Teodor M Atanackovic

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Verified

Teodor verified their affiliation via an institutional email.

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• PhD
• Professor Emeritus at University of Novi Sad

In the constitutive equation of nonlocal elasticity with a kernel of general fractional derivative of Riesz type, the time delay of the form is introduced. We call the function the constitutive equation of time delay. Second‐order terms in expansion is taken into account, and the resulting generalized wave equation is derived. The influence of para...

We analyze the problem of finding the shape of a heavy cantilever that minimizes volume and satisfies certain condition at the free end that corresponds to the so-called longest reach problem. The Pontryagin's maximum principle with cross-sectional area as "control" is used in the optimization procedure. We point out some specific properties of the...

This study presents comparison of our recently formulated two compartmental model with General fractional derivative (GFD) and Korsmeyer-Peppas, Makoid-Banakar and Kopcha diffusion models. We have used our GFD model to study the release of gentamicin in poly (vinyl alcohol)/chitosan/gentamicin (PVA/CHI/Gent) hydrogel aimed for wound dressing in med...

We analyze wave equation for spatially one-dimensional continuum with constitutive equation of non-local type. The deformation is described by a specially selected strain measure with general fractional derivative of the Riesz type. The form of constitutive equation is assumed to be in strain-driven type, often used in nano-mechanics. The resulting...

A novel two-compartment model for drug release was formulated. The general fractional derivatives of a specific type and distributed order were used in the formulation. Earlier used models in pharmacokinetics with fractional derivatives follow as special cases of the model proposed here. As a first application, we used this model to study the relea...

As a part of this study, a mathematical model based on two-compartment pharmacokinetic system with general fractional derivatives was developed. Its aim was to describe the release of BisGMA and TEGDMA monomers from composite core resin over three time periods. For this purpose, five Clearfil Photo Core resin samples were prepared and elution was m...

This study presents a new two compartmental model with, recently defined General fractional derivative. We review that concept of General fractional derivative and use the kernel function that generalizes the classical Caputo derivative in a mathematically consistent way. Next we use this model to study the release of antibiotic gentamicin in poly...

In this paper, the dynamical behavior of the Euler-Bernoulli beam resting on a generalized Kelvin-Voigt-type viscoelastic foundation, subjected to a moving point load, is analyzed. Generalization is done in the sense of fractional derivatives of complex-order type. Mixed initial-boundary value problem is formulated, and the solution is given in the...

We studied a Zener-type model of a viscoelastic body within the context of general fractional calculus and derived restrictions on coefficients that follow from the dissipation inequality, which is the entropy inequality under isothermal conditions. We showed, for a stress relaxation and a wave propagation, that the restriction that follows from th...

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of...

Purpose was to determine the viscoelastic properties of three root canal sealers as a function of temperature, time and frequency using dynamic oscillatory measurements. Methods: Measurements were performed on the dynamic oscillatory rheometer set to temperatures: 25�C, 35�C, 40�C and 65�C. Stress sweep and frequency sweep tests were used in order...

Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality (entropy inequality under isothermal conditi...

We analyze a viscoelastic body in a linear stress state with the distributed order fractional derivative in the constitutive equation. The model was formulated so that it takes into account all derivatives of stress and strain between zero and one. Using a new procedure, we derive restrictions on a model that follow from the Clausius-Duhem inequali...

We analyze the dissipation inequality for the constitutive equation of a complex order fractional Zener model and obtain appropriate thermodynamical restrictions for the wave-type model equation in terms of its Laplace transform. These constraints obtained on the model parameters are less restrictive than the ones known in the previous literature....

We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load F of the column and the length L, find cross‐sectional area A, such that the volume W of the column attains minimal value. This is a classical Clausen problem. Howe...

The problem of finding the shape of initially straight elastic cantilever that minimizes volume, and satisfies certain condition at the free end is analyzed. The recently analyzed longest reach problem follows as a special case. New results are presented about shape of the cantilever, having prescribed volume and having longest reach. The Pontryagi...

