Micro‐Macro Modelling of Piezoelectric Fiber Composites

The present work deals with the numerical modeling of 1‐3 periodic composites made of piezoceramic (PZT) fibers embedded in a soft non‐piezoelectric matrix. We especially focus on predicting the effective co‐efficients of the periodic transversely isotropic piezoelectric fiber composites using representative volume element method (unit cell method). The results which are obtained from the FEM technique are compared with analytical homogenization method for different volume fractions. The effective co‐efficients are obtained for rectangular and hexagonal arrangement of unidirectional piezoelectric fiber composites. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular approximation of the method of periodic components in statistical mechanics of composite materials

A quasi-periodic model is developed for random structures of composites, when the locations of inclusions are given in terms of random deviations from nodes of an ideal periodic lattice. Solution of the stochastic boundary problem of the theory of elasticity is examined for a quasi-periodic component by the method of periodic components, which is reduced to determination of the field of deviations from the known solution for a corresponding periodic composite. The solution is presented for the tensor of effective elastic properties of a quasi-periodic composite in singular approximation of the method of periodic components in terms of familiar solutions for tensors of the effective elastic properties of composites with periodic and chaotic structures and the parameters of the quasi-periodic structure: the coefficient of periodicity and the tensor of the anisotropy of inclusion disorder. A numerical calculation is performed for the effective transversally isotropic elastic properties of unidirectional fibrous composites with different degrees of fiber disorder.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 460–473, July–August, 1997.     

Homogenization of Fatigue and Creep Durability of Composites

The paper is an extension of the theory presented in [4] to fatigue, creep and other types of time‐dependent material behaviour. In the previous paper a multi‐phase periodic composite subjected to prescribed mechanical loads was considered. The asymptotic homogenization was applied for calculation of homogenized macro‐stresses. A nonlocal approximate macro‐strength condition, defined on homogenised stress‐field, was derived from the micro‐strength conditions and their convergence to the approximate macro‐strength condition, as the structure period tends to zero, was proved. This paper presents the analogous results for the macro‐durability of elastic and viscoelastic composites.     

The elastic response for microlayered functionally graded media

In this note microlayered composites having continuously varying macroscopic properties are considered. Such composites are referred to as the functionally graded laminates (FGL). The aim of this contribution is to derive a new averaged model describing the elastic response of the FGL, using the modified tolerance averaging technique, developed for periodic composites and structures by Woźniak and Wierzbicki (2000). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Models of Composite Mechanics in the Problems of Designing Structural Elements in Aerospace Engineering

The present-day trends and development prospects of stochastic averaging methods in composite mechanics are analyzed. The following methods are considered: the traditional one (the medium of comparison is homogeneous), the method of periodic components (the medium of comparison has a regular structure), the numerical method of local approximation with regard for the short-range order in the arrangement and interaction of inhomogeneities, and a synthesis of the periodic-component and local-approximation methods. A detailed procedure is presented for calculating the functionals of second, third, and higher orders of the stochastic problem. The results obtained are used for evaluating, in an explicit form, corrections to the effective elastic moduli found for quasi-isotropic and unidirectional fiber composites by the traditional averaging method and the method of periodic components. Analytical formulas for the second-order moment functions of structural stresses and strains are derived. It is shown that the fields of structural stresses and strains are locally ergodic. A new multilevel approach is proposed for designing composites, which takes into account the effect of structurally technological factors. Thus, unidirectional fiber composites are calculated by a two-level model, layered structures - by a three-level model, and carbon-carbon structures - by a five-level model. A stage-by-stage solution procedure is suggested for the boundary-value problem of micromechanics of composites. For a wide class of composites, the effective elastic moduli are calculated and the strength surfaces are constructed.     

Shakedown analysis of periodic composites with kinematic hardening material model

Lower-bound limit and shakedown analysis of periodic composites with the consideration of kinematic hardening are carried out on the representative volume element level. In combination with homogenization theory, the homogenized macroscopic admissible loading domains are determined. Furthermore, the strengths of periodic composites by using elastic perfectly plastic, unlimited and linear limited kinematic hardening material models are calculated and compared. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A non-conforming finite element for limit analysis of periodic composites

In this work, a non-conforming three-dimensional finite element coupled with direct methods and homogenization technique is presented for the limit analysis of periodic metal-matrix composites. Using this element, which is constructed from bilinear shape functions and enriched by internal second-order polynomials, limit analysis of composite material can be efficiently carried out. Accuracy and overall performance are illustrated through comparison with different structural solid elements in the context of direct as well as incremental methods. It is shown that the limit domain of periodic composites for different fiber distributions and volume fractions provides a foundation for the structural design. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of dynamic laminates

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities (called dynamic materials). The study was initiated by Lurie in his pioneering work of 1997. Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell's system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. In particular, we examine a cell problem for periodic composites as well as the laminate formulae. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell's system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.     

