Effective thermal conductivity of partially saturated porous rocks

The present work is concerned with the determination of the effective thermal conductivity of porous rocks or rock-like composites composed by multiple solid constituents, in partially saturated conditions. Based on microstructure observations, a two-step homogenization scheme is developed: the first step for the solid constituents only, and the second step for the (already homogenized) solid matrix and pores. Several homogenization schemes (dilute, Mori–Tanaka, the effective field method and Ponte Castañeda–Willis technique) are presented and compared in this context. Such methods are allowing: (i) to incorporate in the modellization the physical parameters (mineralogy, morphology) influencing the effective properties of the considered material, and the saturation degree of the porous phase; (ii) to account for interaction effects between matrix and inhomogeneities; (iii) to consider different spatial distributions of inclusions (spherical, ellipsoïdal). An orientation distribution function (ODF) permits simultaneously to incorporate in the modelling the transverse isotropy of pore systems. Appearing as homogeneous at the macroscopic scale, it is showed that the effective conductivity depends on the physical properties of all subsidiary phases (microscopic inhomogeneities). By considering the solution of a single ellipsoïdal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the geometry, shape and spatial distribution of inhomogeneities on the effective thermal conductivity and its dependence with the saturation degree of liquid phase. The predictive capacities of the two-step homogenization method are evaluated by comparison with experimental results obtained for an argillite.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoïdal inhomogeneities

The objective of this work is to investigate the thermal conduction phenomena in transversely isotropic geomaterials or rock-like composites with arbitrary oriented ellipsoïdal inhomogeneities of low aspect ratio. Based on the evaluation of the Green function, we provide here new expressions for the interaction tensor whose knowledge permits to obtain the concentration tensor of the polarization field used itself to evaluate the effective thermal conductivity tensor by homogenization. Some particular cases of the obtained general solution are equally presented, in order to validate the developed formalism. The obtained results are next used to study the effect of matrix anisotropy, pores systems and microstructure-related parameters on the overall effective thermal conductivity in transversely isotropic rocks. A two-step homogenization scheme is developed for the prediction of the initial anisotropy effects and to test the ability of the proposed model in the evaluation of effective thermal conductivity. With the help of an Orientation Distribution Function (ODF) the anisotropy due to the pore systems is also accounted. Numerical applications and comparisons with available experimental data are finally carried out for a partially saturated Opalinus clay and an argillite which are both composed of an argillaceous matrix and multiple solid minerals constituents.     

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.     

Contact analysis in the presence of an ellipsoidal inhomogeneity within a half space

Many materials contain inhomogeneities or inclusions that may greatly affect their mechanical properties. Such inhomogeneities are for example encountered in the case of composite materials or materials containing precipitates. This paper presents an analysis of contact pressure and subsurface stress field for contact problems in the presence of anisotropic elastic inhomogeneities of ellipsoidal shape. Accounting for any orientation and material properties of the inhomogeneities are the major novelties of this work. The semi-analytical method proposed to solve the contact problem is based on Eshelby’s formalism and uses 2D and 3D Fast Fourier Transforms to speed up the computation. The time and memory necessary are greatly reduced in comparison with the classical finite element method. The model can be seen as an enrichment technique where the enrichment fields from the heterogeneous solution are superimposed to the homogeneous problem. The definition of complex geometries made by combination of inclusions can easily be achieved. A parametric analysis on the effect of elastic properties and geometrical features of the inhomogeneity (size, depth and orientation) is proposed. The model allows to obtain the contact pressure distribution – disturbed by the presence of inhomogeneities – as well as subsurface and matrix/inhomogeneity interface stresses. It is shown that the presence of an inclusion below the contact surface affects significantly the contact pressure and subsurfaces stress distributions when located at a depth lower than 0.7 times the contact radius. The anisotropy directions and material data are also key elements that strongly affect the elastic contact solution. In the case of normal contact between a spherical indenter and an elastic half space containing a single inhomogeneity whose center is located straight below the contact center, the normal stress at the inhomogeneity/matrix interface is mostly compressive. Finally when the axes of the ellipsoidal inclusion do not coincide with the contact problem axes, the pressure distribution is not symmetrical.     

