J Elast (2012) 107:105–123DOI 10.1007/s1065901193461
A Fractional Model of Continuum Mechanics
C.S. Drapaca
·
S. Sivaloganathan
Received: 8 January 2010 / Published online: 2 June 2011© Springer Science+Business Media B.V. 2011
Abstract
Although there has been renewed interest in the use of fractional models in manyapplication areas, in reality fractional analysis has a long and distinguished history and canbe traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J. Éc. Polytech. 13:71, 1832), and Riemann (Gesammelte Werke, p. 62, 1876). Recent publications
(Podlubny in Math. Sci. Eng. 198, 1999; Sabatier et al. in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin,2007; Das in Functional fractional calculus for system identiﬁcation and controls, Springer,Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering. Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop nonlocalconstitutive relations (Lazopoulos in Mech. Res. Commun. 33:753–757, 2006). Attemptshave also been made to model microscopic forces using fractional derivatives (Vazquez inNonlinear waves: classical and quantum aspects, pp. 129–133, 2004). Our approach in thispaperdiffersfromprevioustheoreticalwork,inthatwedevelopageneralframeworkdirectlyfrom the classical continuum mechanics, by deﬁning the laws of motion and the stresses using fractional derivatives. The timeliness and relevance of this work is justiﬁed by the surgein interest in applications of fractional order models to biological, physical and economicsystems. The aim of the present paper is to lay the foundations for a new nonlocal modelof continuum mechanics based on fractional order derivatives which we will refer to as thefractional model of continuum mechanics. Following the theoretical development, we apply
C.S. DrapacaDepartment of Engineering Science and Mechanics, The Pennsylvania State University,University Park, PA 16802, USAemail: csd12@psu.eduS. Sivaloganathan (
)Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2J 3G1email: ssivalog@sumathi.uwaterloo.caS. SivaloganathanCentre for Mathematical Medicine, Fields Institute for Mathematical Sciences, Toronto, ON,Canada M5T 3J1
106 C.S. Drapaca, S. Sivaloganathan
this framework to two onedimensional model problems: the deformation of an inﬁnite barsubjected to a selfequilibrated load distribution, and the propagation of longitudinal wavesin a thin ﬁnite bar.
Keywords
Continuum mechanics
·
Fractional calculus
·
Deformable media
Mathematics Subject Classiﬁcation (2000)
74
·
26A33
·
34A08
1 Introduction
Fractional calculus has a long history and has its srcins in the work of Leibniz [33], Liouville [34], and Riemann [44]; however, the past half century has seen a resurgence in
interest (predominantly in the engineering and physics literature) due to its demonstratedapplications in a wide spectrum of ﬁelds ranging from ﬂuid and solid mechanics, controltheory and dynamical systems to signal/image processing, economics and biomathematics[15, 24, 28, 36, 39, 41, 47, 48]. Indeed, it would be fair to say that recent advances in frac
tional calculus are, by and large, modern examples of its application to problems arisingin physics and engineering. For example, fractional derivative models used in signal andimage processing (as well as control theory, dynamical systems, and economics) accountfor the longrange dependence of phenomena and thus result in methods which have bettercurve ﬁtting, ﬁltering, pattern recognition, edge detection, identiﬁcation, stability, controllability, observability and robustness characteristics [15, 36, 47]. Similar models have been
shown to be useful in a rheological context for describing the viscoelastic behaviour of materials, especially polymers [1–3, 23, 45, 51], and very recently, in a biological context for
modelling brain tissue deformation [16, 53]. It is wellknown that the classical equations
of continuum mechanics are not wellsuited to the modelling of many problems of fundamental importance in solid mechanics such as those which involve the formation of cracks,phase transitions, or the presence of inclusions or mixtures. Classical continuum mechanicsis a
local
theory whose fundamental axioms exclude multiphase or mixture materials. Thereason why the classical mechanics framework does not work for these problems is that it isa local theory where the displacement ﬁelds are represented using partial derivatives whichare undeﬁned along discontinuities. In recent years nonlocal methods have been proposedto address the limitations of classical continuum mechanics [4–6, 21, 22, 31, 46, 49, 54].
More recently, it has been shown that fractional calculus can be related to new nonlocalconstitutive laws of elasticity [13, 14, 17, 18, 32, 52] as well as to the mechanics of fractal
media [9–11].
