Konstantinοσ Anastasios Lazopoulos

• Ph.D in Mechanics
• National Technical University of Athens

Fractional Analysis is a mathematical method based on different principles from those governing the well-known mathematical principles of differential and integral calculus. The main difference from ordinary differential analysis lies in its property being a non-local analysis, not a local one. This analysis is essential in studying problems in phy...

Buckling of axially loaded beams is discussed in the context of Λ-fractional analysis and mechanics. An axially compressed cantilever beam is considered in the Λ-fractional space, and the critical load is defined. The variational buckling problem of the simply supported beam is considered in the Λ-fractional space. It is pointed out that the Euler–...

Λ-fractional analysis has already been presented as the only fractional analysis conforming with the Differential Topology prerequisites. That is, the Leibniz rule and chain rule do not apply to other fractional derivatives; This, according to Differential Topology, makes the definition of a differential impossible for these derivatives. Therefore,...

Extending the investigation of the modified Van der Pol equation with its chaotic behaviour into the context of Λ -fractional analysis, the solutions of the Λ-fractional Van der Pol oscillator are studied under the influence of a horizon restricting the region of non-locality. The various solutions to the equation depend upon the horizon, however,...

After defining the fractional Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda $$\end{document}-derivative, having all the prerequisites for corresponding to...

Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the strain or the stre...

Global stability criteria for Λ-fractional Mechanics are established. The fractional extension of a bar under axial loading is discussed. Globally minimizing the total energy function, non-smooth deformations are introduced. The co-existence of phases phenomenon is established in Λ-fractional elasticity. It is pointed out that global minimization o...

Pointing out that Λ-fractional analysis is the unique fractional calculus theory including mathematically acceptable fractional derivatives, variational calculus for Λ-fractional analysis is established. Since Λ-fractional analysis is a non-local procedure, global extremals are only accepted. That means the extremals should satisfy not only the Eul...

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 ca...

Since the global stability criteria for Λ-fractional mechanics have been established, the Λ-fractional beam bending problem is discussed within that context. The co-existence of the phase phenomenon is revealed, allowing for elastic curves with non-smooth curvatures. The variational bending problem in the Λ-fractional space is considered. Global mi...

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...

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence...

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...

Λ-Fractional analysis was introduced to fill up the mathematical gap exhibited in fractional calculus, where the various fractional derivatives fail to fulfill the prerequisites demanded by differential topology. Nevertheless, the various advantages exhibited by the fractional derivatives, and especially their non-local character, attracted the int...

Zener's viscoelastic model is studied with the help of the non-local accurate Λ-fractional mathematical analysis. The model has been applied to the general Λ-fractional visco-elastic beam bending.

Fractional Differential Geometry of curves is discussed, with the help of a new fractional derivative, the Λ-fractional derivative, with the corresponding Λ-fractional space. Λ-Fractional derivative completely conforms with the demands of Differential Topology, for the existence of a differential. Therefore Fractional Differential Geometry is estab...

Answer to some comments presented by ResearchGate

After defining the fractional Λ-derivative, having all the requirements for corresponding to a differential, the fractional Λ-strain is established. Contrary to the common strain, that has a local character, fractional strain access a non-local character, quite important for expressing deformations in non-homogeneous media with microcracks and inho...

Fractional derivatives have non-local character, although they are not mathematical derivatives, according to differential topology.New fractional derivatives satisfying the requirements of differential topology are proposed, that have non-local character. A new space, the Λ-space corresponding to the initial space is proposed,where the derivatives...

Non-local plane elasticity problems are discussed in the context of Λ-fractional linear elasticity theory. Adapting the Λ-fractional derivative along with the Λ-fractional space, where geometry and mechanics are valid in the conventional way, non-local plane elasticity problems are solved with the help of biharmonic functions. Then, the results are...

Introducing the fractional \(\Lambda \)-derivative, with the corresponding \(\Lambda \)-fractional spaces, the fractional beam bending problem is presented. In fact, non-local derivatives govern the beam bending problem that accounts for the interaction of microcracks or materials non-homogeneities, such as composite materials or materials with fra...

