The solution to this equation is

(7.174)

Thus, the solution for fibre is

(7.175)

and the solution for the matrix is

(7.176)

The axial displacement in fibre and matrix can be determined by integrating the first of Equation (7.172) with respect to . In this equation the left hand side is independent of .
Thus,

(7.177)

It should be noted that   and   are constants and  and  are the arbitrary constants of integration. The constants  and  and functions  and  are unknown. These can be determined from specific boundary conditions.

The  displacement in fibre as given in Equation (7.175) should be bounded when . This requires the condition that

(7.178)

Further, the displacements are continuous at the interface of fibre and matrix. This results in

(7.179)

Using Equation (7.178) and Equation (7.179), the continuity conditions become

(7.180)

On equating the terms in  and  in the second of Equation (7.180), we get

(7.181)

This means that the strains in -direction in fibre and matrix are given by same function of . Further, the unknown constant   in fibre and matrix is same. This constant, in fact, is the axial strain. Therefore, it is denoted as .