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Let us consider a lattice model where all particles are displaced in one direction. We assume that the displacement of particles from its equilibrium position is determined by a scalar field. It allows us to describe the properties of the lattice and its continuum limit by using simple equations. The equations of motion for a one-dimensional lattice system of interacting particles have the form ∞ ∂2u (t) + M n = g K(n, m) u (t) u (t) + F (n), (1) ∂t2 n m m=−∞ − mX=6 n   where un(t) = u(n, t) is the displacement of n particle from its equilibrium position, g is the coupling constant for interparticle interactions in the lattice, the terms F (n) characterize an interaction of the particles with the external on-site force. For simplicity, we assume that all particles have the same mass M. The elements K(n, m) of equation (1) describe the interparticle interaction in the lattice. For an unbounded homogeneous lattice, due to its homogeneity, K(n, m) has the form K(n, m)= K(n m). Equations of motion (1) is invariant with respect to its displacement of the lattice as a whole,− provided external forces are absent. It should be noted that the noninvariant terms lead to in the continuous limit [13].

3 Transform Operation for Lattice Models

In order to define the operation that transforms the lattice equations for un(t) into the contin- uum equation for a scalar field u(x, t), we use the methods suggested in [29, 30]: We consider u (t) as Fourier coefficients of some functionu ˆ(k, t) on [ k /2,k /2], then we use the n − 0 0 continuous limit k0 to obtainu ˜(k, t), and finally we apply the inverse Fourier integral transformation to obtain→ ∞u(x, t). Diagrammatically this can be written in the following form.

−1 F∆ F u (t) uˆ(k, t) Lim u˜(k, t) u(x, t) (2) n −−−→ −−−→ −−−→ We performed similar transformations for differential equations to map the lattice equation into an equation for the elastic continuum. We can represent these sets of operations in the form of the following diagrams.

Equation for u (t) F rom Lattice to Continuum Equation for u(x, t) n −−−−−−−−−−−−−−−−−→ transform Inverse Fourier integral transform (3)  x Equation for uˆ(k, t) Equation for u˜(k, t) y −−−−−−−→Limit ∆x→0  Therefore the transformation operation that maps our lattice model into a continuum model is a sequence of the following three actions (for details see [29, 30]): 1. The Fourier series transform : u (t) u (t) =u ˆ(k, t) that is defined by F∆ n →F∆{ n } +∞ uˆ(k, t)= u (t) e−ikxn = u (t) , (4) n F∆{ n } n=−∞ X 2 1 +k0/2 u (t)= dk uˆ(k, t) eikxn = −1 uˆ(k, t) , (5) n k F∆ { } 0 Z−k0/2 where xn = n∆x, and ∆x =2π/k0 is the inter-particle distance. For simplicity we assume that all lattice particles have the same inter-particle distance ∆x.

2. The passage to the limit ∆x 0(k0 ) denoted by Lim :u ˆ(k, t) Lim uˆ(k, t) = u˜(k, t). The functionu ˜(k, t)→ can be derived→∞ fromu ˆ(k, t) in the limit ∆→x 0.{ Note that} → u˜(k, t) is a Fourier integral transform of the field u(x, t), andu ˆ(k, t) is a Fourier series transform of un(t), where we use 2π un(t)= u(xn, t) k0 considering x = n∆x =2πn/k x. n 0 → 3. The inverse Fourier integral transform −1 :u ˜(k, t) −1 u˜(k, t) = u(x, t) is defined F →F { } by +∞ u˜(k, t)= dx e−ikxu(x, t)= u(x, t) , (6) F{ } Z−∞ 1 +∞ u(x, t)= dk eikxu˜(k, t)= −1 u˜(k, t) . (7) 2π F { } Z−∞ Using the suggested notations we can represent diagram (3) in the following form,

