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About this book

This book addresses theoretical and experimental methods for exploring microstructured metamaterials, with a special focus on wave dynamics, mechanics, and related physical properties. The authors use various mathematical and physical approaches to examine the mechanical properties inherent to particular types of metamaterials. These include: • Boundary value problems in reduced strain gradient elasticity for composite fiber-reinforced metamaterials • Self-organization of molecules in ferroelectric thin films • Combined models for surface layers of nanostructures • Computer simulation at the micro- and nanoscale • Surface effects with anisotropic properties and imperfect temperature contacts • Inhomogeneous anisotropic metamaterials with uncoupled and coupled surfaces or interfaces • Special interface finite elements and other numerical and analytical methods for composite structures

Table of Contents

Frontmatter

On Anti-Plane Surface Waves Considering Highly Anisotropic Surface Elasticity Constitutive Relations

Abstract
Within the framework of highly anisotropic surface elasticity model we discuss the propagation of new type of surface waves that are anti-plane surface waves. By the highly anisotropic surface elasticity model we mean the model with a surface strain energy density which depends on incomplete set of second derivatives of displacements. From the physical point of view this model corresponds to a coating made of a family of parallel long fibers which posses bending and extensional stiffness in one direction only. As for other models with surface energy there exist anti-plane surface waves. In the paper the dispersion relation is derived and dependence on the material parameters is analyzed.
Victor A. Eremeyev

Periodic Problem for a Plane Composed of Two-Layer Strips with a System of Longitudinal Internal Inclusions and Cracks

Abstract
The paper considers plane deformation state of a piecewise homogeneous uniformly layered plane of two dissimilar materials, when there is a system of cracks on the midlines of layers made of one material, and layers made of the other material are reinforced by a system of elastic inclusions also located on the midlines. A system of governing equations of the problem is obtained in the form of a system of singular integral and integro-differential equations for the dislocation function of the points of the cracks faces and tangential contact stresses acting on the long sides of the inclusions. The solutions of the obtained systems are constructed by the method of mechanical quadrature. A numerical calculation was carried out and the laws of change in the coefficients of concentration of destructive stresses at the end points of cracks and contact stresses were studied depending on the mechanical and geometric parameters of the problem.
V. N. Hakobyan, A. V. Sahakyan, K. L. Aghayan

An Experimental Model of the Acoustic Wave Propagation Through a Cascading Triple-Periodic Array of Cylindrical Holes

Abstract
We study experimental acoustic properties of the meta-materials made as aluminum parallelepipeds with a crossed periodic system of through round holes. The experiments have been made with the use of industrial ultrasonic flaw detectors by the through-transmission technique in a wide interval of the ultrasonic frequencies. There is performed the analysis of the obtained temporary and spectral characteristics of the through-transmitted signal.
Vladimir M. Zotov, Vitaly V. Popuzin, M. Y. Remizov

Simulation of the Surface Structure of Ferroelectric Thin Films

Abstract
Metropolis and Wang-Landau algorithms are described and illustrated on the base two-dimensional Ising model. The influence of the ferroelectric film thickness and the depolarizing field on the spontaneous polarization and the order parameter of the film has been investigated by means of the Monte-Carlo method. Dependences of the polarization of the thin film on the temperature are calculated at different values of its thickness and the potential well depth of the Lennard-Jones potential. To investigate the geometrical and optical properties of textured coatings the anisotropic three-dimensional model based on the fractal plurality of Julia is used. The developed method allows to determine the values of the model parameters for a number of coating samples of steel sheet obtained under different conditions of their formation. The fractal dimension of the objects obtained on the base of this model is determined.
Olga G. Maksimova, Tatiana O. Petrova, Victor A. Eremeyev, Vladislav I. Egorov, Alexandr R. Baidganov, Olga S. Baruzdina, Andrei V. Maksimov

Diffraction of the High-Frequency Waves by Arrays of Obstacles in the Two-Dimensional Elastic Medium, with Multiple Reflections and Transformations

