Vladyslav V. Danishevskyy

Higher order asymptotic homogenization and wave propagation in periodic composite materials

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Nonlinear Elastic Problems

We begin with a multi-layer composite material, composed of interacting components \({{\varOmega }^{(1)}}\) and \({{\varOmega }^{(2)}}\) (Fig. 8.1).

Composite dynamic models for periodically heterogeneous media

Propagation of elastic waves through discrete and continuous periodically heterogeneous media is studied. A two-scale asymptotic procedure allows us to derive macroscopic dynamic equations applicable at frequencies close to the resonant frequencies of the unit cells. Matching the asymptotic solutions by two-point Padé approximants, we obtain new higher-order equations that describe the dynamic behaviour of the medium both in the low and in the high frequency limits. An advantage of the proposed approach is that all the macroscopic parameters can be determined explicitly in terms of the microscopic properties of the medium. Dispersion diagrams are evaluated and the propagation of transient waves induced by pulse and harmonic loads is considered. The developed analytical models are verified by comparison with data of numerical simulations. For high-contrast media, we can observe an analogy between the propagation of waves in heterogeneous solids and in thin-walled waveguides. It is also shown that different combinations of cell resonances may result in some additional types of waves that do not appear in the classical continuous theory.

Wide Frequency Higher-Order Dynamic Model for Transient Waves in a Lattice

Propagation of transient waves through a periodic elastic lattice is considered. Asymptotic solutions are used to describe the effective dynamic properties of the structure at frequencies close to the continuous and anti-continuous limits. Matching the asymptotic solutions by two-point Pade approximants, we derive a new dynamic equation that is applicable in a wide frequency range. An advantage of the proposed approach is that all the macroscopic parameters can be determined explicitly in terms of the microscopic properties of the medium. Dispersion diagram is evaluated and the propagation of transient waves induced by pulse and harmonic loads is studied. The developed analytical model is verified by comparison with data of numerical simulations.

Elastic waves in periodically heterogeneous two-dimensional media : locally periodic and anti-periodic modes

Propagation of anti-plane waves through a discrete square lattice and through a continuous fibrous medium is studied. In the long-wave limit, for periodically heterogeneous structures the solution can be periodic or anti-periodic across the unit cell. It is shown that combining periodicity and anti-periodicity conditions in different directions of the translational symmetry allows one to detect different types of modes that do not arise in the purely periodic case. Such modes may be interpreted as counterparts of non-classical waves appearing in phenomenological theories. Dispersion diagrams of the discrete square lattice are evaluated in a closed analytical from. Dispersion properties of the fibrous medium are determined using Floquet–Bloch theory and Fourier series approximations. Influence of a viscous damping is taken into account.

Asymptotic Analysis of Perforated Membranes, Plates and Shells

The perforated membranes, plates and shells are widely used in the numerous technical applications, and some examples are shown in the Fig. 7.1.

Models of Composite Materials and Mathematical Methods of Their Investigation

The linear theory of elasticity yields the following relations between the displacements \({{u}_{i}}\), deformations \({{\varepsilon }_{ij}}\) and stress \({{\sigma }_{ij}}\) in a continuous matter.

Elastic and Viscoelastic Properties of Fibre- and Particle-Reinforced Composites

It is known in industry that in order to increase stiffness and loading ability of materials it is suitable to reinforce the material by fibres.

Conductivity of Particle-Reinforced Composites: Analytical Homogenization Approach

In this chapter, we consider the particle-reinforced composites consisting of infinite matrix \({{\varOmega }^{(1)}}\) and spherical inclusions \({{\varOmega }^{(2)}}\), composed of simple cubic (SC) (Fig. 4.1a) and body-centred cubic (BCC) (Fig. 4.1b) lattices.

Conductivity of Fibre Composites: Analytical Homogenization Approach

Application of the multi-scale asymptotic homogenization method allowed us to separate global and local components of the solution and to reduce the input boundary value problem in a multi-connected domain to a recurrent sequence of local problems, considered within a representative unit cell of the composite structure.