Properties for Design of Composite Structures. Neil McCartney

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Название Properties for Design of Composite Structures
Автор произведения Neil McCartney
Жанр Техническая литература
Серия
Издательство Техническая литература
Год выпуска 0
isbn 9781118789780
Properties for Design of Composite Structures - Neil McCartney
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alt="StartFraction partial-differential sigma 12 Over partial-differential x 1 EndFraction plus StartFraction partial-differential sigma 22 Over partial-differential x 2 EndFraction plus StartFraction partial-differential sigma 23 Over partial-differential x 3 EndFraction equals 0 comma"/>(2.121)

      where the stress tensor is symmetric such that

      When using cylindrical polar coordinates (r,θ,z) such that

      x 1 equals r cosine theta comma x 2 equals r sine theta comma x 3 equals z comma(2.124)

      the equilibrium equations (2.120)–(2.122) are written as

      StartFraction partial-differential sigma Subscript r r Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential theta EndFraction plus StartFraction partial-differential sigma Subscript r z Baseline Over partial-differential z EndFraction plus StartFraction sigma Subscript r r Baseline minus sigma Subscript theta theta Baseline Over r EndFraction equals 0 comma(2.125)

      StartFraction partial-differential sigma Subscript r z Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta z Baseline Over partial-differential theta EndFraction plus StartFraction partial-differential sigma Subscript z z Baseline Over partial-differential z EndFraction plus StartFraction sigma Subscript r z Baseline Over r EndFraction equals 0 comma(2.127)

      where the stress tensor is symmetric such that

      sigma Subscript r theta Baseline equals sigma Subscript theta r Baseline comma sigma Subscript r z Baseline equals sigma Subscript z r Baseline comma sigma Subscript theta z Baseline equals sigma Subscript z theta Baseline period(2.128)

      When using spherical polar coordinates (r,θ,ϕ) such that

      x 1 equals r sine theta cosine phi comma x 2 equals r sine theta sine phi comma x 3 equals r cosine theta comma(2.129)

      the equilibrium equations (2.120)–(2.122) are written as

      StartFraction partial-differential sigma Subscript r r Baseline Over partial-differential r EndFraction zero width space zero width space plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential theta EndFraction plus StartFraction 1 Over r s i n theta EndFraction StartFraction partial-differential sigma Subscript r phi Baseline Over partial-differential phi EndFraction plus StartFraction 1 Over r EndFraction left-parenthesis 2 sigma Subscript r r Baseline minus sigma Subscript theta theta Baseline minus sigma Subscript phi phi Baseline plus sigma Subscript r theta Baseline c o t theta right-parenthesis equals 0 comma(2.130)

      StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential r EndFraction zero width space zero width space plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta theta Baseline Over partial-differential theta EndFraction plus StartFraction 1 Over r s i n theta EndFraction StartFraction partial-differential sigma Subscript theta phi Baseline Over partial-differential phi EndFraction plus StartFraction 1 Over r EndFraction left-bracket 3 sigma Subscript r theta Baseline plus left-parenthesis sigma Subscript theta theta Baseline minus sigma Subscript phi phi Baseline right-parenthesis c o t theta right-bracket equals 0 comma(2.131)

      StartFraction partial-differential sigma Subscript r phi Baseline Over partial-differential r EndFraction zero width space zero width space plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta phi Baseline Over partial-differential theta EndFraction plus StartFraction 1 Over r s i n theta EndFraction StartFraction partial-differential sigma Subscript phi phi Baseline Over partial-differential phi EndFraction plus StartFraction 1 Over r EndFraction left-parenthesis 3 sigma Subscript r phi Baseline plus 2 sigma Subscript theta phi Baseline c o t theta right-parenthesis equals 0 comma(2.132)

      where the stress tensor is symmetric such that

      sigma Subscript r theta Baseline equals sigma Subscript theta r Baseline comma sigma Subscript r phi Baseline equals sigma Subscript phi r Baseline comma sigma Subscript theta phi Baseline equals sigma Subscript phi theta Baseline period(2.133)

      2.13 Strain–Displacement Relations

      For the special case when the strain tensor εij is uniform, and on using (2.107), the displacement fields

      StartLayout 1st Row u 1 left-parenthesis x right-parenthesis equals epsilon 11 x 1 plus epsilon 12 x 2 plus epsilon 13 x 3 comma 2nd Row u 2 left-parenthesis x right-parenthesis equals epsilon 12 x 1 plus epsilon 22 x 2 plus epsilon 23 x 3 comma 3rd Row u 3 left-parenthesis x right-parenthesis equals epsilon 13 x 1 plus epsilon 23 x 2 plus zero width space epsilon 33 x 3 comma EndLayout(2.134)

      and

      both lead to the same strain field given by

      StartLayout 1st Row epsilon 11 left-parenthesis x right-parenthesis identical-to StartFraction partial-differential u 1 Over partial-differential x 1 EndFraction equals epsilon 11 comma zero width space zero width space zero width space zero width space epsilon 22 left-parenthesis x right-parenthesis identical-to StartFraction partial-differential u 2 Over partial-differential </div> </div> </div> <hr> <div class=

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