Local effective thermoelastic properties of graded random structure matrix composites

Summary  We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal uncoated or coated inclusions, where the concentration of the inclusions is a function of the coordinates (functionally graded material). Effective properties, such as compliance and thermal expansion coefficient, as well as first statistical moments of stresses in the components are estimated for the general case of inhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green function technique as well as on the generalization of the multiparticle effective field method (MEFM), previously proposed for the research of statistically homogeneous random structure composites. The hypothesis of effective field homogeneity near the inclusions is used; nonlocal effects of overall constitutive relations are not considered. Nonlocal dependences of local effective thermoelastic properties as well as those of conditional averages of the stresses in the components on the concentration of the inclusions are demonstrated. Received 11 November 1999; accepted for publication 4 May 2000     

Effective thermoelastic properties of composites with periodicity in cylindrical coordinates

The aim of this work is to study composites that present cylindrical periodicity in the microstructure. The effective thermomechanical properties of these composites are identified using a modified version of the asymptotic expansion homogenization method, which accounts for unit cells with shell shape. The microscale response is also shown. Several numerical examples demonstrate the use of the proposed approach, which is validated by other micromechanics methods.     

Dispersion and attenuation in thermoelastic multisize particulate composites

This paper deals with the problem of multiple scattering by a random distribution of spherical solid particles in a solid. The material properties of both media are taken as thermoelastic. The radii of the inclusions may be different. The self-consistent method in its variant of the effective medium is used to find the dispersion and attenuation of quasi-elastic, quasi-thermal and shear waves. The single scattering problem required by this technique is solved approximately by means of the Galerkin method applied to an integral equation using the Green function. Numerical results display a characteristic resonance phenomena which appears in the interval where the results are approximately valid, that is, for very long waves down to wavelengths about twice the largest diameter of the spheres. Examples are shown, for composites with two sets of inclusions, which have either a very similar or dissimilar size. Comparisons are made with the elastic counterpart. Among the material properties, the mass density ratio, inclusion to matrix, seems to play an important and simple role. Frequency intervals are distinguished and shown to depend on that ratio, where the attenuation and dispersion of quasi-elastic and P-waves are either very close to each other or not at all. The same applies to shear waves in either composite. The mass density ratio also displays a simple monotonic decreasing behaviour as a function of the frequency at the first attenuation maximum and velocity minimum. These results may be of interest for the nondestructive testing characterization of particulate composites.     

Crack-tip stress fields in functionally graded materials with linearly varying properties

Crack-tip stress fields for a stationary crack along or inclined to the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with Westergaard’s stress function approach. The elastic modulus of the FGM is assumed to vary linearly along the gradation direction. The first six terms for a crack along the direction of property variation and first four terms for a crack inclined to the direction of property variation in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. Using these stress fields, contours of constant maximum shear stress and constant out of plane displacement are generated and the effect of inclination of property gradation direction on these contours is discussed. The strain energy density criterion is applied to obtain critical conditions for crack initiation and the effect of property gradation is discussed. It is shown that the materials with varying properties can offer more resistance to crack propagation and will suppress crack growth in some situations.     

Nonlinear strain properties of particulate composites with a metallic matrix

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 32–37, May, 1991.     

Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions

Thermoelastic behavior of functionally graded particulate materials is investigated with a micromechanical approach. Based on a special representative volume element constructed to represent the graded microstructure of a macroscopic material point, the relation between the averaged strains of the particle and matrix phases is derived with pair-wise particle interactions, and a set of governing equations for the thermoelastic behavior of functionally graded materials is presented. The effective coefficient of thermal expansion at a material point is solved through the overall averaged strain of two phases induced by temperature change under the stress-free condition, and is shown to exhibit a weak anisotropy due to the particle interactions within the graded microstructure. When the material gradient is eliminated, the proposed model predicts the effective coefficient of thermal expansion for uniform composites as expected. If the particle interactions are disregarded, the proposed model recovers the Kerner model. The proposed semi-analytical scheme is consistent and general, and can handle any thermal loading variation. As examples, the thermal stress distributions of graded thermal barrier coatings are solved for two types of thermal loading: uniform temperature change and steady-state heat conduction in the gradation direction.     

