FRACTIONAL RHEOLOGICAL MODELS FOR THERMOMECHANICAL SYSTEMS. DISSIPATION AND FREE ENERGIES. Mauro Fabrizio

Transcription

1 SURVEY PAPER FRACTIONAL RHEOLOGICAL MODELS FOR THERMOMECHANICAL SYSTEMS. DISSIPATION AND FREE ENERGIES Mauro Fabrizio Abstract Within the fractional derivative framework, we study thermomechanical models with memory and compare them with the classical Volterra theory. The fractional models involve significant differences in the type of kernels and predicts important changes in the behavior of fluids and solids. Moreover, an analysis of the thermodynamic restrictions provides compatibility conditions on the kernels and allows us to determine certain free energies, which in turn enables the definition of a topology on the history space. Finally, an analogous analysis is carried out for the phenomenon of heat propagation with memory. MSC 2 : Primary 26A33; Secondary 33E2, 34A8, 34K37, 35R, 6G22 Key Words and Phrases: fractional calculus, thermodynamics, partial differential equations, free energy. Introduction Materials whose constitutive equations can be described by a fractional derivative [5] are of increasing interest in recent years (see [3, 29]). It is well known that materials with such properties can be considered in the class of materials with memory and may describe elastic, fluid and electromagnetic materials, but also other kinds of phenomena, such as heat flux models. In c 24 Diogenes Co., Sofia pp , DOI:.2478/s Download Date 6/3/8 5:26 AM

2 FRACTIONAL RHEOLOGICAL MODELS this paper, we use the Caputo fractional derivative [8, 6] and limit ourselves to the study of thermomechanical and heat flux problems. The first issue, which will be addressed concerns a comparison of the classical Volterra description of materials with memory and the fractional derivative theory [2]. We will see that the two view points are formally similar. Indeed, fractional models are those for which the viscoelastic memory kernel (or relaxation function) K(s) is given by K(s) = k s α, α (, ),k >. (.) We however observe certain different properties for the fractional case. Compared to the Volterra theory for fluids with memory, this new definition of kernel (.) is not L (, ), which implies significant dissimilarity in the solution behavior. The differences are more evident for solid materials. Therefore, the fractional and Volterra models provide view points which are not reconcilable. It is appropriate to observe that, for solid viscoelastic materials, some experimental observations are particularly in agreement with models using fractional derivatives, because of the power law behavior of the stress relaxation function for viscoelastic materials as given by (.) (see [7] [9, 36] [26, 34, 3]). The creep function for such models also has power law form. Such experimental backing has motivated many studies of materials with fading memory given by a fractional derivative, including [6, 22, 4, 4, 7, 3, 23] and in the frequency domain [9, 36]. Many experimental observations on a variety of materials subject to a constant load show plastic behavior, which can be described by the fractional derivative approach. However, this is not predicted by Volterra models, which under constant load describe elastic materials. Moreover, when the load is removed, fractional models predict recovery of a portion of the deformation, unlike for the case of classical viscous fluids. Thus, the fractional derivative approach allows us to describe materials displaying both elastic and viscous/plastic behavior. In the last part of the work, we apply the fractional theory to heat diffusion. It seems natural to generalize the Fourier law and the Cattaneo- Maxwell equation, using a fractional derivative instead of the time derivative. This approach allows us to describe a wider range of phenomena and gives a good description of frequency behavior [35, 2, 28, ]. Finally, we study the dissipation and thermodynamic features, in relation to fractional models. Such an analysis allows us to define free energies associated with these systems, from which it is possible to obtain the natural topology for materials described by fractional methods. Download Date 6/3/8 5:26 AM

