Toward fractional gradient elasticity

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 σしぐま(x) is the stress and εいぷしろん(x) is the strain, with the space variable x and the scale parameter lβべーた2 being dimensionless. The symbol CDa+βべーた is the Caputo derivative of order βべーた with n-1<βべーた<n. The ± sign is kept for generality, as various previous nonfractional gradient elasticity models use either sign (for a comprehensive of nonfractional gradient elasticity models, the reader may consult [18]). The left-sided Caputo fractional derivative [3] of order αあるふぁ>0 for x∈[a, b] is defined by

where n-1<αあるふぁ<n and Ia+αあるふぁ is the left-sided Riemann-Liouville fractional integral of order αあるふぁ>0 defined as

Ia+αあるふぁf(x)=1Γがんま(αあるふぁ)∫axf(z)dz(x-z)1-αあるふぁ, (z>a).

Then using the usual definition of the strain εいぷしろん(x) in terms of the displacement u(x)

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_{αあるふぁ}(x) and f(x) denoting, as usual, the external body force field. The explicit form of the function A_{αあるふぁ}(x) is derived from the conservation law for non-local media using the fractional vector calculus [19]. Substitution of Eq. (4) into (5) gives

where

For the case αあるふぁ=1, the governing fractional differential equation reads

In general, we have Dx1 CDxβべーた+1≠ CDxβべーた+2.

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  CDa+αあるふぁ CDa+βべーた and the derivative  CDa+αあるふぁ+βべーた.

Using Eq. (2.4.6) in [3] of the form

and using Property 2.1, Eq. (2.1.16), in [3],

where αあるふぁ>0 and βべーた>-1, we obtain the relation

where 0<αあるふぁ≤1, n-1<βべーた≤n. This relation explicitly shows a violation of the semigroup property for the Caputo derivative.

Using Eq. (11), we rewrite Eq. (6) in the form

where 0<αあるふぁ<1, 1<βべーた<2 (n=2), or 2<βべーた<3 (n=3) and f_eff(x) is an effective body force defined by

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, f_eff(x)=0). To use these results, we assume that a=0. Let us consider the case 0<αあるふぁ<1, 1<βべーた<2 (i.e., 1<αあるふぁ+1<2=m, 2<βべーた+1<3=n). Then the solution of Eq. (12) has the form

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_{αあるふぁ,βべーた}(z) is the Mittag-Leffler function [3], which is defined by

Note also that E_{1, 1}[z]=e^z. The asymptotic behavior (see Eq. (1.8.27) in [3]) of the Mittag-Leffler function E_{αあるふぁ,βべーた}(z) is

where 0<αあるふぁ<2.

Remark: It should be emphasized that the absence of the external force (f(x)=0) does not imply the vanishing of the effective force f_eff. In general, f_eff(x)≠0 for f(x)=0. Only in the case of commutativity of the Caputo fractional derivatives, i.e., if the semigroup property

( CDa+αあるふぁ CDa+βべーたu)(x)=( CDa+αあるふぁ+βべーたu)(x)

is not violated, the vanishing of the external force f(x)=0 leads to the vanishing of the effective force f_eff(x)=0. It is easy to see that the semigroup property is satisfied if

If we consider Eq. (12) in the case αあるふぁ=1, 1<βべーた<2, f(x)=0, and assume that condition (21) is satisfied, then solution (14) of Eq. (12) has the form

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.

3.1 Sub-gradient elasticity model

The fractional model of sub-gradient elasticity is described by Eq. (24) with αあるふぁ=2 and 0<βべーた<2, i.e.,

The order of the fractional Laplacian (-Δでるた)^{βべーた/2} is less than the order of the first term related to the usual Hooke’s law. As a simple example, we can consider the square of the Laplacian, i.e., βべーた=1. In general, parameter βべーた defines the order of the power law non-locality that characterizes the elastic continuum. The particular solution of Eq. (35) for the point force problem at hand reads

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 (-Δでるた)^{βべーた/2} with 0<βべーた<2 is given by

for large distances |r|»1.

3.2 Super-gradient elasticity model

The fractional model of super-gradient elasticity is described by Eq. (24), with αあるふぁ>2 and βべーた=2. In this case, we have

The order of the fractional Laplacian (-Δでるた)^{αあるふぁ/2} is greater than the order of the first term related to the usual Hooke’s law. Parameter αあるふぁ>2 defines the order of the power law non-locality of the elastic continuum. If αあるふぁ=4, Eq. (41) reduced to Eq. (26). The case 3<αあるふぁ<5 can be considered to correspond as closely as possible (αあるふぁ≈4) to the usual gradient elasticity model of Eq. (26).

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 βべーた and that the corresponding relation (42) does not depend on c_βべーた. The displacement field u(r) for short distances away from the point of load application is determined only by the term with (-Δでるた)^{αあるふぁ/2} (αあるふぁ>βべーた), which can be considered as a fractional counterpart of the usual extra non-Hookean term of gradient elasticity.

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].