### Bifurcating solutions in a constrained minimization problem of elasticity

#### Resumo

We consider the problem of a cylindrically anisotropic disk subject to an imposed displacement on its boundary. In the context of classical linear elasticity, the solution of this problem is not locally injective. This characterizes material overlapping, which is not physically admissible. To prevent this anomalous behavior, we minimize the energy functional of classical linear elasticity subject to the local injectivity constraint. One possible solution of the associated Euler-Lagrange equations, reported in the literature, is radially symmetric. In this work we search for a secondary solution, which is rotationally symmetric. In the region where the constraint is not active, we determine closed-form expressions for the displacement field. In the region where the constraint is active, which is annular, we determine a relation between the components of the displacement field to ensure the imposition of this constraint. The expressions obtained in both regions depend on constants of integration that are determined numerically. In addition, we also determine the inner and outer radii of the active annular region. This research is of interest in the investigation of solids with radial microstructure, such as carbon fibers.

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Aguiar, A. R. Local and global injective solution of the rotationally symmetric sphere problem, Journal of Elasticity, 84:99-129, 2006. DOI: 10.1007/sl0659-006-9058-0.

Aguiar, A. R. and Rocha, L. A. Numerical investigation of bifurcation instability in constrained minimization problem of elasticity, Proceedings of the Ibero-Latin-American Congress on Computational Methods in Engineering, 2020. ISSN: 2675-6269.

Christensen, R. M. Properties of carbon fibers, Journal ofthe Mechanics and Physics of Solids, 42:681-695, 1994. DOI: 10.1016/0022-5096(94)90058-2.

Fosdick, R., Freddi, F. and Royer-Carfagni, G. Bifurcation instability in linear elasticity with the constraint of local injectivity, Journal of Elasticity, 90:99-126, 2008. DOI: 10.1007/sl0659-007-9134-0. 5

Fosdick, R. and Royer-Carfagni, G. The constraint of local injectivity in linear elasticity theory, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 457:2167-2187, 2001. DOI: 10.1098/rspa.2001.0812.

DOI: https://doi.org/10.5540/03.2021.008.01.0380

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