Continuum Mech. Thermodyn. (2009) 20: 429–458 DOI 10.1007/s00161-009-0092-6 ORIGINAL ARTICLE Lidiya Nazarenko · Leonid Khoroshun · Wolfgang H. Müller · Ralf Wille Effective thermoelastic properties of discrete-fiber reinforced materials with transversally-isotropic components Received: 31 October 2007 / Accepted: 10 December 2008 / Published online: 5 February 2009 © Springer-Verlag 2009 Abstract In the present paper, we will illustrate the application of the method of conditional moments by constructing the algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters of inclusions. A special case of two-component matrix composite with randomly distributed unidirectional spheroidal inclusions is considered. To this end it is assumed that the components of the composite show transversally isotropic symmetry of thermoelastic properties and that the axes of symmetry of the thermoelastic properties of the matrix and inclusions coincide with the coordinate axis x 3 . As a numerical example a composite based on carbon inclusions and epoxide matrix is investigated. The dependencies of Young’s moduli, Poisson’s ratios and shear modulus from the concentration of inclusions and for certain values which characterize the shape of inclusions are analyzed. The results are compared and discussed in context with other theoretical predictions and experimental data. Keywords Discrete-fiber composite material · Stochastic structure · Transversally-isotropic components PACS 81.40.Jj 1 Introduction The problem of finding the macroscopic properties of composites from those of its constituents is both practi- cally and theoretically important and currently attracts much attention. The corresponding theory for compos- ites with isotropic components is rather well developed, including different approaches to the problem with varying degrees of mathematical precision and physical relevance. An important direction in the theory of composite materials is the investigation of overall properties of randomly inhomogeneous media. The main three methods for studying randomly inhomogeneous media are based on perturbation theory, ad hoc assumptions to truncate a hierarchy, and variational principles. Pertur- bation theory works well for media whose properties vary only slightly from point to point. If composites are strongly heterogeneous the necessity of using other approximations is inevitable in practice. Thus exact esti- mates were determined for the effective properties of ad hoc models of composites by various authors; rigorous variational bounds were given for the properties of random composites, and precise definitions and explicit “homogenization” formulae were first proposed for properties of periodic composites and then developed for stochastic one’s (“stochastic homogenization” theory). A comparative analysis of different approaches, but by Communicated by V. Berdichevsky L. Nazarenko · W. H. Müller (B ) · R. Wille Technische Universität Berlin, Fakultät V, Lehrstuhl für Kontinuumsmechanik und Materialtheorie (LKM), Einsteinufer 5, 10587 Berlin, Germany E-mail: mcmt@mech2.pi.tu-berlin.de L. Khoroshun S.P. Timoshenko Institute of Mechanics of NAS Ukraine, P. Nesterov Street 3, Kiev 57 03057, Ukraine