Abstract
Materials with constitutive equations expressed in terms of fractional derivatives [47] are of increasing interest in recent years (see [214, 287]). It is well known that such materials can be considered in the class of materials with memory and may describe elastic, fluid, viscoelastic, and electromagnetic materials, but also other kinds of phenomena, such as heat flux models.
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1 Introduction
Materials with constitutive equations expressed in terms of fractional derivatives [47] are of increasing interest in recent years (see [214, 287]). It is well known that such materials can be considered in the class of materials with memory and may describe elastic, fluid, viscoelastic, and electromagnetic materials, but also other kinds of phenomena, such as heat flux models.
The fractional derivative central to the present study is that of Caputo [48, 51]. We consider thermomechanical models with memory within this fractional derivative framework, and compare them with the classical Volterra theory, which is that described and used throughout most of the present work. It emerges that the two viewpoints are formally similar [250]. Indeed, fractional models are those for which the viscoelastic memory kernel (or relaxation function) G(s) is given by
However, in contrast to the Volterra theory for fluids with memory, this kernel is not L 1(0, ∞), which implies significant dissimilarity in the behavior of solutions of the dynamical equations, compared with the traditional theory. The differences are more evident for solid materials. Therefore, the fractional and Volterra models provide viewpoints which are not reconcilable. Various of these differences between the two models are noted in [128].
An analysis of the thermodynamic restrictions provides compatibility conditions on the kernels. These conditions, combined with analogies with the Volterra theory, yield certain free energies, which enable the definition of a topology on the history space. A similar analysis can be carried out for the phenomenon of heat propagation with memory.
The derivation of the minimum free energy in this context is presented in particular detail because it requires careful treatment of the factorization problem.
For solid viscoelastic materials, some experimental observations are particularly in agreement with models using fractional derivatives, because of the power law behavior of the relaxation function given by (23.1.1) (see [50, 167, 202, 280, 293, 296, 303]). The creep function for such models also has a power law form. Such experimental backing has motivated many studies of materials with fading memory given by a fractional derivative, including [26, 48, 140, 200, 214, 249, 261] and in the frequency domain [202, 303] .
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.
The fractional theory can also be applied 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 oscillating behavior [1, 223, 285, 299].
In recent decades, the fractional calculus has been widely used, as indicated by the many mathematical volumes dealing with this topic (e.g., Baleanu et al. [28], Caponetto [53], Caputo [46], Diethelm [95], Hilfer [198], Jiao et al. [214], Kilbas et al. [219], Kyriakova [222], Mainardi [248], McBride [259], Miller and Ross [264], Petras [286], Samko et al. [295], Podlubny [287], Sabatier et al. [294], Torres and Malinowska [309], Ying and Chen [327]), by the many meetings dedicated to it and the plethora of articles appearing in mathematical (e.g., Kilbas and Marzan [218], Heinsalu et al. [196], Luchko and Gorenflo [246]) and non-mathematical journals.
The use of derivative of fractional order has also spread into many other fields of science besides mathematics and physics (e.g., Laskin [229], Naber [275], Baleanu et al. [29], Zavada [328], Baleanu et al. [30], Caputo and Fabrizio [55]) such as biology (e.g., Cesarone et al. [62], Caputo and Cametti [52]), economics (e.g., Caputo [54]), demography (e.g., Jumarie [217]), geophysics (e.g., Iaffaldano [207]) and medicine (e.g., El Sahed [99], Magin [247]). However, its somewhat cumbersome mathematical definition and the consequent complications in the solution of fractional order differential equations have led to some difficulties.
2 Fractional Derivatives
In this section, the original Caputo fractional derivative is introduced, together with certain new fractional derivative formulae without singularities in the kernel.
2.1 The Caputo Fractional Derivative
We begin with an argument which provides motivation for the formula, introduced below [47].
Let f(z) be analytic on an open subset O of the complex plane, where O includes the real axis. Then, an immediate consequence of Cauchy’s Integral Formula, (B.2) is
where f (n)(z) is the nth derivative of f(z), while f ′(z) = f (1)(z). The contour C ⊂ O includes the point z. Let us replace n by \(\alpha \in {\mathbb R}^+\). We take z to be on the real axis, denoted by t, and the contour to be tightly around the branch cut joining the branch point t to infinity. This step forces the exclusion of integrals with α ∈ (1, ∞). The cut is assumed to lie along the semi-infinite interval (−∞, t). The integration variable is changed to τ. We have therefore
where θ is the argument of the denominator, which varies as we move around the contour, anti-clockwise. Below the cut, we have θ = −πα, while above the cut, it is θ = πα. The quantity Γ(⋅) is the Gamma function, given for any β > 0, by
Thus,
Using the standard formula
relation (23.2.1) becomes
The analyticity property assumed for f can be weakened but must ensure that the derivative and integral in (23.2.4) exist.
