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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target="_blank" rel="nofollow" href="#fb3_img_img_231ab5f6-66e1-5357-bf91-86d4ac7885bf.png" alt="StartLayout 1st Row u prime Subscript 1 Baseline equals u 1 cosine phi plus u 2 sine phi comma u 1 equals u prime Subscript 1 Baseline cosine phi minus u prime Subscript 2 Baseline sine phi comma 2nd Row u prime Subscript 2 Baseline equals u 2 cosine phi minus u 1 sine phi comma u 2 equals u prime Subscript 1 Baseline sine phi plus u prime Subscript 2 Baseline cosine phi comma 3rd Row u prime Subscript 3 Baseline equals u 3 comma u 3 equals u prime Subscript 3 Baseline period EndLayout"/>(2.175)

      The stress and strain at any point in a material is a dyadic (an array of ordered vector pairs) or second-order tensor whose value is wholly independent of the coordinate system that is used to describe its components. The second-order stress tensor σ may, therefore, be written as (where summation over values 1, 2 and 3 is implied by repeated lower case suffices)

      where σkl and σ′kl are the stress components referred to the two coordinate systems being considered. On defining m = cos ϕ and n = sin ϕ, it follows from (2.172) that

      Thus, from (2.176) and (2.177), because σ′12=σ′21, σ′13=σ′31 and σ′23=σ′32

      The inverse relationships are obtained by replacing ϕ by −ϕ (i.e. n is replaced by –n) so that

      StartLayout 1st Row epsilon 11 equals m squared epsilon prime Subscript 11 Baseline plus n squared epsilon prime Subscript 22 Baseline minus 2 m n epsilon prime Subscript 12 Baseline comma 2nd Row epsilon 22 equals n squared epsilon prime Subscript 11 Baseline plus m squared epsilon prime Subscript 22 Baseline plus 2 m n epsilon prime Subscript 12 Baseline comma 3rd Row epsilon 33 equals epsilon prime Subscript 33 Baseline comma 4th Row epsilon 23 equals m epsilon prime Subscript 23 Baseline plus n epsilon prime Subscript 13 Baseline equals epsilon 32 comma 5th Row epsilon 13 equals minus n epsilon prime Subscript 23 Baseline plus m epsilon prime Subscript 13 Baseline equals epsilon 31 comma 6th Row epsilon 12 equals m n epsilon prime Subscript 11 Baseline minus m n epsilon prime Subscript 22 Baseline plus left-parenthesis m squared minus n squared right-parenthesis epsilon prime Subscript 12 Baseline equals epsilon 21 comma EndLayout(2.180)

       with inverse relations

      2.17 Transformations of Elastic Constants

      On substituting the stress-strain relations (2.170) into (2.181)

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