Properties for Design of Composite Structures. Neil McCartney

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)

(4.30)

(4.31)

(4.32)

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

(4.33)

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

(4.34)

      (4.35)

      Relations (4.29) and (4.31) then assert that

(4.36)

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

(4.37)

(4.38)

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

(4.39)

      and from (4.36) it follows that

(4.40)

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

(4.41)

      and from (4.36) it follows that

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