Effective elastic properties of matrix composites with transversely-isotropic phases

The present work addresses the problem of calculation of the macroscopic effective elastic properties of composites containing transversely isotropic phases. As a first step, the contribution of a single inhomogeneity to the effective elastic properties is quantified. Relevant stiffness and compliance contribution tensors are derived for spheroidal inhomogeneities. The limiting cases of spherical, penny-shaped and cylindrical shapes are discussed in detail. The property contribution tensors are used to derive the effective elastic moduli of composite materials formed by transversely isotropic phases in two approximations: non-interaction approximation and effective field method. The results are compared with elastic moduli of quasi-random composites.     

Effective elastic moduli of triply periodic particulate matrix composites with imperfect unit cells

In many problems the material may possess a periodic microstructure formed by the spatial repetition of small microstructures, or unit cells. Such a perfectly regular distribution, of course, does not exist in actual cases, although the periodic modeling can be quite useful, since it provides rigorous estimations with a priori prescribed accuracy for various material properties. Triply periodic particulate matrix composites with imperfect unit cells are analyzed in this paper. The multiparticle effective field method (MEFM) is used for the analysis of the perfect and imperfect periodic structure composites. The MEFM is originally based on the homogeneity hypothesis (H1) (see for details [Buryachenko, V.A., 2001. Multiparticle effective field and related methods in micromechanics of composite materials. Appl. Mech. Rev. 54, 1–47]) of effective field acting on the inclusions. In this way the pair interaction of different inclusions is taken directly into account by the use of analytical approximate solution. For perfect periodic structures the hypothesis (H1) is enough for estimation of effective properties. Imperfection of packing necessitates exploring some additional assumption called a closing hypothesis. The next imperfections are analyzed. (A) The probability of location of an inclusion in the center of a unit cell below one (missing inclusion). (B) Some hard inclusions are randomly replaced by the porous (modeling the complete debonding) with some probability. At first, one obtains general explicit integral representations of the effective elastic moduli and strain concentrator factors depending on three numerical solutions: for the perfect periodic structure, for the infinite periodic structure with one imperfection, and for the infinite periodic structure with two arbitrary located imperfections. The method proposed is general; it is not limited by concrete numerical scheme. No restrictions were assumed on both the concrete microstructure and inhomogeneity of stress fields in the inclusions. The inclusions of one kind are assumed to be aligned. The problem (A) is solved at the level of numerical results obtained in the framework of the hypothesis (H1). For the problem (B) the numerical results are obtained if the elastic inclusions (for example hard inclusions) are randomly replaced by another inclusion (for example by the voids modeling the complete debonding). The mentioned problems are solved by three methods. The first one is a Monte Carlo simulation exploring an analytical approximate solution for the binary interacting inclusions obtained in the framework of the hypothesis (H1). The second one is a generalization of the version of the MEFM proposed for the analysis of the perfect periodic particulate composites and based on the choice of a comparison medium coinciding with the matrix. The third method uses a decomposition of the desired solution on the solution for the perfect periodic structure and on the perturbation produced by the imperfections in the perfect periodic structure. All three methods lead to close results in the considered examples; however, the CPU times expended for the solution estimation by Monte Carlo simulation differ by a factor of 1000.     

Generalized thermoelastic functionally graded solid with a periodically varying heat source

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity without energy dissipation (TEWOED). The governing equations of generalized thermoelasticity without energy dissipation (GN model type II) for a functionally graded materials (FGM) (i.e. material with spatially varying material properties)are established. The governing equations are expressed in Laplace–Fourier double transform domain and solved in that domain. Now, the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand is obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking nonhomogeneity parameter suitably. Finally the results obtained are presented graphically to show the effect of nonhomogeneity on displacement, temperature, stress and strain.     

Effective thermoelastic constants of discretely-fibrous composites with anisotropic components

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 10, pp. 21–28, October, 1988.     

Effective moduli of multiphase matrix composites

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 120–125, January–February, 1991.     

Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium, which consists of ahomogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in aperiodic arrayand subjected to inhomogeneous boundary conditions. The hypothesis of effectivefieldhomogeneity near the inclusions is used. The general integral equation obtained reducestheanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusionsinsome representative volume element (RVE) . The integral equation is solved by theFouriertransform method as well as by the iteration method of the Neumann series ( first-orderapproximation) . The nonlocal macroscopic constitutive equation relating the unit cellaverages ofstress and strain is derived in explicit closed forms either of a differential equation ofasecond-order or of an integral equation. The employed of explicit relations fornumericalestimations of tensors describing the local and nonlocal effective elastic properties aswell asaverage stresses in the composites containing simple cubic lattices of rigid inclusions andvoids areconsidered.     

Effective strength parameters of composites in coupled physicomechanical fields

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 124–130, July–August, 1992.     

