticity of defects. These nonsingular dislocation key equations (Burgers formula, MuraWillis equation and PeachKoehler stress formula) have been implemented in the UCLA Discrete Dislocation Dynamics (DD) code called “model” (Model 2014) and used for applications (Po et al. 2014).
In order to model dislocations in cubic crystals, the extension of isotropic simplified incompatible strain gradient elasticity towards anisotropic simplified incompatible strain gradient elasticity has been given by Lazar and Po (2015a, b). Anisotropic incompatible strain gradient elasticity represents an anisotropic gradient elasticity with separable weak nonlocality which is a special version of Mindlin’s anisotropic strain gradient elasticity theory with up to six independent length scale parameters. The framework models materials where anisotropy is twofold, namely the bulk material anisotropy (farfield anisotropy) and a weak nonlocal anisotropy (nearfield anisotropy) relevant at the Ångströmscale. Using Fourier transform, Lazar and Po (2015a, b) have computed the threedimensional elastic Green tensor of anisotropic incompatible strain gradient elasticity as fundamental solution of the twofold anisotropic HelmholtzNavier operator as integral over the unit sphere in Fourier space. Using anisotropic incompatible strain gradient elasticity, Po et al. (2018) have developed a nonsingular theory of threedimensional dislocation loops in anisotropic crystals. The theory is systematically developed as a generalization of the classical anisotropic elasticity theory of dislocation. The nonsingular version of all key equations of anisotropic dislocation theory have been derived as line integrals in terms of the threedimensional elastic Green tensor, including the Burgers displacement equation with isolated solid angle, the PeachKoehler stress equation, the MuraWillis equation for the elastic distortion, and the PeachKoehler force. The anisotropic nonsingular dislocation theory is shown to be in good agreement with molecular statics calculations without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. In particular, the virial stress of an edge dislocation in \(\alpha\)iron obtained from atomistic calculations is in perfect agreement with the nonsingular stress using anisotropic incompatible strain gradient elasticity. Furthermore, the theory of anisotropic incompatible strain gradient elasticity has been used by Seif et al. (2015) to model the atomistically enabled nonsingular anisotropic elastic representation of nearcore dislocation stress fields in \(\alpha\)iron. Using a magnetic bondorder potential to model atomic interactions in iron, molecular statics calculations have been performed, and an optimization procedure has been developed to extract the required length scale parameter. Results showed that the method can accurately replicate the magnitude and decay of the nearcore dislocation stresses even for atoms belonging to the dislocation core itself. Comparisons with the singular isotropic elasticity and anisotropic elasticity theories have shown that the nonsingular anisotropic gradient elasticity theory of dislocations leads to a substantially more accurate representation of the stresses of both screw and edge dislocations near the dislocation core, in some cases showing improvements in accuracy of up to an order of magnitude. Therefore, as shown by Po et al. (2018) and Seif et al. (2015) results for dislocations in anisotropic crystals obtained by using anisotropic incompatible strain gradient elasticity theory are in agreement with atomistic results. The main advantage of those dislocation keyformulas is the absence of any singularity and that they are valid even in the dislocation core region. Until now, for nonsingular fields of straight dislocations in anisotropic crystals the threedimensional dislocationkey equations and the threedimensional elastic Green tensor have been applied using the projection from 3D to 2D. However, the twodimensional elastic Green tensor and the analytical expressions of straight dislocations are still lacking in the literature of anisotropic strain gradient elasticity.
What about dislocations in Mindlin’s first strain gradient elasticity? For the incompatible version of Mindlin’s first strain gradient elasticity, the two Lamé constants and the five strain gradient parameters lead to four characteristic lengths due to the presence of the eigenstrain fields of dislocations. Using the incompatible version of Mindlin’s first strain gradient elasticity, nonsingular and smooth displacement fields of screw and edge dislocations have been given by Delfani and Tavakol (2019) and Delfani et al. (2020), respectively. All nonsingular dislocation fields including elastic strain, stress, and displacement fields of screw and edge dislocations have been computed by Lazar (2021) in the framework of incompatible first strain gradient elasticity of Mindlin type. The elastic fields of screw and edge dislocations have a similar form in simplified incompatible first strain gradient elasticity and in incompatible first strain gradient elasticity of Mindlin type (see, e.g., Lazar (2021, 2022)). Only the shape of the dislocation core of an edge dislocation with asymmetric form due to its inherent asymmetry can be modelled more realistic in incompatible first strain gradient elasticity of Mindlin type (Lazar 2021). Somehow, incompatible first strain gradient elasticity of Mindlin type is more sophisticated than simplified incompatible first strain gradient elasticity. For the isotropic case and the anisotropic case, the stress fields of straight dislocations and dislocation loops computed in the framework of simplified incompatible first strain gradient elasticity are in full agreement with the corresponding stress fields obtained in Eringen’s nonlocal elasticity of Helmholtz type (see, e.g., Eringen (2002); Lazar et al. (2005, 2020)). Thus, simplified incompatible first strain gradient elasticity is a very robust and powerful theory for an efficient modelling of dislocation fields without singularities at small scales. Moreover, the importance of simplified first strain gradient elasticity as nonsingular dislocation continuum theory in comparison with other existing nonsingular dislocation continuum theories has been given in Lazar (2017) and Po et al. (2014). Moreover, the use of nonlocality to describe the elastic fields within defect cores has received a renewed attention (e.g., Lazar and Agiasofitou (2011); Taupin et al. (2014, 2017); Lazar et al. (2020)). Zhang et al. (2016) considered the effects of corespreading dislocation in anisotropic bimaterials. Semicoherent heterophase interfaces with corespreading dislocation structures in magnetoelectroelastic multilayers under external surface loads were investigated in Vattré and Pan (2019).
The purpose of the present work is to derive the nonsingular dislocation keyformulas of straight dislocations in an anisotropic medium using nonsingular twodimensional Green functions of simplified anisotropic first strain gradient elasticity. In A nonsingular dislocation theory based on anisotropic incompatible strain gradient elasticity section, we review the framework of a nonsingular dislocation theory based on anisotropic incompatible strain gradient elasticity. All necessary Green functions, including the twodimensional Green tensor of the twofold anisotropic HelmholtzNavier and the twodimensional \(\varvec{F}\)tensor of generalized plane strain, are derived in Relevant Green functions in nonsingular dislocation theory section. In Dislocation keyequations section, the dislocation keyequations of straight dislocations are computed for generalized plane strain. In Straight dislocations in cubic materials section, the dislocation fields of straight dislocations in cubic materials are given using simplified anisotropic first strain gradient elasticity. The necessary material parameters are given for cubic materials such as aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) computed from a second nearestneighbour modified embeddedatommethod (2NN MEAM) interatomic potential in Material parameters for cubic crystals section. The characteristic length scale of simplified anisotropic first strain gradient elasticity is computed based on the material parameters computed from a 2NN MEAM interatomic potential. As an application, the nonsingular elastic fields of screw and edge dislocations of \(\frac{1}{2} \langle 111\rangle\) Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of \(\frac{1}{2} \langle 110\rangle\) Burgers vector in fcc copper and fcc aluminum are computed and presented in equalvalue contour plots in Elastic distortion and stress fields of screw and edge dislocations in cubic crystals section. Some technical remarks are given in the Appendix.
