In this work, crack initiation and propagation in 2D and 3D highly heterogeneous materials models, such as those obtained by micro-CT imagery of cementitious materials, is investigated for the first time by means of the phase field method. A shifted strain split operator algorithm is proposed to handle unilateral contact within cracks in a very efficient manner. The various advantages of the phase field method for voxel-based models are discussed. More specifically, we show that the resolution related to the initial image and thus to meshes for discretizing the same microstructure does not significantly affect the simulated crack path.