Central European Journal of Physics. Vol.11. No.11. (2013) 1580-1588

Lattice Model with Power-Law Spatial Dispersion
for Fractional Elasticity

Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University, Moscow 119991, Russia

E-mail<EMAIL_ADDRESS>
###### Abstract

A lattice model with a spatial dispersion corresponding to a power-law type is
suggested. This model serves as a microscopic model for elastic continuum with
power-law non-locality. We prove that the continuous limit maps of the
equations for the lattice with the power-law spatial dispersion into the
continuum equations with fractional generalizations of the Laplacian
operators. The suggested continuum equations, which are obtained from the
lattice model, are fractional generalizations of the integral and gradient
elasticity models. These equations of fractional elasticity are solved for two
special static cases: fractional integral elasticity and fractional gradient
elasticity.

PACS: 45.10.Hj; 61.50.Ah; 62.20.Dc

##  1 Introduction

The integration and differentiation of any arbitrary order have a long history
since 1695 [2] \- [5]. The theory of equations with derivatives and integrals
of non-integer orders [4, 5, 6] is powerful tool to describe the behavior of
media and systems with spatial non-locality and memory of a power-law type.
Fractional differential equations have a vast applications in physics and
mechanics and form an area called fractional dynamics [7] \- [16]. The
fractional calculus allows us to formulate a fractional generalization of non-
local elasticity models in two forms: the fractional differential (gradient)
elasticity models (weak power-law non-locality) and the fractional integral
models (strong power-law non-locality). Fractional integral elasticity models
are considered in [17] \- [23].

Lattice models are very important in elasticity theory (see for example [24]
-[29]). In this paper we suggest a lattice model with a power-law spatial
dispersion as a microscopic model of elastic continuum with weak and strong
power-law non-locality. For this purpose, we consider the models of lattices
and their corresponding continuous limits by using the methods suggested in
[30, 31] (see also [32, 33, 34]). In [30, 31] we prove that the continuum
equations with fractional Laplacian in the Riesz form [4, 5, 6] can be
directly derived from lattice models with different types of long-range
interactions. In this paper, we show how the continuous limit for the lattice
with a power-law spatial dispersion gives the corresponding continuum equation
of the fractional integral and gradient elasticity. The continuum equations of
fractional elasticity, which are derived from the lattice model, are solved
for two static cases: fractional integral elasticity and fractional gradient
elasticity.

##  2 Equations for Displacement of Lattice Particles

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

$M\,\frac{\partial^{2}u_{n}(t)}{\partial
t^{2}}=g\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\;K(n,m)\;\Bigl{(}u_{n}(t)-u_{m}(t)\Bigr{)}+F(n),$
(1)

where $u_{n}(t)=u(n,t)$ 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 divergences in the
continuous limit [14].

##  3 Transform Operation for Lattice Models

In order to define the operation that transforms the lattice equations for
$u_{n}(t)$ into the continuum equation for a scalar field $u(x,t)$, we use the
methods suggested in [30, 31]: We consider $u_{n}(t)$ as Fourier series
coefficients of some function $\hat{u}(k,t)$ on $[-k_{0}/2,k_{0}/2]$, then we
use the continuous limit $k_{0}\to\infty$ to obtain $\tilde{u}(k,t)$, and
finally we apply the inverse Fourier integral transformation to obtain
$u(x,t)$. Diagrammatically this can be written in the following form.

$\begin{CD}u_{n}(t)@>{{{\cal
F}_{\Delta}}}>{}>\hat{u}(k,t)@>{\operatorname{Lim}}>{}>\tilde{u}(k,t)@>{{\cal
F}^{-1}}>{}>u(x,t)\end{CD}$ (2)

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.

$\begin{CD}\boxed{Equation\ for\ u_{n}(t)}@>{From\ Lattice\ to\
Continuum}>{}>\boxed{Equation\ for\ u(x,t)}\\\ @V{Fourier\ series\
transform}V{}V@A{Inverse\ Fourier\ integral\ transform}A{}A\\\
\boxed{Equation\ for\ \hat{u}(k,t)}@>{}>{Limit\ \Delta x\to
0}>\boxed{Equation\ for\ \tilde{u}(k,t)}\end{CD}$ (3)

Therefore the transformation operation that maps our lattice model into a
continuum model is a sequence of the following three actions (for details see
[30, 31]):

  1. 1.

The Fourier series transform ${\cal F}_{\Delta}:\quad u_{n}(t)\to{\cal
F}_{\Delta}\\{u_{n}(t)\\}=\hat{u}(k,t)$ that is defined by

$\hat{u}(k,t)=\sum_{n=-\infty}^{+\infty}\;u_{n}(t)\;e^{-ikx_{n}}={\cal
F}_{\Delta}\\{u_{n}(t)\\},$ (4)
$u_{n}(t)=\frac{1}{k_{0}}\int_{-k_{0}/2}^{+k_{0}/2}dk\
\hat{u}(k,t)\;e^{ikx_{n}}={\cal F}^{-1}_{\Delta}\\{\hat{u}(k,t)\\},$ (5)

where $x_{n}=n\Delta x$, and $\Delta x=2\pi/k_{0}$ is the inter-particle
distance. For simplicity we assume that all lattice particles have the same
inter-particle distance $\Delta x$.