A nonsingular dislocation theory based on anisotropic incompatible strain gradient elasticity
Here, we consider the eigendistortion problem of dislocations in an anisotropic crystal. We consider an infinite elastic body in threedimensional space and use the property that the gradient of the displacement field \(\varvec{u}\) can be additively decomposed into an incompatible elastic distortion tensor \(\varvec{\beta }\) and an incompatible plastic distortion (eigendistortion) tensor \(\varvec{\beta }^{\text {P}}\):
The elastic strain tensor, \(\varvec{e}\), is the symmetric part of \(\varvec{\beta }\):
In dislocation theory, the dislocation density tensor, \(\varvec{\alpha }\), is defined in terms of the incompatible plastic distortion tensor (see, e.g., Kröner (1958); deWit (1973a); Lazar (2014))
and can be also expressed in terms of the incompatible elastic distortion tensor
where \(\epsilon _{jkl}\) indicates the LeviCivita tensor. Sometimes, the tensor \(\varvec{\alpha }\) is called the KrönerNye tensor. Moreover, the dislocation density tensor satisfies the Bianchi identity of dislocations
which means that dislocations cannot end inside the body.
Mindlin’s anisotropic first strain gradient elasticity
In Mindlin’s anisotropic first strain gradient elasticity theory (Mindlin 1964, 1968, 1972), the strain energy density for a homogeneous and centrosymmetric^{Footnote 1} material is given by (see also Lazar and Kirchner (2007); Lazar et al. (2022))
where \(\mathbb {C}_{ijkl}\) is the fourthrank constitutive tensor of elastic constants possessing the minor symmetries
and the major symmetry
while \(\mathbb {D}_{ijmkln}\) is the sixthrank constitutive tensor of the gradientelastic constants possesses the minor symmetries
and the major symmetry
For the general anisotropic case (triclinic), the constitutive tensor \(\mathbb {C}_{ijkl}\) has 21 independent elastic constants and the constitutive tensor \(\mathbb {D}_{ijmkln}\) has 171 independent gradientelastic constants (see, e.g., Auffray et al. (2013)). For cubic crystals of point group \(m\overline{3}m\), the constitutive tensor \(\mathbb {C}_{ijkl}\) has 3 independent elastic constants and the constitutive tensor \(\mathbb {D}_{ijmkln}\) has 11 independent gradientelastic constants (see, e.g., Mindlin (1968); Auffray et al. (2013); Lazar et al. (2022); Lazar and Agiasofitou (2023)).
Simplified anisotropic first strain gradient elasticity
In simplified anisotropic first strain gradient elasticity, it is assumed (see also Lazar and Kirchner (2007); Gitman et al. (2010); Lazar and Po (2015b); Po et al. (2018); Polizzotto (2018)) that the sixthrank constitutive tensor \(\mathbb {D}_{ijmkln}\) can be decomposed into the product of the fourthrank constitutive tensor \(\mathbb {C}_{ijkl}\) and a secondrank tensor \(\Lambda _{mn}\) of gradient length scale parameters with units of squared length, that is
Note that Eq. (11) represents the constitutive assumption of simplified anisotropic first strain gradient elasticity. As consequence of the major symmetry (10) and of positive definiteness of \(\mathcal {W}\), the tensor \(\Lambda _{mn}\) must be symmetric and positive definite. The classification of the gradient length scale tensor \(\Lambda _{mn}\) for triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, trigonal, cubic, and isotropic materials has been given in Lazar and Po (2015b); Lazar et al. (2020). For the general anisotropic case (triclinic), the gradient length scale tensor \(\Lambda _{mn}\) has 6 independent gradient length scale parameters. The decomposition (11) represents the separation of two anisotropies present in anisotropic strain gradient elasticity, namely the elastic bulk anisotropy (elastic moduli anisotropy) via \(\mathbb {C}_{ijkl}\) and the anisotropy of the gradient length scale parameters (internal length anisotropy or weak nonlocal anisotropy at small scales) via \(\Lambda _{mn}\). The latter, which is not present in classical anisotropic elasticity, reflects the discrete nature of matter and becomes relevant in the presence of defects at the Ångströmscale as dislocation core anisotropy. The decomposition (11) is not ad hoc because it considers that the gradients \(\partial _m\) and \(\partial _n\) in Eq. (6) give rise to length scale effects via \(\Lambda _{mn}\). In general, the symmetries of the tensors \(\mathbb {C}_{ijkl}\) and \(\Lambda _{mn}\) can be different due to the twofold anisotropy. Also note that the decomposition (11) in strain gradient elasticity corresponds to the twofold anisotropy present in Eringen’s nonlocal elasticity theory, namely the elastic moduli anisotropy of the bulk described by \(\mathbb {C}_{ijkl}\) and the anisotropy of the nonlocality at small scales described by a nonlocal kernel function \(\alpha\) (see Eringen (1978, 2002); Lazar and Agiasofitou (2011); Lazar et al. (2020)). Such a twofold anisotropy can be used to model the anisotropy of the dislocation core in an anisotropic crystal (as mentioned in the Introduction), namely the symmetry of the perfect crystal via \(\mathbb {C}_{ijkl}\) and the lower symmetry of the dislocation core of the imperfect crystal via \(\Lambda _{mn}\).
Using Eq. (11), the strain energy density (6) reduces to
The Cauchy stress tensor \(\varvec{\sigma }\) and the double stress tensor \(\varvec{\tau }\) are given by
An important property of simplified strain gradient elasticity theory is the remarkable fact that the double stress tensor (14) is nothing but the first gradient of the Cauchy stress tensor (13) multiplied by the length scale tensor \(\Lambda _{mn}\) (see also Lazar and Maugin (2005)). This is the result of the decomposition (11). Using the constitutive relations (13) and (14), the strain energy density (12) can be written in the “compact” form
in terms of the stress tensor \(\sigma _{ij}\) and the elastic strain tensor \(e_{ij}\) and their first gradient. The strain energy density (15) has a remarkable symmetry in the stress and elastic strain tensors.