The aim of this study is to analyze viscoelastic properties of direct composite core (Lightcore, Build-it, Clearfil Photo Core, Rebilda). Experiments are preformed and mathematical models developed, based on derivatives of fractional order, to describe the viscoelastic properties of the studied materials. The basic assumption that materials are of...

A variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated, and the invariance of this principle under the action of a local group of symmetries is determined. By the Noether theorem the conservation law for the corresponding fractional Euler–Lagrange equation is o...

By using a Pontryagin’s principle, we study the optimal shape of a rotating nano rod and determine the optimal cross-section that is stable against buckling due to centrifugal forces. We generalize the results of the earlier studies focused on the constant cross-sectional area of nano rods. The problem of the optimal shape of a Bernoulli–Euler rota...

This is a review article which elaborates the results presented in [1], where the variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated and the invariance of this principle under the action of a local group of symmetries is determined. The conservation law for t...

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to...

In this paper, we analyze the restrictions on the coefficients in the constitutive equations of linear Viscoelasticity that follow from the Second Law of Thermodynamics under isothermal conditions. Especially, we analyze the constitutive equations in which fractional derivatives of real and complex order appear. We present the conditions that follo...

We investigate, in the distributional setting, the restrictions on the constitutive equation for a fractional Burgers model of viscoelastic fluid that follow from the weak form of the entropy inequality under isothermal conditions. The results are generalized, from the Burgers model, to an arbitrary class of linear constitutive equations with fract...

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for...

We derive optimality conditions for variational problems of Herglotz type whose Lagrangian depends on fractional derivatives of both real and complex order, and resolve the case of subdomain when the lower bounds of variational integral and fractional derivatives differ. Moreover, we consider a problem of the Herglotz type that corresponds to the c...

We analyze the problem of finding the shape of the tallest column. For the system of equations that determine the optimal shape we construct a variational principle and two new first integrals. From the first integrals we are able to determine, analytically, the size of the cross-sectional area of the optimal column at the bottom, as well as the co...

Static stability problem for axially compressed rotating nano-rod clamped at one and free at the other end is analyzed by the use of bifurcation theory. It is obtained that the pitchfork bifurcation may be either super-or sub-critical. Considering the imperfections in rod's shape and loading, it is proved that they constitute the two-parameter univ...

Static stability problem for axially compressed rotating nano-rod clamped at one and free at the other end is analyzed by the use of bifurcation theory. It is obtained that the pitchfork bifurcation may be either super- or sub-critical. Considering the imperfections in rod’s shape and loading, it is proved that they constitute the two-parameter uni...

We consider several generalizations of the Hamilton principle and study optimality conditions involving Euler-Lagrange equations. We present the concept of the fractional complementary variational principle. Moreover, we propose an expansion formula for the fractional derivative as a tool for solving the Euler-Lagrange equations.

Waves in nonlocal and viscoelastic media are investigated through the constitutive equations containing fractional derivatives of real order in the time and space variables. The distributional framework of the transferred equation enable us to find out distributional solutions for which we show the continuity. The classical solutions can be obtaine...

We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the coefficients in the constitutive equation that follow from the weak form of the dissipation principle are obtained. We formulate a problem of motion of a spatially one dimensio...

The classical wave equation is generalized within fractional framework, by using fractional derivatives of real and complex order in the constitutive equation, so that it describes wave propagation in one dimensional infinite viscoelastic rod. We analyze existence, uniqueness and properties of solutions to the corresponding initial-boundary value p...

We study approximate solutions of CDtβy(t)=f(t,y(t)), separately, for β ∈ (0, 1) and β ∈ (1, 2) with different boundary data, where CDtβ is the Caputo fractional derivative of complex-order. For this purpose we use the expansion formula for such fractional derivatives and prove the existence and the uniqueness of approximate solutions under certain...

