1. Introduction
Differential equations of fractional orders appear in many branches of physics and mechanics. There are numerous solutions to concrete problems collected in the books [
1,
2,
3,
4,
5,
6], for example. Fractional derivatives (FDs) are non-local operators, and, in most applications in mechanics, fractional derivatives, when the independent coordinate is time, are used to model the dissipation and/or memory in the system. If FDs are used to describe memory, physically they represent the (fading) memory of the system. However, when spatial coordinates are used as independent variables, the fractional derivatives model non-local action. As is well known in nano-materials, the non-local action (see [
7]) is an important phenomenon that is used to explain many properties that are characteristic of such materials. In solid mechanics in general, and non-local and nano-mechanics in particular, there are two types of constitutive equations that are used: strain- and stress-driven constitutive equations. In the strain-driven form of constitutive equation, that we use in this work, the stress is determined by action of a non-local operator on strain. In this work we shall formulate relevant equations for the spatially one-dimensional body with a
linear constitutive equation in a specially selected deformation measure (strain) that is non-local. The use of a general fractional derivative of a displacement field
in the form of the so-called
truncated power-law kernel (see [
8]) is the novelty that we propose here. After we formulate the problem in distributional setting, we shall study some special motions, with the emphasis on wave propagation with or without body forces.
2. Mathematical Model
Consider a rod with straight axis. Let
x be a coordinate coinciding with the rod axis. Suppose that the rod occupies a part of the space for which
, with
We state the definitions of the left and right general fractional derivative (GFD) of the Caputo type (see [
9,
10]). Throughout, we assume
The left GFD derivative of the Caputo type of order
is defined as
where
is Euler’s gamma function and
y is assumed to be absolutely continuous, i.e.,
In writing (
1) we used specific kernel
suggested for application in continuum mechanics; see [
11], p. 6. The kernel
is called the truncated power-law kernel and was used earlier for the study of anomalous diffusion in [
8] (Equation (2.35)), [
12] (Equation (
13)) and in [
13] (Equation (
10)) as a friction kernel for the study of lipid motion in a lipid bilayer system. Similarly, the right GFD derivative of the Caputo type of order
with
is defined as
Note that
,
We need the definitions of Riesz-type GFD. Using (
1) and (
2), we define
Equation (
3) is written as
Before we define a new deformation measure, we state the classical strain tensor of linear elasticity (see [
14]) that for the three-dimensional body reads
In (
5), we denoted by
,
components of the displacement vector with respect to a prescribed Cartesian coordinate system with axes
Also, we use
t to denote time. Since here we consider one-dimensional bodies only, (
5) becomes
Here
is the only non-zero component of displacement vector. Also,
is the axis coinciding with the rod axis. Instead of (
6), we shall use the following deformation measure
In (
7) we used
defined by (
4). Also, the derivatives are taken with respect to
We assume that the displacement field is absolutely continuous
in the variable
x (absolutely continuous, i.e.,
. In this case, (
1) and (
2) exist and
with
,
exists too. The deformation measure, or strain,
is non-local. Actually
takes values of the classical strain
at all points of the body, with the weighting function equal to
.
Remark 1. For the case of three-dimensional elasticity theory, one can use the recently defined multidimensional generalized Riesz derivative of the Luchko type, presented in [15], to define generalization of the type (7) for the strain tensor (5). Next we propose the constitutive equation of the rod. We assume that it is given in strain-driven form, so that with (
7) the linear stress-deformation measure relation becomes
where
is the stress and
is a constant (generalized modulus of elasticity). We consider several special cases of (
8).
is the strain measure used in [
16];
, since ;
is the Riesz type of derivative for the Caputo–Fabrizio fractional derivative
if we take
and add the constant
in front [
17,
18].
Suppose that we assume that the displacement field is given as
, with
being arbitrary. This represents the rigid body motion along the axis of the body. Then
Therefore, we conclude that the rigid body motion of the rod, i.e.,
, with
being arbitrary, leads to zero strain. This shows that
can be used as a strain measure. Now we analyze the general problem of motions that result in strain given by (
7), that is equal to zero. To do this, we examine the solution to the equation
in general. We consider
The next Lemma summarizes the result.
Lemma 1. For the only solution to (9) for , , is Proof. Let
be a function defined as
Then
where ∗ denotes the convolution. Equation (
9) now becomes
Since (
10) is an equation represented by convolution of two elements from the space of tempered distribution, one of which is with compact support, we analyze (
10) in the space of tempered distribution
Let
be a regular distribution defined by
is with the compact support. Distribution
defined by
is in
Consequently, the convolution
exists. We now apply the Fourier transform to (
10). Firstly, we have, see [
19], p. 346,
where
Next, by using (
11) in (
10) we obtain
Since
is an entire function in
from (
13) and the uniqueness theorem for the Fourier transform we conclude that
if
or
a.e. Thus,
□
We combine (
4) and (
11) to obtain
where
is given by (
12).