  2. 2.

The passage to the limit $\Delta x\to 0$ ($k_{0}\to\infty$) denoted by
$\operatorname{Lim}:\quad\hat{u}(k,t)\to\operatorname{Lim}\\{\hat{u}(k,t)\\}=\tilde{u}(k,t)$.
The function $\tilde{u}(k,t)$ can be derived from $\hat{u}(k,t)$ in the limit
$\Delta x\to 0$. Note that $\tilde{u}(k,t)$ is a Fourier integral transform of
the field $u(x,t)$, and $\hat{u}(k,t)$ is a Fourier series transform of
$u_{n}(t)$, where we use

$u_{n}(t)=\frac{2\pi}{k_{0}}u(x_{n},t)$

considering $x_{n}=n\Delta x=2\pi n/k_{0}\to x$.

  3. 3.

The inverse Fourier integral transform ${\cal
F}^{-1}:\quad\tilde{u}(k,t)\to{\cal F}^{-1}\\{\tilde{u}(k,t)\\}=u(x,t)$ is
defined by

$\tilde{u}(k,t)=\int^{+\infty}_{-\infty}dx\ e^{-ikx}u(x,t)={\cal
F}\\{u(x,t)\\},$ (6) $u(x,t)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}dk\
e^{ikx}\tilde{u}(k,t)={\cal F}^{-1}\\{\tilde{u}(k,t)\\}.$ (7)

Using the suggested notations we can represent diagram (3) in the following
form,

$\begin{CD}u_{n}(t)@>{From\ Particle\ to\ Field}>{}>u(x,t)\\\ @V{{\cal
F}_{\Delta}}V{}V@A{}A{{\cal F}^{-1}}A\\\ \hat{u}(k,t)@>{}>{\operatorname{Lim\
\ \Delta x\to 0}}>\tilde{u}(k,t)\ .\end{CD}$ (8)

The combination of these three actions ${\cal F}^{-1}$, $\operatorname{Lim}$,
and ${\cal F}_{\Delta}$ allows us to realize the transformation of lattice
models into continuum models [30, 31].

Note that equations (4) and (5) in the limit $\Delta x\to 0$
($k_{0}\to\infty$) are used to obtain the 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 ${\cal F}_{\Delta}$ maps the lattice equations of motion

$M\,\frac{\partial^{2}u_{n}(t)}{\partial
t^{2}}=g\sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}K(n,m)\,\Bigl{(}u_{n}(t)-u_{m}(t)\Bigr{)}+F(n),$ (9)

where $K(n,m)$ satisfies the conditions

$K(n,m)=K(n-m)=K(m-n),\quad\sum^{\infty}_{n=1}|K(n)|^{2}<\infty,$ (10)

into the continuum equation

$M\,\frac{\partial^{2}\hat{u}(k,t)}{\partial
t^{2}}=g\Bigl{(}\hat{K}(0)-\hat{K}(k\Delta x)\Bigr{)}\,\hat{u}(k,t)+{\cal
F}_{\Delta}\\{F(n)\\},$ (11)

where $\hat{u}(k,t)={\cal F}_{\Delta}\\{u_{n}(t)\\}$, $\hat{K}(k\Delta
x)={\cal F}_{\Delta}\\{K(n)\\}$, and ${\cal F}_{\Delta}$ is an operator
notation for the Fourier series transform. Here we use the notation

$\hat{K}(k\Delta x)=\sum^{+\infty}_{\begin{subarray}{c}n=-\infty\\\
n\not=0\end{subarray}}e^{-ikn\Delta x}K(n).$ (12)

Using $K(-n)=K(n)$, the function (12) can be represented by

$\hat{K}(k\Delta x)=\sum^{+\infty}_{n=1}K(n)\left(e^{-ikn\Delta
x}+e^{ikn\Delta x}\right)=2\sum^{+\infty}_{n=1}K(n)\cos\left(k\Delta
x\right),$ (13)

and

$\hat{K}(0)=\sum^{+\infty}_{\begin{subarray}{c}n=-\infty\\\
n\not=0\end{subarray}}K(n)=2\sum^{\infty}_{n=1}K(n).$ (14)

For details see [30, 31] and [14].

