Neil McCartney - Properties for Design of Composite Structures

For loading conditions characterised by the shear stress τ applied to an infinite sample of matrix in the absence of fibre (filling the entire region of space), the solution is given by

(4.82)

(4.83)

A comparison of ( 4.72) and ( 4.82) with ( 4.77) and ( 4.83) indicates that the identification α=τ/2 can be made. It then follows from ( 4.81) that

(4.84)

Substitution into ( 4.72) leads to the following expression for the displacement component uz:

(4.85)

It should be noted that when fibre is introduced, the displacement components are perturbed as a second term appears in expression ( 4.85) for the matrix displacement that is inversely proportional to the radial distance r . This additional term will now be considered when applying Maxwell’s method of estimating the effective properties of a fibre-reinforced composite.

Owing to the use of the far-field in Maxwell’s method for estimating the properties of fibre composites, it is possible to consider multiple fibre reinforcements. Suppose in a cluster of fibres that there are N different types such that for i = 1, …, N , there are ni fibres of radius ai. The properties of the fibres of type i are denoted by a superscript i . The cluster is assumed to be homogeneous regarding the distribution of fibres, and this leads to transverse isotropic effective properties.

For the case of multiple phases, relation ( 4.85) is generalised to the following form:

(4.86)

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

(4.87)

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 the relation

(4.88)

It then follows that ( 4.86) may be written in the form

(4.89)

The coefficients of the 1/ r terms in relations ( 4.87) and ( 4.89) must be identical so that

(4.90)

It then follows on using ( 4.1) that the effective axial shear modulus for the multiphase composite may be written as

(4.91)

Consider a cluster of n cylindrical fibres of the same radius a embedded in an infinite matrix having different properties. The cluster is just enclosed by a cylinder of radius b and the fibre distribution is sufficiently homogeneous for it to lead to transverse isotropic properties for the composite formed by the cluster of fibres and the matrix lying within this cylinder. If the fibre volume fraction of the composite is denoted by V fthen

(4.92)

where V mis the corresponding volume fraction of the matrix.

First, a single cylindrical fibre of radius a is placed in an infinite matrix and the origin of cylindrical polar coordinates ( r , θ , z ) is taken on the axis of the fibre. The system is then subject only to a transverse shear stress applied at infinity. The temperature change ΔT from the stress-free temperature, where the stresses and strains in fibre and matrix are everywhere zero, is also assumed to be everywhere zero.

Consider the displacement field having the form

(4.93)

where kT and μt are the transverse bulk and shear moduli, respectively.

On using the strain–displacement relations (2.142), the corresponding strain field is given by

(4.94)

When ΔT=0, the stress-strain relations ( 4.14)–( 4.17) are written as

(4.95)

(4.96)

(4.97)

(4.98)

where ET=2μt(1+νt). It follows directly from ( 4.94) and ( 4.98) that

(4.99)

From ( 4.94) and ( 4.97) it is clear that

(4.100)

and on summing ( 4.95) and ( 4.96)

(4.101)