Modern Physics Letters B. Vol.28. No.7. (2014) 1450054

General Lattice Model of Gradient Elasticity

Vasily E. Tarasov

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

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

New lattice model for the gradient elasticity is suggested. This lattice model
gives a microstructural basis for second-order strain-gradient elasticity of
continuum that is described by the linear elastic constitutive relation with
the negative sign in front of the gradient. Moreover the suggested lattice
model allows us to have a unified description of gradient models with positive
and negative signs of the strain gradient terms. Possible generalizations of
this model for the high-order gradient elasticity and three-dimensional case
are also suggested.

##  1 Introduction

The two most widely used theories of elastic deformation in solid materials
are a microscopic approach based on the statistical mechanics of lattices [2,
3, 4] and the quantum theory of solid-states [5], and a macroscopic approach
based on the classical continuum mechanics [6, 7]. Continuum elasticity is a
phenomenological theory representing continuum limit of lattice dynamics,
where the length-scales are much larger than inter-atomic distances. Nonlocal
elasticity theory is based on the assumption that the forces between material
points can be at long-range in character, thus reflecting the long-range
character of interatomic and intermolecular forces. In general, the nonlocal
continuum models describe materials whose behavior at any point depends on the
states of all other points in the media, in addition to its own state and the
state of external fields. Such considerations are well-known in solid-state
physics, where the nonlocal interactions between the atoms and molecules are
prevalent in determining the properties of the media and materials.

The theory of nonlocal continuum mechanics was formally initiated by the
papers of Kroner [8] and Eringen, Edelen [9, 10]. Kroner [8] indicated the
relation between nonlocal elasticity theory of materials with long range
cohesive forces. Eringen and Edelen [10] provided derivation of the
constitutive equations for the nonlocal elasticity. Eringen and Kim [11]
described a relation between non-local elasticity and lattice dynamics. Kunin
described the physical aspects of nonlocal elasticity in the book [12], and
studied various problems in Fourier space. In the book [13] Eringen considered
a unified approach to field theories for elastic solids, viscous fluids, and
heat-conducting electromagnetic solids and fluids that include nonlocal
effects. Rogula [14] considered the mathematical aspects of nonlocal
elasticity models, proposed different types of nonlocal constitutive relations
between stress and strain, and applied it to various problems in continuum
mechanics. Non-local continuum mechanics has been treated with two different
approaches [14, 15]: the gradient elasticity theory (weak non-locality) and
the integral non-local theory (strong non-locality). In this paper we discuss
the gradient models of non-locality elasticity. Usually two classes of
gradient models are distinguished by the different signs of the strain
gradient terms in the constitutive relations for the strain $\varepsilon_{ij}$
and the stress $\sigma_{ij}$:

$\sigma_{ij}=\Bigl{(}\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}\Bigr{)}\pm
l^{2}\,\Delta\,\Bigl{(}\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}\Bigr{)},$
(1)

where $\lambda$ and $\mu$ are the Lame coefficients, $l$ is the scale
parameter. If $l^{2}=0$, we have the classical case of the linear elastic
constitutive relations for isotropic case that is the well-known Hooke’s law.

The first class of gradient elasticity models are described by equations (1)
with the positive sign in front of gradient. The main motivation to use this
form of the gradient elasticity is the description of dispersive wave
propagation through heterogeneous media. In many studies, gradient elasticity
models with the positive sign in (1) have been derived from associated lattice
models by the continualization procedure for the response of a lattice [17,
18, 19].

The second class of gradient elasticity models are described by equations (1)
with the negative sign in front of the gradient. The strain gradients in
equation (1) with the negative sign are equivalent to those derived from the
positive-definite deformation energy density, and therefore these models of
the strain gradients are stable.

The positive sign of the strain gradient term in equation (1) makes this term
destabilizing. The corresponding equation for the displacements is unstable
for wave numbers $k>1/l^{2}$ [16, 20, 18, 21]. In dynamics the instabilities
lead to an unbounded growth of the response in time without external work. It
is known the instabilities are related to loss of uniqueness in static
boundary value problems. Instabilities in statics and dynamics for the second-
gradient models with the positive sign are discussed in [22].

At this moment there is the opinion that gradient elasticity models with the
negative sign in equations (1) cannot be obtained from lattice models [15]. It
is usually assumed that this class of the second-gradient models does not have
a direct relationship with discrete microstructure and lattice models [16]. It
was proved that the homogenization (continualization) procedure, which is
considered in [17, 18, 19, 20, 23], uniquely leads to a second-order strain
gradient term that is preceded by a positive sign. The second-gradient model
with negative sign cannot be derived by this homogenization procedure. From a
mathematical point of view it is caused by properties of the Taylor series
that is used in this procedure.

In this article we propose lattice models, that allow us to derive linear
elastic constitutive relations with negative and positive signs. Moreover the
suggested lattice models give unified description of the gradient models with
positive and negative signs of the strain gradient terms. To obtain continuum
equation from the lattice equations we use an approach that is suggested in
[25, 26, 27, 28].

