Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

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)

StartFraction partial-differential sigma 13 Over partial-differential x 1 EndFraction plus StartFraction partial-differential sigma 23 Over partial-differential x 2 EndFraction plus StartFraction partial-differential sigma 33 Over partial-differential x 3 EndFraction equals 0 comma (2.122)

where the stress tensor is symmetric such that

sigma 12 equals sigma 21 comma sigma 13 equals sigma 31 comma sigma 23 equals sigma 32 period (2.123)

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

(2.125)

(2.126)

(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

(2.130)

(2.131)

(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

StartLayout 1st Row u 1 left-parenthesis x right-parenthesis equals epsilon 11 x 1 plus 2 epsilon 12 x 2 plus 2 epsilon 13 x 3 comma 2nd Row u 2 left-parenthesis x right-parenthesis equals epsilon 22 x 2 plus 2 epsilon 23 x 3 comma 3rd Row u 3 left-parenthesis x right-parenthesis equals zero width space epsilon 33 x 3 comma EndLayout (2.135)

both lead to the same strain field given by

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