Effective viscoelastic behavior of particulate polymer composites at finite concentration

Polymeric materials usually present some viscoelastic behavior. To improve the mechanical behavior of these materials, ceramics materials are often filled into the polymeric materials in form of fiber or particle. A micromechanical model was proposed to estimate the overall viscoelastic behavior for particulate polymer composites, especially for high volume concentration of filled particles. The method is based on Laplace transform technique and an elastic model including two-particle interaction. The effective creep compliance and the stress and strain relation at a constant loading rate are analyzed. The results show that the proposed method predicts a significant stiffer response than those based on Mori-Tanaka's method at high volume concentration of particles.     

Dynamic thermomechanical properties of composites based on thermoelastic wave scattering

Summary The dynamic thermomechanical properties of a macroscopically homogeneous and isotropie particle composite are determined by means of an analytical model based on the theory of thermoelastic wave scattering. According to the present method, the energies of the incident and scattered waves are determined in terms of the dynamic stress/strain fields and heat flux/temperature fields, both for an incident shear wave and for an incident one consisting of a linear combination of a pressure and a thermal wave. The basic assumption is that there is no interaction between the respective fields of neighbouring inclusions, i.e. a low volume concentration of the particles and also a wave length of the incident field much greater than the diameter of the smallest sphere which can surround the inclusion. The model is applied to a particle composite containing rigid spherical, thermally insulated inclusions. Results for a wide range of the involved parameters are presented.

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Bestimmung der dynamischen thermodynamischen Eigenschaften von Verbundwerkstoffen aus der thermoelastischen Wellenstreuung

Übersicht Die dynamischen thermomechanischen Eigenschaften makroskopisch homogener und isotroper Partikel-Verbundwerkstoffe werden mit Hilfe eines Modells, das auf der Theorie der Streuung thermoelastischer Wellen beruht, bestimmt. Die Energien der einfallenden und gestreuten Wellen werden ausgedrückt durch die dynamischen Spannungs-Verzerrungsfelder und Wärmefluß-Temperaturfelder, und zwar jeweils für eine einfallende Scherwelle uhd eine Linearkombination aus einfallender thermischer und Druckwelle. Grundlegend ist die Annahme, daß keine Wechselwirkung zwischen den betreffenden Feldern benachbarter Einschlüsse vorhanden ist, d. h. eine geringe Teilchenkonzentration und eine Länge der einfallenden Wellen, die sehr viel größer ist als der Durchmesser der kleinstmöglichen Kugel um den Einschluß. Das Modell wird auf einen Verbund mit starren und wärmeisolierten sphärischen Einschlüssen angewandt und mit Ergebnissen für einen weiteren Parametersatzbereich demonstriert.

     

Effective thermoelastic properties of discrete-fiber reinforced materials with transversally-isotropic components

In the present paper, we will illustrate the application of the method of conditional moments by constructing the algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters of inclusions. A special case of two-component matrix composite with randomly distributed unidirectional spheroidal inclusions is considered. To this end it is assumed that the components of the composite show transversally isotropic symmetry of thermoelastic properties and that the axes of symmetry of the thermoelastic properties of the matrix and inclusions coincide with the coordinate axis x ₃. As a numerical example a composite based on carbon inclusions and epoxide matrix is investigated. The dependencies of Young’s moduli, Poisson’s ratios and shear modulus from the concentration of inclusions and for certain values which characterize the shape of inclusions are analyzed. The results are compared and discussed in context with other theoretical predictions and experimental data.      

Some exact results concerning thermoelastic properties of hollow sphere composites

1.IntroductionAsakindofwidelyusedlowdensitymaterials,compositeconsistingofamatrixanddistributedhollowsphereshaveattractedIncreasingresearchinterest.Relatedworkswhicharefocusedonthetheoreticalpredictionof.theeffectiveelasticmoduliofsuchcompositescanbefoundinLeeandWestmannl'1,QiuandWengl=],HuangandGibsonl31aswellasChercauoietal.HI.HuangandGibson13]havealsocarriedoutanexperimentalinvestigationonmechanicalbehavioursofthecomposites.Thepresentpapercontinuestheseinvestigationsandisfocusedonfindi…     

Effective magnetoelastic properties of laminated composites

Effective physical parameters of a fine-layered medium whose layers exhibit linearized magnetostriction as ferrites are determined. Ferromagnetic materials of cubic system with ferromagnetic resonance are considered __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 36–43, August 2006.     

Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.