Fractional Differential Equations

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

The present literature review focuses on geometrically non-linear free and forced vibrations of shells made of traditional and advanced materials. Flat and imperfect plates and membranes are excluded. Closed shells and curved panels made of isotropic, laminated composite, piezoelectric, functionally graded and hyperelastic materials are reviewed and great attention is given to non-linear vibrations of shells subjected to normal and in-plane excitations. Theoretical, numerical and experimental studies dealing with particular dynamical problems involving parametric vibrations, stability, dynamic buckling, non-stationary vibrations and chaotic vibrations are also addressed. Moreover, several original aspects of non-linear vibrations of shells and panels, including (i) fluid–structure interactions, (ii) geometric imperfections, (iii) effect of geometry and boundary conditions, (iv) thermal loads, (v) electrical loads and (vi) reduced-order models and their accuracy including perturbation techniques, proper orthogonal decomposition, non-linear normal modes and meshless methods are reviewed in depth.

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Non-linear vibrations of shells: A literature review from 2003 to 2013

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory

On the Λ-fractional spherical ballons


 Λ-Fractional Mechanics has already been introduced since it combines non-locality with mathematical analysis. It is well established, that conventional mechanics is not a proper theory for describing various phenomena in micro or nanomechanics. Further, various experiments in viscoelasticity, fatigue, fracture, etc., suggest the introduction of non-local mathematical analysis in their description. Fractional calculus has been used in describing those phenomena. Nevertheless, the well-known fractional derivatives with their calculus fail to generate differential geometry, since the established fractional derivatives do not fulfill the prerequisites of differential topology. A Λ-fractional analysis can generate geometry conforming to the prerequisites of differential topology. Hence Λ-fractional mechanics deals with non-local mechanics, describing the various inhomogeneities in various materials with more realistic rules.


On Λ-Fractional peridynamic mechanics

Abstract Bending deformations are reviewed in the context of strain gradient linear elasticity, considering the complete set of strain gradient components. It is well understood that conventional bending deformations depend on the collective uniaxial extension of axial fibers resulting in the dependence on the curvature of the neutral geometry of various (linear or surface) structures. Nevertheless, the deformation of each fiber depends not only on the local curvature of the neutral geometry but also on the distance of the fiber from the neutral axis. Hence, the strain gradient tensor of the conventional bending strain should include not only components along the neutral axis but also those on the transverse direction. The problems of bending and buckling, along with geometrically non-linear and post-critical behavior, are reviewed in the context of strain gradient elasticity considering not only conventional bending strain but also the complete components of the strain gradient.

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On the strain gradient bending deformations

Fractional differential equations are solved with L-fractional derivatives, using numerical procedures. Two characteristic fractional differential equations are numerically solved. The first equation describes the motion of a thin rigid plate immersed in a Newtonian fluid connected by a massless spring to a fixed point, and the other one the diffusion of gas in a fluid.

A.K. Lazopoulos

Papers

On the Λ-fractional spherical ballons

On Λ-Fractional peridynamic mechanics

On the strain gradient bending deformations

Numerical solutions of differential equations with fractional L-derivative