3 28 M. Fabrizio 2. Materials with memory In this section, we study the differences and similarities between the classical Volterra view point and the fractional derivative approach to describing materials with memory. 2.. Volterra s theory Consider a solid viscoelastic material, linked with reference domain R 3. Its constitutive equation, within the Volterra theory, is given by σ(x, t) =G (x)ε(x, t)+ G (x, s)ε(x, t s)ds, t [,T), (2.) where σ and ε denote the stress and strain tensor respectively. The fourth order tensors G and G characterize the nature of the material. The quantity G ( ) must be positive. In [2], it is shown that the restriction imposed by the laws of thermodynamics takes the form G (x, s)sinωs ds <, ω R ++. (2.2) For a solid material with fading memory, we have G (,s) L (, ), and G(x, s) = s G (x, τ) dτ + G (x). The limit lim G(x, s) =G (x) t yields a finite, positive tensor. From (2.) we obtain the equivalent forms σ(x, t) =G (x)ε(x, t)+ and σ(x, t) =G (x)ε(x, t)+ G (x, s)(ε(x, t s) ε(x, t))ds, t [,T) (2.3) (G(x, s) G (x)) ε(x, t s) ds, t [,T), (2.4) where ε(x, τ) = ε(x, τ). The set of past histories ε t (s) =ε(t s), s (, ), is defined using a free energy functional related to the constitutive equation (2.3) or (2.4). There are in general many choices of such free energies. In this study, we consider only the Graffi free energy, given for a solid by [ G ε(t) ε(t)+ 2 ρψ S (ε(t),ε t ( )) = (2.5) ] G (s)ε(x, t s) ε(x, t)) ε(x, t s) ε(x, t) ds. Download Date 6/3/8 5:26 AM

4 FRACTIONAL RHEOLOGICAL MODELS The set of histories ε t =(ε(t),ε t ( )) admissible for the system, is defined by the space K S (ε t )= { ε t :[, ) Sym(V ); ψ S (ε(t),ε t ( )) < }. (2.6) In the Volterra theory, a viscoelastic fluid is defined by the constitutive equation (see [3]) σ(x, t) = p(x, t)i +2 μ (x, s)(ε(x, t s) ε(x, t))ds, t [,T), where p denotes the pressure, I is the identity tensor and the scalar μ is the viscosity kernel, such that μ(x, s) = s μ(x, τ) dτ L (, ). For such fluids, the restrictions which follow from thermodynamic laws reduce to the condition μ(s)cosωs ds >, ω R. The Graffi-Volterra free energy for a fluid is given by the functional ρψ F (ε(t),ε t ( )) = μ (x, s)(ε(x, t s) ε(x, t)) 2 ds. (2.7) Hence, we define the history space for fluids by the set K F (ε t )= { ε t :[, ) Sym(V ); ψ F (ε(t),ε t ( )) < }. (2.8) 2.2. Fractional derivative view point In many papers, the study of solid viscoelastic materials was explored by means of the well-known Caputo α fractional derivative [5], defined for any α (, ) by C a D α t f(t) = t a f (τ) dτ, (2.9) (t τ) α where a [,t)andγ( ) is the gamma function given for any β>by Γ(β) = r β e r dr. In the classical papers using fractional derivatives, as [22], the constitutive equation of viscoelasticity is defined by σ(x, t) = C(x) t a ε(x, τ) dτ, (2.) (t τ) α where C(x) is a fourth order tensor. We suppose a =, since if necessary it is always possible to extend ε to the interval [,a) by the null tensor. Download Date 6/3/8 5:26 AM

5 2 M. Fabrizio Thus, (2.) can be written as σ(x, t) = or, by a change of variable, αc(x) σ(x, t) = αc(x) t ε(x, t) ε(x, τ) (t τ) +α dτ, (2.) ε(x, t s) ε(x, t) (s) +α ds (2.2) which are equivalent representations of the Caputo derivative. Thus, the definition (2.) may be rewritten using the new notation or C D α t ε(x, t) = C α Dα t ε(x, t) = α t ε(x, t) ε(x, τ) (t τ) +α dτ, (2.3) ε(x, t s) ε(x, t) (s) +α ds. (2.4) These definitions involve the integral of a pseudo difference-quotient of order ( + α). The constitutive equations (2.) or (2.2) allow us to define the domain of definition of these functionals by a fractional Sobolev space, now called a Gagliardo space [6], defined for any x, W α, (, ) (2.5) { } = ε(t) L ε(t) ε(τ) (, ), (t τ) +α L ((,t) (, )), with norm given by ε W α, (2.6) (, ) ( α t ) ε(x, t) ε(x, τ) = ε(t) dt + (t τ) +α dτ dt. In this framework, the constitutive equation of an incompressible viscoelastic fluid is entirely analogous to (2.), (2.) or (2.2), with the formal difference that instead of the tensor C(x), we now have a scalar constant η, which is related to the viscosity of the fluid. So, the constitutive functional is given by the stress T(x, t) = p(x, t)i + σ E (x, t), where p denotes the pressure and σ E the extra-stress defined by σ E (x, t) = 2η t ε(x, τ) dτ (2.7) (t τ) α Download Date 6/3/8 5:26 AM