The general form of the Caputo α fractional derivative, defined for any \(\alpha \in \left ( 0,1\right )\), is given by
where \(a\in \left (-\infty ,t\right )\), f ∈ H 1(a, b), where b > t. We can take a = −∞, since if necessary it is always possible to extend f to the interval \(\left ( -\infty ,a\right ) \) by the zero function. Thus, we have
which agrees with the form (23.2.4). We assume that f(t) → 0, as t tends to −∞. Now, (23.2.6) can be written as
or, by a change of variable,
which are equivalent representations of the Caputo derivative. Thus, the definition (23.2.5) may be rewritten as
Remark 23.2.1
As α tends to zero, we see from (23.2.6) that
If α tends to 1, the integral in (23.2.5) diverges. However, there is a limit which gives a finite value (as must be true, given the derivation of (23.2.4)). Before the limit α → 1 is taken, let us reverse the shrinking of the contour described above, so that the integral exists for all nonnegative powers of the denominator. In particular, as α → 1, we have
2.2 Fractional Derivatives Without Singular Kernels
We present a new definition of fractional derivative with a smooth kernel which takes on two different representations for the temporal and spatial variables, respectively. For the first, operating on time variables, it is natural to use the Laplace transform. For the other representation, related to the spatial variables by a nonlocal fractional derivative, it is more convenient to use the Fourier transform. The interest in this new approach with a regular kernel in terms of spatial variables arose from the perception that there is a class of nonlocal models, which have the ability to describe material heterogeneities and fluctuations on different scales, which cannot be described adequately by classical local theories or by fractional models with singular kernels.
The original definition of fractional derivative appears to be particularly convenient for mechanical and electromagnetic phenomena involving plasticity, fatigue, damage, and electromagnetic hysteresis. When these effects are not present it seems more appropriate to use the new temporal fractional derivative to describe, for example, the behavior of classical viscoelastic materials, thermal media, and electromagnetic systems.
The new nonlocal fractional derivative for spatial variables can describe material heterogeneities and structures with different scales, which cannot be described adequately by classical local theories.
2.2.1 A New Fractional Time Derivative
We refer to the usual Caputo fractional time derivative as UFDt. For order α, it is given by (23.2.5), or in somewhat simplified notation, by
with\(~\alpha \in \left [ 0,1\right ] \) and \(a\in \left ( -\infty ,t\right ) ,~f\in H^{1}(a,b),~b>a\). By replacing the kernel (t − τ)−α with the function \(exp(-\frac {\alpha }{1-\alpha }(t-\tau ))\) and \(\frac {1}{\varGamma (1-\alpha )}\) with \(\frac {M(\alpha )}{1-\alpha }\), we obtain the following definition of a new fractional time derivative, denoted by NFDt, and given by the formula
where M(α) is a normalization coefficient with the property that M(0) = M(1) = 1. According to the definition (23.2.13), the NFDt is zero when f(t) is constant, as is the UFDt. However, in contrast to the UFDt, the kernel in (23.2.13) does not have a singularity for t = τ.
The NFDt can also be applied to functions that do not belong to H 1(a, b). Indeed, the definition (23.2.13) can be formulated also for f ∈ L 1(−∞, b) and for any \(\alpha \in \left ( 0,1\right )\), which can be seen by expressing (23.2.13) in the form
If we put
definition (23.2.13) of NFDt assumes the form
where \({\sigma } \in \left (0,\infty \right )\) and M ∗(σ) is the corresponding normalization term to M(α), and is such that M ∗(0) = M ∗(∞) = 1.
In the context of mechanical models discussed later, for a →−∞, the relation (23.2.14) corresponds to a relaxation function given by
instead of that given by (23.1.1) for the Caputo fractional derivative. This describes the well-known Maxwell material (for example [167]), which is a very simple model of a viscoelastic fluid.
For α → 1, we have σ → 0. Also,
Therefore (see [150] and [197]),
Also, for α → 0 we have σ → +∞. Hence,
If f(a) = 0, then
Thus, (23.2.15) and (23.2.17) indicate that for α = 0 and 1, the NFDt behaves as expected in the integer limit.
Simulations comparing the UFDt and NFDt for particular choices of f(⋅) were described in [56]. These suggested that for α = 0.66 the classical NFDt is very similar to the UFDt. However, for models with α close to 0, we see different behavior. For α = 0.1 differences between NFDt and UFDt become apparent. In particular the classical UFDt is more affected by past history than NFDt, which exhibits rapid stabilization.
For integer n where n ≥ 1, and \(\alpha \in \left [ 0,1\right ] \) the fractional time derivative \(\mathcal {D}_{t}^{(\alpha +n)}f(t)\) of order (n + α) is defined by
Theorem 23.2.1
If the function f(t) is such that
then we have
Proof
Consider the case n = 1. From definition (23.2.18) of \(\mathcal {D} _{t}^{(\alpha +1)}f(t),\) we obtain
By means of an integration by parts, using the property f ′(a) = 0, we obtain
Also,
It is easy to generalize the proof for any n > 1. □
The property asserted in Theorem 23.2.1 is implied by the notation on the left of (23.2.18). Also, note that (23.2.18), together with (23.2.15) and (23.2.16) yield that
which, again, is expected behavior for a non-integer derivative.
In the following, we suppose the function M(α) = 1. We can rewrite the definition (23.2.14) in the form
obtained from (23.2.13) or (23.2.14) with ν = 1∕σ > 0, where V (ν) = (ν + 1)M ∗(1∕ν). Then, we have the following theorem.
Theorem 23.2.2
If the function f ∈ W 1, 1(a, b), then the integral in (23.2.19) exists for t ∈ [a, b] and \(\tilde {\mathcal {D}}_{t}^{(\nu )}f(t)\in L^{1} \left [ a,b\right ] .\)
Proof
Let us write
where \(p_{\nu }(y)=\exp (-vy),\) when 0 < y < b − a, with p ν(y) = 0 when y < 0 or y > b − a. Also, q(y) = f ′(y) when a ≤ y ≤ b. Finally, q(y) = 0 when y < a or y > b. Hence, under the hypotheses of the theorem, the functions p ν , q ∈ L 1(a, b). Then, by the classical results on Lebesgue integrals (see [ 326 ]), the integral (23.2.19) exists almost everywhere in t ∈ [a, b] and \(\tilde {\mathcal {D}}_{t}^{(\nu )}f(t)\in L^{1}\left [ a,b \right ] \).
