On the fractional deformation of a linearly elastic bar

Abstract: Fractional derivatives have non-local character, although they are notmathematical derivatives, according to differential topology. New fractional derivatives satisfying the requirements of differential topology are proposed, that have non-local character. A new space, the Λ-space corresponding to the initial space is proposed, where the derivatives are local. Transferring the results to the initial space through Riemann-Liouville fractional derivatives, the non-local character of the analysis is shown up. Since fractional derivatives have been established, having the mathematical properties of the derivatives, the linearly elastic fractional deformationof an elastic bar is presented. The fractional axial stress along the distributed body force is discussed. Fractional analysis with horizon is also introduced and the deformation of an elastic bar is also presented.


Introduction
Fractional Calculus is a blooming mathematical tool for various applications in physics and engineering. Machado et al. [1] have presented a historical retrogression for fractional calculus originated by Leibnitz [2], Liouville [3] and established by Riemann [4]. The main advantage of fractional calculus is the non-local character of the fractional derivatives, contrary to the common derivatives, that have inherited by their definition a strong local property. Fractional calculus has been an indispensable tool in various branches in physics, mechanics, control theory, engi-*Corresponding Author: Konstantinos A. Lazopoulos: 14 Theatrou Str., Rafina, 19009 Greece; Email: orfeakos74@gmail.com Anastasios K. Lazopoulos: Mathematical Sciences Department, Hellenic Army Academy Vari, 16673 Greece neering and economics [5][6][7][8]. Classical references are the books [9][10][11] including also various applications.
Although fractional calculus was intensively used for the last 50 years, it was referred only to time non-locality Klimek [12]. Lazopoulos [13] turned the attention from time to space, just to include non-local properties in space due to inhomogeneities, microcracks etc. In fact he established the fractional strain with strong non-local character, contrary to the local character of the conventional strain, Truesdell [14]. However, all the well known fractional derivatives have mainly an operative character, instead of a derivative one. All the known derivatives do not satisfy the properties of a derivative demanded by Differential Topology, just to correspond to a fractional differential. So their use was not mathematically established, but it has an ad-hoc character. Lazopoulos [15] trying to fill that gap, proposed the fractional L-derivative, that had already been presented. Nevertheless, that effort was not successful, since again the conditions demanded by differential topology were not satisfied. Lately, Lazopoulos [16] proposed the fractional Λ-derivative, that is a modification of the fractional L-derivative, along with the fractional Λspace where the fractional Λ-derivative behaves in conventional derivative rules. Further following that procedure, the beam bending problem, Lazopoulos [22], along with the fractional Taylor Series, with the fractional variation calculus and branching analysis of an axially compressed rod, Lazopoulos [23], has been presented.
The present work is targeting to the establishment of the axial fractional strain, that acquires non-local character, contrary to the conventional strain that is local. With the establishment of the fractional Λ-derivative and fractional Λ-space, the fractional Λ-strain is proposed, that has non-local character in the initial space, whereas it preserves local character in the fractional Λ-space. A linear elastic bar with given axial displacement is studied, defining the distribution of axial stress along with the distribution of body forces. Further, motivated by peridynamic theory, the horizon of the fractional influence is introduced. Again the deformation of a linearly elastic bar is discussed with fractional horizon.

Basic properties of Fractional Calculus
Fractional Calculus has recently become a branch of pure mathematics with many applications in physics and engineering. Many definitions of fractional derivatives exist. In fact, Fractional Calculus originated by Leibniz, looking for the possibility of defining the derivative d n g dx n when n = 1 2 . The various types of the fractional derivatives exhibit some advantages over the others. Nevertheless they all are non local, contrary to the conventional ones.
The detailed properties of fractional derivatives may be found in Kilbas et al. [17], Podlubny [10], Samko et al. [8]. Starting from Cauchy formula for the n-fold integral of a primitive function f(x) x > 0, n ∈ N and x > 0, n ∈ N we define the left and right fractional integral of f as: In Eqs. (3,4) we assume that is the order of fractional integrals with m < ≤ m + 1, considering Γ(x) = (x − 1)! with Γ( ) Euler's Gamma function.
Thus the left and right Riemann-Liouville (R-L) derivatives are defined by: and The L-FD was first pointed out in Lazopoulos et al. [15] as the derivative that occurs from the Adda [8] definition of a fractional differential (See Lazopoulos et al. [18,19]): where d f (x) is the fractional differential of function f (x) and d a x the fractional differential of x. The first definition of the L-FD was stated as the ratio of the corresponding Caputo derivatives: Nevertheless, in the article we revised the L-FD, using Riemann-Liouville derivatives: Eqs. (8,9) are equivalent when f(a)=0 and f(x) is differentiable.