The purpose of this work is to develop a new model estimate of the fatigue life of a hip prosthesis due to aseptic loosening as a multifactorial phenomenon. The formula developed here is a three-parameter model based on Basquin’s law for fatigue, eccentric compression formula for the compressive stress and torsion in the prosthesis due to the horiz...

The Caputo-Fabrizio fractional derivative is analyzed in classical and distributional settings. The integral inequalities needed for application in linear viscoelasticity are presented. They are obtained from the entropy inequality in a weak form. Moreover, integration by parts, an expansion formula, approximation formula and generalized variationa...

We study the heat conduction with a general form of a constitutive equation containing fractional derivatives of real and complex order. Using the entropy inequality in a weak form, we derive sufficient conditions on the coefficients of a constitutive equation that guarantee that the second law of thermodynamics is satisfied. This equation, in spec...

In this paper, a novel non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete is proposed. The rheological model consists of a linear springpot unit placed in series with a second springpot used for non-linear creep which activates under high stress and temperature. The model paramete...

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Sc...

This study examines the static stability of a heavy axially compressed nanorod. The rod is free at one end, while the other end is rigidly connected to a circular disk positioned on an elastic half space. The critical values of load parameters, i.e., axial force and specific weight, for which the nanorod loses stability are determined by using the...

Two variational problems of finding the Euler–Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex...

In this paper, the problem of determining the dynamic stability boundary (critical value of the axial force) of an axially loaded nonlocal rod of Eringen’s type is considered. The rod is positioned on a viscoelastic foundation of the Pasternak type. Constitutive equations containing fractional derivatives of real and complex order are used to model...

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. T...

Discussions under this title, inspired by the famous Gauguin painting, were held during a Round Table in frames of the International Conference “ Along with the presentations made during this Round Table, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of...

We introduce complex order fractional derivatives in models that describe viscoelastic materials. This can not be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fracti...

Two fractional two-compartmental models are applied to the pharmacokinetics of articaine. Integer order derivatives are replaced by fractional derivatives, either of different, or of same orders. Models are formulated so that the mass balance is preserved. Explicit forms of the solutions are obtained in terms of the Mittag–Leffler functions. Pharma...

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. T...

The axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations...

We study a system of partial differential equations with integer and fractional derivatives arising in the study of forced oscillatory motion of a viscoelastic rod. We propose a new approach considering a quotient of relations appearing in the constitutive equation instead the constitutive equation itself. Both, a rod and a body are assumed to have...

In this study we analyze viscoelastic properties of three flowable (Wave, Wave MV, Wave HV) and one universal hybrid resin (Ice) composites, prior to setting. We developed a mathematical model containing fractional derivatives in order to describe their properties. Isothermal experimental study was conducted on a rheometer with parallel plates. In...

Beck's type column on Winkler type foundation is the subject of the present analysis. Instead of the Bernoulli-Euler model describing the rod, two generalized models will be adopted: Eringen non-local model corresponding to nano-rods and viscoelastic model of fractional Kelvin-Voigt type. The analysis shows that for nano-rod, the Herrmann-Smith par...

We consider vibrations of an elastic rod loaded by axial force of constant intensity and positioned on a viscoelastic foundation of complex order fractional derivative type. The solution to the problem is obtained by the separation of variables method. The critical value of axial force, guaranteeing stability, is determined.

The space-time fractional Zener wave equation, describing viscoelastic materials obeying the timefractional Zener model and the space-fractional strain measure, is derived and analysed. This model includes waves with finite speed, as well as nonpropagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy prob...

We use the expansion formula for the fractional derivatives to reduce the problem of solving non-linear fractional order differential equations arising in mechanics to the problem of solving a system of integer order differential equations. We prove the convergence of the solutions to the reduced integer order systems to the solutions of the origin...

Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscill...

The aim of this study is to promote a model based on the fractional differential calculus related to the pharmacokinetic individualization of high dose methotrexate treatment in children with acute lymphoblastic leukaemia, especially in high risk patients.We applied two-compartment fractional model on 8 selected cases with the largest number (4-19)...

