Fractional Nonlocal Elasticity and Solutions for Straight Screw and Edge Dislocations

Nonlocal elasticity assumes integral stress constitutive equation, takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. In this paper, we propose the nonlocal elasticity theory with the kernel (the weight function in the integral stress constitutive equation) which is the Green function of the Cauchy problem for the fractional diffusion equation and is expressed in terms of the Mainardi function and Wright function being the generalizations of the exponential function. The solutions for the straight screw and edge dislocations in an infinite solid are obtained in the framework of the proposed theory.


INTRODUCTION
At the present time, it is generally recognized that process of deformation occurs at various structural levels with their own scale lengths [1][2][3][4][5]. Therefore, it is necessary to use specific physical concepts and mathematical tools at every structural level, while events progressing at different scale levels are interdependent.
Real crystals contain a large amount of defects. The problem of modeling stress and strain fields produced by imperfections in a crystal has attracted much attention of researchers. Imperfections in an elastic solid, as a rule, are modeled by certain distribution of concentrated forces or by certain distribution of the tensor of plastic strain incompatibility, see, e.g., [6][7][8][9][10][11]. A limitation of classical elastic solutions is that the stress fields have nonphysical singularities and the elastic energy diverges if one does not cut the defect core. The classical solutions cannot describe the situation in the strongly distorted region in the vicinity of the defect. Moreover, the classical elasticity is not valid in the immediate vicinity of the imperfection and fails to explain the phenomena at the atomic scale. This situation has given rise to the regular attempts to improve elastic models of crystal defects.
Recently, great advances have been made in crystal defect research by application of nonlocal elasticity. Several versions of nonlocal continuum mechanics based on various suggestions have been proposed (see [2,3,[12][13][14][15][16][17][18][19][20], among others). Starting from interrelated equations describing elasticity and diffusion, Podstrigach [12] excluded the chemical potential from the constitutive equation for the stress tensor and obtained the stress-strain relation containing spatial and time derivatives. In the infinite medium this relation can be integrated using the Fourier and Laplace integral transforms, and the final result has the nonlocal integral form with exponential kernel. Kröner [13], Krumhansl [14] and Kunin [15,19,20] started from a discrete lattice and interpolated functions of discrete argument by special continuous functions. In this case, the stress at a reference point in the body depends not only on the strain at this point but also on the strains at all other points of the body. Thus, the relation between the stress tensor and the strain tensor has the nonlocal integral form. The interested reader is also referred to the recent book [21] and references therein.
Nonlocal elasticity takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. The nonlocal theory of elasticity makes it possible to apply a unified approach to the study of phenomena at different structural levels. In contrast to many generalized models that improve the description of only one specific type of defects, in the context of nonlocal theory of elasticity one can consider various types of defects from a single point of view. A number of problems solved in the framework of nonlocal theory indicate the power of the theory. More justified results can be obtained, as the nonlocal elasticity is effective in removing nonphysical singularities occurring at dislocations, disclinations, points of applications of singular forces, cracks, etc. It should be emphasized that the theory of nonlocal elasticity indicates its power in the study of point defects [22,23], straight edge [24] and screw [25][26][27] dislocations, straight wedge and twist disclinations [28], circular prismatic and glide dislocation loops [29], circular twist, rotation and wedge disclination loops [30], line cracks [3,31,32], and concentrated forces [33][34][35][36].
The fractional calculus (the theory of integrals and derivatives of noninteger order) has numerous applications in mechanics, physics, geophysics, rheology, engineering, chemistry, geology, biology, bioengineering, finance, and medicine, see [37][38][39][40][41][42][43][44][45][46] among many others. Very interesting line of investigation in the field of nonlocal theories consists in using fractional calculus. Di Paola and Zingales [47] formulated the governing equations in terms of the Marchaud fractional derivative of the displacement field and considered an elastic bar with long-range interaction. In the case of one spatial dimension, Challamel et al. [48] postulated the fractional differential relation between the uniaxial stress and uniaxial strain with symmetric Caputo fractional derivative and studied harmonic wave propagation in a bar. Carpinteri et al. [49] used the fractional Riesz integral in the constitutive equation relating uniaxial strain to the uniaxial stress in a bar. Tarasov [50][51][52][53] considered long-range interactions of lattice particles, introduced the lattice fractional integro-differential operators and transformed a lattice field of some function into the corresponding continuum field function. The generalization of the Frenkel-Kontorova model of dislocation based on fractional-order differences was obtained in [54]. Tarasov and Aifantis [55] studied fractional gradient elasticity based on fractional Laplacian in the Riesz form. The results of Podstrigach [12] were generalized in [56] to space-time nonlocal dependence between stresses and strains based on the fractional calculus approach. Instead of the exponential kernel in the nonlocal constitutive equation for stresses, the kernel being the Wright function was obtained. Sumelka [57,58] introduced fractional deformation gradients and fractional strains and used this approach in nonlocal elasticity [59].
Eringen [2,3,17,[24][25][26] proposed several forms of nonlocal modulus, in particular the nonlocal kernel being the Green function of the Cauchy problem for the diffusion equation. In [60,61] the exponential Helmholtz nonlocal kernel was used to solve nonlocal elasticity problems for nanobeams. Lazar et al. [62] employed Eringen's model of nonlocal elasticity with bi-Helmholtz type kernels which are Green functions of partial differential equations of fourth order in the case of one, two and three spatial dimensions. They also obtained the solutions for straight dislocations. In the present paper, we propose the nonlocal elasticity theory with the kernels being the Green functions of the Cauchy problem for the fractional diffusion equation with the Caputo and Riesz derivatives and obtain the solution for the straight screw and edge dislocations in an infinite solid in the framework of the considered theory.

