This article is concerned with the development and application of a simple continuum theory for rocks that may contain both randomly as well as preferentially oriented plane discontinuity surfaces. The theory stipulates that displacement discontinuities are independently activated on these surfaces as soon as an appropriate yield criterion is fulfilled; these displacement jumps account for the irreversible, ‘plastic’ part of the bulk deformation. In stress space, the critical conditions for the activation of discontinuous slip or opening displacements define an overall yield envelope that could be initially anisotropic, reflecting for example a weakness of certain orientations due to pre-existing joint sets. For the yield conditions studied in this paper, essentially a Coulomb-type friction law and a simple fracture opening condition, the inferred stress-strain response under typical triaxial loading conditions reveals the sensitivity of the two discontinuous deformation modes to the confining pressure. The incipient growth of a geological fold in such a material is modelled as a problem of plate bending. The slip- and opening-modes of deformation are found to be activated typically in the fold intrados and extrados, respectively. Under certain conditions, both modes will be activated simultaneously at the same locality and contribute to the total deformation. Field observations on a well exposed sandstone anticline are reported here, which support this conclusion. The present plasticity model for discontinua can clearly be explored in more detail for realistic distributions of faults and joints taken from field observations. It could also be improved in various ways in its description of the underlying deformation mechanisms. Apart from its interest as a mechanical constitutive model, it can also serve as a point of departure for studies of stress-sensitive, anisotropic permeability distributions in fractured formations.