Fractional Gradient Elasticity from Spatial Dispersion Law

Abstract

Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

1. Introduction

The theory of derivatives and integrals of noninteger orders [1–3] allows us to investigate the behavior of materials and media that are characterized by nonlocality of power-law type. Fractional calculus has a wide application in mechanics and physics (e.g., see [4–14]). Nonlocal elasticity theories in continuum mechanics can be treated with two different approaches [15]: the gradient elasticity theory (weak nonlocality) and the integral nonlocal theory (strong nonlocality). The fractional calculus allows us to formulate a fractional generalization of nonlocal elasticity models in two forms: the fractional gradient elasticity models (weak power-law nonlocality) and the fractional integral nonlocal models (strong power-law nonlocality). The idea to include some fractional integral term in the equations of the elasticity has been proposed by Lazopoulos in [16]. Fractional models of integral nonlocal elasticity are considered in different papers; see, for example, [16–22]. The microscopic models of fractional integral elasticity are also described. For this reason, the fractional integral elasticity models are not discussed here.

This paper focuses on the fractional generalization of gradient elasticity which describes a weak nonlocality of power type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. Complex lattice dynamics has been the subject of continuing interest in the theory of elasticity. As it was shown in [23, 24] (see also [25–27]), the equations with fractional derivatives can be directly connected to lattice models with long-range interactions. In this paper, we consider models of lattices with spatial dispersion and its continuous limits. We define a map of lattice models into continuum models. A connection between the dynamics of lattice system of particles with long-range interactions and the fractional continuum equations is proved by using the transform operation [23, 24]. We make the transformation to the continuous limit and derive the fractional equation, which describes the dynamics of the nonlocal elastic materials. We show how the continuous limit for the lattice with fractional weak spatial dispersion gives the corresponding continuum equation of the fractional gradient elasticity. The continuum equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

2. Lattice Equations

The lattice is characterized by space periodicity. In an unbounded lattice, we can define three noncoplanar vectors 𝐚₁, 𝐚₁, 𝐚₁, such that displacement of the lattice by the length of any of these vectors brings it back to itself. The vectors 𝐚_𝑖, 𝑖=1,2,3, are the shortest vectors by which a lattice can be displaced and be brought back into itself. As a result, all spatial lattice points can be defined by the vector 𝐧=(𝑛₁,𝑛₂,𝑛₃), where 𝑛_𝑖 are integer. If we choose the coordinate origin at one of the sites, then the position vector of an arbitrary lattice site with 𝐧=(𝑛₁,𝑛₂,𝑛₃) is written: 𝐫(𝐧)=³_𝑖=1𝑛_𝑖𝐚_𝑖. (1) In a lattice, the sites are numbered in the same way as the particles, so that the vector 𝐧 is at the same time “number vector” of a corresponding particle.

We assume that the equilibrium positions of particles coincide with the lattice sites 𝐫(𝐧). A lattice site coordinate 𝐫(𝐧) differs from the coordinate of the corresponding particle, when particles are displaced relative to their equilibrium positions. To define the coordinates of a particle, it is necessary to indicate its displacement with respect to its equilibrium positions. We denote the displacement of a particle with vector 𝐧 from its equilibrium position by the vector field 𝐮(𝐧,𝑡).

The equation of motion of lattice particle is 𝑀𝜕²𝑢^𝑘(𝐧,𝑡)𝜕𝑡²=−_𝐦𝐾_𝑘𝑙(𝐧,𝐦)𝑢^𝑙(𝐦,𝑡)+𝐹_𝑘(𝐧,𝑡), (2) where 𝑀 is the mass of particle and 𝐹_𝑘(𝐧,𝑡) are components of the external on-site force. The italics 𝑘,𝑙 are the coordinate indices. We assume the summation over doubly repeated coordinate indices from 1 to 3. The coefficients 𝐾_𝑘𝑙(𝐧,𝐦) describe the interparticle interaction in the lattice. For simplicity, we assume that all particles have the same mass 𝑀.

It is easy see one important property of the coefficients 𝐾_𝑘𝑙(𝐧,𝐦). Assume the lattice to be displaced as a whole: 𝑢^𝑘(𝐧,𝑡)=𝑢^𝑘=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. Then the internal lattice state cannot be changed in case of absence of external forces. As a result, (2) gives _𝐦𝐾_𝑘𝑙(𝐧,𝐦)=_𝐦𝐾_𝑘𝑙(𝐦,𝐧)=0. (3) These conditions should be satisfied for any particle in the lattice, that is, for any vector 𝐧. Equation (3) follows from the conservation of total momentum in the lattice.

For an unbounded homogeneous lattice, due to its homogeneity the matrix 𝐾_𝑘𝑙(𝐧,𝐦) has the form 𝐾_𝑘𝑙(𝐧,𝐦)=𝐾_𝑘𝑙(𝐧−𝐦), (4) where elements of 𝐾_𝑘𝑙(𝐧−𝐦) of (2) are satisfied by condition _𝐦𝐾_𝑘𝑙(𝐧−𝐦)=_𝐧𝐾_𝑘𝑙(𝐧−𝐦)=0. (5) In a simple lattice, each particle is an inversion center, and we have 𝐾_𝑘𝑙(𝐧−𝐦)=𝐾_𝑘𝑙(𝐦−𝐧). (6) Using condition (5), we can represent (2) in the form 𝑀𝜕²𝑢^𝑘(𝐧,𝑡)𝜕𝑡²=−_𝐦𝐾_𝑘𝑙(𝐧,𝐦)𝑢^𝑙(𝐧,𝑡)−𝑢^𝑙(𝐦,𝑡)+𝐹_𝑘(𝐧,𝑡). (7) This equation of motion has the invariance with respect to its displacement of lattice as a whole in case of absence of external forces even if condition (5) is not satisfied. It should be noted that the noninvariant terms lead to the divergences in the continuous limit [13].

Equation of motion (7) is equation for three-dimensional displacement vectors. In this paper, we will use the simplest model to describe the lattice, where all particles are displaced in one direction; we assume that the displacement of particle from its equilibrium position is determined by a scalar rather than a vector. This model allows us to describe the main properties of the lattice using simple equations.

The equations of motion for one-dimensional lattice system of interacting particles have the form 𝑀𝜕²𝑢_𝑛(𝑡)𝜕𝑡²=𝑔^+∞_{𝑚=−∞𝑚≠𝑛}𝐾_𝛼(𝑛,𝑚)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)+𝐹(𝑛), (8) where we use the summation condition over repeated indexes. Here 𝑢_𝑛(𝑡)=𝑢(𝑛,𝑡) are displacements from the equilibrium, 𝑔 is the coupling constant for interparticle interactions in the lattice, and the term  𝐹(𝑛) characterizes an interaction of the particles with the external on-site force.