##  2 Equations of lattice model

Let us consider the vibration of an unbounded homogeneous lattice, such that
all particles are displaced from its equilibrium position in one direction,
and the displacement of particle is described by a scalar field. We consider
one-dimensional lattice system of interacting particles, where the equation of
motion of $n$th particle is

$M\frac{d^{2}u_{n}(t)}{dt^{2}}=g_{2}\,\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq
n\end{subarray}}^{+\infty}\;K_{2}(n,m)\;u_{m}(t)+g_{4}\,\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq n\end{subarray}}^{+\infty}\;K_{4}(n,m)\;u_{m}(t)+F(u_{n}(t)),$ (2)

where $u_{n}(t)$ are displacements from the equilibrium, $g_{2}$ and $g_{4}$
are coupling constants, $F(u_{n})$ is the external on-site force, $K_{2}(n,m)$
and $K_{4}(n,m)$ are the functions with different power-law asymptotic
behavior of the functions

$\hat{K}_{s}(k)=2\sum^{\infty}_{n=1}K_{s}(n,0)\cos(kn),\quad(s=2;4)$ (3)

for $k\to 0$. We will consider interactions terms for which the difference
$\hat{K}_{s}(k)-\hat{K}_{s}(0)$ are asymptotically equivalent to $|k|^{s}$ as
$|k|\to 0$. Note some general properties of $K_{s}(n,m)$, with $s=2;4$. The
conservation law of the total momentum in the lattice (2) in case of absence
of external forces $F(u_{n})=0$ gives

$\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\;K_{2}(n,m)=0,\qquad\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq n\end{subarray}}^{+\infty}\;K_{4}(n,m)=0,$ (4)

for all $n$. For homogeneous unbounded lattice, we have

$K_{2}(n,m)=K_{2}(n-m),\quad K_{4}(n,m)=K_{4}(n-m),$

where elements of $K_{s}(n,m)$ are constrained by the conditions (4), and

$\sum_{m}K_{s}(n-m)=\sum_{n}K_{s}(n-m)=0.$ (5)

For a simple case each particle is an inversion center and
$K_{s}(n-m)=K_{s}(|n-m|)$, where $s=2;4$. Using the condition (4), we can
rewrite equation (2) as

$M\frac{d^{2}u_{n}}{dt^{2}}=g_{2}\,\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\;K_{2}(|n-m|)\;\Bigl{(}u_{n}-u_{m}\Bigr{)}+g_{4}\,\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq
n\end{subarray}}^{+\infty}\;K_{4}(|n-m|)\;\Bigl{(}u_{n}-u_{m}\Bigr{)}+F(u_{n}).$
(6)

In this form of equation of motion the interaction terms are translation
invariant. It should be noted that the noninvariant terms lead to the
divergences in the continuum models [28].

Let us give an effective discrete mass-spring system for the suggested lattice
model (6). In Figure 1, we present the nearest-neighbor and next-nearest-
neighbor interactions only. In general, the functions $K_{2}(|n-m|)$ and
$K_{4}(|n-m|)$ describe long-range interactions with power-law asymptotic of
(3).

n-2n-1nn+1n+2$k^{eff}_{2}$$k^{eff}_{2}$$k^{eff}_{2}$$k^{eff}_{2}$$k^{eff}_{4}$$k^{eff}_{4}$$k^{eff}_{4}$$k^{eff}_{4}$$k^{eff}_{4}$MMMMM$a$$a$Figure
1: Discrete mass-spring system with effective stiffness coefficients
$k^{eff}_{2}=k^{eff}_{2}(g_{2},g_{4})$and
$k^{eff}_{4}=k^{eff}_{4}(g_{2},g_{4})$, the mass $M$ and the distance $a$ that
correspond to the lattice model with coupling constants $g_{2}$ and $g_{4}$.

##  3 From lattice model to continuum model

Let us consider a set of operations [25, 26, 28] that transforms the equations
of motion of the lattice model into a continuum equation for the displacement
field $u(x,t)$. We assume that $u_{n}(t)$ are Fourier coefficients of the
field $\hat{u}(k,t)$ on $[-K_{0}/2,K_{0}/2]$ that is described by the
equations

$\hat{u}(k,t)=\sum_{n=-\infty}^{+\infty}\;u_{n}(t)\;e^{-ikx_{n}}={\cal
F}_{\Delta}\\{u_{n}(t)\\},$ (7)
$u_{n}(t)=\frac{1}{K_{0}}\int_{-K_{0}/2}^{+K_{0}/2}dk\
\hat{u}(k,t)\;e^{ikx_{n}}={\cal F}^{-1}_{\Delta}\\{\hat{u}(k,t)\\},$ (8)

where $x_{n}=n\,a$ and $a=2\pi/K_{0}$ is distance between equilibrium
positions of the lattice particles. Equations (7) and (8) are the basis for
the Fourier transform ${\cal F}_{\Delta}$ and the inverse Fourier series
transform ${\cal F}^{-1}_{\Delta}$. The Fourier transform can be derived from
(7) and (8) in the limit as $a\to 0$ ($K_{0}\to\infty$). In this limit ($a\to
0$ or $K_{0}\to\infty$) the sum becomes the integral, and equations (7) and
(8) become

$\tilde{u}(k,t)=\int^{+\infty}_{-\infty}dx\ e^{-ikx}u(x,t)={\cal
F}\\{u(x,t)\\},$ (9) $u(x,t)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}dk\
e^{ikx}\tilde{u}(k,t)={\cal F}^{-1}\\{\tilde{u}(k,t)\\}.$ (10)

Here we use the lattice function

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

with continuous $u(x,t)$, where

$x_{n}=n\,a=\frac{2\pi n}{K_{0}}\to x.$

We assume that $\tilde{u}(k,t)={\cal L}\hat{u}(k,t)$, where ${\cal L}$ denotes
the passage to the limit $a\to 0$ ($K_{0}\to\infty$), i.e. the function
$\tilde{u}(k,t)$ can be derived from $\hat{u}(k,t)$ in the limit $a\to 0$.
Note that $\tilde{u}(k,t)$ is a Fourier transform of the field $u(x,t)$. The
function $\hat{u}(k,t)$ is a Fourier series transform of $u_{n}(t)$, where we
can use $u_{n}(t)=(2\pi/K_{0})\,u(n\,a,t)$.

