 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 13217 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8152
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| In English (Mech. Solids): | | 5065 |
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| N.I. Ostrosablin, "On the Cauchy Strain Tensor, Compatibility Conditions, and Defining Equations of an Elastic Medium," Mech. Solids. 60 (3), 1625-1639 (2025) |
| Year |
2025 |
Volume |
60 |
Number |
3 |
Pages |
1625-1639 |
| DOI |
10.1134/S0025654424605317 |
| Title |
On the Cauchy Strain Tensor, Compatibility Conditions, and Defining Equations of an Elastic Medium |
| Author(s) |
N.I. Ostrosablin (Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, o.n.ii@yandex.ru) |
| Abstract |
Using the example of four-dimensional equilibrium equations for kinetic stresses in Eulerian rectangular coordinates, it is shown that the operator of the four-dimensional Cauchy strain tensor is conjugate (transposed) to the operator of the equilibrium equations. The same connection
between the operators of the equilibrium equations and the Cauchy strain tensor also holds in the
three-dimensional case. Three variants of the derivation of the conditions for the compatibility of
Cauchy deformations are given. In the four-dimensional case, there are 21 compatibility conditions,
and in the three-dimensional case, there are six Saint-Venant compatibility conditions. It is shown
that the Cauchy strain tensor, both in Eulerian and Lagrangian variables, completely determines the
deformed state of a continuous medium. At the same time, no restrictions on the amount of displacements, deformations or rotations are required. The Lagrange-Green and Euler-Almancy tensors, the
so-called large or finite deformations, and the displacements are expressed using Cesaro formulas in
terms of the Cauchy strain tensor. The defining equations of an elastic continuous medium relate the
Cauchy true stress tensor and the Cauchy strain tensor one to another. Using proper bases in the spaces
of symmetric stress and strain tensors, the de ning relations can be written as six separate independent
equations containing functions of only one argument. For continuous media with crystallographic
symmetries, we can use the bases obtained on the basis of the generalized Hooke’s law. |
| Keywords |
kinetic stresses, Cauchy strain and stress tensors, four-dimensional continuous medium, Lagrangian and Euler variables, compatibility conditions, Lagrange-Green and Euler-Almancy tensors, Cesaro formulas, defining equations, eigenstates |
| Received |
10 September 2024 | Revised |
07 November 2024 | Accepted |
10 November 2024 |
| Link to Fulltext |
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