Kalman Schulgasser

Relationships between the effective properties of transversely isotropic piezoelectric composites

Abstract We consider two-phase fibre-reinforced piezoelectric composites of arbitrary transverse phase geometry wherein both the constitutents and the composite exhibit transverse isotropy. Such a material is described by a total of 10 overall elastic moduli, piezoelectric constants and permittivities. It is shown that five universal relationships which are independent of geometry at given volume fractions connect six of these effective physical constants. The result is a generalization of the relations found by Hill [J. Mech. Phys. Solids 12, 199 (1964)] for the purely elastic case. An additional relationship between three of the other constants is also known to exist.

Relationship between single‐crystal and polycrystal electrical conductivity

The problem of the prediction of the effective electrical conductivity of a polycrystal from the electrical conductivity of a single crystal is considered. It is shown that the familiar Voigt‐Reuss bounds on the behavior of a polycrystal are the very best generally valid bounds that have thus far been proposed and that the various methods that are claimed to predict exact effective conductivity (or narrow bounds) all include implicit restrictions on the internal geometry of the polycrystal. This is accomplished by constructing a series of statistically homogeneous and isotropic polycrystal models for which the effective conductivity can be exactly calculated. It is hence to be expected that no universal relationship between single‐crystal and polycrystal conductivity exists. Experimental evidence is adduced to support this conclusion. The results are also applicable to the analogous problems of thermal conductivity and electrical permittivity.

Sphere assemblage model for polycrystals and symmetric materials

The problem of the prediction of the effective conductivity of a polycrystal given the conductivity of the single crystal is considered in the light of what can be learned from a constructible polycrystal model for which the effective conductivity can be exactly calculated. It is shown that if the only information known about the internal geometry of the polycrystal is that the aggregate is statistically homogeneous and isotropic it is not possible to narrow appreciably the well‐known ‘‘average conductivity‐average resistivity’’ bounds on the effective conductivity. The model also casts some light on the analogous problem for two phase symmetric materials.

On a phase interchange relationship for composite materials

A theorem exists relating the transverse conductivity of a fiber reinforced material in a determinate manner to the conductivity of the composite with the phase properties interchanged. It is shown that no such theorem can exist in the three‐dimensional case, e.g., for a statistically isotropic composite material. However, an inequality is established relating the two effective conductivities.

On the in-plane elastic constants of paper

Abstract It is shown in this paper that the in-plane shear modulus and Poissons ratio of paper can be predicted with reasonable accuracy if the Youngs moduli of the paper in the machine and cross directions are known. The experimentally observed invariance of in-plane shear modulus with respect to sheet orientation is utilised to determine a relationship between the first two coefficients in a Fourier series expansion of the fibre distribution. Using Coxs classic model for paper this permits the sought-after relationships to be determined.

On the conductivity of fiber reinforced materials

A two‐phase material in which the phase boundaries are cylindrical surfaces is considered. A technique exists for finding upper and lower bounds on the effective thermal conductivity (or electrical conductivity, permittivity, or magnetic permeability) of the composite in the direction perpendicular to the generators of the phase boundaries in terms of two different three point correlation functions. It is shown how a phase interchange theorem can be introduced into these bounds enabling us to express them in terms of a single geometrical constant of phase geometry. We determine what range of values of this factor is realizable for real phase geometries, and we show that the bounds thus obtained span exactly all realizable effective conductivities for such composites. Finally, we show that the bounds as expressed here enable us to use a knowledge of the effective conductivity of a composite for one ratio of constituent conductivities to narrow the bounds for some other ratio.

Theory of poroelastic beams with axial diffusion

The governing equations for a transversely isotropic poroelastic rod subjected to axial or transverse loads are formulated for the case of microgeometries which permit fluid motion in the axial direction only. The quasi-static problem of beam bending is analysed and series solutions are found for normal loadings with various mechanical and diffusion boundary conditions. The unique features of the time-dependent behavior of such beams are pointed out.

Thermal expansion of polycrystalline aggregates with texture

Anexact relationship is found between the thermal expansion coefficients and certain compliances of statistically homogeneous, but not isotropic, polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry. The method generalizes some of the results of Hashin, J. Mech. Phys. Solids,32, 149–157 (1984).

Fibre orientation in machine-made paper

Application of theories of the mechanical behaviour of machine-made paper requires statistical knowledge of the fibre orientation, Efficient experimental techniques are now available for obtaining the necessary data. These data are best represented in the form of a one-parameter distribution function. Several such distributions are offered in the literature to represent angular data. Some of the more common distributions are examined and it is shown that the wrapped Cauchy distribution is best able to predict the observed mechanical properties of a typical machine-made paper sheet.

Concerning the effective transverse conductivity of a two-dimensional two-phase material

Abstract A two-phase material in which the phase boundaries are cylindrical surfaces is considered (fibrereinforced material). We consider such a material for which the precise details of phase geometry are not known, and investigate to what extent knowledge of the effective transverse thermal conductivity for one pair of phase conductivities determines the effective conductivity for a geometrically identical material with different phase conductivities. Also considered is whether or not one can hope to obtain narrow bounds on effective conductivity in terms of three-point correlation functions.