The definition (23.2.19) can be generalized by choosing p ν in (23.2.20) to be any function with the properties assigned to the exponential kernel in Theorem 23.2.2, where p ν(0) is a finite number, chosen for convenience to be unity. This latter property ensures that it constitutes a non-singular kernel. It can be shown without difficulty that Theorem 23.2.1 also applies to the generalized definition.
2.2.2 Some Results for Given Histories
It is of interest to see the fractional derivatives of elementary functions according to the new definition (23.2.13). We begin with \(\sin \omega t\) and \(\cos \omega t\). It is convenient to first consider \(f(t) = \exp (i\omega t)\), which combined both of these. In fact, we have
and
where \(E(\alpha )=\frac {M(\alpha )}{1-\alpha }\). We have, from these relations that
where λ is such that
Thus, the new derivative of \(\sin \omega t\) yields a change of phase by amount λ, while the amplitude becomes
Also,
which also exhibits a phase change and the same amplitude variation noted for the case of \(\sin \omega t\).
The new derivative, for an exponential history, has the form
Finally, for a linear history, defined by
we obtain
2.2.3 The Laplace Transform of the NFDt
The Laplace transform of the NFDt, defined by Eq. (23.2.13), will be of interest. We have
One can show that
Because of the convolution form of N(t) in (23.2.13), we have
Similarly,
and, more generally,
2.2.4 Fractional Gradient Operator
We introduce a new concept of fractional gradient, which can describe nonlocal dependence in constitutive equations [324, 325].
Let us consider a set \(\varOmega \in \mathbb {R}^{3}\) and a scalar function \( u(\cdot ):\varOmega \rightarrow \mathbb {R}\). We define the fractional gradient of order \(\alpha \in \left [ 0,1\right ] \) as follows
with x, y ∈ Ω. The nonlocal property of this fractional derivative relates to the integration of y over Ω. A rotationally invariant three-dimensional Normal (Gaussian) distribution has been chosen to describe this nonlocality.
It is easy to prove from definition (23.2.22) that
using the relation
Thus, when α = 1, ∇(α) u(x) loses the nonlocality property. Also, we clearly have
This fractional gradient is easily generalized to the case of a vector u(x), where the gradient is assumed to exist on Ω. We define the fractional gradient of this vector by
Thus, a material with a nonlocal property may be described by fractional constitutive equations. As an example we consider an elastic nonlocal material, defined by the following constitutive equation between the stress tensor T and ∇(α) u(x)
where A is a fourth order symmetric tensor. The nonlocal property is clear from the detailed form
Likewise, we can introduce the fractional divergence, defined for a smooth \( \mathbf {u}(\cdot ):\varOmega \rightarrow \mathbb {R}^{3}\) by
Theorem 23.2.3
From definitions (23.2.22) and (23.2.23), we have for any \(u(\mathbf {x}):\varOmega \rightarrow \mathbb {R}\) , such that
the following identity
Proof
Using (23.2.22), we obtain
Hence, for the boundary condition (23.2.24), the identity (23.2.25) is proved, because
2.2.5 Fourier Transform of the Fractional Gradient and Divergence
For a smooth function \(u(\mathbf {x}):\mathbb {R}^{3}\rightarrow \mathbb {R}\), the Fourier transform of the fractional gradient is defined by
This quantity is given by
where the well-known formula for the Fourier transform of a convolution product has been used. From
we obtain
The Fourier transform of fractional divergence is defined by
from which we have
2.2.6 Fractional Laplacian
In the study of partial differential equations, the Laplacian is of considerable interest. It is therefore of interest to determine the factional Laplacian. Based on the definitions of fractional gradient and divergence, we suggest a representation of the fractional Laplacian for a smooth function f(x): \(\varOmega \rightarrow \mathbb {R}^{3},~\)such that \(\left . \nabla f(\mathbf {x)}\cdot \mathbf {n}\right \vert _{\partial \varOmega }=0,~\) of the form
By the use of Theorem 23.2.3, we have
Suppose that
Then, we can extend the function f(x) to \({\mathbb R}^3\) by taking it to be zero on \(\mathbb {R}^{3}\backslash \varOmega \). This allows us to consider the Fourier transform
Finally, if α = 1 we obtain from (23.2.26)
2.2.7 Memory Operators
Fractional derivatives are memory operators which usually represent dissipation of energy [128, 248, 287] or damage [55] in the medium, as in the case of anelastic media or diffusion in porous media. In general, they are in agreement with the Second Law of Thermodynamics [109, 128].
Their validity rests not only on the fact that they represent appropriately a variety of phenomena, but also, in the case of the Caputo derivative, because they have the “elegant and rigorous property” that when the order of differentiation is integer, they coincide with the classic derivative of that order. However, this property is not relevant to the effects they represent in physical phenomena. It may be that using other differential operators, possibly simpler but without this property, one may obtain the same results as for Caputo fractional derivatives.
The effects of the fractional memory formalism for the new fractional derivative (NFDt), compared with the Caputo derivative (UFDt) on a linear trend are presented in [56].
A distributed order fractional memory operator may be introduced, which is simpler and easier to handle than the Caputo derivative [326]. It is defined by
where g(α) is a weight function and 0 < a < b < 1. We now take the Laplace transform of (23.2.27). Following the method of Caputo [45, 46, 56], one may interchange the order of integration of α and t. Thus, we obtain
By virtue of the convolution form of the integral over τ, we have, as in (23.2.21),
which represents the filtering properties of the operator and is simpler than that obtained using the Caputo derivative.
