Fractional Euler–Bernoulli Beam Theory Based on the Fractional

Vol. 134 (2018) ACTA PHYSICA POLONICA A No. 2 Fractional Euler–Bernoulli Beam Theory Based on the Fractional Strain–Displacement Relation and its Application in Free Vibration, Bending and Buckling Analyses of Micro/Nanobeams Zaher Rahimia;∗, Samrand Rash Ahmadia and W. Sumelkab aMechanical Engineering Department, Urmia University, Urmia, Iran bPoznan University of Technology, Institute of Structural Engineering, Piotrowo 5, 60-965 Poznan, Poland (Received May 12, 2018; in final form July 4, 2018) Applications of fractional calculus in the constitutive relation lead to the fractional derivatives models. They are stately generalization of the integer derivatives models — this general form makes fractional derivatives models more flexible and suitable to describe properties and behavior of different materials/structures. In the present work, the general strain deformation gradient has been presented by using the modified conformable fractional derivatives definition. Within this approach the fractional Euler–Bernoulli beam theory has been formulated and applied to the analysis of free vibration, bending and buckling of micro/nanobeams which exhibit strong scale effect. DOI: 10.12693/APhysPolA.134.574 PACS/topics: fractional derivatives model, fractional Euler–Bernoulli theory, free vibration, bending, buckling, nanobeam 1. Introduction structural mechanics and introduced nonlocal Kirchhoff– love plate theory. Atanackovic and Stankovic [9] modified the kinematics strain–displacement relationship with an Fractional calculus has become an exciting new math- alternative nonlocal formulation and by using the Ca- ematical method of solution of diverse problems in math- puto fractional derivatives generalized wave equation in ematics, science, and engineering [1–3]. Recent advances nonlocal elasticity. Lazopoulos [10] assumed that strain of fractional calculus are dominated by modern exam- energy density depends not only on the local strain but ples of applications in differential and integral equations, also on a fractional derivative of the strain. Carpinteri physics, signal processing, fluid mechanics, viscoelastic- et al. [11] by means of attenuation function of strain and ity, mathematical biology and electrochemistry [4]. the Caputo fractional derivatives introduced a fractional Applications of fractional calculus have been brought calculus approach to the nonlocal elasticity. to applied mechanics’ problems and leads to fractional In the present work, the general strain deformation derivatives models (FDMs). Many authors pointed out gradient has been presented by using modified con- that derivatives and integrals of non-integer order are formable fractional derivatives definition (CFDD). The very suitable for the description of properties of various fractional Euler–Bernoulli beam theory (FEBBT) has real materials [5]. These models are the generalized form been presented based on this general form and has two of classical models, therefore, they are able to describe free parameters: fractional parameter (which control the the behavior of materials better than integer derivatives displacement’s derivative in strain-displacement relation) models (IDMs). For instance, Challamel et al. [6] by us- and length scale parameter to consider size effects in the ing fractional derivatives (FDs) generalized the Eringen micron and the sub-micron scales. Finally, vibration of nonlocal theory and showed that the optimized fractional clamped-clamped (C-C) microbeams under axial force derivative model has a perfect matching with the disper- and free vibration, bending and buckling of nanobeam sive wave properties of the Born–Kármán model of lattice have been studied by the theory. It should be empha- dynamics and is better than the classical Eringen the- sized that compared to the previous papers [12–14] the ory. Also Demir et al. [7] studied vibration of viscoelastic novelty lays in the definition of the constitutive relation beam that obeys a fractional differentiation constitutive which is based on the fractional strain, as mentioned, law and stated that FDs are useful for describing the whereas in [12, 13] the constitutive relation is based on occurrence of vibrations in engineering practice. FDMs the general form of the Eringen nonlocal elasticity the- have been introduced in different problems, Sumelka [8] ory, and in [14] the formulation is based on the fractional applied fractional calculus to a classical problem of the strain energy. This paper has been divided into four sections: 1. Us- ing fractional Taylor series expansion and the CFDD, the ∗corresponding author; e-mail: [email protected] general form of the deformation gradient has been introduced. 2. The FEBBT based on the general form of (574) Fractional Euler–Bernoulli Beam Theory Based on the Fractional Strain–Displacement Relation. 