##  4 Weak Spatial Dispersion of Power-Law Type

In lattice models, dispersion is associated with different properties of the
wave such as its frequency, 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 $\hat{K}(|{\bf k}|)$ on the wave vector ${\bf
k}$. This dependence is caused by non-local interactions in the elastic
continuum. The spatial dispersion leads to a non-local connection between the
stress tensor $\sigma_{kl}$ and the strain tensor $\varepsilon_{kl}$. The
tensor $\sigma_{kl}$ at any point ${\bf r}$ of the continuum is not uniquely
defined by the values of $\varepsilon_{kl}$ at this point. It also depends on
the values of $\varepsilon_{kl}$ at neighboring points ${\bf r}^{\prime}$,
located near the point ${\bf r}$. 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 ${\bf r}$,
which causes an additional shift of the particles in neighboring and more
distant points ${\bf r}^{\prime}$ in some neighborhood region. Therefore, the
properties of the continuum, and hence the stress tensor field $\sigma_{kl}$
depend on the values of strain tensor field $\varepsilon_{kl}$ not only in a
selected point, but also in some neighborhood region. The size $R_{0}$ of the
area of the mutual influence are usually of the order of the interparticle
distance in the lattice. The wavelength, $\lambda$, of elastic waves is
several orders larger than the size of this region, so for a region of size
$R_{0}$ the values of the field of the elasticity wave do not change. In other
words, the wavelength $\lambda$ usually holds $kR_{0}\sim R_{0}/\lambda\ll 1$.
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 $\hat{K}(|{\bf k}|)$ for small values of $k=|{\bf k}|$ 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 Taylor series
can be very useful for approximating non-integer power-law functions [37].
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.

For an isotropic linear medium with the weak spatial dispersion the function
$\hat{K}(|{\bf k}|)$ can be represented in the form

$\hat{K}(|{\bf k}|)\approx\hat{K}(0)+\sum^{N}_{j=1}a_{\alpha_{j}}|{\bf
k}|^{\alpha_{j}},$ (15)

where the frequency dispersion is neglected, and thus $\hat{K}(0)$, and
$a_{\alpha_{j}}$ ($j=1,...,N$) do not depend on the frequency $\omega$.

If $\alpha_{j}=j$ for all $j\in\mathbb{N}$, we can use the usual Taylor’s
formula. In this case we have a well-known weak spatial dispersion. In
general, we should use a fractional generalization of the Taylor’s series [4,
5, 38, 39, 40, 41, 42]. 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 $\hat{K}(|{\bf k}|)$ will be the
best approximation.

We consider models of lattices with weak spatial dispersion and their
continuous limits by using the methods suggested in [30, 31]. In this limit we
obtain the fractional Laplacian in the Riesz’s form since the inverse
Fourier’s integral transform of $|{\bf k}|^{\alpha}$ gives the fractional
Laplacian $(-\Delta)^{\alpha/2}$. Note that some equivalence of lattice-type
networks with long-range interactions and continuum models for elasticity
theory is considered in [43], where the Marchaud fractional derivatives are
used.

For $\alpha>0$ and $x\in\mathbb{R}^{n}$, the fractional Laplacian in the
Riesz’s form is defined in terms of the Fourier transform ${\cal F}$ by

$((-\Delta)^{\alpha/2}f)(x)={\cal F}^{-1}\Bigl{(}|k|^{\alpha}({\cal
F}f)(k)\Bigr{)}.$ (16)

For $\alpha>0$, the fractional Laplacian in the Riesz’s form usually is
defined in the form of the hyper-singular integral by

$((-\Delta)^{\alpha/2}f)(x)=\frac{1}{d_{n}(m,\alpha)}\int_{\mathbb{R}^{n}}\frac{1}{|z|^{\alpha+n}}(\Delta^{m}_{z}f)(z)\,dz,$

where $m>\alpha$, and $(\Delta^{m}_{z}f)(z)$ is a finite difference of order
$m$ of a function $f(x)$ with a vector step $z\in\mathbb{R}^{n}$ and centered
at the point $x\in\mathbb{R}^{n}$:

$(\Delta^{m}_{z}f)(z)=\sum^{m}_{k=0}(-1)^{k}\frac{m!}{k!(m-k)!}\,f(x-kz).$

The constant $d_{n}(m,\alpha)$ is defined by

$d_{n}(m,\alpha)=\frac{\pi^{1+n/2}A_{m}(\alpha)}{2^{\alpha}\Gamma(1+\alpha/2)\Gamma(n/2+\alpha/2)\sin(\pi\alpha/2)},$

where

$A_{m}(\alpha)=\sum^{m}_{j=0}(-1)^{j-1}\frac{m!}{j!(m-j)!}\,j^{\alpha}.$

Note that the hyper-singular integral $((-\Delta)^{\alpha/2}f)(x)$ does not
depend on the choice of $m>\alpha$. The Fourier transform ${\cal F}$ of the
fractional Laplacian is given by

$({\cal F}(-\Delta)^{\alpha/2}f)(k)=|k|^{\alpha}({\cal F}f)(k).$

This equation is valid for Lizorkin space [4, 5] and the space
$C^{\infty}(\mathbb{R}^{n})$ of infinitely differentiable functions on
$\mathbb{R}^{n}$ 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 $\Delta x\to 0$, the lattice equations

