From pharmaceutical to mining or traveling desert dunes to earthquakes, granular materials are at the heart of many industries and natural phenomena. Improving the efficiency of the machines handling them or, constructing safer buildings requires a critical understanding of their behavior. However, this is not a straightforward task as opposed to what one might think due to the abundance of particulate matter. From a fundamental point of view, it has been only recently realized that they cannot be easily classified as a solid or liquid or even a gas as they are able to mimic all of these states under slightly different conditions. The challenge of the scientific research today, is to establish the link between the collective behavior and properties of individual particles composing granular materials.Such a relation would enable to characterize them with only a few parameters in contrast to billions of particles typically found in practice. In the first part of this thesis, we study the mechanical behavior of idealized sphere packings with discrete element simulations. The polydispersity and coefficient of friction of the particles are varied systematically to characterize their influence on the macroscopic stress strain response. In isotropically deformed packings, the critical volume fraction marking the transition from a solid to fluid like state increases with polydispersity, whereas it decreases with the coefficient of friction. The coordination number, i.e. average number of contact per particle, is discontinuous at this density. During decompression it drops from its isostatic value to zero and obeys a power law at higher volume fractions. The effect of polydispersity on the pressure is determined by the ratio of critical volume fraction and the contact density which is equal to the trace of the fabric times a correction factor that depends only on the moments of the particle size distribution. Using the micromechanical definition of the stress tensor, we derive an incremental constitutive model for the pressure which includes changes of fabric. With one fit parameter the linear regime of lower pressure is described, while with two parameters, the model captures well the non-linear pressure evolution in isotropically deformed polydisperse, frictionless and frictional packings. Anisotropic deformations are studied with triaxial test simulations. The shear strength of the packings is measured by the deviatoric stress ratio which first increases then saturates with increasing particle coefficient of friction. Volumetric strain also depends on the particle friction albeit in a non monotonic way. The maximum compaction after which packings start to dilate, is achieved at a relatively small coefficient of friction. The stress strain response depends indirectly on the polydispersity which determines initial packing conditions. When initially the volume fraction is fixed, the pressure as well as the shear strength decrease with polydispersity. The opposite is observed when the initial pressure is imposed, although the effect of polydispersity on the stress-strain behavior is less significant in this case. Finally, a hypoplastic constitutive model is calibrated with simulation results and the resulting material coefficients are related to particle properties. Most granular materials are amorphous and disordered as realized up to now. However,crystal structures can be built by placing uniform particles on a regular lattice. The second part of the thesis is about pattern transformation in two-dimensional granular crystals composed of bi-disperse soft and hard cylindrical particles. We show with experiments and simulations that upon uniaxial compression the particles undergo structural rearrangements from an initial square to hexagon-like lattice. It is found that the characteristics of the transformation strongly depend on the size ratio of the particles rather than their material properties. If the ratio is small enough the transformation is homogeneous and practically reversible. The band structure of the granular crystal changes due to the pattern transformation. Using a linearized contact force model, we compute the dispersion relation at different levels of deformation and show that band gaps open and close as the structure of the crystal changes.This could find applications in tunable acoustic devices such as filters or vibration isolators. In short, this thesis concerns the mechanics of granular materials subject to different modes of deformation. The constitutive behavior of disordered sphere packings and pattern transformation in regular arrays of cylinders have been studied.