Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 62

It is useful to extend the methodology used by Maxwell [1] to estimate the electrical conductivity of a cluster of fibres embedded in an infinite matrix so that fibre-reinforced composites can be considered. The following description modifies the approach of Maxwell so that it applies to the analogous problem of thermal conductivity, and to a multiphase system of fibres. Hasselman and Johnson [4] considered the same problem where the fibres were all the same type size and where there is an interfacial thermal resistance between the fibers and the matrix.

Consider a single fibre of radius a embedded in an infinite matrix. A set of cylindrical polar coordinates (r,θ,z) is introduced having origin on the axis of the fibre. The temperature distribution in the fibre and surrounding matrix must satisfy Laplace’s equation, namely

(4.3)

The transverse thermal conductivity of the fibre is denoted by κTf whereas the matrix is assumed isotropic having thermal conductivity κm. On the external boundary r→∞ a temperature distribution is imposed that would lead, in a homogeneous matrix material under steady-state conditions, to the following temperature distribution having a uniform gradient α

(4.4)

At the fibre/matrix interface, the following boundary conditions are imposed

(4.5)

The temperature distributions in the fibre and matrix, denoted by Tf and Tm, respectively, must satisfy (4.3) and the boundary conditions (4.5) so that

(4.6)

(4.7)

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.