The condition of the static equilibrium is given by the EulerLagrange equation and reads as
In terms of the Cauchy stress and double stress tensors, Eq. (16) reduces to
Using Eq. (14), Eq. (17) simplifies to
where
is a scalar anisotropic Helmholtz operator. Using Eqs. (1) and (13), Eq. (18) can be cast in the following twofold anisotropic inhomogeneous HelmholtzNavier equation for the displacement vector
with the anisotropic Navier operator
The “sourceterm” in Eq. (20) is given by the plastic distortion tensor \(\varvec{\beta }^{\text {P}}\). Equation (20) is an inhomogeneous partial differential equation of fourth order and can be written as system of two partial differential equations (Lazar 2014), namely an anisotropic inhomogeneous HelmholtzNavier equation for the displacement vector \(\varvec{u}\):
where the “sourceterm” in Eq. (22) is given by the classical plastic distortion tensor \(\varvec{\beta }^{\text {P},0}\) and an inhomogeneous Helmholtz equation for the plastic distortion tensor \(\varvec{\beta }^{\text {P}}\):
Moreover, the dislocation density tensor (3) also satisfies an inhomogeneous Helmholtz equation
where \(\varvec{\alpha }^0\) denotes the classical dislocation density tensor.
Relevant Green functions in nonsingular dislocation theory
In this section, all twodimensional Green functions necessary in nonsingular dislocation theory of straight dislocation are given.
Twodimensional Green tensor of the twofold anisotropic HelmholtzNavier operator
First, we derive the twodimensional Green tensor of the twofold anisotropic HelmholtzNavier Eq. (22) which is a partial differential equation of fourth order. The twodimensional Green tensor of the twofold anisotropic HelmholtzNavier operator \(L L_{ik}\) is defined by
where \(\varvec{x} \in \mathbb {R}^2\). In Eq. (25), \(\delta _{ij}\) is the Kronecker symbol and \(\delta (.)\) is the twodimensional Dirac deltafunction.
Since the HelmholtzNavier operator \(L L_{ik}\) is the product of the Helmholtz operator L and the Navier operator \(L_{ik}\), the corresponding Green tensor of the HelmholtzNavier Eq. (25) can be written as the convolution of the Green function \(G^L\) of the anisotropic Helmholtz equation and the “classical” Green tensor \(G^0_{ij}\) of the anisotropic Navier operator, that is
with \(G^L\) and \(G^0_{ij}\) satisfying, respectively:
where the anisotropic Helmholtz operator L and the anisotropic Navier operator \(L_{ik}\) are given by
Here \(*\) denotes the spatial convolution and \(\varvec{\Lambda }\) is a symmetric \(2\times 2\) matrix:
The corresponding inverse matrix \(\varvec{\Lambda }^{1}={\text {adj}\, \varvec{\Lambda }}/{\text {det}\, \varvec{\Lambda }}\) is given by
Equation (26) reveals that the Green function \(G^L\) plays the role of an anisotropic regularization function for the singular Green tensor of classical elasticity, \(G^0_{ij}\). The twodimensional anisotropic Green function \(G^L\) of Eq. (27) reads (see Lazar and Agiasofitou (2011); Lazar et al. (2020))
where \(K_0\) denotes the modified Bessel function of order 0. Notice that Eq. (33) possesses an independent anisotropy due to the tensor \(\Lambda _{mn}\) with 3 independent components \(\Lambda _{11}\), \(\Lambda _{12}\), \(\Lambda _{22}\) describing anisotropic length scale effects in the \(x_1x_2\) plane. In order that \(\Lambda _{mn}\) with \(m,n=1,2\) is positive definite, it is necessary and sufficient that the following inequalities are satisfied (see Lazar and Agiasofitou (2011); Lazar and Po (2015b); Lazar et al. (2020))
The twodimensional Green tensor of the Navier operator in classical anisotropic elasticity is given by Lazar (2021)
where \(\gamma\) is the Euler constant (\(\gamma \approx 0.57721\ldots\)).
Solution using the method of Fourier transform
The twodimensional Fourier transform of Eq. (25) gives for the Green tensor in Fourier space \(\hat{G}_{kj}(\varvec{k})\):
where
is the Navier operator in Fourier space. Now, if we define the twodimensional unit vector
then the solution of Eq. (36) in Fourier space is given by
In Eq. (39), we have introduced the function \(\lambda (\varvec{\kappa })\)
The twodimensional Green tensor in real space is obtained by the twodimensional inverse Fourier transform of Eq. (39)
In Eq. (41), \(\text {d}\hat{V}=k\, \text {d}k\, \text {d}\phi\) indicates the twodimensional volume element in Fourier space in polar coordinates, and \(\phi\) (\(0<\phi \le 2\pi\)) is an appropriate polar angle scanning a unit circle \(\varvec{\kappa }^2=1\). The twodimensional unit vector \(\varvec{\kappa }(\phi )\) varies with \(\phi\) and can be expressed as
where the unit vectors \(\hat{\varvec{e}}_1\) and \(\hat{\varvec{e}}_2\) represent a Cartesian basis in the two dimensional plane.^{Footnote 2}
Integration in k is performed using the relations
and
where \(G^{a, b}_{c, d}( )\) is the Meijer Gfunction (see Erdélyi et al. (1981); Gradshteyn and Ryzhik (2000)). Hence, Eq. (41) can be expressed as:
It can be seen that the Green tensor (45) of the HelmholtzNavier operator is a sum of the Green tensor (35) of the Navier operator and a gradient part given in terms of the Meijer Gfunction. The twodimensional Green tensor (45) is an integral over the unit circle in Fourier space, whereas the threedimensional Green tensor is an integral over the unit sphere in Fourier space (see also Lazar and Po (2015b)).
The asymptotics of the Meijer Gfunction with the above values is
where \(z=\frac{(\varvec{\kappa }\cdot \varvec{x})^2}{4 \lambda ^2(\varvec{\kappa })}\) and \(\psi ^{(0)}\) is the digamma function. The logarithmic singularity of the Green tensor (35) of the Navier operator is removed (regularized) in the Green tensor (45) of the HelmholtzNavier operator by the near field of the Meijer Gfunction (see Eq. (46)). The integrand (bracket) of Eq. (45) is plotted in Fig. 1. Therefore, the integrand of the Green tensor (45) of the HelmholtzNavier operator is finite and nonsingular, namely
The Meijer Gfunction in Eq. (45) can be expressed in terms of elementary functions, suitable for numerical manipulation and implementation, as
where the hyperbolic sine integral function is given by
and the hyperbolic cosine integral function is given by
Note that \(\text {Chi}(z)\) has a branch cut discontinuity in the complex z plane running from \(\infty\) to 0, whereas \(\text {Shi}(z)\) has no branch cut discontinuity.