Equation of motion is
where
f denotes the prescribed body forces and
denotes density. We take body force in the form of friction force, proportional to the Caputo fractional derivative of displacement
with respect to time, i.e.,
with
The form of the body force is taken to be proportional to the Caputo fractional derivative of displacement in order to be able to model viscous force
and purely elastic resistance force
Then, the equation of motion becomes
To (
17), we prescribe the following initial condition
Remark 2. Note thatwhereis modified Riesz potential (cf. [20]). It reduces to the classical Riesz potential for 3. Notations
In this Section, we present definitions that we shall use in the sequel. The Fourier transform of a tempered distribution
is the tempered distribution
that we denote by
It is defined by
where
The inverse Fourier transform is
where
is a homeomorphism of
onto
Operations
and
are the inverse of each other; see [
21,
22,
23,
24].
For the case where
f has the classical Fourier transform
We denote by
the regular distribution defined by function
Then
and
In the case where
then
a.e. See [
23].
Suppose that
and consider
defined as
Then, as
converges in
to a function
and
converges in
to
The functions
f and
F are connected by the formulae
for almost all values of
x; see [
25], p. 69. Then,
is defined as the Fourier transform of
Fourier transform is a linear isometry of
onto
and
see [
23], p. 148 and [
22], p. 216.
Let
and suppose that for some
,
for
sufficiently large. Then
where
; see [
23], p. 159 and 202.
From Fubini’s theorem, it is easily seen that f holds the following additional condition: belongs to then
The GFD derivative of the Caputo type of order
, of a function
f,
,
and for
is defined by (
1) and (
2). The definitions of the order
GFD of the Caputo type (
1) and (
2) can be extended on
as follows. For
f,
we have (see (
14)),
where
From the definition, it follows that
does not exist for every
For the case where
we have
Consequently, using Definition 1 we extended the operator onto and for we determine that is a regular distribution.
The Laplace transform of
can be defined as
where
(cf. [
22,
24]).
4. Solutions to the Cauchy Problem (17), (18)
To write the relevant system of equations in the distributional form, we note that the system of equations in dimensionless form, describing motion of one-dimensional continuum, consists of the equation of motion, the constitutive equation, and the strain definition (geometrical equation) defined for
, and reads
where
,
,
, and
denote the stress, displacement, and strain at the point
x and time
respectively. The initial and boundary conditions associated with (
22) are
Since we will consider the above equation with initial data over
(and
), we put for the displacement
where
H is Heviside distribution, and we consider
u as a distribution. This implies
where
, so that the distributional form to (
22) becomes
with
,
and where we used
and
.
The equation in
which corresponds to (
17) and (
18) is
where
,
,
and
Also,
and
denote the partial derivatives in the sense of distributions. We first sought the solutions to (
24) that are regular distributions
. Our main result is the following theorem.
Theorem 1. Let . Suppose that and have Fourier transforms and , respectively, such that , and , , are regular distributions or measures. Then the distributions u given by:
Proof. First we look for solutions
that are regular tempered distributions defined by the function
,
,
By applying the Fourier and Laplace transforms to (
24) (cf.
Section 3), we obtain
where
,
Consequently,
The inverse Laplace transform of (
30), with (
31), is given in [
26], Equation (
38), and reads
which proves (
25). To obtain other forms of the solution, we consider the following special cases:
Let
Then, we have
with
where
Now the solution corresponding to
becomes
Since
and
exist for every tempered distribution, the distribution
u given by (
32) is a solution to (
24) if and only if
This follows from Definition 1 and the fact that the functions
and
are bound on
Let
,
In this case, we have
The inverse Laplace transform of (
33) is
Note that when
, we have in (
35)
So, theorem is proved. □
Remark 3. Suppose that In this case, we have We have this case treated in [16]. The inverse Laplace transform giveswhere and ; see [27], p. 171. From (36), it follows that , could only be a regular distribution or a measure (distribution of order zero) because the products in the both two addends of (36) have to exist. Finally,which is the result presented in [16]. 5. Numerical Examples
(A) As a first specific example, we take generalization of the problem treated in [
16]. Let
and
Then
so that (
32) becomes
where
Since
is even, we have from (
38)
In the
Figure 1 and
Figure 2 we show solution given by (
39) for the same set of parameters except for values of
It is seen that increase in
leads to a decrease in the amplitude of the propagating wave. However, an increase in
leads to the increase in speed of propagation of the maximum of the wave. This is a rather unexpected effect of
.
(B) As a second example we take
,
, for
Since
the Equation (
32) becomes
or
Equation (
41) shows that in this case we have the classical wave equation with the speed of propagation
B.
(C) Next, we take
,
,
,
and
,
. By solving (
25), we obtain the result shown in
Figure 3.
(D) Finally we present the solution to (
35). We take
,
,
, for
,
,
Since
the Equation (
35) becomes
From the
Figure 3 and
Figure 4, we conclude that, for the case where other parameters have fixed values, the order of fractional derivative that models external dissipation
has small influence on the waves at the beginning of motion. It decreases the amplitude of waves for larger times.