3. Transform Operations for Lattice Equations

Let us define the operation that transforms the lattice equations for 𝑢_𝑛(𝑡) into the continuum equation for a scalar field 𝑢(𝑥,𝑡). In order to obtain continuum equation from the lattice equations, we assume that 𝑢_𝑛(𝑡) are Fourier coefficients of some function 𝑢(𝑘,𝑡). We define the field 𝑢(𝑘,𝑡) on [−𝑘₀/2,𝑘₀/2] by the equation 𝑢(𝑘,𝑡)=^+∞_𝑛=−∞𝑢_𝑛(𝑡)𝑒^{−𝑖𝑘𝑥𝑛}=ℱ_Δ𝑢_𝑛(𝑡,𝑢_𝑛)(𝑡)=1𝑘₀^+𝑘0/2_−𝑘0/2𝑑𝑘𝑢(𝑘,𝑡)𝑒^{𝑖𝑘𝑥𝑛}=ℱ_Δ⁻¹{𝑢(𝑘,𝑡)}, (9) where 𝑥_𝑛=𝑛𝑑 and 𝑑=2𝜋/𝑘₀ is the interparticle distance. For simplicity, we assume that all particles have the same interparticle distance 𝑑. Equations (9) can be used to obtain the Fourier transform in the limit 𝑑→0 (𝑘₀→∞). Then change the sum to an integral, and (9) become 𝑢(𝑘,𝑡)=^+∞_−∞𝑑𝑥𝑒^{−𝑖𝑘𝑥}𝑢(){𝑢(,𝑢()=1𝑥,𝑡=ℱ𝑥,𝑡)}𝑥,𝑡2𝜋^+∞_−∞𝑑𝑘𝑒^𝑖𝑘𝑥𝑢(𝑘,𝑡)=ℱ⁻¹{𝑢(𝑘,𝑡)}. (10) We replace the discrete function 𝑢_𝑛(𝑡)=2𝜋𝑘₀𝑢𝑥_𝑛,𝑡 (11) by continuous field 𝑢(𝑥,𝑡) considering 𝑥_𝑛=𝑛𝑑=2𝜋𝑛/𝑘₀→𝑥. We assume that 𝑢(𝑘,𝑡)=ℒ𝑢(𝑘,𝑡), where ℒ denotes the passage to the limit 𝑑→0 (𝑘₀→∞). Here 𝑢(𝑘,𝑡) is a Fourier transform of the field 𝑢(𝑥,𝑡) and 𝑢(𝑘,𝑡) is a Fourier series transform of 𝑢_𝑛(𝑡), where we use 𝑢_𝑛(𝑡)=(2𝜋/𝑘₀)𝑢(𝑛𝑑,𝑡). The function 𝑢(𝑘,𝑡) can be derived from 𝑢(𝑘,𝑡) in the limit 𝑑→0.

As a result, we define the map from a lattice model into a continuum model by the transform operation 𝑇, which is the combination [23, 24] 𝑇=ℱ⁻¹ℒℱ_Δ of the following operations:(1)the Fourier series transform: ℱ_Δ∶𝑢_𝑛(𝑡)⟶ℱ_Δ𝑢_𝑛(𝑡)=𝑢(𝑘,𝑡), (12) (2)the passage to the limit 𝑑→0: ℒ∶𝑢(𝑘,𝑡)⟶ℒ{𝑢(𝑘,𝑡)}=𝑢(𝑘,𝑡), (13) (3)the inverse Fourier transform: ℱ⁻¹∶𝑢(𝑘,𝑡)⟶ℱ⁻¹{𝑢(𝑘,𝑡)}=𝑢(𝑥,𝑡). (14)

The similar transformations can be performed for differential equations to map the lattice equation into an equation for the elastic continuum. Therefore, the operation 𝑇=ℱ⁻¹ℒℱ_Δ allows us to realize transformation of lattice models of interacting particles into continuum models [23, 24].

Let us consider the Fourier series transform of the interaction term.

Proposition 1. Let 𝐾_𝛼(𝑛,𝑚) be such that conditions 𝐾_𝛼(𝑛,𝑚)=𝐾_𝛼(𝑛−𝑚)=𝐾_𝛼(𝑚−𝑛_∞𝑛=1),𝐾_𝛼(𝑛)²<∞ (15) hold. Then the Fourier series transform ℱ_Δ maps the term ^+∞_{𝑚=−∞𝑚≠𝑛}𝐾_𝛼(𝑛,𝑚)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡), (16) where 𝑢_𝑛=𝑢_𝑛(𝑡) is a position of the 𝑛th particle, into the term ℱ_Δ^+∞_{𝑚=−∞𝑚≠𝑛}𝐾_𝛼(𝑛,𝑚)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)=𝐾_𝛼(0)−𝐾_𝛼(𝑘𝑑)𝑢(𝑘,𝑡), (17) where 𝐾_𝛼(𝑘𝑑)=ℱ_Δ𝐾_𝛼(𝑛),𝑢(𝑘,𝑡)=ℱ_Δ𝑢_𝑛(𝑡). (18)

Proof. To derive the Fourier series transform of the interaction term (16), we multiply (16) by exp(−𝑖𝑘𝑛𝑑) and sum over 𝑛 from −∞ to +∞. Then ^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛−𝑢_𝑚=^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛−^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑚. (19) Using conditions (15), we introduce the notations 𝐾_𝛼(𝑘𝑑)=^+∞_{𝑛=−∞𝑛≠0}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛),𝑢(𝑘,𝑡)=^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝑢_𝑛(𝑡). (20)(21) Using 𝐾_𝛼(−𝑛)=𝐾_𝛼(𝑛), the function (20) can be represented by 𝐾_𝛼(𝑘𝑑)=^+∞_𝑛=1𝐾_𝛼(𝑛)𝑒^{−𝑖𝑘𝑛𝑑}+𝑒^{𝑖𝑘𝑛𝑑}=2^+∞_𝑛=1𝐾_𝛼(𝑛)cos(𝑘𝑑). (22) From (22), we can see that 𝐾_𝛼(𝑘𝑑) is a periodic function 𝐾_𝛼(𝑘𝑑+2𝜋𝑚)=𝐾_𝛼(𝑘𝑑), (23) where 𝑚 is an integer. Using (21) and (20), the first term on the right-hand side of (19) gives ^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛=^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝑢_𝑛^+∞_{𝑚′𝑚=−∞′≠0}𝐾_𝛼𝑚^′=𝑢(𝑘,𝑡)𝐾_𝛼(0). (24) Here we use (15) and 𝐾_𝛼(𝑚^′+𝑛−𝑛)=𝐾_𝛼(𝑚^′) and 𝐾_𝛼(0)=^+∞_{𝑛=−∞𝑛≠0}𝐾_𝛼(𝑛)=2^∞_𝑛=1𝐾_𝛼(𝑛). (25) Using 𝐾_𝛼(𝑚,𝑛^′+𝑚)=𝐾_𝛼(𝑛^′), the second term on the right-hand side of (19) has the form ^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑚=^+∞_𝑚=−∞𝑢_𝑚^+∞_{𝑛=−∞𝑛≠𝑚}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼()=𝑛−𝑚_{+∞𝑚=−∞}𝑢_𝑚𝑒^{−𝑖𝑘𝑚𝑑+∞}_{𝑛′𝑛=−∞′≠𝑛}𝑒^{−𝑖𝑘𝑛′𝑑}𝐾_𝛼𝑛^′=𝑢(𝑘,𝑡)𝐾_𝛼(𝑘𝑑). (26) Equations (24) and (26) give the expression 𝐾_𝛼(0)−𝐾_𝛼(𝑘𝑑)𝑢(𝑘,𝑡), (27) where 𝐾_𝛼(𝑘𝑑) is defined by (20).

Let us give the statement that describes the Fourier transform of the lattice equations.

Proposition 2. The Fourier series transform ℱ_Δ maps the lattice equations of motion 𝑀𝜕²𝑢_𝑛(𝑡)𝜕𝑡²=𝑔^+∞_{𝑚=−∞𝑚≠𝑛}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)+𝐹(𝑛), (28) where 𝐾_𝛼(𝑛−𝑚) satisfies conditions (15), into the continuum equation 𝑀𝜕²𝑢(𝑘,𝑡)𝜕𝑡²=𝑔𝐾_𝛼(0)−𝐾_𝛼(𝑘𝑑)𝑢(𝑘,𝑡)+ℱ_Δ{𝐹(𝑛)}, (29) where 𝑢(𝑘,𝑡)=ℱ_Δ{𝑢_𝑛(𝑡)}, 𝐾_𝛼(𝑘𝑑)=ℱ_Δ{𝐾_𝛼(𝑛)}, and ℱ_Δ is an operator notation for the Fourier series transform.

