Properties for Design of Composite Structures. Neil McCartney

Чтение книги онлайн.

СКАЧАТЬ p₀ must satisfy the relation

(3.20)

      3.3.2 Applying Maxwell’s Methodology to Isotropic Multiphase Particulate Composites

      Owing to the use of the far-field in Maxwell’s methodology, it is again possible to consider multiple types of spherical reinforcement. 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 are denoted by a superscript (i).

      Relations (3.19) for nonzero stresses in the matrix are generalised to

(3.21)

      where p0i is the pressure at the particle/matrix interface when an isolated particle of species i is placed in infinite matrix material. From (3.20) the following value of p0i−p is obtained

      (3.22)

      It then follows that the stress distribution in the matrix at large distances from the discrete cluster of particles shown in Figure 3.1(a) is approximately given by

(3.23)

      where the volume fractions Vpi of particles of type i defined by (3.1) have been introduced.

      When (3.23) is applied to a single sphere of radius b, having the effective properties of a composite representing the multiphase cluster of particles embedded in matrix material (see Figure 3.1(b)), the exact matrix stress distribution, for given values of keff and αeff, is

(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,

СКАЧАТЬ