6 FRACTIONAL RHEOLOGICAL MODELS... 2 η t v(x, τ)+( v(x, τ)) T = (t τ) α dτ, where v is the fluid velocity, or σ E (x, t) = 2αη t ε(x, t) ε(x, τ) (t τ) +α dτ. (2.8) Remark 2.. However, an apparent difference between solids and fluids is located in the coefficient α (, ). When α is close to, the model well represents a viscoelastic solid. When α iscloseto, wehavea visco-elastic fluid. Remark 2.2. Another important feature of a solid is the existence of only one null strain ε (or reference configuration) such that the space of histories is a subset of { } GS t = ε t ( ) :[, ) Sym(V ); ε t L (, ); lim ε t (s) =ε s where, from (2.) or (2.), if ε t (s) =ε we have σ(ε t (s)) =. For a fluid, the set of histories belongs to G t F = { ε t ( ) :[, ) Sym(V ); ε t L (, ) }. The main difference between the Volterra and fractional derivative models, is evident in the study of solid materials, when we examine the stress behavior for t. Indeed, in the context of the Volterra theory, if the system is subject to a constant strain ε, then stress will tend to G ε. This can be proved, if we consider a constant process ε (x) fort>t,so that we obtain the following limit lim σ(x, t) = (2.9) t G (x)ε (x)+ lim G (x, s)(ε(x, t s) ε (x))ds = G (x)ε (x), t>t. t t t On the other hand, in the fractional theory we find that the stress will go to. Indeed, αc(x) lim t t ε (x) ε(x, τ) (t τ) +α dτ =, t > t. In other words, in this case, the material undergoes a kind of plastic deformation [9]. Download Date 6/3/8 5:26 AM

7 22 M. Fabrizio Remark 2.3. It is worthwhile to observe that, by memory effects, Volterra sought to describe the properties related to dislocation phenomena. However, it is known that the model proposed in (2.) or (2.3) is not capable of describing plastic effects produced by dislocations. 3. Thermodynamics We need to study the compatibility of fractional derivative models with thermodynamics. Only isothermal processes will be considered, so that the Second Law of Thermodynamics reduces to the Dissipation Principle ρ(x) Ψ(x, t) σ(x, t) ε(x, t), (3.) where Ψ denotes a free energy and ρ is the density. We have from (3.) that on any cyclic process of period T = 2π ω, (ω R++ ), T σ(x, t) ε(x, t) dt. (3.2) In particular, for periodic strain processes of the form ε(x, t) =ε (x)cosωt + ε 2 (x)sinωt, (3.3) it follows from (3.2) (see [2], [3]) that for all ε,ε 2 Sym(V ), ( ) ε sin ωs ds C (s) +α ε + ε 2 C (s) +α ε 2 + ε C CT (s) +α ε 2 cos ωs ds, for all ω R +. (3.4) The second term can vary arbitrarily in sign and magnitude for different choices of ε and ε 2, so that it can be concluded that the tensor C is symmetric. Also, in the first term, the quantity εcε must have a definite signature for the inequality to be obeyed in a simple manner. We choose εcε so that the condition (3.4) becomes (s) +α sin ωs ds for all ω R+. (3.5) Remark 3.. For a fluid defined in (2.8), we obtain from the Second Law the same inequality (3.5). Alternatively, if (2.7) is used, we have the equivalent condition cos ωs ds for all ω R. (s) α Download Date 6/3/8 5:26 AM