Fractional Calculus is a robust mathematical tool with many applications in science and physics. Nevertheless fractional derivatives fail to fulfill the properties of the common derivatives since they do not correspond to differentials. Hence, their use in geometrical and physical problems is questionable. In the present article, a new fractional d...

The present correction concerns the work “On Fractional Modelling of ViscoelasticMechanical Systems”MRC,(2016), 78a, pp. 1–5. Of course the central point of that publication was the introduction of the L-Fractional derivative, instead of the other Fractional Derivatives that do not correspond to differential and are not allowed to participate in th...

Basic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been s...

Following the concepts of fractional differential and Leibnitz's L-Fractional Derivatives, proposed by the author [1], the L-fractional chain rule is introduced. Furthermore, the theory of curves and surfaces is revisited, into the context of Fractional Calculus. The fractional tangents, normals, curvature vectors and radii of curvature of curves a...

Clarifying the geometry of the fractional tangent space of a curve, fractional differential geometry of curves has already been revisited, Lazopoulos and Lazopoulos (2015), defining also the curvature vector. Fractional bending of a beam is introduced, applying Euler–Bernoulli bending principle. The proposed theory is implemented to the bending def...

The strain gradient elasticity is enriched with fractional derivatives of the strain, contributing for a more accurate description of the non-local stress–strain response, introducing a better description of the influence of microstructure (local and non-local), since in some respects the fractal texture of the material is also introduced. Using th...

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions...

Bending deformations of various surface structures, are studied within the context of a special theory of gradient elasticity (referred to as the GRADELA model), with a focus on plates and shells. In particular, the governing equilibrium equations for plates and non-linear shallow shells, along with the corresponding boundary conditions.

Since stress fibers have micro-size dimensions, their biomechanical behavior should demand mechanical models conforming with gradient strain deformation theories. In particular, the torsion and the stretching of stress fibers are discussed into the context of strain gradient elasticity theory and their size effects. It is proven for the torsion pro...

Adherent cells change their orientation to substrate stretching. This is a mechanochemical process involving a mechanical stretching signal and cytoskeletal remodeling. Stress fibers, which are bundles of actin filaments are aligned with the long axis of the cells. Let us recall that the long axis of a cell is directed towards the highest contracti...

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 (l...

The torsion problem of elastic bars of any cross-sections is discussed, into the context of strain gradient elasticity. It is proven that torsion problem is feasible only for the bars with circular cross-sections. For the other bars (with non-circular cross sections), the non-classical boundary conditions are not satisfied.

Considering the influence of the microstructure, the Timoshenko beam model is revisited, invoking Mindlin's strain gradient strain energy density function. The equations of motion are derived and the bending equilibrium equations are discussed. The adopted strain energy density function includes new terms. Those terms introduce the strong effect of...

Simple models for upper pharyngeal obstruction, describing the sleep apnea syndrome are proposed. Stability is discussed, of two and three individualized elements, with and without elastic connections, interacting with the steady flow. Considering the flow as the controlling parameter, critical steady state flows are located and their post-critical...

The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedur...

The critical loads for five non-conservative problems are defined under the context of gradient elasticity theory of a beam. The first problem deals with the stability of a gradient elastic beam compressed by a follower force (Beck's problem) and the second deals with the stability of a gradient elastic beam compressed by a force with a fixed line...

Singularity theory is applied for the study of the characteristic three-dimensional tensegrity-cytoskeleton model after adopting an incompressibility constraint. The model comprises six elastic bars interconnected with 24 elastic string members. Previous studies have already been performed on non-constrained systems; however, the present one allows...

It is well documented in a variety of adherent cell types that in response to anisotropic signals from the microenvironment cells alter their cytoskeletal organization. Previous theoretical studies of these phenomena were focused primarily on the elasticity of cytoskeletal actin stress fibers (SFs) and of the substrate while the contribution of foc...