Micromechanics of Electro- and Magneto-active Soft Composites

We study the coupled behavior in soft active microstructured materials undergoing large deformations in the presence of an external electric or magnetic field. We focus on the role of the microstructures on the coupled behavior, and examine the phenomenon in the composites with (a) periodic composites with rectangular and hexagonal periodic unit cells, and (b) in composites with the random distributions of active particles embedded in a soft matrix. We show that for these similar microstructures exhibit very different responses in terms of the actuation, and the coupling phenomenon. Next, we consider the macroscopic and microscopic instabilities in the active composites. We show that the external field has a significant influence of the instability phenomena, and can stabilize or destabilize the composites depending on the direction relative to composite geometry. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effects of external disturbances on the performance and chaotic behaviour of industrial FCC units

The dynamic behaviour of an industrial Type IV fluid catalytic cracking for the production of gasoline unit is investigated for a case where the air feed temperature is periodically forced. The investigation concentrates on the behaviour of the system for a case of bistability for the autonomous system with special emphasis on the effect of forcing on the periodic attractor of the autonomous system. When the centre of forcing is very close to the homoclinical termination point of the autonomous periodic attractor, period-doubling mechanism and Type 1 intermittency have been identified as the routes to chaos for this six-dimensional (6D) system. Chaotic behaviour occurs at very low forcing amplitudes which simulate small disturbances that are unavoidable in the operation of any industrial unit. While in certain ranges of the values of the forcing amplitudes the output amplitudes of the forced system are higher than their counterparts in the autonomous system, other regions show the opposite behaviour. Average gasoline yield in the bistability region for the attractor resulting from the forcing of the autonomous periodic attractor is much higher than that resulting from forcing the autonomous static attractor. This yield is very close to that obtained with the optimum steady state which is unstable and requires prohibitively high values of controller gains to be stabilized.     

具有时滞的周期Logistic方程的持续性与周期解

这篇文章的目的是讨论一类具有多个时滞的周期Logistic方程的动力学行为.证明 了在某些条件下系统是持续的,建立了全局吸引周期解的充分条件.     

A model of immune system with time-dependent immune reactivity

In this paper we deal with the Marchuk model of an immune system. Among the main parameters of the model are the coefficients which describe the state of infected organism and the rate of production of antibodies. In the classical model these coefficients are constants. We consider the case when these coefficients are time-dependent. In particular, we are interested in the case of periodic coefficients which can describe periodic changes of the immune reactivity due to periodic changes of the environment. We examine the asymptotic behaviour of solutions. Under some assumptions we prove that the solutions tend to periodic functions. We also present the results of numerical simulations to illustrate the behaviour of solutions.     

Simulation of a long slender structure in turbulent flow

The dynamic behaviour of a long slender structure in turbulent flow can vary considerably, depending on the particular excitation. Regimes from simple static deformation, over periodic to irregular and chaotic behaviour are observed. An example is the self-excited periodic motion of a long slender structure in cross flow. In such cases, the fluid flow and the structure movement are strongly coupled. The paper presents a robust partitioned coupling approach based on the Immersed Boundary Method which integrates the geometrically exact Cosserat rod model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic study of the macroscopic behaviour of a solid-fluid mixture

In the framework of homogenization theory we study a mixture of an elastic solid and a viscous compressible fluid with periodic structure and its limit behaviour as the period tends to zero Existence, uniqueness and convergence theorems are given. The limit behaviour is viscoelastic.     

C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C¹ and C² maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 1881]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.     

The extremal index,hitting time statistics and periodicity

The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.     

Sensitivity to perturbations in variable structure systems

Variable structure systems are systems for which the behaviour switches between several configurations, each of them modelled by a system of ODEs. The behaviour of perturbations around trajectories of variable structure systems is studied and a sensitivity formula is derived. This formula consists of a chain of monodromy matrices and corrections which are rank-1 updates of the unity matrix. Applications include the adaptation of classical shooting algorithms for computing periodic solutions to piecewise smooth systems.     

Numerical aspects on computational homogenization of epoxy/glass composites

Numerical aspects of two-scale modeling of epoxy/glass composites are presented. The homogenization process is carried out under consideration of periodic boundary constraints (PBC) of the representative volume element (RVE) due to the periodic structure of glassfiber reinforced epoxy systems. The introduction of artificial constraints for computing macro-stresses and macro-moduli is presented by giving the modified algorithmic treatment of a two-scale approach using PBC. The proposed algorithm is applied to an ISO 527 epoxy/glass test specimen. The results of computations considering or not considering interphases and interfaces within the composite are compared. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time asymptotics for solutions of the Burgers equation with a periodic force

We consider the Burgers equation with a periodic force which presents a simplified model for turbulence. We are interested in the asymptotic behaviour of solutions for . This problem has been studied by Sinai who uses a probabilistic and very technical approach. Using methods from spectral theory we get similar results. This functional analytic approach gives an easier proof. For certain initial data (periodic or some random perturbations of those) we show time-convergence towards a deterministic periodic limit solution related to the ground state of a certain Schr?dinger operator. Received June 10, 1998