The thermomagnetomechanical instability of the critical state in hard superconductors has been investigated, taking into account the dependence of critical current density j_c on magnetic field value H in a sample. The conditions for emergence of the thermomagnetomechanical instability have been found for an arbitrary critical state model. Samples with various geometries have been considered under different conditions of external cooling. The criteria obtained have been compared with the analogous ones found on the basis of Bean's critical state model (1914). It is shown that the dependence of j_c on H can reduce noticeably the critical state stability threshold.