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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Superscript 5 Baseline zero width space zero width space plus left-parenthesis 2 plus 2 nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline left-parenthesis a slash r right-parenthesis cubed right-bracket sine theta cosine 2 phi comma EndLayout right-brace midline-horizontal-ellipsis a less-than r less-than infinity period"/>(3.38)

      The representation is identical in form to that used by Christensen and Lo [9] although they used a definition of ϕ that differs from that used here by an angle of π/4. This difference has no effect on the approach to be followed. It follows from (3.35)–(3.38) that the continuity conditions (3.34) are satisfied if the following four independent relations are satisfied

      StartLayout 1st Row 2 upper A Subscript p Baseline plus 12 nu Subscript p Baseline upper C Subscript p Baseline equals 2 gamma zero width space zero width space minus 3 upper B Subscript m Baseline zero width space zero width space plus left-parenthesis 10 minus 8 nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline comma 2nd Row upper A Subscript p Baseline plus left-parenthesis 7 minus 4 nu Subscript p Baseline right-parenthesis upper C Subscript p Baseline equals gamma plus upper B Subscript m Baseline plus left-parenthesis 2 minus 4 nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline comma 3rd Row mu Subscript p Baseline left-bracket upper A Subscript p Baseline minus 3 nu Subscript p Baseline upper C Subscript p Baseline zero width space zero width space right-bracket equals mu Subscript m Baseline left-bracket gamma plus 6 upper B Subscript m Baseline minus 2 left-parenthesis 5 minus nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline right-bracket comma 4th Row mu Subscript p Baseline left-bracket upper A Subscript p Baseline plus left-parenthesis 7 plus 2 nu Subscript p Baseline right-parenthesis upper C Subscript p Baseline zero width space zero width space right-bracket equals mu Subscript m Baseline left-bracket gamma minus 4 upper B Subscript m Baseline zero width space zero width space plus left-parenthesis 2 plus 2 nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline right-bracket comma EndLayout right-brace(3.39)

      and it can then be shown that

      As Cp=0, it follows from (3.35) and (3.36) that both the strain and stress distributions in the particle are uniform.

      3.4.2 Application of Maxwell’s Methodology

      To apply Maxwell’s methodology to a cluster of N particles embedded in an infinite matrix, the stress distribution in the matrix at large distances from the cluster is considered. The perturbing effect in the matrix at large distances from the cluster of particles is estimated by superimposing the perturbations caused by each particle, regarded as being isolated, and regarding all particles to be located at the origin. The properties of particles of type i will again be denoted by a superscript (i).

      The stress distribution at very large distances from the cluster is then given by the following generalisation of relations (3.38)

      where from (3.40), for i = 1, …, N,

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