Given \(G_{ij}\) as an integral over the unit circle in Fourier space, its gradient is obtained as:
The asymptotics of the Meijer Gfunction with the above values are
and
The “classical” 1/rsingularity is removed (regularized) in the integrand of Eq. (51) due to the near field of the Meijer Gfunction (see Eq. (53)). Therefore, the integrand of Eq. (51) is nonsingular, namely zero at the origin (see Fig. 2).
The Meijer Gfunction in Eq. (51) can be expressed in terms of elementary functions, suitable for numerical manipulation and implementation, as
Twodimensional \(\varvec{F}\)tensor in strain gradient anisotropic elasticity
The socalled \(\varvec{F}\)tensor has been introduced by Kirchner (1984) (see also Lazar and Kirchner (2013); Po et al. (2018); Lazar et al. (2020)). The twodimensional \(\varvec{F}\)tensor is defined by (see also Lazar (2021))
where the twodimensional Green function of the Laplace operator reads (see, e.g., Vladimirov (1971))
with
where \(r=\sqrt{x_1^2+x_2^2}\).
Using the twodimensional Fourier transform, Eq. (55) becomes
The twodimensional inverse Fourier transform gives the twodimensional \(\varvec{F}\)tensor
Therefore, the integrand of the \(\varvec{F}\)tensor (59) is finite and nonsingular, namely
From a numerical viewpoint, it is noteworthy that the Green tensor (45), its gradient (51), and the \(\varvec{F}\)tensor (59) are even functions of \(\varvec{k}\). Hence the integral over the unit circle appearing in their expressions can be expressed by twice the integral over a semicircle.
Twodimensional Green function of the anisotropic LaplaceHelmholtz equation
The Green function of the anisotropic LaplaceHelmholtz equation is defined by
and in Fourier space it becomes
Using the twodimensional inverse Fourier transform, the Green function of the anisotropic LaplaceHelmholtz equation is obtained for general anisotropy
and for the isotropic or cubic case with only one length scale parameter \(\ell\), it reduces to
Dislocation keyequations
In this section, we derive expressions for dislocation keyequations of straight dislocation from the general 3D dislocation keyequations.
General case
In the nonsingular theory of dislocations, which is based on simplified anisotropic strain gradient elasticity, the 3D dislocation keyequations read (Po et al. 2018; Lazar and Po 2018a)

anisotropic MuraWillislike equation for the nonsingular elastic distortion tensor
$$\begin{aligned} \beta _{ik}=\epsilon _{knr}\mathbb {C}_{jmln} \partial _m G_{ij}*\alpha ^0_{lr} \end{aligned}$$(65) 
anisotropic Burgerslike equation for the nonsingular displacement vector
$$\begin{aligned} u_i=\partial _k G^{\Delta L}*\beta ^{\text {P},0}_{ik}\epsilon _{knr}\mathbb {C}_{jklm}F_{mnij}*\alpha ^0_{lr} \end{aligned}$$(66) 
anisotropic Blin’slike formula for the elastic interaction energy
$$\begin{aligned} W_{(AB)}&=\int _{\mathbb {R}^3}\epsilon _{qns}\mathbb {C}_{psit}\epsilon _{tkr}\mathbb {C}_{jmlk}\left( F_{mnij}*\alpha ^{0(A)}_{lr}\right) \, \alpha ^{0(B)}_{pq}\, \text {d} V \end{aligned}$$(67) 
anisotropic PeachKoehlerlike stress equation for the Cauchy stress tensor
$$\begin{aligned} \sigma _{pq}= \mathbb {C}_{pqik}\epsilon _{knr}\mathbb {C}_{jmln} \partial _m G_{ij}*\alpha ^0_{lr} \end{aligned}$$(68) 
PeachKoehler force for a dislocation density in the stress field of another dislocation
$$\begin{aligned} \mathcal{F}^{\text {PK}}_k=\int _{\mathbb {R}^3} \epsilon _{kji}\sigma _{lj}\alpha ^0_{li}\, \text {d} V\, . \end{aligned}$$(69)
Moreover, the dislocation density tensor and the plastic dislocation tensor are given by
and
where it can be seen that the Green function \(G^L\) plays the role as a dislocation spreading function (Lazar 2014).
Generalized plane strain of straight dislocations
Now, we consider straight dislocations with line direction parallel to the \(x_3\)axis belonging to the framework of generalized plane strain which is 2D elasticity consisting of plane strain and antiplane strain. In general, the plane strain and antiplane strain fields do not decouple due to the anisotropy. Only for an orthotropic system, the plane strain and antiplane strain fields are separable. In generalized plane strain, all dislocation fields are independent of the variable \(x_3\), all derivatives with respect to the \(x_3\)axis vanish, \(\partial _3=0\) and \(\varvec{x}\in \mathbb {R}^2\). Therefore, all dislocation fields depend only on \(x_1\) and \(x_2\) and are twodimensional fields.