We can state that a lattice model transforms into a continuum model by the
combination ${\cal F}^{-1}{\cal L}\ {\cal F}_{\Delta}$ of the following
operation.
The Fourier series transform:

${\cal F}_{\Delta}:\quad u_{n}(t)\to{\cal
F}_{\Delta}\\{u_{n}(t)\\}=\hat{u}(k,t).$ (11)

The passage to the limit $a\to 0$:

${\cal L}:\quad\hat{u}(k,t)\to{\cal L}\\{\hat{u}(k,t)\\}=\tilde{u}(k,t).$ (12)

The inverse Fourier transform:

${\cal F}^{-1}:\quad\tilde{u}(k,t)\to{\cal
F}^{-1}\\{\tilde{u}(k,t)\\}=u(x,t).$ (13)

These operations allow us to get a continuum model from the lattice model [25,
26, 28].

##  4 Lattice model with nearest-neighbor interaction

Let us derive the usual elastic equation from the lattice model with the
nearest-neighbor interaction with coupling constant $g_{2}=K$ by the method
suggested in [25, 26, 28]. We will use equations (2) with

$\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\;K_{2}(n,m)\;u_{m}(t)=u_{n+1}(t)-2u_{n}(t)+u_{n-1}(t),\quad
K_{4}(n,m)=0,$ (14)

where the term $K_{2}(n,m)$ describes the nearest-neighbor interaction.

We can give the following statement regarding the lattice model with the
nearest-neighbor interaction and the corresponding continuum equation that is
obtained in the limit $a\to 0$.

Proposition 1. In the continuous limit ($a\to 0$) the lattice equations of
motion

$M\frac{d^{2}u_{n}(t)}{dt^{2}}=K\cdot\Bigl{(}u_{n+1}(t)-2u_{n}(t)+u_{n-1}(t)\Bigr{)}+F(u_{n}(t))$
(15)

are transformed by the combination ${\cal F}^{-1}{\cal L}\ {\cal F}_{\Delta}$
of the operations (11-13) into the continuum equation:

$\frac{\partial^{2}u(x,t)}{\partial
t^{2}}=C^{2}_{e}\,\frac{\partial^{2}u(x,t)}{\partial
x^{2}}+\frac{1}{\rho}f(u),$ (16)

where $C^{2}_{e}=E/\rho={K\,a^{2}}/M$ is a finite parameter, and
$f(u)=F(u)/(A\,a)$.

Proof. To derive the equation for the field $\hat{u}(k,t)$, we multiply
equation (15) by $\exp(-ik\,n\,a)$, and summing over $n$ from $-\infty$ to
$+\infty$. Then

$M\,\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}\frac{d^{2}u_{n}}{dt^{2}}=K\cdot\sum^{+\infty}_{n=-\infty}\,e^{-ik\,n\,a}\,\Bigl{(}u_{n+1}-2u_{n}+u_{n-1}\Bigr{)}+\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}F(u_{n}).$
(17)

The first term on the right-hand side of (17) is

$K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-ik\,n\,a}K_{2}(n,m)u_{n}=K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-ik\,n\,a}\,\Bigl{(}u_{n+1}-2u_{n}+u_{n-1}\Bigr{)}=$
$=K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-iknd}u_{n+1}-2K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-iknd}u_{n}+K\cdot\sum^{+\infty}_{n=-\infty}\ e^{-ik\,n\,a}u_{n-1}=$
$=e^{ik\,a}\,K\cdot\sum^{+\infty}_{m=-\infty}\
e^{-ik\,m\,a}u_{m}-2K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-ik\,n\,a}u_{n}+e^{-ikd}K\cdot\sum^{+\infty}_{j=-\infty}\
e^{-ik\,j\,a}u_{j}.$

Using the definition of $\hat{u}(k,t)$, we obtain

$K\cdot\sum^{+\infty}_{n=-\infty}\
e^{-ik\,n\,a}K_{2}(n,m)u_{n}=K\cdot\Bigl{(}e^{ikd}\hat{u}(k,t)-2\hat{u}(k,t)+e^{-ik\,a}\hat{u}(k,t)\Bigr{)}=$
$=K\cdot\Bigl{(}e^{ik\,a}+e^{-ik\,a}-2\Bigr{)}\hat{u}(k,t)=2K\cdot\Bigl{(}\cos\left(k\,a\right)-1\Bigr{)}\,\hat{u}(k,t)=-4K\cdot\sin^{2}\left(\frac{k\,a}{2}\right)\hat{u}(k,t).$
(18)

Substitution of (18) into (17) gives

$M\frac{\partial^{2}\hat{u}(k,t)}{\partial
t^{2}}=-4K\cdot\sin^{2}\left(\frac{k\,a}{2}\right)\hat{u}(k,t)+\mathcal{F}_{\Delta}\\{F\left(u_{n}(t)\right)\\}.$
(19)

For $a\to 0$, the asymptotic behavior of the sine is
$\sin(k\,a/2)=k\,a/2+O((k\,a)^{3})$, then

$-4\,\sin^{2}\left(\frac{k\,a}{2}\right)=-\left(k\,a\right)^{2}+O((k\,a)^{4}).$

Using the finite parameter $C^{2}_{e}=K\,a^{2}/M$, the transition to the limit
$a\to 0$ in equation (19) gives

$\frac{\partial^{2}\tilde{u}(k,t)}{\partial
t^{2}}=-C^{2}_{e}k^{2}\tilde{u}(k,t)+\frac{1}{M}{\cal F}\\{F(u)\\},$ (20)

where

$\rho=\frac{M}{A\,a},\qquad E=\frac{K\,a}{A},\qquad
C^{2}_{e}=\frac{E}{\rho}=\frac{K\,a^{2}}{M}.$ (21)

The inverse Fourier transform ${\cal F}^{-1}$ of (20) has the form

$\frac{\partial^{2}{\cal F}^{-1}\\{\tilde{u}(k,t)\\}}{\partial
t^{2}}=-C^{2}_{e}{\cal F}^{-1}\\{k^{2}\tilde{u}(k,t)\\}+\frac{1}{\rho}f(u),$

where $f(u)=F(u)/(A\,a)$ is the force density. Then we can use ${\cal
F}^{-1}\\{\tilde{u}(k,t)\\}=\tilde{u}(x,t)$, and the connection between
derivatives and its Fourier transform: ${\cal
F}^{-1}\\{k^{2}\tilde{u}(k,t)\\}=\partial^{2}u(x,t)/\partial x^{2}$. As a
result, we obtain the continuum equation (16). This ends the proof.