FRACTIONAL NONLOCAL ELASTICITY
For the static case with vanishing body forces, the basic equations for a linear isotropic nonlocal elastic solid are [3,17,26]: where x and x are the reference and running points, t and  are the nonlocal and classical stress tensors, u is the displacement vector, e the linear strain tensor,  and  are Lamé constants,  and  denote the gradient operators with respect to x and x, respectively, I stands for the unit tensor. The volume integral in Eq. (2) is over the region occupied by the solid. The nonlocal kernel (|x -x|, ) depends on the distance |x -x| between the reference and running points and includes the parameter  connected with a characteristic length ratio a/l, where a is an internal characteristic length and l is an external characteristic length [2,3]. The nonlocal kernel (|x -x|, ) is a delta sequence and in the classical elasticity limit   0 becomes the Dirac delta function.
In this case, the nonlocal stress tensor is a solution of the corresponding Cauchy problem [3,26] ( , ) In the present paper, we consider the nonlocal kernel (|x -x|, ) as the Green function of the Cauchy problem for the fractional diffusion operator Here is the gamma function. The Laplace transform rule for the Caputo derivative (12) has the following form: where the asterisk denotes the transform, s is the Laplace transform variable.
The Riesz derivative (-) /2 f (x) being the fractional generalization of the classical Laplace operator, obtained for  = 2, can be defined by its Fourier trans- x is a vector of space variables,  is a vector of transform variables.

One-Dimensional Case
Equations (10) and (11) are rewritten as where the tilde denotes the Fourier transform,  is the transform variable.
Inverting the integral transforms, we get the nonlocal kernel Here E  (z) is the Mittag-Leffler function in one parameter  (see Appendix A). When  = 2, taking into account the relation [45] 2 where M(; x) is the Mainardi function (see Appendix A), we obtain the nonlocal kernel in the form Due to Eq. (A10), for  = 1 which coincides with the corresponding equations of [3].
Another particular case of (22) is obtained for   0. Due to Eq. (A9), we get The exponential Helmholtz kernel was considered in [60,61].