Material multipole mechanics of elastic dielectric composites

The overall mechanical and electrical behaviors of elastic dielectric composites are investigated with the aid of the concept of material multipoles. In particular, by introducing a statistical continuum material multipole theory, the effects of the electric-elastic interaction and the microstructure (size, shape, orientation,...) of inhomogeneous particles on the overall behaviors of the composites can be obtained. A basic solution for an ellipsoidal elastic inhomogeneity with electric polarization in an infinite elastic dielectric medium is first given, which shows that classical Eshelby ’s elastic solution is modified by the presence of electric-elastic interaction. The overall macroscopic constitutive relations and their overall macroscopic material parameters accounting for electroelastic interaction effect are then derived for the elastic dielectric composites. Some quantitative calculations on the problems with statistical anisotropy, the shape effect and the electric-elastic interaction are finally given for dilute composites.     

The averaged mechanical properties of prismatic lattice formed by ellipsoidal inclusions

Summary The objective of this paper is to evaluate the averaged elastic properties of 3-D grained composites in which identical inclusions form a prismatic network interacting with the matrix material. The inclusions are of ellipsoidal shape with transverse circular sections located at the nodes of a doubly-periodic lattice with an orthogonal elementary cell. When the arrays of inclusions are set at equal spacings in normal directions through the thickness of the matrix, the material formed is an anisotropic composite with tetragonal symmetry at planes transverse to the fiber axis. The longitudinal and transverse elastic and shear moduli as well as the longitudinal Poisson's ratios of such composites are evaluated in this paper. The averaged properties are studied in terms of the aspect ratio and volume fraction of the inclusions as well as the relative rigidity of the constituent phases. Employing the Eshelby's theory for the stress field around a single ellipsoidal inhomogeneity, which is surrounded by the effective anisotropic material, and considering the Mori-Tanaka's concept for the mutual interaction of the neighboring inclusions, we may evaluate the averaged elastic properties of grained composites with aligned ellipsoidal inclusions at finite concentrations. The results provided in a closed-form solution concern the stiffness of 3-D grained composites with parallely dispersed ellipsoidal inclusions forming a prismatic network inside the principal material. It is shown that the stiffness is affected by both the geometry of the inclusions and their concentration. The use of different composite models in the analysis shows that intense variations of stiffness occur mainly in hard composites weakened by soft ellipsoidal inclusions. These findings come in full verification with experimental or theoretical results from the literature. Received 10 February 1998; accepted for publication 27 November 1998     

The effective medium and the average field approximations vis-à-vis the Hashin-Shtrikman bounds. II. The generalized self-consistent scheme in matrix-based composites

The present Part II of this two-part study is concerned with the average field approximation (AFA), and the effective medium approximation (EMA) in two-phase matrix-based dielectric composites through the use of an auxiliary configuration in which a particle of the inclusion phase is first surrounded by some matrix material, and then embedded in the effective medium. Those models will be referred as the generalized self-consistent scheme-average field approximation (GSCS-AFA), and the generalized self-consistent scheme-effective medium approximation (GSCS-EMA). We show that there are four types of the GSCS-AFA and a single type of the GSCS-EMA. In this paper the application of those models to dielectric composites with isotropic constituents and an inclusion phase that consists of randomly oriented ellipsoidal particles will be studied. The analytical solution of the auxiliary problem, which consists of an ellipsoidal particle confocally surrounded by a matrix shell and embedded in the effective medium, is achieved by means of ellipsoidal harmonics. Our results show that the effective property predictions of the GSCS-EMA and GSCS-AFA for the considered systems differ from each other, and more importantly, out of the four GSCS-AFA models, three of them violate the Hashin-Shtrikman bounds. The predictions of the GSCS-EMA obey the bounds. It is then shown that the version of the GSCS-AFA which obeys the Hashin-Shtrikman bounds for an inclusion phase with randomly oriented ellipsoids will violate them in the case of a particle shape which is not simply connected. Moreover, it turns out that the SCS-AFA studied in Part I also violates the Hashin-Shtrikman bounds in that case; the EMA, as expected, owing to its realizability property, continues to obey the bounds. Among the AFA and EMA in matrix-based composites, the GSCS-EMA therefore stands out as the method to be recommended.     

Differential schemes for the elastic characterisation of dispersions of randomly oriented ellipsoids

The paper deals with the elastic characterisation of dispersions of randomly oriented ellipsoids: we start from the theory of strongly diluted mixtures and successively we generalise it with a differential scheme. The micro-mechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent elastic moduli of the overall composite materials. This micromechanical technique has been explicitely developed for describing embeddings of randomly oriented not spherical objects. In particular, this study has been applied to characterise media with different shapes of the inclusions (spheres, cylinders and planar inhomogeneities) and for special media involved in the mixture definition (voids or rigid particles): an accurate analysis of all these cases has been studied yielding a set of relations describing several composite materials of great technological interest. The differential effective medium scheme (developed for generally shaped ellipsoids) extends such results to higher values of the volume fraction of the inhomogeneities embedded in the mixture. For instance, the analytical study of the differential scheme for porous materials (with ellipsoidal zero stiffness voids) reveals a universal behaviour of the effective Poisson ratio for high values of the porosity. This means that Poisson ratio at high porosity assumes characteristic values depending only on the shape of the inclusions and not on the elastic response of the matrix.     

