We analyze the transition from the nonlinear elastic behavior to the elasto-plastic one. A specific class of what we call hyperbolic elasticity arises from theoretical considerations as a straight consequence of the yield criterion invariance on plasticity level. The latter property of fixed yield residual plasticity is observed experimentally for many geomaterials. We superimpose this nonlinear elastic (or hyperelastic) hypothesis on the plastic constitutive relation. Curiously, we found that the hyperbolic nonlinearity, introduced through the Standard Generalized Material formalism, affects the residual yield surface shape. In particular, the initially linear surface curves to a quadratic one, establishing correspondence between Mohr–Coulomb and Hoek–Brown (or alternatively between Drucker–Prager and Pan–Hudson) criteria. Our first conclusion is that one possible justification of the empirical fitted Hoek–Brown criterion widely used in geoscience is the material’s hyperelastic nature. We compare further elasto-plastic responses of standard tests for Drucker–Prager constitutive relation with both linear and nonlinear elastic phases. Most notably, the nonlinear case distinguishes itself by the dilatancy saturation and accommodation phenomena during cyclic loading in the triaxial compression test. This rather complex experimentally observed behavior emerges straight from the nonlinear unloading property in the elasto-plastic description.