$M\,\frac{\partial^{2}u_{n}(t)}{\partial
t^{2}}=g\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\;K(n-m)\;\Bigl{(}u_{n}(t)-u_{m}(t)\Bigr{)}+F(n),$
(17)

with weak spatial dispersion of the form (15) gives the fractional continuum
equation

$\frac{\partial^{2}u(x,t)}{\partial
t^{2}}=-\sum^{N}_{j=1}G_{\alpha_{j}}\,((-\Delta)^{\alpha_{j}/2}u)(x,t)+\frac{1}{\rho}\,f(x),$
(18)

with the fractional Laplacian $(-\Delta)^{\alpha_{j}/2}$ of order
$\alpha_{j}$. Here the variables $x$ and $\Delta x$ are dimensionless,
$f(x)=F(x)/A|\Delta x|$, $\rho=M/(A|\Delta x|)$, $A$ is the cross-sectional
area of the medium, and

$G_{\alpha_{j}}=\frac{g\,a_{\alpha_{j}}\,|\Delta
x|^{\alpha_{j}}}{M}\quad(j=1,...,N)$ (19)

are finite parameters.

Proof. Let us describe the main points in the proof of this statement. Here we
use the methods from [30, 31] and [14]. The Fourier series transform ${\cal
F}_{\Delta}$ of equation (17) gives (11). After division by the cross-
sectional area of the medium $A$ and the inter-particle distance $|\Delta x|$,
the limit $\Delta x\to 0$ for equation (11) gives

$\frac{\partial^{2}}{\partial
t^{2}}\hat{u}(k,t)=\sum^{N}_{j=1}\frac{g\,|\Delta
x|^{\alpha_{j}}}{M}\;\hat{\mathcal{K}}_{\alpha_{j},\Delta}(k)\;\hat{u}(k,t)+\frac{1}{\rho}\mathcal{F}_{\Delta}\\{f(n)\\},$
(20)

where $\rho=M/A|\Delta x|$ is the mass density, $f(n)=F(n)/A|\Delta x|$ is the
force density, $|\Delta x|$ is the inter-particle distance, $A$ is the cross-
sectional area of the material, and

$\hat{\mathcal{K}}_{\alpha_{j},\Delta}(k)=-a_{\alpha_{j}}|k|^{\alpha_{j}}.$

Here we use (15), with $G_{\alpha_{j}}$ ($j=1,...,N$) are finite parameters
that are defined by (19). The expression
$\hat{\mathcal{K}}_{\alpha_{j},\Delta}(k)$ can be considered as a Fourier
series transform of the interaction term. Note that $ga_{\alpha_{j}}\to\infty$
for the limit $\Delta x\to 0$, if $G_{\alpha_{j}}$ are finite parameters.

In the limit $\Delta x\to 0$, equation (20) gives

$\frac{\partial^{2}\tilde{u}(k,t)}{\partial
t^{2}}=\sum^{N}_{j=1}G_{\alpha_{j}}\,\hat{\mathcal{K}}_{\alpha_{j}}(k)\,\tilde{u}(k,t)+\frac{1}{\rho}\mathcal{F}\\{f(x)\\},$
(21)

where
$\hat{\mathcal{K}}_{\alpha_{j}}(k)=\operatorname{Lim}\hat{\mathcal{K}}_{\alpha_{j},\Delta}(k)=-a_{\alpha_{j}}|k|^{\alpha_{j}}$.
The inverse Fourier transform of (20) gives (18). Here, we use the connection
between the Riesz fractional Laplacian and its Fourier transform [4, 5, 6] in
the form $|k|^{\alpha_{j}}\longleftrightarrow(-\Delta)^{\alpha_{j}/2}$.

To illustrate this Statement we give a few examples.

Example 1. If we can use the weak spatial dispersion in the form

$\hat{K}(k)\approx\hat{K}(0)+a_{2}\,k^{2},$ (22)

then we obtain the well-known one-dimensional equation for elastic continuum

$\frac{\partial^{2}u(x,t)}{\partial t^{2}}=G_{2}\,\Delta
u(x,t)+\frac{1}{\rho}\,f(x),$ (23)

where

$G_{2}=\frac{g\,a_{2}\,|\Delta x|^{2}}{M\,A}=\frac{E}{\rho},$

Here $E=K|\Delta x|/A$ is Young’s modulus, $K=ga_{2}$ is the spring stiffness,
and $\rho=M/A|\Delta x|$ is the mass density. The corresponding dispersion
relation is $\omega^{2}(k)=G_{2}k^{2}$.