For generalized plane strain of dislocations, Eqs. (70) and (71) become
and
For generalized plane strain of dislocations, the 2D dislocationkey Eqs. (65)–(69) reduce to

anisotropic MuraWillislike equation for the nonsingular elastic distortion tensor
$$\begin{aligned} \beta _{ik}=\epsilon _{kn3}\mathbb {C}_{jmln} \partial _m G_{ij}*\alpha ^0_{l3}\, ,\qquad i,j,l=1,2,3\quad \qquad k,m,n=1,2 \end{aligned}$$(74) 
anisotropic Burgerslike equation for the nonsingular displacement vector
$$\begin{aligned} u_i=\partial _2 G^{\Delta L}*\beta ^{\text {P},0}_{i2}\epsilon _{kn3}\mathbb {C}_{jklm}F_{mnij}*\alpha ^0_{l3} \end{aligned}$$(75)with \(i,j,l=1,2,3\) and \(k,m,n=1,2\)

anisotropic Blin’slike formula for the elastic strain energy
$$\begin{aligned} W_{(AB)}&=\int _{\mathbb {R}^3}\epsilon _{3ns}\mathbb {C}_{psit}\epsilon _{tk3}\mathbb {C}_{jmlk}\left( F_{mnij}*\alpha ^{0(A)}_{l3}\right) \, \alpha ^{0(B)}_{p3} \,\text {d} V \end{aligned}$$(76)with \(i,j,l,p=1,2,3\) and \(k,m,n,s,t=1,2\)

anisotropic PeachKoehlerlike stress equation equation for the Cauchy stress tensor
$$\begin{aligned} \sigma _{pq}=\mathbb {C}_{pqik} \epsilon _{kn3} \mathbb {C}_{jmln} \partial _m G_{ij}*\alpha ^0_{l3} \end{aligned}$$(77) 
PeachKoehler force
$$\begin{aligned} \mathcal{F}^{\text {PK}}_k=\int _{\mathbb {R}^3} \epsilon _{kj3}\sigma _{lj}\alpha ^0_{l3}\, \text {d} V\, . \end{aligned}$$(78)
Dislocation keyequations of straight dislocations
Now, we consider straight dislocations with line direction parallel to the \(x_3\)axis and defect surface in the \(x_1 x_3\) half plane for negative \(x_1\) (\(x_2=0\), \(x_1<0\)). For a straight dislocation with Burgers vector \(b_i\) located at \((x_1,x_2)=(0,0)\), the classical dislocation density and the classical plastic distortion are given by (see also deWit (1973b); Mura (1987))
and
which possesses a discontinuity at \(x_2=0\) for \(x_1<0\). Here H(.) denotes the Heaviside step function.
If we substitute Eqs. (79) and (80) into Eqs. (72) and (73), respectively, we obtain for the dislocation density of a straight dislocation
and for the plastic distortion of a straight dislocation
where \(\varvec{x}^{\prime }=(X,x_2)\). In general, the dislocation density tensor defines the dislocation core region and determines the shape and size of the dislocation core (see also Hartley and Mishin (2005); Lazar (2013, 2017)). For that reason, one can call \(\alpha _{ij}\) the dislocation core tensor. The dislocation density (81) is only nonzero in the dislocation core. Equation (81) models the dislocation core in the \(x_1x_2\)plane with anisotropic shape (core anisotropy) depending on the 3 length scale parameters \(\Lambda _{11}\), \(\Lambda _{22}\) and \(\Lambda _{12}\). The dislocation density (81) gives with 3 length scale parameters \(\Lambda _{11}\), \(\Lambda _{22}\) and \(\Lambda _{12}\) a rotated elliptical dislocation core shape (see Fig. 3) and with 2 length scale parameters \(\Lambda _{11}\) and \(\Lambda _{22}\) an elliptical dislocation core shape (see Fig. 4). For \(\Lambda _{11}=\Lambda _{22}\) and \(\Lambda _{12}=0\), the dislocation core has a circular shape (see below).
Substituting Eq. (79) into Eq. (74), the nonsingular elastic distortion tensor of a straight dislocation in an infinite anisotropic medium reads as
and explicitly it becomes using Eq. (51)
Equation (84) represents the gradientextension of the BarnettSwanger (Barnett and Swanger 1971) formula for the elastic distortion of a straight dislocation in classical anisotropic elasticity. In Eq. (84), \(\hat{L}^{1}_{ij}(\varvec{\kappa })\) describes the bulk anisotropy and \(\lambda ^2(\varvec{\kappa })\) describes the core anisotropy.
Using Eq. (79), the Cauchy stress tensor of a straight dislocation (77) becomes
Using Eq. (51), it reads as
Using Eqs. (79) and (80), the displacement vector of a straight dislocation (75) reduces to
where the \(\varvec{F}\)tensor is given in Eq. (59). Eq. (87) is the gradientextension of the displacement field of a straight dislocation in classical anisotropic elasticity (see Lazar (2021)).
Using Eq. (79), the elastic interaction energy per unit length of two straight dislocations with Burgers vectors \(b^{(A)}_l\) and \(b^{(B)}_p\) reads
If we use Eq. (79), the PeachKoehler force per unit length of a straight dislocation with Burgers vector \(b_l\) in a stress field \(\sigma _{jl}\) reads as
Straight dislocations in cubic materials
Let us consider straight dislocations in cubic materials. For cubic symmetry, \(\Lambda _{11}=\Lambda _{22}=\ell ^2\) and \(\Lambda _{12}=0\), the dislocation density (81) and the plastic distortion (82) simplify to
The dislocation density (90) is plotted in Fig. 5 and gives the shape and size of the dislocation core of a straight dislocation in cubic crystals. Due to only one length scale parameter \(\ell\) the dislocation core possesses a circular shape. Such a shape of the dislocation core of straight dislocations in cubic crystals is in good agreement with experimental results (see, e.g., Kret et al. (2000); Hartley and Mishin (2005)). The plastic distortion (91) is nonsingular, smooth and finite as it can be seen in Fig. 6.
The elastic distortion tensor (84) and the Cauchy stress tensor (86) reduce to
and
respectively.
The displacement vector (87) becomes
Simplified anisotropic strain gradient elasticity provides a regularization of the classical singular dislocation fields for cubic crystals with one regularization parameter \(\ell\) in terms of Meijer Gfunctions and modified Bessel functions leading to a nonsingular nearfield in the dislocation core.
Material parameters for cubic crystals
In Mindlin’s first strain gradient elasticity theory, the elastic constants and the gradientelastic constants are characteristic material parameters which can be computed from interatomic potentials (see, e.g., Admal et al. (2017); Po et al. (2019)) or via ab initio DFT calculations (see, e.g., Shodja et al. (2018)). For some important cubic materials such as Al, Cu, Fe and W, the 3 elastic constants and 11 gradientelastic constants have been computed using a second nearestneighbour modified embeddedatommethod (2NN MEAM) interatomic potential (Admal et al. 2017; Po et al. 2019; Lazar et al. 2022).
We consider a cubic crystal with centrosymmetry. Let the Cartesian coordinate axes \(x_1\), \(x_2\) and \(x_3\) coincide with the cubic crystal directions [100], [010] and [001], respectively. For cubic crystals, the fourthrank constitutive tensor \(\mathbb {C}_{ijkl}\), which is the tensor of elastic constants, is given by (see, e.g., Dederichs and Leibfried (1969); Bacon et al. (1980); Lazar et al. (2022))
with
where \(\varvec{e}^{(1)}\), \(\varvec{e}^{(2)}\), \(\varvec{e}^{(3)}\) are the (orthogonal) unit vectors of the cubic system. Because the coordinate system coincide with the cubic system, it yields \(\delta _{ijkl}=1\) if \(i=j=k=l\) and \(\delta _{ijkl}=0\) otherwise (Dederichs and Leibfried 1969). In Eq. (95), \(C_{11}\), \(C_{12}\) and \(C_{44}\) are the 3 independent elastic constants of a cubic crystal.
