The following is a description of a new computer‐simulation technique for the quasi‐static particle mechanics of granular media and its application to the dilatant simple shear of periodic 2D arrays of nearly rigid frictional disks. By means of supercomputer implementation, we have investigated the effects of (Coulomb) particle friction and polydispersity on Reynolds dilatancy and plasticity of a granular mass subject to a constant isotropic confining pressure. The simulation employs deforming periodic‐cell boundary conditions, with subcells arranged to reduce the contact‐search effort, together with a quasi‐static motion algorithm which converts an assumed global deformation to a local forcing of particle motion. Nearly identical shearing dilatancy s = dε V /dγ is obtained for samples of 56, 132, and 306 polydisperse disks under identical conditions. Our computed dilatancy s=0.81 for the monodisperse assemblage obtained in this study is larger than the Reynolds‐type prediction of 0.5 for isotropic assemblages but is in good agreement with the theoretical value 0.87 which we obtain by a proper accounting for crystalline anisotropy. Our value of 0.35 for the polydisperse assemblage is in fair agreement with Kishino’s (1983) experimental value 0.4, Thornton and Barnes’ (1986) simulation value 0.3, and Bardet and Proubet’s (1990a, 1990b) simulation value of 0.37. In accordance with the prediction of Reynolds (1885) and with the results from previous experiments and simulations, it is found that particle–particle friction has no significant effect on dilatancy. We also present results on the shear and normal stresses, as well as the particle‐contact topology. The rheological behavior is found to be roughly in accord with the Reynolds–Rowe stress dilatancy picture of plastic yield and generally exhibits noncoaxiality of stRess and strain‐rate tensors. Both mono‐ and polydisperse disk asseMblages eventualLy te.d to devorm at constant rolume (area) at roughly the same solid fraction φ≊0.65 and at the geometric bond percolation threshold for particle contacts.