### Methods of Mathematical-Physics Applied to Polymer Composites

#### Abstract

The aim of this lecture note is twofold:

1. To provide a detailed analytical derivation of the Kerner’s classical homogenization formulae

because the original paper by E.H. Kerner is particularly concise and difficult to understand.

Moreover, in the past, the geometrical assumption the model uses to justify the mathematical

treatment of the homogenizing process was subject to great criticism. For this reason, the original

paper was critically re-examined, the homogenizing procedure newly stated as weight average

of matrix/particle and matrix/matrix stress distribution and the equations of the model were

re−derived accordingly. This critical effort confirmed that Kerner’s original formulae are correct.

2. To set−up a novel experimental method to predict the elastic properties of MgCl

their derived Ziegler−Natta catalysts which are spherical in shape and have an average diameter

of few tenths of a micron. The novel method is derived by inverting Kerner’s classical model

which describes a particulate filled material consisting of micro-spheres randomly dispersed in a

matrix and perfectly bound to the matrix.

[

1. To provide a detailed analytical derivation of the Kerner’s classical homogenization formulae

because the original paper by E.H. Kerner is particularly concise and difficult to understand.

Moreover, in the past, the geometrical assumption the model uses to justify the mathematical

treatment of the homogenizing process was subject to great criticism. For this reason, the original

paper was critically re-examined, the homogenizing procedure newly stated as weight average

of matrix/particle and matrix/matrix stress distribution and the equations of the model were

re−derived accordingly. This critical effort confirmed that Kerner’s original formulae are correct.

2. To set−up a novel experimental method to predict the elastic properties of MgCl

_{2}particles andtheir derived Ziegler−Natta catalysts which are spherical in shape and have an average diameter

of few tenths of a micron. The novel method is derived by inverting Kerner’s classical model

which describes a particulate filled material consisting of micro-spheres randomly dispersed in a

matrix and perfectly bound to the matrix.

[

**DOI**: 10.1685/SELN08003] About DOI#### Full Text:

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