Properties for Design of Composite Structures. Neil McCartney

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Properties for Design of Composite Structures - Neil McCartney

Properties for Design of Composite Structures - Neil McCartney

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equals epsilon 22 comma epsilon 33 left-parenthesis x right-parenthesis identical-to StartFraction partial-differential u 3 Over partial-differential x 3 EndFraction equals epsilon 33 comma 2nd Row epsilon 12 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 1 Over partial-differential x 2 EndFraction plus StartFraction partial-differential u 2 Over partial-differential x 1 EndFraction right-parenthesis equals epsilon 12 comma epsilon 13 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 1 Over partial-differential x 3 EndFraction plus StartFraction partial-differential u 3 Over partial-differential x 1 EndFraction right-parenthesis equals epsilon 13 comma 3rd Row epsilon 23 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 2 Over partial-differential x 3 EndFraction plus StartFraction partial-differential u 3 Over partial-differential x 2 EndFraction right-parenthesis equals epsilon 23 period EndLayout"/>(2.136)

      When using cylindrical polar coordinates (r,θ,z), the displacement components ur,uθ,uz are related to the Cartesian components u1,u2,u3 as follows

      StartLayout 1st Row u Subscript r Baseline equals u 1 cosine theta plus u 2 sine theta comma 2nd Row u Subscript theta Baseline equals u 2 cosine theta minus u 1 sine theta comma 3rd Row u Subscript z Baseline equals u 3 comma EndLayout(2.137)

      having inverse

      StartLayout 1st Row u 1 equals u Subscript r Baseline cosine theta minus u Subscript theta Baseline sine theta comma 2nd Row u 2 equals u Subscript theta Baseline cosine theta plus u Subscript r Baseline sine theta comma 3rd Row u 3 equals u Subscript z Baseline comma EndLayout(2.138)

      and the strain–displacement relations are given by

      StartLayout 1st Row epsilon Subscript r r Baseline equals StartFraction partial-differential u Subscript r Baseline Over partial-differential r EndFraction comma epsilon Subscript theta theta Baseline equals StartFraction 1 Over r EndFraction left-parenthesis u Subscript r Baseline plus StartFraction partial-differential u Subscript theta Baseline Over partial-differential theta EndFraction right-parenthesis comma epsilon Subscript z z Baseline equals StartFraction partial-differential u Subscript z Baseline Over partial-differential z EndFraction comma 2nd Row epsilon Subscript r theta Baseline equals 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 comma epsilon Subscript theta z Baseline equals 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 comma epsilon Subscript r z Baseline equals 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 period EndLayout(2.139)

      When using spherical polar coordinates (r,θ,ϕ), the displacement components ur,uθ,uϕ are related to the Cartesian components u1,u2,u3 as follows:

      StartLayout 1st Row u Subscript r Baseline equals sine theta cosine phi u 1 plus sine theta sine phi u 2 plus cosine theta u 3 comma 2nd Row u Subscript theta Baseline equals cosine theta cosine phi u 1 plus cosine theta sine phi u 2 minus sine theta u 3 comma 3rd Row u Subscript phi Baseline equals cosine phi u 2 minus sine phi u 1 comma EndLayout(2.140)

      having inverse

      StartLayout 1st Row u 1 equals sine theta cosine phi u Subscript r Baseline plus cosine theta cosine phi u Subscript theta Baseline minus zero width space zero width space sine phi u Subscript phi Baseline comma 2nd Row u 2 equals sine theta sine phi u Subscript r Baseline plus cosine theta sine phi u Subscript theta Baseline plus cosine phi u Subscript phi Baseline comma 3rd Row u 3 equals cosine theta u Subscript r Baseline minus sine theta u Subscript theta Baseline comma EndLayout(2.141)

      and the strain–displacement relations are given by

      2.14 Constitutive Equations for Anisotropic Linear Thermoelastic Solids

      As we are concerned in this book with various types of composite material, it is necessary to define a set of constitutive relations that will form the basis for the development of theoretical methods for predicting the behaviour of anisotropic materials. Consider a general homogeneous infinitesimal strain εkl (applied to a unit cube of the composite material) defined in terms of the displacement vector uk and the position vector xk by (2.107), namely,

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