Fractional mechanical model for the dynamics of non-local continuum

Abstract

In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both model shapes and natural frequency of the non-local systems are then studied.

Appendix: Properties of fractional operators

Fractional calculus might involve very cumbersome calculations that can hardly be tackled by hand. In some cases, the composition properties and the Leibniz rule may be helpful and are reported in the following.

1.1 A.1 Leibniz rule

In this section, we present the extension of the classical Leibniz rule

$$D^n \left( {f_{} g} \right) = \sum\limits_{j = 0}^n {\left( {\begin{array}{c} n \\ j\end{array}} \right)D^{n - j} \left( f \right)D^j \left( g \right)}$$

((A.1))

to fractional derivatives. It is an important rule because it consents to transform fractional derivatives of many functions in series forms, without performing any burdensome integration. Indeed, given two functions f and g analytic in the interval $\left[ {a,b} \right]$, ([35], p. 280), the generalized Leibniz rule is expressed in the form

$$\left( {{\cal D}_{a + }^\gamma f_{} g} \right)\left( x \right) = \sum\limits_{j = 0}^\infty {\left( {\begin{array}{c} \gamma \\ j\end{array}} \right)\left( {{\cal D}_{a + }^{\gamma - j} f_{} } \right)\,\left( x \right)\,g^{\left( j \right)} \left( x \right)}$$

((A.2))

where $\left( {\begin{array}{c} \gamma \\ j\end{array}} \right)$ are the binomial coefficients and $g^{\left( j \right)}$ is the derivative of integer order j. One must note that the presence of the integer derivative of the function $g\left( x \right)$ can be fruitful from the computational perspective. Indeed, the knowledge of the jth classical derivative of $g\left( x \right)$, combined with the fractional derivative of a constant, for example, $f\left( x \right) = 1$, gives quite easily a series representation of the fractional derivative of $g\left( x \right)$ in the form

$$\left( {{\cal D}_{0 + }^\gamma 1_{} g} \right)\left( x \right) = \sum\limits_{j = 0}^\infty {\left( {\begin{array}{c} \gamma \\ j\end{array}} \right)\frac{{x^{j - \gamma } }}{{\Gamma \left( {j - \gamma + 1} \right)}}\,g^{\left( j \right)} \left( x \right)}$$

((A.3))

with the particular choice $a = 0$. The last expression suggests that any fractional derivative, if it exists, has a series representation.

1.2 A.2 Compositions rules

Some useful properties dealing with fractional operators will be indicated; for readability’s sake, we will report only some simple composition rules referring to [26, 35] for further relations and rigorous proofs. Let us suppose the existences of the integrals and derivatives involved in the following relations, indicated with $\gamma _1 > 0$ and $\gamma _2 > 0$, then, the properties

$$I_{a + }^{\gamma _1 } I_{a + }^{\gamma _2 } f = I_{a + }^{\gamma _1 + \gamma _2 } f = I_{a + }^{\gamma _2 } I_{a + }^{\gamma _1 } f$$

((A.4))

$$I_{b - }^{\gamma _1 } I_{b - }^{\gamma _2 } f = I_{b - }^{\gamma _1 + \gamma _2 } f = I_{b - }^{\gamma _2 } I_{b - }^{\gamma _1 } f$$

((A.5))

on the composition of different order integrals are valid. On the contrary, commutation between fractional integration and differentiation is not straightforward and needs some introductory remarks. In ordinary calculus it is well known that performing first the integral of a function and then a derivative, $\frac{d}{{dx}}\int_a^x {f\left( x \right)dx = f\left( x \right)} ,$ the original function is obtained.

But, if one changes the operations order, i.e, $\int_a^x {\frac{d}{{dx}}f\left( x \right)dx \ne f\left( x \right)}$ the result is different because of the presence of a constant. For higher order derivatives and integrals, in the same way $\frac{{d^n }}{{dx^n }}I_{a + }^n f = f$ holds, but $I_{a + }^n \frac{{d^n }}{{dx^n }}f$ differs from the function by a polynomial of order $n - 1$. This simple argument is valid also in the case of fractional operators. Then, the equality

$$D_{a + }^\gamma I_{a + }^\gamma f = f\left( x \right)$$

((A.6))

is always valid and, conversely,

$$I_{a + }^\gamma D_{a + }^\gamma f = f\left( x \right)$$

((A.7))

has been shown [35, p. 43–45] to be true only for those functions having

$$\frac{{d^k }}{{dx^k }}I_{a + }^{1 - \left\{ \gamma \right\}} f\left( a \right) = 0 {\rm for} for k = 1,2,\ldots ,\left[ \gamma \right]$$

((A.8))

Also the simultaneous application of integration and differentiation can be simplified in the following way with $f\left( x \right) \in Leb_1 \left( {\left[ {a,b} \right]} \right){\rm{:}}$

$$D_{a + }^{\gamma _2 } I_{a + }^{\gamma _1 } f = I_{a + }^{\gamma _1 - \gamma _2 } f,\,\,\,\,\,\,\,\,\,\,{\rm{Re}}\gamma _1 > {\rm{Re}}\gamma _2$$

((A.9))

On the contrary, if $f\left( x \right)$ does not satisfy Eq. (A8), the relation

$$\left( {I_{a + }^\gamma D_{a + }^\gamma f} \right)\left( x \right) = f\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{\left( {x - a} \right)^{\gamma - k - 1} }}{{\Gamma \left( {\gamma - k} \right)}}} \frac{d}{{dx^{n - k - 1} }}^{n - k - 1} I_{a + }^{1 - \left\{ \gamma \right\}} f\left( a \right)$$

((A.10))

holds.

In particular, the composition rules involving classical derivatives read

$$\frac{{d^n }}{{dx^n }}\left( {{\cal D}_{a + }^\gamma f} \right)\left( x \right) = \left( {{\cal D}_{a + }^{\gamma + n} f} \right)\left( x \right)$$

((A.11))

$$\frac{{d^n }}{{dx^n }}\left( {{\cal D}_{b - }^\gamma f} \right)\left( x \right) = \left( { - 1} \right)^n \left( {{\cal D}_{b - }^{\gamma + n} f} \right)\left( x \right)$$

((A.12))

$$\left( {{\cal D}_{a + }^\gamma \frac{{d^n }}{{dx^n }}f\left( x \right)} \right)\left( x \right) = \left( {{\cal D}_{a + }^{\gamma + n} f} \right)\left( x \right) - \sum\limits_{j = 0}^{n - 1} {\frac{{\left( {x - a} \right)^{j - \gamma - n} }}{{\Gamma \left( {1 + j - \gamma - n} \right)}}} f^{\left( j \right)} \left( a \right) $$

((A.13))