Abstract
In this paper we derive and solve nonlocal elasticity a model describing the elastic behavior of composite materials, involving the fractional Laplacian operator.
In dimension one we consider in (($\mathcal{D}$)) the case of a nonlocal elastic rod restrained at the ends, and we completely solve the problem
showing the existence of a unique weak solution and providing natural sufficient conditions under which this solution is actually a classical solution of the problem. For the model (($\mathcal{D}$)) we also perform numerical simulations and a parametric analysis,
in order to highlight the response of the rod, in terms of displacements and strains, according to different values of the mechanical
characteristics of the material. The main novelty of this approach is the extension of the central difference method by the numerical estimate
of the fractional Laplacian operator through a finite-difference quadrature technique.
For higher dimensions
1 Introduction
In the last years the nonlocal elasticity theory has been used in wider and wider engineering applications involving small-size devices and/or materials with marked microstructures. The driving force for developing advanced materials comes from society’s call of large composite structures having solid mechanical features. This increasing need requires new engineering composite materials that work reliably and safely at the frontiers of cutting edge technologies, and pushes towards material systems in support of energy sustainability, [8].
The key tool in this theory is the analysis of the nonlocal interactions among different locations of the body, in terms of elastic central long-range body forces proportional to the volumes or masses for composite solids. In other words, in nonlocal phenomena, the displacement field u is not just reverting on its infinitesimal average, but instead it is influenced by its values at many scales, according to an integral average of the entire solid. As a feature of certain materials, nonlocality has been acknowledged in the literature since many years ago. In [7] it is shown that the elastic response of a material which presents distributed defects is necessarily nonlocal. Also in plasticity and in damage phenomena nonlocal behavior naturally arises, see for example [6, 24]. It is also worth noting that the homogenization of a composite with periodic microstructure produces a material with a nonlocal behavior, see [9].
The nonlocal effects are captured, in the equilibrium equations, by an integral term which is the resultant of all the long-range interactions. Hence, the corresponding equilibrium problem is ruled by one or more integro-differential equations in terms of u, which can be often tackled with certain tools of fractional calculus, cf. [12, 18, 19].
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real-powers
of differentiation operators, generalizing the concept of integer-powers derivatives.
An important point is that the fractional derivative of order
In this paper we derive and solve nonlocal elasticity models for composite materials involving the fractional Laplacian operator,
since it is particularly useful in modeling materials exhibiting
nonlocal behaviors caused by the effects of long range interactions among particles.
We adopt the formalization of nonlocal elasticity proposed in [21, 22], which is particularly appealing because
it allows us to use a continuum approach as in classical elasticity. More specifically, in the context of civil and mechanical engineering,
it is of great interest to consider the nonlocal phenomena occurring in those elastic media which can be modeled as points connected
each other by springs. Unlike what has been done up to now in the literature, concerning the 1-dimensional case,
the use of the fractional Laplacian represents a new different approach to face these problems.
One of the main advantages of this strategy is the possibility to generalize the analysis easily to higher dimensions
In the present paper, the fractional Laplacian operator
along any function φ in
is the Fourier transform of
In the case
We assume throughout the paper that
For (($\mathcal{D}$)) we prove the existence of a unique weak solution and we provide natural sufficient conditions under which the weak solution is actually a classical solution of (($\mathcal{D}$)), and vice versa we investigate when classical solutions of (($\mathcal{D}$)) are weak solutions, cf. Section 2 for further details and comments.
In Section 3 we treat a nonlinear model for nonlocal elastic ideal infinite
N-dimensional rods for any
with
where
In passing from a road of finite length to one of infinite length, we assume that the potential V is a measurable positive weight, with
the property that there exist
Here and in the following, for any exponent
Moreover, K is a positive measurable weight for which there exist real numbers
Let
When
by virtue of [10, Corollary 8.9]. From now on we assume that (1.3)–(1.4) hold, unless otherwise specified.
Theorem 1.1.
