A theory is suggested for rock cutting resistance based on a model of cutting in a brittle mode, i.e. with a latent internal sliding crack. The theory considers any plastic, viscous and other irreversible strains for a body subjected to cutting;. and also any spatial configuration of the cutter. The maximum cutting force in relation to cutting depth, cutting tool shape, initial stressed state, and rock properties is worked out using the invariant Γ-integral. It is established that the maximum cutting force in a brittle mode is only governed by the sliding viscosity of the rock and it does not depend on its other elastic and strength characteristics. A simpler equation derived by the authors previously emerges in the particular case of a plane-stressed or plane-strained sliding viscosity of solids from the results of experiments with cutting by any cutting tool of spatial configuration. Some results are given for the experiments of cutting typical sedimentary rocks.