8 FRACTIONAL RHEOLOGICAL MODELS Free energies In this section, we consider the free energies in the fractional theory. Let us first consider a solid, described by the equation (2.2). The free energy is a functional Ψ S which satisfies the inequality (3.) ρ(x) Ψ S (x, t) σ(x, t) ε(x, t). (4.) For simplicity, let us take the C(x) to be a scalar quantity. The internal power can be written as σ(x, t) ε(x, t) = αc(x) ε(x, t s) ε(x, t) (s) +α ds ε(x, t) = αc(x) ε(x, t s) ε(x, t) (s) +α d (ε(x, t s) ε(x, t))ds (4.2) dt + αc(x) ε(x, t s) ε(x, t) (s) +α d (ε(x, t s) ε(x, t))ds. ds Let us define a free energy by the functional Ψ S αc(x) (ε(x, t s) ε(x, t)) 2 (x, t) = 2ρ(x) (s) +α ds. (4.3) Comparing with (2.5), we see that it is the Graffi free energy for fractional derivative models. The inequality (4.) is satisfied because by (4.2) we obtain ρ(x) Ψ S (x, t) =σ(x, t) ε(x, t)+ αc(x) d ds (ε(x, t s) ε(x, t))2 2 (s) +α ds = σ(x, t) ε(x, t) αc(x)( + α) 2 (ε(x, t s) ε(x, t)) 2 (s) 2+α ds. (4.4) Hence ρ(x) Ψ S (x, t) =σ(x, t) ε(x, t) D(x, t), (4.5) where D(x, t) denotes the rate of dissipation αc(x)( + α) (ε(x, t s) ε(x, t)) 2 D(x, t) = 2 (s) 2+α ds. The set of histories HS t available with this model is defined by HS t = { ε t :[, ) Sym(V ); Ψ S (ε(t),ε t ( )) < }. For a viscoelastic fluid, we use the constitutive equation (2.8), which is more convenient than (2.7). The internal power is given by σ E (x, t) ε(x, t) = αη t ε(x, t) ε(x, τ) (t τ) +α dτ ε(x, t) Download Date 6/3/8 5:26 AM

9 24 M. Fabrizio = αη t ε(x, t) ε(x, τ) (t τ) +α d (ε(x, t) ε(x, τ))dτ (4.6) dt = αη d 2( dt +(+α) t t (ε(x, t) ε(x, τ)) 2 (t τ) +α dτ (ε(x, t) ε(x, τ)) 2 (t τ) 2+α dτ ). Let us assume that the free energy is given by the functional Ψ F (x, t) = αη t (ε(x, t) ε(x, τ)) 2 2ρ(x) (t τ) +α dτ, (4.7) or using the variable s = t τ, in the equivalent form Ψ F (ε t (x)) = αη (ε(x, t) ε(x, t s)) 2 2ρ(x) (s) +α ds. (4.8) This is the Graffi free energy for fluids, which of course has a similar form to (4.3) for solids. We may define the set of histories HF t available with this model by H t F = { ε t :[, ) Sym(V ); Ψ F (ε(t),ε t ( )) < }. The rate of dissipation D(x, t) is given by or D(x, t) = D(x, t) = αη( + α) t (ε(x, t) ε(x, τ)) 2 2ρ(x) (t τ) 2+α dτ, (4.9) αη( + α) (ε(x, t) ε(x, t s)) 2 2ρ(x) (s) 2+α ds. Remark 4.. Hence for fluids, we see by comparing (2.7) and (4.9), that there is a similarity between the Volterra and fractional expressions. Indeed, the two free energies coincide if the kernel of (2.7) is defined by μ αη( + α) (x, s) =. (4.) 2s+α Observe however that the kernel (4.) is not an element of L (, ) L 2 (, ). Download Date 6/3/8 5:26 AM

