Rock mass has a complex internal structure. Under the action of external loads, incompatible deformation develops in rock mass that makes the concept of elementary volume and the Saint Venant's condition of compatible deformation problematic. The incompatibility of the deformation indicates the non-coincidence of the internal and the external metrics and the breaking of the Euclidean structure of the space. Therefore, the use of differential geometry to describe the incompatible deformation is natural. At present, the expression of non-Euclidean models is not concise and compact; the constitutive relation is not complete and the evolution of incompatibility parameters is absent. In this study, with the help of the orthogonal frame method, we used the generalized distortion tensor, torsion tensor, and Riemann tensor to describe the incompatible deformations. The used geometric parameters have clear physical meaning and can be used as thermodynamic variables. By constructing Helmholtz free energy and using irreversible thermodynamics, we obtained the constitutive equations. For completing the constitutive equations, the evolution equations of the used geometric parameters are derived. In this manner, the non-Euclidean description of the incompatible deformation extends the classical models of deformable bodies.