Estimation of crack density due to fragmentation of brittle ellipsoidal inhomogeneities embedded in a ductile matrix

Summary An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading. In the state of completed fragmentation the inhomogeneity is replaced by a void with zero stiffness. Thus, the problem of estimating the energy release reduces to the eigenstrain problem solved by Eshelby. The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. The application to the geological system of garnet inhomogeneities embedded in a quartz matrix is considered.     

Equivalent inhomogeneity method for evaluating the effective elastic properties of unidirectional multi-phase composites with surface/interface effects

A new technique is presented for evaluating the effective properties of linearly elastic, multi-phase unidirectional composites. Various effects on the fiber/matrix interfaces (perfect bond, homogeneously imperfect interfaces, uniform interphase layers) are allowed. The analysis of nano-composite materials based on the Gurtin and Murdoch model of material surface is also included. The basic idea of the approach is to construct a circular inhomogeneity in an infinite plane whose effects on the displacements and stresses at distant points are the same as those of a finite cluster of inhomogeneities (fibers of circular cross-section) arranged in a pattern representative of the composite material in question. The elastic properties of the equivalent inhomogeneity then define the effective elastic properties of the material. The volume ratio of the composite material is found after the size of the equivalent circular inhomogeneity is defined in the course of the solution procedure. This procedure is based on a semi-analytical solution of a problem of an infinite plane containing a cluster of non-overlapping circular inhomogeneities subjected to loading at infinity. The method works equally well for periodic and random composites and – importantly – eliminates the necessity for averaging either stresses or strains. New results for nano-composite materials are presented.     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

Evaluation of the effective elastic moduli of tetragonal fiber-reinforced composites based on Maxwell’s concept of equivalent inhomogeneity

Maxwell’s concept of an equivalent inhomogeneity is employed for evaluating the effective elastic properties of tetragonal, fiber-reinforced, unidirectional composites with isotropic phases. The microstructure induced anisotropic effective elastic properties of the material are obtained by comparing the far-field solutions for the problem of a finite cluster of isotropic, circular cylindrical fibers embedded in an infinite isotropic matrix with that for the problem of a single, tetragonal, circular cylindrical equivalent inhomogeneity embedded in the same isotropic matrix. The former solutions precisely account for the interactions between all fibers in the cluster and for their geometrical arrangement. The solutions to several example problems that involve periodic (square arrays) composites demonstrate that the approach adequately captures microstructure induced anisotropy of the materials and provides reasonably accurate estimates of their effective elastic properties.     

Elastic properties of inhomogeneous transversely isotropic rocks

The problem to determine the effective elastic moduli and velocities of elastic wave propagation in transversely isotropic solid containing aligned spheroidal inhomogeneities (solid grains, vugs and micro-cracks) has been solved using the self-consistent scheme known as effective medium approximation (EMA). Since a solution of so-called one-particle problem is a base for each self-consistent method, we solved this problem as a first step for spheroidal inhomogeneity in a transversely isotropic medium. In contrast to the known solution of this problem by Lin and Mura we obtained the expressions for the strain field inside inclusion in the explicit form (without quadratures). The obtained solution was used then in the symmetric variant of the EMA where each component of the system was considered as spheroid with its own aspect ratio. This approach was applied to simulate the properties of the rocks containing isolated pores and micro-cracks. For connected fluid-filled pores we used the anisotropic variant of the Gassmann theory. The results of the calculations, obtained for the effective elastic moduli, have been compared with the experimental data and theoretical simulations of the other authors. Unlike many other rock mechanics theories, EMA approximation gives correct elastic moduli values even in the nondilute concentration of inhomogeneities. The comparison of the experimental data for oriented crack system with the EMA predictions indicates their good correspondence.     

