Fractional Euler–Bernoulli beams: Theory, numerical study and experimental validation

Abstract

In this paper the classical EULER-BERNOULLI beam (CEBB) theory is reformulated utilising fractional calculus. Such generalisation is called fractional Euler-Bernoulli beams (FEBB) and results in non-local spatial description. The parameters of the model are identified based on AFM experiments concerning bending rigidities of micro-beams made of the polymer SU-8. In experiments both force as well as deflection data were recorded revealing significant size effect with respect to outer dimensions of the specimens. Special attention is also focused on the proper numerical solution of obtained fractional differential equation.

Strength measurement and rupture mechanisms of a micron thick nanocrystalline MoS2 coating using AFM based micro-bending tests

“…While this leads to complications for nanoindenters and in situ Scanning Electron Microscope experiments (SEM), Atomic Force Microscopy (AFM) can be a good solution as higher resolution in force, deflection, and accurate loading location can be achieved. Micro-bending tests using AFM on submicron beams fabricated by FIB [31] and chemical and wet etching processes [32][33][34][35] have allowed the study of stiffness, bending strength, fracture toughness, and even fatigue toughness of materials.…”

Numerical Study for Fractional Euler-Lagrange Equations of a Harmonic Oscillator on a Moving Platform

“…Fractional calculus was regarded for a long time only from a mathematical point of view. Recently, the fractional calculus becomes a very useful and powerful tool in many branches of sciences and engineering [1][2][3]. Several definitions of noninteger order operators have been introduced and used; such as the Riemann-Liouville integral and differential operators, the Caputo derivatives, the Hadamard integral, the Weyl integral, etc.…”

One-dimensional dispersion phenomena in terms of fractional media

“…This paper presents the analysis of the dispersive behaviour of a 1D solid undergoing axial vaibrations, using the Fractional Continuum Mechanics approach proposed by Sumelka [28,30]. The formulation uses the fractional RieszCaputo derivatives in the definition of the longitudinal deformation which, together with the Young modulus, leads to two additional model parameters: the size of the non-local surrounding l f and the order of the fraction derivative α.…”

“…In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach proposed by Sumelka [28,30], and Sumelka et al [31]. This proposed formulation introduces non-locality in the spatial variable using Riesz-Caputo (RC) fractional derivative [32,33], and introduces two phenomenological/material parameters: 1) the order of fractional continua α; and 2) the size of non-local surrounding l f .…”