As a result, we prove that the lattice equations (15) in the limit $a\to 0$
give the continuum equation with derivatives of second order only. This
conclusion agrees with the results of [24], where the relation

$\exp\,i\left(-i\,a\,\frac{\partial}{\partial x}\right)\,u(x,t)=u(x+a,t)$

and the representation of (15) by pseudo-differential equation are used.

##  5 From general lattice model to gradient elasticity model

Let us consider the lattice model that is described by (2), where the terms
$K_{s}(n,m)$ with $s=2$ and $s=4$ satisfy the conditions

$K_{s}(n,m)=K_{s}(|n-m|),\qquad\sum^{\infty}_{n=1}|K_{s}(n)|^{2}<\infty.$ (22)

To describe gradient elasticity models, we consider the inter-particle
interactions, that are described by $K_{s}(n)$ ($s=2$ or $s=4$) of the
following special type. We assume that the function

$\hat{K}_{s}(k)=\sum^{+\infty}_{\begin{subarray}{c}n=-\infty\\\
n\not=0\end{subarray}}e^{-ikn}K_{s}(n)=2\sum^{\infty}_{n=1}K_{s}(n)\cos(kn),$
(23)

satisfies the condition

$\lim_{k\rightarrow 0}\frac{\hat{K}_{s}(k)-\hat{K}_{s}(0)}{|k|^{s}}=A_{s},$
(24)

where $0<|A_{s}|<\infty$. Condition (24) means that

$\hat{K}_{s}(k)-\hat{K}_{s}(0)=A_{s}|k|^{s}+R_{s}(k),$ (25)

for $k\rightarrow 0$, where $\lim_{k\rightarrow 0}\ R_{s}(k)/|k|^{s}=0$. This
also means that, we can consider arbitrary functions $K_{s}(|n-m|)$ for which
$\hat{K}_{s}(k)-\hat{K}_{s}(0)$ are asymptotically equivalent to $|k|^{s}$ as
$|k|\to 0$.

As an example of the interaction terms $K_{s}(|n-m|)$, which give the
continuum equations of gradient elasticity models, we consider the function

$K_{s}(|n-m|)=\frac{(-1)^{|n-m|}}{2\,\Gamma(s/2+1+|n-m|)\,\Gamma(s/2+1-|n-m|)}.$
(26)

We use $2$ in the denominator to cancel with $2$ from equations (23). The
terms $K_{s}(|n-m|)$ are considered for $n\neq m$, i.e. $|n-m|\neq 0$. For
$s=2j$, we have $K_{s}(|n-m|)=0$ for all $|n-m|\geq j+1$. The function
$K_{s}(n-m)$ with even value of $s=2j$ describes an interaction of the
$n$-particle with $2\,j$ particles with numbers $n\pm 1$ . . . $n\pm j$. To
represent properties of (26), we can consider the function

$f_{K}(x,y)=\operatorname{Re}[K_{y}(x)]=\frac{\operatorname{Re}[(-1)^{|x|}]}{2\,\Gamma(y/2+1+|x|)\,\Gamma(y/2+1-|x|)}$
(27)

of two continuous variables $x$ and $y>0$. Note that
$\operatorname{Re}[(-1)^{|x|}]=(-1)^{|x|}$ for integer $x=n-m$. The plots of
the function (27) are presented by Figures 2 and 3 for different ranges of $x$
and $y$. This function decays rapidly with growth $x$ and $y$. The function
(27) defines the interaction terms $K_{s}(|n-m|)$ by the equation
$K_{s}(|n-m|)=f_{K}(|n-m|,s)$.

Using an inverse relation to (23) with $\hat{K}_{s}(k)=|k|^{s}$ that has the
form

$K_{s}(n)=\frac{1}{\pi}\int^{\pi}_{0}k^{s}\,\cos(n\,k)\,dk$

we get another example of $K_{s}(|n-m|)$ in the form

$K_{s}(|n-m|)=\frac{\pi^{s}}{s+1}\,_{1}F_{2}\left(\frac{s+1}{2};\frac{1}{2},\frac{s+3}{2};-\frac{\pi^{2}\,(n-m)^{2}}{4}\right),$
(28)

where $\,{}_{1}F_{2}$ is the Gauss hypergeometric function (see Chapter II in
[41]). Note that the interactions with (28) for $s=2$ and $s=4$ are long-range
interactions of $n$-particle with all other particles ($m\in\mathbb{N}$). It
is easy to see that expression (28) is more complicated than (26).

For $s=2$, we can also use the long-range interactions in the following two
forms

$K_{2}(|n-m|)=\frac{(-1)^{|n-m|}}{(n-m)^{2}},\quad
K_{2}(|n-m|)=\frac{1}{|n-m|^{\alpha}},\quad(\alpha\geq 3).$ (29)

A main advantage of the interaction in the forms (26) and (28) is a
possibility to use for other generalizations for the case of the high-order
gradient elasticity by using arbitrary integer values of $s$ and the
fractional generalization of gradient elasticity by non-integer values of $s$.