The analysis of thermoplastic characteristics of special polymer sulfur composite

Specific chemical environments step out in the industry objects. Portland cement composites (concrete and mortar) were impregnated by using the special polymerized sulfur and technical soot as a filler (polymer sulfur composite). Sulfur and technical soot was applied as the industrial waste. Portland cement composites were made of the same aggregate, cement and water. The process of special polymer sulfur composite applied as the industrial waste is a thermal treatment process in the temperature of about 150–155 \(^{\circ }\hbox {C}\). The result of such treatment is special polymer sulfur composite in a liquid state. This paper presents the plastic constants and coefficients of thermal expansion of special polymer sulfur composites, with isotropic porous matrix, reinforced by disoriented ellipsoidal inclusions with orthotropic symmetry of the thermoplastic properties. The investigations are based on the stochastic differential equations of solid mechanics. A model and algorithm for calculating the effective characteristics of special polymer sulfur composites are suggested. The effective thermoplastic characteristics of special polymer sulfur composites, with disoriented ellipsoidal inclusions, are calculated in two stages: First, the properties of materials with oriented inclusions are determined, and then effective constants of a composite with disoriented inclusions are determined on the basis of the Voigt or Rice scheme. A brief summary of new products related to special polymer sulfur composites is given as follows: Impregnation, repair, overlays and precast polymer concrete will be presented. Special polymer sulfur as polymer coating impregnation, which has received little attention in recent years, currently has some very interesting applications.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

Spheroidal inhomogeneity in a transversely isotropic piezoelectric medium

Summary Piezoelectric material containing an inhomogeneity with different electroelastic properties is considered. The coupled electroelastic fields within the inclusion satisfy a system of integral equations solved in a closed form in the case of an ellipsoidal inclusion. The solution is utilized to find the concentration of the electroelastic fields around an inhomogeneity, and to derive the expression for the electric enthalpy of the electroelastic medium with an ellipsoidal inclusion that is relevant for various applications. Explicit closed-form expressions are found for the electroelastic fields within a spheroidal inclusion embedded in the transversely isotropic matrix. Results are specialized for a cylinder, a flat rigid disk and a crack. For a penny-shaped crack, the quantities entering the crack propagation criterion are found explicitly. Received 17 February 2000; accepted for publication 9 May 2000     

Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes

A multiscale modeling approach is utilized to evaluate the contribution of irregularly shaped three-dimensional pores to the overall elastic properties of carbon/carbon composites. The degree of anisotropy of a carbon matrix depends on nanotexture, which is defined by manufacturing conditions. Elastic properties of the matrix are predicted assuming a Fisher distribution of orientations of graphene planes with respect to the pyrolytic carbon deposition direction. X-ray computed microtomography is employed to identify pores in a sample of carbon/carbon composite. The pores have highly irregular shapes so that micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable. Thus, the cavity compliance tensor of an individual pore is found numerically by finite element method, and then used in a micromechanical modeling procedure. Examples of pores in isotropic and transversely isotropic pyrolytic carbon matrices are considered. The accuracy of pore approximation by ellipsoidal shapes is evaluated.     

Stress intensity factor of an elliptical crack as influenced by a spherical inhomogeneity

The interaction between an elliptical crack and a spherical inhomogeneity embedded in a three-dimensional solid subject to uniaxial tension is investigated. Both the inhomogeneity and the solid are isotropic but have different elastic moduli. The Eshelby's equivalent inclusion method is applied together with the principle of superposition. An approximate solution for the stress intensity factor is obtained by an approach that expands the distance between the center of the crack and inhomogeneity in series. The local stress field can be increased or decreased depending on the relative modulus of the spherical inhomogeneity and matrix. If the inhomogeneity modulus is larger than that of the matrix, a reduction in the stress intensity factor prevails. Displayed numerically are results to exhibit the influence of inhomogeneity and its distance to the crack.     

Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix

This paper presents two different analytical methods to investigate the magneto-mechanical coupling effect for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. First, the magnetoelastic solution is expressed in terms of magnetoelastic Green's function that can be decoupled into elastic Green's function and magnetic Green's function. Second, the problem is analyzed by the equivalent inclusion method, and then, the formulation of the inhomogeneity problem can be decoupled into an elastic problem and a magnetic inhomogeneity problem connected by some eigenstrain and eigenmagnetic fields. For the piezomagnetic composites with a non-piezomagnetic matrix, these two solutions are completely equivalent each other though they are obtained by means of two different methods. Moreover, based upon the unified energy method, the effective magnetoelastic moduli of the composites are expressed explicitly in terms of phase properties and volume fractions. Then the dilute and Mori–Tanaka schemes are discussed, respectively. Finally, the calculations are made to predict the effective magnetoelastic moduli and illustrate the performance of each model.