As an example we may consider the simple case g(α) = 1, which gives
This can be written as follow
3 The Fractional Derivative Memory Model
Fractional derivatives and their connection with power law relaxation functions are now discussed. Constraints imposed by thermodynamics are derived, and the Graffi–Volterra free energy for fractional derivative models is introduced.
3.1 Power Laws and Fractional Derivatives
Let us assume that the viscoelastic memory kernel or relaxation function \(\mathbb {G}(s)\) is given by the power law form (23.1.1)
where Γ(⋅) is the gamma function and \(\mathbb {C}\) is a fourth order tensor. Relaxation functions of this type are discussed briefly in [167, page 32], where references to older works are given. We have
The property \(\mathbb {G}_\infty = \mathbf {0}\) in the classical Volterra theory corresponds to that for viscoelastic fluids. In the fractional model though the kernel (23.3.1) is not L 1(0, ∞), we will see (Remark 23.3.1 below) that this is true for values of α close to 1. However, for solid viscoelastic materials, some experimental observations are in approximate agreement with predictions based on (23.3.1) [167, 280, 293, 296], notably the property that the loss angle [167] is independent of frequency, as indicated by (23.5.2) below.
It follows from (23.3.1) that
The creep function for materials characterized by (23.3.1) or (23.3.3) also has a power law form [167].
Observe from (23.3.1) that there are no dimensional parameters in the theory, other than the overall coefficient \(\mathbb {C}\) and the time dimensional quantity s. This implies that various quantities can be simply determined to within a multiplying constant, by means of dimensional analysis.
It is in fact advisable to introduce an extra parameter ξ with the dimension of time. This can usually be absorbed into \(\mathbb {C}\) or η, if the parameter α is no being explicitly varied. For solids, we put
so that \(\mathbb {C}_1\) has dimensions of stress. For fluids, we have
where η 1 has dimensions of stress.
The use of the Caputo fractional derivative, defined for any \(\alpha \in \left ( 0,1\right ) \) by any of the forms (23.2.6)–(23.2.9), is equivalent to adopting (23.3.1) as the relaxation function of the material. This model may be applied to both fluids and solids.
We follow here the classical papers [249] on fractional derivatives in defining the constitutive equation of viscoelasticity by
Let us take a = −∞, since if necessary it is always possible to extend E to the interval \(\left ( -\infty ,a\right ) \) by the null tensor. Thus, (23.3.5) can be written as
or, by a change of variable,
which are equivalent representations of the Caputo derivative. Using the notation of Sect. 23.2.1, relations (23.3.5)–(23.3.7) may be put in the form
The constitutive equations (23.3.6) or (23.3.7) allow us to define the domain of definition of these functionals by a fractional Sobolev space, now called a Gagliardo space [142], defined for any x ∈ Ω,
with norm given by
In this framework, the constitutive equation of an incompressible viscoelastic fluid is entirely analogous to (23.3.5), (23.3.6) or (23.3.7), 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
where p denotes the pressure and T E the extra-stress defined by
where v is the fluid velocity. Alternative forms are given by
Remark 23.3.1
Note that when α is close to 0, the model well represents a viscoelastic solid. When α is close to 1, we have a viscoelastic fluid. These features of the model are implied by the (23.3.8), in view of the properties of the Caputo fractional derivative given by (23.2.10) and (23.2.11), leading to
where we have used (23.3.4).
Remark 23.3.2
Another important feature of a solid is the existence of only one null strain E 0 (or reference configuration) such that the space of histories is a subset of
where, from (23.3.5) or (23.3.6), if E t(s) = E 0 we have T(E t(s)) = 0.
For a fluid, the set of histories belongs to
The main difference between the Volterra and fractional derivative models, is evident in the study of solid materials, if we examine stress behavior for t →∞. Indeed, in the context of the Volterra theory, if the system is subject to a constant strain E 0, then stress will tend to G ∞ E 0. This can be proved by assuming a constant strain E 0(x) for t > t 0, so that the following limit is obtained
On the other hand, in the fractional theory we find that the stress will go to zero. Indeed,
In this case, the material undergoes a kind of plastic deformation [49].
Remark 23.3.3
It is interesting to observe that Volterra sought to describe properties related to dislocation phenomena in terms of memory effects. However, it is now known that the model with the standard constitutive relation ((8.1.5), for example) is not capable of describing plastic effects produced by dislocations.
4 Thermodynamical Constraints and Free Energies
The issue of compatibility of fractional derivative models with thermodynamics is explored in this section. Only isothermal processes will be considered, so that the Second Law of Thermodynamics reduces to the Dissipation Principle
where ψ denotes a free energy and ρ is the mass density. This is equivalent to property P3 given by (16.1.28). We have from (23.4.1) that on any cyclic process of period T = 2π∕ω with \(\omega \in {{\mathbb R}}^{++}\),
In particular, for periodic strain processes of the form
it follows from (23.4.2) (see [108, 124]) that for all E 1, E 2 ∈ Sym(V ),
The second term can vary arbitrarily in sign and magnitude for different choices of E 1 and E 2, so that it can be concluded that the fourth order tensor \(\mathbb {C}\) is symmetric. Also, in the first term, the quantity \(\mathbf {E} \cdot \mathbb {C} \mathbf {E}\) must have a definite signature for the inequality to be obeyed in a simple manner. We choose \(\mathbf {E} \cdot \mathbb {C} \mathbf {E} \geq 0\) so that \(\mathbb {C}\) is at least positive semidefinite. Thus, the condition (23.4.3) becomes
Remark 23.4.1
For a fluid defined by (23.3.10), we obtain from the second law the same inequality (23.4.4). Alternatively, if (23.3.9) is used, the equivalent condition emerges
We observe that (23.4.5) has the form (16.1.12) for the case of a fractional derivative relaxation function. It will be confirmed that this property is actually true in Remark 23.5.1.
