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k Subscript upper T Baseline Over k Subscript upper T Baseline plus mu Subscript t Baseline EndFraction StartFraction upper B Over r squared EndFraction minus StartFraction 2 k Subscript upper T Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction 3 upper C r squared plus StartFraction 6 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta period"/>(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.

      4.5.2 Stress Field in the Absence of Fibre

      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)

      4.5.3 Displacement and Stress Fields in Fibre

      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)

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