The stress characteristics in the anisotropic bicrystal and tricrystal specimens were analyzed using the anisotropic elastic model, orthotropic Hills model and rate-dependent crystallographic model. The finite element analysis results show that non-uniform stresses are induced by the grain boundary. For bicrystal specimens in different crystallographic orientations, there exist stress concentrations and high stress gradients nearby the boundaries. The activation and slipping of the slip systems are dependent on the crystallographic orientations of the grains and also on the relative crystallographic orientations of the two adjoining grains. For the tricrystal specimens, there is not always any stress concentrations in the triple junction, and the concentration degree depends on the relative crystallographic orientations of the three grains. Different from the bicrystal specimens, there may be or no stress concentration in the vicinity of grain boundaries for the tricrystal specimens, which depends on the relative crystallographic orientations of the three grains. The stress concentration near to the grain boundaries and triple junction can be high enough for the local plastic deformation, damage and voiding or cracking even when the whole specimen is still under the elastic state.It can be further concluded that homogeneous assumption for polycrystalline materials is not suitable to study the detailed meso- or micro-mechanisms for damaging and fracturing.