Piecewise Homogeneous Medium Research Articles

Analysis of the decay of edge effects in laminated materials on the basis of a representative element

A model and design model for representative-element analysis of the decay of the Saint-Venant edge effects in laminated materials are proposed. The decay of the edge effect in a matrix with laminated covering (a composite material of irregular structure) is considered for the case of symmetric deformation. The source of the edge effect is simulated by a piecewise-constant surface periodic load normal to the layers. This load is local for the working domain and is varied within a boundary section that is commensurable with the characteristic size of the heterogeneity of the material structure. The equations of the linear theory of elasticity, a model of a piecewise-homogeneous medium, and quantitative criteria for identification of edge effects are used. A discrete model of the problem and its solution are constructed within the concept of basic schemes. Data on the zone of the edge effect and the character of its decay for a representative element of the material are analyzed.

The simulation of electromagnetic scattering in piecewise homogeneous media using unstructured grids

This paper presents a method for simulating, in the time domain, the scattering of electromagnetic waves in piecewise homogeneous media. The method employs an explicit Taylor–Galerkin approach, implemented on a general unstructured triangular grid, with the electric and magnetic fields approximated in a continuous piecewise linear fashion. All boundary and material interface conditions are weakly applied in an integral form, following a characteristic decomposition of the solution. A number of simple examples, for which exact scattering width distributions are available, are included to demonstrate the numerical performance of the proposed procedure.

Calculation of stability of laminated composite coatings in tribotechnics

The stability of laminated coatings in tribotechnics under compression loads is investigated on the basis of the three-dimensional linearized theory of stability within the framework of a model of piecewise-homogeneous medium. The general formulation of the problem is given, and the solution of a particular example is considered.

Surface Instability of Laminated Coatings upon Inelastic Deformation

The problem of surface instability of laminated coatings with inelastic properties is considered within the framework of a model of piecewise-homogeneous media on the basis of the three-dimensional linearized theory of stability. A general statement of the problem is formulated and the basic characteristic equations are derived. The solutions of particular problems are obtained for elastoplastic and viscoelastic models of solids.

Effect of geometric and mechanical characteristics of antifriction coatings on their stability

We consider the problem of the stability of layered antifriction coatings on the basis of three-dimensional linearized stability theory, within the framework of a piecewise-homogeneous medium subjected to small precritical strains. We have formulated the problem and constructed the basic characteristic equations. We have done a numerical study of the effect of the basic mechanical and geometric characteristics of the layers on the surface instability of the coating, and we give recommendations for designing coatings with optimal parameters.

Asymptotic Distribution of Eigenfunctions and Eigenvalues of the Basic Boundary-Contact Oscillation Problems of the Classical Theory of Elasticity

Abstract The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained.

Existence of solution for a free boundary problem in a nonlinear piecewise homogeneous medium

A free boundary problem arising from the bidimensional thermal modelling of aluminium electrolytic cells is studied. The medium is assumed piecewise homogeneous and nonlinear. A fixed domain method is proposed which leads to a weak formulation of the problem. Existence of weak solution is proved by regularizing the contact condition between the homogeneous subdomains and passing to the limit.

Elastodynamic boundary element method for piecewise homogeneous media

A general higher-order formulation for the time domain elastodynamic direct boundary element method is presented for computing the transient displacements and stresses in multiply connected two-dimensional solids. The displacement and traction interpolation functions are linear in time and quadratic in space. All integrations are analytical, and are expressed in terms of twelve basic recurring integrals. Causality is ensured by integrating only over the dynamically active parts of each element, and the algorithm presented is time-marching and implicit. The use of analytical integrations allows both unbounded and bounded domain problems to be solved without having to introduce special enclosing elements. All of these improved features allow for a formulation that is very efficient and accurate. The stability and accuracy of the elastodynamic boundary element algorithm is demonstrated by solving several example problems and comparing the results with available analytical and numerical solutions. © 1998 John Wiley & Sons, Ltd.

Scattering theorems for time-harmonic electromagnetic waves in a piecewise homogeneous medium

We consider the scattering of time-harmonic electromagnetic plane waves by a piecewise homogeneous obstacle. In the interior of the obstacle there exists a core which may be a dielectric, a perfect conductor or an imperfect conductor. First, we obtain integral representations for the scattered fields incorporating all the boundary and transmission conditions. These representations are then used to derive the corresponding electric far field patterns. We prove reciprocity and scattering theorems and study the spectrum of the electric far field operator for these problems. Finally, we prove the optical theorem, that is a connection of the electric far field pattern to the scattering cross-section.

Composites with interlaminar imperfections: substantiation of the bounds for failure parameters in compression

Abstract The problem of failure caused by internal instability is considered within the scope of the exact statement formulated for cases of compressible and incompressible, elastic and elastic-plastic, orthotropic and isotropic layers in compression along interfacial defects. This exact statement is based on the model of a piecewise homogeneous medium and equations of the three-dimensional linearized theory of deformable bodies stability (TLTDBS). To estimate critical loads, upper and lower bounds are suggested for laminated composites with defects by utilizing results for rigid-connected and sliding layers. Numerical investigations for composites with an elastic-plastic matrix have shown that the suggested bounds present reasonably, fair results for particular cases of composites.

