target="_blank" rel="nofollow" href="#fb3_img_img_b66d7340-b367-5f6c-ad6c-aa6d7abdb198.png" alt="upper V Subscript f Baseline equals StartFraction n a squared Over b squared EndFraction equals 1 minus upper V Subscript m Baseline comma"/>(4.92)
where Vm is 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.
4.5.1 Representation for Displacement Strain and Stress Distributions
Consider the displacement field having the form
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
When ΔT=0, the stress-strain relations (4.14)–(4.17) are written as
where ET=2μt(1+νt). It follows directly from (4.94) and (4.98) that
From (4.94) and (4.97) it is clear that