Proof. To derive the equation for the field 𝑢(𝑘,𝑡), we multiply (28) by exp(−𝑖𝑘𝑛𝑑) and sum over 𝑛 from −∞ to +∞. Then ^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝜕²𝜕𝑡²𝑢_𝑛(𝑡)=𝑔^+∞_𝑛=−∞^+∞_{𝑚=−∞𝑚≠𝑛}𝑒^{−𝑖𝑘𝑛𝑑}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛−𝑢_𝑚+^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝐹(𝑛). (30) Using (21), the left-hand side of (30) has the form ^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝜕²𝑢_𝑛(𝑡)𝜕𝑡²=𝜕²𝜕𝑡^2+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝑢_𝑛(𝑡)=𝜕²𝑢(𝑘,𝑡)𝜕𝑡². (31) The second term of the right-hand side of (30) is ^+∞_𝑛=−∞𝑒^{−𝑖𝑘𝑛𝑑}𝐹(𝑛)=ℱ_Δ{𝐹(𝑛)}. (32) The Fourier series transform ℱ_Δ maps the interaction term (16) into expression (17). As a result, we obtain (30) in the form (29), where ℱ_Δ{𝐹(𝑛)} is an operator notation for the Fourier series transform of 𝐹(𝑛).

4. Fractional Weak Spatial Dispersion

4.1. Weak Spatial Dispersion

Spatial dispersion is the dependence of 𝐾_𝛼(|𝐤|) on the wave vector 𝐤 that leads to nonlocal properties of the continuum. The spatial dispersion gives nonlocal connection between the stress tensor 𝜎_𝑘𝑙 and the strain tensor 𝜀_𝑘𝑙. The tensor 𝜎_𝑘𝑙 at any point 𝐫 of the continuum is not uniquely defined by the values of 𝜀_𝑘𝑙 at this point. It also depends on the values of 𝜀_𝑘𝑙 at neighboring points 𝐫^′, located near the point 𝐫.

A nonlocal constitutive relation between the stress 𝜎_𝑘𝑙 and the strain 𝜀_𝑘𝑙 can be understood on the basis of analysis of a lattice model. The particles of the lattice oscillate about their equilibrium positions and interact with each other. The equations of oscillations of the lattice particles with the local (nearest-neighbor) interaction give the partial differential equation of integer orders in the continuum limit [23, 24]. Note that the lattice with nonlocal (long-range) interactions in the continuous limit can give fractional partial differential equations for nonlocal continuum [23, 24].

Qualitatively describing the process we can say that the fields of the elastic wave move particles from their equilibrium positions at a given point 𝐫, which causes an additional shift of the particles in neighboring and more distant points 𝐫^′ in some neighborhood. Therefore, the properties of the medium, and hence the stress tensor field 𝜎_𝑘𝑙, depend on the values of strain tensor field 𝜀_𝑘𝑙 not only in a selected point but also in its neighborhood.

The size of the area in which the kernel 𝐾_𝛼(|𝐤|) is significant is determined by the characteristic length of interaction 𝑅₀. The size 𝑅₀ of the area is usually of the order of the lattice constant. Wavelength 𝜆 of elastic wave is several orders larger than the size of this region, so the values of the field of elasticity wave do not change for a region of size 𝑅₀. In other words, the wavelength 𝜆 usually holds 𝑘𝑅₀∼𝑅₀/𝜆≪1. In such lattice, the spatial dispersion is weak. To describe the lattice dynamics, it is enough to know the dependence of the function 𝐾_𝛼(|𝐤|) only for small values 𝑘=|𝐤| and we can replace this function by the Taylors polynomial series. For an isotropic linear medium, we use 𝐾_𝛼(𝑘)=𝐾_𝛼(0)+𝑎₁𝑘+𝑎₂𝑘²+⋯. (33) Here we neglect a frequency dispersion, and so 𝐾_𝛼(0), 𝑎₁, 𝑎₂ do not depend on the frequency 𝜔.

4.2. Fractional Taylor Series Approach

The weak spatial dispersion in the media with power-law type of nonlocality cannot be described by the usual Taylor approximation. The fractional Taylor series is very useful for approximating noninteger power-law functions [28]. For example, the usual Taylor series for the nonlinear power-law function 𝐾_𝛼(𝑘)=𝑎₀+𝑎_𝛼𝑘^𝛼 (34) has infinitely many terms for noninteger 𝛼.

If we use the fractional Taylor's formula (see Appendix A) we get a finite number of terms. For example, Taylor's series in the Odibat-Shawagfeh form that contains the Caputo fractional derivative ^𝐶_𝑎𝐷^𝛼_𝑘 has two terms for (34). Using ^𝐶₀𝐷^𝛼_𝑘𝑘^𝛽=Γ𝛽+1Γ𝑘𝛽−𝛼+1^𝛽−𝛼,𝑘>0,𝛼>0,𝛽>0 (35) for the case 𝛽=𝛼, we get ^𝐶₀𝐷^𝛼_𝑘𝑘^𝛼=Γ(𝛼+1),^𝐶₀𝐷^𝛼_𝑘^𝑛𝑘^𝛼=0. (36) As a result, we have ^𝐶₀𝐷^𝛼_𝑘𝐾_𝛼(0)=Γ(𝛼+1_𝐶0),𝐷^𝛼_𝑘^𝑛𝐾_𝛼(0)=0,(𝑛≥2) (37) and the fractional Taylor’s series approximation of function (34) is exact.

4.3. Weak Spatial Dispersion of Power-Law Types

We consider properties of the lattice with weak spatial dispersion that is described by the function 𝐾_𝛼(|𝐤|) of a noninteger power-law type. In the continuous limit this model gives a model of continuum with power-law nonlocality.

The Fourier series transform ℱ_Δ of the interaction term (16) is defined by (17), where 𝐾_𝛼(|𝐤|)=^+∞_{𝑛=−∞𝑛≠0}𝑒^{−𝑖𝑘𝑛}𝐾_𝛼(𝑛)=2^∞_𝑛=1𝐾_𝛼(𝑛)cos(𝑛|𝐤|), (38) and 𝑢(𝑘,𝑡)=ℱ_Δ{𝑢_𝑛(𝑡)}. If the function 𝐾_𝛼(|𝐤|) is given, then 𝐾_𝛼(𝑛) can be defined by 𝐾_𝛼(𝑛)=1𝜋^𝜋₀𝐾_𝛼(|𝐤|)cos(𝑛|𝐤|)𝑑|𝐤|. (39)

The weak spatial dispersion will be called 𝛼₁-type, if the function (38) satisfies condition lim_|𝐤|→0𝐾_𝛼(|𝐤|)−𝐾_𝛼(0)|𝐤|^𝛼1=𝑎_𝛼1, (40) where 𝛼₁>0 and 0<|𝑎_𝛼1|<∞. The weak spatial dispersion (and the interparticle interaction in the lattice) will be called 𝛼=(𝛼₁,𝛼₂)-type, if the function 𝐾_𝛼(|𝐤|) satisfies conditions (40) and lim_|𝐤|→0𝐾_𝛼(|𝐤|)−𝐾_𝛼(0)−𝑎_𝛼1|𝐤|^𝛼1|𝐤|^𝛼2=𝑎_𝛼2, (41) where 𝛼₂>𝛼₁>0 and 0<|𝑎_𝛼2|<∞.

