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

Time response of fractional automatic control systems

Projectile motion is studied using fractional calculus. Specifically, a newly defined fractional derivative (the Leibniz L-derivative) and its successor (Λ-fractional derivative) are used to describe the motion of the projectile. Experimental data were analyzed in this study, and conclusions were made. The results of well-established fractional derivatives were also compared with those of L-derivative and Λ-fractional derivative, showing the many advantages of these new derivatives.

Fractional Derivatives and Projectile Motion

Living adherent cells change their orientation in response to substrate stretching such that their cytoskeletal components reorganize in a new direction. To study this phenomenon, we model the cytoskeleton as a planar system of elastic cables and struts both pinned at their endpoints to a ∞at ∞exible substrate. Tensed (pre-strained) cables represent actin stress flbers, whereas compression-bearing struts represent microtubules. We assume that in response to uniaxial substrate stretching the model reorients and deforms into a new conflguration that minimizes its total potential energy. Using the Maxwell’s global stability criterion, we flnd global minima conflgurations during static extension and compression of the substrate. Based on these results, we predict reorientation during cyclic stretching of the substrate. We flnd that in response to cyclic stretching cells either reorient transversely to the direction of stretching, or exhibit multiple conflgurations symmetrically distributed relative to the direction of stretching. These predictions are consistent with experimental data on living cells from the literature.

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A model of cytoskeletal reorientation in response to substrate stretching

Infant hydrocephalus is a clinically abnormal clinical state with an accumulation of fluid in cavities (ventricles) deep within the brain. Hence, pressure is increased inside the skull. The ventricles widen due to the excess fluid applying pressure upon the (parenchyma) brain tissues. The infant brain tissue is described by a biomechanics model as a hyperelastic, Λ-fractional viscoelastic material, trying to describe the various conditions developing hydrocephalus. Λ-fractional continuum mechanics is applied with time variables due to viscosity and space fractional variables due to porosity. The simultaneous influence of both the viscosity and porosity of the membrane material (parenchyma) increases the cerebrospinal fluid’s pressure, causing the fluid’s accumulation in the brain.

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The Λ-Fractional Hydrocephalus Model

Fractional derivatives can express anomalous diffusion in brain tissue. Various brain diseases such as Alzheimer’s disease, multiple sclerosis, and Parkinson’s disease are attributed to the accumulation of proteins in axons. Discrete swellings along the axons cause other neuro diseases. To model the propagation of voltage in axons with all those causes, a fractional cable geometry has been adopted. Although a fractional cable model has already been presented, the non-existence of fractional differential geometry based on the well-known fractional derivatives raises questions. These minute parts of the human neural system are modeled as cables that function with a non-uniform cross-section in the fractional realm based upon the Λ-fractional derivative (Λ-FD). That derivative is considered the unique fractional derivative generating differential geometry. Examples are presented so that fruitful conclusions can be made. The present work is going to help medical and bioengineering scientists in controlling various brain diseases.

https://www.mdpi.com/2075-1680/12/2/136/pdf?version=1674979360

A.K. Lazopoulos

Papers

Time response of fractional automatic control systems

Fractional Derivatives and Projectile Motion

A model of cytoskeletal reorientation in response to substrate stretching

The Λ-Fractional Hydrocephalus Model

On Λ-Fractional Derivative and Human Neural Network