A. N. Basu

Self‐consistent T‐matrix solution for the effective elastic properties of noncubic polycrystals

The multiple scattering theory has been a powerful tool in estimating the effective elastic properties of polycrystalline substances and different types of composite materials in terms of the component elastic properties. Both perturbative and self‐consistent solutions within the framework of multiple scattering theory have been developed for cubic polycrystals by R. Zeller and P. H. Dederichs [Phys. Status. Solidi B 55, 831 (1973)]. Recently we have suggested [T. R. Middya, Mala Paul, and A. N. Basu, J. Appl. Phys. 58, 4095 (1985)] a perturbative method of calculation for all the noncubic polycrystals up to orthorhombic symmetry. Although the method has been quite successful in dealing with noncubic polycrystals with low anisotropy factors, it becomes too complex to handle the high anisotropy cases. Moreover, the results for many crystals for such cases with perturbation carried up to second order are inconsistent as they fall outside the well‐known Hashin–Shtrikman (H–S) bounds. In order to overcome thi...

Self‐consistent T‐matrix solution for the effective elastic properties of perfectly disordered multiphase solids

A self‐consistent solution for the effective elastic properties of polycrystalline and perfectly disordered multiphase composites has been discussed by using the T‐matrix method under certain suitable approximations. Compared to the existing formulas these new relations for the disordered composites are very useful in practical situations for a quick and more accurate estimate of the effective elastic properties, in particular for a case where the composite has components with widely different values of the elastic constants. For comparison we have discussed the results based on Kroner’s theory which also purports to solve the same problem. It is found that the two solutions do not agree. To resolve the difference we take help of Hill’s exact solution of the composite problem when the components have equal rigidities. It is found that while Kroner’s theory is inconsistent with the exact result the present self‐consistent solution analytically reproduces it. Another interesting finding of the present inves...

Multiple scattering theoretical and computer simulated dynamic model approaches to effective elastic properties of randomly disordered composites

In the present investigation we have generalized the result obtained in the preceding paper [J. Appl. Phys. 59, 2368 (1985)] to treat composites containing crystallites of different crystal symmetries and arbitrary values of the elastic constants. This result has been used to discuss the existing theories intended to solve the same problem. An interesting finding of this analysis is that the present approach provides a single framework to discuss all the previous results which are particular cases of the general formulas derived here. Earlier all these cases were treated separately employing different approaches. Next we apply our results to six different binary composites and we find that our predictions compare quite favorably with experiment. For comparison we also calculate the same quantities by simulating the composite on a computer by the dynamic method originally developed by [T. R. Middya, A. N. Basu, and S. Sengupta [J. Appl. Phys. 57, 1844 (1985)] to discuss monocomponent polycrystal elastic pr...

Static and computer‐simulated dynamic model approaches to effective elastic properties of noncubic polycrystals

Based on the static deformation scheme envisaged by the effective medium theory developed by Zeller and Dederichs [Phys. Status Solidi B 55, 831 (1973)] formally similar to that of the quantum mechanical multiple scattering method, we have deduced a general expression for determining the effective elastic properties of any single component polycrystalline substance. From these general formulas using appropriate symmetry for the component crystallite the expressions for cubic, hexagonal, tetragonal, trigonal, and orthorhombic polycrystals may be easily derived. Two sets of approximate formulae are given and their ranges of validity discussed. For comparison with this static approach, we have also calculated the same quantities by simulating the polycrystal on a computer using the dynamic model developed by Middya, Basu, and Sengupta [J. Appl. Phys. 57, 1844 (1985)]. The results obtained by these two approaches based on entirely different assumptions are remarkably close to each other and to experiment for ...

Elastic properties of a computer‐simulated polycrystalline aggregate of a single component

A simple procedure involving sound velocities in component crystallites has been developed to calculate the effective elastic properties of a hypothetical aggregate which is macroscopically homogeneous but has fluctuations on a microscopic scale, with the underlying assumption that the sound wave can distinguish between the different crystallites oriented at random. For comparison we have also employed the effective medium theory based on a static deformation scheme to calculate the effective elastic properties of the same hypothetical aggregate. Both the procedures use the identical set of input data. Although the assumptions on which the above dynamic and static methods are based are entirely different the results obtained from them for seventeen different aggregates are remarkably close to each other and also to experiments done on polycrystalline specimens using ultrasonic velocities. Finally, a critical discussion of the results and their consequences is presented.

