Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials

In this chapter the dynamic problems for piezoelectric nanosized bodies with account for coupled damping and surface effects are considered. For these problems we propose new mathematical model which generalizes the models of the elastic medium with damping in sense of the Rayleigh approach and with surface effects for the cases of piezoelectric materials. Our model of attenuation and surface effects has coupling properties between mechanical and electric fields, both for the damping terms and constitutive equations for piezoelectric materials on the surface. For solving the problems stated the finite element approximations are discussed. A set of effective finite element schemes is examined for finding numerical solutions of week statements for nonstationary problems, steady-state oscillation problems, modal problems and static problems within the framework of modelling of piezoelectric nanosized materials with damping and surface effects. For transient and harmonic problems, we demonstrate that the proposed models allow the use of the mode superposition method. In addition, we note that for transient and static problems we can use efficient finite element algorithms for solving the systems of linear algebraic equations with symmetric quasi-definite matrices both in the case of uncoupled surface effects and in the case of coupled surface effects.

The paper is devoted to the calculation of the reflection and the transmission coefficients, when a plane longitudinal wave is incident on a three-dimensional grating with a periodic array of rectangular cracks in the elastic material. In the one-mode frequency range the problem is reduced to a system of integral equations, which can be solved for various sizes of the cracks to give an explicit representation for the wave field inside the cracked structure, as well as the values of the reflection and the transmission coefficients.

The formation of polymer coating on a solid substrate is investigated by means of computer simulation (Monte-Carlo method). The sticking coefficient depending on different factors affecting the adhesion of monomer units is calculated. Mechanical properties are stimulated on the base of the hybrid discrete-continuous model, which describes the system consisting of flexible substrate and polymer coating. At different temperatures and intermolecular interactions constants, the dependencies of Young modulus on the deformation degree are calculated. Ferroelectric properties of the polymer coating depending on frequency and amplitude of external electric field, temperature and interchain interactions are investigated.

In the present paper we study the problem on image identification for clusters of linear cracks located inside an unbounded elastic medium, by using a circular Ultrasonic echo-method. The parameters to be reconstructed are the number of cracks, their size, location and the slope of each defect. The scanning is performed by an ultrasonic transducer of a fixed frequency placed at a certain distance in a far-zone, which can generate an ultrasonic wave incident to the system of cracks at arbitrary angle. The input data, used for the reconstruction algorithm, is taken as the back-scattered amplitudes measured in the echo method for the full circular interval of the scanning angle. The diffraction of the elastic waves is studied in the scalar approximation. The proposed numerical algorithm is tested on some examples with clusters of cracks whose position and geometry are known a priori.

The classical diffraction problem of high-frequency waves emitted from a point source in an elastic medium with a void or a system of voids is considered. The voids are bounded by arbitrary smooth surfaces. Single reflection cases of longitudinal and transverse waves are studied by taking into account their transformations on the boundary surface. Double reflection cases are investigated for two different possibilities of transformations of elastic waves: the longitudinal wave transformed to the transverse one, and vice versa. The developed method of the research is based on the evaluation of diffraction integrals by means of multidimensional stationary-phase method. The novel approach allows us to obtain the leading asymptotic term of the diffracted displacement field as a closed-form expression in the cases of single and double reflections, which corresponds to the geometrical theory of diffraction (GTD).

The chapter presents mathematical modelling and computer design of effective properties of anisotropic porous elastic materials with a nanoscale random structure of porosity. This integrated approach includes the effective moduli method of composite mechanics, the simulation of representative volumes with stochastic porosity and the finite element method. In order to take into account nanoscale sizes of pores, the Gurtin-Murdoch model of surface stresses is used at the borders between material and pores. The general methodology for determination of effective mechanical properties of porous composites is produced for a two-phase bulk (mixture) composite with special conditions for stresses discontinuities at the phase interfaces. The mathematical statements of boundary value problems and the resulting formulas to determine the complete set of effective stiffness moduli of the two-phase composites with arbitrary anisotropy and with surface stresses are described; the generalized problem definitions are formulated and the finite element approximations are given. It is used, that the homogenization procedures for porous composites with surface effects can be considered as special cases of the corresponding procedures for the two-phase composites with interphase stresses if the moduli material of the second phase (nanoinclusions) are negligibly small. These approaches have been implemented in the finite element package ANSYS for a model of nanoporous silicon with cubic crystal system for various values of surface moduli, porosity and number of pores. Model of representative volume was built in the form of a cube, evenly divided into cubic solid finite elements, some of which had been declared as pores. Surface stresses on the boundaries between material and pores were modeled by shell finite elements with the options of membrane stresses. It has been noted that the magnitude of the area of the interphase boundaries has influence on the effective moduli of the porous materials with nanosized stochastic structure.

