Neil McCartney - Properties for Design of Composite Structures

On substituting for σzz using ( 4.100), it then follows that

(4.102)

On subtracting ( 4.95) and ( 4.96),

(4.103)

It follows from ( 4.94), on addition and subtraction, that

(4.104)

(4.105)

Relations ( 4.102) and ( 4.104) then assert that

(4.106)

Relations ( 4.103) and ( 4.105) assert that

(4.107)

The addition and subtraction of ( 4.106) and ( 4.107) leads to the results

(4.108)

(4.109)

From ( 4.94) and ( 4.98), it follows that

(4.110)

It is easily shown that the stress field given by relations ( 4.108)–( 4.110) satisfies automatically the following equilibrium equations for any values of the parameters A , B , C and D (see (2.125)–(2.127))

(4.111)

(4.112)

where use has been made of ( 4.99) and the fact that σzz is independent of z .

It follows from ( 4.93) and ( 4.108)–( 4.110) that, when the matrix occupies the whole of space and is subject to a transverse shear stress τ at infinity, the resulting displacement and stress field is given by

(4.113)

The displacement and stress fields within the fibre must be bounded as r→0 and it follows from ( 4.93) and ( 4.108)–( 4.110) that

(4.114)

(4.115)

(4.116)

(4.117)

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)