u (t) From Particle to Field u(x, t) n −−−−−−−−−−−−−−→ − F∆ F 1 (8)  x uˆ(k, t) u˜(k, t) . y −−−−−−−→Lim ∆x→0  −1 The combination of these three actions , Lim, and ∆ allows us to realize the transfor- mation of lattice models into continuum modelsF [29, 30]. F Note that equations (4) and (5) in the limit ∆x 0 (k ) are used to obtain the → 0 → ∞ Fourier integral transform equations (6) and (7), where the sum is changed by the integral. Let us give the statement that describes the Fourier series transform for the equations for displacement of lattice particles (1). The Fourier series transform ∆ maps the lattice equations of motion F ∞ ∂2u (t) + M n = g K(n, m) u (t) u (t) + F (n), (9) ∂t2 n m m=−∞ − mX=6 n   where K(n, m) satisfies the conditions ∞ K(n, m)= K(n m)= K(m n), K(n) 2 < , (10) − − | | ∞ n=1 X into the continuum equation ∂2uˆ(k, t) M = g Kˆ (0) Kˆ (k∆x) uˆ(k, t)+ F (n) , (11) ∂t2 − F∆{ }   3 whereu ˆ(k, t) = u (t) , Kˆ (k∆x) = K(n) , and is an notation for the F∆{ n } F∆{ } F∆ Fourier series transform. Here we use the notation

+∞ Kˆ (k∆x)= e−ikn∆xK(n). (12) n=−∞ Xn=06

Using K( n)= K(n), the (12) can be represented by − +∞ +∞ Kˆ (k∆x)= K(n) e−ikn∆x + eikn∆x =2 K(n)cos(k∆x) , (13) n=1 n=1 X  X and +∞ ∞ Kˆ (0) = K(n)=2 K(n). (14) n=−∞ n=1 Xn=06 X For details see [29, 30] and [13].

4 Weak Spatial Dispersion of Power-Law Type

In lattice models, dispersion is associated with different properties of the wave such as its fre- quency, wavelength, wave-number, amplitude and others. Spatial dispersion is the dependence of the elastic waves on the wave vector. In the model that is described by equation (1) the spatial dispersion means the dependence of the kernel Kˆ ( k ) on the wave vector k. This de- pendence is caused by non-local interactions in the elastic| continuu| m. The spatial dispersion leads to a non-local connection between the stress tensor σkl and the strain tensor εkl. The tensor σkl at any point r of the continuum is not uniquely defined by the values of εkl at this ′ point. It also depends on the values of εkl at neighboring points r , located near the point r. Qualitatively describing the process in a lattice with spatial dispersion implies that the fields of the elastic wave moves 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 region. Therefore, the properties of the continuum, and hence the stress tensor field σkl depend on the values of strain tensor field εkl not only in a selected point, but also in some neighborhood region. The size R0 of the area of the mutual influence are usually of the order of the interparticle distance in the lattice. The wavelength, λ, of elastic waves is several orders larger than the size of this region, so for a region of size R0 the values of the field of the elasticity wave do not change. In other words, the wavelength λ usually holds kR0 R0/λ 1. In such a lattice the spatial dispersion is weak. ∼ ≪ To describe the dynamics of the lattice it is enough to know the dependence of the function Kˆ ( k ) for small values of k = k only. Therefore we can replace this function by the Taylor polynomial.| | The weak spatial| dispersion| in the media with a power-law type of non-locality cannot be described by the usual Taylor approximation. The fractional can be very useful for approximating non-integer power-law functions [36]. This is due to the fact that the usual Taylor series for the power-law function has an infinite number of terms. Using the fractional Taylor’s formula we obtain a finite number of terms.