The proposed Λ-fractional space
The main disadvantage of the existing fractional derivatives is that they fail to satisfy Leibnitz and composition rules. Therefore they cannot correspond to a differential. As a consequence, the use of the various fractional derivatives in mathematical, geometrical and physical problems is questionable. On the other hand, the proposed formulation yields derivatives corresponding to differentials, fulfilling all the necessary conditions demanded by a derivative according to Differential Topology. Hence, it could be feasible to create a fractional differential geometry theory, not in the original coordinates but in a coordinate system which will be defined below. Let us consider the R-L derivative, of a function f(x): Hence, Eq. (9) may define the Λ-fractional derivative with: The -Riemann-Liouville fractional derivative is expressed as a common derivative of the ( -1)-Riemann-Liouville fractional derivatives, hence as a common derivative corresponds to a differential. Although for the present purposes the R-L derivative might be satisfactory, variational problems might be raised with its use, since Noether's variational theory might be questionable and the adoption of that derivative might not satisfy geometrical and physical demands, Atanackovic et al. [4]. The Λ-Fractional Derivative introduced by Lazopoulos [16], might be adopted without the problems that the R-L exhibits. The Λ-Fractional Derivative has been defined by: Introducing the new variable, and imposing F (X) = D −1 (f (x (X))), the L-fractional derivative, Eq. (9) may be expressed as a common derivative of the function F(X) with respect to X. Hence the analysis has been transferred to plane (X, F(X)) with the conventional differential analysis. Then the results may be transferred back to the initial variables. Indeed, the transferring may be performed through the relation, Following that procedure, various fractional mathematical analysis areas, such as Fractional Differential Geometry, Fractional Field Theory, Fractional differential equations etc., may be established. The proposed Λ-fractional derivative satisfies Leibnitz's rule for the product of the I 1−a f (x) and I 1−a y(x). Indeed, Furthermore, the fractional derivative of a composite function is defined by, The linearity properties along with Leibnitz's and composition rules, Eqs. (16,17), satisfy all the Differential Topology conditions, Chillingworth [18], for the existence of a differential and further the existence of Fractional Differential Geometry. In fact the non-local mathematical analysis is established with derivatives having non-local character. Yet, some of the most established books in Fractional Analysis are referred [8][9][10], just to compare the present version of the Fractional Analysis to the existing one. Further, the analysis is simplified when 0 < < 1. In that case, the -multiple integral is defined by with corresponding definitions for the right fractional integrals and derivatives [9]. Moreover, the Λ-fractional derivative is defined by, In addition the results from the Λ-space may be transferred to the initial space through the relation, Defining as X = a Ix 1− x and F(X) = a I 1− x f (x), the Λ-FD is defined as a conventional derivative in the fractional space (X, F(X)). In fact the Fractional Differential Geometry is defined as a conventional differential geometry in the Λ-fractional space, (X, F(X)).
Just to clarify the ideas, let us work as an example on the function, Then the Λ-fractional plane (X,F(X)) is defined by Further considering from Eq. (23), Eq. (25) yields Therefore the curve in the original plane (x, f(x)) shown in Figure 1 corresponds to the fractional plane (space) shown in Figure 2, for = 0.6 Further, the derivative with M = (2 − 3 + 2 )XΓ(1 − ). For X 0 = 0.6 and = 0.6, the derivative in the fractional plane is equal to D(F(X 0 )) = 1.1580.
Since the tangent space Y(X) of the curve at a point X 0 is defined by the line, In the original plane (x, f(x)) the corresponding tangent space is defined by the curve that will be built as follows: In the present case, for the function Thus the fractional tangent space g(x) in the original space (x, f(x)) is defined as Hence, at X 0 = 0.6 for = 0.6, x 0 = 0.81 the tangent space is defined by for = 0.6 at x = 0.81

The fractional deformation of a linearly elastic bar
Let us consider a bar of length L, linearly elastic, with elastic modulus E, following the fractional analysis, see Lazopoulos [16]. If u(x) and σ(x) are the displacement and the stress at a point x of the bar in the original space, the Here f(x) might express the x coordinate, the displacement u(x) or the stress σ(x) functions corresponding to the l X, the displacement l U(X) and and the stress l Σ(X) functions in the left Λ-Fractional space. The units dimension of the function f(x) in the initial space is increased during its transformation F(X) in the Λ-space by L 1− where L is the physical unit of the length in the initial space. Let us point out that: Further, the displacement function U(x) in the left fractional space Λ is defined by, Therefore, the fractional stress l Σ(X) may be defined through the linear elastic low, Furthermore, the linear momentum equation demands for equilibrium, where l P (︁ l X )︁ is the body force per unit length in the left Λ-space. Transferring the problem to the initial space, the stress and the distributed body force may be defined by the low expressed by Eq. (21), Having defined the stress and the distributed body force for the left fractional space, the procedure is repeated for the right Λ-space, defining the right stress and body force in the original space. Indeed, for the right part of the rod (x,L) the corresponding Λ-space is defined by, where again f(x) might be the x coordinate, the displacement u(x) and the stress σ(x) functions corresponding to the r X, r U(X) and r Σ(X) functions in the right Λ-Fractional space. Indeed, Further, the displacement function U(x) in the right fractional space Λ is defined by, Therefore, the fractional stress r Σ(X) may be defined through the linear elastic low, Furthermore, the equilibrium equation demands for equilibrium, Transferring the stresses from the Λ-space to the initial space the Eq. (21) should be recalled.
Also the corresponding equation for the right Λ-space, should be mentioned. Then, the stress is defined as the mean value of the left and right stress. Just the same is valid for the body forces. The following application shows the methodology clarifying the various steps of the present procedure.

