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The paper describes the modeling process of the magnetomechanical properties of magnetorheological elastomers. Unlike the primary sources in which a viscoelastic model is commonly used, in this work, the elastic–plastic model with linear hardening was adopted. Its parameters were determined from data obtained in an experiment. Measurement data encompassed a range of various strain rates and amplitudes, as well as a range of magnetic field values. Model parameters as a function of a magnetic field were obtained as a result of identification. The correctness of data correlation was shown by comparing hysteresis loops τ ( γ ) . Satisfactory consistency between experimental and model research was achieved on the assumption that the model was applied only to higher strain rates, above the boundary value γ ˙ A = 0.025 ( 1 / s ) .