Properties for Design of Composite Structures. Neil McCartney
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3 upper C r squared minus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta comma 2nd Row epsilon Subscript theta theta Baseline identical-to StartFraction 1 Over r EndFraction StartFraction partial-differential u Subscript theta Baseline Over partial-differential theta EndFraction plus StartFraction u Subscript r Baseline Over r EndFraction equals left-parenthesis upper A minus StartFraction k Subscript upper T Baseline minus mu Subscript t Baseline Over k Subscript upper T Baseline plus mu Subscript t Baseline EndFraction StartFraction upper B Over r squared EndFraction plus StartFraction k Subscript upper T Baseline plus mu Subscript t Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction 3 upper C r squared minus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta comma 3rd Row epsilon Subscript r theta Baseline identical-to one-half left-parenthesis StartFraction 1 Over r EndFraction StartFraction partial-differential u Subscript r Baseline Over partial-differential theta EndFraction plus StartFraction partial-differential u Subscript theta Baseline Over partial-differential r EndFraction minus StartFraction u Subscript theta Baseline Over r EndFraction right-parenthesis equals left-parenthesis upper A plus StartFraction k Subscript upper T Baseline Over k Subscript upper T Baseline plus mu Subscript t Baseline EndFraction StartFraction upper B Over r squared EndFraction plus StartFraction 3 k Subscript upper T Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction upper C r squared plus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis sine 2 theta comma 4th Row epsilon Subscript z z Baseline identical-to StartFraction partial-differential u Subscript z Baseline Over partial-differential z EndFraction equals 0 comma epsilon Subscript r z Baseline identical-to one-half left-parenthesis StartFraction partial-differential u Subscript r Baseline Over partial-differential z EndFraction plus StartFraction partial-differential u Subscript z Baseline Over partial-differential r EndFraction right-parenthesis equals 0 comma epsilon Subscript theta z Baseline identical-to one-half left-parenthesis StartFraction partial-differential u Subscript theta Baseline Over partial-differential z EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential u Subscript z Baseline Over partial-differential theta EndFraction right-parenthesis equals 0 period EndLayout"/>(4.94)
When ΔT=0, the stress-strain relations (4.14)–(4.17) are written as
where ET=2μt(1+νt). It follows directly from (4.94) and (4.98) that
From (4.94) and (4.97) it is clear that
and on summing (4.95) and (4.96)
On substituting for σzz using (4.100), it then follows that
On subtracting (4.95) and (4.96),
It follows from (4.94), on addition and subtraction, that