Example 2. If the spatial dispersion law has the form

$\hat{K}(k)\approx\hat{K}(0)+a_{2}\,k^{2}+a_{4}\,k^{4},$ (24)

then we derive the equation of the gradient elasticity [35] as

$\frac{\partial^{2}u(x,t)}{\partial t^{2}}=G_{2}\,\Delta
u(x,t)-G_{4}\,\Delta^{2}u(x,t)+\frac{1}{\rho}\,f(x),$ (25)

where the constant of phenomenology model is associated with the lattice
constants by

$G_{4}=\frac{g\,a_{4}\,|\Delta x|^{4}}{M\,A}=\frac{a_{4}\,E\,|\Delta
x|^{2}}{a_{2}\,\rho}.$

The correcponding dispersion relation is thus
$\omega^{2}(k)=G_{2}k^{2}+G_{4}k^{4}$. Note that the parameter $l^{2}$ of the
gradient elasticity is related with the coupling constants of the lattice by
the equation

$l^{2}=\frac{\left|a_{4}\right|\,|\Delta x|^{2}}{|a_{2}|}.$ (26)

The sign of the second-gradient term is defined by
$\operatorname{sgn}(a_{4}/a_{2})$. 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 [35]. The proposed lattice model radically changes the
situation.

Example 3. If we can use the fractional spatial dispersion law in the form

$\hat{K}(k)\approx\hat{K}(0)+a_{2}\,k^{2}+a_{\alpha}\,k^{\alpha},$ (27)

then the fractional elasticity equation is

$\frac{\partial^{2}u(x,t)}{\partial t^{2}}=G_{2}\,\Delta
u(x,t)-G_{\alpha}\,(-\Delta)^{\alpha/2}u(x,t)+\frac{1}{\rho}\,f(x),$ (28)

where

$G_{\alpha}=\frac{g\,a_{\alpha}\,|\Delta
x|^{\alpha}}{M\,A}=\frac{a_{\alpha}\,E\,|\Delta x|^{\alpha-2}}{a_{2}\,\rho}.$

The correcponding dispersion relation is
$\omega^{2}(k)=G_{2}k^{2}+G_{\alpha}|k|^{\alpha}$. Equation (28) define the
fractional elasticity model for the one-dimensional case ($x\in\mathbb{R}$).

##  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

$\hat{K}({\bf k})=\hat{K}(0)+\sum^{N}_{j=1}a_{\alpha_{j}}\,|{\bf
k}|^{\alpha_{j}}.$ (29)

Then the continuum equation for fractional elasticity model has the form

$\frac{\partial^{2}u({\bf r},t)}{\partial
t^{2}}=-\sum^{N}_{j=1}c_{j}\,((-\Delta)^{\alpha_{j}/2}u)({\bf
r},t)+\frac{1}{\rho}\,f({\bf r}),$ (30)

where we use $c_{j}$ as a new notation for the constants instead of
$G_{\alpha_{j}}$ used in the one-dimensional case. Note that ${\bf r}$ and
$r=|{\bf r}|$ are dimensionless.

### Static equation and its solution

Let us consider the statics ($\partial u({\bf r},t)/\partial t=0$, i.e.
$u({\bf r},t)=u({\bf r})$) in the suggested fractional elasticity model. Then
equation (30) gives

$\sum^{N}_{j=1}c_{j}\,((-\Delta)^{\alpha_{j}/2}u)({\bf
r})=\frac{1}{\rho}f({\bf r}).$ (31)

Equation (30) has a particular solution $u({\bf r})$ for the case
$\alpha_{N}>1$ and $c_{N}\neq 0$ (see Section 5.5.1. pages 341-344 in [6]).
The particular solution is represented in the form of the convolution of the
functions $G^{n}_{\alpha}({\bf r})$ and $f({\bf r})$ as follow

$u({\bf r})=\frac{1}{\rho}\,\int_{\mathbb{R}^{n}}G^{n}_{\alpha}({\bf r}-{\bf
r}^{\prime})\,f({\bf r}^{\prime})\,d^{n}{\bf r}^{\prime},$ (32)

where $n=1,2,3$ and the function $G^{n}_{\alpha}({\bf r})$ is the Green
function that is given by

$G^{n}_{\alpha}({\bf r})=\int_{\mathbb{R}^{n}}\left(\sum^{N}_{j=1}c_{j}|{\bf
k}|^{\alpha_{j}}\right)^{-1}\ e^{+i({\bf k},{\bf r})}\,d^{n}{\bf k},$ (33)

where $\alpha=(\alpha_{1},...,\alpha_{N})$. The Green function (33) can be
simplified by using the relation (Lemma 25.1 of [4, 5]) of the form

$\int_{\mathbb{R}^{n}}e^{i({\bf k},{\bf r})}\,f(|{\bf k}|)\,d^{n}{\bf
k}=\frac{(2\pi)^{n/2}}{|{\bf
r}|^{(n-2)/2}}\int^{\infty}_{0}f(\lambda)\,\lambda^{n/2}\,J_{n/2-1}(\lambda|{\bf
r}|)\,d\lambda.$ (34)

Here $J_{\nu}$ 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
[6]) in the form of an integral with respect to one parameter $\lambda$,

$G^{n}_{\alpha}({\bf r})=\frac{|{\bf
r}|^{(2-n)/2}}{(2\pi)^{n/2}}\int^{\infty}_{0}\left(\sum^{N}_{j=1}c_{j}\lambda^{\alpha_{j}}\right)^{-1}\lambda^{n/2}\,J_{(n-2)/2}(\lambda|{\bf
r}|)\,d\lambda,$ (35)

where $n=1,2,3$ and $\alpha=(\alpha_{1},...,\alpha_{m})$, and $J_{(n-2)/2}$ is
the Bessel function of the first kind, which can be represented as
$J_{1/2}(z)=\sqrt{2/\pi z}\,\sin(z)$ for the 3-dimensional case.

### Thomson’s problem for fractional integral and gradient elasticity

If we have the dispersion law in the form

$\hat{K}(|{\bf k}|)\approx\hat{K}(0)+a_{\alpha}|{\bf k}|^{\alpha}+a_{2}|{\bf
k}|^{2},$ (36)

where $\alpha>0$, then we obtain the fractional elasticity equation

$c_{2}\Delta u({\bf r})-c_{\alpha}\,((-\Delta)^{\alpha/2}u)({\bf
r})+\frac{1}{\rho}\,f({\bf r})=0,$ (37)

where

$c_{2}=\frac{E}{\rho}=\frac{g\,a_{2}\,|\Delta x|^{2}}{M},\quad
c_{\alpha}=\frac{g\,a_{\alpha}\,|\Delta x|^{\alpha}}{M}.$ (38)

If $\alpha=4$ we have the well-known static equation of the gradient
elasticity [35]:

$c_{2}\,\Delta u({\bf r})-c_{4}\Delta^{2}u({\bf r})+\frac{1}{\rho}\,f({\bf
r})=0,$ (39)

where

$c_{4}=\pm\,l^{2}\,\frac{E}{\rho}=\frac{g\,a_{4}\,|\Delta x|^{4}}{M}.$ (40)

The second-gradient term is preceded by the sign that is defined by
$\operatorname{sgn}(g\,a_{4})$, where $g\,a_{2}>0$.

Equation (37) with $n=3$ has the particular solution [6] of the form

$u({\bf r})=\frac{1}{\rho}\,\int_{\mathbb{R}^{3}}G^{3}_{\alpha}({\bf r}-{\bf
r}^{\prime})\,f({\bf r}^{\prime})\,d^{3}{\bf r}^{\prime},$ (41)

where the Green type function is given by

$G^{3}_{\alpha}({\bf r})=\frac{|{\bf
r}|^{-1/2}}{(2\pi)^{3/2}}\int^{\infty}_{0}\left(c_{\alpha}\lambda^{\alpha}+c_{2}|\lambda|^{2}\right)^{-1}\lambda^{3/2}\,J_{1/2}(\lambda|{\bf
r}|)\,d\lambda.$ (42)

Here $J_{1/2}$ is the Bessel function of the first kind.

Let us consider W. Thomson (1848) problem [36] 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 at distances
$|{\bf r}|$, which are larger compared to the size of the region, we can
assume that the force is applied at a point, i.e.

$f({\bf r})=f_{0}\,\delta({\bf r})=f_{0}\,\delta(x)\delta(y)\delta(z).$ (43)

Then the displacement field $u({\bf r})$ of fractional elasticity has a simple
form of the particular solution that is proportional to the Green s function

$u({\bf r})=\frac{f_{0}}{\rho}\,G^{n}_{\alpha}({\bf r}).$ (44)

As a result the displacement field for the force that is applied at a point
(43) has the form

$u({\bf r})=\frac{1}{2\pi^{2}}\frac{f_{0}}{\rho\,|{\bf
r}|}\,\int^{\infty}_{0}\frac{\lambda\,\sin(\lambda|{\bf
r}|)}{c_{\alpha}\lambda^{\alpha}+c_{2}\lambda^{2}}\,d\lambda.$ (45)

We can distinguish the following two cases: (1) Weak power-law spatial
dispersion with $\alpha<2$; (2) Weak power-law spatial dispersion with
$\alpha>2$. This is due to the fact that in nonlocal elasticity theory usually
distinguish the following two cases: (1) Fractional integral elasticity
($\alpha<2$); (2) Fractional gradient elasticity ($\alpha>2$).

