Properties for Design of Composite Structures. Neil McCartney

Baseline right-parenthesis EndFraction comma 2nd Row sigma Subscript theta theta Superscript m Baseline equals sigma Subscript phi phi Superscript m Baseline equals negative p plus StartFraction p b cubed Over 2 r cubed EndFraction left-parenthesis StartFraction 1 slash k Subscript m Baseline plus 3 slash left-parenthesis 4 mu Subscript m Baseline right-parenthesis Over 1 slash k Subscript eff Baseline plus 3 slash left-parenthesis 4 mu Subscript m Baseline right-parenthesis EndFraction minus 1 right-parenthesis minus StartFraction 3 upper Delta upper T b cubed Over 2 r cubed EndFraction StartFraction alpha Subscript eff Baseline minus alpha Subscript m Baseline Over 1 slash k Subscript eff Baseline plus 3 slash left-parenthesis 4 mu Subscript m Baseline right-parenthesis EndFraction comma EndLayout right-brace b less-than r less-than infinity period"/>(3.24)

      Maxwell’s methodology asserts that, at large distances from the cluster, the stress distributions (3.23) and (3.24), and hence the coefficients of p and ΔT, are identical leading to the following ‘mixtures’ rules for the functions 1/[1/k+3/(4μm)] and α/[1/k+3/(4μm)], respectively,

(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