4 For an isotropic linear medium with the weak spatial dispersion the function Kˆ ( k ) can be | | represented in the form N Kˆ ( k ) Kˆ (0) + a k αj , (15) | | ≈ αj | | j=1 X ˆ where the frequency dispersion is neglected, and thus K(0), and aαj (j = 1, ..., N) do not depend on the frequency ω. If α = j for all j N, we can use the usual Taylor’s formula. In this case we have j ∈ a well-known weak spatial dispersion. In general, we should use a fractional generalization of the Taylor’s series [3, 4, 37, 38, 39, 40, 41]. The orders of the fractional Taylor series approximation should be correlated with the orders of power-laws of weak spatial dispersions, which are experimentally determined. In this case the fractional Taylor series approximation of Kˆ ( k ) will be the best approximation. | | We consider models of lattices with weak spatial dispersion and their continuous limits by using the methods suggested in [29, 30]. In this limit we obtain the fractional Laplacian in the Riesz’s form since the inverse Fourier’s integral transform of k α gives the fractional Laplacian | | ( ∆)α/2. Note that some equivalence of lattice-type networks with long-range interactions and continuum− models for elasticity theory is considered in [42], where the Marchaud fractional derivatives are used. For α > 0 and x Rn, the fractional Laplacian in the Riesz’s form is defined in terms of the ∈ by F (( ∆)α/2f)(x)= −1 k α( f)(k) . (16) − F | | F   For α > 0, the fractional Laplacian in the Riesz’s form usually is defined in the form of the hyper-singular integral by

α/2 1 1 m (( ∆) f)(x)= α+n (∆z f)(z) dz, − d (m, α) Rn z n Z | | m where m>α, and (∆z f)(z) is a finite difference of order m of a function f(x) with a vector step z Rn and centered at the point x Rn: ∈ ∈ m m! (∆mf)(z)= ( 1)k f(x kz). z − k!(m k)! − Xk=0 − The constant dn(m, α) is defined by π1+n/2A (α) d (m, α)= m , n 2αΓ(1 + α/2)Γ(n/2+ α/2) sin(πα/2) where m m! A (α)= ( 1)j−1 jα. m − j!(m j)! j=0 X − Note that the hyper-singular integral (( ∆)α/2f)(x) does not depend on the choice of m>α. − The Fourier transform of the fractional Laplacian is given by F ( ( ∆)α/2f)(k)= k α( f)(k). F − | | F 5 This equation is valid for Lizorkin space [3, 4] and the space C∞(Rn) of infinitely differentiable functions on Rn with compact support.

5 Fractional Elasticity Equation from Lattice Model

In the continuous limit the equations for lattices with weak spatial dispersion of a power-law type gives the continuum equation for the fractional elasticity model. Statement. In the continuous limit ∆x 0, the lattice equations → ∞ ∂2u (t) + M n = g K(n m) u (t) u (t) + F (n), (17) ∂t2 n m m=−∞ − − mX=6 n   with weak spatial dispersion of the form (15) gives the fractional continuum equation

∂2u(x, t) N 1 = G (( ∆)αj /2u)(x, t)+ f(x), (18) ∂t2 − αj − ρ j=1 X

αj /2 with the fractional Laplacian ( ∆) of order αj. Here the variables x and ∆x are dimen- sionless, f(x) = F (x)/A ∆x , −ρ = M/(A ∆x ), A is the cross-sectional area of the medium, and | | | | αj g aα ∆x G = j | | (j =1, ..., N) (19) αj M are finite parameters.

Proof. Let us describe the main points in the proof of this statement. Here we use the methods from [29, 30] and [13]. The Fourier series transform of equation (17) gives (11). F∆ After division by the cross-sectional area of the medium A and the inter-particle distance ∆x , the limit ∆x 0 for equation (11) gives | | → N ∂2 g ∆x αj 1 uˆ(k, t)= | | ˆ (k)u ˆ(k, t)+ f(n) , (20) ∂t2 M Kαj ,∆ ρF∆{ } j=1 X where ρ = M/A ∆x is the mass density, f(n) = F (n)/A ∆x is the force density, ∆x is the inter-particle distance,| | A is the cross-sectional area of the| material,| and | |

