Properties for Design of Composite Structures. Neil McCartney
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Название: Properties for Design of Composite Structures

Автор: Neil McCartney

Издательство: John Wiley & Sons Limited

Жанр: Техническая литература

Серия:

isbn: 9781118789780

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СКАЧАТЬ 2 sigma Subscript r theta Baseline Over r EndFraction equals 0 comma"/>(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

      4.5.3 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

      StartLayout 1st Row u Subscript r Superscript f Baseline equals minus left-parenthesis upper A Subscript f Baseline r plus upper C Subscript f Baseline r cubed right-parenthesis cosine 2 theta comma u Subscript z Superscript f Baseline equals 0 comma 2nd Row u Subscript theta Superscript f Baseline equals left-parenthesis upper A Subscript f Baseline r plus StartFraction 2 k Subscript upper T Superscript f Baseline plus mu Subscript 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 cubed right-parenthesis sine 2 theta comma EndLayout(4.114)

      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)–(СКАЧАТЬ