There exists a threshold
For
where f is a perturbation of class
Theorem 1.2.
There exists
If
Theorem 1.2 is not contained in Theorem 1.1 even when
The paper is organized as follows. In Section 2 we first derive the balance law equation for a nonlocal elastic 1-dimensional finite rod in terms of a linear combination of the standard Laplacian operator and the fractional Laplacian operator and attain the model (($\mathcal{D}$)). The existence of a unique weak solution of (($\mathcal{D}$)) is proved in Theorem 2.1, while in Theorem 2.2 we provide sufficient conditions under which the weak solution of (($\mathcal{D}$)) is also a classical solution of (($\mathcal{D}$)). Finally, in Theorem 2.3 we show that every classical solution of (($\mathcal{D}$)) is a weak solution of (($\mathcal{D}$)).
Section 3 is devoted to the study of the existence of (weak) solutions for (($\mathcal{P}_{\lambda}$)) and contains the proof of Theorem 1.1, while in Section 4 we prove Theorem 1.2 for the existence of (weak) solutions for (($\mathcal{P}$)).
In Section 5 we provide numerical simulations of the nonlocal elastic rod modeled in (($\mathcal{D}$)). Moreover, a parametric analysis is performed with the aim of understanding the role of the mechanical characteristics of the material, with a special attention to their local and nonlocal effects.
Finally, in Section 6 we resume the obtained results and present an overview of comments and conclusions on the validity of the proposed models, both from a theoretical point of view and from an applicative perspective related to the numerical simulations developed in Section 5.
2 A linear model for a nonlocal elastic 1-dimensional rod
In the present section we introduce the problem of a 1-dimensional solid (a rod) with properties of nonlocal elasticity, according to the definition of [21]. In doing this, we use the approach proposed in [38].
Let us consider the rod centered at zero of length
It is assumed that the constitutive law of the material is given by
where
Furthermore, E is the Young modulus of the material and k is a positive
constant typical of the material. Moreover,
It is worth noting that the same expression for the constitutive law (except for the absence of the term containing
On the other hand, the balance law is
where
where g is in
and the strain
Since the displacement u is also required to be
Hence, (2.2) in terms of u becomes
with
where Γ denotes the Euler function.
The restriction
The first term on the left-hand side of (2.4) stands for the contribution coming from the classical
elasticity theory. Indeed, taking
Before proceeding, let us recall some definitions and properties. Let s be a real number such that
Following [27] and [28, Theorem 2.1], we say that
are the forward and backward Caputo fractional derivatives of order
Moreover, the forward and backward
Riemann–Liouville fractional derivatives of order
Now fix
Therefore,
Similarly,
and so
Thus, since u verifies the boundary conditions in (2.4), we get immediately in I
We note in passing that the above relation between the Riemann–Liouville and the Caputo
fractional derivatives of order
We next show that any solution u of (2.4) satisfies in I
where
Since
By (1.1) for
Integrating twice by parts, by the boundary conditions in (2.4) and the fact that
In other words, by the Fubini theorem,
Without loss of generality, let us assume that
where
Clearly, for any
Consider
where
Since (2.9) holds in a set having one accumulation point in Ω, it follows that (2.9) is valid in the entire open connected set Ω by the identity principle of holomorphic functions. Thus, (2.8) becomes
by the Euler identity and the fact that
Therefore, from (2.7) we get
as claimed.
In conclusion, if
with
It is interesting to note that, in the case
A somewhat related approach, which takes inspiration from [14] and [19], has been used also in the recent paper [38] to provide
a physical interpretation of (2.11) when
We recall that u indicates both the displacement in I as well as its canonical extension to the entire
By the Poincaré inequality
for all
Similarly,
by virtue of the Plancherel theorem.
The embedding
Inequality (2.13) is a direct consequence of [17, Proposition 2.2 and remark after Theorem 2.4].
We say that uis a weak solution of (($\mathcal{D}$))
if
for all
A classical solution u of (($\mathcal{D}$)) is a function
We are now able to prove
Theorem 2.1.