4.1 The Graffi–Volterra Free Energy
We now consider a particular free energy within fractional derivative theory. Let us first take the case of a solid, described by Eq. (23.3.7). This functional is denoted by ψ S. Any free energy functional must satisfy the inequality (23.4.1). Thus, we must have
For simplicity, let us take the \(\mathbb {C}(\mathbf {x})\) to be a scalar quantity C(x). Using the identity
we have
Let us assume that ψ S(x, t) is given by
This is the Graffi–Volterra free energy for fractional derivative models. It is discussed in a more general context in Sects. 17.3.1 and 10.1.1, where it is shown to be the only free energy that is a single-integral quadratic form. The inequality (23.4.6) is satisfied because by (23.4.7) we obtain
Hence
where D(x, t) ≥ 0 denotes the rate of dissipation, given by
The set of histories \(\mathcal {H}_{S}^{t}\) available with this model is defined by
For a viscoelastic fluid, we use the constitutive Eq. (23.3.10), which is more convenient than (23.3.9). This gives
The Graffi–Volterra for fluids is given in this case by
or, using the variable s = t − τ, the equivalent form
Of course, it has a similar form to (23.4.8) for solids. We may define the set of histories \(\mathcal {H}_{F}^{t}\) available with this model by
The rate of dissipation D(x, t) is given by
or
5 Frequency-Domain Quantities for Scalar Fractional Derivative Materials
In the next two sections, we deal, for simplicity, with the scalar theory, for which the relaxation function and strain history are scalar quantities. Also, we generally omit the space variable x.
5.1 Complex Modulus for the Fractional Derivative Model
It follows from (23.3.3) and (16.1.7)2 that the quantity \(G^\prime _+(\omega )\) does not exist for fractional derivative forms, while \({\widetilde G}_+(\omega )\), given by (16.1.7)1, is finite. Therefore, the complex modulus, defined by (16.1.9)2, is finite. Recalling (23.3.1) and (23.3.2), we see that it is given by
where the coefficient k corresponds to \(\mathbb {C}\) (or C) and η in Sect. 23.3. From dimensional analysis, we can determine that
where c is a dimensionless constant to be determined. Putting z = iω and rotating it to a point on the positive real axis, the integral can be evaluated in terms of the Gamma function. Rotating back, we find that ( [167, page 33])
The real and imaginary parts of M(ω) have the form
Remark 23.5.1
The positivity of G c(ω) is clear from (23.5.1), for \(\omega \in {\mathbb R}^+\) and therefore for all real ω. This confirms the thermodynamic constraint (16.1.12)1.
The loss angle γ, defined by the relationship
is given by
which is independent of ω, as noted earlier.
It is apparent from (23.5.1) that the only singularity in G +(ω) is a branch cut from the origin to infinity. Thus, minimal states for power law materials are singletons, in accordance with Proposition 16.5.2. The branch cut must lie in Ω +. Apart from this constraint, we can choose it to be in any direction. Each choice yields a different function. We choose it to be along the positive imaginary axis. An expression for the minimum free energy of this material is derived in Chap. 14. Thus, we can reach the negative real axis, without crossing singularities, by a rotation e −iπ of ω. This gives
which has a branch cut along the negative imaginary axis.
From (16.1.21)4, we see that the frequency-domain version of the constitutive relation has the form
The last forms are obtained by moving the contour to closely surround the cut on the negative imaginary axis. The first term of (23.5.4)2 becomes the integral over [0, −i∞) on the right side of the negative imaginary axis, while in the second term becomes the integral over (−i∞, 0] on the left side. Relation (16.1.17) has been invoked in writing the last equation.
Using (16.1.13), (23.5.1)2 and (23.5.3), we deduce that the function H is defined over \({\mathbb R}\) by
Using (16.1.23)1, we can write T(t) in the form
The penultimate form is obtained by transforming the integrals according to the changes described in relation to (23.5.4), while the final form uses (16.1.17).
5.2 The Work Function for Fractional Derivative Materials
Relation (17.3.19) for the work function becomes
which involves an integrable singularity. Using (23.5.5), relations (17.3.19) become
The basic property \(\dot W(t) = T(t) \dot E(t)\) can be shown using (23.5.6)1, (23.5.7), (16.1.18)2 and the evenness of H(ω).
6 The Minimum Free Energy for Fractional Derivative Models
We now derive the form for the minimum free energy and the corresponding rate of dissipation for fractional derivative materials. These are for general histories. Simple explicit formulae are also given for sinusoidal and exponential histories. The derivations are for the scalar case.
6.1 General Form of the Minimum Free Energy
The factors H ±(ω) of H(ω) have the form
for ω > 0. The phase λ(ω) remains to be determined. On the complex plane, H ±(ω) are analytic continuations of these quantities, except at their singularities, which are now described.
The singularities of H +(ω) are chosen to be along the positive imaginary axis, similarly to G +(ω) as described before (23.5.3). Then H −(ω) is the complex conjugate of this function with singularities consisting of a branch cut along the negative imaginary axis.
