Um estudo do desempenho de elementos finitos da elasticidade plana

Abstract

This work makes a study of the performance of rectangular finite elements in plane elasticity. Four-, eight- and nine-node elements formulated using the isoparametric formulation and the strain gradient notation (SGN) are compared when applied to the solution of different problems. The influence of spurious terms, called parasitic shear and which are responsible for the modeling error known as shear locking, is investigated. For good performance of the elements, modeling errors should be avoided. The elimination of spurious terms in isoparametric elements is accomplished using numerical techniques reduced, uniform or selective, according to the finite element employed. In the high-order elements, in this case, the eight- and nine-node elements, these techniques function only partially. By eliminating spurious terms, reduced integration techniques also eliminate legitimate terms that make up the equation by introducing null energy spurious modes, that is, another modeling error. Spurious terms in strain gradient elements are precisely identified because the strain gradient notation is a physically interpretable notation, which reveals the physical meanings of the polynomial coefficients. Thus, the spurious terms can be removed from the deformation polynomials, making the elements free from modeling errors. It is emphasized that spurious terms are correctly eliminated, that is, without the introduction of spurious modes of zero energy. These procedures are described in this work and validated in the numerical features presented. In this numerical analysis, we also investigate the influence of the aspect ratio of the elements, the geometry of the mesh used, and the variation of the Poisson coefficient in the approximation of the results, as well as the convergence with the uniform and non- uniform refining of the mesh. The analysis shows the equivalence between the isoparametric formulation with complete integration and the Strain gradient notation with spurious terms in elements with four, eight and nine nodes. For the four-node element, uniform reduced integration was shown to be ineffective in solving the proposed problems, while the selective reduced integration of the isoparametric formulation showed equivalence in the results to strain gradient notation with no spurious terms. For higher order elements, eight and nine nodes, strain gradient notation showed better performance than the isoparametric formulation when the aspect ratio of the element is close to one, for problems with Poisson coefficient close to zero. When the Poisson coefficient is increased, the results of strain gradient notation become better also for elements whose base is more elongated with respect to height too.