ˆ (k)= a k αj . Kαj ,∆ − αj | |

Here we use (15), with Gαj (j = 1, ..., N) are finite parameters that are defined by (19). The ˆ expression αj ,∆(k) can be considered as a Fourier series transform of the interaction term. Note that gaK for the limit ∆x 0, if G are finite parameters. αj →∞ → αj In the limit ∆x 0, equation (20) gives → ∂2u˜(k, t) N 1 = G ˆ (k)˜u(k, t)+ f(x) , (21) ∂t2 αj Kαj ρF{ } j=1 X 6 where ˆ (k) = Lim ˆ (k) = a k αj . The inverse Fourier transform of (20) gives (18). Kαj Kαj ,∆ − αj | | Here, we use the connection between the Riesz fractional Laplacian and its Fourier transform [3, 4, 5] in the form k αj ( ∆)αj /2. | | ←→ − To illustrate this Statement we give a few examples. Example 1. If we can use the weak spatial dispersion in the form

Kˆ (k) Kˆ (0) + a k2, (22) ≈ 2 then we obtain the well-known one-dimensional equation for elastic continuum ∂2u(x, t) 1 = G ∆u(x, t)+ f(x), (23) ∂t2 2 ρ where g a ∆x 2 E G = 2 | | = , 2 M A ρ

Here E = K ∆x /A is Young’s modulus, K = ga2 is the spring stiffness, and ρ = M/A ∆x is | | 2 2 | | the mass density. The corresponding dispersion relation is ω (k)= G2k . Example 2. If the spatial dispersion law has the form

Kˆ (k) Kˆ (0) + a k2 + a k4, (24) ≈ 2 4 then we derive the equation of the gradient elasticity [34] as ∂2u(x, t) 1 = G ∆u(x, t) G ∆2u(x, t)+ f(x), (25) ∂t2 2 − 4 ρ where the constant of phenomenology model is associated with the lattice constants by

4 2 g a4 ∆x a4 E ∆x G4 = | | = | | . M A a2 ρ

2 2 4 The correcponding dispersion relation is thus ω (k) = G2k + G4k . Note that the parameter l2 of the gradient elasticity is related with the coupling constants of the lattice by the equation a ∆x 2 l2 = | 4| | | . (26) a | 2| The sign of the second-gradient term is defined by sgn(a4/a2). Note that earlier it was thought that a phenomenological model of the gradient elasticity with a minus sign does not have the appropriate microscopic model, and it is thus considered one of its main weaknesses [34]. The proposed lattice model radically changes the situation. Example 3. If we can use the fractional spatial dispersion law in the form

Kˆ (k) Kˆ (0) + a k2 + a kα, (27) ≈ 2 α then the fractional elasticity equation is ∂2u(x, t) 1 = G ∆u(x, t) G ( ∆)α/2u(x, t)+ f(x), (28) ∂t2 2 − α − ρ

7 where α α−2 g aα ∆x aα E ∆x Gα = | | = | | . M A a2 ρ The correcponding dispersion relation is ω2(k) = G k2 + G k α. Equation (28) define the 2 α| | fractional elasticity model for the one-dimensional case (x R). ∈ 6 Solution of Fractional Elasticity Equations

Using the same methods as above, we can derive a general model of three-dimensional lattice with fractional weak spatial dispersion of the form

N Kˆ (k)= Kˆ (0) + a k αj . (29) αj | | j=1 X Then the continuum equation for fractional elasticity model has the form

∂2u(r, t) N 1 = c (( ∆)αj /2u)(r, t)+ f(r), (30) ∂t2 − j − ρ j=1 X where we use cj as a new notation for the constants instead of Gαj used in the one-dimensional case. Note that r and r = r are dimensionless. | | Static equation and its solution Let us consider the statics (∂u(r, t)/∂t = 0, i.e. u(r, t) = u(r)) in the suggested fractional elasticity model. Then equation (30) gives

N 1 c (( ∆)αj /2u)(r)= f(r). (31) j − ρ j=1 X Equation (30) has a particular solution u(r) for the case α > 1 and c = 0 (see Section N N 6 5.5.1. pages 341-344 in [5]). The particular solution is represented in the form of the n of the functions Gα(r) and f(r) as follow