10 FRACTIONAL RHEOLOGICAL MODELS Differential systems 5.. Viscoelastic solid Let R 3 be a smooth bounded domain of a linear viscoelastic solid, whose constitutive equation is given by the fractional model (2.2). The initial boundary value problem is defined by the differential system in the domain Q = (,T) = ρ (x) 2 u(x, t) 2 (5.) α C(x) ε(x, t s) ε(x, t) (s) +α ds + ρ (x)f(x, t), where ρ (x) denotes the density, u(x, t) the displacement such that ε = 2 ( u + u T )andf(x, t) the body forces. The initial conditions are u(x, t) u(x, ) = u (x), = v (x), (5.2) t= along with the boundary conditions u(x, t) = u (x). (5.3) Using the definition of fractional derivative given in (2.3), equation (5.) can be rewritten in the form ρ (x) 2 u(x, t) 2 = (C Dt α C(x)ε(x, t)) + ρ (x)f(x, t). (5.4) Now, our purpose is to obtain an energy theorem for the problem (5.)- (5.3). To this end, we multiply (5.4) by u(x,t). Then, after an integration on Q = (,T), we obtain = T ( T ρ (x) 2 u(x, t) 2 (C D α t C(x)ε(x, t)) u(x, t) u(x, t) dx dt (5.5) + ρ (x)f(x, t) ) u(x, t) dx dt. Hence, using the divergence theorem and the boundary condition (5.3), it follows from (5.5) that T ( ) u(x, t) 2 ρ (x) dx dt (5.6) 2 T ( = C(x) ( C Dt α ε(x, t) ) ) ε(x, t) u(x, t) + ρ (x)f(x, t) dx dt. Download Date 6/3/8 5:26 AM

11 26 M. Fabrizio Then, from (4.4) we obtain T ( ( u(x, ) t) 2 ρ (x) +Ψ S (x, t)) dx dt 2 T u(x, t) ρ (x)f(x, t) dx dt. (5.7) Finally, carrying out the time integration, we find that ( ( u(x, ) t) 2 ρ (x) +Ψ S (x, t)) dx (5.8) 2 ρ (x) ( v (x) 2 +Ψ S (x, ) ) T u(x, t) dx + ρ (x)f(x, t) dx dt Viscoelastic fluids The initial boundary value problem for a viscoelastic incompressible fluid described by the velocity v(x, t), the pressure p(x, t) and the constant density ρ, is defined by the differential system ρ v(x, t) = p + η t v(x, τ) (t τ) α dτ + ρ f(x, t) (5.9) v(x, t) = (5.) with initial and boundary conditions v(x, ) = v (x), v(x, t) =. (5.) We again seek an energy theorem. It follows from (5.9) and (5.) that T d ρ v 2 (x, t)dxdt dt T ( η t ) = v(x, τ) (t τ) α dτ v(x, t)+ρ f(x, t) v(x, t) dx dt. By virtue of (4.6) and (5.) T d dt ρ v 2 (x, t)dxdt Download Date 6/3/8 5:26 AM

12 FRACTIONAL RHEOLOGICAL MODELS = T ( d dt +( + α) t t (ε(x, t) ε(x, τ)) 2 (t τ) +α dτ (ε(x, t) ε(x, τ)) 2 (t τ) 2+α dτ T + ρ f(x, t) v(x, t) dx dt which yields, using the notation of (4.7), = T T d dt ( d t ρ v 2 (x, t)+ dt ( ρ v 2 (x, t)+ψ F (ε t (x)) ) T dxdt Hence, we have (ε(x, t) ε(x, τ)) 2 (t τ) +α dτ ) dx dt ) dx dt (5.2) ρ f(x, t) v(x, t)dxdt. ( ρ v 2 (x, t)+ψ F (ε t (x)) ) dx (5.3) ( ρ v 2 (x, ) + Ψ F (ε (x)) ) T dx + 6. Heat equation ρ f(x, t) v(x, t) dx dt. In the previous papers [35,, 5], the fractional method was used to describe heat conduction, motivated by experimental findings such as those reported in [2, 28]. Heat conductivity in a fractional model provides a convenient approximation of the diffusive interface and gives a suitable estimate of the thermal conductivity. We introduce the First Law of Thermodynamics for a rigid heat conductor ė(x, t) = q + r, (6.) where e is the internal energy, q the heat flux and r the heat supply. The classical Fourier constitutive equation is given by q(x, t) = k(θ(x, t)) θ(x, t), (6.2) where q is the heat flux, θ the heat gradient and k(θ) > thethermal conductivity. It is well known, that aiming at obtaining a finite wave speed propagation, Cattaneo [] proposed the modified constitutive relation γ q(x, t) =q(x, t)+k(θ(x, t)) θ(x, t), γ >, (6.3) now called the Cattaneo equation. Download Date 6/3/8 5:26 AM