Assume that
in
Proof.
The form
is bilinear and symmetric in
where
where
is linear and continuous in
Theorem 2.2.
Suppose that the assumptions of Theorem 2.1 are satisfied.
Assume furthermore that V and f are also continuous in
Proof.
Let u be a weak solution of (($\mathcal{D}$)),
satisfying the assumptions of the theorem. Since
Combining these facts, we get at once that
We claim that actually
since
For all
Since u satisfies the identity (2.14) and
where
Theorem 2.3.
Suppose that the assumptions of Theorem 2.1 are satisfied. Then every classical solution u of (($\mathcal{D}$)) is a weak solution of (($\mathcal{D}$)).
Proof.
Let u be a classical solution of (($\mathcal{D}$)), so that
Since u is a classical solution of (($\mathcal{D}$)), multiplying the equation in (($\mathcal{D}$)) by any
Since
3 A nonlinear model for infinite nonlocal elastic N-dimensional rods
In [43] (see in particular the details of the proof of Lemma 1) a precise expression for
the nonlocal operator
From an exact mathematical point of view, problem (2.11) for an ideal infinite rod becomes
In particular, (2.10) is converted into
Therefore every solution of (3.1) satisfies
In this section, we extend the linear model (3.1) to the nonlinear version (($\mathcal{P}_{\lambda}$)) given in the Introduction.
First recall that the space
by the Plancherel theorem.
The fractional Laplacian operator
see [17, Lemma 3.5], where
Furthermore, for all φ,
where for simplicity in notation
The embedding
Clearly
From here on we assume that the potential V is a positive weight satisfying condition (1.3) given in the Introduction.
Assumption (1.3) guarantees
that the embedding
is the weighted Lebesgue space related to the positive potential V, endowed with the norm
Similarly, the weighted Lebesgue space
has norm
By (1.2) and [36, Lemma 2.6] the embedding
for all
In the same way also the weighted Lebesgue space
is equipped with the norm
We note in passing that the embedding
By (1.4) the embedding
In the special case
Consequently,
Similarly, when
and
In conclusion,
for all
Problems (($\mathcal{P}_{\lambda}$)) and (($\mathcal{P}$))
can be weakly solved in
for all
Clearly
If
Problem (($\mathcal{P}_{\lambda}$)) has a variational structure and the underlying Euler–Lagrange functional
for all
In order to find a solution of (($\mathcal{P}_{\lambda}$)), we intend to apply the mountain pass theorem of Ambrosetti and Rabinowitz [1] to the functional
where
Lemma 3.1.
For any
Proof.
Fix
Taking
for all
Let
Thus
From the proof of Lemma 3.1 it is evident that if
Fix
Clearly,
We recall that
as
Before proving the relatively compactness of the Palais–Smale sequences for
Lemma 3.2.
It results
Proof.
Fix
Therefore,
Therefore there exists
Clearly
and this would contradict (3.12). In conclusion,
Consider now the path
and letting
Now, we are ready to prove the Palais–Smale condition at level
Lemma 3.3.
There exists
Proof.
Take
Hence, (3.11) yields at once that as
Therefore,
(3.14)
Furthermore, by (3.11)
Thus, by (3.14) and (3.15) we have
We first assert that
Otherwise
which is impossible by Lemma 3.2 and proves assertion (3.17).
Moreover,
By (3.14) and the fact that
since (3.14) gives that
and by (3.14) and the celebrated Brézis–Lieb lemma, see [11],
(3.20)
as
Therefore, by (3.19) we have the main formula
which, in particular, yields by (3.20) that
by (3.10) and the facts that
In particular,
By (3.21) and (3.7), for all
We claim that there exists
which is impossible by (3.22). This proves the claim.
In conclusion, there exists
and so by (3.21)
as required. ∎
Proof of Theorem 1.1.
