Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 47

In a well-mixed cluster of N types of isotropic reinforcement embedded in and perfectly bonded to an infinite isotropic matrix (see Figure 3.1(a)), there are ni spherical particles of radius ai, i = 1, …N. Particle properties of type i, which may differ from those of other types, are denoted by a superscript i. The cluster of all particle types may be just enclosed by a sphere of radius b and the particle distribution is homogeneous leading to isotropic effective properties of the composite formed by the cluster and matrix lying within this sphere. The particles must not be distributed regularly or packed so densely that the effective properties are anisotropic. The volume fractions of particles of type i within the enclosing sphere of radius b are given by

Figure 3.1 (a) Discrete particle model and (b) smoothed effective medium model of a particulate composite having spherical reinforcements embedded in infinite matrix material.

(3.1)

where Vm is the volume fraction of matrix. For just one type of particle, with n particles of radius a within the enclosing sphere of radius b, the particulate volume fraction Vp is such that

(3.2)

Whatever the nature and arrangement of spherical particles in the cluster, Maxwell’s methodology considers the far-field when replacing the isotropic discrete particulate composite shown in Figure 3.1(a), that can just be enclosed by a sphere of radius b, by a homogeneous effective composite sphere of the same radius b embedded in the matrix as shown in Figure 3.1(b). There is no restriction on sizes, properties and locations of particles except that the equivalent effective medium is homogeneous and isotropic. Composites having statistical distributions of both size and properties can clearly be analysed.