Journal of Engineering Mechanics

The foundations of classical homogenization techniques, which aim at predicting the overall behavior of heterogeneous materials from that of their constituents, are reviewed. After introductory definitions and a methodological preamble, attention is focused on linear elasticity, for which the basic principles of estimating and bounding the overall properties are introduced and illustrated. In this context, special recourse is made for that to the solution of the inclusion and inhomogeneity problems as reported by Eshelby in 1957. Approaches proposed recently to account in a better way for the structural morphology of the considered materials are briefly mentioned. The case of linear elasticity with eigenstrains is then discussed: several applications, including heterogeneous thermoelasticity, can be investigated within this framework. Finally, outlines of nonlinear micromechanics are briefly reported from a historical point of view: from rate-independent elastoplasticity to nonlinear elasticity and viscoplasticity, examples of a fruitful interaction between the search for new estimates and the derivation of rigorous bounds are given and the crucial question of the description of intraphase heterogeneity is emphasized. Viscoelastic coupling and rate-dependent effects are briefly discussed in conclusion.

Upscaling methods aim at representing the evolution of a given physical process in a given heterogeneous medium by an equivalent macroscopic continuous behavior. In this paper, we recall the main features of the method of homogenization by multiscale asymptotic expansions. To illustrate the method, a few illustrative examples are revisited concerning heat transfer in composite materials (memory effects due to highly different conductivities of the components and the effect of contact thermal resistance in between the components) and fluid flow through rigid porous media (transient flow and flow in noninertial porous media).

The mechanical properties of composite, elastic solids are determined using the method of volume averaging. Volume-averaged forms of Hooke’s law and Cauchy’s equation are developed for each phase. When the condition of local displacement equilibrium is valid, the volume-averaged displacement vectors are essentially equal and the spatially smoothed equations for each phase can be added to obtain a one-equation model of a two-phase system. This one-equation model contains area integrals of the spatial deviation displacement vector, and a closure problem is developed to determine this vector. The closed form for the stress-deformation relation contains effective coefficients to be determined by the solution of the closure problem.

Our recent progress in the analysis and reconstruction of real porous media are surveyed and some emphasis is put on the two-field and on the grain reconstruction methods. Then, after a short summary of our previous analyses of transports, two recent developments are presented, namely, the determination of the resistivity index and of the dispersion tensor in multiphase flow.

Double-porosity materials were introduced as models for oil and gas reservoirs having both storage and transport porosities, and were at first usually treated as static mechanical systems in order to study the flow patterns of fluids during reservoir pump down. Because fluid withdrawal normally increases the effective stress acting on the reservoir, it also turns out to be important to study the geomechanics of the reservoir and how changing fluid pressure affects the solid compaction and fluid permeability of these systems. At the microscale, the mechanical properties of the solid constituents and their distribution in space determine the overall macromechanics of the reservoir system. For systems containing two porosities and two types of solid constituents, exact results for all but one (which may be taken as the overall drained bulk modulus of the system) of the mechanical constants can be derived when the constitutents’ properties are known using methods developed in this paper. For multiporosity systems, closure of the system of equations remains an open question, although it is clear that the system can always be closed by the addition of further macroscale measurements.

This paper considers a saturated porous medium in which the matrix is a cracked solid. Progressive crack closure is responsible for an overall nonlinear poroelastic behavior. The state equations of nonlinear poroelasticity are derived in a differential form within a micromechanical framework. When a hydraulic connection exists between the cracks and the pores of the porous space, the tangent drained stiffness tensor as well as the tangent Biot tensor and modulus are shown to depend on Terzaghi effective stress. Estimates for these coefficients as functions of Terzaghi effective stress are then derived with the tools of homogenization for disordered media. They are based on a crack closure criterion giving the condition for a crack to be closed under a given macroscopic stress state, depending on its aspect ratio. In the case of an isotropic orientation of cracks, it is shown that the influence of cracks on the overall poroelastic properties is governed by the crack density parameter which characterizes the distribution of aspect ratios. Conversely, an experimental methodology for the determination of the distribution of aspect ratios from the measurement of the macroscopic compliance is proposed.

The homogenization method is used to study the properties of the strength criterion of a granular material in the dry, saturated, and unsaturated situations. Adopting a periodic description of the granular material at the microscopic level, the main features of the up-scaling technique are recalled. Then, a general definition of the strength criterion at the macroscopic scale is given in the framework of yield design homogenization theory. This approach makes it possible to find again classical properties of the strength criterion as a function of the properties of the granular material at the microscopic level. Furthermore, some experiments are presented in order to assess the accuracy of the model used to describe the unsaturated granular medium at the microscopic level.

Key developments in the concept of effective stress are surveyed with an emphasis on the ensemble averaging based micromechanical approach reported by Didwania and de Boer in 1999. Various assumptions underlying the effective stress for special cases like rigid/poroelastic solid material saturated by incompressible fluid under homogeneous conditions are elucidated. It is shown that for inhomogeneous saturated porous material the concept of effective stress needs to be generalized by introducing higher gradients of porosity, pore pressure, etc.

