Properties for Design of Composite Structures. Neil McCartney

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СКАЧАТЬ 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

(3.40)

      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)

(3.41)

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

(3.42)

For the isolated sphere of radius b having the effective properties of the particulate composite cluster as illustrated in Figure 3.1(b), it follows that the stress field in the matrix at large distances is described exactly by relations of the type (3.38) where the coefficient Dm is replaced by D¯m having the value determined by the relation

(3.43)

      where μeff is the effective shear modulus of the isotropic particulate composite. It then follows from (3.38) and (3.40) that the exact matrix stress distribution, for a given value of μeff, is

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