Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 64

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

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

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

(4.19)

The equilibrium equations for the fibres and matrix in the absence of body forces are (see (2.125)–(2.127))

(4.20)

(4.21)

(4.22)