13 28 M. Fabrizio 6.. Fractional Cattaneo equation We can generalize (6.3) by replacing the time derivative left-hand side by a fractional derivative. This yields γ C Dα t q(x, t) =q(x, t)+k(θ(x, t)) θ(x, t), (6.4) where α> is the order of the derivative. Then the constitutive equation assumes the form γα t q(x, t) q(x, τ) (t τ) +α dτ = q(x, t)+k(θ(x, t)) θ(x, t). (6.5) For this fractional problem, given the similarity with fluid models, the case where 2 <α< appears especially convenient (see [], [2]). We need to study the restriction resulting from the Second Law of Thermodynamics for a heat rigid conductor, which is given by the inequality η(x, t) ė(x, t) θ(x, t) + q(x, t) θ(x, t), (6.6) θ2 where the internal energy e and the entropy η are state functions. The state σ depends on the particular choice of constitutive equations. Introducing the free energy ψ = e θη, we obtain the following approximation of (6.6) ψ ė + η θ θ q θ. (6.7) In this framework, we consider the constitutive equation (6.4) or (6.5), from which we have, for smooth fields, q θ = γ C Dα t q(x, t) q k(θ(x, t)) θ(x, t) q, (6.8) where γd C t α q(x, t) q = γα t q(x, t) q(x, τ) (t τ) +α dτ q(x, t) (6.9) ( γα t t ) = (t τ) +α dτ q(x, τ) q2 (x, t) dτ q(x, t). (t τ) +α Therefore, γα t q(x, τ) dτ q(x, t) (6.) (t τ) +α ( γα( + α) d t q 2 (x, t, τ) t q 2 ) (x, t, τ) = dτ (2 + α) dτ, 2 dt (t τ) 2+α (t τ) 3+α where q(x, τ) = t τ q(x, s)ds. The free energy is assumed to be given by the sum of two terms ψ(θ, q(x, ) =ψ (θ)+ψ ( q(x, ), Download Date 6/3/8 5:26 AM

14 where FRACTIONAL RHEOLOGICAL MODELS ψ ( q(x, ) = γα( + α) t q 2 (x, t, τ) dτ, 2 (t τ) 2+α while the internal energy e and the entropy η are functions only of θ(x, t), e(x, t) =e(θ(x, t)), Then, from (6.7), we have ψ (θ) θ = e(θ) θ Finally, by virtue of (6.2), (6.9) and (6.), η(x, t) =η(θ(x, t)). + η(θ), (6.) ψ θ q θ. (6.2) θ q(x, t) θ(x, t) = (6.3) = θ γ C D α t q(x, t) q(x, t) k θ (θ(x, t)) θ(x, t) θ(x, t) = d dt ψ γα ( q(x, )) ( + Therefore, where ( + α)(2 + α) 2 t t (t τ) +α dτ q2 (x, t) q 2 (x, t, τ) (t τ) 3+α dτ) k θ (θ(x, t)) θ(x, t) θ(x, t). ψ ( q(x, )) = θ q(x, t) θ(x, t) D( q(x, )), D( q(x, )) = ( + α)(2 + α) t + 2 γα t (t τ) +α dτ q2 (x, t) (6.4) q 2 (x, t, τ) (t τ) 3+α dτ + k θ (θ(x, t)) θ(x, t) θ(x, t) denotes the rate of dissipation. The differential system related to equations (6.) and (6.4) is given by e θ θ(x, t) = q + r, γα t q(x, t) q(x, τ) (t τ) +α dτ = q(x, t)+k(θ(x, t)) θ(x, t). Download Date 6/3/8 5:26 AM

