Neil McCartney - Properties for Design of Composite Structures

For the case of multiple phases, relation ( 4.27) 1is generalised to the following form

(4.55)

where

(4.56)

The cluster of all types of fibre is now considered to be enclosed in a cylinder of radius b such that the volume fraction of fibres of type i within the cylinder of radius b is given by Vfi=niai2/b2. The volume fractions must satisfy relation ( 4.1) 2namely

(4.57)

It then follows that ( 4.27) may be written in the form

(4.58)

When the result ( 4.55) is applied to a single fibre of radius b having effective properties corresponding to the multiphase cluster of fibres it follows that

(4.59)

where the parameter ϕ¯ is defined by

(4.60)

The coefficients of the 1/ r terms in relations ( 4.59) and ( 4.58) must be identical so that

(4.61)

It then follows from ( 4.56) and ( 4.60) that for any values of the parameters ε, σT and ΔT

(4.62)

Clearly the following relations must be satisfied

(4.63)

(4.64)

(4.65)

On using ( 4.1), these relations are now expressed as

(4.66)

(4.67)

(4.68)

It should be noted that the quantity αT+νAαA is the transverse thermal expansion coefficient for plane strain conditions such that uz=0. On using ( 4.1), relation ( 4.66) may also be expressed as

(4.69)

On using ( 4.1) and ( 4.66), relations ( 4.67) and ( 4.68) may be written as

(4.70)

(4.71)

The following analysis applies to an isolated cylindrical fibre perfectly bonded to an infinite matrix, subject to a uniform temperature change Δ T and to a uniform axial shear stress. The fibre and matrix are regarded as transverse isotropic solids so that the stress-strain relations are given by ( 4.14)–( 4.17).

A solution is now sought of the following form

(4.72)

where A , α and β are constants to be determined. On differentiating the displacement field using the strain-displacement relations (2.139), it follows that

(4.73)

(4.74)

It follows directly from ( 4.14)–( 4.17), ( 4.73) and ( 4.74) that

(4.75)

In addition, it follows from ( 4.17), ( 4.73) and ( 4.74) that

(4.76)

It is easily shown that the stress field automatically satisfies the equilibrium equations ( 4.20)–( 4.22) for any values of the parameters A , α and β.

The following continuity conditions must be satisfied at the interface r = a between fibre and matrix

(4.77)

(4.78)

It then follows from ( 4.76) that

(4.79)

and from ( 4.72) that

(4.80)

The substitution of ( 4.80) into ( 4.79) then leads to

(4.81)