Explicit elasticity-conductivity connections for composites with anisotropic inhomogeneities

Explicit elasticity-conductivity connections for anisotropic heterogeneous materials were recently derived and verified experimentally by the present authors. The constituents were assumed isotropic, so that the overall anisotropy was due solely due to nonrandom orientations of inhomogeneities. Motivated by the materials science applications that deal with strongly anisotropic inclusions, we derive alternative elasticity-conductivity connections that cover these cases. They hold for a broad class of orientation distributions—up to three families of parallel inhomogeneities forming arbitrary angles with each other, with moderate orientation scatter allowed in each of the families. In the case of the isotropic inhomogeneities, this form has substantially better accuracy than the approximate connections derived earlier. The results are compared with experimental data on fiber-reinforced plastics. The agreement is quite good for the entire set of the effective anisotropic constants, and it is achieved without any fitting parameters.     

Contact of rough surfaces: A simple model for elasticity, conductivity and cross-property connections

Two rough plates with multiple contacts of diverse microgeometries are pressed against one another. Two types of contacts can be distinguished: Hertzian ones and “welded” areas. We find that for circular contacts (1) the two types produce the same effect on the incremental stiffness of the interface and on the effective conductivity across it if their contact areas are the same; (2) for both contact types, the compliance and the conductivity are controlled by the same microstructural parameter, where S_k is the kth contact area; (3) the explicit cross-property connection is established that gives the (incremental) stiffness in terms of the conductivity; it holds for an arbitrary mixture of Hertzian and welded contacts and does not require any knowledge of their microgeometries. Whereas the two types of contacts produce the same effect on the incremental stiffness, the Hertzian contacts cause non-linearities (the incremental stiffness increases with loading). The non-linearity is controlled by the microstructural parameter that is sensitive to contrasts of curvatures of the contacting parts. Model predictions are generally in good agreement with experimental data on conductivities of rough metal surfaces and stiffnesses of rough rock surfaces.     

Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

On the effect of interactions of inhomogeneities on the overall elastic and conductive properties

A simple method of estimating the effect of inhomogeneity interactions on the overall properties (elastic and conductive) is developed. It is formulated in terms of property contribution tensors that give the contribution of an inhomogeneity to the overall properties. The method can be viewed as further development of the approach of Rodin and Hwang (1991) and Rodin (1993) that generalized the method of analysis of crack interactions (Kachanov, 1987) to inhomogeneities. We also extend the method to the conductive properties. Considering the effect of interactions on the property contribution tensors on the example of pores we find that this effect is generally moderate, at most (even when pores touch one another) – in contrast with the effect on local fields. On example of two spheres, we compare the interaction effects on the elastic and the conductive properties, and discuss the impact of interactions on the cross-property connections.     

Effective properties of elastic periodic composite media with fibers

A series solution to obtain the effective properties of some elastic composites media having periodically located heterogeneities is described. The method uses the classical expansion along Neuman series of the solution of the periodic elasticity problem in Fourier space, based on the Green's tensor, and exact expressions of factors depending on the shape of the inclusions. Some properties of convergence of the solution are presented, more specifically concerning the elasticity tensor of the reference medium, showing that the convergence occurs even for empty fibers. The solution is extended for rigid inclusions. A comparison is made with previous exact solutions for a fiber composite made of cylindrical fibers with circular cross-sections and with previous estimates. Different examples are presented for new situations concerning the study of fiber composites: composites with elliptic cross-sections and multi-phase fibrous composites.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Carbonate-salt-based composite materials for medium- and high-temperature thermal energy storage

This paper discusses composite materials based on inorganic salts for medium- and high-temperature thermal energy storage application. The composites consist of a phase change material （PCM）, a ceramic material, and a high thermal conductivity material. The ceramic material forms a microstructural skeleton for encapsulation of the PCM and structural stability of the composites; the high thermal conductivity material enhances the overall thermal conductivity of the composites. Using a eutectic salt of lithium and sodium carbonates as the PCM, magnesium oxide as the ceramic skeleton, and either graphite flakes or carbon nanotubes as the thermal conductivity enhancer, we produced composites with good physical and chemical stability and high thermal conductivity. We found that the wettability of the molten salt on the ceramic and carbon materials significantly affects the microstructure of the composites.     

