Properties for Design of Composite Structures. Neil McCartney

4.27), and the corresponding stress distribution is specified by (4.33), (4.39)–(4.44). The stress-strain relations (4.14)–(4.17) and the equilibrium equations (4.20)–(4.22) are satisfied exactly. The boundary and interface conditions (4.28) are also satisfied exactly.

      4.3.3 Solution in the Absence of Fibre

      For the general loading conditions characterised by the parameters ε, σT and ΔT and applied to an infinite sample of matrix in the absence of fibre (filling the entire region of space) the solution is given by

      (4.53)

(4.54)

      where the parameter Am is given by (4.46). It should be noted that the axial stress in the matrix given by (4.54)₂ is identical to that in the matrix when the fibre is present (see (4.44)). It should also be noted that when the fibre is introduced the radial displacement function is perturbed as a second term appears in (4.27)₁ that is inversely proportional to the radial distance r. This additional term will now be considered when applying Maxwell’s method of estimating the properties of a fibre-reinforced composite.

      4.3.4 Applying Maxwell’s Approach to Multiphase Fibre Composites

      Owing to the use of the far-field in Maxwell’s method for estimating the properties of fibre composites, it is possible to consider multiple fibre reinforcements. Suppose in a cluster of fibres that there are N different types such that for i = 1, …, N there are ni fibres of radius ai. The properties of the fibres of type i are denoted by a superscript i. The cluster is assumed to be homogeneous regarding the distribution of fibres, and leads to transverse isotropic effective properties.

      For the case of multiple phases, relation (4.27)₁ is generalised to the following form

(4.55)

      where

(4.56)

      The cluster of all types of fibre is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibres of type i within the cylinder of radius b is given by Vfi=niai2/b2. The volume fractions must satisfy relation (4.1)₂ namely

      (4.57)

      It then follows that (4.27) may be written in the form

(4.58)

      When the result (4.55) is applied to a single fibre of radius b having effective properties corresponding to the multiphase cluster of fibres it follows that

(4.59)

      where the parameter ϕ¯ is defined by

(4.60)

      The coefficients of the 1/r terms in relations (4.59) and (4.58) must be identical so that