Lemmas 3.1 and 3.3
guarantee that there exists
4 The model (𝒫 )
In this section
From now on we assume that the positive weights V and K satisfying (1.3)–(1.4) are also radial
and we continue to use the notation (3.10). Hence all the properties (3.5)–(3.7) are still valid. The perturbation
Problem (($\mathcal{P}$)) has a variational structure and solutions of (($\mathcal{P}$)) correspond to the critical points of the underlying Euler–Lagrange functional
for all
for all
In order to find the critical points of J, we intend to apply both the mountain pass theorem of [1] and the Ekeland variational principle given in [20]. For a wide selection of applications of critical point theory to fractional elliptic differential problems we refer to the recent monograph [32].
Lemma 4.1.
There exist three positive numbers α, δ and ρ such that
In particular,
Proof.
The proof is similar to that of Lemma 3.1. By the continuity of the embedding
for all
Take
Thus
The last part of the lemma is trivial. ∎
We say that
as
Lemma 4.2.
The functional J satisfies the Palais–Smale condition in
Proof.
Let
for all k. Hence, the sequence
(4.4)
by [30, Proposition I.1], being
For all
Therefore,
Now, by (4.2) it results that
and by (4.4) it follows that
being
since
Therefore, using again (4.5) and (4.6), we have
and so
as
Similarly, as
In conclusion, from the fact that
as required. ∎
Fix
Clearly
Lemma 4.2 yields that J satisfies the Palais–Smale condition in
admits a convergent subsequence in
Proof of Theorem 1.2.
First we prove that (($\mathcal{P}$)) admits a nontrivial radial mountain pass solution
Let us now assume that fis nontrivial, that is
being f nontrivial.
Indeed, by density, there exists
The claim is proved taking
Now, for any
This implies that for a fixed
By the Ekeland variational principle and Lemma 4.1 there exists a sequence
for all
Since
where
Therefore, u and v are two nontrivial critical points of
We claim that u and v are two critical points of J in the entire
The Hilbert space
where
The claim is a consequence of [16, Proposition 3.1], since
Indeed, fixed
since
Therefore, for all
for all
As a final comment to the proof of Theorem 1.2 we remark that
even if Theorem 1.1 and its Corollary 1.2 in [40, Chapter IV] are stated for functions of class
5 Numerical simulations for (𝒟 )
In this section we provide numerical computations for problem (($\mathcal{D}$)) which has completely been solved from a pure analytical point of view in Section 2.
Before starting, we recall that model (($\mathcal{D}$)) describes the displacement
where
Due to the extreme difficulty in defining an explicit analytical
expression for u, numerical methods could help in finding approximated solutions of the problem.
In particular, we look for a discrete form of the problem seeking the values
The terms in the equation of (($\mathcal{D}$)) are evaluated as
follows.
The second derivative of the solution
To estimate
where the weights
A simple truncation of the sum in (5.1) at a finite value
since
Hence, the discrete version of the equation in (($\mathcal{D}$)) is transformed into the problem of finding the values
The goal is to find the zeros of the above system of n functions in n variables. To this end we use the Python programming language. In particular, we employ the function “fsolve”, based on Powell’s hybrid method as implemented in MINPACK (see [33]).
The main ideas of the Powell hybrid method, as reported in [25], are briefly resumed in the following. Problem (5.2) can be rewritten as
Start from an estimate
where
where γ is a suitable positive parameter, and
instead of recalculating J.
We underline that the choice of the Powell hybrid method is motivated by the fact that it could be employed to perform numerical simulations also for more general problems than (($\mathcal{D}$)), as (($\mathcal{P}_{\lambda}$)) and (($\mathcal{P}$)), since it is particularly suited also for nonlinear systems.
5.1 The case of a discontinuous force f
In this section we apply the numerical procedure described before to two cases, taking somehow inspiration from [19, 14].
Case study #1.
We consider the rod loaded by two forces having equal magnitude and opposite direction (and therefore the resultant force is zero) applied close to the mid-span, as shown in Figure 1.
Let A be the cross section area of the rod.