Abstract
Within the geometric theory of diffraction, the problem of the propagation of ultrasonic waves through array of obstacles in an infinite two-dimensional elastic medium is investigated. A tonal impulse of a time-harmonic longitudinal or transverse plane elastic high-frequency wave of several wave-lengths is introduced through the array of obstacles, and in a certain domain inside the elastic medium the through-transmitted wave with arbitrary reflections and transformations is received. Some integral representations for displacement in the reflected waves are written out on the basis of the Kirchhoff physical diffraction theory. With the use of an asymptotic estimate of multiple diffraction integrals by the multidimensional stationary phase method we have written out explicitly the geometric-theory approximation for displacements in the multiply reflected and transformed waves.
Nikolay V. Boyev

The Mixed Boundary Conditions Problem of Layered Composites with Meta-Surfaces in Electro Elasticity

Abstract
The influence of taking into account the presence of roughness of both the external mechanically free surfaces, and the internal surfaces connecting various media, on the propagation of a high-frequency wave signal in a multilayer waveguide is investigated. In order to solve the quasistatic problem of the coupled electroelastic (magnetoelastic, thermoelastic) fields, the joints of the rough surfaces at the composites are simulated as meta-surfaces. In different models of the connection of thick piezoelectric layers, in the zone of the connection of their surfaces, the thin geometrically and physically inhomogeneous multilayer zone, which is equivalent to the meta surface with the dynamic loads, virtually arises. Taking into account the known principles of wave formation and propagation of high-frequency (short-wave) wave signals, as well as the magnitude of the surface roughness, hypotheses of magneto (electro, thermo) elastic layered systems are introduced (hypothesis MELS—Magneto Elastic Layered Systems). Proper selection of the surface exponential functions (SEF) in hypotheses, in equations and in thermodynamic relationships of the problem ensures that the surface roughness is taken into account. The introduction of hypothesis MELS allows modeling of the mathematical boundary-value problem of the contact of rough surfaces of continuous media with related physical and mechanical fields. This approach also makes it easy to calculate the equivalent dynamic electro-mechanical loads on the simulated meta-surface at the interface of the media. The following examples have been analyzed: (i) the propagation of the signal of an elastic shear wave in the case of the connection of rough surfaces of two piezoelectric layers with another thin piezoelectric layer, (ii) the propagation of an electroelastic wav(e in a single-shaped piezoelectric layer, the surfaces roughness of which is filled with an isotropic dielectric or ideal conductor, (iii) the propagation of high-frequency shear elastic waves on interface of isotropic elastic half-spaces with canonical surface protrusions.
Ara S. Avetisyan

A Comparative Analysis of Wave Properties of Finite and Infinite Cascading Arrays of Cracks

Abstract
Reflection and transmission coefficients in the problems of the normal plane wave incidence on the system of finite and infinite periodic arrays of cracks in an elastic body are determined. We propose a method permitting to solve the scalar diffraction problem for both single crack and any finite number of cracks with arbitrary lattice geometry. Under the condition of one-mode frequency regime the problem is reduced to a discretization of the basic integral equation holding on the boundary of the scatterers located in one horizontal waveguide. A semi-analytical method developed earlier for diffraction problems on infinite periodic crack arrays permits a comparative analysis of the properties of the main external parameters for a finite periodic system of cracks, where the solution of the boundary integral equations is numerically constructed, and we obtain explicit analytical representations for the wave field at the boundary of the obstacles. The analysis of the properties of the scattering coefficients depending on the physical parameters is carried out for three diffraction problems: a finite periodic system in a scalar formulation, an infinite periodic system in a scalar formulation, an infinite periodic system in a plane problem of the elasticity theory.
Vitaly V. Popuzin, M. Y. Remizov, Mezhlum A. Sumbatyan, Michele Brigante

Homogenization of Piezoelectric Composites with Internal Structure and Inhomogeneous Polarization in ACELAN-COMPOS Finite Element Package

