The behaviour of granular materials during silo filling is investigated with a polar hypoplastic model. The model is an alternative to elastoplastic constitutive models and can capture the salient features of granular bodies with consideration of shear localization. It is of the rate type, incrementally non-linear, and is established within the framework of a Cosserat continuum. It takes into account Cosserat rotations and couple stresses using the mean grain diameter as a characteristic length. Numerical simulations of silo filling with dry dense sand are performed with a Cosserat type approach implemented in a finite element method for model silos with parallel and slightly convergent walls. The silo walls are smooth, rough, and very rough. Because of the presence of a characteristic length in the constitutive model, the numerical results are independent of the spatial discretization. The results show that polar boundary conditions with consideration of Cosserat rotations and wall roughness realistically describe the interface behaviour between the silo wall and the silo fill. The Cosserat effect, which is manifested by the appearance of grain rotations, is only significant in the wall shear zone. A quantitative comparison between the numerical calculations and the model tests carried out in a plane strain bin and hopper shows satisfactory agreement. The advantages of the approach used are outlined.