Equations of the non-classical Cosserat continuum theory, which takes into account independent rotations along with the translational motion of the microstructure particles of the material, are generalized to describe irreversible plastic deformation under the action of intense mechanical disturbances that exceed the elastic limits. The mathematical model is formulated as a variational inequality for a differential operator, hyperbolic by Friedrichs, with one-sided inequality-type constraints on the variable functions. Based on the integral generalization of the variational inequality, a class of admissible discontinuous solutions with neutral and dissipative shock waves is analyzed. A priori estimates are obtained that guarantee the uniqueness and continuous dependence “in the small” by time of solutions of the Cauchy problem and boundary value problems with dissipative boundary conditions, including discontinuous solutions.
Keywords: moment continuum, elasticity, plasticity, strong discontinuity condition, neutral discontinuity, dissipative discontinuity, variational inequality
Received December 31, 2025
Revised January 21, 2026
Accepted January 26, 2026
Funding Agency: This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2026-735).
Vladimir Mikhailovich Sadovskii, Dr. Phys.-Math. Sci., Prof., Corresponding member of the RAS, Institute of Computational Modelling of the Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, 660036 Russia, e-mail: sadov@icm.krasn.ru
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Cite this article as: V.M. Sadovskii. Analysis of the mathematical model of a moment continuum that takes into account the irreversible deformation of a structurally inhomogeneous material. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2026, vol. 32, no. 1, pp. 224–235.
