Search

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the compartments susceptible (S); incubated - infected yet not infectious (C), infected and infectious (I), and recovered - immune (R). An infection is 'visible' only when an individual is in state I. Upon infection, an individual performs the transition pathway S to C to I to R to S remaining in each compartments C, I, and R for certain random waiting times, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We obtain formulae for the endemic equilibrium and a condition of its existence for cases when the waiting time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. We implement a simple multiple random walker's approach (microscopic model of Brownian motion of Z independent walkers) with random SCIRS waiting times into computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random walker's approach offers an appropriate microscopic description for the macroscopic model., 26 pages, 11 figures

We introduce a modified SIR model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a finite time interval. In the first part, we introduce a random life time or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and cannot be captured by standard SIR models., Comment: 10 pages, 13 Figures

An online control by the magnetic method is considered as a nondestructive evaluation approach to detect the variation of the microstructure. The magnetic model used for each phase is based on a magneto-mechanical coupling model, which is characterized, on the one hand, by the influence of applied field on the magnetic susceptibility and magnetostriction; on the other hand, it is characterized by the effect of mechanical stress on magnetization of a material. In order to predict the macroscopic behavior correctly, this model takes not only account for the multiphased state of dual-phase steels for each phase separately but also for the heterogeneity of stress and magnetic field through a self-consistent localization-homogenization scheme. The proposed multiscale approach is based on the hypothesis of domain energy balance, including the localization step, the local constitutive law application, evaluation of the volumetric fraction of martensite, and the homogenization step. Results are discussed and compared with experimental data from the literature.

Reliable monitoring of solution crystallization processes is important to provide further insight into process dynamics and to improve process control in the regimen of Process Analytical Technology (PAT), e.g. as the case studied here: detection of crystallization of the anhydrous and monohydrate forms of Citric Acid (CA). To set up the relationship between acoustic emission (AE) and crystallization form, two experiments (monohydrate and anhydrous citric acid) were carried out to specify the features and origins of the different acoustic signals emitted during batch cooling solution crystallization processes. Two kinds of AE experimental variables convey information about the development of crystallization processes: frequency and acoustic energy variables. The experimental results show notably that though it has less acoustic bursts, the acoustic activity generated by the crystallization of the monohydrate form of CA actually releases more acoustic energy than the crystallization of anhydrous form. It is also shown that the form of the crystallization is associated with the percentage of absolute energy. The proportion of the absolute energy [150-700 KHz] released by CA

International audience; Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two onedimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked.

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics., 19 pages, 8 figures. Continuum Mech. Thermodyn. (2021)

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process `space-time Mittag-Leffler process&rsquo, We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a &ldquo, well-scaled&rdquo, diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel&rsquo, solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

In this paper, we study nonlocal random walk strategies generated with the fractional Laplacian matrix of directed networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete-time Markovian process with transition probabilities between nodes expressed in terms of powers of the Laplacian matrix. We analyze the elements of the transition matrices and their respective eigenvalues and eigenvectors, the mean first passage times and global times to characterize the random walk strategies. We apply this approach to the study of particular local and nonlocal ergodic random walks on different directed networks; we explore circulant networks, the biased transport on rings and the dynamics on random networks. We study the efficiency of a fractional random walker with bias on these structures. Effects of ergodicity loss which occur when a directed network is not any more strongly connected are also discussed., 15 pages, 9 figures

International audience; We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze `generalized space-time fractional diffusion' in the infinite $\it d$-dimensional integer lattice $\it \mathbb{Z}^d$. We obtain in the diffusion limit a `macroscopic' space-time fractional diffusion equation. Classical CTRW models such as with Laskin's fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.

