Properties for Design of Composite Structures. Neil McCartney

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СКАЧАТЬ V Subscript m Baseline Over 1 slash k Subscript m Baseline plus 3 slash left-parenthesis 4 mu Subscript m Baseline right-parenthesis EndFraction comma"/>(3.25)

(3.26)

      On using (3.1), the result (3.25) may be written as

(3.27)

      so that the effective bulk modulus of the multiphase particulate composite may instead be obtained from a ‘mixtures’ relation for the quantity 1/(k+43κm). On using (3.1) and (3.27), the effective bulk modulus may be estimated using

(3.28)

      It follows from (3.26) and (3.27) that the corresponding relation for effective thermal expansion is

(3.29)

      The bounds for effective bulk modulus of multiphase isotropic composites derived by Hashin and Shtrikman [6, Equations (3.37)–(3.43)] and the bounds derived by Walpole [7, Equation (26)] are identical and may be expressed in the following simpler form having the same structure as the result (3.27) derived using Maxwell’s methodology

(3.30)

      where the parameters kmin and μmin are the lowest values of bulk and shear moduli of all phases in the composite, respectively, whereas kmax and μmax are the highest values.

The bounds for effective thermal expansion involve the effective bulk modulus, and the specification of bounds is complex and beyond the scope of this chapter. An analysis has been undertaken showing numerically, for a very wide range of parameter values, that the effective thermal expansion obtained using Maxwell’s methodology lies between the absolute bounds for all volume fractions of spherical particle distributions that are consistent with isotropic properties.

      3.4 Shear Modulus

      3.4.1 Spherical Particle Embedded in Infinite Matrix Material Subject to Pure Shear Loading

      For a state of pure shear, and in the absence of thermal effects, the displacement field of a homogeneous sample of material referred to a set of Cartesian coordinates (x1,x2,x3) has the form

      (3.31)

      and the corresponding strain and stress components are given by

      (3.32)

      (3.33)

      The parameters γ and τ are the shear strain (half the engineering shear strain) СКАЧАТЬ