Conceived as a potential alternative to the classical design methods employed for analyzing the stability of jointed rock structures, the homogenization approach advocated in this contribution stems from the intuitive idea that, from a macroscopic point of view, a rock mass intersected by a network of joint surfaces may be perceived as a homogeneous medium. The strength criterion of the latter can be theoretically determined from the knowledge of the failure conditions of the individual constituents, namely, rock matrix and joint interfaces. It turns out that, as could be expected, this criterion is of the anisotropic cohesive frictional type, as shown by the closed-form expressions obtained in the case of two mutually orthogonal joint families. While it appears that such a homogenization concept is well suited for densely fractured rock masses, a significant “scale effect” may prevail in the case of structures involving a relatively low number of joints. It is conjectured that a possible way to capture such a scale effect, while still keeping advantage of the homogenization approach, is to adopt a description of the fractured rock mass as a Cosserat or micropolar continuum. This is achieved by proposing a formulation of the macroscopic failure condition in terms of stresses and couple stresses. Such a generalized homogenization method is then applied to a simple illustrative example.

Highly filled composite material systems exhibit, in triaxial compression, a composite strength that is greater than either the weaker particulate or matrix strength. This is due to an amplification of the local confinement in the matrix activating frictional mechanism. The paper quantitatively addresses this increase of the friction coefficient of a matrix reinforced by rigid inclusions using assorted means of nonlinear micromechanics. The approach is based on a nonlinear elastic representation of a Drucker–Prager type frictional strength behavior of the matrix at failure. The key to success of the homogenization procedure relies on the appropriate definition of effective strains in the matrix, to capture local confinement effects and shear effects in the connected matrix phase. It is shown that an effective strain concept based on linear volume averaging (i.e., classical secant method) leads to overestimate the inclusion effects; while an effective strain concept based on quadratic volume averaging (i.e., modified secant method) provides a more realistic representation of shear strains and local confinement effects that develop in triaxial compression in the matrix. Finally, a combination of these two methods leads to a mixed secant method, which gives a relative friction increase of $\sqrt{1+1.5f_I}$ (volume fraction $f_I)\text{.}$ This estimate accurately predicts the experimentally observed frictional behavior of unleached and leached cement-based mortars, composed of a cement paste matrix and rigid sand inclusions.

The phase of void growth is of great importance in the comprehension of ductile fracture of metals. One way to quantify it is to obtain the behavior equation of a porous material, the matrix of which has already known behavior equation. In the present paper, we address the case of a Bingham matrix and we propose a behavior equation for a porous Bingham viscoplastic material. We use the methods introduced by Rice and Tracey in 1969 and Gurson in 1977.

A general three-dimensional micromechanical approach to modeling anisotropic damage of brittle materials such as concrete, rocks, or certain ceramics is presented. Damage is analyzed as a direct consequence of microcracks growth. Following a rigorous scale change methodology, the macroscopic free energy of the microcracked medium is built considering either open and closed microcracks. Moreover, the microcracks opening/closure criterion as well as the moduli recovery conditions (unilateral effects) are addressed in stress-based and strain-based formulations. An alternative derivation of the homogenized properties, based on the well-known Eshelby method, is also presented and extended here to closed cracks. From the micromechanical analysis, an energy-based yield condition is formulated and illustrated in various stress subspaces. Assuming that the normality rule applies, we then present the damage evolution law and the rate form of the constitutive model. The main capabilities and advantages of the micromechanical model are illustrated through various examples in which material microstructure evolutions are presented.

We recently found that mineralized tissues (mineralized tendons and bones), at an observation scale of some microns, are open isotropic hydroxyapatite crystal foams which are reinforced unidirectionally by (organic) collagen molecules. The collagen reinforcement is mechanically activated by crosslinks between collagen assemblies and hydroxyapatite. With this morphology in mind, we develop in this paper a continuum micromechanics model for the ultrastructural stiffness of mineralized tissues. The homogenization is achieved in two steps: At a scale of some hundred nanometers, the isotropic crystal foam is represented as a two-phase polycrystal composed of a hydroxyapatite crystal phase and a nonminerally phase filling the intercrystalline space. At a scale above of some five to ten micrometers, the polycrystal plays the role of a connected matrix, in which a collagen inclusion phase is embedded. The effective stiffness of this phase is determined by the tight links between collagen and hydroxyapatite. The input for the model are the mineral volume fraction and the collagen volume fraction, which are species and tissue-type specific. Then, on the basis of four intrinsic micromechanical stiffness constants, the model is able to predict the full ultrastructural stiffness tensor of mineralized tissues, from low-mineralized turkey leg tendon to highly anisotropic human bones, and high-mineralized isotropic ear bones of whales. This is shown on the basis of a large data set compiled in the Appendix.