Properties for Design of Composite Structures. Neil McCartney

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Название Properties for Design of Composite Structures
Автор произведения Neil McCartney
Жанр Техническая литература
Серия
Издательство Техническая литература
Год выпуска 0
isbn 9781118789780
Properties for Design of Composite Structures - Neil McCartney
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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

      Relations (4.103) and (4.105) assert that

      The addition and subtraction of (4.106) and (4.107) leads to the results

      sigma Subscript theta theta Baseline equals 2 mu Subscript t Baseline left-parenthesis upper A zero width space zero width space plus StartFraction 6 k Subscript upper T Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction upper C r squared minus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta period(4.109)

      From (4.94) and (4.98), it follows that

      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))

      StartFraction partial-differential sigma Subscript r r Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential theta EndFraction plus StartFraction sigma Subscript r r Baseline minus sigma Subscript theta theta Baseline Over r EndFraction equals 0 comma(4.111)

      StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta theta Baseline Over partial-differential theta EndFraction plus StartFraction 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)

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