It is undeniable that the fractional calculus has emerged as a powerful new mathematicalmethod of solution in a variety of problems arising in the mathematical and physical sciences. However, the dramatic increase in research publications in the ﬁeld, makes it impossible to give a comprehensive review of its role in the modelling of discrete and continuoussystems. Oldham and Spanier [39] is, by all accounts, the ﬁrst book dedicated to fractionalcalculus and its applications in engineering and science. We refer the reader to [24] for moreextensive reviews of the fractional calculus and its applications. Apart from its use in practical applications, more recent fundamental considerations have led to the introduction of fractional calculus in constitutive laws in an attempt to develop nonlocal stressstrain relations[21, 22, 32].Ontheotherhand,attemptshavealsobeenmadetomodelthemicroscopic
forces using fractional derivatives [13, 17, 52]. Our approach in this paper differs from pre
vious theoretical work, in that we develop a general framework, deﬁning the laws of motion
A Fractional Model of Continuum Mechanics 107
and the stresses using fractional derivatives. This relaxes the constraint of differentiabilityimposed on the displacement ﬁelds by the classical theory, and extends the class of allowable displacements to continuous but not necessarily differentiable ﬁelds. Since fractionalderivatives are nonlocal, this new theory, that we call the
fractional model of continuummechanics
offers an elegant and uniﬁed way to study most problems involving continua,including those aforementioned. The model builds upon concepts from the classical theoryof continuum mechanics, the peridynamic model [49], and Carpentieri et al. work [9–11]
on the mechanics of fractal media. The timeliness and relevance of this work is justiﬁed bythe surge in interest in applications of fractional order models to biological, physical andeconomic systems.The paper is structured as follows. In Sect. 2 we brieﬂy review some theoretical resultsof classical continuum mechanics, followed by a section on fractional calculus (Sect. 3).In Sect. 4 we introduce our new fractional model of continuum mechanics and show theconnection between this and classical continuum mechanics. In Sect. 5 we present applications of our theoretical framework to two model problems: (1) the deformation of an inﬁnitebar subjected to a selfequilibrated load distribution and (2) the propagation of longitudinalwaves in a thin ﬁnite bar. The paper ends with a section of conclusions and proposed futurework.
2 Classical Continuum Mechanics
A
continuous medium
is a body which completely ﬁlls the space that it occupies, leaving nopores or empty spaces, so that its properties are describable by continuous functions [25].We assume that a continuous medium occupies at time
t
=
0 a region
which is asubset of
R
3
and, at time
t >
0, occupies a region
t
∈
R
3
, where
and
t
are assumed tobe bounded, open, and connected.
is called the
reference conﬁguration
of the body, and,for every
t >
0,
t
is called the
current conﬁguration
of the body. It is important to mentionhere that even if the reference conﬁguration is usually chosen to be an undeformed one, it isnot essential it be that way [7] (this allows, for example, the modelling of growth processesin materials with residual stresses; for more on this topic see the review article of [35] andreferences therein).Let P be an arbitrary particle of the body,
X
=
(X
K
)
K
=
1
,
3
be the position of the particleat
t
=
0 and
x
=
(x
k
)
k
=
1
,
3
be the position of the same particle P at
t >
0.
Deﬁnition 1
The
motion
of a body is determined by the position
x
of the material pointsin space as a function of the reference position
X
and the time
t
, and hence is a family of functions
χ
(
·
,t)
:
→
χ
(,t)
=
t
deﬁned for every
t >
0 by
x
=
χ
(
X
,t).
(2.1)
Deﬁnition 2
For a ﬁxed
t >
0, the function
χ
deﬁned by (2.1) is called a
deformation
of thereference conﬁguration
. The vector ﬁeld
U
(
X
,t)
=
χ
(
X
,t)
−
X
is called the
displacement vector
in the conﬁguration of reference.In the classical theory of continuum mechanics the following two assumptions hold:1. For every
t >
0,
χ
(
·
,t)
is a smooth onetoone map of every material point of
onto
t
.2. There exists a unique inverse of (2.1) at least locally if and only if the Jacobian of (2.1)
given by
J
=
(
∂x
k
∂X
K
)
k,K
=
1
,
3

is not identically zero.
108 C.S. Drapaca, S. Sivaloganathan
Together, they are known as the
axiom of continuity
, which expresses the
indestructibility of matter
(no region of positive, ﬁnite volume is deformed into one of zero or inﬁnite volume,i.e., 0
< J <
∞
) and also the
impenetrability of matter
(one portion of matter never penetrates into another) [20]. It is important to notice that the axiom of continuity is not valid formultiphase, fractured, or mixtures of materials (for more on this topic see [43]).
