Isotropic Constituents Research Articles

On the micromechanics of fibrous piezoelectric composites

Fibrous piezoelectric composites with transversely isotropic constituents are considered. For two-phase systems which are effectively transversely isotropic, the composite cylinder assemblage model is adopted, and exact expressions are derived for nine out of the ten effective constants which characterize the composite aggregate. The moduli are given in terms of closed form simple formulae and have been obtained by invoking recently established theorems in piezoelectric composites. In multiphase systems with transversely isotropic constituents we show that some of the effective moduli can be exactly determined if the constituents possess the same transverse shear modulus. They are again given by closed form simple formulae. Finally, for two-phase fibrous systems, phase-interchange relationships for some of the moduli are derived.

Exact Results Concerning the Local Fields and Effective Properties in Piezoelectric Composites

This paper consists of two parts: (a) a concise summary and discussion is given of the recent contributions of the author in the micromechanics of piezoelectric composites. The underlying theme here is the derivation of exact connections for the local fields and effective moduli of heterogeneous piezoelectric solids. Composites of arbitrary phase geometry as well as fibrous systems are considered. (b) New results are presented on the effective behavior of fibrous piezoelectric systems. Fibrous composites with transversely isotropic constituents and cylindrical microgeometry are considered. The exact connections of the author (Benveniste (1993), Proc. R. Soc., Series A, Vol. 441, pp. 59-81) are extended to include the most generally possible overall symmetry of the composite aggregate. The other category of the new findings concerns exact expressions for the effective thermal terms of fibrous systems which possess the same shear modulus GT.

Polar optical oscillations of layered semiconductor structures in the long-wavelength limit

Polar optical oscillations coupled to unretarted electric fields are discussed for the long-wavelength limit with application to layered semiconductor structures (quantum wells, superlattice, etc.). A Lagrangian formalism is adopted for the deduction of the equations of both mechanical and electrical quantities. The obtained equations bear the form of second-order coupled differential equations for the fundamental quantities, displacement field u and electric potential г. Matching boundry conditions are rigorously derived from the equations and interpreted physically. The particular case of materials belonging to the cubic symmetry is discussed with special application to the double heterostructure. Some comments are also made about the case of isotropic constituent materials. We have thus settled a theory for long- wavelength oscillations taking into account dispersion up to quadratic terms in the wave vector (through the introduction of medium internal stresses) with the aim of avoiding some problems which have been detected and discussed in earlier treatments of this subject.

Elastoplastic constitutive relations for fiber-reinforced solids

In this paper, we make use of a procedure for estimating the effective properties of nonlinear composite materials, proposed recently by Ponte Castañeda (1991, J. Mech. Phys. Solids39, 45–71), to study the effective constitutive behavior of ductile, fiber-reinforced composites. Both estimates and rigorous bounds are obtained for the effective energy functions of multiple-phase, fiber composites with general ductile behaviors (in the context of deformation theory of plasticity) for the isotropic constituent phases. The resulting expressions for the energy functions may be differentiated in a straightforward manner to obtain corresponding estimates for the anisotropic effective stress-strain relations. Explicit calculations are carried out for the case of an aluminum-matrix composite reinforced with boron fibers. The results reveal some interesting features distinguishing the constitutive behavior of ductile-matrix, fiber-reinforced composites from that of linear-elastic, fiber-reinforced composites. One such feature is the strong coupling between the dilatational and distortional modes for the ductile fiber composites. Finally, comparisons are made with available experimental data.

A New Variational Principle and Its Application to Nonlinear Heterogeneous Systems

A new variational principle is proposed for estimating the effective properties of nonlinear heterogeneous systems. The variational principle expresses the effective energy-density function of a given nonlinear heterogeneous dielectric in terms of an infinite-dimensional optimization problem involving the effective energy-density functions of the class of linear comparison materials with variable dielectric cofficients. The variational principle can alternatively be used in an approximate fashion—by optimizing over finite-dimensional subspaces of the original space of comparison dielectric coefficients—to obtain bounds on the effective dielectric properties of nonlinear composite materials with isotropic constituent phases in given proportions. The application of the procedure is illustrated by computing an exact estimate for the effective energy-density function of a nonlinear laminated dielectric material (which has overall anisotropic effective behavior). It is also demonstrated that the Talbot–Willis nonlinear extension of the Hashin–Shtrikman variational principle can be generated from the new variational principle by estimating the effective energy of the linear comparison material (in the new variational principle) directly from the Hashin–Shtrikman variational principle for linear materials. However, the new variational principle is more general in the sense that it can be used in conjunction with other bounds and estimates for the linear comparison material to yield corresponding bounds and estimates for nonlinear composites.