1 n ′ ′ n ′ u(r)= Gα(r r ) f(r ) d r , (32) ρ Rn − Z n where n =1, 2, 3 and the function Gα(r) is the Green function that is given by

N −1 n αj +i(k,r) n Gα(r)= cj k e d k, (33) Rn | | j=1 ! Z X where α =(α1, ..., αN ). The Green function (33) can be simplified by using the relation (Lemma 25.1 of [3, 4]) of the form (2π)n/2 ∞ i(k,r) k nk n/2 r e f( ) d = (n−2)/2 f(λ) λ Jn/2−1(λ ) dλ. (34) Rn | | r | | Z | | Z0 8 Here Jν 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 (34), the Green function (33) can be represented (see Theorem 5.22 in [5]) in the form of an integral with respect to one parameter λ,

−1 r (2−n)/2 ∞ N Gn(r)= | | c λαj λn/2 J (λ r ) dλ, (35) α (2π)n/2 j (n−2)/2 | | 0 j=1 ! Z X where n = 1, 2, 3 and α = (α1, ..., αm), and J(n−2)/2 is the Bessel function of the first kind, which can be represented as J1/2(z)= 2/πz sin(z) for the 3-dimensional case. p Thomson’s problem for fractional integral and gradient elasticity If we have the dispersion law in the form

Kˆ ( k ) Kˆ (0) + a k α + a k 2, (36) | | ≈ α| | 2| | where α> 0, then we obtain the fractional elasticity equation 1 c ∆u(r) c (( ∆)α/2u)(r)+ f(r)=0, (37) 2 − α − ρ where E g a ∆x 2 g a ∆x α c = = 2 | | , c = α | | . (38) 2 ρ M α M If α = 4 we have the well-known static equation of the gradient elasticity [34]: 1 c ∆u(r) c ∆2u(r)+ f(r)=0, (39) 2 − 4 ρ where E g a ∆x 4 c = l2 = 4 | | . (40) 4 ± ρ M

The second-gradient term is preceded by the sign that is defined by sgn(g a4), where g a2 > 0. Equation (37) with n = 3 has the particular solution [5] of the form

1 3 ′ ′ 3 ′ u(r)= Gα(r r ) f(r ) d r , (41) ρ R3 − Z where the Green type function is given by

−1/2 ∞ r −1 G3 (r)= | | c λα + c λ 2 λ3/2 J (λ r ) dλ. (42) α (2π)3/2 α 2| | 1/2 | | Z0  Here J1/2 is the Bessel function of the first kind. Let us consider W. Thomson (1848) problem [35] for the fractional elasticity model described by equation (37). This problem implies that we should determine the deformation of an infinite elastic continuum, when a force is applied to a small region in it. If we consider the deformation

9 at distances r , which are larger compared to the size of the region, we can assume that the | | force is applied at a point, i.e.

f(r)= f0 δ(r)= f0 δ(x)δ(y)δ(z). (43)

Then the displacement field u(r) of fractional elasticity has a simple form of the particular solution that is proportional to the Greens function f u(r)= 0 Gn(r). (44) ρ α As a result the displacement field for the force that is applied at a point (43) has the form 1 f ∞ λ sin(λ r ) u(r)= 0 | | dλ. (45) 2π2 ρ r c λα + c λ2 | | Z0 α 2 We can distinguish the following two cases: (1) Weak power-law spatial dispersion with α < 2; (2) Weak power-law spatial dispersion with α > 2. This is due to the fact that in nonlocal elasticity theory usually distinguish the following two cases: (1) Fractional integral elasticity (α< 2); (2) Fractional gradient elasticity (α> 2). For the fractional integral elasticity, the order of the fractional Laplacian is less than the order of the term related to Hooke’s law. For the fractional gradient elasticity, the order of the fractional Laplacian is greater then the order of the Hooke’s term.