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.     

Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Explicit expressions of the Barnett-Lothe tensors for anisotropic materials

The three Barnett-Lothe tensorsS, H, andL appear frequently in the real form solutions of two-dimensional anisotropic elasticity problems. Explicit expressions for the components of these tensors are presented for general anisotropic materials. The special cases of monoclinic materials with the plane of material symmetry at x₃=0, x₂=0, and x₁=0 are then deduced. For monoclinic materials with the symmetry plane at x₂=0 or x₁=0, the locations of image singularities for the Green's functions for a half-space have a special geometry.     

Stress concentration and effective stiffness of aligned fiber reinforced composite with anisotropic constituents

The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim, the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. The number of fibers is sufficient to account for the micro structure statistics of composite. A new method based on the multipole expansion technique is developed to obtain the exact series solution for the micro stress field. The method combines the principle of superposition, technique of complex potentials and some new results in the theory of special functions. A proper choice of potentials and new results for their series expansions allow one to reduce the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations. This reduction provides high numerical efficiency of the developed method. Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress fields.     

Dynamics of asymmetrical crack propagation in composite materials

When a crack appears in composite materials, the fibrous system will form bridges, and the crack propagates asymmetrically as a rule. A dynamic model of an asymmetrical crack propagation is considered and investigated by applying the self-similar functions. The formulation involves the development of a Riemann–Hilbert problem. The analytical solution of an asymmetrical propagation crack of composite materials under the action of variable moving loads and unit-step moving loads is obtained.     

A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media

An arbitrarily curved three-dimensional anisotropic thin interphase between two anisotropic solids is considered. The purpose of this study is to model this interphase as a surface between its two neighbouring media by means of appropriately devised interface conditions on it. The analysis is carried out in the setting of unsteady heat conduction and dynamic elasticity, and makes use of the simple idea of a Taylor expansion of the relevant fields in thin regions. It consists of a generalization of a previous study by Bövik [1994. On the modelling of thin interface layers in elastic and acoustic scattering problems. Q. J. Mech. Appl. Math. 47, 17-42] which was confined to the isotropic setting. The remarkable feature of the presently derived anisotropic interface model is that formally it has a more compact form than that of Bövik's isotropic version. This is achieved by a judicious choice of surface differential operators which have been used in the derivation, and makes possible to show that several previously known classical interface models are recovered as special cases of the one obtained in this study, once suitable assumptions are made on the magnitude of the conductivity and elasticity tensors of the interphase.     

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part I: Closed form expression for the effective higher-order constitutive tensor

It is shown that second-order homogenization of a Cauchy-elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient (Mindlin) elastic material. This result is valid for both plane and three-dimensional problems and extends earlier findings by Bigoni and Drugan [Bigoni, D., Drugan, W.J., 2007. Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753] from several points of view: (i) the result holds for anisotropic phases with spherical or circular ellipsoid of inertia; (ii) the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material; (iii) a perfect energy match is found between heterogeneous and equivalent materials (instead of an optimal bound). The constitutive higher-order tensor defining the equivalent Mindlin solid is given in a surprisingly simple formula. Applications, treatment of material symmetries and positive definiteness of the effective higher-order constitutive tensor are deferred to Part II of the present article.     

Procedures for construction of anisotropic elastic-plastic property closures for face-centered cubic polycrystals using first-order bounding relations

Microstructure-sensitive design (MSD) is a novel mathematical framework that facilitates a rigorous consideration of the material microstructure as a continuous design variable in the engineering design enterprise [Adams, B.L., Henrie, A., Henrie, B., Lyon, M., Kalidindi, S.R., Garmestani, H., 2001. Microstructure-sensitive design of a compliant beam. J. Mech. Phys. Solids 49(8), 1639-1663; Adams, B.L., Lyon, M., Henrie, B., 2004. Microstructures by design: linear problems in elastic-plastic design. Int. J. Plasticity 20(8-9), 1577-1602; Kalidindi, S.R., Houskamp, J.R., Lyons, M., Adams, B.L., 2004. Microstructure sensitive design of an orthotropic plate subjected to tensile load. Int. J. Plasticity 20(8-9), 1561-1575]. MSD employs spectral representations of the local state distribution functions in describing the microstructure quantitatively, and these in turn enable development of invertible linkages between microstructure and effective properties using established homogenization (composite) theories. As a natural extension of the recent publications in MSD, we provide in this paper a detailed account of the methods that can be readily used by mechanical designers to construct first-order elastic-plastic property closures. The main focus in this paper is on the crystallographic texture (also called Orientation Distribution Function or ODF) as the main microstructural parameter controlling the elastic and yield properties of cubic (fcc and bcc) polycrystalline metals. The following specific advances are described in this paper: (i) derivation of rigorous first-order bounds for the off-diagonal terms of the effective elastic stiffness tensor and their incorporation in the MSD framework, (ii) delineation of the union of the property closures corresponding to both the upper and lower bound theories resulting in comprehensive first-order closures, (iii) development of generalized and readily usable expressions for effective anisotropic elastic-plastic properties that could be applied to all cubic polycrystals, and (iv) identification of the locations of readily available or easily processable ODFs (e.g. textures that are produced by rolling, drawing, etc.) on the property closures. It is anticipated that the advances communicated in this paper will make the mathematical framework of MSD highly accessible to the mechanical designers.     

A two-scale method for identifying mechanical parameters of composite materials with periodic configuration

In this paper, a two-scale method (TSM) is presented for identifying the mechanics parameters such as stiffness and strength of composite materials with small periodic configuration. Firstly, a formulation is briefly given for two-scale analysis (TSA) of the composite materials. And then a two-scale computation formulation of strains and stresses is developed by displacement solution with orthotropic material coefficients for three kinds of such composites structures, i.e., the tension column with a square cross section, the bending cantilever with a rectangular cross section and the torsion column with a circle cross section. The strength formulas for the three kinds of structures are derived and the TSM procedure is discussed. Finally the numerical results of stiffness and strength are presented and compared with experimental data. It shows that the TSM method in this paper is feasible and valid for predicting both the stiffness and the strength of the composite materials with periodic configuration.The project supported by the Special Funds for Major State Basic Research Project (2005CB321704) and the National Natural Science Foundation of China (10590353 and 90405016). The English text was polished by Yunming Chen.     

Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transforms

A viscoplastic approach using the Fast Fourier Transform (FFT) method for obtaining local mechanical response is utilized to study microstructure-property relationships in composite materials. Specifically, three-dimensional, two-phase digital materials containing isotropically coarsened particles surrounded by a matrix phase, generated through a Kinetic Monte Carlo Potts model for Ostwald ripening, are used as instantiations in order to calculate the stress and strain-rate fields under uniaxial tension. The effects of the morphology of the matrix phase, the volume fraction and the contiguity of particles, and the polycrystallinity of matrix phase, on the stress and strain-rate fields under uniaxial tension are examined. It is found that the first moments of the stress and strain-rate fields have a different dependence on the particle volume fraction and the particle contiguity from their second moments. The average stresses and average strain-rates of both phases and of the overall composite have rather simple relationships with the particle volume fraction whereas their standard deviations vary strongly, especially when the particle volume fraction is high, and the contiguity of particles has a noticeable effect on the mechanical response. It is also found that the shape of stress distribution in the BCC hard particle phase evolves as the volume fraction of particles in the composite varies, such that it agrees with the stress field in the BCC polycrystal as the volume of particles approaches unity. Finally, it is observed that the stress and strain-rate fields in the microstructures with a polycrystalline matrix are less sensitive to changes in volume fraction and contiguity of particles.