for Thermoelastic Analysis
When considering an isolated cylindrical fibre embedded in matrix material, it is convenient to introduce a set of cylindrical polar coordinates (r,θ,z) where the origin lies on the axis of the fibre. All fibres are assumed to be made of transverse isotropic solids where properties are isotropic in the plane normal to the fibre axes. When using cylindrical polar coordinates, transverse isotropic solids (see (2.199)) are characterised by stress-strain relations of the form
Superscript ‘f’ is used to denote anisotropic fibre properties and subscript ‘m’ is used to denote the properties Em, νm, μm and αm of an isotropic matrix such that Em=2μm(1+νm). If the fibres are also isotropic then
The equilibrium equations for the fibres and matrix in the absence of body forces are (see (2.125)–(2.127))
4.3.1 Properties Defined from Axisymmetric Distributions
The following analysis applies to an isolated cylindrical fibre of radius a that is perfectly bonded to an infinite matrix, subject to a uniform temperature change ΔT, where the system is subject to a uniform axial strain ε and a uniform transverse stress σT. The equilibrium equations (4.20)–(4.22) for the fibre and matrix, assuming symmetry about the fibre axis so that stress components are independent of θ and the shear stresses σrθ and σθz are zero, then reduce to the form
In regions away from the loading mechanism it is reasonable to assume that
where ε is the axial strain applied to the composite. A solution is now sought of the following classical Lamé form