Bending of strain gradient elastic thin plates is studied, adopting Kirchhoff’s theory of plates. Simple linear strain gradient elastic theory with surface energy is employed. The governing plate equation with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of t...

It is well documented in literature that under plane substrate stretching adherent cells reorganize their actin cytoskeleton by reorienting their stress fibres in one or two distinct directions, depending upon the magnitude of the substrate strain and the contractile mechanism of the cell. Since the cell is a quite deformable body, previous theoret...

Singularity theory is applied for the study of constrained tensegrity systems. Previous studies have already been performed on non-constrained systems; however, the present one allows for general non-symmetric equilibrium configurations. A modified T3 tensegrity model comprising seven rigid bars, three elastic cables and one rotational spring is co...

It is well documented that directed motion of cells is influenced by substrate stiffness. When cells are cultured on a substrate of graded stiffness, they tend to move from softer to stiffer regions--a process known as durotaxis. In this study, we propose a mathematical model of durotaxis described as an elastic stability phenomenon. We model the c...

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 flat flexible substrate. Tensed (pre-strained) cables repres...

It is well known that substrate stretching reorganizes the actin cytoskeleton of an adherent cell. Experiments have proved that the stress fibers are reoriented into one or two distinct directions. It is further pointed out that reorientation of the stress fibers phenomena are observed with quite high strains, where linear elasticity theory is not...

Equilibrium of a bar under uniaxial tension is considered as optimization problem of the total potential energy. Uniaxial deformations are considered for a material with linear constitutive law of strain second gradient elasticity. Applying tension on an elastic bar, necking is shown up in high strains. That means the axial strain forms two homogen...

Is classification of the singularities of the potential, concerning the homogeneous deformations in Finite Elasticity, an important material property? The present study demonstrates that the answer to the question is positive. Since the type of singularity prescribes Maxwell's sets in the neighborhood of a singularity, the emergence of multiphase s...

Singularity analysis is performed for homogeneous deformations of any hyper-elastic, constrained anisotropic material, under any type of conservative quasi-static loading. Critical conditions for branching of the equilibrium paths are defined and their post-critical behavior is discussed. Classification of the simple (cuspoids) and compound (umbili...

Stability studies of a T3 tensegrity structure are performed. This structure is composed of three slender struts interconnected by six nonlinear elastic tendons and is prestressed. The struts are governed by linear constitutive laws and are allowed to buckle. Since tensegrity is used for modeling structures with quite large deformations, for exampl...

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is intr...

Piece-wise homogeneous three-dimensional deformations in incompressible materials in finite elasticity are considered. The emergence of discontinuous strain fields in incompressible materials is studied via singularity theory. Since the simplest singularities, including Maxwell’s sets, are the cusp singularities, cusp conditions for the total energ...

A general method for the study of piece-wise homogeneous strain fields in finite elasticity is proposed. Critical homogeneous deformations, supporting strain jumping, are defined for any anisotropic elastic material under constant Piola–Kirchhoff stress field in three-dimensional elasticity. Since Maxwell’s sets appear in the neighborhood of singul...

Stability studies of a tensegrity structure, used as a model for cell deformability, are performed. This structure is composed by six slender struts interconnected by 24 linearly elastic tendons and is prestressed. The tendons and the struts are governed by linear constitutive laws. The struts are allowed to buckle. Since the deformations are large...

It is well documented that in response to substrate stretching adhering cells alter their orientation. Generally, the cells reorient away from the direction of the maximum substrate strain, depending upon the magnitude of the substrate strain and the state of cell contractility. Theoretical models from the literature can describe only some aspects...

Stability studies of a tensegrity structure, used as a model for cell deformability, are performed. This structure is composed of 6 slender struts interconnected by 24 linearly elastic cables. The cables and the struts are governed by linear constitutive laws. The struts are allowed to buckle. Adapting experimental evidence, the struts have already...

The experimental method of caustics is applied to buckling behaviour of straight cracked columns. Two major parameters of the problem may be defined through the caustics method: the SIF and the high strain internal region. The knowledge of these parameters may be used to improve predictions of the buckling behaviour of cracked columns.

