It follows directly from (4.14)–(4.17), (4.73) and (4.74) that
(4.75)
In addition, it follows from (4.17), (4.73) and (4.74) that
(4.76)
It is easily shown that the stress field automatically satisfies the equilibrium equations (4.20)–(4.22) for any values of the parameters A, α and β.
The following continuity conditions must be satisfied at the interface r = a between fibre and matrix
(4.77)
(4.78)
It then follows from (4.76) that
(4.79)
and from (4.72) that
(4.80)
The substitution of (4.80) into (4.79) then leads to
(4.81)
4.4.2 Solution in the Absence of Fibre
For loading conditions characterised by the shear stress τ applied to an infinite sample of matrix in the absence of fibre (filling the entire region of space), the solution is given by
(4.82)
(4.83)
A comparison of (4.72) and (4.82) with (4.77) and (4.83) indicates that the identification α=τ/2 can be made. It then follows from (4.81) that
(4.84)
Substitution into (4.72) leads to the following expression for the displacement component uz: