Fractional Nonlocal Model Research Articles

Nonlinear bending and postbuckling analysis of FG nanoscale beams using the two-phase fractional nonlocal continuum mechanics

The geometrically nonlinear bending and postbuckling of nanoscale beams are investigated herein according to the two-phase fractional nonlocal continuum model. It is considered that the beams have been made from functionally graded materials (FGMs), and the Bernoulli-Euler beam model is employed for their modeling. The variational form of the governing fractional equation is obtained first by means of an energy approach. Thereafter, a novel numerical solution method is proposed named as fractional variational differential quadrature method (FVDQM). In FVDQM, which is applied to the variational statement of the problem in a direct way, a combination of the differential quadrature method and matrix operators is utilized. The efficiency of the proposed fractional nonlocal model is evaluated by molecular dynamics (MD) simulations. Selected numerical results are given to explore the influences of fractional order, nonlocality, length-to-thickness ratio and FG index on the nonlinear bending and postbuckling responses of FG nanobeams with various types of boundary conditions.

A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam

The present study aims to analyze the buckling behavior of Euler–Bernoulli nanobeam in conjunction with a novel fractional nonlocal model namely conformable fractional nonlocal model. Fractional models are getting more popular among the researchers because of its applicability and fixability to handle many complex physical phenomena which are not possible to model with integer operators. Also, the main advantage of fractional models over integer model is its applicability to handle both the integer and noninteger operators which makes it much more flexible in term of real-world application. In this regards, the nonlocal constitutive relation is developed in conjunction with conformable fractional derivatives and fractional strain energy to analyze the buckling behavior of Euler–Bernoulli Nanobeam. In this study, the Simply Supported-Simply Supported (SS), Clamped-Simply Supported, and Clamped-Clamped boundary conditions are taken into the investigation with the help of the Differential Quadrature Method (DQM). Critical buckling load parameters are computed for the SS, CS, and CC boundary conditions from generalized eigenvalue problem. Graphical, as well as tabular results, are calculated by using MATLAB programmes and effects of various parameters such as fractional parameter, nonlocal parameter, aspect ratio on critical buckling load parameters extensively studied.

The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model

The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model

Fractional strain energy and its application to the free vibration analysis of a plate

Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and provides a new approach to non-local mechanics. In this study, a theoretical consideration on a new fractional non-local model is presented based on existence of fractional strain energy. It has two additional free parameters compared to classical local mechanics: (1) a fractional parameter which controls the strain gradient order in the strain energy relation and makes the model more flexible to describe physical phenomena, and (2) a non-local parameter to consider small scale effects in micron and sub-micron scales. The model has been used to obtain a fractional non-local plate theory. Free vibrations of a rectangular simply supported (S–S–S–S) plate has been investigated and the meaning of different parameters, such as fractional and non-local parameters, has been shown. The non-linear governing equation has been solved by the Galerkin method. A simple form of the governing equation and the numerical solution is an advantage of this fractional non-local model. Furthermore, the fractional nonlocal theory is contrasted with the Eringen nonlocal theory to show that fractional one enables to obtain much wider class of solutions.

Vibration analysis of FG nanobeams on the basis of fractional nonlocal model: a variational approach

The nonlocal vibrations of Euler–Bernoulli nanobeams are studied in this paper within the framework of fractional calculus. It is assumed that the material properties are functionally graded in the thickness direction and are estimated using the power-law function. Hamilton’s principle is applied to drive the fractional equation of motion which is then solved based on a new numerical approach named as variational finite difference method (VFDM). VFDM is formulated by the finite difference method (FDM) and matrix differential/integral operators. Since the method is directly applied to the variational form of governing equation, it is advantageous over existing approaches used for the fractional nonlocal models. The effects of nonlocality, fractional parameters and gradient of material on the fundamental frequencies of nanobeams subject to fully clamped, fully simply supported and clamped-simply supported boundary conditions are analyzed through illustrative examples.

A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams

The application of fractional calculus in fractional models (FMs) makes them more flexible than integer models inasmuch they can conclude all of integer and non-integer operators. In other words FMs let us use more potential of mathematics to modeling physical phenomena due to the use of both integer and fractional operators to present a better modeling of problems, which makes them more flexible and powerful. In the present work, a new fractional nonlocal model has been proposed, which has a simple form and can be used in different problems due to the simple form of numerical solutions. Then the model has been used to govern equations of the motion of the Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT). Next, free vibration of the Timoshenko and Euler-Bernoulli simply-supported (S-S) beam has been investigated. The Galerkin weighted residual method has been used to solve the non-linear governing equations.