Isotropic Inclusions Research Articles

Plane Wave-Perturbative Method for Evaluating the Effective Speed of Sound in 1D Phononic Crystals

A method for calculating the effective sound velocities for a 1D phononic crystal is presented; it is valid when the lattice constant is much smaller than the acoustic wave length; therefore, the periodic medium could be regarded as a homogeneous one. The method is based on the expansion of the displacements field into plane waves, satisfying the Bloch theorem. The expansion allows us to obtain a wave equation for the amplitude of the macroscopic displacements field. From the form of this equation we identify the effective parameters, namely, the effective sound velocities for the transverse and longitudinal macroscopic displacements in the homogenized 1D phononic crystal. As a result, the explicit expressions for the effective sound velocities in terms of the parameters of isotropic inclusions in the unit cell are obtained: mass density and elastic moduli. These expressions are used for studying the dependence of the effective, transverse and longitudinal, sound velocities for a binary 1D phononic crystal upon the inclusion filling fraction. A particular case is presented for 1D phononic crystals composed of W-Al and Polyethylene-Si, extending for a case solid-fluid.

A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems

The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to far-field operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß–Newton method and show numerical experiments in the special case of star-shaped obstacles.

Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.

Effective viscoelastic properties of micro-cracked heterogeneous materials

The aim of this paper is to model the effective viscoelastic properties of micro-cracked masonry based on a homogenization approach. The masonry is modeled as a mixture of a micro-cracked viscoelastic mortar matrix and isotropic elastic brick inclusions. Firstly, the effect of micro-cracks on the effective viscoelastic properties of mortar is considered at micro-scale. Secondly, brick inclusions are added in the micro-cracked mortar matrix at mesoscale to simulate effective viscoelastic properties of masonry. The modified Maxwell viscoelastic model is employed for noncracked, micro-cracked mortar matrix and micro-cracked masonry. The use of the same modified Maxwell model of the noncracked mortar to model the viscoelastic properties of the micro-cracked mortar is an approximation. This approximation is carried out in the short- and the long-term behaviors in the Laplace–Carson space. It is validated in transient situation with exact solutions obtained from the inverse Laplace–Carson transform for simple loading conditions. This technique allows avoiding the complexity of the inverse Laplace–Carson transform. The isotropic case with random orientation distribution of open micro-cracks in mortar and with spherical brick inclusions is considered.

Оценка методом самосогласования эффективной теплопроводности текстурированного композита с трансверсально изотропными эллипсоидальными включениями

Оценка методом самосогласования эффективной теплопроводности текстурированного композита с трансверсально изотропными эллипсоидальными включениями

Micromechanics Analysis of Elastic Modulus of Alumina-Carbon Refractories

Based on the micromechanics theory, Alumina-carbon refractories were regarded as resin-carbon bonded composites, including alumina, graphite and pores derived from particles packing gaps and phenolic resin pyrolysis. Graphite was regarded as isotropic spherical inclusions; particles packing gaps and phenolic resin pyrolysis pores were regarded as pore phase all together. Applying Mori-Tanaka multi-phase spherical inclusion method, firstly, elastic constants of resin-carbon phase were computed reversely by the elastic constants known alumina-carbon refractories, alumina and graphite, and then the effective elastic modulus of alumina-carbon refractories were estimated by the calculated elastic constants of resin-carbon and other raw materials. The results show that: the predicted elastic modulus by Mori-Tanaka model are higher than the experimental measurement values; resin carbon residue and pores have a great influence on effective elastic modulus of alumina-carbon refractories.

Colloidal particles in blue phase liquid crystals.

We study the effect of disorder on the phase transitions of a system already dominated by defects. Micron-sized colloidal particles are dispersed chiral nematic liquid crystals which exhibit a blue phase (BP). The colloids are a source of disorder, disrupting the liquid crystal as the system is heated from the cholesteric to the isotropic phase through the blue phase. The colloids act as a preferential site for the growth of BPI from the cholesteric; in high chirality samples BPII also forms. In both BPI and BPII the colloids lead to localised melting to the isotropic, giving rise to faceted isotropic inclusions. This is in contrast to the behaviour of a cholesteric LC where colloids lead to system spanning defects.

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i) is positive definite only when the discrepancy tensor is negative defined; (ii) the non-local material symmetries are the same of the discrepancy tensor, and (iii) the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthotropic matrix.