If we had chosen different straight line branch cuts, the resulting factors would differ from those we determine here by a constant phase factor. This can be seen from (23.6.1) by considering a transformed complex plane with typical point \(\omega _1 = \omega \exp (i \delta )\), where δ is a constant. It is relevant that λ(ω) is later shown to be independent of ω. The phase factor δ would not affect the resulting formula for the minimum free energy.
The branch cuts described are simple consequences of the factors ω η in (23.6.1). The product of H ±(ω) will lead to branch cuts in H(ω). The factors of H(ω) obey the following relationships, for general complex ω:
from which it follows that H +(ω) (H −(ω)) must be real on the negative (positive) imaginary axis; indeed, this property applies to any point on the imaginary axis where these factors exist. Thus,
The discontinuity in H −(ω) across the branch cut along the imaginary axis is given by the discontinuity in
Thus, the discontinuity is determined by H(ω), divided by a real quantity of the form \(H_+(r \exp (-i\frac {\pi }{2}))\). Therefore, the singularities in H −(ω) are determined by those in H(ω). Similar observations apply to the singularities of H +(ω).
Proposition 23.6.1
The phase λ(ω) is independent of ω.
Proof
The phase factors e ±iλ(ω) are continued analytically to the whole complex plane. By the argument just outlined, they cannot contribute new singularities over and above those determined by H(ω), which is independent of λ(ω). Thus, we conclude that the phase factors will yield no singularities and must therefore be entire functions. However, this means either that they contribute essential singularities at infinity, which must be excluded in the same way as singularities on the finite plane, or they are constant. Therefore, the quantity λ is independent of ω. □
The factorization (16.1.14) clearly allows us to replace H ±(ω) by − H ±(ω).
For \(\omega \in {\mathbb R}^{++}\), we have
where, from (23.6.2),
We put
so that \({\theta } = \arg (\omega )\). The behavior of H +(ω) as ω approaches the positive imaginary axis from the first and second quadrants, respectively, are given by
Similarly, the behavior of H −(ω) as it approaches the negative imaginary axis from the fourth and third quadrants, respectively, are given by
In the light of (23.6.4), the limiting values (23.6.5) and (23.6.6) reduce to
and
Let us use (23.6.3) to extend H ±(ω) to \(\omega \in {\mathbb R}\). We take θ = −π to obtain H +(ω), and θ = π for H −(ω). This gives, for ω < 0,
These formulae are consistent with (22.1.8). Note that on multiplying the factors in (23.6.3) and (23.6.7) together, we obtain (23.5.5) for \(\omega \in {\mathbb R}\).
The quantity \(p^t_-(\omega )\) has the form
on using (23.6.3) and (23.6.7). The singularities of \(E^t_{r+}(\omega )\) are in \(\mathbb {C}^{(+)}\). Recalling the position of the cut in H −(ω), we see that the integrations over the real axis can be moved into the lower half-plane to closely surround the branch cut, yielding
where \(E^t_{r+}(-i r_1)\) is a real quantity given by
Using
and Cauchy’s integral formula, we obtain
Using (16.1.17), we can also write (23.6.9) in the form
The quantity η is defined by (23.6.1)3.
The minimum free energy may be written in the form (see (11.9.14))
where the equilibrium term vanishes, since G ∞ = 0. We must have
where G(u) is the relaxation function. Thus, we have
Recalling (23.3.2), we see that the properties (17.3.7) are valid in this case. Relation (23.6.11) is confirmed since
by virtue of the formula, for complex z ( [168, page 285]),
together with the relations (23.2.2) and (23.2.3). We now seek to determine the rate of dissipation corresponding to the minimum free energy. Relation (16.1.18)2 yields that
so that, from (23.6.9), we have
Using (23.6.12) and the expression (23.6.9)3 for κ, the last term of (23.6.13) becomes
by virtue of (23.5.6). Consequently, recalling (16.1.28), it follows that the rate of dissipation is given by the negative of the first term on the right-hand side of (23.6.13), where we have used (16.1.17). These relations can be deduced also from (16.4.12). The quantity D m(t) can be written in the form (17.3.11) with
6.2 The Minimum Free Energy for Simple Histories
Free energies and rates of dissipation for sinusoidal and increasing exponential histories are discussed in [15, 16]. Sinusoidal histories are useful in many practical contexts, though the total dissipation and the work function, defined by (16.1.30), are infinite. Increasing exponential histories provide a simple example where all quantities are finite. Also, the algebra involved is similar to, though simpler than, the sinusoidal case.
6.2.1 Sinusoidal Histories
Formulae relating to general materials for sinusoidal histories are presented in [15, 16] and earlier papers. Also, for exponential histories, similar general results are introduced in [16]. Here, we consider the specific cases (23.6.10) and (23.6.14) directly for the relevant forms of the strain history.
Consider a history and current value (E t, E(t)) defined by
where E 0 is an amplitude and \(\overline E_0\) its complex conjugate. The quantities \(E^t_+\) and \(\dot E^t_+\) have the form
From (17.6.8)2, we find that
where r is real. The final form of (23.5.6), together with (23.6.12) and (23.6.17) give
Any real algebraic quadratic form in E(t) or real functional quadratic form in E t(s) can be written in the form (16.11.3), denoted by V . Recalling (17.6.12) we introduce the abbreviated notation
Thus, we can write ψ m(t), given by (23.6.10), as
where, by dimensional arguments, it follows that
The dimensionless quantities ρ and χ are to be determined. We find, using (23.6.17), that
The following integrals, together with (23.6.12), will be required in the calculations below. For complex y, z, we have
Relation (23.6.21)1 is given in [168, page 289], while (23.6.21)2 is a special case of a result in [168, page 285]. The latter can also be obtained by differentiating (23.6.12) with respect to z.