Abstract
The paper presents the current version of the finite element package ACELAN-COMPOS with the focus on its capabilities for solving the homogenization problems for piezoelectric composites with inhomogeneous polarization of piezoceramic phase. We describe the basic version of the effective moduli method, as well as the simplified theoretical approaches for taking into account the inhomogeneous polarization in the finite element solution of the homogenization problems. We provide the brief description of the main features of the ACELAN-COMPOS package, which we use for solving the described problems. The results of the numerical solution of the homogenization problems for porous piezoceramic composites demonstrate the importance of taking into account the inhomogeneous polarization field for the effective moduli determination.
T. E. Gerasimenko, N. V. Kurbatova, D. K. Nadolin, A. V. Nasedkin, A. A. Nasedkina, P. A. Oganesyan, A. S. Skaliukh, A. N. Soloviev

Three-Dimensional Problems of Harmonic Wave Propagation in an Elastic Layer

Abstract
In the present paper three-dimensional problem of propagation of elastic waves in a waveguide is considered, when several different boundary conditions are realized on the surfaces of the waveguide. We then establish the conditions where surface waves are permissible.
Mels V. Belubekyan, Vagharshak M. Belubekyan

The Experimental Study of a Metamaterial with a Triple-Periodic Microstructure on the Epoxy Base

Abstract
We perform an experimental investigation of the acoustic properties of a metamaterial consisting of triple periodic system of metallic spheres, coated with an epoxy resin. Comparisons are given with the replacement of the wave propagation medium from an epoxy base to water and ice. Then we consider the case of aluminum specimens with holes filled with water and ice. A detailed analysis of the experimental data is performed.
Vladimir M. Zotov, Vitaly V. Popuzin, Alexander E. Tarasov

On the Equations of the Surface Elasticity Model Based on the Theory of Polymeric Brushes

Abstract
Motivating by theory of polymers, in particular, by the models of polymeric brushes we present here the homogenized (continual) two-dimensional (2D) model of surface elasticity. A polymeric brush consists of an system of almost aligned rigid polymeric chains. The interaction between chain links are described through Stockmayer potential, which take into account also dipole-dipole interactions. The presented 2D model can be treated as an highly anisotropic 2D strain gradient elasticity. The surface strain energy contains both first and second derivatives of the surface field of displacements. So it represents an intermediate class of 2D models of the surface elasticity such as Gurtin-Murdoch and Steigmann-Ogden ones.
Roman A. Gerasimov, Tatiana O. Petrova, Victor A. Eremeyev, Andrei V. Maximov, Olga G. Maximova

Finite Element Study of Ceramic Matrix Piezocomposites with Mechanical Interface Properties by the Effective Moduli Method with Different Types of Boundary Conditions

Abstract
The paper deals with the problem of finding the effective moduli of a ceramic matrix composite with surface stresses on the interphase boundaries. The composite consists of a PZT ceramic matrix, elastic inclusions and interface boundaries. It is assumed that the interface stresses depend on the surface strains according to the Gurtin–Murdoch model. This model describes the size effects and contributes to the total stress-strain state only for nanodimensional inclusions. The homogenization problem was set and solved with the help of the effective moduli method for piezoelectric composites with interface boundaries and finite-element technologies used for simulating the representative volumes and solving the resulting boundary-value electroelastic problems. Here in the effective moduli method, different combinations of linear first-kind boundary conditions and constant second-kind boundary conditions for mechanical and electric fields were considered. The representative volume consisted of cubic finite elements with the material properties of the matrix or inclusions and also included the surface elements on the interfaces. Bulk elements were supplied with the material properties of the matrix or inclusions, using a simple random method. In the numerical example, the influence of the fraction of inclusions, the interface stresses and boundary conditions on the effective electroelastic modules were analysed.
G. Iovane, A. V. Nasedkin