This paper presents an analytical investigation on the buckling and post-buckling behavior of rotating nanorods subjected to axial compression and clamped at both ends. The nonlinear governing equations are derived based on the classical Euler–Bernoulli theory and Eringen's nonlocal elasticity model. The critical load parameters such as angular vel...

We study stability of a rotating compressed nano-rod clamped at one and free at the other end. By using the Euler method of adjacent equilibrium configuration, we determine critical values for angular speed and compressive force. Post-critical shape of the rod is obtained by solving the non-linear system of equilibrium equations for several values...

This chapter reviews several generalizations of the heat conduction equation using the integer-order derivatives. It describes the existence and uniqueness of a generalized solution to the initial-boundary value problem, considering distributions depending on two variables. The chapter defines the model of the fractional Jeffreys-type heat conducti...

This chapter discusses linear and nonlinear vibrations of a single-degree-of-freedom mechanical system with fractional derivative type dissipation. Since the Laplace transform is continuously used in the framework of S'+ and K'+, the chapter assumes that all functions and distributions belong to K'+, so that their Laplace transforms are analytic fu...

This chapter first considers the impact of a viscoelastic body, which slides without the presence of friction, against the rigid wall. During collisions followed by a capture of the colliding bodies, large amounts of energy may be dissipated in a very short period of time; so, the dissipation in the viscoelastic rod is not enough to cease the motio...

This chapter commences with a discussion on notation and definitions of the fractional calculus theory. It considers equality almost everywhere, as well as the integration in the sense of Lebesgue. The section on Laplace transform of a function considers a particular class of Laplace transformable functions, the Fourier transform of a tempered dist...

This chapter reviews some basic properties of fractional integrals and derivatives like the Riemann-Liouville fractional integrals and derivatives and Riemann-Liouville fractional integrals and derivatives on the real half-axis, which will be needed later in the analysis of concrete problems. The chapter presents the definition of Caputo fractional...

This chapter presents the equations that will be treated and presenting their backgrounds. The framework of the generalized functions can be used for the analysis, which gives flexibility in calculations. Solutions of equations will always be interpreted as the functions of polynomial or exponential growth. The chapter presents similarity transform...

This chapter begins by introducing some necessary definitions and mathematical preliminaries of the fractional calculus theory. It then describes Laplace transform of a function. Spaces of distributions are considered next. Finally, a fundamental solution of a linear partial integro-differential operator with constant coefficients is derived.

This chapter is devoted to the analysis of a problem of forced oscillations of a body attached to a viscoelastic rod. Analysis is conducted in the cases when the rod is considered to be heavy (i.e. the mass of a rod is comparable with the mass of a body) as well as when the rod is considered to be light (i.e. the mass of a rod is negligible in comp...

This chapter presents some results for variational problems in which the Lagrangian density involves derivatives of real (fractional) order, as well as a generalization of the classical Hamilton principle of mechanics in which Hamilton action integral is minimized within the specified set of functions and with respect to the order of the derivative...

This chapter presents constitutive models of a viscoelastic body containing fractional derivatives of stress and strain. Isothermal processes and spatially one-dimensional cases are only considered. Constitutive models should satisfy restrictions that follow from the second law of thermodynamics, i.e. the entropy inequality. The restriction followi...

This chapter presents the results from various papers related to the formulation and solution of various problems from the mechanics of the deformable body, especially from the lateral vibrations of the viscoelastic rods. It considers the lateral vibrations of a viscoelastic rod described by the fractional Kelvin-Voigt moment-curvature relation. It...

This chapter first presents the (one-dimensional) wave equation, which describes the motion, i.e. the change of displacement during time t at point x, of an elastic medium. It describes the wave equation for viscoelastic infinite media described by the fractional Zener and linear fractional models. The chapter then considers the wave equation for t...

We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of a solution for a class of linear viscoelastic solids is given in terms of a convolution integral. Restrictions...

This book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton-type principles. Mathematical rigor will...