15 22 M. Fabrizio 6.2. Fractional Fourier equation Another model which extends equation (6.2) is given as follows: q(x, t) = C Dα t θ(x, t) = k(x) = k(x)α t t θ(x, τ) (t τ) α dτ θ(x, t) θ(x, τ) (t τ) +α dτ. (6.5) Under the same assumptions as the for previous case in relation to the internal energy e and entropy η, we study the inequality (6.2) in the context of (6.5). Consider the internal product q(x, t) θ(x, t) = k(x)α θ = k(x)α ( t from which ( = k(x)α d dt t (t τ) +α dτ θ2 (x, t) k(x)α t t θ(x, t) θ(x, τ) (t τ) +α dτ θ(x, t) (6.6) t ) θ(x, τ) dτ θ(x, t) (t τ) +α θ(x, τ) dτ θ(x, t) (6.7) (t τ) +α θ 2 (x, t, τ) dτ (2 + α) (t τ) 2+α t θ 2 (x, t, τ) dτ (t τ) 3+α where θ(x, t, τ) = t τ θ(x, s)ds. Finally, if we put ˆψ(θ, θ) = ˆψ (θ) + ˆψ ( θ), it follows from (6.6) and (6.7), d dt ˆψ ( θ(x, t)) q(x, t) θ(x, t)+d( θ(x, t)), (6.8) θ where the free energy ˆψ ( θ) is defined by ˆψ ( θ) = k(x)α while the dissipation D( θ) is D( θ(x, t)) = γα ( t t θ 2 (x, t, τ) dτ, (t τ) 2+α θ 2 ) (x, t) (+α)(2+α) t θ 2 (x, t, τ) dτ dτ. (t τ) +α 2 (t τ) 3+α Finally, we consider the differential system associated to the constitutive equation (6.6). From (6.) and the assumption that e depends only on θ(x, t), we obtain the heat equation k(x)α e θ θ(x, t) = t θ(x, t) θ(x, τ) (t τ) +α dτ + r(x, t). (6.9) ), Download Date 6/3/8 5:26 AM

16 FRACTIONAL RHEOLOGICAL MODELS Acknowledgements This paper was inspired by a talk with M. Caputo and S. Rionero during a research workshop held in Vietri in April 23. Moreover, I would like to express my gratitude to F. Mainardi for his useful explanations and suggestions. References [] O.P. Agrawal, Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dynamics 38 (24), [2] G. Amendola, M. Fabrizio and M. Golden, Thermodynamics of Materials with Memory. Theory and Applications. Springer, New York (22); [3]G.Amendola,M.Fabrizio,J.M.Golden,B.Lazzari,Freeenergiesand asymptotic behaviour for incompressible viscoelastic fluids. Appl. Anal. 88 (29), [4] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. of Rheology 27, No 3 (983), 2 2. [5] M. Caputo, Linear model of dissipation whose Q is almost frequency independent-ii. Geophys. J. R. Astr. Soc. 3 (967), ; Reprinted in: Fract. Calc. Appl. Anal., No (28), 3 4; fcaa/. [6] M. Caputo, Vibrations of a thin viscoelastic layer with a dissipative memory. J. Acoustical Soc. of America 56 (974), [7] M. Caputo, Mean fractional-order-derivatives differential equations and filters. Annali dell Universita di Ferrara 4 (995), [8] M. Caputo, The Green function of the diffusion of fluids in porous media with memory. Atti Acc. Naz. Lincei, Rend. Lincei Mat. Appl. 7 (996), [9] M. Caputo, A model for the fatigue in elastic materials with frequency independent Q. J. Acoustical Soc. of America 66 (979), [] C. Cattaneo, Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3 (948), 83. [] M. Fabrizio, G. Gentili, J.M. Golden, The minimum free energy for a class of compressible viscoelastic fluids. Advances in Differential Equations 7 (22), [2] M. Fabrizio, B. Lazzari, R. Nibbi, Thermodynamics of non-local materials: Extra fluxes and internal powers, Continuum Mech. Thermodyn. 23 (2), [3] M. Fabrizio, A. Morro, Dissipativity and irreversibility of electromagnetic systems. Math.Mod.Meth.Appl.Sc. (2), Download Date 6/3/8 5:26 AM