The applied forces
The values adopted for the parameters are
The values obtained for the displacements u and the strain ϵ are shown in Figure 2.
The obtained results are in good agreement with those reported in [19].
Case study #2.
As before, let us consider the rod of length
The force
The values adopted for the parameters are
The values obtained for the displacements u and the strain ϵ are shown in Figure 4.
The obtained results are in good agreement with those reported in [14].
5.2 The case of a continuous force f
Let us consider the distributed continuous external force
schematized in Figure 5.
The following values have been used
Two constant levels for the function V have been considered, namely
We remark that when
5.3 Parametric analysis
In this subsection we perform a parametric investigation on the problem (($\mathcal{D}$)) in order to highlight the different response of the rod under the action of the continuous external force f introduced in Section 5.2, both in the nonlocal and in the purely local setting. The mechanical characteristics are
Moreover, we recall that
Changing 𝜷 𝟏 and 𝜷 𝟐 .
First, we have considered the behavior of the rod according to different values of the parameter
If
where
and the error function is defined as
The obtained displacements and strains are shown in Figure 7.
Note that the graphics corresponding to the case
As can be observed, for a fixed value of k, for example
Moreover, the different behavior of the strains, induced by nonlocality, can be clearly observed.
Indeed, if
Under the action of a continuous external force f, it is particularly interesting to consider also the contribute of an effective positive constant potential V. The solution of problem (5.3) is now given by
where
The results, obtained for
Considerations similar to those of the case
Changing s .
The analysis has been enlarged with numerical simulations concerning the variation of the fractional
exponent s. In this context, the values
The results are shown, in terms of displacements u and strains ϵ,
in Figure 9 for the case
It is interesting to note an evident increase of the strain ϵ
near the end points and at the mid-span of the rod, when s is close to
Changing k .
A further investigation we present here concerns the effects on u and ϵ
deriving from different values of k. Such effects are schematized in Figure 11.
In particular, we note that the value
In this case, we obtain a solution which does not coincide with the one obtained in the local case, and this difference may be used to assess the influence of the fractional Laplacian in problem (($\mathcal{D}$)). As it is shown in Figure 11, the displacements and the corresponding strains increase as the value of k decrease.
Concerning Figure 11, we recall that the purely local case refers to problem (5.3).
Changing V .
Finally, the action of the restoring force V has been studied and the corresponding results are presented in Figure 12.
As it was expected, since the term
6 Conclusions and perspectives
Composite materials are acquiring an important role in the development of innovative solutions which are environmentally sustainable and energetically efficient. The possibility to design the characteristics of these composites, using the properties of the single phases and their mutual arrangement, is precious and leads to significant outcomes. However, the resulting material, when considered at the macro-scale, is often characterized by nonlocal properties, in the sense that the stress at a point depends not only on the strain at the point but also on the strain at distant points. For materials exhibiting nonlocal behavior, the models of the classical mechanics are not sufficient and a new investigative approach turns to be necessary.
In this paper, following the approach proposed by Eringen, a nonlocal constitutive model for a rod has been obtained.
The equation governing the problem contains a term involving the fractional Laplacian operator
which accounts for the nonlocal part of the response.
It has been shown that the problem of the rod subjected to external forces has a unique weak solution,
and under reasonable conditions the weak solution is actually a classical solution of the problem.
Moreover, the model has been extended to a nonlinear multi-dimensional problem depending on a real parameter λ,
for which nontrivial solutions have been determined for all λ behind a threshold
In order to estimate the solution of problem (($\mathcal{D}$)), numerical simulations have been produced. In particular, the problem has been discretized by mean of finite differences. The fractional Laplacian term has been evaluated following the approach due to Huang and Oberman. The obtained results have been validated by comparison with the exact solution known in the purely local case and with previous results in the literature. The numerical investigations have highlighted the importance of the role of the parameters in the nonlocal model, especially for what concerns the order s of the fractional Laplacian and the interactions between the local and nonlocal nature of the material.
