The analytical and numerical modeling of the thermomechanical behavior of composite materials has become important over the past decades due to the application of such materials in many fields. The models typically assume that the different length scales present in the composite and its constituents are sufficiently well spread that sequential homogenization can be applied without problems. The present master thesis focusses on a special, highly idealized case in which two such length scales are only weakly separated, a polycrystalline matrix reinforced by fibers of similar size to the matrix grains being studied. On the one hand, the effective elastic behavior is described by onestep models, which concurrently account for the inhomogeneities and the resolved grain structure of the matrix. On the other hand, twostep approaches are used, in which the matrix is homogenized first and the overall response of the composites is described by fibers embedded in the resulting effective matrix material. The grains are idealized as regular hexagons, the fibers being positioned in their centers or in the triple points. Five different types of elastic, fiberlike inhomogeneities are studied by unit cell models and, in addition, HashinStrikman bounds are evaluated. The phase geometries are modeled and meshed with ABAQUS/CAE, random orientations of the grains are generated by a Python script, boundary conditions of the model and loadcases are obtained with the program package MedTool, and linear finite element simulations are performed with finite element program ABAQUS/Standard. Finally, the elasticity tensors and elastic moduli of all cases are evaluated with MedTool and analyzed, typically in terms of ensemble averages. The results show that the relative differences between the singlestep and twostep macroscopic elasticity tensors are smaller in the "normal part" of the elasticity tensor whereas larger differences are present in its "shear part". Ranges of the von Mises stresses and first principal stresses in fibers and matrix are considerably larger for onestep than for twostep homogenization. As to the HashinStrikman bounds, effective moduli evaluated in this work with the isotropized, homogenized matrix behavior either fall within the bounds or closely approach the lower or the upper bounds, for composited reinforced by stiff fibers or composites reinforced by inhomogeneities more compliant than the matrix, respectively. The closest transversally isotropic elasticity tensors obtained from the ensemble averages of the onestep and twostep results, however, in many cases do not fulfill the bounds. Taken together, the results show that sequential homogenization entails a loss of accuracy in situations where length scales are weakly spread.