Similarly we define the weak spatial dispersion and the interaction in the lattice of the 𝛼=(𝛼₁,…,𝛼_𝑁)-type. For the weak spatial dispersion of the 𝛼=(𝛼₁,…,𝛼_𝑁)-type, the function 𝐾_𝛼(|𝐤|) can be represented in the form 𝐾_𝛼(|𝐤|)=𝐾_𝛼(0)+^𝑁_𝑗=1𝑎_𝛼𝑗|𝐤|^𝛼𝑗+𝑅_𝛼^(𝑁)(|𝐤|), (42) where 0<𝛼₁<𝛼₂<⋯<𝛼_𝑁 and lim_|𝐤|→0𝑅_𝛼^(𝑁)(|𝐤|)|𝐤|^𝛼𝑁=0. (43) As a result, we can use the following approximation for weak spatial dispersion: 𝐾_𝛼(|𝐤|)≈𝐾_𝛼(0)+^𝑁_𝑗=1𝑎_𝛼𝑗|𝐤|^𝛼𝑗. (44)

If 𝛼_𝑗=𝑗 for all 𝑗∈ℕ, we can use the usual Taylor’s formula. In this case, we have the usual case of the weak spatial dispersion. In general, we should use a fractional generalization of Taylor's series (see Appendix A). If the orders of the fractional Taylor series approximation will be correlated with the type of weak spatial dispersion, then the fractional Taylor series approximation of 𝐾_𝛼(|𝐤|) will be exact. In the general case 0<𝛼_𝑗+1−𝛼_𝑗<1, we can use the fractional Taylor’s formula in the Dzherbashyan-Nersesian form (see Appendix A). For the special cases 𝛼_𝑗=𝑗𝛼₁, where 𝛼₁<1 and/or 𝛼_𝑗=𝛼+𝑗, we could use the other kind of the fractional Taylor’s formulas.

5. Fractional Gradient Elasticity Equation for Continuum

In the continuous limit the equation for lattice with the interaction of the 𝛼-type gives the equation for continuum of the fractional gradient model.

Proposition 3. In the continuous limit the lattice equation of motion 𝑀𝜕²𝑢_𝑛(𝑡)𝜕𝑡²&=𝑔^+∞_{𝑚=−∞𝑚≠𝑛}𝐾_𝛼(𝑛−𝑚)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)+𝐹(𝑛) (45) with the weak spatial dispersion of the 𝛼-type gives the fractional continuum equation of the form 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²=−^𝑁_𝑗=1𝐺_𝛼𝑗(−Δ)^𝛼𝑗/2𝑢()+1𝑥,𝑡𝜌𝑓(𝑥), (46) where (−Δ)^𝛼𝑗/2 is the fractional Laplacian of order 𝛼_𝑗 in Riesz’s form (see Appendix B), the variables 𝑥 and 𝑑=𝑑 are dimensionless, 𝑓(𝑥)=𝐹(𝑥)/(𝐴𝑑), 𝜌=𝑀/(𝐴𝑑), and 𝐺_𝛼𝑗=𝑔𝑎_𝛼𝑗𝑑^𝛼𝑗𝑀,𝑗=1,…,𝑁 (47) are finite parameters.

Proof. The Fourier series transform ℱ_Δ of (45) gives (29). After division by the cross-sectional area of the medium 𝐴 and the interparticle distance 𝑑, the limit 𝑑→0 for (29) gives 𝜕²𝜕𝑡²𝑢(𝑘,𝑡)=^𝑁_𝑗=1𝑔𝑑^𝛼𝑗𝑀𝒦_𝛼𝑗,Δ(𝑘)𝑢()+1𝑘,𝑡𝜌ℱ_Δ𝑓(𝑛), (48) where 𝜌=𝑀/(𝐴𝑑) is the mass density, 𝑑 is the interparticle distance, 𝑓(𝑛)=𝐹(𝑛)/(𝐴𝑑), and 𝒦_𝛼𝑗,Δ(𝑘)=−𝑎_𝛼𝑗|𝑘|^𝛼𝑗−𝑅_𝛼^(𝑁)(𝑘𝑑)𝑑^−𝛼𝑗. (49) Here we use (44), and 𝐺_𝛼𝑗 (𝑗=1,…,𝑁) are finite parameters that are defined by (47). Note that 𝑅_𝛼^(𝑁) satisfies the condition lim_𝑑→0𝑅_𝛼^(𝑁)(𝑘𝑑)𝑑^𝛼𝑁=0. (50) The expression for 𝒯_𝛼𝑗,Δ(𝑘) can be considered as a Fourier transform of the interaction term (see Proposition 1). Note that 𝑔𝑎_𝛼𝑗→∞ for the limit 𝑑→0, if 𝐺_𝛼𝑗 are finite parameters.
In the limit 𝑑→0, (48) gives 𝜕²𝑢(𝑘,𝑡)𝜕𝑡²=^𝑁_𝑗=1𝐺_𝛼𝑗𝒯_𝛼𝑗(𝑘)𝑢()+1𝑘,𝑡𝜌ℱ𝑓(𝑥), (51) where 𝒦_𝛼𝑗(𝑘)=ℒ𝒦_𝛼𝑗,Δ(𝑘)=−𝑎_𝛼𝑗|𝑘|^𝛼𝑗,𝑢(𝑘,𝑡)=ℒ𝑢(𝑘,𝑡). (52) The inverse Fourier transform of (51) has the form 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²=^𝑁_𝑗=1𝐺_𝛼𝑗𝒯_𝛼𝑗(𝑥)𝑢()+1𝑥,𝑡𝜌𝑓(𝑥), (53) where 𝒯_𝛼𝑗(𝑥)=ℱ⁻¹𝒦_𝛼𝑗(𝑘)=−𝑎_𝛼𝑗(−Δ)^𝛼𝑗/2. (54) Here, we use the connection between the Riesz fractional Laplacian (−Δ)^𝛼/2 and its Fourier transform (see Appendix B and [1–3]): ℱ(−Δ)^𝛼/2𝑢(𝐫)(𝐤)=|𝐤|^𝛼𝑢(𝐤) (55) in the form |𝑘|^𝛼𝑗⟷(−Δ)^𝛼𝑗/2. (56) Substitution of (54) into (53) gives the continuum equation (46).

Equations (46) and (47) give the close relation between the discrete microstructure of lattice with weak spatial dispersion of power-law type and the fractional gradient models of weak nonlocal continuum.

Let us consider the special case 𝛼_𝑗=𝑗 for integer 𝑗∈ℕ. If the function 𝐾_𝛼(𝑘) has the form 𝐾_𝛼(𝑘)≈𝐾_𝛼(0)+𝑎₂𝑘², (57) then we get the well-known equation 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²=𝐺₂()+1Δ𝑢𝑥,𝑡𝜌𝑓(𝑥). (58) Here 𝐺₂=𝑔𝑎₂𝑑²=𝐸𝑀𝐴𝜌, (59) where 𝐸=𝐾𝑑/𝐴 is Young’s modulus, 𝐾=𝑔𝑎₂ is the spring stiffness, and 𝜌=𝑀/(𝐴𝑑) is the mass density.

If we can use the spatial dispersion law in the form 𝐾_𝛼(𝑘)≈𝐾_𝛼(0)+𝑎₂𝑘²+𝑎₄𝑘⁴, (60) then we have the equation of the gradient elasticity as 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²=𝐺₂Δ𝑢(𝑥,𝑡)−𝐺₄Δ²𝑢()+1𝑥,𝑡𝜌𝑓(𝑥), (61) where 𝛼_𝑗=𝑗, 𝐺₄=𝑔𝑎₄𝑑⁴=𝑎𝑀𝐴⁴𝐸𝑑²𝑎₂𝜌,𝑎_𝑗=𝜕^𝑗𝐾_𝛼(𝑘)𝜕𝑘^𝑗_𝑘=0. (62) The scale parameter 𝑙² of the gradient elasticity is connected with the coupling constants of the lattice by the equation 𝑙²=𝑎₄𝑑²𝑎₂. (63) The second-gradient term is preceded by the sign that is defined by sgn(𝑎₄/𝑎₂).

Similarly, we can consider more general model of lattice with fractional weak spatial dispersion of 𝛼=(𝛼₁,…,𝛼_𝑁)-type 𝐾_𝛼(𝐤)=𝐾_𝛼(0)+^𝑁_𝑗=1𝑎_𝛼𝑗|𝐤|^𝛼𝑗. (64) Then the continuum equation for fractional gradient model has the form 𝜕²𝑢(𝐫,𝑡)𝜕𝑡²=−^𝑁_𝑗=1𝑐_𝑗(−Δ)^𝛼𝑗/2𝑢()+1𝐫,𝑡𝜌𝑓(𝐫), (65) where we use new notation for the constants, 𝑐_𝑗=𝐺_𝛼𝑗. Note that 𝐫 and 𝑟=|𝐫| are dimensionless.