The analysis carried out allows us to enlarge the present knowledge about always more efficient and energetically sustainable structural systems, employed in technical fields, trough the of linear and nonlinear models describing the elastic behavior of innovative composite materials. The obtained results could be useful in the identification of the mechanical characteristics of a composite material exhibiting a nonlocal behavior and in developing models to fit experimental data. The proposed approach seems to open new perspectives for the investigation and the analysis of other interesting engineering problems, concerning effective applications. The procedure appears to be particularly fruitful in the field of the mechanics of nanomaterials, plane problems with eventual radial symmetry (circular plates, pipes) and nonlinear systems.
Funding statement: The first author was partly supported by the Italian Project Caratterizzazione di modelli e sviluppo di codici di calcolo per il comportamento visco-termo-elastico di materiali compositi per l’edilizia sostenibile, l’efficienza energetica e la sostenibilità ambientale, under the auspices of Proposta Progettuale “Promozione della ricerca e dell’Innovazione” (UM12024L002 POR Umbria FSE 2007–2013 – n. 10949). The fourth author was partly supported by the Italian MIUR project titled Variational and perturbative aspects of nonlinear differential problems (201274FYK7) and is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM–GNAMPA Projects 2015 titled Modelli ed equazioni nonlocali di tipo frazionario (Prot_2015_000368) and 2016 titled Problemi variazionali su varietà Riemanniane e gruppi di Carnot (Prot_2016_000421).
References
[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar
[2] T. M. Atanackovic and B. Stankovic, Generalized wave equation in nonlocal elasticity, Acta Mech. 208 (2009), 1–10. 10.1007/s00707-008-0120-9Search in Google Scholar
[3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–714. 10.1016/j.na.2015.06.014Search in Google Scholar
[4] G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 977–1009. 10.1007/s00030-012-0193-ySearch in Google Scholar
[5] G. Autuori and P. Pucci, Entire solutions of nonlocal elasticity models for composite materials, in preparation. 10.3934/dcdss.2018020Search in Google Scholar
[6] J. L. Bassani, A. Needleman and E. Van der Giessen, Plastic flow in a composite: A comparison of nonlocal continuum and discrete dislocation predictions, Int. J. Solids Struct. 8 (2001), 833–853. 10.1016/S0020-7683(00)00059-7Search in Google Scholar
[7] Z. P. Bazant, Why continuum damage is nonlocal: Micromechanics arguments, J. Eng. Mech. 117 (1991), 1070–1087. 10.1061/(ASCE)0733-9399(1991)117:5(1070)Search in Google Scholar
[8] P. W. R. Beaumont, On the problems of craking and the question of structural integrity of engineering composite materials, Appl. Compos. Mater. 21 (2014), 5–43. 10.1007/s10443-013-9356-1Search in Google Scholar
[9] M. J. Beran and J. J. McCoy, Mean field variations in a statistical sample of heterogeneous linearly elastic solids, Int. J. Solids Struct. 6 (1970), 1035–1054. 10.1016/0020-7683(70)90046-6Search in Google Scholar
[10] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar
[11] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490. 10.1090/S0002-9939-1983-0699419-3Search in Google Scholar
[12] A. Carpinteri, P. Cornetti and A. Sapora, Static-kinematic fractional operators for fractal and nonlocal solids, Z. Angew. Math. Mech. 89 (2009), 207–217. 10.1002/zamm.200800115Search in Google Scholar
[13] W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Amer. 115 (2004), 1424–1430. 10.1121/1.1646399Search in Google Scholar
[14] P. Cornetti, A. Carpinteri, A. G. Sapora, M. Di Paola and M. Zingales, An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus, XIX Congress AIMETA (Ancona 2009), Italian Association for Theoretical and Applied Mechanics, Ancona (2009), 14–17. Search in Google Scholar
[15] G. Cottone, M. Di Paola and M. Zingales, Elastic waves propagation in 1D fractional nonlocal continuum, Phys. E 42 (2009), 95–103. 10.1016/j.physe.2009.09.006Search in Google Scholar