In this paper we explore the evolution of transport capacity on networks with stochastic incidence of damage and accumulation of faults in their connections. For each damaged configuration of the network, we analyze a Markovian random walker that hops over weighted links that quantify the capacity of transport of each connection. The weights of the links in the network evolve due to randomly occurring damage effects that reduce gradually the transport capacity of the structure. We introduce a global measure to determine the functionality of each configuration and how the system ages due to the accumulation of damage that cannot be repaired completely. Then, by assuming a minimum value of the functionality required for the system to be "alive", we explore the statistics of the lifetimes for several realizations of this process in different types of networks. Finally, we analyze the characteristic longevity of such a system and its relation with the "complexity" of the network structure. One finding is that systems with greater complexity live longer. Our approach introduces a model of aging processes relating the reduction of functionality with the accumulation of "misrepairs" and the lifetime of a complex system., Comment: 14 pages, 7 figures

A non-Markovian counting process, the `generalized fractional Poisson process' (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters $00$ and a time scale parameter. Generalizations to Laskin's fractional Poisson distribution and to the fractional Kolmogorov-Feller equation are derived. We develop a continuous time random walk subordinated to a GFPP in the infinite integer lattice $\mathbb{Z}^d$. For this stochastic motion, we deduce a `generalized fractional diffusion equation'. In a well-scaled diffusion limit this motion is governed by the same type of fractional diffusion equation as with the fractional Poisson process exhibiting subdiffusive $t^{\beta}$-power law for the mean-square displacement. In the special cases $\alpha=1$ with $0, Comment: 27 pages, 4 figures. Accepted for publication in Physica A. arXiv admin note: text overlap with arXiv:1906.09704

We survey the 'generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges $00$ and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in $d$ dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems., Comment: 38 Pages, 4 Figures

In this paper, we explore the reduction of functionality in a complex system as a consequence of cumulative random damage and imperfect reparation, a phenomenon modeled as a dynamical process on networks. We analyze the global characteristics of the diffusive movement of random walkers on networks where the walkers hop considering the capacity of transport of each link. The links are susceptible to damage that generates bias and aging. We describe the algorithm for the generation of damage and the bias in the transport producing complex eigenvalues of the transition matrix that defines the random walker for different types of graphs, including regular, deterministic, and random networks. The evolution of the asymmetry of the transport is quantified with local information in the links and further with non-local information associated with the transport on a global scale such as the matrix of the mean first passage times and the fractional Laplacian matrix. Our findings suggest that systems with greater complexity live longer., Comment: 23 pages, 7 figures. arXiv admin note: text overlap with arXiv:1909.01493

Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt to real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with integer IID interarrival times which converge in well-scaled continuous-time limits to the Prabhakar-generalized fractional Poisson process. These processes exhibit non-Markovian features and long-time memory effects. We recover for special choices of parameters the discrete-time versions of classical cases, such as the fractional Bernoulli process and the standard Bernoulli process as discrete-time approximations of the fractional Poisson and the standard Poisson process, respectively. We derive difference equations of generalized fractional type that govern these discrete time-processes where in well-scaled continuous-time limits known evolution equations of generalized fractional Prabhakar type are recovered. We also develop in Montroll–Weiss fashion the “Prabhakar Discrete-time random walk (DTRW)” as a random walk on a graph time-changed with a discrete-time version of Prabhakar renewal process. We derive the generalized fractional discrete-time Kolmogorov–Feller difference equations governing the resulting stochastic motion. Prabhakar-discrete-time processes open a promising field capturing several aspects in the dynamics of complex systems.

The algorithm is developed to model two-dimensional dynamic processes in a nonlocal square lattice on the basis of the shift operators. The governing discrete equations are obtained for local and nonlocal models. Their dispersion analysis reveals important differences in the dispersion curve and in the sign of the group velocity caused by nonlocality. The continuum limit allows to examine possible auxetic behavior of the material described by the nonlocal discrete model.

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type $L^{\frac{\alpha}{2}}$ where $L$ indicates a `simple' Laplacian matrix. We refer such walks to as `Fractional Random Walks' with admissible interval $0 \alpha$ (recurrent for $d\leq\alpha$) of the lattice. As a consequence for $0, Comment: 28 pages, 5 figures. HAL Id : hal-01543330, version 1

A new mathematical model for non-equilibrium evaporation/condensation including boiling effect is proposed. A simplified differential-algebraic system of equations is obtained. A code to solve numerically this differential-algebraic system has been developed. It is designed to solve both systems of equations with and without the boiling effect. Numerical calculations of ammonia-water systems with various initial conditions, which correspond to evaporation and/or condensation of both components, have been performed. It is shown that, although the system evolves quickly towards a quasi equilibrium state, it is necessary to use a non-equilibrium evaporation model to calculate accurately the evaporation/condensation rates, and consequently all the other dependent variables., 20 pages, 6 figures

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions L\'evy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions. In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like L\'evy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks., Comment: 33 pages, 5 figures

International audience; In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen's model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen's nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.