Proposition 2. The lattice equations

$M\frac{d^{2}u_{n}}{dt^{2}}=g_{2}\sum_{\begin{subarray}{c}m=-\infty\\\ m\neq
n\end{subarray}}^{+\infty}\,K_{2}(|n-m|)\;[u_{n}-u_{m}]+g_{4}\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq n\end{subarray}}^{+\infty}\,K_{4}(|n-m|)\;[u_{n}-u_{m}]+F(u_{n}),$ (30)

where $g_{2}$ and $g_{4}$ are coupling constants, $K_{2}(|n-m|)$ and
$K_{4}(|n-m|)$ are defined by (26), $u_{n}=u_{n}(t)$, are transformed by the
combination ${\cal F}^{-1}{\cal L}\ {\cal F}_{\Delta}$ of the operations
(11-13) into the continuum equation:

$\frac{\partial^{2}u(x,t)}{\partial
t^{2}}-G_{2}\frac{\partial^{2}u(x,t)}{\partial
x^{2}}+G_{4}\frac{\partial^{4}u(x,t)}{\partial
x^{4}}-\frac{1}{\rho}f\left(u(x,t)\right)=0,$ (31)

where

$G_{2}=\frac{g_{2}\,a^{2}}{4M},\qquad G_{4}=\frac{g_{4}a^{4}}{48M}$ (32)

are finite parameters, $\rho=M/(A\,a)$ is the mass density and
$f(u)=F(u)/(A\,a)$ is the force density.

Proof. To derive the equation for the field $\hat{u}(k,t)$, we multiply
equation (30) by $\exp(-ik\,n\,a)$, and summing over $n$ from $-\infty$ to
$+\infty$. Then

$M\,\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}\frac{d^{2}}{dt^{2}}u_{n}(t)=\sum^{+\infty}_{n=-\infty}\
\sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}\sum_{s=2;4}e^{-ik\,n\,a}g_{s}\,K_{s}(|n-m|)\
[u_{n}-u_{m}]+\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}F(u_{n}).$ (33)

The left-hand side of (33) gives

$\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}\frac{\partial^{2}u_{n}(t)}{\partial
t^{2}}=\frac{\partial^{2}}{\partial
t^{2}}\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}u_{n}(t)=\frac{\partial^{2}\hat{u}(k,t)}{\partial
t^{2}},$ (34)

where $\hat{u}(k,t)$ is defined by (7). The second term of the right-hand side
of equation (33) is $\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}F(u_{n})={\cal
F}_{\Delta}\\{F(u_{n})\\}$. The first term on the right-hand side of (33) is

$\sum^{+\infty}_{n=-\infty}\ \sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)\,[u_{n}-u_{m}]=$
$=\sum^{+\infty}_{n=-\infty}\ \sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)\,u_{n}-\sum^{+\infty}_{n=-\infty}\sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)\,u_{m}.$ (35)

Using (7) and (22), the first term in r.h.s. of (35) gives

$\sum^{+\infty}_{n=-\infty}\ \sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)\,u_{n}=\sum^{+\infty}_{n=-\infty}e^{-ik\,n\,a}u_{n}\sum^{+\infty}_{\begin{subarray}{c}m^{\prime}=-\infty\\\
m^{\prime}\not=0\end{subarray}}K_{s}(m^{\prime})=\hat{u}(k,t)\hat{K}_{s}(0),$
(36)

where

$\hat{K}_{s}(k\,a)=\sum^{+\infty}_{\begin{subarray}{c}n=-\infty\\\
n\not=0\end{subarray}}e^{-ik\,n\,a}K_{s}(n)={\cal F}_{\Delta}\\{K_{s}(n)\\}.$
(37)

The second term in r.h.s. of (35) gives

$\sum^{+\infty}_{n=-\infty}\ \sum^{+\infty}_{\begin{subarray}{c}m=-\infty\\\
m\not=n\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)u_{m}=\sum^{+\infty}_{\begin{subarray}{c}n=-\infty\\\
n\not=m\end{subarray}}e^{-ik\,n\,a}K_{s}(|n-m|)\
\sum^{+\infty}_{m=-\infty}u_{m}=$
$=\sum^{+\infty}_{\begin{subarray}{c}n^{\prime}=-\infty\\\
n^{\prime}\not=0\end{subarray}}e^{-ik\,n^{\prime}\,a}K_{s}(n^{\prime})\
\sum^{+\infty}_{m=-\infty}u_{m}e^{-ik\,m\,a}=\hat{K}_{s}(k\,a)\ \hat{u}(k,t).$
(38)

As a result, equation (33) has the form

$M\frac{\partial^{2}\hat{u}(k,t)}{\partial
t^{2}}=\sum_{s=2;4}\Bigl{(}\hat{K}_{s}(0)-\hat{K}_{s}(k\,a)\Bigr{)}\,\hat{u}(k,t)+{\cal
F}_{\Delta}\\{F(u_{n})\\},$ (39)

where ${\cal F}_{\Delta}\\{F(u_{n})\\}$ is an operator notation for the
Fourier series transform of $F(u_{n})$.

