Asymptotical Mechanics of Composites by Igor V. Andrianov Jan Awrejcewicz & Vladyslav V. Danishevskyy

Author:Igor V. Andrianov, Jan Awrejcewicz & Vladyslav V. Danishevskyy
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.46)

we derive

(5.47)

where

A is a typical size of inclusions (see Fig. 5.7), ; is a non-dimensional inclusions’ size, . Parameter is related to fibres’ volume fraction c as follows: .

Then, expression (5.47) is substituted into Eq. (5.44). Evaluating the integrals, we obtain the approximate formulas for effective elastic moduli

(5.48)

(5.49)

For the case of composite with voids , one obtains

(5.50)

Formulae (5.48)–(5.50) are the analogous of Maxwell formula for conductivity.

The derived solutions for and have the same analytical form. Evidently, this is caused by symmetry of the input problem (5.29), (5.30). Taking into account the next terms in series (5.27) will allow specifying the expressions (5.48), (5.49) and, consequently, will involve differences between them. Formulas (5.48), (5.49) are obtained on the assumption of small inclusions (). However, they also correctly represent asymptotic behaviour of effective moduli at .

When inclusions are large, and especially if the properties of the components are essentially different, evaluating effective characteristics may become more complicated. In this case, the solution depends strongly upon interactions of neighbouring fibres. Rapid variations of physical fields concentrate in narrow gaps between inclusions. Quantitative and in many instances qualitative behaviours of effective moduli are governed principally by the specific geometry of the microstructure of particular composite materials.

Here, we derive an asymptotic solution of the cell problem (5.40), (5.41). Our approach is related to the idea of lubrication approximation. We assume a half of non-dimensional distance between two neighbouring fibres as a natural small parameter

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.