Deﬁnition 3
Given a deformation we deﬁne at each point of
its ﬁrst order spatial gradientas
F
=∇
X
χ
=
∂x
k
∂X
K
k,K
=
1
,
3
(2.2)which is called the
(local) deformation gradient
.We denote by
∇
X
·=
(
∂∂X
1
·
,
∂∂X
2
·
,
∂∂X
3
·
)
. Thus we have:
χ
(
X
+
U
,t)
=
χ
(
X
,t)
+
F
(
X
,t)
U
+
o(
U
),
as
U
→
0
,
(2.3)or, equivalently,
u
=
FU
(2.4)i.e.,
F
(
X
,t)
U
is the ﬁrst order linear correction to
χ
(
X
,t)
for the particles located at
X
+
U
close to
X
, where
u
=
χ
(
X
+
U
,t)
−
χ
(
X
,t)
is the displacement vector in the actual conﬁguration.It is easy to see that
J
=
det
F
, and, since 0
< J <
∞
, any deformation gradient
F
is aninvertible tensor with positive determinant
J
, i.e.,
F
∈
InvLin
+
(
R
3
)
. Also, since
J
measuresthe local change of volume we can say that a deformation conserves the volume of a materialregion or is
isochoric
if and only if
J
=
det
F
=
1
.
(2.5)The Eulerian forms of the global equations of motion of a deformed body occupyingdomain
t
and of mass density
ρ(
x
,t)
which is subjected to a contact force given by asurface traction
t
(
x
,t)
acting over the surface area of
∂
t
and a body force
b
(
x
,t)
actingover the volume of
t
are given by:
Principle of Linear Momentum
∂
t
t
da
+
t
b
dv
=
d dt
t
d
x
dt ρdv
⇔
t
∇
x
·
T
+
b
−
ρd
2
x
dt
2
dv
=
0
,
(2.6)where
T
is the socalled
Cauchy stress tensor
.
Principle of Angular Momentum
∂
t
x
×
t
da
+
t
x
×
b
dv
=
d dt
t
x
×
d
x
dt
ρdv.
(2.7)
A Fractional Model of Continuum Mechanics 109
The corresponding local forms of the equations of motion are respectively:
∇
x
·
T
+
b
−
ρd
2
x
dt
2
=
0
,
(2.8)following from (2.6), and respectively:
T
=
T
T
,
(2.9)(i.e., the Cauchy stress tensor is symmetric) following from (2.7) and (2.8). In the above
equations we denote by
∇
x
·=
(
∂∂x
1
·
,
∂∂x
2
·
,
∂∂x
3
·
)
.
3 Fractional Calculus Results
In this section we will present some fractional calculus results taken from [27, 38, 42, 48]
that will be used in our fractional model of continuum mechanics.
Deﬁnition 4
If
f
: [
a,b
] →
R
is a continuous function (i.e.,
f
∈
C(
[
a,b
]
)
),
α
∈
(
−∞
,
1
)
,and
Ŵ(s)
=
∞
0
e
−
t
t
s
−
1
dt
is the gamma function, then:1.
I
αa
+
f(t)
=
1
Ŵ(α)
t a
f(τ)(t
−
τ)
1
−
α
dτ,
(3.1)is called the
leftsided RiemannLiouville fractional integral of order
α
, and2.
I
αb
−
f(t)
=
1
Ŵ(α)
bt
f(τ)(τ
−
t)
1
−
α
dτ,
(3.2)is called the
rightsided RiemannLiouville fractional integral of order
α
, while3.
D
αa
+
f(t)
=
d
m
dt
m
I
m
−
αa
+
f(t),
if
α
∈
(m
−
1
,m), m
=
1
,
2
,
3
...,
d
m
dt
m
f(t),
if
α
=
m, m
=
1
,
2
,
3
...,I
−
αa
+
f(t),
if
α <
0
,
(3.3)is called the
leftsided RiemannLiouville fractional derivative of order
α
. Similarly, wecan introduce the
rightsided RiemannLiouville fractional derivative of order
α
.4.
a
+
R
αb
−
f(t)
=
1
Ŵ(α)
ba
f(τ)

t
−
τ

1
−
α
dτ
=
I
αa
+
f(t)
+
I
αb
−
f(t),
(3.4)is called the
Riesz integral
.For simplicity, we will focus only on the leftsided fractional integrals and derivatives.Some of the proprieties of fractional integrals and derivatives that we will need further arelisted in the following proposition: