Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.     

Bounds for the effective conductivity and elastic moduli of fully-disordered multicomponent materials

Communicated by R. Kohn     

Bounds for effective elastic moduli of disordered materials

Recently P.H. Dederichs and R. Zeller (1973) have developed a formal theory of the bounds of odd order n for the effective elastic moduli of linearly elastic, disordered materials. The bounds are established by use of statistical information given in terms of correlation functions up to order n (= 1, 3, 5,…). This theory is extended to include the bounds of even order n. It is indicated how these bounds can be made optimum under the given statistical information. The results for bounds of even and odd order are obtained in forms which resemble Neumann series, containing multiple integrals up to order (n?1). These integrals can be calculated for certain materials which are classified in terms of a gradual statistical homogeneity, isotropy and disorder. Materials which possess these properties up to the correlation functions of nth order are called overall grade n materials. The optimum bounds for overall grade 2 and grade 3 materials are given explicitly. Optimum bounds for materials which are of grade ∞ in homogeneity and isotropy (i.e. (statistically) perfectly homogeneous and isotropic) and, at the same time, disordered of grade 2 or 3 are also derived. Those for grade 2 in disorder are the Z. Hashin and S. Shtrikman's (1963) bounds. Those for grade 3 are the narrowest, explicit bounds so far derived for random elastic materials. They contain within themselves the so-called self-consistent elastic moduli.     

Bounds and estimates for elastic constants of random polycrystals of laminates

In order to obtain formulas providing estimates for elastic constants of random polycrystals of laminates, some known rigorous bounds of Peselnick, Meister, and Watt are first simplified. Then, some new self-consistent estimates are formulated based on the resulting analytical structure of these bounds. A numerical study is made, assuming first that the internal structure (i.e., the laminated grain structure) is not known, and then that it is known. The purpose of this aspect of the study is to attempt to quantify the differences in the predictions of properties of the same system being modelled when such internal structure of the composite medium and spatial correlation information is and is not available.     

Estimates for the elastic moduli of random trigonal polycrystals and their macroscopic uncertainty

Explicit expressions of the upper and lower estimates on the macroscopic elastic moduli of random trigonal polycrystals are derived and calculated for a number of aggregates, which correct our earlier results given in Pham [Pham, D.C., 2003. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct. 40, 4911–4924]. The estimates are expected to predict the scatter ranges for the elastic moduli of the polycrystalline materials. The concept of effective moduli is reconsidered regarding the macroscopic uncertainty of the moduli of randomly inhomogeneous materials.     

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin-Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.     

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhniki, No. 5, pp. 144–150, September–October, 1973.     

Effective elastic moduli of third-order composite materials

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 118–123, November–December, 1990.     

Connexions between the moduli for anistropic elastic materials

Summary For homogeneous isotropic elastic materials there are simple interrelations connecting Young's modulus, Poisson's ratio, the rigidity modulus and the modulus of compression. However for anisotropic materials the situation is quite different. Young's modulus is a function of direction and Poisson's ratio and the rigidity modulus are functions of pairs of orthogonal directions. Here some simple universal connexions between the moduli for various directions are simply derived for general anisotropic materials. No particular symmetry is assumed in the material.     

On the overall moduli of non-linear elastic composite materials

From the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.     

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.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 123–130, September 2004.     

Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.     

Scaling function, anisotropy and the size of RVE in elastic random polycrystals

This article is focused on the identification of the size of the representative volume element (RVE) in linear elastic randomly structured polycrystals made up of cubic single crystals. The RVE is approached by setting up stochastic Dirichlet and Neumann boundary value problems consistent with the Hill(-Mandel) macrohomogeneity condition. Within this framework we introduce a scaling function that relates the single crystal anisotropy to the scale of observation. We derive certain exact characteristics of the scaling function and postulate others based on detailed calculations on copper, lithium, tantalum, magnesium oxide and antimony-yttrium. In deriving the above, we make use of the fact that cubic crystals and polycrystals have a uniquely determined scale-independent bulk modulus. It turns out that the scaling function is exact in the single crystal anisotropy. A methodology to develop a material selection diagram that clearly separates the microscale from the macroscale is proposed. The proposed scaling function not only bridges the length scales but also unifies the treatment of a wide spectrum of cubic crystals. Although the scope of this article is restricted to aggregates made up of cubic-shaped and cubic-symmetry single crystals, the concept of the scaling function can be generalized to other crystal shapes and classes as well as to scaling of other elastic/inelastic properties.     

The effective elastic moduli of a random packing of spheres

The effective incremental elastic moduli of a random packing of identical elastic spheres are derived. Although the method of derivation is suitable for any initial deformed configuration that induces compressive forces between any spheres in contact, specific results are given only for the cases in which this corresponds to that of a hydrostatic compression or a uniaxial compression. For additional simplicity it is assumed that the spheres are either infinitely rough or perfectly smooth.