Using the series (see Sec.5.4.8.12 in [29]) of the form

$\sum^{\infty}_{n=1}\frac{(-1)^{n}}{\Gamma(\nu+1+n)\Gamma(\nu+1-n)}\cos(nk)=\frac{2^{2\nu-1}}{\Gamma(2\nu+1)}\sin^{2\nu}\left(\frac{k}{2}\right)-\frac{1}{2\Gamma^{2}(\nu+1)},$

where $\nu>-1/2$ and $0<k<2\pi$, we get for the function (23) of the form (26)
the equation

$\hat{K}_{s}(a\,k)-\hat{K}_{s}(0)=\frac{2^{s-1}}{\Gamma(s+1)}\sin^{s}\left(\frac{a\,k}{2}\right)=\frac{1}{2\Gamma(s+1)}\,|a\,k|^{s}+O(k^{s+2}).$
(40)

Here we use $\nu=s/2$ and $\sin(k/2)=k/2+O(k^{3})$. Note that $2$ in the
denominator of (26) cancels with $2$ from equation (23) in front of the sum
from zero to infinity. The limit $k\to 0$ gives

$\lim_{k\to
0}\frac{\hat{K}_{s}(k)-\hat{K}_{s}(0)}{|k|^{s}}=\frac{1}{2\Gamma(s+1)},$ (41)

and we have $A_{s}=1/(2\Gamma(s+1))$.

The Fourier series transform ${\cal F}_{\Delta}$ of (30) gives (39). We will
be interested in the limit $a\rightarrow 0$. Using (40), equation (39) can be
written as

$\frac{\partial^{2}}{\partial
t^{2}}\hat{u}(k,t)-\frac{g_{2}\,a^{2}}{M}\;\hat{\mathcal{T}}_{2,\Delta}(k)\;\hat{u}(k,t)-\frac{g_{4}a^{4}}{M}\;\hat{\mathcal{T}}_{4,\Delta}(k)\;\hat{u}(k,t)-\frac{1}{M}\mathcal{F}_{\Delta}\\{F\left(u_{n}(t)\right)\\}=0,$
(42)

where

$\hat{\mathcal{T}}_{s,\Delta}(k)=-\frac{1}{2\Gamma(s+1)}|k|^{s}+a^{2}\,O(|k|^{s+2}).$
(43)

In the limit $a\rightarrow 0$, using

$\hat{\mathcal{T}}_{s}(k)={\cal
L}\hat{\mathcal{T}}_{s,\Delta}(k)=-\frac{1}{2\Gamma(s+1)}|k|^{s}\quad(s=2;4),$
(44)

we get

$\hat{\mathcal{T}}_{2}(k)={\cal
L}\hat{\mathcal{T}}_{2,\Delta}(k)=-\frac{1}{4}|k|^{2},\quad\hat{\mathcal{T}}_{4}(k)={\cal
L}\hat{\mathcal{T}}_{4,\Delta}(k)=-\frac{1}{48}|k|^{4}.$ (45)

The passage to the limit $a\rightarrow 0$ for the third term of (42) gives
${\cal F}_{\Delta}F(u_{n})\rightarrow{\cal L}{\cal F}_{\Delta}F(u_{n})$. Then

${\cal L}{\cal F}_{\Delta}\\{F(u_{n})\\}={\cal F}\\{{\cal L}F(u_{n})\\}={\cal
F}\\{F({\cal L}u_{n})\\}={\cal F}\\{F(u(x,t))\\},$ (46)

where we use ${\cal L}{\cal F}_{\Delta}={\cal F}{\cal L}$.

As a result, equation (42) in the limit $a\rightarrow 0$ gives

$\frac{\partial^{2}}{\partial
t^{2}}\tilde{u}(k,t)-G_{2}\;\hat{\mathcal{T}}_{2}(k)\;\tilde{u}(k,t)-G_{4}\;\hat{\mathcal{T}}_{4}(k)\;\tilde{u}(k,t)-\frac{1}{M}\mathcal{F}\\{F\left(u(x,t)\right)\\}=0,$
(47)

where $\tilde{u}(k,t)={\cal L}\hat{u}(k,t)$, and we use finite parameters
$G_{2}$ and $G_{4}$, that are defined by (32).

The inverse Fourier transform of (47) is

$\frac{\partial^{2}}{\partial
t^{2}}u(x,t)-G_{2}\;\mathcal{T}_{2}(x)\;u(x,t)-G_{4}\;\mathcal{T}_{4}(x)\;u(x,t)-\frac{1}{\rho}f\left(u(x,t)\right)=0,$
(48)

where the finite parameters $G_{2}$ and $G_{4}$ are defined by (32). Using
(48) the operators $\mathcal{T}_{2}(x)$ and $\mathcal{T}_{4}(x)$ are defined
by

$\mathcal{T}_{2}(x)=\mathcal{F}^{-1}\\{\hat{\mathcal{T}}_{2}(k)\\}=+\frac{\partial^{2}}{\partial
x^{2}},\quad\mathcal{T}_{4}(x)=\mathcal{F}^{-1}\\{\hat{\mathcal{T}}_{4}(k)\\}=-\frac{\partial^{4}}{\partial
x^{4}}.$ (49)

Here, we have used the connection between the derivatives of the second and
fourth orders and their Fourier transforms
$k^{2}\longleftrightarrow-\partial^{2}/\partial x^{2}$ and
$k^{4}\longleftrightarrow+\partial^{4}/\partial x^{4}$, and equations (45) .

As a result, we obtain continuum equations (31). This ends the proof.

Proposition 2 illustrates the close relation between the discrete
microstructure and the gradient non-local continuum. Let us consider special
cases of the suggested model.

Lattice equations (30) have two parameters $g_{2}$ and $g_{4}$. The
corresponding equation (31) for the elastic continuum has two finite
parameters $G_{2}$ and $G_{4}$. If we use $g_{2}=4\,K$ and $g_{4}=0$, then
$G_{2}=C^{2}_{e}=K\,a^{2}/M$, $G_{4}=0$, and we get equation (16). If we
assume that $g_{2}=4\,K$ and $g_{4}=-4\,K$, then $G_{2}=C^{2}_{e}=K\,a^{2}/M$,
$G_{4}=C^{2}_{e}\,a^{2}/48$ and we get the equation

$\frac{\partial^{2}u(x,t)}{\partial
t^{2}}=C^{2}_{e}\,\frac{\partial^{2}u(x,t)}{\partial
x^{2}}+\frac{a^{2}\,C^{2}_{e}}{12}\,\frac{\partial^{4}u(x,t)}{\partial
x^{4}}+\frac{1}{\rho}f(u),$ (50)

where $C_{e}=\sqrt{E/\rho}$ is the elastic bar velocity. Equation (50) can
also be derived by the homogenization procedure [17, 18, 19].

