Isotropic Spherical Inclusions Research Articles

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

Investigation of ultrasonic backscatter using three-dimensional finite element simulations.

Theoretical models are commonly used to describe ultrasonic backscattering in polycrystalline materials. However, although a full multiple scattering formalism has been derived, due to the difficulty in evaluation, currently only the single and double scattering effects have been evaluated. Three-dimensional finite element (3D FE) models have recently been demonstrated to be capable of predicting ultrasonic attenuation in polycrystalline materials and thereby show great potential in overcoming this limitation. In this paper, the application of 3D FE models is extended to the backscatter problem. First, longitudinal-to-longitudinal backscattering amplitudes from single grains are predicted, where the setup and configuration of the finite element (FE) model are verified with an isotropic spherical inclusion for which an exact solution is available. Subsequently, backscatter in terms of the root-mean-square noise levels in two different pulse-echo scenarios is investigated; the first is an idealised configuration with plane wave transmission and point reception; the second represents a more realistic finite-size transducer acting with the same apodization in both transmission and reception. Comparisons of FE predictions and approximate theoretical solutions within a range of validity show good agreement; however, the results demonstrate that 3D FE is useful where the simple Independent Scatterer models break down. As computing power increases, 3D FE is an increasingly viable tool to further the understanding of wave propagation in polycrystalline materials.

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.

Effect of Interface Anisotropy on Elastic Wave Propagation in Particulate Composites

ABSTRACTThe scattering of time harmonic plane longitudinal and transverse elastic waves in a composite consisting of randomly distributed identical isotropic spherical inclusions embedded in an isotropic matrix with anisotropic interface layers is examined. The interface region is modeled as a spherically isotropic shell of finite thickness with five independent elastic constants. The Frobenius power series solution method is utilized to deal with the interface anisotropy and the effect of random distribution of particulates in the composite medium is taken into account via a recently developed generalized self-consistent multiple scattering model. Numerical values of phase velocities and attenuations of coherent plane waves as well as the effective elastic constants are obtained for a moderately wide range of frequencies, particle concentrations, and interface anisotropies. The numerical results reveal the significant dependence of phase velocities and effective elastic constants on the interface properties. They show that interface anisotropy can moderately depress the effective phase velocities and the elastic moduli, but leave effective attenuation nearly unaffected, especially at low and intermediate frequencies. Limiting cases are considered and good agreements with recent solutions have been obtained.

Effect of an inhomogeneous interphase on the thermal expansion coefficient of a particulate composite

This work is a study of the thermal properties of composite materials containing isotropic spherical inclusions surrounded by an inhomogeneous interphase, embedded in an isotropic matrix. By modelling the thermal and mechanical properties of the interphase as continuous radial functions, a replacement technique is used to derive a differential equation which models the thermal expansion coefficient of the composite. The results are generalised for a wide class of functions describing the inhomogeneous properties of the interphase. When the mechanical properties of the interphase are described by a power law profile a closed form solution is obtained for various thermal expansion profiles. The results are displayed graphically against various other theoretical models and with experimental results. A comparison of the results for different values of the problem parameters is also presented. The present model showed good agreement with the experimental results of Holliday and Robinson [Holliday L, Robinson J. Review: the thermal expansion of composites based on polymers. J Mater Sci 1973;8:301–11].

Effective elastic moduli of periodic granular composite with transversely isotropic phases

We find a rigorous solution describing the macroscopically uniform stress state of a periodic granular composite with transversely isotropic phases. The structure of the composite is modeled by a cube containing a finite number of arbitrarily arranged and oriented, transversely isotropic spherical inclusions. This provides the model with a flexible means of describing the microstructure. Applying periodic vector solutions and local expansion formulas reduces the initial boundary-value problem to a system of linear algebraic equations. By averaging the solution over the unit cell, we derived exact finite expressions for the components of the effective stiffness tensor. The numerical data presented help to evaluate the efficiency of the method and the limits of applicability of available approximate theories.

Stress Concentration at Interacting Spherical Inclusions in a Transversely Isotropic Body

An elastic-equilibrium problem is rigorously solved for a transversely isotropic body containing a finite number of arbitrarily arranged and oriented transversely isotropic spherical inclusions or an infinite periodic array of such inclusions. The solution is found based on the superposition principle for general solutions of single-particle problems and the addition theorems for partial vector solutions of the elastic equilibrium equations. The numerical results presented allow assessing the influence of matrix anisotropy on the stress concentration between inclusions

THERMAL STRESS-FOCUSING EFFECT IN A TRANSVERSELY ISOTROPIC SPHERICAL ZIRCONIA INCLUSION IN AN INFINITE ELASTIC MEDIUM CAUSED BY IMPACT COOLING

When an isotropic infinite elastic medium with a transversely isotropic spherical zirconia inclusion is suddenly subjected to a uniform temperature fall, stresswavesoccur at the interface of the spherical inclusion the moment thermal impact is applied. The stress wave in a transversely isotropic spherical inclusion proceeds radially inward to the center of the inclusion. The wave may accumulate at the center and cause very large stress magnitudes, even though the initial thermal stress should be relatively small. This phenomenon is called the stress-focusing effect. The stress wave in an inclusion may be affected by phase transformation during the tempering process. The stress-focusing effects are analyzed in an exact manner, and a clear indication of the mechanism of thermal and phase transformational stress-focusing effects in a transversely isotropic spherical Zirconia inclusion in the isotropic infinite elastic domain is given.