17 222 M. Fabrizio [4] C. Friedrich. Mechanical stress relaxation in polymers: Fractional integral model versus fractional differential model. J. Non-Newtonian Fluid Mech. 46 (993), [5] J.D. Gabano and T. Poinot, Fractional modelling applied to heat conductivity and diffusivity estimation. Phys. Scr. T 36 (29), # 45 (6pp), doi:.88/3-8949/29/t36/45/. [6] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7 (958), [7] N. Heymans, J.C. Bauwens, Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33 (994), [8] N. Heymans, Constitutive equations for polymer viscoelasticity derived from hierarchical models in cases offailure of time-temperature superposition. Signal Process. 83 (23), [9] S. Holm, S.P. Näsholm, A causal and fractional all-frequency wave equation for lossy media. J. of the Acoustical Soc. of America 3 (2), [2] V.V. Kulish, J.L. Lage, Fractional-diffusion solutions for transient temperature and heat transfer. ASME J. Heat Transfer 22 (2), [2] F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 5, No 4 (22), 72 77; DOI:.2478/s ; [22] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College, London (2). [23] H.R. Metzler, A. Blumen, T.F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with solutions. J. Phys. A 28 (995), [24] H.R. Metzler, W. Schick, H.G. Kilian, and T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 3 (995), [25] H.R. Metzler, T.F. Nonnenmacher, Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plast. 9 (23), [26] P.G. Nutting, A new general law deformation. J. Franklin Inst. 9 (92), [27] G. Pagnini, A. Mura, F. Mainardi, Two-particle anomalous diffusion: Probability density functions and self-similar stochastic processes. Phil.Trans.R.Soc.A37, No 99 (23), # 2254; doi:.98/rsta Download Date 6/3/8 5:26 AM

18 FRACTIONAL RHEOLOGICAL MODELS [28] I. Petras, B.M. Vinagre, L. Dorcak, V. Feliu, Fractional digital control of a heat solid: Experimental results. In: International Carpathian Control - Conference, Malenovice, 22. [29] I. Podlubny, Fractional Differential Equations. Academic Press, New York (998). [3] Z. Jiao, Y.Q. Chen, I. Podlubny, Distributed Order Dynamic Systems. Springer, New York - London - Heidelberg (22). [3] P.E. Rouse and K. Sittel, Viscoelastic properties of dilute polymer solutions. J. App. Phys. 24 (953), [32] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science, Amsterdam (993). [33] H. Schiessel, A. Blumen, Mesoscopic pictures of the sol-gel transition: Ladder models and fractal network. Macromolecules 28 (994), [34] G. Scott-Blair, An application of the theory of quasi-properties to the treatment of anomalous stress-strain relation. Phys. Mag. 4 (949), [35] D. Sierociuk, A. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, T. Skovranek, Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. A Math. Phys. Eng. Sci. 37, No 99 (23), # 2246; doi:.98/rsta [36] P.D. Spanos and G.I. Evangelatos, Response of a non-linear system with restoring forces governed by fractional derivatives-time domain simulation and statistical linearization solution. Soil.Dyn.Earthquake Eng. 3 (99), Department of Mathematics, University of Bologna Via Piazza Porta S. Donato 5, I 427 Bologna, ITALY mauro.fabrizio@unibo.it Received: September 6, 23 Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 7, No (24), pp ; DOI:.2478/s Download Date 6/3/8 5:26 AM