In this paper, an analytical form of the Unit Step Function (or Heaviside Step function) is presented. This important function constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics. In particular, Heaviside Step Function is performed in a very simple manner by the use of a finite number of standard operations, as the summation of six inverse tangent functions. The novelty of this work when compared with other analytical expressions, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Beta function, Logistic function. Besides, it is neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

In this paper, the author derives an explicit form of Heaviside Step Function, which evidently constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics.In particular, this special function is exhibited in a very simple manner as a summation of four inverse tangent functions. The novelty of this work is that the proposed exact formulae are not performed in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also are neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence.Hence, this formula may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

In this paper, a further mathematical analysis of some previous results of the author’s ongoing investigation regarding Prandtl’s equations is accomplished. In particular, the objective of the present work is the performance of an approximate explicit solution to Prandtl’s system of equations for a laminar, isothermal, incompressible and steady boundary layer flow of a Newtonian fluid, over an infinite horizontal flat plate.

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula 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. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Dynamic mechanical analysis (DMA) is a versatile technique that complements the information given by the more traditional thermal analysis techniques such as differential scanning calorimetry (DSC), thermogravimetric analysis (TGA), and thermal mechanical analysis (TMA). Dynamic constants such as storage modulus, loss modulus, and loss factor are temperature dependent and provide information about interfacial bonding between reinforced fibre and polymer matrix of composite material. To study the above mentioned properties at the glass transition region, for unidirectional fibrous composites reinforced with continuous fibers a reliable model was applied. In particular, the composite material was considered as composed of three phases with the intermediate phase between matrix and fibres, the interphase, to have variable properties depending on those of main phases and the mode of preparation of the overall material. The glass transition temperature is defined as the point at which the specific volume versus temperature curve changes abruptly slope marking the region between rubbery polymer and glassy polymer nature. Hence, the behaviour of unidirectional fibrous composites was investigated at this region. Examination of the glass transition temperature, which constitutes an upper limit for the structurally important glassy region through the loss factor, was performed by its consideration as a combination of glass transition temperature of matrix and interphase.

In this paper, an upper bound of the longitudinal elastic modulus of unidirectional fibrous composites is proposed according to strength of materials approach, on the premise that the fiber is much stiffer than the matrix. In the mathematical derivations, the concept of boundary interphase between fiber and matrix was also taken into account and the main objective of this work is the attainment of an upper bound for the interphase stiffness with respect to fiber concentration by volume. The novel element here is that the authors have not taken into consideration any specific variation law to approximate the interphase modulus. The theoretical results arising from the proposed formula were compared with those obtained from some reliable theoretical models as well as with experimental data found in the literature, and a satisfactory agreement was observed.

In this paper, a geometric body-centered model to simulate the periodic structure of unidirectional fibrous composites is presented. To this end, three prescribed configurations are introduced to predict in a deterministic manner the arrangement of internal and neighboring fibers inside the matrix. Thus, three different representative volume elements (RVEs) are established. Furthermore, the concept of the interphase has been taken into account, stating that each individual fiber is encircled by a thin layer of variable thermomechanical properties. Next, these three unit cells are transformed in a unified manner to a coaxial multilayer cylinder model. This advanced model includes the influence of fiber contiguity in parallel with the interphase concept on the thermomechanical properties of the overall material. Then, by the use of this model, the authors propose explicit expressions to evaluate the longitudinal and transverse thermal conductivity of this type of composite. The theoretical predictions were compared with experimental results, as well as with theoretical values yielded by some reliable formulae derived from other workers, and a reasonable agreement was found.

In this paper, an icosahedral non – body centered model to simulate the periodic structure of homogeneous particulate composites, by predicting the particle arrangement, is presented. This model has yielded three different variations which correspond at three different deterministic particle configurations. In addition, the concept of boundary interphase between matrix and inclusions was taken into account. Thus, the influence of particle vicinity was examined in parallel with the interphase concept on the thermomechanical properties of the overall material. Next, by the use of this model the authors derived a closed – form expression to estimate the thermal conductivity of this type of composites. To test the validity of the model, the theoretical values arising from the proposed formula were compared with other theoretical predictions obtained from several accurate formulae found in the literature and an adequate accordance was observed.