6. Solution of Fractional Gradient Elasticity Equation

6.1. Plane Wave Solution

Let us consider the plane waves 𝑢(𝐫,𝑡)=𝑒^{−𝑖𝜔𝑡}𝑢(𝐫). Then (65) gives ^𝑁_𝑗=1𝑐_𝑗(−Δ)^𝛼𝑗/2𝑢(𝐫)−𝜔²𝑢(𝐫)=1𝜌𝑓(𝐫). (66)

We apply the Fourier method to solve fractional equation (66), which is based on the relation ℱ(−Δ)^𝛼/2𝑢(𝐫)(𝐤)=|𝐤|^𝛼𝑢(𝐤). (67) Applying the Fourier transform ℱ to both sides of (66) and using (67), we have (𝐤)=1ℱ𝑢)(𝜌^𝑁_𝑗=1𝑐_𝑗|𝐤|^𝛼𝑗−𝜔²⁻¹ℱ𝑓(𝐤). (68)

The fractional analog of the Green function (see Section 5.5.1. in [3]) is given by 𝐺^𝑛_𝛼(𝐫)=ℱ^−1𝑁_𝑗=1𝑐_𝑗|𝐤|^𝛼𝑗−𝜔²⁻¹(𝐫)=_ℝ𝑛^𝑁_𝑗=1𝑐_𝑗|𝐤|^𝛼𝑗−𝜔²⁻¹𝑒^{+𝑖(𝐤,𝐫)}𝑑^𝑛𝐤, (69) where 𝛼=(𝛼₁,…,𝛼_𝑚).

The following relation _ℝ𝑛𝑒^{𝑖(𝐤,𝐫)}𝑓(|𝐤|)𝑑^𝑛𝐤=(2𝜋)^𝑛/2|𝐫|^{(𝑛−2)/2∞0}𝑓(𝜆)𝜆^𝑛/2𝐽_𝑛/2−1(𝜆|𝐫|)𝑑𝜆 (70) holds (see Lemma 25.1 of [1, 2]) for any suitable function 𝑓 such that the integral in the right-hand side of (70) is convergent. Here 𝐽_𝜈 is the Bessel function of the first kind. As a result, the Fourier transform of a radial function is also a radial function.

Using relation (70), the Green function (69) can be represented (see Theorem 5.22 in [3]) in the form of the integral with respect to one parameter: 𝐺^𝑛_𝛼(𝐫)=|𝐫|^(2−𝑛)/2(2𝜋)^𝑛/2×^∞₀^𝑁_𝑗=1𝑐_𝑗𝜆^𝛼𝑗−𝜔²⁻¹𝜆^𝑛/2𝐽_(𝑛−2)/2(𝜆|𝐫|)𝑑𝜆, (71) where 𝑛=1,2,3 and 𝛼=(𝛼₁,…,𝛼_𝑚) and 𝐽_(𝑛−2)/2 is the Bessel function of the first kind.

For the 3-dimensional case, we use 𝐽_1/2(𝑧)=2(𝑧).𝜋𝑧sin (72) Then we have 𝐺³_𝛼(𝐫)=12𝜋²|𝐫|^∞₀^𝑁_𝑗=1𝑐_𝑗𝜆^𝛼𝑗−𝜔²⁻¹𝜆sin(𝜆|𝐫|)𝑑𝜆. (73) For the 1-dimensional case, we use 𝐽_−1/2(𝑧)=2(𝑧).𝜋𝑧cos (74) Then we have (see Theorem 5.24 in [3] pages 345-346) the function 𝐺¹_𝛼(𝐫)=1𝜋^∞₀^𝑚_𝑠=1𝑎_𝛼𝑗𝜆^𝛼𝑗−𝜔²⁻¹cos(𝜆|𝐫|)𝑑𝜆. (75)

If 𝛼_𝑁>1 and 𝑐_𝑁≠0, then (66) (see, e.g., Section 5.5.1. pages 341–344 in [3]) has a particular solution 𝑢(|𝐫|). Such particular solution is represented in the form of the convolution of the functions 𝐺^𝑛_𝛼(|𝐫|) and 𝑓(|𝐫|) as follows: 𝑢(𝐫)=1𝜌_ℝ^𝑛𝐺^𝑛_𝛼𝐫−𝐫^′𝑓𝐫^′𝑑^𝑛𝐫^′, (76) where the Green function 𝐺^𝑛_𝛼(𝐫) is given by (71).

In 3-dimensional case, the function 𝑓(|𝐫|) does not depend on the angles. Therefore, we can use the spherical coordinates and then reduce the integration 𝑑³𝐫^′ in (76) to 𝑑𝑟=𝑑|𝐫| by integrating with respect to the angles 𝑢(𝑟)=4𝜋𝜌_ℝ𝐺³_𝛼𝐫−𝐫^′𝑓𝑟^′𝑟^′2𝑑𝑟^′, (77) where 𝑟=|𝐫| and 𝑟^′=|𝐫^′|.

6.2. Static Solution

Let us consider the statics (𝜕𝑢(𝐫,𝑡)/𝜕𝑡=0, i.e., 𝑢(𝐫,𝑡)=𝑢(𝐫)) in the suggested fractional gradient elasticity model. We can consider the fractional partial differential equation (66) with 𝜔²=0 and 𝑐₁≠0, when 𝑁≥1, and also the case where 𝛼₁<3, 𝛼_𝑁>1, 𝑁≥1, 𝑐₁≠0, 𝑐_𝑁≠0, 𝛼_𝑁>⋯>𝛼₁>0, which is given by ^𝑁_𝑗=1𝑐_𝑗(−Δ)^𝛼𝑗/2𝑢(𝐫)=1𝜌𝑓(𝐫). (78) Equation (78) has the following particular solution (see Theorem 5.23 in [3]) that is represented in the form of the convolution of the functions as 𝑢(𝐫)=1𝜌_ℝ^𝑛𝐺^𝑛_𝛼𝐫−𝐫^′𝑓𝐫^′𝑑^𝑛𝐫^′ (79) with the Green function 𝐺^𝑛_𝛼(𝐫)=|𝐫|^(2−𝑛)/2(2𝜋)^{𝑛/2∞0𝑁}_𝑗=1𝑐_𝑗𝜆^𝛼𝑗−1𝜆^𝑛/2𝐽_(𝑛−2)/2(𝜆|𝐫|)𝑑𝜆, (80) where 𝑛=1,2,3 and 𝛼=(𝛼₁,…,𝛼_𝑚).

These particular solutions allow us to describe static fields in the elastic continuum with the weak spatial dispersion of 𝛼=(𝛼₁,…,𝛼_𝑁)-type.

7. Fractional Weak Spatial Dispersion of (𝛼,𝛽)-Type

7.1. Fractional Gradient Elasticity Equation for Dispersion of (𝛼,𝛽)-Type

If we have the dispersion law in the form 𝐾_𝛼(|𝐤|)≈𝑎_𝛼|𝐤|^𝛼+𝑎_𝛽|𝐤|^𝛽+𝐾_𝛼(0), (81) where 𝛼>1, 𝛽<3, and 0<𝛽<𝛼, then we have the fractional gradient elasticity equation 𝑐_𝛼(−Δ)^𝛼/2𝑢(𝐫)+𝑐_𝛽(−Δ)^𝛽/2𝑢(𝐫)=1𝜌𝑓(𝐫), (82) where 𝑐_𝛼=𝑔𝑎_𝛼𝑑^𝛼𝑀,𝑐_𝛽=𝑔𝑎_𝛽𝑑^𝛽𝑀. (83) If 𝛼=4 and 𝛽=2, we have the well-known equation of the gradient elasticity [15]: 𝑐₂Δ𝑢(𝐫)−𝑐₄Δ²𝑢(𝐫)+1𝜌𝑓(𝐫)=0, (84) where 𝑐₂=𝐸𝜌=𝑔𝑎₂𝑑²𝑀,𝑐₄=±𝑙²𝐸𝜌=𝑔𝑎₄𝑑⁴𝑀. (85) The second-gradient term is preceded by the sign that is defined by sgn(𝑔𝑎₄), where 𝑔𝑎₂>0.

