It should be noted that any temperature T0 can be added to the solution defined by (4.6) and (4.7) without affecting the satisfaction of the interface conditions (4.5). It should also be noted that when considering an infinite medium having a uniform temperature gradient, it is inevitable that temperatures lower than absolute zero will be encountered. This is not a matter for concern as the use of an infinite medium is simply a mathematical construct, designed to enable a specific method of estimating effective properties of the composite enclosed by the cylinder of radius b.
4.2.2 Maxwell’s Methodology for Estimating Transverse Conductivity
The first stage is to consider the perturbing effect of an isolated cluster of parallel fibres, having different sizes and properties as described in Section 4.1, in the infinite matrix at very large distances from the cluster. The transverse thermal conductivity of the fibres of type i is denoted by κTf(i). The cluster is assumed to be homogeneous regarding the distribution of fibres, and leads to an effective transverse thermal conductivity κTeff for the composite. The perturbing effect of the fibre cluster is estimated by superimposing the perturbations caused by each fibre and recognising that, at large enough distances, all the fibres in the cluster can considered as located at the origin that is situated at the centre of one of the fibres in the cluster. Thus, for the case of multiple phases, the temperature distribution (4.7) for the matrix is generalised to the form
The cluster of all types of fibre is now considered to be enclosed in a cylinder of radius b such that the volume fraction Vfi of fibres of type i within the cylinder of radius b is given by (4.1). On using (4.7) the temperature distribution in the matrix outside the cylinder of radius b is then given by
The second stage is to equate the perturbation terms in (4.8) and (4.9) (i.e. the second terms on the right-hand side) so that, on using (4.1), the effective thermal conductivity of the multiphase cluster of fibres may be obtained from the relation
The effective thermal conductivity of the multiphase fibre composite is then given by
The bounds for the conductivity of a multiphase composite are given by Torquato [5] which may be written in the following simpler form, having the same structure as the result (4.10) derived using Maxwell’s methodology
where κmin is the lowest value of conductivities for all phases, whereas κmax is the highest value.
The effective axial thermal conductivity of the unidirectional composite is given by the following mixtures rule
The validity of this relationship arises from the fact that the temperature distribution is such that there is no heat flow across the fibre/matrix interfaces. The temperature is linear in the axial direction having the same gradient in both fibre and matrix.
4.3 The Basic Equations for Thermoelastic Analysis
When considering an isolated cylindrical fibre embedded in matrix material, it is convenient to introduce a set of cylindrical polar coordinates (r,θ,z) where the origin lies on the axis of the fibre. All fibres are assumed to be made of transverse isotropic solids where properties are isotropic in the plane normal to the fibre axes. When using cylindrical polar coordinates, transverse isotropic solids (see (2.199)) are characterised