Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 77

The displacement and stress fields within the matrix must be bounded as r→∞ and it follows from (4.93), (4.108)–(4.110) and (4.113) that

(4.118)

(4.119)

(4.120)

(4.121)

The unknown coefficients Af,Cf,Bm,Dm are found by imposing continuity conditions at the interface between the isolated fibre and the matrix. The continuity conditions are given by

(4.122)

(4.123)

(4.124)

(4.125)

On imposing the continuity conditions (4.122)–(4.125), it follows that

(4.126)

and that

(4.127)

From (4.126), it follows on addition that

(4.128)

and on subtraction that

(4.129)

The substitution of (4.129) into (4.128) then leads to

(4.130)

From (4.127) it follows on addition that

(4.131)

and on subtraction that

(4.132)

The substitution of (4.132) into (4.131) then leads to

(4.133)

It now follows from (4.129) and (4.132) that

(4.134)

and from (4.130), (4.133) and (4.134) that

(4.135)

(4.136)

As Cf=0, it follows from (4.115)–(4.117) that the stress and strain fields are uniform within the fibre. From (4.133) and (4.134)

(4.137)

On substituting (4.136) into (4.137) to eliminate Dm, it can be shown that

(4.138)

It should be noted that the displacement and stress fields in the fibre and the matrix can now be calculated. It is clear from (4.118)–(4.121) that at large distances from the fibre the perturbations of the displacement and stress fields arising from the presence of the fibre are characterised by the values of the parameters Bm and Dm which are related according to relation (4.134) and depend on fibre properties. It is also clear that the far-field is insensitive to the actual location of the fibre. This means that a cluster of weakly interacting fibres can easily be considered, and this is the basis of Maxwell’s method, which is now described.