Coupled 2D electric, magnetic, and mechanical fields in dielectrics with cracks and thin inclusions

We apply methods of the theory of thermoelastic deformation of bodies with thin inclusions to study magnetoelectroelastic media with thin inhomogeneities. We construct the integral equations of the problem and propose an efficient numerical procedure of the boundary-element method for their solution. It is established that the fields of mechanical stresses, electric displacements, and magnetic induction near the tip (front) of a thin inclusion, the mathematical model of which can be based on the coupling principle for continua of different dimensions, have a square-root singularity. For the complete description of asymptotic relations for stresses, displacements, and other quantities near the front of an inhomogeneity, we introduce 10 generalized intensity factors for stresses, electric displacements, and magnetic induction. We obtain analytical solutions of the problem for the limiting cases of a permeable crack and a rigid electroconductive inclusion in a magnetoelectroelastic medium. Results of the numerical analysis of intensity factors for an elastic isotropic inclusion in an anisotropic magnetoelectroelastic medium are presented.

Interface Parameters of Composite Materials with an Elliptical Cross-Section Fiber Bundle

How to design the interfacial properties is a significant fundamental issue in the field of the composite materials, while little work was concerned with the mechanical design of the interface for the fiber reinforced polymer. In the present work, a fiber bundle embedded in the matrix was described as a transversely isotropic material. Based on the imperfect interface conditions, the interface parameters were derived to satisfy the neutral conditions for the composite materials reinforced by the elliptical cross-section fiber bundle. It is found that the interface parameter is not always associated with the applied loading in the case of the anti-plane shear. In the state of equal-biaxial tension, the normal interface parameter is merely related to the mechanical properties of components except for the shape of the fiber bundle, but independent of the loading magnitude. In the other cases of pure shear and uniaxial tension, the neutral interface does not exist except that the fiber bundle has a circular cross-section. It is also found that the interface parameters can be expressed in the forms similar to that for an isotropic inclusion by using Kolosov constant in the in-plane deformations.

Tensor spherical harmonics theories on the exact nature of the elastic fields of a spherically anisotropic multi-inhomogeneous inclusion

Eshelby's theories on the nature of the disturbance strains due to polynomial eigenstrains inside an isotropic ellipsoidal inclusion, and the form of homogenizing eigenstrains corresponding to remote polynomial loadings in the equivalent inclusion method (EIM) are not valid for spherically anisotropic inclusions and inhomogeneities. Materials with spherically anisotropic behavior are frequently encountered in nature, for example, some graphite particles or polyethylene spherulites. Moreover, multi-inclusions/inhomogeneities/inhomogeneous inclusions have abundant engineering and scientific applications and their exact theoretical treatment would be of great value. The present work is devoted to the development of a mathematical framework for the exact treatment of a spherical multi-inhomogeneous inclusion with spherically anisotropic constituents embedded in an unbounded isotropic matrix. The formulations herein are based on tensor spherical harmonics having orthogonality and completeness properties. For polynomial eigenstrain field and remote applied loading, several theorems on the exact closed-form expressions of the elastic fields associated with the matrix and all the phases of the inhomogeneous inclusion are stated and proved. Several classes of impotent eigenstrain fields associated to a generally anisotropic inclusion as well as isotropic and spherically anisotropic multi-inclusions are also introduced. The presented theories are useful for obtaining highly accurate solutions of desired accuracy when the constituent phases of the multi-inhomogeneous inclusion are made of functionally graded materials (FGMs).

Elastic composite with negative stiffness inclusions in antiplane strain

An elastic composite consisting of an isotropic matrix with isotropic cylindrical inclusions of negative shear modulus is considered in the antiplane strain approximation. We show that the effective moduli of a composite with negative stiffness inclusions are defined uniquely and can be determined by using the conventional elastic solution where the positive moduli are replaced with the negative ones. We compute the effective shear modulus of the composite with interacting negative stiffness inclusions using the differential self-consistent method. We found that for concentrations of inclusions below a certain critical value the composite is stable and has positive effective shear modulus that is higher than that of the matrix. Thus negative stiffness inclusions stiffen the matrix, as has been suggested by R. Lakes. However, as soon as the concentration of inclusion exceeds the critical value the effective modulus of the composite abruptly drops to a negative value and the composite becomes unstable. Correspondingly, the value of the negative effective stiffness depends upon the stabilising device. The critical concentration depends upon the value of the negative modulus of inclusions; there exists a value of negative shear modulus at which the critical concentration is zero, i.e. the composite is always unstable.

인장 하중을 받는 무한 고체에 포함된 다수의 다이아몬드 형 함유체 문제 해석을 위한 체적 적분방정식법

A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in unbounded isotropic elastic solids containing multiple interacting isotropic or anisotropic diamond-shaped inclusions subject to remote uniaxial tension. The method is applied to two-dimensional problems involving long parallel diamond-shaped cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of the inclusions. The effects of the number of isotropic or anisotropic diamond-shaped inclusions and of the various fiber volume fractions for the circular inclusions circumscribing its respective diamond-shaped inclusion on the stress field at the interface between the matrix and the central inclusion are also investigated in detail. The accuracy and efficiency of the method are examined through comparison with results obtained using the finite element method.