Two-Dimensional Axisymmetric Case
In the polar coordinates in the axisymmetric case The two-fold Fourier transform in the case of axial symmetry is reduced to the Hankel transform of the zeroth order with respect to the radial coordinate r [45,64]. Hence where the hat denotes the Hankel transform, and For  = 1 and  = 2, taking into account integral (B1) from Appendix B, we get When   0 and  = 2, evaluating the corresponding integral (see Eq. (B2)), we arrive at Both kernels (31) and (32) were considered by Eringen [3,26].

Three-Dimensional Central Symmetric Case
In the spherical coordinates in the central symmetric case After employing the Laplace transform and threefold Fourier transform (see Appendix C), we obtain the expression for the nonlocality kernel proposed by Eringen [3,26].

SCREW DISLOCATION IN THE FRAMEWORK OF FRACTIONAL NONLOCAL ELASTICITY
In what follows, we restrict ourselves to the case  = 2. The classical elasticity stress tensor for the straight screw dislocation has one nonzero component [6,7] , , 2 where b is the Burgers vector,  is the Lamé constant.
In the framework of the proposed fractional nonlocal elasticity, taking into account the expressions for components of the Laplacian of the stress tensor (see Appendix D) we obtain that the component t z of the nonlocal stress tensor satisfies the equation under initial condition 0: .
The Laplace transform with respect to the parameter  and the Hankel transform of the first order with respect to the radial coordinate r give After inverting integral transform we get Consider several particular cases of the solution (43). For  = 1, taking into account Eqs. (A2) and (B3), we arrive at the solution which coincides with that obtained by Eringen [25,26] for the exponential nonlocal modulus. When   0, taking into consideration Eqs. (A3) and (B5), we have The solution (45) was also considered by Eringen [3,26] for the nonlocal modulus being the modified Bessel function. Figure 1 shows the dependence of nondimensional stress component on nondimensional spatial coordinate  = r/ /2 . We also present the classical local solution 1/.

EDGE DISLOCATION IN THE FRAMEWORK OF FRACTIONAL NONLOCAL ELASTICITY
The stress tensor associated with the straight edge dislocation in an infinite solid in polar coordinates has the nonzero components [6,7] sin , sin , cos , where b is the Burgers vector,  is the Poisson ratio.
From the Cauchy problem (15), (16) with  = 2, taking into consideration the components of the Laplacian of the stress tensor in polar coordinates (see Appendix D), we obtain the following system of equations for determining T rr , T  and T r : The system of coupled equations (48) [3,24].
For   0, using integrals (B5) and (B6), we get Figure 2 shows the dependence of nondimensional stress component rr t =  /2 t rr /B on nondimensional distance  = r/ /2 for  = /2 and different values of the order  of the derivative. We also present the classical local solution 1/.

CONCLUDING REMARKS
In the present paper, we have considered the generalization of Eringen's theory of nonlocal elasticity in which the nonlocal modulus in integral constitutive equation for the stress tensor is the Green function of the Cauchy problem for fractional diffusion equation with the Caputo derivative of the order 0 <   1 with respect to the nonlocality parameter  and the fractional generalization of the Laplace operator with respect to the spatial coordinates of order 1    2. Eringen proposed several nonlocal moduli which in the framework of fractional nonlocal elasticity are obtained as particular cases of the Wright function and the Mainardi function. The stress fields caused by the straight screw and edge dislocations in an infinite solid are investigated. The studied solution for the screw dislocation in the case 0 <   1 interpolates the appropriate results of Eringen obtained for different nonlocal kernels associated with the values  = 0 and  = 1. The solution (56) for the edge dislocation corresponds to the solution (45) for the screw dislocation. The components of the stress tensor reach a maximum at some distance from the dislocation line. The point of the maximum and the maximum value of stress depends on the order of fractional derivative ; the values of maximum are consistent with estimations of Eringen. In the general case, the proposed theory can be used for better matching the theory of elasticity and the atomic lattice theory. Another particular case of the Mittag-Leffler function is The Mittag-Leffler function E  (z) appears in the inverse Laplace transform The Wright function in two parameters  and  is defined as [37,63,65]