Fractional integral elasticity model The fractional integral elasticity model is described by equation (37) with α< 2 of the form 1 c ∆u(r) c (( ∆)α/2u)(r)+ f(r)=0, (0 <α< 2). (46) 2 − α − ρ The order of the fractional Laplacian ( ∆)α/2 is less than the order of the first term related to the usual Hooke’s law. Note that the continuum− equation (46) of fractional integral elasticity is derived from the lattice equations with weak spatial dispersion in the form (36) with α< 2. The particular solution of equation (46) for the force that is applied at a point (43) is the displacement field f ∞ λ sin(λ r ) u(r)= 0 | | dλ (α< 2). (47) 2π2ρ r c λ2 + c λα | | Z0 2 α Using Section 2.3.1 in the book [43], we can obtain the asymptotic behavior for (47) for

∞ C (α) C (α) u(r) 0 + k ( r ), (48) ≈ r 3−α r (2−α)(k+1)+1 | |→∞ | | Xk=1 | | where f0 π C0(α)= 2 Γ(2 α) sin α , (49) 2π ρcα − 2   f ck ∞ C (α)= 0 2 z(2−α)(k+1)−1 sin(z) dz. (50) k −2π2 ρck+1 α Z0 10 As a result, the displacement field for the force that is applied at a point in the continuum 3−α with this type of fractional weak spatial dispersion is given by u(r) C0(α)/ r , where 0 <α< 2, on the long distance r 1. The asymptotic behavior r≈ 0 for the| | fractional | | ≫ | | → integral elasticity does not depend on the parameter α.

Fractional gradient elasticity model The fractional gradient elasticity model is described by the equation 1 c ∆u(r) c (( ∆)α/2u)(r)+ f(r)=0, (α> 2). (51) 2 − α − ρ This may be derived from the lattice model with the fractional weak spatial dispersion in the form (36) with α> 2. The order of the fractional Laplacian ( ∆)α/2 is greater than the order of the first term related to the Hooke’s law. If α = 4 equation− (51) become the equation (39). Therefore the case 3 <α< 5 can be considered as close as possible (α 4) to the usual gradient elasticity (39). The continuum equation (51) of fractional gradient elasticity≈ is derived from the lattice equations with weak spatial dispersion in the form (36) with α> 2. The particular solution of equation (51) for the force that is applied at a point (43) is the displacement field f ∞ λ sin(λ r ) u(r)= 0 | | dλ (α> 2). (52) 2π2ρ r c λ2 + c λα | | Z0 2 α The asymptotic behavior r for the fractional gradient elasticity does not depend on the parameter α. The asymptotic| |→∞ behavior of the displacement field u(r) for r 0 is given by | |→ f Γ((3 α)/2) r 0 r α−3 u( ) α 2 − , (2 <α< 3), (53) ≈ 2 π √πρcα Γ(α/2) · | | f u(r) 0 , (α> 3). (54) ≈ 1−3/α 3/α 2παρc2 cα sin(3π/α)

Note that the asymptotic behavior for 2 <α< 3 does not depend on c2. The displacement field u(r) of the short distance is determined only by term with ( ∆)α/2 (α > 2) which can thus be considered as a fractional non-locality of the gradient type.−

7 Conclusion

We demonstrate the close relation between the discrete microstructure of a lattice with weak spatial dispersion of a power-law type and the fractional integral and gradient elasticity models. We prove that fractional elasticity models can be directly derived from the lattice models with power-law spatial dispersion. It has been shown that a characteristic feature of the behavior of a fractional non-local continuum is the spatial power-tails of non-integer orders. We assume that fractional elastic media (and plasma-like media with power-law spatial dispersion [36]) should demonstrate a universal behavior in space by analogy with the universal behavior of low-loss dielectrics in time [44, 45, 46, 47, 48]. Note that the universal behavior in time can be connected with the interaction between the particles and the environment [49].