We analyze the “fractional continuum limit” and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton’s (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian\( - ( - \Delta )^{\tfrac{\alpha } {2}} \) with 0 n - 1 \) being always greater than n−1 characterizing a regular lattice with local interparticle interactions. Furthermore, we study in this setting anomalous diffusion generated by this Laplacian which is the source of Levi flights in n-dimensions. In the limit of “large scaled times” ∼ t/rα >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t-n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Levi parameter α.

International audience; The potential of Acoustic Emission (AE) for controlling crystallization processes is investigated. The sensing technology is successfully applied to monitor the batch cooling crystallization of citric acid (CA) in water. The solvent-mediated phase transition between the anhydrous and the monohydrated forms of CA is clearly detected from the recorded acoustic measurements. A tremendous amount of acoustic data is recorded by the equipment, and the analysis of the data is focused on the evaluation of AE as a new sensor for monitoring the basic steps of the crystallization processes (i.e., nucleation, growth, phase transition, etc.) A time- and frequency-domain analysis is presented which shows the wealth of the technique. It is finally concluded that AE allows very early detection of nucleation events, provides a means of monitoring the development of the crystallization process and allows monitoring phase transition phenomena obtained through cooling. It is thus suggested that acoustic emission could be valuable in the development of new crystallization monitoring and control strategies: this is all the more interesting that the acoustic piezo-sensor is non-intrusive and does not require any sampling of the slurry, two features which are of tremendous importance in the field of cooling crystallization processes.

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and infinite lattice in n=1,2,3,..n=1,2,3,.. dimensions. The present model which is based on Hamilton's variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels., arXiv admin note: text overlap with arXiv:1604.01725

Continuum material models can be enriched with additional gradients in order to model phenomena that are driven by processes at lower levels of observation. For a systematic comparison of various gradient-enriched continua, dispersion analysis may be used. In this contribution, we will explore the occurence of critical wave lengths and its implications for the material stability. In particular, we will present a unifying theorem that permits to assess the stability of elastic, hardening and softening gradient-enriched continua by means of a critical wave length analysis, whereby the upper or lower bound nature of the critical wave length indicates whether the model is stable or unstable.

We develop a general procedure for solving the first and second fundamental problems of the theory of elasticity for cases where boundary conditions are prescribed on a spherical surface, using Love's general solution of the elastostatic equilibrium equations in terms of three scalar harmonic functions. It is shown that this general solution combined with a methodology by Brenner paves an elegant way to determine the three harmonic functions in terms of the boundary data. Thus, with this general scheme, solution of any such boundary-value problem is reducible to a routine exercise thereby providing some 'economy of effort'. Furthermore, we develop a similar general scheme for thermoelastic problems for cases when temperature type boundary conditions are prescribed on a spherical surface. We then illustrate the application of the procedure by solving a number of problems concerning rigid spherical inclusions and spherical cavities. In particular, apart from furnishing alternative solutions to the known problems, we demonstrate the use of this general procedure in solving the problem of interaction of a rigid spherical inclusion with a concentrated moment and that of a concentrated heat source situated at an arbitrary point outside the inclusion. We also derive closed-form expressions for the net force and the net torque acting on a rigid spherical inclusion embedded into an infinite elastic solid under an ambient displacement field characterized by an arbitrary-order polynomial in the Cartesian coordinates. To the best of our knowledge, these results are new.