The chapter presents the mathematical models for thermopiezomagnetoelectric composite materials of an arbitrary anisotropy classes for computer engineering finite element package ACELAN. These homogenization models are based on the method of effective moduli with different boundary conditions, the approaches for generation of representative volumes with specified properties and the finite element method. Important features of ACELAN package also include the original models of irreversible processes of polarization and repolarization for polycrystalline ferroelectric materials. In this paper the software architecture of ACELAN package and its ability for creation of representative volumes with different structures are also described.

We discuss here the mechanics of thin three-layered plates and shallow shells with thin soft core. Recently such thin-walled structures are widely used in engineering, among examples are laminated glasses and photovoltaic panels. We briefly consider layer-wise and first-order shear deformable plates and shells theories in order to model these structures.

The Ray method is applied to study the propagation of a high-frequency plane wave through a triple-periodic system of the spherical obstacles. The initial plane wave is taken as a superposition of spherical waves, which in discretization are reduced to a system of waves, each of them being studied by the Ray method in a local formulation. On the first step, we calculate the geometric parameters of the trajectory of each ray transmitted through the system of spherical obstacles, which is a spatial broken polyline. On the second step, we calculate the wave characteristics, by using methods of the short-wave diffraction.

In the present chapter we consider both computer and natural experimental approaches for the wave propagation through an elastic material with the doubly-periodic system of holes. The numerical study is performed by applying the Boundary Integral Equation method with further discretization to the algebraic system by the Boundary Element Method. A wide range of numerical experiments is conducted for different setups of the doubly periodic system, varying distances, sizes of the holes and their locations. The influence of hole cross-sections on the wave-transmission coefficient is examined by considering different star-like shapes. Natural experiments are based on the ultrasonic testing performed for the steel and plastic materials with the system of small holes. The experimental data are analyzed from the point of their spectral characteristics as well as the amplitude-time dependence.

In this work an integrated approach has been proposed for the determination of the effective mechanical and temperature properties of thermoelastic periodic brick masonry wall with various porous structures. According to the classical method of determining effective moduli of composites, in order to describe internal micro- or macrostructure, we consider a representative volume cell, which enables us to describe effective properties of the equivalent homogeneous anisotropic material. The problems for representative cells are simulated and analyzed as thermoelastic boundary value problems, using special programs in APDL language for ANSYS finite element package. The post processing of the solution gives averaged characteristics of the stress–strain state and thermal flux fields that allow computing the effective moduli of the composite. The proposed method has been applied to several examples of periodic masonry with porous, hollow and porous–hollow bricks. A periodic part of masonry with porous, hollow and porous–hollow bricks was chosen as a representative cell with thermoelastic tetrahedral and hexahedral finite elements. In order to take into account the porosity of the bricks in the masonry, using similar approaches we have preliminary solved at the microlevel the problems of the effective moduli detection for the porous bricks as thermoelastic composite bodies with random porosity structures. After that at the macrolevel the material of porous brick was considered as a homogeneous body with its own effective properties. The results of numerical experiments showed that the structures of the representative cells and porosity could significantly affect the values of the effective moduli for the considered brick masonry walls.

This chapter considers the eigenvalue problems for nanodimensional piezoelectric bodies with voids and with account for uncoupled mechanical and electric surface effect. The piezoelectric body is examined in frictionless contact with massive rigid plane punches and covered by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin-Nunziato model is used. For modelling the nanodimensional effects the theory of uncoupled surface stresses and dielectric films is applied. The weak statements for considered eigenvalue problem are given in the extended and reduced forms. By using methods of functional analysis, the discreteness of the spectrum, completeness of the eigenfunctions and orthogonality relations are proved. A minimax principle for natural frequencies is constructed which has the properties of minimality, similar to the well-known minimax principle for problems with pure elastic media. As a consequence of the general principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric body with voids change, are established. All of these results have been determined for both problems with and without account for surface effects.

A review on models for the statics of out-of-plane deformable pantographic fabrics is presented, along with a model describing the dynamics of in-plane-only deformable pantographic fabrics. We discuss those models able to describe the mechanical exotic properties conferred by the peculiar microstructure possessed by pantographic metamaterials, when three-dimensional deformations and in-plane dynamics are separately involved. For each approach, model formulation and modelling assumptions are discussed along with the presentation of numerical solutions in exemplary cases, and no attempt is made to model damage and failure phenomena.