With the aid of (23.6.21)1, we find that
These integrals can also be evaluated using (23.6.21). With the use of results derivable from (23.6.21)1, we find that ρ in (23.6.20)1 has the form
Also, using (23.6.21)2, one can show that the quantity χ in (23.6.20)2 is given by
Observe that
The rate of dissipation given by (23.6.14)2 is now considered. Let us put
Then
in the notation (23.6.19), where from (23.6.17) and (23.6.12),
In the rightmost form of (23.6.22), D 1 and D 2 are dimensionless constants, which are now determined. The coefficient in this term emerges from dimensional analysis. Relation (23.6.22)1 becomes
which gives
From (23.6.15) and (23.6.18), we see that the rate of input of mechanical energy is given by
By virtue of (16.1.28), we must have
which are easily confirmed. Relation (16.1.29) involves divergent quantities, namely \({{\mathcal D}}(t)\) and W(t), for sinusoidal histories [15, 16].
6.2.2 Exponential History
Consider a history and current value (E t, E(t)) given by
where E 1 is a constant amplitude. Some of the formulae for this history may be obtained from the sinusoidal case by simple substitutions [16]. However, we present direct derivations here. Instead of (23.6.16), we have
The stress function becomes
on using (23.6.12). The minimum free energy ψ m(t), given by (23.6.10)1, has the form
where A γ is given by
with the aid of (23.6.21) and the integral ( [168, page 289])
This formula is in fact in a similar category to (23.6.21) but where the parameters are in a different range. We conclude that
It follows from (23.6.9)3, (23.6.12), and (23.6.14) that
Finally, from (23.6.23) and (23.6.24) we find that
and (16.1.28) is obeyed. Indeed, we can also consider (16.1.29) in the case of exponential histories, since there are no convergence difficulties. We write ψ m(t), given by (23.6.25), and the integrated forms of (23.6.26) and (23.6.27), as
Thus, (16.1.29) is also clearly obeyed.
6.2.3 The Physical Free Energy
The physical free energy, discussed in several different contexts in the present work is defined by the property that its associated rate of dissipation is the true rate for the material. We tentatively identify the physical free energy of a fractional derivative material, as the minimum free energy by virtue of the following argument.
Two functionals have been identified as being associated with such materials, the Graffi–Volterra and the minimum free energies. The Graffi–Volterra functional is a degenerate version of the full two variable quadratic form discussed in earlier chapters. As such, it cannot be included as a free energy with deeper physical meaning, though it is a simple functional with the correct positivity properties and therefore very useful as a mathematical tool.
Invoking the property P4 introduced in (18.2.1), we see that the work function cannot be a valid free energy. Also, it has degenerate features somewhat similar to the Graffi–Volterra functional. Therefore, the minimum free energy, given by the elementary explicit expression (23.6.10), is the only non-degenerate free energy functional associated with the material. It must therefore be identified as the physical free energy of this material. Furthermore, the physical rate of dissipation has the form (23.6.14).
7 Application to Viscoelastic Systems
We now consider the dynamical equations for viscoelastic solids and fluids within the framework of fractional derivative models. An energy theorem is proved in both of these cases.
7.1 Viscoelastic Solids
Let \(\varOmega \subset {\mathbb R}^{3}\) be a smooth bounded domain of a linear viscoelastic solid, whose constitutive equation is given by the fractional model with constitutive relation given by (23.3.7). The initial boundary value problem is defined by the differential system in the domain Q = Ω × (0, T) by
where ρ 0(x) denotes the mass density, u(x, t) the displacement such that \(\mathbf {{E} =}\frac {1}{2}\mathbf {(}\nabla \mathbf {u} +\nabla {\mathbf {u}}^{T})\) and f(x, t) the body forces. The initial conditions are
along with the boundary conditions
where u 0 and v 0 are given functions.
Using the definition of fractional derivative given in (23.3.8), Eq. (23.7.1) can be rewritten in the form
Now, our purpose is to obtain an energy theorem for the problem (23.7.1) and (23.7.2). To this end, we multiply (23.7.3) by the first time derivative of u(x, t). Then, after an integration on Q = Ω × (0, T), we obtain
where d x is the three-dimensional space volume element. Hence, using the divergence theorem and the boundary condition (23.7.2), it follows from (23.7.4) that
Then, from (23.4.6) or (23.4.9), we obtain
where Ψ S(x, t) is the Graffi–Volterra free energy functional for solids, given by (23.4.8). Finally, carrying out the time integration, we find that
It is easily checked that the same inequalities hold for any free energy for the system. This in effect means any positive functional obeying (23.4.6) or (23.4.9). A similar observation applies to the case of fluids, which is now discussed.
7.2 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 0, is defined by the differential system
with initial and boundary conditions
We again seek an energy theorem. It follows from (23.7.5)3 and (23.7.6), together with a standard step involving integration by parts, that
Using the inequality (23.4.6), we obtain
Hence, we have
8 Application to Rigid Heat Conductors
In this section, we use the fractional method to describe the behavior of heat conductors. The literature on this topic, including [1, 141, 299], is motivated by experimental findings such as those reported in [223, 285].