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length.We obtain explicit expressions for this fractional Laplacianmatrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional derivative) on the finite periodic string.In this approach we introduce two material parameters, the particle mass $\mu$ anda frequency $\Omega\_{\alpha}$. The requirement of finiteness of the the total mass and total elastic energy in the continuum limit (lattice constant $h\rightarrow 0$) leads to scaling relations for the two parameters, namely$\mu \sim h$ and $\Omega\_{\alpha}^2\sim h^{-\alpha}$.The present approach can be generalized to define lattice fractional calculus on periodic lattices in full analogy to the usual `continuous' fractional calculus., Comment: arXiv admin note: text overlap with arXiv:1412.5904

International audience; In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pade approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

The purpose of this paper is an investigation of the possible role of supplementary cementitious materials (SCMs) on water transport kinetics and mechanical properties of hydrated lime (CL90) and Portland cement (PC) mortars. The properties of hydrated lime are significantly different from those of cement and therefore modifying fresh and hardened properties of these mortars are vital for mortar/substrate optimisation in masonry construction. The parameters investigated in this paper often are the main barriers to the use of hydrated lime in construction practice. The results show that transfer sorptivity and time to dewater freshly-mixed hydrated lime mortars can be modified when binder is partially replaced with SCMs. Compressive strength of CL90 mortars is increased systematically with the increased replacement levels of SCMs and the results are supported with the microstructural images. The ability to modify the water transport kinetics and mechanical properties allows compatibility between the mortar and the substrate unit in masonry construction.El objetivo de este artículo es investigar el papel de los materiales cementantes suplementarios (SCMs) en la cinética de transporte del agua y en las propiedades mecánicas de los morteros de cal hidratada (CL90) y cemento Portland. Las propiedades de la cal hidratada son significativamente diferentes a las del cemento y por lo tanto el control de las propiedades de los morteros frescos y endurecidos es fundamental en la optimización mortero/substrato en albañilería. Los parámetros estudiados en este trabajo son a menudo las principales barreras para el uso de la cal hidratada en la práctica de la construcción. Los resultados indican que la absortividad y el tiempo necesario para deshidratar morteros de cal hidratada recién mezclados pueden ser controlados cuando el conglomerante es parcialmente remplazado por SCMs. La resistencia a compresión de los morteros CL90 aumenta sistemáticamente con el nivel de sustitución de SCM. Las imágenes microestructurales realizadas, confirman estos resultados. La posibilidad de manipular la cinética de transporte de agua y las propiedades mecánicas permite la compatibilidad entre el mortero y la unidad de sustrato en albañilería.

International audience; In this paper, we show that two numerical methods, specifically the finite difference method and the finite element method applied to continuous beam dynamics problems, can be asymptotically investigated by some kind of enriched continuum approach (gradient elasticity or nonlocal elasticity). The analysis is restricted to the vibrations of elastic beams, and more specifically the computation of the natural frequencies for each numerical method. The analogy between the finite numerical approaches and the equivalent enriched continuum is demonstrated, using a continualization procedure, which converts the discrete numerical problem into a continuous one. It is shown that the finite element problem can be transformed into a system of finite difference equations. The convergence rate of the finite numerical approaches is quantified by an equivalent Rayleigh's quotient. We also present some exact analytical solutions for a first-order finite difference method, a higher-order finite difference method or a cubic Hermitian finite element, valid for arbitrary number of nodes or segments. The comparison between the exact numerical solution and the approximated nonlocal approaches shows the efficiency of the continualization methodology. These analogies between enriched continuum and finite numerical schemes provide a new attractive framework for potential applications of enriched continua in computational mechanics.

In this brief note, we have derived an approximate analytical expression for the Rayleigh wave speed for an isotropic half-space using Lanczos's approximation. This expression is simpler than the exact expressions reported earlier by a number of authors while at the same is very accurate, with an error not exceeding 0.45%. The same approximation has been also used to derive an approximate analytical expression for an incompressible orthotropic half-space. The latter has been shown to be within a maximum error not exceeding 0.05%. In many theoretical and practical applications, such approximate formulas might prove more useful and convenient to manipulate with.

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem ( Eshelby, 1957 ), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions.

