Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

Subscript r z Superscript m Baseline left-parenthesis infinity comma z right-parenthesis equals 0 comma 3rd Row sigma Subscript r r Superscript m Baseline left-parenthesis a comma z right-parenthesis equals sigma Subscript r r Superscript f Baseline left-parenthesis a comma z right-parenthesis comma 4th Row sigma Subscript r z Superscript m Baseline left-parenthesis a comma z right-parenthesis equals sigma Subscript r z Superscript f Baseline left-parenthesis a comma z right-parenthesis comma 5th Row u Subscript r Superscript m Baseline left-parenthesis a comma z right-parenthesis equals u Subscript r Superscript f Baseline left-parenthesis a comma z right-parenthesis comma 6th Row u Subscript z Superscript m Baseline left-parenthesis upper R comma z right-parenthesis equals u Subscript z Superscript f Baseline left-parenthesis upper R comma z right-parenthesis period EndLayout right-brace"/>(4.28)

On differentiating the displacement field, it follows, on using (2.139) on setting uθ≡0, that.

(4.29)

epsilon Subscript r z Superscript f Baseline equals epsilon Subscript theta z Superscript f Baseline equals epsilon Subscript r theta Superscript f Baseline identical-to 0 comma (4.30)

(4.31)

epsilon Subscript r z Superscript m Baseline equals epsilon Subscript theta z Superscript m Baseline equals epsilon Subscript r theta Superscript m Baseline identical-to 0 period (4.32)

It follows directly from (4.17), (4.30) and (4.32) that for fibre and matrix

StartLayout 1st Row sigma Subscript r z Superscript f Baseline equals sigma Subscript theta z Superscript f Baseline equals sigma Subscript r theta Superscript f Baseline identical-to 0 comma 2nd Row sigma Subscript r z Superscript m Baseline equals sigma Subscript theta z Superscript m Baseline equals sigma Subscript r theta Superscript m Baseline identical-to 0 period EndLayout right-brace (4.33)

On subtracting (4.14) and (4.15), it follows that for fibre and matrix

epsilon Subscript theta theta Superscript f Baseline minus epsilon Subscript r r Superscript f Baseline equals StartFraction 1 Over 2 mu Subscript t Superscript f Baseline EndFraction left-parenthesis sigma Subscript theta theta Superscript f Baseline minus sigma Subscript r r Superscript f Baseline right-parenthesis comma (4.34)

epsilon Subscript theta theta Superscript m Baseline minus epsilon Subscript r r Superscript m Baseline equals StartFraction 1 Over 2 mu Subscript m Baseline EndFraction left-parenthesis sigma Subscript theta theta Superscript m Baseline minus sigma Subscript r r Superscript m Baseline right-parenthesis period (4.35)

Relations (4.29) and (4.31) then assert that

StartLayout 1st Row sigma Subscript theta theta Superscript f Baseline equals sigma Subscript r r Superscript f Baseline comma 2nd Row sigma Subscript theta theta Superscript f Baseline equals sigma Subscript r r Superscript f Baseline plus StartFraction 2 a squared Over r squared EndFraction phi period EndLayout right-brace (4.36)

On substituting (4.33) and (4.34) into the equilibrium equations (4.20)–(4.22) it follows that

StartFraction partial-differential sigma Subscript r r Superscript f Baseline Over partial-differential r EndFraction equals 0 comma StartFraction partial-differential sigma Subscript z z Superscript f Baseline Over partial-differential z EndFraction equals 0 comma (4.37)

StartFraction partial-differential zero width space sigma Subscript r r Superscript m Baseline Over partial-differential r EndFraction equals StartFraction 2 a squared phi Over r cubed EndFraction comma StartFraction partial-differential sigma Subscript z z Superscript m Baseline Over partial-differential z EndFraction equals 0 period (4.38)

On integrating (4.38)₁ subject to the condition (4.28)₁,

sigma Subscript r r Superscript m Baseline equals sigma Subscript upper T Baseline minus StartFraction a squared Over r squared EndFraction phi comma (4.39)

and from (4.36) it follows that

sigma Subscript theta theta Superscript m Baseline equals sigma Subscript upper T Baseline plus StartFraction a squared Over r squared EndFraction phi period (4.40)

On integrating (4.37)₁ using (4.39) together with the continuity condition (4.28)₃

sigma Subscript r r Superscript f Baseline equals sigma Subscript upper T Baseline minus phi comma (4.41)

and from (4.36) it follows that

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