A Far-Field Asymptotic Analysis in the High-Frequency Diffraction by Cracks

Abstract
On the basis of recently obtained asymptotic solutions of integral equations by the Wiener-Hopf method for diffraction by a straight finite-length crack in a linear elastic medium, we study the properties of the far-zone scattered field at high frequencies for (1) - anti-plane problem in a homogeneous medium, (2) - anti-plane problem for an interface crack, and (3) - in-plane problem in a homogeneous medium.The method proposed is founded on a high-frequency solution of the basic integral equation of the scattering problem. Then we develop an explicit analytical representation for the leading asymptotic term, by estimating the far-field behavior of the relevant integrals with high oscillations by the method of stationary phase. This allows us to obtain the final form of the scattered field in an explicit analytical form as some quadratures.
M. Y. Remizov

3d Propagation of Ultrasonic Waves Through a System of Defects in an Elastic Material, with Arbitrary Reflections and Transformations

Abstract
In frames of the three-dimensional problem, we study a short wavelength diffraction of elastic waves by a system of voids in the elastic medium. The defects are bounded by arbitrary smooth surfaces. The problem is reduced to a classical diffraction problem for high-frequency waves irradiated from a point source in the elastic medium by the system of voids located in this medium. We consider multiple reflections with various possible transformations of elastic waves. To study the problem, a special method is proposed, which is based on the asymptotic estimate of the diffraction integrals by the multidimensional stationary phase method. On the basis of the developed method, we obtain in explicit form the leading asymptotic term of the displacements in the diffracted field, for arbitrary cases of multiple reflections (longitudinal wave to longitudinal one and transverse wave to transverse one) and transformations (longitudinal wave to transverse one and transverse wave to longitudinal one), at the points of mirror reflections. The obtained explicit expressions for the displacements agree with the Geometrical Diffraction Theory (GDT) for elastic waves.
Nikolay V. Boyev, Mezhlum A. Sumbatyan, Vittorio Zampoli

Numerical Investigation of Effective Moduli of Porous Elastic Material with Surface Stresses for Various Structures of Porous Cells

Abstract
The chapter deals with the model problem of finding the effective moduli of a nanoporous elastic material, in which the surface stresses are defined on the pore surface to reflect the size effect using the Gurtin–Murdoch model. One cell of a porous material in the form of a cube with one pore located in the center is considered. The objective of the study is to assess the influence of the pore shape and the magnitude of the scale factors on the effective moduli of the composite material. The homogenization problem is formulated within the framework of the effective moduli method, and to find its solution, the finite element method and the ANSYS software package are used. In the finite element model, the surface stresses are taken into account by membrane elements covering the pore surfaces and conformable with the finite element mesh of bulk elements. Numerical experiments carried out for pores of cubic and spherical shapes show the cumulative significant effect of pore geometry and scale factors on the effective elastic moduli.
A. V. Nasedkin, A. S. Kornievsky

Electro-Magneto-Elastic Coupled Waves in Piezoactive Periodic Structures

Abstract
Based on the complete set of Maxwell’ electrodynamics equations and the theory of elasticity the two-dimensional equations have obtained describing the coupled wave process in piezoactive electro-magneto-elastic (MEE) structure and allowing solution of a new class of problems, in particular, the problems of propagation and internal resonance of electro-magneto-elastic waves in periodic MEE structures. For longitudinal lattice vibrations of oppositely polarized MEE periodic superlattice the effect of phonon–photon polariton is investigated with a full three-phase coupling between elastic, electromagnetic fields. The results show that the new coupled phonon–photon polariton exhibits properties different from piezoelectric or piezomagneticpolaritons.
Karen B. Ghazaryan, Davit G. Piliposyan, Gayane T. Piliposian

Non-linear Dynamics of Pantographic Fabrics: Modelling and Numerical Study

Abstract
In this work, the dynamical behavior of a pantographic sheet undergoing sinusoidal (in time) imposed displacement is numerically investigated. The used model has been largely exploited to analyse the quasi-static behavior of pantographic materials. Here we propose to use a non-linear generalization of such a model for the description of a pantographic material dynamical behavior.
Marco Laudato, Emilio Barchiesi
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