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

isbn:

СКАЧАТЬ phi overbar Over 2 mu Subscript m Baseline r EndFraction comma"/>(4.59)

      where the parameter ϕ¯ is defined by

      The coefficients of the 1/r terms in relations (4.59) and (4.58) must be identical so that

      phi overbar identical-to sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper V Subscript f Superscript i Baseline phi Subscript i Baseline period(4.61)

      It then follows from (4.56) and (4.60) that for any values of the parameters ε, σT and ΔT

      StartLayout 1st Row StartStartFraction left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript eff Baseline right-parenthesis epsilon plus left-parenthesis alpha Subscript upper T Superscript eff Baseline plus nu Subscript upper A Superscript eff Baseline alpha Subscript upper A Superscript eff Baseline right-parenthesis upper Delta upper T minus alpha Subscript m Baseline left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis upper Delta upper T plus one-half left-parenthesis StartFraction 1 Over k Subscript upper T Superscript eff Baseline EndFraction minus StartFraction 1 Over k Subscript upper T Superscript m Baseline EndFraction right-parenthesis sigma Subscript upper T Baseline OverOver StartFraction 1 Over k Subscript upper T Superscript eff Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction equals 2nd Row sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper V Subscript f Superscript i Baseline StartStartFraction left-parenthesis nu Subscript m Baseline minus nu Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline right-parenthesis epsilon plus left-parenthesis alpha Subscript upper T Superscript f left-parenthesis i right-parenthesis Baseline plus nu Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline alpha Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline right-parenthesis upper Delta upper T minus alpha Subscript m Baseline left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis upper Delta upper T plus one-half left-parenthesis StartFraction 1 Over k Subscript upper T Superscript left-parenthesis i right-parenthesis Baseline EndFraction minus StartFraction 1 Over k Subscript upper T Superscript m Baseline EndFraction right-parenthesis sigma Subscript upper T Baseline OverOver StartFraction 1 Over k Subscript upper T Superscript left-parenthesis i right-parenthesis Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction period EndLayout(4.62)

      Clearly the following relations must be satisfied

      sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper V Subscript f Superscript i Baseline StartStartFraction nu Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline minus nu Subscript m Baseline OverOver StartFraction 1 Over k Subscript upper T Superscript f left-parenthesis i right-parenthesis Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction equals StartStartFraction nu Subscript upper A Superscript eff Baseline minus nu Subscript m Baseline OverOver StartFraction 1 Over k Subscript upper T Superscript eff Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction comma(4.64)

      sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper V Subscript f Superscript i Baseline StartStartFraction left-parenthesis alpha Subscript upper T Superscript f left-parenthesis i right-parenthesis Baseline plus nu Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline alpha Subscript upper A Superscript f left-parenthesis i right-parenthesis Baseline right-parenthesis minus alpha Subscript m Baseline left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis OverOver StartFraction 1 Over k Subscript upper T Superscript f left-parenthesis i right-parenthesis Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction equals StartStartFraction left-parenthesis alpha Subscript upper T Superscript eff Baseline plus nu Subscript upper A Superscript eff Baseline alpha Subscript upper A Superscript eff Baseline right-parenthesis minus alpha Subscript m Baseline left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis OverOver StartFraction 1 Over k Subscript upper T Superscript eff Baseline EndFraction plus StartFraction 1 Over mu Subscript m Baseline EndFraction EndEndFraction period(4.65)

      On using (4.1), these relations are now expressed as