The inverse elastic tensor \(\mathbb {C}^{1}_{ijkl}\), which is the tensor of elastic compliances \(\mathbb {S}_{ijkl}\equiv \mathbb {C}_{ijkl}^{1}\), reads as
with (see Hirth and Lothe (1982); Wooster (1978))
This tensor is defined by the property (Teodosiu 1982)
and therefore
For cubic crystals of point group \(m\overline{3}m\) (cubic hexoctahedral), the sixthrank constitutive tensor \(\mathbb {D}_{ijmkln}\) in Mindlin’s first strain gradient elasticity is given by (see, e.g., Lazar et al. (2022); Lazar and Agiasofitou (2023))
with
Here, \(a_1,\dots ,a_{11}\) are the 11 gradientelastic constants of a cubic crystal with centrosymmetry and \(\delta _{ijklmn}=1\) if \(i=j=k=l=m=n\) and \(\delta _{ijklmn}=0\) otherwise.
The tensor \(\Lambda _{mn}\) can also be estimated using the tensor \(\mathbb {S}_{ijkl}=\mathbb {C}^{1}_{ijkl}\). Multiplying both sides of Eq. (11) by \(\mathbb {S}_{ijkl}\), we obtain \(\mathbb {S}_{ijkl}\mathbb {D}_{ijmkln}=6\Lambda _{mn}\), and therefore, the tensor \(\Lambda _{mn}\) can be given in terms of the two constitutive tensor \(\mathbb {D}_{ijmkln}\) and \(\mathbb {S}_{ijkl}\) as (Po et al. 2018)
For cubic materials, the gradient length scale tensor \(\Lambda _{mn}\) reads (Po et al. 2018)
where \(\ell\) is the characteristic length scale of simplified anisotropic strain gradient elasticity for cubic materials. The gradient length scale of cubic materials can be computed directly from the fourthrank constitutive tensor \(\mathbb {C}_{ijkl}\) and the sixthrank constitutive tensor \(\mathbb {D}_{ijmkln}\) using Eqs. (105) and (106) leading to the formula:
If we substitute Eqs. (97) and (103) into Eq. (107), we obtain the following formula for the characteristic length in terms of the 3 elastic constants and the 11 gradientelastic constants
Equation (108) gives an atomistic determination of the characteristic length \(\ell\) from numeric values of the elastic and gradientelastic constants computed from interatomic potentials or via ab initio DFT. Using the elastic and gradientelastic constants of Al, Cu, Fe and W given in Table 1, the characteristic length \(\ell\) of simplified anisotropic strain gradient elasticity is computed using Eq. (108) and reported in Table 2.
Elastic distortion and stress fields of screw and edge dislocations in cubic crystals
In order to illustrate the applicability of simplified anisotropic first strain gradient elasticity, we compute the elastic distortion and stress fields of both screw and edge dislocations in cubic crystals. The elastic distortion is computed according to the formula (92). The elastic distortion is dimensionless. The stress field is computed according to the stress formula (93). Stresses are in units of eV/Å\({}^3\). We choose bcc iron (Fe) and fcc copper (Cu) because they are crystals with high anisotropy of the elastic constants (see Table 3) and bcc tungsten (W) and fcc aluminum (Al) because they are crystals which are nearly isotropic with respect to the elastic constants (see Table 3). Therefore, the change of the dislocation fields from isotropic \(\Longrightarrow\) anisotropic behaviour can be seen by comparing the dislocation fields for bcc: tungsten (W) \(\Longrightarrow\) iron (Fe) and for fcc: aluminum (Al) \(\Longrightarrow\) copper (Cu). We present the dislocation fields in contour plots in order to see the characteristic shape of the far and nearfields of straight dislocations and the influence of the anisotropy of cubic crystals.
BCC Fe
In bcc Fe, we consider dislocations with Burgers vector \(\varvec{b}=a/2\, \langle 111\rangle\). The Burgers vector reads \(b=\sqrt{3}/2\, a=2.482\, {\text{\AA }}\). The elastic constants and the corresponding length of bcc Fe have been taken from Tables 1 and 2. Using these material constants, we compute the elastic distortion and stress fields in the plane orthogonal to infinite straight edge and screw \(1/2[111](\overline{1}10)\) dislocations, which have line directions along the \([\overline{1}\overline{1}2]\) and [111] axes, respectively.
First, we give the plots of the nonsingular elastic distortion components of a screw dislocation in bcc Fe using simplified anisotropic first strain gradient elasticity in Fig. 7. In Fig. 7, it can be seen that a screw dislocation in bcc Fe has pronounced elastic distortion components \(\beta _{zx}\) and \(\beta _{zy}\) (only these components are nonzero in the isotropic case); the other four components \(\beta _{xx}\), \(\beta _{yy}\), \(\beta _{xy}\) and \(\beta _{yx}\) are weak.
The plots of the nonsingular stress components of a screw dislocation in bcc Fe using simplified anisotropic first strain gradient elasticity are given in Fig. 8. It can be seen in Fig. 8 that a screw dislocation in bcc Fe possesses pronounced stress components \(\sigma _{xz}\) and \(\sigma _{yz}\) (only these components are nonzero in the isotropic case); the others are weak. The anisotropic stress components \(\sigma _{xz}\) and \(\sigma _{yz}\) (see Fig. 8e, f) do not differ greatly in form from the corresponding isotropic fields. In Fig. 8a–d, it is interesting to observe that in contrast to the isotropic case there are four additional but weaker stress components \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\) and \(\sigma _{xy}\) (in comparison with the other two anisotropic components) in the anisotropic case reflecting the effects of anisotropy. In particular, in Fig. 8c, it can be seen that the stress component \(\sigma _{zz}\) of a [111] screw dislocation in bcc Fe possesses a threefold symmetry.
The plots of the six nonsingular elastic distortion components of an edge dislocation in bcc Fe using simplified anisotropic first strain gradient elasticity are given in Fig. 9.