820 異方性球状介在物中に生ずる瞬間相変態ひずみに伴う応力焦点化現象

820 異方性球状介在物中に生ずる瞬間相変態ひずみに伴う応力焦点化現象

THERMAL STRESS-FOCUSING EFFECT IN A TRANSVERSELY ISOTROPIC SPHERICAL INCLUSION EMBEDDED IN AN ISOTROPIC INFINITE ELASTIC MEDIUM

When an isotropic infinite elastic medium with a transversely isotropic spherical inclusion is suddenly subjected to a uniform temperature rise, stress waves occur at the interface of spherical inclusion at the moment thermal impact is applied. The stress wave in a transversely isotropic inclusion proceeds radially inward to the center of the inclusion. The wave may accumulate at the center and cause very large stress magnitudes even though the initial thermal stress should be relatively small. This phenomenon is called the stress-focusing effect. The stress wave in an infinite elastic medium proceeds radially to infinity. This paper analyzes, in an exact manner, the effects of these waves using the ray integrals. The results give a clear indication of the mechanism of the thermal stress-focusing effect in a transversely isotropic inclusion embedded in the isotropic infinite elastic medium.

Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites

This work is devoted first to the derivation of the temperature field in an infinite medium constituted of an n-layered isotropic spherical inclusion, embedded in a matrix subjected to a uniform temperature gradient at infinity, under assumptions of no coupling between mechanical and thermal effects and of steady state conditions. These results lead to an estimate of the effective thermal conductivity coefficient of an n-layered inclusion-reinforced material. The second objective of this work is the derivation of the thermoelastic stress and strain fields in the same basic configuration but now, a coupling between stress and thermal properties is considered and the infinite matrix is stress free and subjected to a uniform change of temperature. These results and the solution of the same problem with a hydrostatic loading are used to estimate the effective thermal expansion coefficient and the specific heats for heterogeneous inclusion-reinforced materials.

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

The effective properties of piezoelectric composite materials are very important in engineering. In this paper, the closed-form solutions of the constraint-strain and the constraint-electric-field of a transversely isotropic spherical inclusion in an infinite non-piezoelectric matrix are obtained. The dilute solutions of piezoelectric composite materials with transversely isotropic spherical inclusions are also given. The solutions in the paper can be readily utilized in analysis and design of piezoelectric composite materials or smart materials and smart structures.

A variational principle of the electromagnetic wave diffraction problem and its applications

This Note is devoted to a new variational formulation of the diffraction problem of plane monochromatic electromagnetic waves on an isolated anisotropic dielectric inclusion in a homogeneous isotropic dielectric medium. The proposed variational functional has a clear physical meaning: it is proportional to the forward amplitude of the scattering field and its imaginary part coincides with the scattering cross-section of the given inclusion. An approximate solution of the diffraction problem for spherical anisotropic inclusions is obtained with the help of this variational formulation. It is assumed that the wave field inside the inclusion is a plane wave with unknown amplitude and wave number. The latter are found from the stationarity of the variational functional of the problem. The comparison of the exact and approximate solutions in the case of an isotropic spherical inclusion is presented. The possible area of application of this approximation is self-consistent schemes of the solution of the many-particle problems for composite media.

Statistical approach to light scattering from deformed textured heterogeneous polymer materials

AbstractA statistical theory of light scattering from deformed isotropic and textured heterogeneous polymer materials is formulated. Two types of textured structures are analyzed: assemblies of optically isotropic and anisotropic rods and a spatially anisotropic distribution of isotropic spherical inclusion centers. The small‐angle Hv light‐scattering patterns are calculated. The appearance of scattering from isotropic rods and spheres in deformed materials has been demonstrated. The changes of the Hv scattering patterns as a function of elongation and strucuture parameters are discussed. © 1993 John Wiley & Sons, Inc.

Formula omitted] inclusion-based micromechanical modelling

We derive the elastic strain and stress fields in an infinite medium constituted of an n- layered isotropic spherical inclusion, embedded in a matrix subjected to uniform stress or strain conditions at infinity. Thanks to the separation of the isotropic and the deviatoric parts, the average strain and stress are calculated in each phase and in the n- layered inclusion, and are used for the expression of a strain condition in order to extend the so-called three-phase model to a (n + l)- phase model.

Alternative Formulation of Elastic Fields due to Inhomogeneous Isotropic Spherical Inclusions Undergoing Shape-Conserving Volume Change

In this Note alternative formulae are derived for the elastic fields due to homogeneous initial strains in an isotropic spherical inclusion embedded in an isotropic infinite medium, assuming a shape conserving volume change of the inclusion. The bulk modulus of the inclusion and the shear modulus of the matrix are the only physically relevant elastic constants necessary to describe analytically displacements, strains, and stresses in the inclusion and the matrix.

Thermal Conductivities of Composite Materials

A field solution for long wavelengths of the equation of heat conduction is obtained for a composite material with orthorhombic symmetry. This yields exact expressions for the three principal thermal conductivities. They are valid for any number of ortho rhombic constituents and even for continuously varying thermal conductivities within the elementary cell. For a lamellar composite, the average thermal conductivities of the material are expressed directly as integrals over the space-dependent thermal conductivi ties. For a filamentary composite, such explicit expressions are obtained only in the Wigner-Seitz approximation and only for a stepwise variation of the thermal conductivities within the ele mentary cell. In the same manner, the average thermal conduc tivity of a composite with cubic symmetry, e.g. isotropic spherical inclusions in a cubic lattice, is calculated. Analogies with previous results on diffusion coefficients and elastic shear moduli are discussed.