Integrability of the equilibrium and compatibility equations for a viscoplastic medium with negative strain rate sensitivity

The equilibrium and compatibility equations for viscoplastic medium with an arbitrary material function relating the stress intensity to the strain rate intensity is considered. A general form of the function ensuring complete integrability of two-dimensional equations has been found. The obtained function has an N-shaped (spinodal) graph and in particular cases corresponds to a linearly viscous liquid and perfectly plastic solid. A change of the strain rate sensitivity sign corresponds to a change in the type of the system and passing over the discontinuity line in a solid. The obtained function provides decoupling of the operator in a pair of two-dimensional subspaces where the equations are exactly linearized. The results of this study allows us to extend the class of integrable problems to so-called “active materials” (or “materials with internal dynamics”), which have aroused considerable interest.