We analyze the infinite space solutions of the three-dimensional inhomogeneous wave equation (the ‘retarded potentials’ or ‘causal propagators’) for ellipsoidal sources and for sources of arbitrary shapes. The ‘short-time characteristics’ of the retarded potential for a spatially inhomogeneous source density of δ -shaped time profile is considered. It is found that, the short-time characteristics is governed by the spatial inhomogeneity of the source density in the immediate vicinity of a spacepoint. Surface integral representations are derived for spatial inhomogeneous source regions of ellipsoidal symmetry . For spherical sources these integral representations yield closed form solutions for the retarded potentials. We find that the wave field inside a spherical source consists of an incoming and outgoing spherical wave package, whereas the external wave field consists of an outgoing spherical wave package only. Characteristic runtime and superposition effects are discussed. Moreover, a numerical technique based on Gauss quadrature is applied to generate the wave field for a cubic source. The integral representations derived for the retarded potentials of inhomogeneous ellipsoidal sources are consistent with results previously derived by the authors for the Helmholtz potentials of homogeneous ellipsoids and ellipsoidal shells [Michelitsch, T.M., Gao, H., Levin, V.M., 2003. On the dynamic potentials of ellipsoidal shells. Q. J. Mech. Appl. Math. 56 (4), 629]. The derived solutions are crucial for many problems of wave propagation and diffraction theory as they may occur in materials science. As an example we give a formulation for the solution of the retarded Eshelby inclusion problem due to spatially and temporally varying eigenfields in the elastic isotropic infinite medium.

The solutions of many dynamical problems as wave propagation in electrodynamics, acoustics or elasticity very often require the solution of inhomogeneous Helmholtz equations (determination of dynamic potentials) for ellipsoidal source regions. As in the case of the static (Newtonian) potentials a compact representation of the dynamic potentials in terms of one-dimensional integrals is highly desirable. Due to the mathematical complexity of the problem for ellipsoids such a representation seems not to have been reported in the literature so far. In this paper we close this gap for the dynamic potential of an ellipsoidal shell for internal spacepoints. The derived solution of the inside region can easily be used to find the solution for the outside region by applying Ivory's theorem. In the static limit classical results of Ferrers and Dyson for the Newtonian potential of inhomogeneous ellipsoids are reproduced.

A piezoelectric medium of transversely isotropic symmetry with continuous fiber inclusion parallel to the axis of symmetry is considered. The problem is equivalent to a two-dimensional ‘ quasi-plane ’ piezoelectric medium containing a 2D inclusion. The inclusion is assumed to undergo a spatially uniform δ ( t )-type time domain transformation. The continuous fiber has elliptical, circular and arbitrary cross-sections. The solutions of the inclusion problem is expressed by scalar potentials . In the time domain two of these functions correspond to the retarded potential integrals of the inclusion. Their frequency domain representation which we shall call the ‘ dynamic potentials of the inclusion ’ are also considered. Integral formulae are derived for continuous fiber inclusions with elliptical cross-sections. Known closed-form solutions are reproduced for circular fibers. For fibers with arbitrary cross-sections a numerical method based on Gauss quadrature is applied. High accuracy and efficiency of the numerical method is confirmed. Characteristic superposition and runtime effects for the inclusions are found.

The dynamic Eshelby inclusion problem for an ellipsoidal inclusion in a threedimensional infinite elastic isotropic medium is considered. The dynamic Eshelby tensor is expressed in terms of solutions of the Helmholtz equation (Helmholtz potentials). A compact formulation for the components of the dynamic Eshelby tensor is derived for the inside region of the inclusion. For spheroidal inclusions, one-dimensional integrals similar to the elliptic integrals are obtained. The approach leads to closed-form expressions in cases such as spheres and continuous fibres coinciding with those given in 1990 by Mikata & Nemat-Nasser and in 2002 by Michelitsch et al . by employing other techniques. For the inside region of the inclusion, the static limit is performed in closed form and coincides with Eshelby’s classical 1957 result.

The dynamic potentials of a two–dimensional (2D) quasi–plane piezoelectric infinite medium of transversely isotropic symmetry containing an inclusion of arbitrary shape is derived in terms of scalar solutions of the 2D Laplace and Helmholtz equations. Closed–form expressions for the space–frequency representation of this dynamic potential are obtained for the case when the spatial source distribution is characterized by a region occupied by a circular inclusion embedded in a quasi–plane transversely isotropic matrix. The results are used to solve the dynamic Eshelby problem of a circular inclusion (plane region with the same material characteristics as the matrix) undergoing uniform eigenstrain and eigenelectric field. In contrast to the static case, the dynamic electroelastic fields inside the circular inclusion are non–uniform in the space–frequency representation. The derived dynamic piezoelectric potentials are basic quantities for the description of the dynamic properties of micro–inhomogeneous quasi–plane piezoelectric material systems (e.g. fibre–reinforced piezocomposites).

