T Baseline plus StartFraction a squared Over r squared EndFraction phi period"/>(4.40)
On integrating (4.37)1 using (4.39) together with the continuity condition (4.28)3
and from (4.36) it follows that
The substitution of (4.29)3, (4.41) and (4.42) into (4.16) leads to
It is clear from (4.41) that σzzf is a constant, automatically satisfying (4.37)2. The substitution of (4.31)3, (4.39) and (4.40) into (4.16), applied to the matrix, leads to
thus automatically satisfying (4.38)2. The substitution of (4.29), (4.31), (4.39)–(4.44) into relation (4.14), applied to the fibre and matrix, leads to
where kTf is the plane strain bulk modulus for the transverse isotropic fibre and kTm is the plane strain bulk modulus of the isotropic matrix, defined by (see (2.202))
It should be noted that
where km is the bulk modulus of the matrix defined by (see (2.205))
It now only remains to determine the constant ϕ, which can be specified on applying the remaining condition (4.28)5, because the conditions (4.28)2, (4.28)4 and (4.28)6 are automatically satisfied by (4.25) and (4.33). It follows from (4.26), (4.27), (4.28)5, (4.45) and (4.46) that