In general, the coupling constants $g_{2}$ and $g_{4}$ are independent.
Therefore the coupling constant $g_{4}$ may differ from the constant
$g_{2}=4\,K$. If the relation of stress and displacement of the form
$\varepsilon(x,t)=\partial u(x,t)/\partial x$ is used, and the continuum
equation (31) is expresses as

$\rho\,\frac{\partial^{2}u(x,t)}{\partial
t^{2}}=\frac{\partial\sigma(x,t)}{\partial x}+f(u),$

where $\rho=M/(A\,a)$, then the constitutive relation can be represented by

$\sigma=E\,\left(\frac{g_{2}}{4\,K}\,\varepsilon-\frac{g_{4}\,a^{2}}{48K}\,\frac{\partial^{2}\varepsilon}{\partial
x^{2}}\right),$ (51)

where we use $E=K\,a/A$. Therefore, using the correspondence principle, we
will assume $g_{2}=4\,K$. The second-gradient parameter $l$ is defined by the
relation

$l^{2}=\frac{|g_{4}|\,a^{2}}{48\,K},$ (52)

where the sign in front of the factor $l^{2}$ in the constitutive relation is
determined by the sign of the coupling constant $g_{4}$. If the constant
$g_{4}$ is positive then we get the second-gradient model with negative sign.
As a result the second-gradient model with positive and negative signs

$\sigma=E\left(\varepsilon-\operatorname{sgn}(g_{4})\,l^{2}\,\frac{\partial^{2}\varepsilon}{\partial
x^{2}}\right)$ (53)

can be derived from a microstructure of lattice particles by suggested
approach. The proposed model as shown above uniquely leads to second-order
strain gradient terms that are preceded by the positive and negative signs. It
should be noted that positive value of coupling constant $g_{4}$ of lattice
model can lead to effective stiffness coefficient of the next-nearest-neighbor
interaction with non-convex elastic energy potentials in the effective
discrete mass-spring system. The strain gradients in continuum equation with
the negative sign are equivalent to those derived from the positive-definite
deformation energy density, and therefore these continuum models are stable.
The lattice models with negative value of coupling constant $g_{4}$ of lattice
model leads to the continuum equation with the positive sign in front of the
parameter $l^{2}$. This continuum equation is unstable for wave numbers
$k>1/l^{2}$ The instability leads to an unbounded growth of the response in
time without external work.

Figure 2: Plot of the function (27) for the range $x\in[0,4]$ and $y\in[0,6]$.

Figure 3: Plot of the function (27) for the range $x\in[2,4]$ and $y\in[1,3]$.

##  6 Possible extensions of general lattice model

The suggested lattice model can be generalized and extended for the high-order
gradient elasticity and for three-dimensional lattice models. Let us give some
details about these generalizations.

We can consider a generalization of the suggested lattice model by using the
sum of the functions (26) with the even value $s$. Using the functions (26)
with $s=6$ and other even values, we can consider the lattice models for high-
order gradient elasticity [16, 30, 23]. We can state that the lattice
equations

$M\frac{d^{2}u_{n}(t)}{dt^{2}}=\sum^{N}_{j=1}g_{2j}\sum_{\begin{subarray}{c}m=-\infty\\\
m\neq
n\end{subarray}}^{+\infty}\,K_{2j}(|n-m|)\;\Bigl{(}u_{n}(t)-u_{m}(t)\Bigr{)}+F(u_{n}),$
(54)

where $g_{2j}$ ($j=1,...,N$) are coupling constants, and $K_{2j}(|n-m|)$ are
defined by (26), are transformed by the combination ${\cal F}^{-1}{\cal L}\
{\cal F}_{\Delta}$ of the operations (11-13) into the continuum equation

$\frac{\partial^{2}u(x,t)}{\partial
t^{2}}+\sum^{N}_{j=1}(-1)^{j}\,G_{2j}\frac{\partial^{2j}u(x,t)}{\partial
x^{2j}}-\frac{1}{\rho}f(u)=0,$ (55)

where $G_{2j}=g_{2j}\,a^{2j}/(2\,\Gamma(2j+1))$, $(j=1,...,N)$ are finite
parameters. The proof of this statement is similar to proof of Proposition 2.

The suggested one-dimensional lattice model for second-gradient elasticity can
also be generalized for the three-dimensional case. We may consider a three-
dimensional lattice that is described by the equations

$\frac{d^{2}u^{k}_{\bf n}}{dt^{2}}=\sum_{\begin{subarray}{c}{\bf m}:{\bf
m}\neq{\bf n}\end{subarray}}\;K^{kl}_{2}({\bf n}-{\bf m})\;\left(u^{l}_{\bf
n}-u^{l}_{\bf m}\right)+\sum_{\begin{subarray}{c}{\bf m}:{\bf m}\neq{\bf
n}\end{subarray}}\;K^{kl}_{4}({\bf n}-{\bf m})\;\left(u^{l}_{\bf n}-u^{l}_{\bf
m}\right)+F^{k}(u_{\bf n}),$ (56)

where ${\bf n}=(n_{1},n_{2},n_{3})$, $k,l=1,2,3$ and we assume a sum over
repeated index $l=1,2,3$. In the model (56) the coupling constants are
included in the tensors $K^{kl}_{s}({\bf n}-{\bf m})=K^{kl}_{s}({\bf m}-{\bf
n})$ that are distinguished by different power-law asymptotic behavior. We
assume that the functions $\hat{K}^{kl}_{s}({\bf k})-\hat{K}^{kl}_{s}(0)$ are
asymptotically equivalent to $k_{i}\,k_{j}$ and $k_{i}\,k_{j}\,|{\bf k}|^{2}$
for $s=2$ and $s=4$ respectively, where