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis. The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem. The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.

The elastic Eshelby tensor that plays a key role in elastic inclusion problems [1] is extended to the electroelastic case. For electroelastic media this analogue of the Eshelby tensor is an operator that consists of four tensors of fourth, third and second ranks. The explicit expressions are derived for the components of these tensors for a spheroidal inclusion in hexagonal (transversely isotropic) media. The obtained results may be useful for a treatment of inclusion and inhomogeneity problems of the coupled electroelasticity.

T M Michelitsch, B A Collet, A P Riascos,A F Nowakowski and F C G A Nicolleau,J. Phys. A: Math. Theor. 50 (2017) 055003, doi:10.1088/1751-8121/aa5173; International audience; We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions.The fractional random walk dynamics is governed by a master equation involving {\it fractional powers of Laplacian matrices $L^{\frac{\alpha}{2}}$}where $\alpha=2$ recovers the normal walk.First we demonstrate thatthe interval $0

International audience; The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finite number $N$ of identical particles is introduced. We suggest a ''fractional elastic harmonic potential", and obtain the $N$-periodic fractional Laplacian operator in the form of a power law matrix function for the finite chain ($N$ arbitrary not necessarily large) in explicit form. In the limiting case $N\rightarrow \infty$ this fractional Laplacian matrix recovers the fractional Laplacian matrix of the infinite chain.The lattice model contains two free material constants, the particle mass $\mu$ and a frequency $\Omega_{\alpha}$.The ''periodic string continuum limit" of the fractional lattice model is analyzed where lattice constant $h\rightarrow 0$ and length $L=Nh$ of the chain (''string") is kept finite: Assuming finiteness of the total mass and total elastic energy of the chain in the continuum limit leads to asymptotic scaling behavior for $h\rightarrow 0$ of the two material constants, namely $\mu \sim h$ and $\Omega_{\alpha}^2 \sim h^{-\alpha}$. In this way we obtain the $L$-periodic fractional Laplacian (Riesz fractional derivative) kernel in explicit form. This $L$-periodic fractional Laplacian kernel recovers for $L\rightarrow\infty$ the well known 1D infinite space fractional Laplacian (Riesz fractional derivative) kernel. When the scaling exponent of the Laplacian takes integers, the fractional Laplacian kernel recovers, respectively, $L$-periodic and infinite space (localized) distributional representations of integer-order Laplacians.The results of this paper appear to be useful for the analysis of fractional finite domain problems for instance in anomalous diffusion (Levy flights), fractional Quantum Mechanics, and the development of fractional discrete calculus on finite lattices.

International audience; We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders of the displacement field leading by application of Hamilton's variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. ``Anomalous'' dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.

International audience; We demonstrate that the fractional Laplacian (FL) is the principal characteristic operator of harmonic systems with {\it self-similar}interparticle interactions. We show that the FL represents the ``{\it fractional continuum limit}'' of a discrete ``self-similar Laplacian" which is obtained by Hamilton's variational principle from a discrete spring model.We deduce from generalized self-similar elastic potentials regular representations for the FL which involve convolutions of symmetric finite difference operators of even orders extending the standard representation of the FL.Further we deduce a regularized representation for the FL $-(-\Delta)^{\frac{\alpha}{2}}$ holding for $\alpha\in \R \geq 0$.We give an explicit proof that the regularized representation of the FL gives for integer powers $\frac{\alpha}{2} \in \N_0$ a distributional representation of the standard Laplacian operator $\Delta$ including the trivial unity operator for $\alpha\rightarrow 0$. We demonstrate that self-similar {\it harmonic} systems are {\it all} governed in a distributional sense by this {\itregularized representation of the FL} which therefore can be conceived as characteristic footprint of self-similarity.