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Finite Incompatible Deformations in Elastic Solids: Relativistic Approach

Lobachevskii Journal of Mathematics Aims and scope

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Abstract

The Lagrangian approach for modeling of relativistic elastic solid with distributed defects is developed. The influence of defects on stress-strain state of the solid is taken into account within differential-geometric framework, which allows to describe the corresponding deformation incompatibility in terms of material connection curvature. The specifity of the developed approach is in four-dimensional relativistic formulation of body-tubes and higher-dimensional formulation for rigged body-tubes represented as total spaces of some principal bundles. This approach makes it possible to account the micropolar and micromorphic behavior of relativistic material particles. The generalized deformation is considered as embedding of a smooth manifold that represents rigged body-time into another smooth manifold that represents rigged space-time. Specific technique for constructing of rigged space-time in terms of fiber bundles is developed. The unified variation formulation is suggested for wide range of micromorphic solids. Particular forms for Lagrangian density aligned with general covariance principle are proposed. The considered approach may be applied for the modelling of accretion processes in large-scaled astrophysical objects.

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Notes

  1. In what follows the conventional designation \(\overrightarrow{{{{a}}}{{{b}}}}\) for the value \(\textrm{vec}({{{a}}},\>{{{b}}})\) will be used. The notation for structure and its underlying set will not be distinguished typographically

  2. The family \((\mathfrak{e}^{\alpha})_{\alpha=0}^{3}\) is basis for \(\mathcal{V}^{\ast}\), dual to \((\boldsymbol{e}_{\alpha})_{\alpha=0}^{3}\).

  3. Note that the curve is given in a special parameterization. From general viewpoint, one can distinguish between paths and curves, where the former are smooth mappings of an real axis interval into a manifold, while the latter are images of paths, i.e., sets of points. In what follows, we will not focus on this.

  4. With assumption that \(\dim\mathfrak{B}=3\). In the case if \(\dim\mathfrak{B}<3\) further reasoning is required.

  5. Usually natural coordinates are listed in the following order: first are coordinates on the base manifold and second are coordinates on the model fiber. However, dealing with body time manifold is is appropriate to use the reverse order, which is consistent with coordinates on Minkowski spacetime (the first coordinate is temporal).

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Funding

The study was partially supported by the Government program (contract #AAAA-A20-120011690132-4) and partially supported by the grant RSF No. 22-21-00457. Russian Science Foundation. https://rscf.ru/en/project/22-21-00457/

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    S. Lychev & D. Bout

  2. Bauman Moscow State Technical University, 105005, Moscow, Russia

    K. Koifman

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  1. S. Lychev
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  2. K. Koifman
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  3. D. Bout
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Correspondence to S. Lychev, K. Koifman or D. Bout.

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(Submitted by A. M. Elizarov)

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Lychev, S., Koifman, K. & Bout, D. Finite Incompatible Deformations in Elastic Solids: Relativistic Approach. Lobachevskii J Math 43, 1908–1933 (2022). https://doi.org/10.1134/S1995080222100250

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