Next, we give the plots of the nonsingular stress components of an edge dislocation in bcc Fe using simplified anisotropic first strain gradient elasticity in Fig. 10. It is seen from Fig. 10 that in the case of anisotropy the components of stress \(\sigma _{xz}\) and \(\sigma _{yz}\) of an edge dislocation are weak in comparison with the other four components \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\) and \(\sigma _{xy}\), but they are not zero as in the isotropic case. The anisotropic stress components \(\sigma _{xx}\), \(\sigma _{yy}\) and \(\sigma _{xy}\) (see Fig. 10a, b and d) do not differ greatly in form from the corresponding isotropic fields which is not the case for the stress component \(\sigma _{zz}\) (see Fig. 10c). Due to the anisotropy, we have the appearance of two additional but weaker stress components (in comparison with the other anisotropic components), namely the \(\sigma _{xz}\) (Fig. 10e) and \(\sigma _{yz}\) (Fig. 10f).
As it can be seen in Figs. 7, 8, 9, and 10, the elastic distortion and stress fields of both screw and edge dislocations are nonsingular. The symmetry of the (nonsingular) elastic strain fields (symmetric part of the elastic distortion, see Eq. (2)) of a screw dislocation (Fig. 7) and an edge dislocation (Fig. 9) using simplified anisotropic first strain gradient elasticity are in agreement with the symmetry of the (singular) elastic strain fields of a screw and an edge dislocation given in the framework of classical anisotropic elasticity by Yoo and Loh (1971). Moreover, the symmetry of the (nonsingular) stress fields of a screw dislocation (Fig. 8) and an edge dislocation (Fig. 10) using simplified anisotropic first strain gradient elasticity agree with the symmetry of the (singular) stress fields of a screw and an edge dislocation given in the framework of classical anisotropic elasticity by Baštecká (1965) and Yoo and Loh (1971)(see also Steeds (1973)) and with the symmetry of the (nonsingular) stress fields of a screw and an edge dislocation given in the framework of nonlocal anisotropic elasticity by Lazar et al. (2020).
BCC W
In bcc W, we consider dislocations with Burgers vector \(\varvec{b}=a/2\, \langle 111\rangle\). The Burgers vector reads \(b=\sqrt{3}/2\, a=2.741\, {\text{\AA }}\). The elastic constants and the corresponding length of bcc W have been taken from Tables 1 and 2. Using these material constants, we compute the elastic distortion and stress fields in the plane orthogonal to infinite straight edge and screw \(1/2[111](\overline{1}10)\) dislocations, which have line directions along the \([\overline{1}\overline{1}2]\) and [111] axes, respectively.
The plots of the nonsingular elastic distortion components of a screw dislocation in bcc W using simplified anisotropic first strain gradient elasticity are given in Fig. 11. It can be seen that only the components \(\beta _{zx}\) and \(\beta _{zy}\) give a nonzero contribution since W is isotropic with respect to the elastic constants (see Table 3). The plots of the nonsingular stress components of a screw dislocation in bcc W using simplified anisotropic first strain gradient elasticity are given in Fig. 12. It can be seen that only the components \(\sigma _{xz}\) and \(\sigma _{yz}\) give a nonzero contribution since W is isotropic with respect to the elastic constants (see Table 3).
The plots of the four nonsingular elastic distortion components of an edge dislocation in bcc W using simplified anisotropic first strain gradient elasticity are given in Fig. 13. It can be seen from Fig. 13 that only the four components \(\beta _{xx}\), \(\beta _{yy}\), \(\beta _{xy}\) and \(\beta _{yx}\) give a nonzero contribution because W is isotropic. The plots of the nonsingular stress components of an edge dislocation in bcc W using simplified anisotropic first strain gradient elasticity are given in Fig. 14. It can be seen from Fig. 14 that only the four components \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\) and \(\sigma _{xy}\) give a nonzero contribution since W is isotropic. The shape of the plots of the stresses of an edge dislocation in W given in Fig. 14 agree with the plots given in Yoo and Loh (1970) (see also Steeds (1973)).
If we compare the dislocation fields of screw and edge dislocations in bcc iron in BCC Fe section and in bcc tungsten in BCC W section we observe some characteristic differences. On the one hand, bcc iron is strongly anisotropic and gives 6 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors for both screw and edge dislocations. On the other hand, bcc tungsten is isotropic and gives 4 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors of an edge dislocation and 2 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors of a screw dislocation. The nearfields of the elastic strain and the Cauchy stress fields are nonsingular due to the regularization and show a characteristic shape, namely they are zero at the dislocation line.
FCC Cu
In fcc Cu, we consider dislocations with Burgers vector \(\varvec{b}=a/2\, \langle 110\rangle\). The Burgers vector reads \(b=a/\sqrt{2} = 2.556\, {\text{\AA }}\). The elastic constants and the corresponding length of fcc Cu have been taken from Tables 1 and 2. Using these material constants, we compute the elastic distortion and stress fields in the plane orthogonal to infinite straight edge and screw \(1/2[1\overline{1}0](111)\) dislocations, which have line directions along the \([\overline{1}\overline{1}2]\) and \([1\overline{1}0]\) axes, respectively.
For a screw dislocation in fcc Cu, the plots of the two nonsingular elastic distortion components are given in Fig. 15. The plots of the corresponding two nonsingular stress components of a screw dislocation in fcc Cu using simplified anisotropic first strain gradient elasticity are given in Fig. 16. The shape and symmetry of the (nonsingular) stress fields of a screw dislocation (Fig. 16) using simplified anisotropic first strain gradient elasticity agree with the symmetry of the (singular) stress fields of a screw dislocation given in the framework of classical anisotropic elasticity by Yoo and Loh (1970) (see also Steeds (1973)).
Using simplified anisotropic first strain gradient elasticity, the plots of the six nonsingular elastic distortion components of an edge dislocation in fcc Cu are given in Fig. 17. The plots of the six nonsingular stress components of an edge dislocation in fcc Cu using simplified anisotropic first strain gradient elasticity are given in Fig. 18. It is seen from Fig. 18 that in the case of anisotropy the components of stress \(\sigma _{xz}\) and \(\sigma _{yz}\) of an edge dislocation are weak in comparison with the other four components \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\) and \(\sigma _{xy}\), but they are not zero as in the isotropic case (see also Steeds (1973)). In particular, the component \(\sigma _{zz}\) shows a characteristic shape due to the elastic anisotropy.
As it can be seen in Figs. 15, 16, 17, and 18, the elastic distortion and stress fields of both screw and edge dislocations are nonsingular.
FCC Al
In fcc Al, we consider dislocations with Burgers vector \(\varvec{b}=a/2\, \langle 110\rangle\). The Burgers vector reads \(b=a/\sqrt{2}= 2.863\, {\text{\AA }}\). The elastic constants and the corresponding length of fcc Cu have been taken from Tables 1 and 2. Using these material constants, we compute the elastic distortion and stress fields in the plane orthogonal to infinite straight edge and screw \(1/2[1\overline{1}0](111)\) dislocations, which have line directions along the \([\overline{1}\overline{1}2]\) and \([1\overline{1}0]\) axes, respectively.
