for the stress field (3.44) in the matrix lying outside the sphere having radius b of effective medium (see Figure 3.1(b)) involves terms proportional to r−3 and r−5, but only terms involving r−3 are used when applying Maxwell’s methodology, as seen from (3.41). This means that, in contrast to the cases for the effective bulk modulus, thermal expansion coefficient and thermal conductivity, the resulting estimate for the effective shear modulus does not lead to an exact matrix stress distribution in the region b<r<∞ outside the sphere of effective medium, and consequently estimates for effective shear modulus are likely to be less accurate than those for other effective properties.
The results, discussed previously for various effective properties, are remarkable as one might expect Maxwell’s methodology to be accurate only for sufficiently low volume fractions of reinforcement. The reason is that the methodology involves the examination of the stress, displacement or temperature fields in the matrix at large distances from the cluster of particles, and assumes that the perturbing effect of each particle can be approximated by locating them at the same point. The nature of this approximation is such that interactions between particles are negligible, and it would be expected that resulting effective properties will be accurate only for low volume fractions, as originally suggested by Maxwell [3]. In view of compelling evidence presented in this chapter, based on a wide variety of considerations, a major conclusion is that results for two-phase composites derived using Maxwell’s methodology are not limited to small particulate volume fractions and can be used with confidence using typical volume fractions often encountered in practice.
References
1 1. Hashin, Z. (1983). Analysis of composite materials - A survey. Journal of Applied Mechanics 50: 481–505.
2 2. Torquato, S. (2002). Random Heterogeneous Materials. New York: Springer-Verlag.
3 3. Maxwell, J.C. (1873). A Treatise on Electricity and Magnetism, 1st e (3rd e, 1892). Chapter 9, (Vol. 1, Art. 310-314, pp. 435–441). Oxford: Clarendon Press.
4 4. McCartney, L.N. and Kelly, A. (2008). Maxwell’s far-field methodology applied to the prediction of properties of multi-phase isotropic particulate composites. Proceedings of the Royal Society A464: 423–446.
5 5. Hashin, Z. and Shtrikman, S. (1962). A variational approach to the theory of the effective magnetic permeability of multiphase materials. Journal of Applied Physics 33 (10): 3125–3131.
6 6. Hashin, Z. and Shtrikman, S. (1963). A variational approach to the theory of the elastic behaviour of multiphase composites. Journal of the Mechanics and Physics of Solids 11: 127–140.
7 7. Walpole, L.J. (1966). On bounds for the overall elastic moduli of inhomogeneous systems – I. Journal of the Mechanics and Physics of Solids 14: 151–162.
8 8. Love, A.E.H. (1944). A Treatise on the Mathematical Theory of Elasticity. Chapter XI, 4th ed. New York: Dover Publications.
9 9. Christensen, R.M. and Lo, K.H. (1979). Solutions for the effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids 27: 315–330.
10 10. Sangani, A.S. and Acrivos, A. (1983). The effective conductivity of a periodic array of spheres. Proceedings of the Royal Society of London A386: 263–275.
11 11. Arridge, R.G.C. (1992). The thermal expansion and bulk modulus of composites consisting of arrays of spherical particles in a matrix, with body or face centred cubic symmetry. Proceedings of the Royal Society of London A438: 291–310.
12 12. Bonnecaze, R.T. and Brady, J.F. (1990). A method for determining the effective conductivity of dispersions of particles. Proceedings of the Royal Society of London A430: 285–313.
13 13. Torquato, S. (1990). Bounds on the thermoelastic properties of suspension of spheres. Journal of Applied Physics 67: 7223–7227.
14 14. Cohen, I. and Bergman, D.J. (2003). Effective elastic properties of periodic composite medium. Journal of the Mechanics and Physics of Solids 51: 1433–1457.
4 Maxwell’s Methodology for the Prediction of Effective Properties of Unidirectional Multiphase Fibre-reinforced Composites
Overview:
The methodology developed by Maxwell when estimating the effective electrical conductivity of an isotropic particulate composite is used to estimate many of the effective thermoelastic properties of a fibre-reinforced composite. A detailed description is first given of Maxwell’s methodology applied to the thermal conduction problem, extending the approach to deal with multiphase fibres having different sizes and properties, and assuming perfect thermal contact at the fibre/matrix interfaces. It is noted that a published result, based on Maxwell’s methodology, for two-phase systems having fibres of the same size, but including interfacial thermal resistance, is in error. A stress and displacement formulation that is radial in nature is used in conjunction with Maxwell’s methodology to estimate values for many of the effective properties of a unidirectional fibre-reinforced composite. A method of applying Maxwell’s methodology to the estimation of the effective shear modulus of a unidirectional fibre-reinforced composite is also given. A key aspect of the results obtained, when using Maxwell’s methodology to estimate some of the thermoelastic properties of a unidirectional composite is that they correspond to those derived from the composite cylinders assemblage model, which, in turn, correspond to one of the bounds obtained when using variational methods. This correspondence indicates that the results obtained using Maxwell’s methodology are not restricted to low fibre volume fractions. For each effective property, it is shown that the resulting formulae for the case of just two phases, having fibre reinforcements of the same size, may 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 obtained by variational methods are upper or lower bounds. These conditions differ in some cases from those that have been given in the literature.
4.1 Introduction
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