11 References [1] B. Ross, ”A brief history and exposition of the fundamental theory of fractional ”, Lecture Notes in Mathematics. Vol.457. (1975) 1-36. [2] J.T. Machado, V. Kiryakova, F. Mainardi, ”Recent History of ”, Com- munications in Nonlinear Science and Numerical Simulations. Vol.16. (2011) 1140-1153. [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Applications (Nauka i Tehnika, Minsk, 1987). [4] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993). [5] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differ- ential Equations (Elsevier, Amsterdam, 2006). [6] A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Me- chanics (Springer, New York, 1997). [7] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000). [8] R. Metzler, J. Klafter, ”The random walk’s guide to anomalous diffusion: a fractional dynamics approach” Physics Reports, Vol.339. (2000) 1-77. [9] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (Eds.) Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering (Springer, Dor- drecht, 2007). [10] A.C.J. Luo, V.S. Afraimovich (Eds.), Long-range Interaction, Stochasticity and Fractional Dynamics (Springer, Berlin, 2010). [11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010). [12] J. Klafter, S.C. Lim, R. Metzler (Eds.), Fractional Dynamics. Recent Advances (World Scientific, Singapore, 2011). [13] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011). [14] J.A. Tenreiro Machado, ”Fractional dynamics of a system with particles subjected to impacts”, Communications in Nonlinear Science and Numerical Simulations, vol.16. No.12. (2011) 4596-4601. [15] V.E. Tarasov, ”Review of some promising fractional physical models”, International Jour- nal of Modern Physics B. Vol.27. No.9. (2013) 1330005. (32 pages) [16] K.A. Lazopoulos, ”Non-local continuum mechanics and fractional calculus”, Mechanics Research Communications. Vol.33. No.6. (2006) 753-757. [17] A. Carpinteri, P. Cornetti, A. Sapora, ”Static-kinematic fractional operators for fractal and non-local solids”, Zeitschrift f¨ur Angewandte Mathematik und Mechanik. Applied Mathematics and Mechanics. Vol.89. No.3. (2009) 207-217. [18] A. Carpinteri, P. Cornetti, A. Sapora, ”A fractional calculus approach to nonlocal elastic- ity”, European Physical Journal Special Topics. Vol.193. (2011) 193-204. [19] A. Sapora, P. Cornetti, A. Carpinteri, ”Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach”, Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.1. (2013) 63-74.

12 [20] G. Cottone, M. Di Paola, M. Zingales, ”Elastic waves propagation in 1D fractional non- local continuum”, Physica E. Vol.42. No.2. (2009) 95-103. [21] G. Cottone, M. Di Paola, M. Zingales, ”Fractional mechanical model for the dynamics of non-local continuum”, Advances in Numerical Methods. Lecture Notes in Electrical Engi- neering. Vol.11. Chapter 33. (2009) 389-423. [22] P. Cornetti, A. Carpinteri, A. Sapora, M. Di Paola, M. Zingales, ”An explicit me- chanical interpretation of Eringen non-local elasticity by means of fractional calculus”, http://www.dipmat.univpm.it/aimeta2009/Atti [23] I.A. Kunin, Media with Microstructure Vol. I and Vol.II. (Springer-Verlag, Berlin, New York, 1982 and 1983). [24] J.A. Krumhansl, ”Generalized continuum field representation for lattice vibrations”, in Lattice Dynamics, R.F. Wallis (Ed.), (Pergamon, London, 1965) pp.627-634. (http://venus.usc.edu/PAPERS/MultiScaleMechanics/Krumhansl-63.pdf) [25] A.C. Eringen, B.S. Kim, ”Relation between non-local elasticity and lattive dynamics”, Crystal Lattice Defects. Vol.7. (1977) 51-57. [26] M. Ostoja-Starzewski, ”Lattice models in micromechanics”, Applied Mechanics Reviews. Vol.55. No.1. (2002) 35-60. [27] H. Askes, A. Metrikine, ”Higher-order continua derived from discrete media: continuali- sation aspects and boundary conditions”, International Journal of Solids and Structures. Vol.42. No.1. (2005) 187-202. [28] M. Charlotte, L. Truskinovsky, ”Lattice dynamics from a continuum viewpoint”, Journal of the Mechanics and Physics of Solids. Vol.60. No.8. (2012) 1508-1544. [29] V.E. Tarasov, ”Continuous limit of discrete systems with long-range interaction”, Journal of Physics A. Vol.39. No.48. (2006) 14895-14910. (arXiv:0711.0826) [30] V.E. Tarasov, ”Map of discrete system into continuous”, Journal of . Vol.47. No.9. (2006) 092901. (24 pages) (arXiv:0711.2612) [31] V.E. Tarasov, G.M. Zaslavsky, ”Fractional dynamics of coupled oscillators with long-range interaction”, Chaos. Vol.16. No.2. (2006) 023110. (13 pages) (nlin.PS/0512013) [32] N. Laskin, G.M. Zaslavsky, ”Nonlinear fractional dynamics on a lattice with long-range interactions”, Physica A. Vol.368. (2006) 38-54. (nlin.SI/0512010). [33] V.E. Tarasov, G.M. Zaslavsky, ”Fractional dynamics of systems with long-range inter- action”, Communications in Nonlinear Science and Numerical Simulation. Vol.11. No.8. (2006) 885-898. (arXiv:1107.5436) [34] H. Askes, E.C. Aifantis, ”Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results”, International Journal of Solids and Structures. Vol.48. No.13. (2011) 1962- 1990. [35] L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Oxford, 1986). [36] V.E. Tarasov, J.J. Trujillo, ”Fractional power-law spatial dispersion in electrodynamics”, Annals of Physics. Vol.334. (2013) 1-23. [37] G.H. Hardy, ”Riemann’s form of Taylor series” Journal of the London Mathematical So- ciety. Vol.20. (1945) 45-57.

