## 1 Introduction
Additionally, analytical expressions for the effective properties have been found. Apart from their theoretical importance, exact analytical expressions prove to be very useful for the validation of numerical results. Simple closed analytical expressions for the effective elastic coefficients were found in refs. [10; 19] for square and hexagonal cell distributions, respectively. The AHM was applied to a fiber-reinforced composite with a cylindrical shape periodically distributed in the matrix. Both, matrix and fiber, had transversely isotropic properties and they were in perfect contact. The formulas obtained are also valid for isotropic constituents. The effective moduli of a periodic elastic composite material reinforced by straight parallel circular fibers in a square array were determined in ref. [3]. The composite is made of a transversely isotropic material with axial, tangential and normal imperfect contacts at the interface. The investigation was carried out by adopting the Semi-Analytical Method (SAFEM) which is based on the FEM and the AHM. Quadrilateral quadratic Lagrange functions were used as shape functions in the FEM formulation. In ref. [6], the effective properties of composites materials with fibers of different geometries (circular, square and rhombic) distributed in a square array were studied and three different FEM approaches were implemented: 4-node quadrilaterals, 8-node quadrilaterals and 12-node quadrilaterals. An extension of this SAFEM for coupled field problems was reported in ref. [20]. In addition, a three-dimensional SAFEM was implemented in ref. [21] to calculate the effective properties of periodic elastic-reinforced nanocomposites. Different inclusions were considered, such as discs, ellipsoids, spheres, carbon nanotubes and carbon nanowires. The effective thermoelastic properties in composite materials have also been extensively studied. A simple closed form of the effective thermoelastic properties was obtained in ref. [8]. Three-phase unidirectional transversely isotropic composites were studied. Composites with hexagonal and square arrangements of cylindrical fibers were considered. The effective elastic properties and thermal expansion coefficients of unidirectional cylindrical fiber composites with imperfect interface conditions on the basis of the generalized Self Consistent Scheme model were evaluated in ref. [5]. Imperfect interface conditions were defined in terms of spring type coupling constants relating interface displacement discontinuities with tractions. In ref. [22], the thermal analysis of ceramic matrix composites (CMC) were performed. A Representative Volume Element model with periodic boundary conditions and a full-size model with the actual thickness were developed to investigate the temperature field, the heat flux field, and the effective thermal conductivity of the CMC. Ref. [23] considers the thermoelastic response of a composite body reinforced by coated cylindrical fibers and oriented in different directions. Lower bounds of the effective moduli were obtained, which provide a more accurate estimation of the overall composite properties. In ref. [2], a unit cell model was used to predict the effective thermo-mechanical properties of three-phase coated unidirectional cylindrical fibers through homogenization techniques for different fiber volume fractions. The numerical approach was based on the FEM. Longitudinal and transversal effective thermo-mechanical coefficients were calculated with the FEM and compared with analytical solutions based on the AHM. In the present study, thermoelastic effective properties are obtained for a thermoelastic isotropic or transversely isotropic fiber-reinforced composite. The composite is formed by fibers periodically distributed within a matrix and exhibits perfect contact at the interface between the fiber and matrix, which means that tractions and displacements are continuous across the interface [24]. The results are obtained by means of the SAFEM. The local problems obtained by the AHM are solved through the FEM.