Linear Elastic Composites with Statistically Oriented Spheroidal Inclusions

The purpose of this chapter is to critically review some results that our groups obtained in previous works, which were devoted to the investigation of the elastic properties of composite materials with a statistical distribution of spheroidal inclusions. These studies were motivated by our interest in the description of mechanical properties of fibre-reinforced biological tissues (such as articular cartilage), starting from the internal structure of these tissues. After an introduction to tensor algebra, which defines the notation and clarifies the mathematical framework adopted in the chapter, we present, in a covariant setting inspired by Differential Geometry, Walpole’s representation of isotropic and transversely isotropic secondand fourth-order tensors, along with its properties. Hence, starting from Eshelby’s seminal work on the problem of an inclusion in an infinite matrix, we briefly review the theories developed by Hill, Walpole and Weng for the determination of the overall elasticity tensor of materials with one or more inclusion phases. Then, we discuss in detail the cases of composite materials with aligned spheroidal inclusions and with statistically oriented spheroidal inclusions. Emphasis is put on extending Walpole’s formula to the case of inclusions aligned according to some probability density of orientation, both in the transversely isotropic and the isotropic case.