13 [38] M.M. Dzherbashyan, A.B. Nersesian, ”The criterion of the expansion of the functions to Dirichlet series” Izvestiya Akademii Nauk Armyanskoi SSR. Seriya Fiziko-Matematicheskih Nauk. Vol.11 No.5. (1958) 85-108. in Russian. [39] M.M. Dzherbashyan, A.B. Nersesian, ”About application of some integro-differential op- erators”, Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences). Vol. 121. No.2. (1958) 210-213. in Russian. [40] J.J. Trujillo, M. Rivero, B. Bonilla, ”On a Riemann-Liouville generalized Taylor’s formula”, Journal of Mathematical Analysis and Applications. Vol. 231. No.1. (1999) 255-265. [41] Z.M. Odibat, N.T. Shawagfeh, ”Generalized Taylors formula”, Applied Mathematics and Computation. Vol.186. No.1. (2007) 286-293. [42] M. Di Paola, M. Zingales, ”Long-range cohesive interactions of non-local continuum faced by fractional calculus” International Journal of Solids and Structures. Vol.45. No.21. (2008) 5642-5659. [43] H. Bateman, A. Erdelyi, Tables of integral transforms Volume 1. (New York, McGraw-Hill, 1954). or (Moscow, Nauka, 1969) in Russian. [44] A.K. Jonscher, ”The universal dielectric response”, Nature. Vol.267. (1977) 673-679. [45] A.K. Jonscher, Universal Relaxation Law (Chelsea Dielectrics, London, 1996). [46] A.K. Jonscher, ”Low-loss dielectrics”, Journal of Materials Science. Vol.34. No.13. (1999) 3071-3082. [47] V.E. Tarasov, ”Universal electromagnetic waves in dielectrics”, Journal of Physics: Con- densed Matter. Vol.20. No.17. (2008) 175223. (arXiv:0907.2163) [48] V.E. Tarasov, ”Fractional equations of Curie-von Schweidler and Gauss laws”, Journal of Physics: Condensed Matter. Vol.20. No.14. (2008) 145212. (arXiv:0907.1837) [49] V.E. Tarasov, ”The fractional oscillator as an open system”, Central European Journal of Physics. Vol.10. No.2. (2012) 382-389.

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