Application of fractional deformation of a bar
Let us consider a bar of length L and elastic modulus E, on the axis xϵ(0, L). The displacement field of the bar is defined by: where a ≤ 1, since the discussed problem is considered in linear elasticity where the displacement field is considered as infinitesimal.
Since the derivation is valid in the Λ-space, all the corresponding functions have to be transferred in that fractional space. For the left fractional space the coordinate, see Eq. (34), Further, the corresponding left fractional displacement function in the Λ-space is defined, see Eq. (35) by, Since the axial stress in the left Λ-space is defined by Eq. (36) with the right stress function in the Λ-space defined by the function: where: The stress function in the Λ-space is the mean value of the right and left values. Therefore, Further, the distributed body forces P(X) in the fractional Λ-space are defined through the function  For the specific values, L = 3, = 0.7, the stress function in the Λ-space with respect to the initial x axis has been shown in Figure 5.
Transferring the stresses into the initial space: Recalling Eq. (11) the stress σ(x) in the initial space might be found. Omitting the various computations performed with the help of Mathematica computing algebra pack, the stress has been computed for L = 3 and = 0.7. Further the distributed body force in the initial space has also been computed through, Again the various computations performed with the help of Mathematica computing algebra pack, the stress has been computed for L = 3 and = 0.7. Although the body force exhibits a singularity, the fractional stress is smooth. Indeed, Further, the displacement function U(x) in the right fractional space Λ is defined by, Therefore, the fractional stress r Σ(X) may be defined through the linear elastic low, Furthermore, the linear momentum equation demands for equilibrium, Transferring the stresses from the Λ-space to the initial space the eq. (21) should be recalled.
Also the corresponding equation for the right Λ-space, should be mentioned. Then, the stress is defined as the mean value of the left and right stress. Just the same is valid for the body forces.

Application of fractional deformation with horizon
Let us consider a bar of length L and elastic modulus E, on the axis x. The displacement field of the bar is defined by, where a ≪ 1, since the discussed problem is considered in linear elasticity where the displacement field is considered as infinitesimal.
Since the derivation is valid in the Λ-space, all the corresponding functions have to be transferred in that fractional space. For the left fractional space the coordinate X, see Eq. (56), Further, the corresponding left fractional displacement function in the Λ-space with horizon δ is defined, see Eq. (57) by, Since the axial stress in the left Λ-space is defined by Eq. (58) Further, the right fractional displacement function in the Λ-space, Eq. (63) is defined by, with the right stress function in the Λ-space defined by the function, The stress function in the Λ-space is the mean value of the right and left values. Therefore, Further, the distributed body forces P(X) in the fractional Λ-space are defined through the function For the specific values, δ = 0.2, = 0.7, the stress function in the Λ-space with respect to the initial x axis has been shown in Figure 9.  Transferring the stresses into the initial space, we have: Recalling Eq. (11) the stress σ(x) in the initial space might be found. Omitting the various computations performed with the help of Mathematica computing algebra pack, the stress has been computed for δ = 0.2 and = 0.7. Figure 11: The distribution of stresses in the initial space with respect to the initial x axis Further the distributed body force in the initial space has also been computed through, Again the various computations performed with the help of Mathematica computing algebra pack, the stress has been computed for δ = 3 and = 0.7. Figure 12: Distribution of body forces in the initial space with respect to the initial x axis

The 3D fractional equilibrium problem
Let us point out that geometry exists only on the fractional Λ-space. Therefore all the conventional mechanics rules are valid in that space. The various results may be transferred as simple functions in the initial space. Let us consider the 3D displacement vector u(x) in the initial space. Then the strain function in the fractional Λ-space may be defined by Eq. (33) Further, the corresponding X vector in the fractional Λspace is expressed by: Inserting X from the Eq. (77) into Eq. (78), the fractional Λspace displacement vector U(X) is defined. Hence the fractional linear strain function is given by: Further the geometrical shape of the body b(x) may be transferred to the fractional Λ-space as B(X). In addition any function, like stresses, forces etc may be transferred to the fractional Λ-space. In the Λ-fractional space the derivatives are local and everything works following the conventional procedure. The linear elastic Hook's law is valid with, Solving that problem in the fractional Λ-space, the various results Q(X) may be pulled back like functions to the original space, using the equation,

Conclusion and further research
Introducing the Λ-fractional derivative satisfying the properties of a derivative in the Λ-fractional space, demanded by differential topology, the deformation of an elastic bar governed by fractional analysis has been presented. The fractional stresses along with the fractional body forces have been defined for a given displacement field of the bar. The same problem has been discussed for fractional deformation of the bar with horizon. Again the fractional stresses along with fractional body forces have been defined for the same displacement field. The next step is the discussion of fractional continuum mechanics and fractional dynamics demanding a sound definition and development of fractional differential geometry.