The classical Fourier constitutive equation is given by
where q is the heat flux , g the temperature gradient and k c > 0 is the thermal conductivity, which may be space dependent. In order to obtain a finite wave propagation speed, Cattaneo [59] proposed the modified constitutive relation
This is the Cattaneo–Maxwell or, for brevity the Cattaneo equation. It can be put in the form of an integral equation, describing a material with thermal memory
Relation (23.8.1) can be expressed in terms of the NFDt, given by (23.2.13) or (23.2.14), where
Thus, (23.8.1)1 can be written as
where
8.1 UFDt Fractional Cattaneo Equation
Let us replace the NFDt in (23.8.2) by the UFDt as defined by (23.2.12) or (23.2.6). Thus, we put
giving
where, recalling (9.1.2), we introduce \({\overline {\mathbf {g}}}_r^t(s)\), defined as
to emphasize an analogy between heat flow and viscoelasticity whereby \(-{\overline {\mathbf {g}}}(t)\), q(t) correspond to E(t), T(t). It can be concluded from this and (23.4.8) that the functional
is the Graffi-Volterra free energy for fractional Cattaneo models, with corresponding rate of dissipation deduced from (23.4.10) to be
Indeed, the minimum free energy for isotropic materials of this kind can be written down immediately from (23.6.9) and (23.6.10), replacing \(\dot E^t(u) \dot E^t(v)\) by g t(u) ⋅g t(v). Thus, we obtain
The quantity η is defined by (23.6.1). The coefficient k 1 replaces \(\mathbb {C}\) in Sect. 23.3 or k in Sects. 23.5 and 23.6. Similarly, the corresponding rate of dissipation is deduced from (23.6.14)3 by replacing \(\dot E^t(u)\) with −g t(u), yielding
8.2 The NFDt Model
Analogous results can be derived for the NFDt model given by (23.8.2), which is of course the Cattaneo equation. Free energies for this relationship are those for a simple memory function k(s) described in (23.8.1). This has one decaying exponential and goes to zero at large times. The Graffi-Volterra free energy corresponding to this form is given by
while the corresponding rate of dissipation is
Also, for a relaxation function consisting of one decaying exponential, the minimum free energy has the form
These are a special case of the Day free energy and rate of dissipation where the relaxation function goes to zero at large times.
9 Application to Electromagnetic Systems
We now explore some properties of two electromagnetic bodies, which are characterized by constitutive equations expressed as fractional models. The form of a particular free energy will be derived in both cases. These are examples or generalizations of the Graffi–Volterra free energy discussed in Sect. 23.4.1. This section is based on [19].
9.1 Visco-Ferromagnetic Materials
Let us consider a visco-ferromagnetic material characterized by the following constitutive equation
where the magnetic induction B(x, t) and the magnetic field are H(x, t) are defined for any point x ∈ Ω, the smooth bounded domain occupied by the material; moreover, the quantity C(x) is a second-order positive tensor, defined for any point x ∈ Ω.
This equation is expressed in terms of the α −Caputo fractional derivative (23.2.5); see also [47]
Using this definition, Eq. (23.9.1) assumes the following form
Taking a = −∞ and carrying a time integration by parts, (23.9.1) assumes the more useful form
where H (r) denotes the relative history of the magnetic field.
The dissipation law states that, for any simple electromagnetic material, there exists at least one state functional, denoted by ψ(x, t, and referred to as a free energy, which satisfies the following fundamental requirement:
By introducing the rate of dissipation \(\mathcal {D}(\mathbf {x},t)\geq 0\), the dissipation law can be written as
Taking account of (23.9.1), we can apply an integration by parts to the scalar product on the right-hand side of both Eqs. (23.9.2) and (23.9.3); the expression so derived allows us to identify the following functional as a free energy:
This is a particular example of the Graffi-Volterra free energy. The related rate of dissipation is given by
9.2 Nonlocal Visco-Ferromagnetic Materials
The behavior of nonlocal visco-ferromagnetic materials can be described by means of a constitutive equation expressed in terms of the fractional operator \(\mathcal {M}_{\mathbf {x}}^{\beta }\) of order \(\beta \in (\frac {1}{2} ,1)\), acting on a function f(x), introduced in [56] and expressed by
where x, y ∈ Ω.
By replacing f(x) with C(x, y)H(x, τ), we can introduce the following new constitutive equation for the ferromagnetic induction
where the scalar α, assumed to be in the interval \(\left ( 0,\frac {1}{ 2}\right ) \), is the degree of the Caputo fractional derivative, denoted by \( D_{t}^{\alpha }\), while the second-order tensor C(x, y) is taken to be symmetric in x and y and positively defined.
We note that, for nonsimple materials characterized by (23.9.4), the magnetic induction B(x, t) at any fixed point x ∈ Ω depends on the values of the magnetic field H(y, t) ∀y ∈ Ω.
The total internal power can be obtained by integrating over Ω the scalar product in the right-hand side of both Eqs. (23.9.2) and (23.9.3); thus, by virtue of (23.9.4), it is given by
and the global free energy has the form:
If we assume H(⋅, τ) = 0 for τ ≤ a, the inequality (23.9.2), by using (23.9.5) and (23.9.6), becomes
whence, with a time integration by parts, it follows that
This inequality is satisfied by the following functionals:
which gives the global free energy ψ(Ω, t), and
which is the expression for the related global rate of dissipation \(\mathcal {D} (\varOmega ,t)\).
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Amendola, G., Fabrizio, M., Golden, J. (2021). Fractional Derivative Models of Materials with Memory. In: Thermodynamics of Materials with Memory. Springer, Cham. https://doi.org/10.1007/978-3-030-80534-0_23
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-80533-3
Online ISBN: 978-3-030-80534-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)