$\hat{K}^{kl}_{s}({\bf k})=\sum_{{\bf n}}\;e^{-i{\bf k}\,{\bf
n}}\;K^{kl}_{s}({\bf n}).$

To get continuum equation, we consider the field $u_{\bf n}(t)$ as Fourier
coefficients of the function $\hat{u}({\bf k},t)$, where ${\bf
k}=(k_{1},k_{2},k_{3})$, by

$\hat{u}^{k}({\bf k},t)=\sum_{{\bf n}}\;u^{k}_{\bf n}(t)\;e^{-i{\bf k}{\bf
r}_{\bf n}},$

where ${\bf r}({\bf n})={\bf r}_{\bf n}=\sum^{3}_{i=1}n_{i}\,{\bf a}_{i}$,
with the translational vectors ${\bf a}_{i}$ of the lattice. In three-
dimensional lattice model for second-gradient elasticity, we should consider
interaction terms $K^{kl}_{s}({\bf n}-{\bf m})$ that satisfy the conditions

$\lim_{k_{i},k_{j}\to 0}\frac{\hat{K}^{kl}_{2}({\bf
k})-\hat{K}^{kl}_{2}(0)}{k_{i}\,k_{j}}=A^{kl}_{ij}(2),\quad\lim_{k_{i},k_{j},|{\bf
k}|\to 0}\frac{\hat{K}^{kl}_{4}({\bf
k})-\hat{K}^{kl}_{4}(0)}{k_{i}\,k_{j}\,|{\bf
k}|^{2}}=A^{kl}_{ij}(4),\quad(i,j=1,2,3),$ (57)

where $A^{kl}_{ij}(s)$ are the coupling constants. In the continuous limit
($|{\bf a}_{i}|\to 0$), the three-dimensional lattice (56) gives the continuum
equations in the form

$\frac{\partial^{2}u^{k}({\bf r},t)}{\partial
t^{2}}-G^{kl}_{ij}(2)\frac{\partial^{2}u^{l}({\bf r},t)}{\partial
x_{i}\partial x_{j}}+G^{kl}_{ij}(4)\frac{\partial^{4}u^{l}({\bf
r},t)}{\partial x_{i}\partial x_{j}\partial x_{m}\,\partial
x_{m}}-\frac{1}{\rho}\,f^{k}(u({\bf r},t))=0,$ (58)

where we assume a sum over repeated indices $i,j,l,m\in\\{1,2,3\\}$, and

$G^{kl}_{ij}(2)=\frac{|{\bf a}_{i}|\,|{\bf a}_{j}|}{M}\,A^{kl}_{ij}(2),\quad
G^{kl}_{ij}(4)=\sum^{3}_{m=1}\frac{|{\bf a}_{i}|\,|{\bf a}_{j}|\,|{\bf
a}_{m}|^{2}}{M}\,A^{kl}_{ij}(4),$ (59)

where no summation over repeated indices. We can consider the case with
$G^{kl}_{ij}(4)=l^{2}\,G^{kl}_{ij}(2)$, where $G^{kl}_{ij}(2)=C_{ikjl}$ can be
considered as a stiffness tensor and $l^{2}$ is the scale parameter. For
isotropic case, we have
$C_{ijkl}=\lambda\,\delta_{ij}\delta_{kl}+\mu\,(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
In general equation (58) describes anisotropic gradient continuum. A more
detailed description of the three-dimensional lattice model (56) will be made
in the following article.

##  7 Conclusion

In this paper lattice models for strain-gradient elasticity of continuum are
suggested. The first advantage of the suggested lattice models is a
possibility to consider these models as a microstructural basis of unified
description of gradient models with positive and negative signs of the strain
gradient terms. A second advantage of the proposed model is that it can be
easily generalized to the case of the high-order gradient elasticity by using
the other even values of $s$. Using (26) with positive integer $s=2j$, we have
non-local interaction of the lattice particle, that gives the derivatives of
integer order $2j$ in the continuum equation. Three-dimensional lattice models
and the correspondent continuum equation can also be formulated as (56) and
(58). The third advantage of the proposed form of the interaction is that the
lattice equations can be used not only for the integer but also for fractional
values of the parameter $s$. Therefore the suggested general lattice model can
be extended on the fractional nonlocal case. The suggested types (26) and (28)
of inter-particle interactions in the lattice can be used for non-integer $s$.
If we consider interaction terms defined by (26) and (28) with non-integer
$s=\alpha$, then we will get continuum equations with the Riesz fractional
derivatives [42] of orders $s=\alpha$ by the methods suggested in [25, 26].
The lattice models with long-range interactions of the types (26) and (28)
with non-integer $s=\alpha$, can serve as microscopic models for elastic
continuum with power-law non-locality. It allows us to derive fractional
generalizations of gradient elasticity by using a microscopic approach [31,
32]. We also assume that the suggested lattice model can be generalized to get
discrete (lattice) models for dislocations in the gradient elasticity
continuum [33, 34, 35, 36, 37, 38, 39, 40], and then it will be possible to
extend them to the fractional non-local case. The suggested lattice models
with long-range interactions, which are suggested for the gradient elasticity
continuum, can be important to describe the non-local elasticity of materials
at micro and nano scales [43, 44, 45, 46], where the interatomic and
intermolecular interactions are prevalent in determining the properties of
these materials.

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