Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 60

Chapter 3 described how Maxwell’s methodology [1, 2] can be applied to the estimation of the effective thermoelastic and conduction properties of isotropic spherical particulate composites. It was emphasised that Maxwell’s methodology can provide good estimates of effective properties for a wide range of composites having practical interest. The objective of this chapter is to apply Maxwell’s methodology to the prediction of good estimates of the effective properties of unidirectionally fibre-reinforced composites, where the fibres of circular cross section can be of different types and have different radii. Although some concepts regarding Maxwell’s methodology are repeated, they are included here so that the chapter is more self-contained, and as the actual details of the method differ slightly. The analysis used differs significantly from that in Chapter 3.

The estimation of the thermoelastic and conduction properties of fibre-reinforced composites has been studied in detail in the literature over many years. Hashin [3] has given a very detailed review of many estimation methods, including the use of the well-known composite cylinders assemblage method, and methods based on variational techniques that lead to upper and lower bound estimates of properties. Key aspects are that the composite cylinders assemblage results correspond to one of the bounds obtained using variational methods, and that 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.