Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 68

Owing to the use of the far-field in Maxwell’s method for estimating the properties of fibre composites, it is possible to consider multiple fibre reinforcements. Suppose in a cluster of fibres that there are N different types such that for i = 1, …, N there are ni fibres of radius ai. The properties of the fibres of type i are denoted by a superscript i. The cluster is assumed to be homogeneous regarding the distribution of fibres, and leads to transverse isotropic effective properties.

For the case of multiple phases, relation (4.27)₁ is generalised to the following form

(4.55)

where

(4.56)

The cluster of all types of fibre is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibres of type i within the cylinder of radius b is given by Vfi=niai2/b2. The volume fractions must satisfy relation (4.1)₂ namely

(4.57)

It then follows that (4.27) may be written in the form

(4.58)

When the result (4.55) is applied to a single fibre of radius b having effective properties corresponding to the multiphase cluster of fibres it follows that

(4.59)

where the parameter ϕ¯ is defined by

(4.60)

The coefficients of the 1/r terms in relations (4.59) and (4.58) must be identical so that

(4.61)

It then follows from (4.56) and (4.60) that for any values of the parameters ε, σT and ΔT

(4.62)

Clearly the following relations must be satisfied

(4.63)

(4.64)

(4.65)

On using (4.1), these relations are now expressed as

(4.66)

(4.67)

(4.68)

It should be noted that the quantity αT+νAαA is the transverse thermal expansion coefficient for plane strain conditions such that uz=0. On using (4.1), relation (4.66) may also be expressed as

(4.69)

On using (4.1) and (4.66), relations (4.67) and (4.68) may be written as

(4.70)

(4.71)