Properties for Design of Composite Structures. Neil McCartney
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where kp= λp+23μp and μp are the bulk and shear moduli, respectively, for the particulate reinforcement, and where αp is the corresponding thermal expansion coefficient. Clearly the strain and stress distributions within the particle are both uniform. For the matrix region it can be shown that
where km= λm+23μm and μm are the bulk and shear moduli, respectively, for the matrix, and where αm is the corresponding thermal expansion coefficient. The stress component σrr is automatically continuous across r = a having the value −p0. As the displacement component ur must also be continuous across this interface, the value of p0 must satisfy the relation
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
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
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
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