In this paper, the author obtains an analytic exact form of the unit step function, which is also known as Heaviside function and constitutes a fundamental concept of the Operational Calculus. Particularly, this function is equivalently expressed in a closed form as the summation of two inverse trigonometric functions. The novelty of this work is that the exact representation which is proposed here is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Therefore it may be much more appropriate in the computational procedures which are inserted into Operational Calculus techniques.

In this paper, the thermal conductivity coefficients of unidirectional composites have been estimated. In the present analysis, a model which takes into account the influence of the boundary interphase has been proposed. The thermal conductivity coefficients of the composite were calculated by using this concept to determine the role of the interphase which mainly depends on the quality of adhesion between fibers and matrix. The interphase matter is the part of the polymer matrix lying on the close vicinity of the fiber bounding surface. It has been primarily assumed that the interphase is inhomogeneous with thermophysical properties varying from the fiber surface to the matrix. Five laws of variation have been taken into consideration in order to derive closed form expressions for the thermal conductivity coefficients in which the role of the interphase layer occurs, possessing properties different from those of the constituent phases. A thermal analysis method known in the literature has been adopted in order to find the extent of the interphase layer. According to this viewpoint, theoretical expressions in order to calculate the thermal conductivity coefficients of the glass fiber reinforced composite materials were derived taking into consideration the interphase layer.

In this paper, a model to find the approximate equations for determining the elastic constants of unidirectional fiber – reinforced composite materials in terms of the constituent material properties is described. The novelty of this work is that the fibers are considered to be transversely isotropic. To simulate the microstructure of the composite, we will take into account the concept of interphase with the concurrent assumption that the fibers are parallel to the line formed by the centers of the bases of a three – phase cylinder model, having a uniform distribution inside the matrix without agglomeration. The results were compared with the respective values of some reliable theoretical models as well as with experimental data obtained from other researchers, and they were found to be in reasonable agreement.