Neil McCartney

Properties for Design of Composite Structures

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f Baseline cosine 2 theta comma"/>(4.115)

      sigma Subscript theta theta Superscript f Baseline equals 2 mu Subscript t Superscript f Baseline left-parenthesis upper A Subscript f Baseline zero width space zero width space plus StartFraction 6 k Subscript upper T Superscript f Baseline Over k Subscript upper T Superscript f Baseline minus mu Subscript t Superscript f Baseline EndFraction upper C Subscript f Baseline r squared right-parenthesis cosine 2 theta comma(4.116)

      4.5.4 Displacement and Stress Fields in Matrix

      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

      sigma Subscript r r Superscript m Baseline equals 2 mu Subscript m Baseline left-parenthesis minus StartFraction tau Over 2 mu Subscript m Baseline EndFraction plus StartFraction 2 k Subscript upper T Superscript m Baseline Over k Subscript upper T Superscript m Baseline plus mu Subscript m Baseline EndFraction StartFraction upper B Subscript m Baseline Over r squared EndFraction plus StartFraction 3 upper D Subscript m Baseline Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta comma(4.119)

      sigma Subscript theta theta Superscript m Baseline equals 2 mu Subscript m Baseline left-parenthesis StartFraction tau Over 2 mu Subscript m Baseline EndFraction zero width space zero width space minus StartFraction 3 upper D Subscript m Baseline Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta comma(4.120)

      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

      u Subscript theta Superscript f Baseline equals u Subscript theta Superscript m Baseline on r equals a comma(4.123)

      sigma Subscript r r Superscript f Baseline equals sigma Subscript r r Superscript m Baseline on r equals a comma(4.124)

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

      and that

      From (4.126), it follows on addition that