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
$\alpha<2$ of the form

$c_{2}\Delta u({\bf r})-c_{\alpha}((-\Delta)^{\alpha/2}u)({\bf
r})+\frac{1}{\rho}\,f({\bf r})=0,\quad(0<\alpha<2).$ (46)

The order of the fractional Laplacian $(-\Delta)^{\alpha/2}$ 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
$\alpha<2$. The particular solution of equation (46) for the force that is
applied at a point (43) is the displacement field

$u({\bf r})=\frac{f_{0}}{2\pi^{2}\rho\,|{\bf
r}|}\,\int^{\infty}_{0}\frac{\lambda\,\sin(\lambda|{\bf
r}|)}{c_{2}\lambda^{2}+c_{\alpha}\lambda^{\alpha}}\,d\lambda\quad(\alpha<2).$
(47)

Using Section 2.3.1 in the book [44], we can obtain the asymptotic behavior
for (47) for

$u({\bf r})\approx\frac{C_{0}(\alpha)}{|{\bf
r}|^{3-\alpha}}+\sum^{\infty}_{k=1}\frac{C_{k}(\alpha)}{|{\bf
r}|^{(2-\alpha)(k+1)+1}}\quad(|{\bf r}|\to\infty),$ (48)

where

$C_{0}(\alpha)=\frac{f_{0}}{2\pi^{2}\,\rho\,c_{\alpha}}\,\Gamma(2-\alpha)\,\sin\left(\frac{\pi}{2}\alpha\right),$
(49)
$C_{k}(\alpha)=-\frac{f_{0}c^{k}_{2}}{2\pi^{2}\,\rho\,c^{k+1}_{\alpha}}\int^{\infty}_{0}z^{(2-\alpha)(k+1)-1}\,\sin(z)\,dz.$
(50)

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 $u({\bf r})\ \approx\ \,C_{0}(\alpha)/|{\bf r}|^{3-\alpha}$, where
$0<\alpha<2$, on the long distance $|{\bf r}|\gg 1$. The asymptotic behavior
$|{\bf r}|\to 0$ for the fractional integral elasticity does not depend on the
parameter $\alpha$.

### Fractional gradient elasticity model

The fractional gradient elasticity model is described by the equation

$c_{2}\Delta u({\bf r})-c_{\alpha}((-\Delta)^{\alpha/2}u)({\bf
r})+\frac{1}{\rho}\,f({\bf r})=0,\quad(\alpha>2).$ (51)

This may be derived from the lattice model with the fractional weak spatial
dispersion in the form (36) with $\alpha>2$. The order of the fractional
Laplacian $(-\Delta)^{\alpha/2}$ is greater than the order of the first term
related to the Hooke’s law. If $\alpha=4$ equation (51) become the equation
(39). Therefore the case $3<\alpha<5$ can be considered as close as possible
($\alpha\ \approx\ 4$) 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 $\alpha>2$.

The particular solution of equation (51) for the force that is applied at a
point (43) is the displacement field

$u({\bf r})=\frac{f_{0}}{2\pi^{2}\rho\,|{\bf
r}|}\,\int^{\infty}_{0}\frac{\lambda\,\sin(\lambda|{\bf
r}|)}{c_{2}\lambda^{2}+c_{\alpha}\lambda^{\alpha}}\,d\lambda\quad(\alpha>2).$
(52)

The asymptotic behavior $|{\bf r}|\to\infty$ for the fractional gradient
elasticity does not depend on the parameter $\alpha$. The asymptotic behavior
of the displacement field $u({\bf r})$ for $|{\bf r}|\to 0$ is given by

$u({\bf r})\ \approx\
\frac{f_{0}\,\Gamma((3-\alpha)/2)}{2^{\alpha}\,\pi^{2}\sqrt{\pi}\,\rho\,c_{\alpha}\,\Gamma(\alpha/2)}\,\cdot\,|{\bf
r}|^{\alpha-3},\quad(2<\alpha<3),$ (53) $u({\bf r})\ \approx\
\frac{f_{0}}{2\pi\,\alpha\,\rho\,c^{1-3/\alpha}_{2}\,c^{3/\alpha}_{\alpha}\,\sin(3\pi/\alpha)},\quad(\alpha>3).$
(54)

Note that the asymptotic behavior for $2<\alpha<3$ does not depend on $c_{2}$.
The displacement field $u({\bf r})$ of the short distance is determined only
by term with $(-\Delta)^{\alpha/2}$ ($\alpha>2$) 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 [37]) should demonstrate a universal behavior in
space by analogy with the universal behavior of low-loss dielectrics in time
[45, 46, 47, 48, 49]. Note that the universal behavior in time can be
connected with the interaction between the particles and the environment [50].

## References

  * [1]
  * [2] B. Ross, ”A brief history and exposition of the fundamental theory of fractional calculus”, Lecture Notes in Mathematics. Vol.457. (1975) 1-36.
  * [3] J.T. Machado, V. Kiryakova, F. Mainardi, ”Recent History of Fractional Calculus”, Communications in Nonlinear Science and Numerical Simulations. Vol.16. (2011) 1140-1153.
  * [4] S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Applications (Nauka i Tehnika, Minsk, 1987).
  * [5] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993).
  * [6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).
  * [7] A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics (Springer, New York, 1997).
  * [8] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
  * [9] R. Metzler, J. Klafter, ”The random walk’s guide to anomalous diffusion: a fractional dynamics approach” Physics Reports, Vol.339. (2000) 1-77.
  * [10] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (Eds.) Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007).
  * [11] A.C.J. Luo, V.S. Afraimovich (Eds.), Long-range Interaction, Stochasticity and Fractional Dynamics (Springer, Berlin, 2010).
  * [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010).
  * [13] J. Klafter, S.C. Lim, R. Metzler (Eds.), Fractional Dynamics. Recent Advances (World Scientific, Singapore, 2011).
  * [14] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011).
  * [15] 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.
  * [16] V.E. Tarasov, ”Review of some promising fractional physical models”, International Journal of Modern Physics B. Vol.27. No.9. (2013) 1330005. (32 pages)
  * [17] K.A. Lazopoulos, ”Non-local continuum mechanics and fractional calculus”, Mechanics Research Communications. Vol.33. No.6. (2006) 753-757.
  * [18] A. Carpinteri, P. Cornetti, A. Sapora, ”Static-kinematic fractional operators for fractal and non-local solids”, Zeitschrift für Angewandte Mathematik und Mechanik. Applied Mathematics and Mechanics. Vol.89. No.3. (2009) 207-217.
  * [19] A. Carpinteri, P. Cornetti, A. Sapora, ”A fractional calculus approach to nonlocal elasticity”, European Physical Journal Special Topics. Vol.193. (2011) 193-204.
  * [20] 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.
  * [21] 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.
  * [22] 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 Engineering. Vol.11. Chapter 33. (2009) 389-423.
  * [23] P. Cornetti, A. Carpinteri, A. Sapora, M. Di Paola, M. Zingales, ”An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus”, http://www.dipmat.univpm.it/aimeta2009/Atti
  * [24] I.A. Kunin, Media with Microstructure Vol. I and Vol.II. (Springer-Verlag, Berlin, New York, 1982 and 1983).
  * [25] 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)
  * [26] A.C. Eringen, B.S. Kim, ”Relation between non-local elasticity and lattive dynamics”, Crystal Lattice Defects. Vol.7. (1977) 51-57.
  * [27] M. Ostoja-Starzewski, ”Lattice models in micromechanics”, Applied Mechanics Reviews. Vol.55. No.1. (2002) 35-60.
  * [28] H. Askes, A. Metrikine, ”Higher-order continua derived from discrete media: continualisation aspects and boundary conditions”, International Journal of Solids and Structures. Vol.42. No.1. (2005) 187-202.
  * [29] 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.
  * [30] 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)
  * [31] V.E. Tarasov, ”Map of discrete system into continuous”, Journal of Mathematical Physics. Vol.47. No.9. (2006) 092901. (24 pages) (arXiv:0711.2612)
  * [32] 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)
  * [33] 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).
  * [34] V.E. Tarasov, G.M. Zaslavsky, ”Fractional dynamics of systems with long-range interaction”, Communications in Nonlinear Science and Numerical Simulation. Vol.11. No.8. (2006) 885-898. (arXiv:1107.5436)
  * [35] 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.
  * [36] L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Oxford, 1986).
  * [37] V.E. Tarasov, J.J. Trujillo, ”Fractional power-law spatial dispersion in electrodynamics”, Annals of Physics. Vol.334. (2013) 1-23.
  * [38] G.H. Hardy, ”Riemann’s form of Taylor series” Journal of the London Mathematical Society. Vol.20. (1945) 45-57.
  * [39] 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.
  * [40] M.M. Dzherbashyan, A.B. Nersesian, ”About application of some integro-differential operators”, Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences). Vol. 121. No.2. (1958) 210-213. in Russian.
  * [41] 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.
  * [42] Z.M. Odibat, N.T. Shawagfeh, ”Generalized Taylor s formula”, Applied Mathematics and Computation. Vol.186. No.1. (2007) 286-293.
  * [43] 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.
  * [44] H. Bateman, A. Erdelyi, Tables of integral transforms Volume 1. (New York, McGraw-Hill, 1954). or (Moscow, Nauka, 1969) in Russian.
  * [45] A.K. Jonscher, ”The universal dielectric response”, Nature. Vol.267. (1977) 673-679.
  * [46] A.K. Jonscher, Universal Relaxation Law (Chelsea Dielectrics, London, 1996).
  * [47] A.K. Jonscher, ”Low-loss dielectrics”, Journal of Materials Science. Vol.34. No.13. (1999) 3071-3082.
  * [48] V.E. Tarasov, ”Universal electromagnetic waves in dielectrics”, Journal of Physics: Condensed Matter. Vol.20. No.17. (2008) 175223. (arXiv:0907.2163)
  * [49] 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)
  * [50] V.E. Tarasov, ”The fractional oscillator as an open system”, Central European Journal of Physics. Vol.10. No.2. (2012) 382-389.