Equation (82) is the fractional partial differential equation (78) with 𝑛=3, and such equation has the particular solution [3] of the firm 𝑢(𝐫)=1𝜌_ℝ³𝐺³_𝛼,𝛽𝐫−𝐫^′𝑓𝐫^′𝑑³𝐫^′, (86) where the Green type function is given by 𝐺³_𝛼,𝛽(𝐫)=|𝐫|^−1/2(2𝜋)^3/2∞0𝑐_𝛼𝜆^𝛼+𝑐_𝛽|𝜆|^𝛽−1𝜆^3/2𝐽_1/2(𝜆|𝐫|)𝑑𝜆. (87) Here 𝐽_1/2 is the Bessel function of the first kind.

7.2. Point Load Problem for Fractional Gradient Elasticity

Let us consider point load problem for an infinite elastic continuum (see pages 25-26 in [29]) and determine a deformation of an infinite gradient continuum, when a force is applied to a small region in it. We consider this Thomson's problem for nonlocal elastic continuum with fractional weak spatial dispersion of the form (81). If we consider the deformation at distances |𝐫|, which are larger than the size of the region, then we can assume that the force is applied at a point. In this case, we have 𝑓(𝐫)=𝑓₀𝛿(𝐫)=𝑓₀𝛿(𝑥)𝛿𝑦𝛿(𝑧). (88) Then the displacement field 𝑢(𝐫) of fractional gradient elasticity has a simple form of the particular solution (79) that is proportional to the Greens function 𝑢(𝐫)=𝑓₀𝜌𝐺^𝑛_𝛼(𝐫), (89) where 𝐺^𝑛_𝛼(𝑧) is given by (80). Therefore, the displacement field (86) for the force that is applied at a point (88) has the form 𝑢(𝐫)=12𝜋²𝑓₀𝜌|𝐫|^∞₀𝜆sin(𝜆|𝐫|)𝑐_𝛼𝜆^𝛼+𝑐_𝛽𝜆^𝛽𝑑𝜆. (90)

From a mathematical point of view, there are two special cases: (1) fractional weak spatial dispersion of (𝛼,𝛽)-type with 𝛼=2 and 0<𝛽<2; (2) fractional weak spatial dispersion of (𝛼,𝛽)-type with 𝛼≠2, 𝛼>𝛽, and 0<𝛽<3.

From the point of view of the nonlocal elasticity theory, it is useful to distinguish the two following particular cases:(i)subgradient elasticity (𝛼=2 and 0<𝛽<2),(ii)supergradient elasticity (𝛼>2 and 𝛽=2).

Note that for the first case the order of the fractional Laplacian is less than the order of the first term related to the usual Hooke’s law. In the second case the order of the fractional Laplacian is greater than the order of the first term related to the Hooke law. The names of the sub- and supergradient elasticity caused by the analogy with the names of anomalous diffusion [6–8] such as subdiffusion and superdiffusion.

7.3. Subgradient Elasticity Model

The subgradient elasticity is characterized by the fractional weak spatial dispersion of (𝛼,𝛽)-type with 𝛼=2 and 0<𝛽<2. Fractional model of nonlocal continuum with this spatial dispersion is described by (82) with 𝛼=2 and 0<𝛽<2, given by 𝑐₂Δ𝑢(𝐫)−𝑐_𝛽(−Δ)^𝛽/2𝑢(𝐫)+1𝜌𝑓(𝐫)=0,0<𝛽<2. (91) The order of the fractional Laplacian (−Δ)^𝛽/2 is less than the order of the first term related to the usual Hooke’s law. As a simple example, consider the square of the Laplacian; that is, 𝛽=1.

The particular solution of (91) for the force that is applied at a point (88) is the displacement field 𝑢(𝐫)=12𝜋²𝑓₀𝜌|𝐫|^∞₀𝜆sin(𝜆|𝐫|)𝑐₂𝜆²+𝑐_𝛽𝜆^𝛽𝑑𝜆. (92)

Using equation (1) of Section 2.3 in the book in [30, 31], we obtain the following asymptotic behavior for 𝑢(|𝐫|) with 0<𝛽<2, when |𝐫|→∞: 𝑢𝐫=𝑓(||)⁰2𝜋²𝜌|𝐫|^∞₀𝜆sin(𝜆|𝐫|)𝑐₂𝜆²+𝑐_𝛽𝜆^𝛽≈𝐶𝑑𝜆₀𝛽|𝐫|^3−𝛽+^∞_𝑘=1𝐶_𝑘𝛽|𝐫|^{(2−𝛽)(𝑘+1)+1}, (93) where 𝐶₀𝛽=𝑓₀2𝜋²𝜌𝑐_𝛽Γ𝜋2−𝛽sin2𝛽,𝐶_𝑘𝛽𝑓=−⁰𝑐^𝑘₂2𝜋²𝜌𝑐_𝛽^𝑘+1∞₀𝑧^{(2−𝛽)(𝑘+1)−1}sin(𝑧)𝑑𝑧. (94)

As a result, the displacement field for the force that is applied at a point in the continuum with this type of fractional weak spatial dispersion is given by 𝑢(𝐫)≈𝐶₀𝛽|𝐫|^3−𝛽,0<𝛽<2 (95) on the long distance |𝐫|≫1.

7.4. Supergradient Elasticity Model

The supergradient elasticity is characterized by the fractional weak spatial dispersion of (𝛼,𝛽)-type with 𝛼>2 and 𝛽=2. For the nonlocal continuum with the weak spatial dispersion of the (𝛼,𝛽)-type, where 𝛼>𝛽>0, 0<𝛽<3 and 𝛼≠2, the displacement field for the fractional gradient model which is described by (82) includes two parameters (𝛼,𝛽). As an example of the nonlocal continuum with this type of spatial dispersion, we highlight the case of supergradient elasticity, where 𝛽=2 and 𝛼>2. In this case (82) has the form 𝑐₂Δ𝑢(𝐫)−𝑐_𝛼(−Δ)^𝛼/2𝑢(𝐫)+1𝜌𝑓(𝐫)=0,(𝛼>2). (96) The order of the fractional Laplacian (−Δ)^𝛼/2 is greater than the order of the first term related to the Hooke law. If 𝛼=4, (96) becomes (84). Therefore, the case 3<𝛼<5 can be considered as close as possible (𝛼≈4) to the usual gradient elasticity (84).

For the displacement field that is described by (82), where 𝛼>𝛽>0, 0<𝛽<2 and 𝛼≠2 and the force 𝑓(𝐫) is applied at a point (88), we have the following asymptotic behavior: 𝑢𝐫≈𝑓(||)⁰Γ2−𝛽sin𝜋𝛽/22𝜋²𝜌𝑐_𝛽⋅1|𝐫|^3−𝛽,(|𝐫|⟶∞). (97) We note that this asymptotic behavior |𝐫|→∞ does not depend on the parameter 𝛼. The field on the long distances is determined only by term with (−Δ)^𝛽/2 (𝛼>𝛽) that can be interpreted as a fractional nonlocal “deformation” of Hooke's law.

We note the existence of a maximum for the function 𝑢(|𝐫|)⋅|𝐫| in the case 0<𝛽<2<𝛼.