A multiple scattering theoretical approach to the effective thermal conductivity of disordered solids and its dependence on phase geometry

The authors have estimated the effective thermal conductivity of a polycrystal with orthorhombic, hexagonal, trigonal and tetragonal symmetries, in terms of the principal thermal conductivities of an individual crystal, within the framework of multiple scattering theory. In addition the self-consistent values and different bounds for the effective thermal conductivity are computed for various shapes of the crystallites. It is found that the optical potential bounds depend strongly on the shape of the crystallite. It may be mentioned that the results computed for thermal conductivity are equally applicable to the electrical conductivity, magnetic permeability and dielectric constants for such a medium.

Phase-geometry-dependent bounds and self-consistent results for effective thermal conductivity of a multiphase composite using the multiple scattering theoretical approach

Phase-geometry-dependent third-order optimised bounds, optical potential bounds and self-consistent effective thermal conductivity of multiphase composites have been computed in the framework of multiple scattering theory. The optical potential bound is the strictest of them and depends considerably on phase geometry. Self-consistent results always lie between the bounds. The results obtained indicate a distinct improvement over other existing calculations. Finally an application of the method to the calculation of thermal conductivity of a technologically important composite has been considered.

Self-consistent T-matrix calculation of the pressure derivative of elastic constants for polycrystals

Abstract The self-consistent T -matrix approach has been quite successfully employed for the evaluation of the effective elastic properties of various physical systems with randomly varying mechanical properties including polycrystals. But so far no attempt has been made to apply the method for the estimation of non-linear mechanical properties of random systems. The purpose of the present work is to develop a T -matrix method of calculation for the evaluation of the pressure derivative of the elastic constants of polycrystalline substances in terms of those of the single crystal data. Finally the results obtained by this approach for eight polycrystalline substances are compared with experiment and other existing calculations.

Self‐consistent T‐matrix solution and computer‐simulated velocity averaging approaches for the effective elastic constants of monoclinic polycrystals

The self‐consistent T‐matrix solution envisaged by the effective‐medium approach [R. Zeller and P. H. Dederichs, Phys. Status Solidi B 55, 831 (1973)] has, in general, led to a considerable clarification of our understanding of the mechanical properties of a variety of disorder systems including polycrystals. Specifically, the relevant formulations have been developed for cubic polycrystals by Zeller and Dederichs, and for hexagonal, tetragonal, trigonal, and orthorhombic polycrystals by T. R. Middya and A. N. Basu [J. Appl. Phys. 59, 2368 (1986)]. The present work on monoclinic polycrystals is a sequel to our previous work. We have developed the complete set of equations within the framework of the effective medium theory which delivers in a self‐consistent manner the effective elastic constants of a monoclinic polycrystal in terms of those of the single crystal data. For comparison we have also evaluated the same quantities for each polycrystal by the computer simulation employing the velocity averaging...

Multiple‐scattering theoretic approach to the thermal expansion of inhomogeneous materials

Expression for the effective thermal expansion coefficient (TEC) of statistically homogeneous and isotropic, inhomogeneous material has been derived within the framework of the multiple‐scattering theory. Then, from the general expression, effective TEC is obtained in the single‐grain scattering approximation, for this type of materials, consisting of piecewise homogeneous phases (grains). For simplicity, only spherical grains are considered. Previous exact results for two‐phase composites and polycrystals with crystallites having a preferred axis (e.g., tetragonal, trigonal, and hexagonal) are shown to follow from this approach in the single‐grain scattering approximation. Apart from this, it is shown that the effective TEC for general multiphase composites and polycrystals can be obtained in a self‐consistent way. Finally, the self‐consistent solutions thus obtained have been employed to calculate the effective TEC of polycrystals belonging to different symmetry classes, and the results are found to com...