Properties for Design of Composite Structures. Neil McCartney
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the other bound is obtained simply by interchanging the fibre and matrix properties and the volume fractions of fibre and matrix. This suggests that the bound that corresponds with the composite cylinders assemblage result is likely to be the nearest estimate to the actual value of the property that is being estimated.
A detailed study of the method used by Maxwell has revealed that his methodology can also be applied to the estimation of many other properties of fibre-reinforced composite materials that are reinforced with cylindrical fibres. The principal objective of this chapter is to show how Maxwell’s methodology can be applied to the estimation of many of the effective properties of a composite reinforced with homogeneous cylindrical fibres. The methodology of Maxwell is easily extended so that assemblies of multiphase cylindrical fibres having a range of radii and properties may be considered.
Section 4.2 provides a detailed description of Maxwell’s methodology applied to the thermal conduction problem, extending the approach to deal with multiphase cylindrical fibres having different sizes, and including the effects of thermal resistance at the fibre/matrix interfaces. It is noted that the results of Johnson and Hasselman [4] for two-phase systems having cylindrical fibres of the same size, but including interfacial thermal resistance, are in fact in error. Section 4.3 uses a radial stress and displacement formulation in conjunction with Maxwell’s methodology to estimate values for the effective properties of a fibre-reinforced composite. Sections 4.4 and 4.5 consider the method of applying Maxwell’s methodology to the estimation of the effective shear axial and transverse moduli of a fibre-reinforced composite. For each effective property, the resulting formulae for the case of just two phases, having cylindrical reinforcements of the same size, will be expressed as a mixtures estimate plus a correction term that is used to derive the conditions that determine whether the extreme values of properties are upper or lower bounds. These formulae differ in some cases from those that have been given in the literature.
4.2 General Description of Maxwell’s Methodology Applied to Thermal Conductivity
Consider an isolated cluster of parallel fibre reinforcements, embedded in an isotropic infinite matrix, as shown in Figure 4.1(a). There are N different reinforcement types such that for i = 1, …, N, there are ni cylindrical fibres of radius ai. The properties and geometry of the fibres of type i, that may differ from those of other types, are denoted by a superscript i. The cluster of all types of fibre is just enclosed by a cylinder of radius b and the parallel fibre distribution is sufficiently homogeneous for it to lead to transverse isotropic properties for the composite formed by the cluster of fibres and the matrix lying within this cylinder as shown in Figure 4.1(b). The volume fraction of fibres of type i within the cylinder of radius b is given by
Figure 4.1 (a) Discrete fibre model and (b) smoothed effective medium model of a fibre-reinforced composite embedded in infinite matrix material.
where Vm is the corresponding volume fraction of matrix.
If there is only one type of fibre, and there are n fibres of radius a within the cylinder of radius b, then the fibre volume fraction of the composite is denoted by Vf such that
It should be noted that by selecting all fibres to have the same properties but different sizes, composites having only a statistical distribution of fibre sizes can be considered. Similarly, by selecting all fibres to have the same size but different properties, composites having only a statistical distribution of properties can be considered. Clearly composites having statistical distributions of both size and properties can also be considered.
4.2.1 Temperature Distribution for an Isolated Fibre
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
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
At the fibre/matrix interface, the following boundary conditions are imposed
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