Abstract
The use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.
1 Introduction
The use of fractional derivatives and integrals [1–3] allows us to investigate the behavior of material processes and systems that are characterized by power law non-locality, power law long-term memory, and fractal properties. Fractional calculus has emerged as a powerful tool that has a wide range of applications in mechanics and physics (e.g., [4–13]).
Non-local effects in elasticity theory have been treated with two different approaches: the gradient elasticity theory (weak non-locality) and the integral elasticity theory (strong non-locality). The fractional calculus can then be used to establish a fractional generalization of non-local elasticity in two forms: the fractional gradient elasticity theory (weak non-locality) and the fractional integral elasticity theory (strong non-locality).
Some developments in framework and derivation of corresponding results for the fractional integral elasticity have been made in [14–16]. This has not been done, however, for gradient elasticity (for a recent review of the subject, one may consult [17, 18]). An extension of the phenomenological theory of gradient elasticity using the Caputo and Riesz spatial derivatives of non-integer order is suggested in the present article.
In Section 2, a phenomenological fractional generalization of one-dimensional gradient elasticity is discussed using the Caputo derivative to include gradient effects in the constitutive equation for the stress. The corresponding fractional differential equation for the displacement is solved analytically and expressed in terms of the Mittag-Leffler functions. In order that the stress field be equilibrated, the material should develop an internal force that should be added to the externally applied body force field.
In Section 3, a fractional generalization of gradient elasticity is discussed using the Riesz derivative (in particular, the fractional Laplacian in the Riesz form). Analytical solutions of the corresponding fractional differential equation are obtained for two cases: the sub-gradient and the super-gradient elasticities (in analogy to sub-diffusion and super-diffusion cases) for a continuum carrying a point load. Asymptotic expressions are derived for the displacement field near the point of application of the external load. They may or may not be singular depending on the order of the fractional derivative used.
2 Fractional gradient elasticity based on the Caputo derivative
In this section, we suggest a fractional generalization of the gradient elasticity model that includes the Caputo derivative of non-integer order. For this one-dimensional model, we derive a general solution of the corresponding fractional differential equation for the displacement. We demonstrate how to overcome the difficulties caused by the unusual properties of fractional derivatives. An alternative fractional gradient elasticity model for a three-dimensional case using the Riesz fractional derivative (in the form of a fractional Laplacian) is discussed in Section 3.
2.1 Fractional gradient elasticity equation
Let us consider the constitutive relation for a one-dimensional fractional gradient elasticity model that is based on the Caputo derivative in the form
where
where n-1<
Then using the usual definition of the strain
we obtain the fractional stress displacement equation in the form
In view of the fractional vector calculus framework, we can derive the fractional equation of equilibrium in the form
with the given functions A
where
For the case
In general, we have
2.2 Solution of the fractional gradient elasticity equation
Let us make use of the explicit form concerning the violation of the semigroup property for the Caputo derivative that gives the relationship between the product
Using Eq. (2.4.6) in [3] of the form
and using Property 2.1, Eq. (2.1.16), in [3],
where
where 0<
Using Eq. (11), we rewrite Eq. (6) in the form
where 0<
Eq. (12) is a nonhomogeneous fractional differential equation with constant coefficients.
The solutions to equations of this type are given by theorem 5.16 in [3] (see also theorem 5.13 in [3] for the homogeneous case, feff(x)=0). To use these results, we assume that a=0. Let us consider the case 0<
where
and
The arbitrary real constants C0, C1, and C2 in the case of the Caputo fractional derivatives are defined by the values of the integer-order derivatives u(0), u(1)(0), and u(2)(0).
Here, E
Note also that E1, 1[z]=ez. The asymptotic behavior (see Eq. (1.8.27) in [3]) of the Mittag-Leffler function E
where 0<
Remark: It should be emphasized that the absence of the external force (f(x)=0) does not imply the vanishing of the effective force feff. In general, feff(x)≠0 for f(x)=0. Only in the case of commutativity of the Caputo fractional derivatives, i.e., if the semigroup property
is not violated, the vanishing of the external force f(x)=0 leads to the vanishing of the effective force feff(x)=0. It is easy to see that the semigroup property is satisfied if
If we consider Eq. (12) in the case
For this solution to be admissible, it should be checked if the function given by Eq. (22) satisfies the conditions u(3)(0)=u(4)(0)=u(5)(0)=0. To verify these conditions, we use Eq. (1.8.22) of [3] in the form
The conditions given by Eq. (21) are not satisfied for the function given by Eq. (22). Thus, the solution is not admissible for the fractional one-dimensional gradient elasticity model considered herein. Therefore, we should take into account the effective force defined in Eq. (18) for the solution given by Eq. (14) to describe the fractional one-dimensional gradient elasticity model correctly.
