A Fractional Model of Continuum Mechanics

A Fractional Model of Continuum Mechanics
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  J Elast (2012) 107:105–123DOI 10.1007/s10659-011-9346-1 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. Poly-tech. 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 calcu-lus: 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 appli-cations in science and engineering. Late fundamental considerations have led to the intro-duction of fractional calculus in continuum mechanics in an attempt to develop non-localconstitutive 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 us-ing 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 non-local 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, USAe-mail: csd12@psu.eduS. Sivaloganathan (  )Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2J 3G1e-mail: 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 one-dimensional model problems: the deformation of an infinite barsubjected to a self-equilibrated 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], Li-ouville [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 long-range dependence of phenomena and thus result in methods which have bettercurve fitting, filtering, pattern recognition, edge detection, identification, stability, control-lability, 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 ma-terials, especially polymers [1–3, 23, 45, 51], and very recently, in a biological context for modelling brain tissue deformation [16, 53]. It is well-known that the classical equations of continuum mechanics are not well-suited to the modelling of many problems of funda-mental 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 multi-phase 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 non-local 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 non-localconstitutive 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 sci-ences. However, the dramatic increase in research publications in the field, makes it impos-sible 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 prac-tical applications, more recent fundamental considerations have led to the introduction of fractional calculus in constitutive laws in an attempt to develop non-local stress-strain rela-tions[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 allow-able displacements to continuous but not necessarily differentiable fields. Since fractionalderivatives are non-local, 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 applica-tions of our theoretical framework to two model problems: (1) the deformation of an infinitebar subjected to a self-equilibrated 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 one-to-one 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 pene-trates into another) [20]. It is important to notice that the axiom of continuity is not valid formulti-phase, 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 con-figuration.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 so-called  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  left-sided Riemann-Liouville 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  right-sided Riemann-Liouville 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  left-sided Riemann-Liouville fractional derivative of order   α . Similarly, wecan introduce the  right-sided Riemann-Liouville 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 left-sided fractional integrals and derivatives.Some of the proprieties of fractional integrals and derivatives that we will need further arelisted in the following proposition:
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