Multiple site self consistent scheme

From the solution of the two heterogeneous inclusions problem, we established a class of multiple site self-consistent schemes for which an elementary heterogeneous cell is defined. This cell is composed of an inclusion and its first neighbours and located in the homogeneous equivalent medium. This set of inclusions represents simultaneously the geometrical anisotropy (shape effect) and the spatial distribution of inclusions. The knowledge of the concentration tensor which links the local mean deformation in the inclusion to the macroscopical one, permits the determination of the overall elastic properties of the composite. These properties depend on the elastic properties of fibre and matrix, the volumic fraction of inclusions and their spatial distribution. The proposed model is developed for isotropic constituents and a cubic spatial repartition of the spherical inclusions. In this particular case, the obtained results are compared with the classical models (Voigt, Reuss, Kröner). We especially discuss the effect of the spatial distribution of inclusions on the anisotropy of the macroscopic elastic behaviour. For an heterogeneity ratio of about 100 and a volumic fraction of spheres equal to 0.5, the anisotropy factor (equal to unity for isotropy) reaches the significant value of 1.35.

A new approach to the application of Mori-Tanaka's theory in composite materials

This paper is a reconsideration and reformulation of the Mori-Tanaka's theory in its application to the computation of the effective properties of composites. Previous applications of the theory in this context continued to be linked with eigenstrain, equivalent inclusion, and back stress concepts, and many times involved energy considerations. In this paper we adopt the ‘direct approach’ of defining and computing effective moduli. By elucidating the nature of the approximation in applying Mori—Tanaka's theory to composites insofar as the ‘concentration-factor’ tensors are concerned, we achieve a straightforward exposition and interpretation of the method which are different than those existing in previous formulations. The analysis is given for two-phase composites with anisotropic elastic constituents and an inclusion phase consisting of aligned or randomly oriented ellipsoidal particles. The derived simple expressions for the predicted stiffness and compliance tensors permit a proof of the self-consistency of the method, a discussion of the predictions' relation to the Hashin-Shtrikman bounds in the case of isotropic constituents and randomly oriented ellipsoidal particles, and finally a derivation of some new results in randomly cracked bodies with penny-shaped cracks.

An Annular Crack in Layered Composites with Transversely Isotropic Constituents

An Annular Crack in Layered Composites with Transversely Isotropic Constituents

On: “Long‐wave elastic anisotropy in transversely isotropic media” by James G. Berryman (GEOPHYSICS, May 1979, p. 896–917).

Berryman shows elegantly that the “inequality [Formula: see text] is true for any horizontally stratified, homogeneous material whose constituent layers are isotropic…” However, the final clause of this sentence “…, i.e., any homogeneous, transversely isotropic material,” is, if taken at face value, misleading. It is clear from the proof in the section “A fundamental inequality” that this statement is only shown to hold for lamellated media with isotropic lamellae, and that Berryman chooses arbitrarily and without any warning the phrase homogeneous, transversely isotropic to stand as a synonym for what Backus (1962) painstakingly describes as “smoothed, transversely isotropic, long‐wave equivalent (STILWE).” In view of the fact that even within the context of exploration seismics transverse isotropy can be due to causes other than horizontal stratification with isotropic constituents (e.g., schists can be intrinsically anisotropic, anisotropy might be due to preferential orientation of sandgrains or joints), I believe this choice to be unfortunate. It leads the unsuspecting reader to assume a wider applicability of the fundamental inequality than Berryman really intends to claim, and thus makes it unnecessarily difficult to understand this significant contribution.

Elastic Constants of Fiber-Reinforced Composites With Transversely Isotropic Constituents

Several models for calculating elastic constants of fiber-reinforced composites with transversely isotropic constituents are discussed. Numerical results are obtained from simple explicit expressions and compared with recently published computer solutions of a hexagonal lattice model.

Thermal and Thermoelastic Properties of Isotropic Composites

The results of theoretical estimates are given for several thermal and thermoelastic constants of macroscopically isotropic composite materials. The composites considered consist of random mixtures of N isotropic constituents, at least ( N — 1) of which are presumed to be distributed in a particulate fashion, with the particles roughly spherical in shape. The results are based on "self-consistent" calcula tions, details of which are given in appendices. Equations (some of which are old, but are repeated for con venience) are given for the following quantities: elastic constants, thermal expansion coefficients, heat capacities (at constant volume and constant pressure), thermal conductivity, and, finally, Gruneisen and related constants.