The asymptotic behavior of the displacement field 𝑢(|𝐫|) for |𝐫|→0 is given by 𝑢𝐫≈𝑓(||)⁰Γ((3−𝛼)/2)2^𝛼𝜋²√𝜋𝜌𝑐_𝛼Γ()⋅1𝛼/2|𝐫|^3−𝛼,(),𝑢𝐫≈𝑓1<𝛼<2(||)⁰Γ((3−𝛼)/2)2^𝛼𝜋²√𝜋𝜌𝑐_𝛼Γ()⋅|𝐫|𝛼/2^𝛼−3,(),𝑢𝐫≈𝑓2<𝛼<3(||)⁰2𝜋𝛼𝜌𝑐_𝛽^1−3/𝛼𝑐_𝛼^3/𝛼(),(),sin3𝜋/𝛼𝛼>3 (98)(99)(100) where we use Euler's reflection formula for Gamma function. The asymptotic relation (98) is not directly related to the supergradient case. Note that the above asymptotic behavior does not depend on the parameter 𝛽, and relations (98)-(99) do not depend on 𝑐_𝛽. The displacement field 𝑢(|𝐫|) on the short distances is determined only by term with (−Δ)^𝛼/2 (𝛼>𝛽) that can be considered as a fractional nonlocal “deformation” of the gradient term.

8. Conclusion

A lattice model with spatial dispersion of power-law type is suggested. Gradient elasticity is considered as a phenomenological theory representing continuum limit of lattice dynamics, where the length scales are much larger than interatomic distances. In the continuum limit we derive continuum equations with spatial derivatives of noninteger order 𝛼. The correspondent continuum equations describe fractional generalization of gradient elasticity (the supergradient elasticity model) for 𝛼>2 and a special form of fractional integral elasticity (the subgradient elasticity model) for 0<𝛼<2. The suggested lattice model with spatial dispersion can be considered as a microscopic basis for the fractional nonlocal elastic continuum. We can note that a fractional nonlocal continuum model can be obtained from different microscopic or lattice models [32, 33]. The main advantage of the suggested approach is that we can use the Taylor series in the wave vector space instead of Taylor expansion in a coordinate space. It allows us to use these models as a microstructural basis of unified description of fractional (and integer) gradient models with positive and negative signs of the strain gradient terms. The suggested approach can be generalized for three-dimensional case of gradient elasticity. The proposed lattice model can also be easily generalized for the case of the high-order gradient elasticity and the correspondent fractional extension by using the next terms of fractional Taylor series. The suggested lattice models with long-range interactions can be important to describe the nonlocal elasticity of materials at microscale and nanoscales [34–36], where the interatomic and intermolecular interactions are prevalent in determining the properties of these materials.

Appendices

A. Fractional Taylor Formula

A.1. Riemann-Liouville and Caputo Derivatives

The left-sided Riemann-Liouville derivatives of order 𝛼>0 are defined by ^RL𝐷^𝛼_𝑎+𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛×^𝑥_𝑎𝑓𝑥^′𝑑𝑥^′𝑥−𝑥^{′𝛼−𝑛+1},(𝑛=[𝛼]+1). (A.1) We can rewrite this relation in the form ^RL𝐷^𝛼_𝑎+𝑓(𝑥)=𝑑𝑑𝑥^𝑛𝐼^𝑛−𝛼_𝑎+𝑓(𝑥), (A.2) where 𝐼^𝛼_𝑎+ is a left-sided Riemann-Liouville integral of order 𝛼>0: 𝐼^𝛼_𝑎+𝑓(𝑥)=1Γ(𝛼)^𝑥_𝑎𝑓𝑥^′𝑑𝑥^′𝑥−𝑥^′1−𝛼,(𝑥>𝑎). (A.3)

The Caputo fractional derivative of order 𝛼 is defined by ^𝐶𝐷^𝛼_𝑎+𝑓(𝑥)=𝐼^𝑛−𝛼_𝑎+𝑑𝑑𝑥^𝑛𝑓(𝑥), (A.4) where 𝐼^𝛼_𝑎+ is a left-sided Riemann-Liouville integral (A.3) of order 𝛼>0. In (A.16) we use 0<𝛼<1 and 𝑛=1. The main distinguishing feature of the Caputo fractional derivative is that, like the integer order derivative, the Caputo fractional derivative of a constant is zero.

Note also that the third term in (A.16) involves the fractional derivative of the fractional derivative, which is not the same as the 2𝛼 fractional derivative. In general, ^𝐶𝐷^𝛼𝐶_𝑎+𝐷^𝛼_𝑎+𝑓(𝑥)≠^𝐶𝐷^2𝛼_𝑎+𝑓(𝑥). (A.5) Then the coefficients of the fractional Taylor series can be found in the usual way, by repeated differentiation. This is to ensure that the fractional derivative of order 𝛼 of the function (𝑥−𝑎)^𝛼 is a constant. The repeated fractional derivative of order 𝛼 gives zero. Then the coefficients of the fractional Taylor series can be found in the usual way, by repeated differentiation.

A.2. Fractional Taylor's Series in the Riemann-Liouville Form

Let 𝑓(𝑥) be a real-value function such that the derivative (^RL𝐷^𝛼+𝑚_𝑎+𝑓)(𝑥) is integrable. Then the following analog of Taylor formula holds (see Chapter 1. Section 2.6 [1, 2]): 𝑓(𝑥)=^𝑚−1_𝑗=0^RL𝐷^𝛼+𝑗_𝑎+𝑓(𝑎+)Γ()𝛼+𝑗+1𝑥−𝑎^𝛼+𝑗+𝑅_𝑚(𝑥),(𝛼>0), (A.6) where 𝐷^𝛼+𝑗_𝑎+ are left-sided Riemann-Liouville derivatives and 𝑅_𝑚(𝑥)=𝐼^𝛼+𝑚_𝑎+^RL𝐷^𝛼+𝑚_𝑎+𝑓(𝑥). (A.7)

A.3. Riemann Formal Version of the Generalized Taylor’s Series

The Riemann formal version of the generalized Taylor’s series [37–39] is 𝑓(𝑥)=^+∞_𝑚=−∞^RL𝐷_𝑎^𝛼+𝑚𝑓𝑥₀Γ(𝛼+𝑚+1)𝑥−𝑥₀^𝛼+𝑚, (A.8) where ^RL𝐷^𝛼_𝑎 for 𝛼>0 is the Riemann-Liouville fractional derivative and ^RL𝐷^𝛼_𝑎=𝐼_𝑎^−𝛼 for 𝛼<0 is the Riemann-Liouville fractional integral of order |𝛼|.

A.4. Fractional Taylor's Series in the Trujillo-Rivero-Bonilla Form

The Trujillo-Rivero-Bonilla form of generalized Taylor's formula [40] is 𝑓(𝑥)=^𝑚_𝑗=0𝑐_𝑗Γ𝛼()𝑗+1𝑥−𝑎^{(𝑗+1)𝛼−1}+𝑅_𝑚(𝑥,𝑎), (A.9) where 𝛼∈[0;1] and 𝑐_𝑗=Γ(𝛼)(𝑥−𝑎)^1−𝛼RL𝐷^𝛼_𝑎^𝑗𝑓(𝑥),𝑅)(𝑎+_𝑚(𝑥,𝑎)=^RL𝐷^𝛼_𝑎^𝑚+1𝑓(𝜉)Γ))()((𝑚+1𝛼+1𝑥−𝑎^(𝑚+1)𝛼,𝜉∈[𝑎;𝑥]. (A.10)

A.5. Fractional Taylor's Series in the Dzherbashyan-Nersesian Form

Let 𝛼_𝑘(𝑘=0,1,…,𝑚) be increasing sequence of real numbers such that 0<𝛼_𝑘−𝛼_𝑘−1≤1,𝛼₀=0,𝑘=1,2,…,𝑚. (A.11)

We introduce the following notation [41, 42] (see also Section 2.8 in [1, 2]): 𝐷^(𝛼𝑘)=𝐼^{1−(𝛼𝑘−𝛼𝑘−1)}₀₊𝐷^{1+𝛼𝑘−1}₀₊. (A.12) In general, 𝐷^(𝛼𝑘)≠^RL𝐷^𝛼𝑘₀₊. Fractional derivative 𝐷^(𝛼𝑘) differs from the Riemann-Liouville derivative ^RL𝐷^𝛼𝑘₀₊ by finite sum of power functions since (see equation (2.68) in [3]) 𝐼^𝛼₀₊𝐼^𝛽₀₊≠𝐼^𝛼+𝛽₀₊. (A.13)

