Arch. Mech., 70, 3, pp. 207–239, Warszawa 2018 SEVENTY YEARS OF THE ARCHIVES OF MECHANICS Elastic properties of a unidirectional composite reinforced with hexagonal array of ﬁbers T. CZAPLINSKI 1) , P. DRYGAŚ 2) , S. GLUZMAN 3) , V. MITYUSHEV 4) , W. NAWALANIEC 4) , G. ZIETEK 1) 1) Department of Mechanics, Materials Science and Engineering Faculty of Mechanicsl Engineering Wrocław University of Science and Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland e-mails: tomasz.czaplinski@nobosolutions.com, grazyna.zietek@pwr.edu.pl 2) Department of Diﬀerential Equations and Statistics Faculty of Mathematics and Natural Sciences University of Rzeszów Pigonia 1, 35-959 Rzeszów, Poland e-mail: drygaspi@ur.edu.pl 3) Bathurst 3000 Toronto M6B 3B4 Ontario, Canada e-mail: simon.gluzman@gmail.com 4) Faculty of Mathematics, Physics and Technology Pedagogical University Podchorążych 2, 30-084 Kraków, Poland e-mails: mityu@up.krakow.pl, wnawalaniec@gmail.com It is difficult to measure the transverse shear modulus of the ﬁbrous composites. Thus, theoretical investigations by means of analytical and numerical techniques are paramount. In particular, they are important for the regime with high-concentration of ﬁbers. We apply general techniques to study the mechanical properties of unidirectional ﬁbers with a circular section embedded into the matrix and organized into the hexagonal array. Our theoretical considerations are designed to include two regimes, of low and high concentrations of inclusions. The former regime is controlled by Hashin–Shtrikman lower bounds, while the latter is controlled by square-root singularity. We derived the analytical formulae for the eﬀective shear, Young and bulk moduli in the form of the rational expressions valid up to O(f 7 ) by the method of functional equations. The obtained formulae contains elastic constants of components in a symbolic form as well as the concentration f . The general scheme based on the asymptotically equivalent transformations is developed to extend the obtained analytical formulae to the critical concentration of touching ﬁbers. A com- parison with the numerical FEM is performed for all concentrations of inclusions. Good agreement is achieved for all available concentrations. Key words: eﬀective elastic properties, hexagonal array, asymptotically equivalent transformation. Copyright c  2018 by IPPT PAN