3 Fractional gradient elasticity based on the Riesz derivative
An alternative fractional gradient elasticity model may be obtained using the Riesz fractional derivative. In this case, it turns out that a three-dimensional treatment is possible due to available results on the fractional Laplacian of the Riesz type. The corresponding fractional gradient elasticity governing equation can then be considered in the form
where r∈R3 and r=|r| are dimensionless and (-
where k denotes the wave vector. If
where
Eq. (24) is the fractional partial differential equation that has the particular solution (section 5.5.1 in [3]) of the form
where the Green-type function
is given (see lemma 25.1 of [1, 2]) by the following equation:
Here,
Let us consider, as an example, the W. Thomson (1848) problem [20] for the present model of fractional gradient elasticity. Determine the deformation of an infinite elastic continuum, when a concentrated force is applied to a small region of it. To solve this problem, we consider distances |r|, which are large in comparison with the size of the region (neighborhood) of load application. In other words, we can suppose that the force is applied at a point. In this case, we have
Then the displacement field u(r) of fractional gradient elasticity has a simple form given by the particular solution
where
For this solution of the fractional gradient elasticity equation (24) with
This asymptotic behavior |r|→∞ does not depend on parameter
From a mathematical point of view, there are two special cases: (i) fractional power law-weak non-locality with
Sub-gradient elasticity (
α =2 and 0<β <2).Super-gradient elasticity (
α >2 andβ =2).
In the sub-gradient elasticity model, the order of the fractional derivative is less than the order of the term related to the usual Hooke’s law. The order of the fractional derivative in the super-gradient elasticity equation is larger than the order of the term related to the Hooke’s law. The terms “sub-gradient” and “super-gradient” elasticity are used in analogy to the terms commonly used for anomalous diffusion [6–8]: sub-diffusion and super-diffusion.
3.1 Sub-gradient elasticity model
The fractional model of sub-gradient elasticity is described by Eq. (24) with
The order of the fractional Laplacian (-
The following asymptotic behavior for Eq. (36) can be derived using section 2.3.1 in [21] of the form
where
As a result, the displacement field generated by the force that is applied at a point in the elastic continuum with the fractional non-locality described by the fractional Laplacian (-
for large distances |r|»1.
3.2 Super-gradient elasticity model
The fractional model of super-gradient elasticity is described by Eq. (24), with
The order of the fractional Laplacian (-
The asymptotic behavior of the displacement field u(|r|) for |r|→0 in the case of super-gradient elasticity is given by
Note that the above asymptotic behavior does not depend on parameter
A generalization of the phenomenological theory of gradient elasticity accomplished by including the Caputo and Riesz spatial derivatives of non-integer order is suggested in this article. Related lattice models with spatial dispersion of power law type as microscopic models of the fractional elastic continuum described by Eq. (24) were proposed in [22]. Using the approach suggested in [23, 24], Eq. (24) has been derived from the equations of lattice dynamics with power law spatial dispersion. We can point out that a phenomenological fractional gradient elasticity model can be obtained from different microscopic or lattice models. In addition, we note that the model of fractional gradient elastic continuum has an analogue in the plasma-like dielectric material with power law spatial dispersion [25]. It can be considered as a common or universal behavior of plasma-like and elastic materials in space by analogy with the universal behavior of low-loss dielectrics in time [26–28].
Acknowledgments
V.E.T. thanks Professor Juan J. Trujillo for valuable discussions of applications of fractional models in elasticity theory and the Aristotle University of Thessaloniki for its support and kind hospitality in July 2013. The financial support of the ERC-13 (IL-GradMech-AMS) grant, through the Greek-Secretariat of Research and Technology, is gratefully acknowledged.
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Article note: This article was first submitted to the J. of Elasticity in July 2013. It was returned for revisions in early February 2014. Since the authors did not agree with some of the reviewers’ comments, it was decided to follow the present path, partly due to the long time elapsed between actual completion and potential publication. It is noted, in this connection, that other articles on the topic (e.g. Ref. [22]) were directly motivated by the present work which, partly due to the aforementioned circumstances, may not have been properly quoted, even though it was the first article where the authors (upon the invitation of the first one/VET by the second one/ECA) attempted to address the extension of gradient elasticity to the fractional case (see also arXiv:1307.6999). This may have also occurred for other related articles that may have appeared or been submitted before the publication of the present one.
©2014 by Walter de Gruyter Berlin/Boston
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