A mixed dynamic problem of the theory of elasticity for a piecewise homogeneous plane

We study the process of splitting of a compressed piecewise homogeneous medium under high-speed motion of a wedge with the formation of a crack of unknown extent ahead of the wedge. The motion of the wedge occurs along the interface between physico-mechanical properties of the piecewise homogeneous plane. We carry out numerical studies. We obtain asymptotic representations of the elastic potentials at the limiting velocities of motion of the wedge. Two figures. One table. Bibliography: 6 titles.

A time domain unstructured grid approach to the simulation of electromagnetic scattering in piecewise homogeneous media

The simulation of the scattering of plane electromagnetic waves by obstacles constructed from piecewise homogeneous material media is considered. The governing equations are Maxwell's equations, which are expressed in a conservation form. The solution is obtained on a grid consisting of an unstructured assembly of linear triangular elements in 2D, by modifying a technique developed originally for the solution of the compressible Euler equations. The computed solution is used to determine the radar scattering width of the obstacle.

A nonconforming finite element approximation of the transport equation in quadrangular and hexagonal geometries

This paper introduces a new finite element approximation for multi-dimensional transport problems in piecewise homogeneous media. The transport equation is solved using a Galerkin technique with polynomial basis functions in space-angle variables derived from asymptotic transport theory. The phase space is partitioned into cells consistent with the geometry and having each an elemental expansion which is not a tensor product. Improved accuracy may be obtained by multiplying the number of cells or/and increasing the polynomial degree. Numerical results on 1D and 2D reference problems in square geometry show a good agreement with other approximate methods.

Surface instability of a homogeneous half-space coupled with a finite number of laminas

In the context of the model of a piecewise-homogeneous medium in three-dimensional formulation we study the problem of the surface loss of stability in laminated semi-bounded media with a finite number of laminas. For the study we invoke a version of three-dimensional stability theory constructed for small pre-critical deformations when the pre-critical state is determined from the geometrically linear theory. To construct the resolvent characteristic equations we use the matrix representation of the basic relations.Using a computer we carry out a numerical study of the stability of a homogeneous half-space coupled to various numbers of laminas, and we conduct a comparative analysis of the results.

Solving spectral problems in piecewise homogeneous media

The problems of finding natural oscillation frequencies of heterogeneous waveguides and intricately shaped membranes are studied. A numerical-and-analytic method is proposed, which consists in developing a solution in each subregion as a series in basis functions followed by matching on the boundary lines. The results of particular computations are presented. Bibliography: 3 titles.

A boundary integral method for calculating displacement fields in three-dimensional viscoelastic piecewise-homogeneous media. II. Matrix representation solutions

A method of calculating stationary displacement fields in three-dimensional viscoelastic media, consisting of homogeneous regions with curvilinear separation boundaries, is developed on the basis of the boundary integral method. As a result of using the fundamental solutions corresponding to the sum of converging and diverging waves, the field prolongation through a homogeneous region of the medium reduces to simple multiplication of stress-strain matrices. This makes it possible to construct an effective calculation scheme of the displacement field at any point of the medium, similar to the matrix scheme used in seismology for planar stratified media.

Three-dimensional problems of wall-rock stability in the vicinity of a cleaned working with control exercised over the stress-strain state of the mass

As the depth of mining operations increases, control over the stress-stain state (SSS) of the rock mass to ensure safe and effective exploitation of coal deposits is becoming a basic underground procedural process. An arbitrary change in the SSS of the mass under conditions where it is acted upon by large compressive loads may lead, in turn, however, to local buckling and fracture of the free surface of the rock in the near-face zone of a cleaned working during the mining of mineral resources. Individual classes of rock-stability problems that arise in the three-diumensional statement are formulated in this paper on the basis of analysis of actual conditions and technology for the mining of mineral resources at great depths. The rock mass is treated as a inhomogeneous semi-restrained medium of laminar structure; these studies were therefore conducted within the framework of the model of a piecewise-homogeneous medium on the basis of approaches developed previously. 8 refs., 2 figs., 2 tabs.

Three-dimensional grained composites of extreme thermal properties

A new class of two-phase composites possessing optimal macroscopic tensor of heat conductance is found, in which non-identical inclusions are spaced periodically in a matrix making a piecewise homogeneous medium with a microstructure. Geometrical forms of these inclusions minimizing the total thermal energy can be determined numerically after the analytical investigation of a specific inverse boundary problem by the usage of the Newton volume potential. Grained materials of extreme coefficients of heat expansion are also described in the case of cubic symmetry. These results are obtained by the application of the equi-strength concept used before in the theory of elasticity.

Stability of laminated composite material in compression along a microcrack

The stability problem for an arbitrary laminated structure in compression along interlayer cracks was formulated. This formulation used the model of a piecewise-homogeneous medium and the basic relations of the three-dimensional linearized theory of deformable body stability (TLTDBS). Also introduced were the terms - microcrack, macrocrack, and structural crack, depending on the relationship between the crack length and the thicknesses of the layers. The present article is devoted to numerical solution of the stability problem for one of the laminated structures examined in the case of compression along a microcrack.

Method of R-functions in problems with natural boundary conditions

A comparative analysis is conducted of solutions of two model problems in electrostatics obtained with the help of general and partial structures, i.e., satisfying or not satisfying natural boundary conditions. The problem is solved for composite piecewise-homogeneous media with natural boundary conditions, including large contrast of physical characteristics.