Abstract
The unsteady biaxial tension-compression of a viscous rectangular beam infinite in the third direction is investigated. For the case in which the technological process of tension-compression in two directions is alternating, in particular, periodic (in time), we are talking about the reduction of the body. On the main stress-strain state, a picture of small perturbations is superimposed, which can be two-dimensional in the plane of the section of the beam, and can be three-dimensional. In either of these cases, linearized stability problems are posed in terms of perturbations. These problems are a kind of analogues of the Orr–Sommerfeld problem, which is formulated for the case in which the main flow is a one-dimensional plane-parallel shear in the layer. The analysis of the formulated problem is based on the method of integral relations, which is developed in the paper for unsteady basic processes. Sufficient integral (energy) stability estimates in the sense of Lyapunov and asymptotic stability are developed, from which exponential estimates follow. The domains of stability with respect to two-dimensional and three-dimensional perturbations are depicted on the plane of the two dimensionless parameters, whose meaning is that they are the Reynolds numbers constructed from two typical geometric sizes of the system.
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Georgievskii, D.V. Stability with Respect to Energetic Measures for Biaxial Tension–Compression of a Beam with Rectangular Cross-Section. Russ. J. Math. Phys. 28, 333–341 (2021). https://doi.org/10.1134/S1061920821030067
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DOI: https://doi.org/10.1134/S1061920821030067