The generalized Taylor’s formula [41, 42] is 𝑓(𝑥)=^𝑚−1_𝑘=0𝑎_𝑘𝑥^𝛼𝑘+𝑅_𝑚(𝑥),(𝑥>0), (A.14) where 𝑎_𝑘=𝐷^(𝛼𝑘)𝑓(0)Γ𝛼_𝑘,𝑅+1_𝑚(𝑥)=1Γ𝛼_𝑚+1^𝑥₀(𝑥−𝑧)^𝛼𝑚−1𝐷^(𝛼𝑘)𝑓(𝑧)𝑑𝑧. (A.15)

A.6. Fractional Taylor's Series in the Odibat-Shawagfeh Form

The fractional Taylor series is a generalization of the Taylor series for fractional derivatives, where 𝛼 is the fractional order of differentiation, 0<𝛼<1. The fractional Taylor series with Caputo derivatives [43] has the form 𝑓(𝑥)=𝑓(𝑎)+^𝐶𝐷^𝛼_𝑎+𝑓(𝑎)Γ()()𝛼+1𝑥−𝑎^𝛼+^𝐶𝐷^𝛼𝐶_𝑎+𝐷^𝛼_𝑎+𝑓(𝑎)Γ()()2𝛼+1𝑥−𝑎^2𝛼+⋯, (A.16) where ^𝐶𝐷^𝛼_𝑎+ is the Caputo fractional derivative of order 𝛼.

B. Riesz Fractional Derivatives and Integrals

Fractional integration and fractional differentiation in the 𝑛-dimensional Euclidean space ℝ^𝑛 can be defined as fractional powers of the Laplace operator. For 𝛼>0 and “sufficiently good” functions 𝑓(𝑥), 𝑥∈ℝ^𝑛, the fractional Laplacian in Riesz's form (the Riesz fractional derivative) is defined in terms of the Fourier transform ℱ by (−Δ)^𝛼/2𝑓(𝑥)=ℱ⁻¹|𝑘|^𝛼ℱ𝑓(𝑘). (B.1) The Riesz fractional integration is defined by 𝐈^𝛼_𝑥𝑓(𝑥)=ℱ⁻¹|𝑘|^−𝛼ℱ𝑓(𝑘). (B.2)

The Riesz fractional integration can be realized in the form of the Riesz potential defined as Fourier's convolution of the form 𝐈^𝛼_𝑥𝑓(𝑥)=_ℝ𝑛𝐾_𝛼(𝑥−𝑧)𝑓(𝑧)𝑑𝑧,(𝛼>0), (B.3) where the function 𝐾_𝛼(𝑥) is the Riesz kernel. If 𝛼>0 and 𝛼≠𝑛,𝑛+2,𝑛+4,…, the function 𝐾_𝛼(𝑥) is defined by 𝐾_𝛼(𝑥)=𝛾_𝑛⁻¹(𝛼)|𝑥|^𝛼−𝑛. (B.4) If 𝛼≠𝑛,𝑛+2,𝑛+4,…, then 𝐾_𝛼(𝑥)=−𝛾_𝑛⁻¹(𝛼)|𝑥|^𝛼−𝑛ln|𝑥|. (B.5) The constant 𝛾_𝑛(𝛼) has the form 𝛾_𝑛(𝛼)=2^𝛼𝜋^𝑛/2Γ(𝛼/2)Γ)),()((𝑛−𝛼/2𝛼≠𝑛+2𝑘,𝑛∈ℕ,−1^{(𝑛−𝛼)/2}2^𝛼−1𝜋^𝑛/2Γ𝛼2Γ[]1+𝛼−𝑛2,𝛼=𝑛+2𝑘. (B.6)

Obviously, the Fourier transform of the Riesz fractional integration is given by ℱ𝐈^𝛼_𝑥𝑓(𝑥)=|𝑘|^−𝛼ℱ𝑓(𝑘). (B.7) This formula is true for functions 𝑓(𝑥) belonging to Lizorkin's space. The Lizorkin space of test functions on ℝ^𝑛 is a linear space of all complex-valued infinitely differentiable functions 𝑓(𝑥) whose derivatives vanish at the origin: Ψ=𝑓(𝑥)∶𝑓(𝑥)∈𝑆ℝ^𝑛,𝐷^𝐧_𝑥𝑓(0)=0,|𝐧|∈ℕ (B.8) where 𝑆(ℝ^𝑛) is the Schwartz test-function space. The Lizorkin space is invariant with respect to the Riesz fractional integration. Moreover, if 𝑓(𝑥) belongs to the Lizorkin space, then 𝐈^𝛼_𝑥𝐈^𝛽_𝑥𝑓(𝑥)=𝐈_𝑥^𝛼+𝛽𝑓(𝑥), (B.9) where 𝛼>0 and 𝛽>0.

For 𝛼>0, the fractional Laplacian in Riesz's form can be defined in the form of the hypersingular integral by (−Δ)^𝛼/2𝑓(𝑥)=1𝑑_𝑛(𝑚,𝛼)_ℝ^𝑛1|𝑧|^𝛼+𝑛Δ^𝑚_𝑧𝑓(𝑧)𝑑𝑧, (B.10) where 𝑚>𝛼 and (Δ^𝑚_𝑧𝑓)(𝑧) is a finite difference of order 𝑚 of a function 𝑓(𝑥) with a vector step 𝑧∈ℝ^𝑛 and centered at the point 𝑥∈ℝ^𝑛: Δ^𝑚_𝑧𝑓(𝑧)=^𝑚_𝑘=0(−1)^𝑘𝑚!()!𝑓().𝑘!𝑚−𝑘𝑥−𝑘𝑧 (B.11) The constant 𝑑_𝑛(𝑚,𝛼) is defined by 𝑑_𝑛()=𝜋𝑚,𝛼^1+𝑛/2A_𝑚(𝛼)2^𝛼Γ()Γ()(),1+𝛼/2𝑛/2+𝛼/2sin𝜋𝛼/2 (B.12) where 𝐴_𝑚(𝛼)=^𝑚_𝑗=0(−1)^𝑗−1𝑚!!𝑗𝑗!𝑚−𝑗^𝛼. (B.13) Note that the hypersingular integral ((−Δ)^𝛼/2𝑓)(𝑥) does not depend on the choice of 𝑚>𝛼.

If 𝑓(𝑥) belongs to the space of “sufficiently good” functions, then the Fourier transform ℱ of the fractional Laplacian in Riesz's form is given by ℱ(−Δ)^𝛼/2𝑓(𝑘)=|𝑘|^𝛼ℱ𝑓(𝑘). (B.14) This equation is valid for the Lizorkin space [1, 2] and the space 𝐶^∞(ℝ^𝑛) of infinitely differentiable functions on ℝ^𝑛 with compact support.

The fractional Laplacian in Riesz's form yields an operator inverse to the Riesz fractional integration for a special space of functions. The formula (−Δ)^𝛼/2𝐈^𝛼_𝑥𝑓(𝑥)=𝑓(𝑥),(𝛼>0) (B.15) holds for “sufficiently good” functions 𝑓(𝑥). In particular, (B.15) for 𝑓(𝑥) belongs to the Lizorkin space. Moreover, this property is also valid for the Riesz fractional integration in the frame of 𝐿_𝑝-spaces: 𝑓(𝑥)∈𝐿_𝑝(ℝ) for 1⩽𝑝<𝑛/𝑎 (see Theorem 26.3 in [1, 2]).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author expresses his gratitude to Professor Elias C. Aifantis for valuable discussions of fractional gradient elasticity and to Professor Juan J. Trujillo for valuable discussions of applications of fractional models in elasticity theory.