타원 섬유가 포함된 복합재료에서의 탄성 해석

체적 적분방정식법(Volume Integral Equation Method)이라는 새로운 수치해석 방법을 이용하여, 서로 상호작용을 하는 등방성 또는 이방성 타원 함유체를 포함하는 등방성 무한고체가 정적 인장하중을 받을 때 무한고체 내부에 발생하는 응력분포 해석을 매우 효과적으로 수행하였다. 즉, 등방성 기지에 다수의 등방성 또는 이방성 타원 함유체의 중심이 1) 정사각형 배열 형태 또는 2) 정육각형 배열 형태로 포함되어 있는 경우에, 다양한 타원을 포함하는 원형 실린더 함유체의 체적비에 대하여, 중앙에 위치한 타원 함유체와 등방성 기지의 경계면에서의 인장응력 분포의 변화를 구체적으로 조사하였다. 또한, 체적 적분방정식법을 이용한 해를 유한요소법을 이용한 해 및 해석해와 비교해 봄으로서, 체적 적분방정식법을 이용하여 구한 해의 정확도를 검증하였다. A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solids containing interacting multiple isotropic or anisotropic elliptical inclusions subject to remote uniaxial tension. The method is applied to two-dimensional problems involving long parallel elliptical cylindrical inclusions. A detailed analysis of stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of the inclusions. Effects of the number of isotropic or anisotropic elliptical inclusions and various fiber volume fractions for the circular inclusion circumscribing its respective elliptical inclusion on the stress field at the interface between the matrix and the central inclusion are also investigated in detail. The accuracy and efficiency of the method are examined through comparison with results obtained from analytical and finite element methods. The method is shown to be very accurate and effective for investigating the local stresses in composites containing isotropic or anisotropic elliptical fibers.

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.

A finite element study on the hardness of carbon nanotubes-doped diamond-like carbon film

Abstract

Wave dynamics by a plane wave on a half-space metamaterial made of plasmonic nanospheres: a discrete Wiener–Hopf formulation

A rigorous analytical solution for the description of wave dynamics originated at the interface between a homogeneous half-space and a half-space metamaterial made by arrayed plasmonic nanospheres is presented. The solution is cast in terms of an exact analytical representation obtained via a discretized Wiener–Hopf (WH) technique assuming that each metallic nanosphere is described by the single dipole approximation. The solution analytically provides and describes the wave species in the metamaterial half-space, their modal wavenumbers and launching coefficients at the interface. It explicitly satisfies the generalized Ewald–Oseen extinction principle for periodic structures, and it also provides a simple analytical solution for the reflection coefficient from the half-space. The paper presents a new WH formulation for this class of problems for the first time, and describes the analytical solution, which is also tested against a purely numerical technique. While only the case of orthogonal plane wave incidence and isotropic inclusions is considered here, the method can be easily generalized to the oblique incidence and anisotropic constituent (tensorial polarizability) cases.

Generalized self‐consistent like method for mechanical degradation of fibrous composites

AbstractThis paper deals with homogenization of heterogeneous, fibre‐reinforced material. One of the main points of the work consists in taking into account the breakage of the fibres, to compute the effective behaviour of the damaged composite. Furthermore, the possible yielding of the matrix material is considered. The problem is solved for the coupled thermo‐mechanical field, that is all material characteristics can be temperature dependent and thermal loads can be simulated. The method is based on a self consistent type scheme, suitably extended for the case at hand. The effective behaviour is obtained by considering isotropic elastic‐brittle cylindrical inclusions surrounded by a shell of elasto‐plastic matrix material immersed in an infinite medium endowed with the effective properties. Usually in the framework of the self consistent scheme, the homogenised material behaviour is obtained with a symbolic approach. On the contrary, in this work the solution is found in a non‐classical way, as we use the finite element method to find the solution of the thermo‐mechanical problem as a function of the effective material characteristics.

Elastic analysis of a half-plane containing an inclusion and a void using a mixed volume and boundary integral equation method

A mixed volume and boundary integral equation method is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to a traction-free boundary. A detailed analysis of the stress field is carried out for three different geometries of the problem. It is demonstrated that the method is very accurate and effective for investigating local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.

A Wideband Uniplanar Polarization Independent Left-Handed Metamaterial

A simplified structure made of periodic array of cross cut-wire pairs (CCWPs), printed on only one side of a dielectric substrate, is introduced. This simple negative refractive index (NRI) metamaterial is responsive to arbitrarily linearly polarized incident waves, which supports both symmetric and antisymmetric resonance modes, offering a very wide double-negative (DNG) passband with low losses. The superior performance of this 2-D isotropic inclusion is investigated both numerically and experimentally. The influence of the structure parameters on the magnetic response is studied in detail.