Considerações sobre o comportamento de calotas esféricas abatidas compostas de materiais elásticos

Abstract

In this study, the behavior of a shell with a geometry described by a shallow spherical cap was analyzed, where the boundary condition involved clamping the circumference that forms its base. Both linear elastic and hyperelastic materials were used, represented by the Hooke and Neo-Hookean constitutive models, respectively. The relationships between strains and displacements were determined using the nonlinear shell theories of Novozhilov and Donnell- Mushtari-Vlasov (DMV), both adapted for spherical shells. To derive the equations based on energy models, the Rayleigh-Ritz method applied to the Potential Energy Functional was utilized. In this method, approximate functions were defined using trigonometric functions for the circumferential direction and the Legendre polynomial of the first kind for the meridional direction. The results showed that the natural frequency values were consistent with reference literature, and the nonlinear behavior obtained from the frequency versus amplitude relationship aligned with existing studies. Additionally, the static behavior was examined under a uniformly distributed load on the shell surface with a constant direction (dead load), producing curves similar to those reported in the literature for the shallow cap type considered. Finally, a parametric analysis of static and dynamic responses for different angles of the shallow cap was conducted, revealing changes in behavior with increasing angle. Moreover, it was noted that while physical nonlinearity has minimal impact on linear results, it significantly affects nonlinear results.