The plots of the nonsingular elastic distortion components of a screw dislocation in fcc Al using simplified anisotropic first strain gradient elasticity are given in Fig. 19. It can be seen that only the components \(\beta _{zx}\) and \(\beta _{zy}\) give a nonzero contribution since Al is nearly isotropic with respect to the elastic constants (see Table 3). The plots of the nonsingular stress components of a screw dislocation in fcc Al using simplified anisotropic first strain gradient elasticity are given in Fig. 20. It can be seen that only the components \(\sigma _{xz}\) and \(\sigma _{yz}\) give a nonzero contribution since Al is nearly isotropic with respect to the elastic constants (see Table 3). The shape of the plots of the stresses of a screw dislocation in Al given in Fig. 20 agree with the plots given in Yoo and Loh (1970) (see also Steeds (1973)).
The plots of the four nonsingular elastic distortion components of an edge dislocation in fcc Al using simplified anisotropic first strain gradient elasticity are given in Fig. 21. It can be seen from Fig. 21 that only the four components \(\beta _{xx}\), \(\beta _{yy}\), \(\beta _{xy}\) and \(\beta _{yx}\) give a nonzero contribution because Al is nearly isotropic. The plots of the nonsingular stress components of an edge dislocation in fcc Al using simplified anisotropic first strain gradient elasticity are given in Fig. 22. It can be seen from Fig. 22 that only the four components \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\) and \(\sigma _{xy}\) give a nonzero contribution since Al is nearly isotropic.
If we compare the dislocation fields of screw and edge dislocations in fcc copper in FCC Cu section and in fcc aluminum in FCC Al section we observe some characteristic differences. On the one hand, fcc copper is strongly anisotropic and gives 6 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors for an edge dislocation and 2 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors for a screw dislocation with a characteristic “rotated and deformed” shape. On the other hand, fcc aluminum is nearly isotropic and gives 4 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors of an edge dislocation and 2 nonzero dislocation fields for the elastic strain and the Cauchy stress tensors of a screw dislocation. The nearfields of the elastic strain and the Cauchy stress fields are nonsingular due to the regularization and show a characteristic shape, namely they are zero at the dislocation line.
Conclusions
In this work, we have presented a nonsingular dislocation theory of straight dislocations in anisotropic materials using simplified anisotropic first strain gradient elasticity. This theory is a simplification of Mindlin’s anisotropic first strain gradient elasticity theory based on the key intuition that it is possible to approximate the anisotropy of the constitutive tensor of rank six, \(\mathbb {D}_{ijmkln}\), as given in Eq. (11) because the classical anisotropy of the constitutive tensor of rank four, \(\mathbb {C}_{ijkl}\), is dominant even within the defects core region. In other words, the theory approximates the gradient anisotropy and retains the full classical anisotropy. We showed in previous work that this is an excellent approximation, in good agreement with atomistic calculations without any fitting constant (see, e.g., Po et al. (2018)). In the framework of simplified anisotropic first strain gradient elasticity, all necessary Green tensor functions, being nonsingular, are derived. Interesting to note that the twodimensional Green tensor of the twofold anisotropic HelmholtzNavier operator is given as sum of a classical part and a part given in terms of a Meijer Gfunction. For generalized plane strain of dislocations, the twodimensional dislocation keyequations (anisotropic MuraWillislike equation for the nonsingular elastic distortion tensor, anisotropic Burgerslike equation for the nonsingular displacement vector, anisotropic Blin’slike formula for the elastic strain energy, anisotropic PeachKoehlerlike stress equation, PeachKoehler force) have been derived in terms of twodimensional Green tensor of the twofold anisotropic HelmholtzNavier operator. Furthermore, the twodimensional dislocation keyequations are specified to straight dislocations in anisotropic media and, in particular, in cubic crystals. All relevant material parameters are computed for bcc and fcc cubic crystals such as iron (Fe), tungsten (W), copper (Cu) and aluminum (Al) from a second nearestneighbour modified embeddedatommethod (2NN MEAM) interatomic potential. As representative application, the elastic distortion and stress fields of screw and edge dislocations of \(\frac{1}{2} \langle 111\rangle\) Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of \(\frac{1}{2} \langle 110\rangle\) Burgers vector in fcc copper and fcc aluminum have been computed and presented in contour plots, showing that the obtained dislocation fields are nonsingular.
Availability of data and materials
No datasets were generated or analysed during the current study.
Change history
21 April 2024
In the original publication the vertical line symbol in the equations were incorrect. The article has been updated to rectify the errors.
Notes
Due to the centrosymmetry, there is no coupling between \(e_{ij}\) and \(\partial _m e_{kl}\).
In the numerical evaluation of integrals over the unit circle it is convenient to consider a local reference system such that \(\hat{\varvec{e}}_1=\varvec{x}/\Vert \varvec{x}\Vert\).
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Acknowledgements
Markus Lazar gratefully acknowledges the grant from the Deutsche Forschungsgemeinschaft (Grant number LA1974/42).
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Open Access funding enabled and organized by Projekt DEAL. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project LA1974/42.
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M.L. wrote the main manuscript. G.P. prepared the figures 520. All authors reviewed the manuscript.
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Appendix A: Simplified anisotropic strain gradient elasticity for cubic crystals
Appendix A: Simplified anisotropic strain gradient elasticity for cubic crystals
For cubic crystals, the simplified anisotropic strain gradient elasticity can be obtained from Mindlin’s anisotropic strain gradient elasticity as special case if we assume the following values for the gradientelastic constants
Using the relations (109), the constitutive tensor of rank six, Eq. (103), reduces to
and the double stress tensor (14) becomes
It is worth noting that the double stress tensor (111) is nothing but the gradient of the Cauchy stress tensor (13) with one length scale parameter \(\ell\). The double stress tensor (111) is much simpler than the expression of the double stress tensor for cubic crystals in Mindlin’s anisotropic strain gradient elasticity theory (see Lazar et al. (2022)). In this way, simplified anisotropic strain gradient elasticity has the meaning of an effective and robuste generalized continuum theory with approximate symmetry for nonsingular dislocations.
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Lazar, M., Po, G. Nonsingular straight dislocations in anisotropic crystals. J Mater. Sci: Mater. Theory 8, 5 (2024). https://doi.org/10.1186/s41313024000577
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DOI: https://doi.org/10.1186/s41313024000577