[16] D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó, A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones 19 (2000), 1–17. 10.4067/S0716-09172000000100001Search in Google Scholar
[17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar
[18] M. Di Paola, G. Failla, A. Pirrotta, A. Sofi and M. Zingales, The mechanically based non-local elasticity: An overview of main results and future challenges, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), Article ID 20120433. 10.1098/rsta.2012.0433Search in Google Scholar
[19] M. Di Paola and M. Zingales, Long-range cohesive interactions of nonlocal continuum faced by fractional calculus, Int. J. Solids Struct. 45 (2008), 5642–5659. 10.1016/j.ijsolstr.2008.06.004Search in Google Scholar
[20] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. 10.1016/0022-247X(74)90025-0Search in Google Scholar
[21] A. C. Eringen, Vistas of nonlocal continuum physics, Int. J. Eng. Sci. 30 (1992), 1551–1565. 10.1016/0020-7225(92)90165-DSearch in Google Scholar
[22] A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972), 233–248. 10.1016/0020-7225(72)90039-0Search in Google Scholar
[23] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170. 10.1016/j.na.2013.08.011Search in Google Scholar
[24] M. G. D. Geers, R. de Borst and T. Peijs, Mixed numerical-experimental identification of non-local characteristics of random-fibre-reinforced composites, Compos. Sci. Technol. 59 (1999), 1569–1578. 10.1016/S0266-3538(99)00017-2Search in Google Scholar
[25] K. L. Hiebert, An evaluation of mathematical software that solves systems of nonlinear equations, ACM Trans. Math. Softw. 8 (1982), 5–20. 10.1145/355984.355986Search in Google Scholar
[26] Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference-quadrature approach, SIAM J. Numer. Anal. 52 (2014), 3056–3084. 10.1137/140954040Search in Google Scholar
[27] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1–15. Search in Google Scholar
[28] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006. Search in Google Scholar
[29] K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (2013), 563–591. 10.1007/s00220-012-1621-xSearch in Google Scholar
[30] P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334. 10.1016/0022-1236(82)90072-6Search in Google Scholar
[31] V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd ed., Grundlehren Math. Wiss. 342, Springer, Berlin, 2011. 10.1007/978-3-642-15564-2Search in Google Scholar
[32] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems. With a foreword by Jean Mawhin, Encyclopedia Math. Appl. 162, Cambridge University Press, Cambridge, 2016. 10.1017/CBO9781316282397Search in Google Scholar
[33] J. J. Moré, B. S. Garbow and K. E. Hillstrom, User Guide for MINPACK-1, Technical Report ANL-80-74, Argonne National Laboratory, Argonne, 1980. 10.2172/6997568Search in Google Scholar
[34] S. Neukamm and I. Velcic, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Math. Models Methods Appl. Sci. 23 (2013), 2701–2748. 10.1142/S0218202513500449Search in Google Scholar
[35]
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractionalp-Laplacian in
[36] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), 1529–1566. 10.1016/j.jde.2014.05.023Search in Google Scholar
[37] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. Search in Google Scholar
[38] A. Sapora, P. Cornetti and A. Carpinteri, Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 63–74. 10.1016/j.cnsns.2012.06.017Search in Google Scholar
[39] S. A. Silling, Origin and Effect of Nonlocality in a Composite, Sandia Report SAND2013-8140, Sandia National Laboratories, Albuquerque, 2014. 10.2172/1147358Search in Google Scholar
[40] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. Search in Google Scholar
[41] V. E. Tarasov, Fractional gradient elasticity from spatial dispersion law, Condens. Matter Phys. 2014 (2014), Article ID 794097. 10.1155/2014/794097Search in Google Scholar
[42] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S 7 (2014), 857–885. 10.3934/dcdss.2014.7.857Search in Google Scholar
[43] Q. Yang, F. Liu and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model. 34 (2010), 200–218. 10.1016/j.apm.2009.04.006Search in Google Scholar
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