575 the strain has been presented which has two free param- can be integer and non-integer number between 0 and eters: fractional parameter to control the displacement 1 (0 < α ≤ 1) [9] and leads to the appearance of the derivative in the constitutive relation and the length fractional calculus in the equation of the motion. scale parameter to consider the size effects in micron The general form of strain presented already by and sub-micron scales. 3. To demonstrate the func- Sumelka [8, 18] and Atanackovic and Stankovic [9] where tionality of FEBBT the non-dimensional frequency of based on the Riesz–Caputo definition in which the inte- the microbeams under axial force has been compared gral form makes the numerical solution of the governing with the experimental data and EBT. 4. Bending, equation difficult. In the present work, the general form buckling and vibration of nanobeams has been studied has been presented based on modified CFDD and un- based on the FEBBT. like previous works, its usability has been studied from two part of view: 1. geometrical view, by using fractional 2. Basic definitions and tools Taylor series expansion method, 2. continuum mechanics view. In addition, the use of the modified CFDD makes In this section, the basic definitions that are neces- the numerical solution of the governing equation simpler. sary for the theory are introduced namely the fractional derivatives definition and the fractional Taylor series 3.1. Geometrical derivation of the general strains expansion. FDs allow us to make a generalization of integer derivatives, and therefore in this part by using FTSE we present 2.1. Modified conformable fractional a general form of deformation gradient and improve the derivatives definition strain tensor by assuming a general form as below. Modified CFDD is a definition, which has been pre- Consider a two-dimensional deformation of an in- sented by Khalil et al. [15] and Tallafha et al. [16] is: finitesimal rectangular material element with dimensions Let f; g : R ! R. Then modified CFDD is dxα by dyα where 0 < α ≤ 1 but with this proviso that @αf ddαef(x) value of alpha satisfies the small deformation theory. It Dα(f)(x) = = jxj(dαe−α) : (1) α dαe means that the value of α must not be too small — af- @x jdxj ter the deformation rectangular material element should According to the modified CFDD, its application to the take the form of a rhombus. Based on small deformation multi-variable functions (Appendix A) reads: theory and fractional Taylor series expansion we have: α α dαe u (x + dx; y) ≈ u(x; y) + 1 @ u jdxjα with similar expan- α @ g (dαe−α) @ g(x; y) α @xα Dx (g)(x; y) = = jxj ; sions for all other terms — for α =1 the classical form is @xα j@xjdαe obtained of course. @αg @dαeg(x; y) Next, based on Eq. (1): Dα(g)(x; y) = = jyj(dαe−α) ; (2) y @yα dαe 1 @u j@yj u (x + dx; y) ≈ u(x; y) + jxj1−α jdxjα ; (4) where α 2 (0; 1]; f is (n + 1)-differentiable at x > 0 and α @x dαe is the smallest integer greater than or equal to α. so from the geometry of the deformed element 1 @αv α ( α @xα jdxj 1) we have 2.2. Fractional Taylor series expansion s α 2 α 2 α 1 @ u α 1 @ v α Fractional Taylor series expansion (FTSE) is a gener- A0B0= jdxj + jdxj + jdxj ≈ α @xα α @xα alization of the one presented in [17]. Assume that f is an infinitely α−differentiable func- 1 @αu 1 + jdxjα : (5) tion, for some 0 < α ≤ 1 at a neighborhood of a point α @xα x0. Then f has the fractional power series expansion in 1 @αv α 1 @αx α Note that when α ! 0 then α @xα jdxj or α @xα jdxj ! the form 1 @αv α 1 α (k) kα 1 α @xα jdxj << 1 since 0 < α ≤ 1:. The normal X (Dx f) j(x − x0)j f(x) = 0 ; strain in the x-direction of the rectangular element is de- αkk! 0 fined by: x − R1/α < x < x +R1/α; R > 0; (3) (3) A0B0 − AB 0 0 " = ; (6) where (Dα f)(k) means the application of the fractional x AB x0 k times derivative. and knowing that AB = dxα, we have α α−1 1 @ u "x = (l ) : (7) 3. General form of strain in terms α @xα of fractional calculus Similarly, the normal strain in the y-direction and z- direction becomes α α The general form of strains is obtained by general- α−1 1 @ v 1 α−1 @ w "y = (l ) ;"z = l ; (8) ization of the displacement derivatives in the strain- α @yα α @zα displacement relation, namely the first derivatives are where lα−1 is the length scale parameter similarly like in substituted with derivatives of order α. Parameter α the classical non-local gradient methods and fractional 576 Zaher Rahimi, Samrand Rash Ahmadi, W. Sumelka nonlocal Kirchhoff theory [8]. The introduction of the tions are R and C, respectively. The regular motion of length scale, allows finally to obtain dimensionless quan- the material body B can be written as tity and lets one to consider the size-dependence effects x = @(X; t) or xi = @i(X; t); (12) for micron and sub-micron scales.

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