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
(4.95)
(4.96)
(4.97)
(4.98)
where ET=2μt(1+νt). It follows directly from (4.94) and (4.98) that
