Abstract
The paper deals with the problems of the tensor element of volume (area) of an M-cell for a manifold immersed in a “plane” multidimensional space. The corresponding considerations imply the choice of the reper orientation, which determines the volume (area) element. The latter circumstance is of exceptional importance in micropolar theories of elasticity. The indicated theories can be correctly developed only within the framework of the pseudotensor formalism. This is especially true for the theory of hemitropic elastic continua. The governing information from the algebra of pseudotensors is presented. Manifolds given by Gaussian parametrization are considered. The concept of an M-cell on a manifold and its orientation is introduced. An algorithm for comparing the orientations of M-dimensional cells is described. The concept of the tensor element of volume (area) for the M- cell is determined. The formula for transformation a natural volume element to an invariant volume element is shown. Important applications for the mechanics of the micropolar continuum are noted and discussed.
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Notes
Several names are used in the literature for permutation symbols: discriminant tensor, permutation tensor, alternating tensor, Levi-Civita symbols.
This circumstance is implicitly used in the overwhelming number of publications devoted to micropolar elasticity, where the presentation is carried out in terms of absolute tensors [13–15]. In the case when the parameters of the material are sensitive to transformations of specular reflections and inversions of space, i.e., for semi-isotropic (hemitropic) media, a correct description is possible only when using the algebra of pseudotensors [3, 4].
Then many objects on an M-dimensional manifold can be interpreted as vectors in a “plane” space with a natural Euclidean metric.
In manuals on vector analysis and finite-dimensional spaces [18–20], a transition matrix transposed to (3.7) is sometimes introduced. We follow the definition given in [18].
According to more archaic terminology (see [11]), the term “the extension of the M-cell” can also be used.
A literary search shows that the idea of using pseudo-invariant elements of volume and area in determining the force stress tensor tik, apparently belongs to J. A. Schouten (see [11, c. 201—204]).
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Funding
This work was carried out within the framework of a state assignment (state registration No. АААА-А20-120011690132-4) and with the support of the Russian Foundation for Basic Research (project No. 20-01-00666).
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Translated by I. K. Katuev
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Murashkin, E.V., Radaev, Y.N. ON THEORY OF ORIENTED TENSOR ELEMENTS OF AREA FOR A MICROPOLAR CONTINUUM IMMERSED IN AN EXTERNAL PLANE SPACE. Mech. Solids 57, 205–213 (2022). https://doi.org/10.3103/S0025654422020108
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DOI: https://doi.org/10.3103/S0025654422020108