A T-3 tensegrity structure composed by three struts and six elastic cables is considered. Adopting delay convention, stability of this model is studied. Two kinds of simple instabilities are investigated. The first is concerned with the global (overall) instability of the model and the second with the local-Euler-buckling of the struts. Compound in...

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts a...

A gradient strain elasticity theory of plates is developed for the study of non-linear problems. The existence of intrinsic (material) length modifies Von Karman's non-linear equations for plates. The theory is applied to the study of the buckling behavior of a long rectangular plate under uniaxial compression and small lateral load, supported on a...

Localized bending and buckling of long beam-like straight films due to the change of stiffness is presented. A two-phase beam model is developed. The one phase is considered of infinite stiffness (rigid deformation). The localized phase is studied and the rigid deformation is defined. This kind of two–phase deformations may take place in thin surfa...

A gradient strain beam model is proposed for studying the buckling behavior of thin structures, such as films and long beams supported on a rigid foundation. Since conventional methods hardly help in studying those problems, the multiple scales perturbation method is recalled yielding a localized solution.

For a pseudo elastic bar model with internal variable, we derive non-uniform solutions for the axial extension problem, including the spinoidal region. The variable may represent damage distribution. The theory is applied to the necking problem of an extended pseudo-elastic bar.

In a recent paper the authors have incorporated the notion of discontinuous internal variables into the theory of finite elasticity in order to model certain coexistence of phases phenomena encountered in the deformation of solids. For example, phase changes in polymeric solids may be associated with changes in material properties induced by damage...

The eversion problem of incompressible elastic cylinders is studied in the context of Finite Elasticity theory with non-smooth internal strain, Lazopoulos and Ogden [1]. Considering Mooney-Rivlin like materials with radially symmetric internal strain, non-classical eversion solutions of incompressible cylinders are obtained exhibiting the coexisten...

In this paper, a modified theory of nonlinear elasticity in which the strain-energy function depends on discontinuous internal variables is proposed. Specifically, the internal variables are allowed to be discontinuous across one or more surfaces. The objective is to model nonclassical phenomena in which two or more material phases are separated by...

For a thick incompressible hyperelastic plate under biaxial thrust, flexural and constitutive (homogeneous) instabilities are considered. In the present study their interaction (coupling) is discussed, when the critical conditions coincide for both kinds of instabilities. With the help of branching theory, especially the Fredholm Alternative theore...

The compound branching problem of elastic systems is discussed with neither elimination of the passive coordinates nor orthogonalization of the energy function. Critical conditions will be defined and the post-critical equilibrium paths are described. Formulae are derived that are proper for numerical techniques such as the Finite Element method. A...

Branching analysis for the homogeneous deformations of a compressible elastic unit cube under dead loading is performed. Critical conditions for branching of the equilibrium paths are derived and the post-critical equilibrium paths are described. Special attention is given to the compound branching.

Branching of a thick incompressible plate of hyperelastic material under uniaxial thrust is studied. The acceptable barreling second order displacement field is described, corresponding to the first order lower flexure mode. An application, clarifying the theory, is presented for a Mooney-Rivlin material

A bifurcation analysis for the equilibrium path of an axially loaded pinned-pinned beam with a non-convex stress-strain law is presented. Adopting global stability criteria and Maxwell’s law of equal area, the equilibrium configurations of a beam exhibiting the coexistence of phases phenomenon are studied. These states are similar to the elastic-pl...

In this paper branching analysis of elastic systems expressed in generalized coordinates will be discussed. The critical points and the post-critical paths will be described for simple but especially for double branching problems without requiring elimination of passive coordinates or orthogonalization of the energy function. The method is an indis...

The buckling problem of a cantilever column is discussed when the axial loading device introduces friction forces. Adopting global stability criteria and applying the bifurcation theory to the present non-conservative eigenvalue problem, the critical load is defined and the post-critical behavior is investigated.Die Ausknickung einer eingespannten...