We modify the expansion formula introduced in [T.M. Atanackovic, B. Stankovic, An expansion formula for fractional derivatives and its applications, Fract Calc. Appl. Anal. 7 (3) (2004) 365-3781 for the left Riemann-Liouville fractional derivative in order to apply it to various problems involving fractional derivatives. As a result we obtain a new...

In this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation f...

This study presents a new two compartmental model that contains fractional derivatives of different order. The model is so formulated that the mass balance is preserved. In this way we give an answer on a claim that such a model is not possible. The generalization that includes nonlinear terms and fractional order dynamics between compartments is a...

The aim of this study is to develop fractional derivative models for the assessment of viscoelastic properties related to handling characteristics of dental resin composites belonging to two classes: flowable (Revolution Formula 2 and Filtek Ultimate) and posterior "bulk-fill" flowable base (Smart Dentin Replace). Rheological measurements on all ma...

We study forced oscillations of a rod with a body attached to its free end so that the motion of a system is described by two sets of equations, one of integer and the other of the fractional order. To the constitutive equation we associate a single function of complex variable that plays a key role in finding the solution of the system and in dete...

We propose a new method for finding solution of Bagley-Torvik equation based on recently derived expansion formula for fractional derivatives. The case of nonlinear equations of Bagley-Torvik type is also discussed.

A fractional nonlocal elasticity model is presented in this Note. This model can be understood as a possible generalization of Eringenʼs nonlocal elastic model, with a free non-integer derivative in the stress–strain fractional order differential equation. This model only contains a single length scale and the fractional derivative order as paramet...

The objective of this study is to introduce a modified incremental technique that leads to improved marginal adaptation and to develop a mathematical model that explains the results obtained. The technique proposed is a two-step incremental technique that reduces volume of a resin that is polymerized at each step and eliminates the central point in...

The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient ∂x and fractional time derivative instead of the first order partial time derivative ∂t. The existence of the...

This paper deals with optimal shapes against buckling of an elastic nonlocal small-scale Pflüger beams with Eringen’s model for constitutive bending curvature relationship. By use of the Pontryagin’s maximum principle the optimality condition in form of a depressed quartic equation is obtained. The shape of the lightest (having the smallest volume)...

The aim of this study was to examine the lifespan or number of cycles to failure of tapered rotary nickel-titanium (Ni-Ti) endodontic instruments. Simulated root canals with different curvatures were used to determine a relation between canal curvature and instrument lifespan. Using a novel mathematical model for the deformation of pseudoelastic Ni...

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of c...

Complementary variational principles for a class of fractional boundary value problems are formulated. They are used for the error estimates of solutions for a general mechanical problems, first Painlevé equation also given in the form with fractional derivatives and in the task of image regularization.

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of c...

We give the existence of solutions to the linear differential equation with fractional derivatives which are real numbers and two different types of initial conditions. Relations between these initial conditions is considered.

Objectives: The purpose of this study was to examine cycle fatigue of Ni-Ti rotary endodontic instruments subjected to artificial root canals with different curvatures and to determine a mathematical model connecting lifespan of the instrument with the curvature to which it is subjected. Methods: We used a new mathematical model for the deformati...

The aims of this study were to examine the ultimate strength of the resto-red maxillary incisors with composite resin, dental amalgam and glassionomer cement as a transitional restoration. Fifty-six extracted human maxillary central incisors with intact and carious dentin were used. The control group consisted of eight unrestored teeth with intact...

This study presents a new nonlinear two compartmental model and its application to the evaluation of valproic acid (VPA) pharmacokinetics in human volunteers after oral administration. We have used literature VPA concentrations. In the model, the integer order derivatives are replaced by derivatives of real order often called fractional order deriv...

We study waves in a rod of finite length with a viscoelastic constitutive equation of distributed fractional order type for the special choice of weight functions. Prescribing boundary conditions on displacement and stress, we obtain, as special solutions, cases corresponding to creep and forced oscillations. In solving system of differential and i...