Incompatible Deformations in Relativistic Elasticity

• Published in: Lobachevskii journal of mathematics 2023-06, Vol.44 (6), p.2352-2397
• Main Authors: Lychev, S. A., Koifman, K. G., Pivovaroff, N. A.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Crusts; Curvature; Defects; Deposition; Differential geometry; Evolution; Geometry; Manifolds (mathematics); Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Neutron stars; Probability Theory and Stochastic Processes; Relativity; Spacetime; Vector spaces
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080223060343

The work develops differential-geometric methods for modeling incompatible deformations and stresses caused by them in hyperelastic supermassive bodies within the framework of special and general relativity. Particular attention is paid to a unified geometric language used both for modeling distributed defects and gravitational interaction. It is shown that finite incompatible deformations that do not evolve in time can be formalized in terms of the curvature or torsion of the connection on the 3D material manifold obtained as the image of the projection of the body world tube, while the gravitational interaction is formalized in terms of the curvature and torsion of 4D space-time manifold. To describe the evolution of the material composition of a body, which occurs, for example, as a result of accretion, instead of a 3D material manifold, one should consider a body-tube in a 4D material space. In this case, from a formal mathematical point of view, the description of distributed defects and gravity becomes equivalent, and both, material manifold and space-time manifold are derived from a single fundamental object, which is a four-dimensional topological vector space. The developed approach can be used to model the processes of evolution of the stress-strain state of neutron star crusts and their local spontaneous destruction.