Análise não linear utilizando uma abordagem isogeométrica de sólido-casca

Abstract

Shells are curved three-dimensional structures whose thickness is significantly smaller than their in-plane dimensions. They are widely used in various engineering fields, including aerospace, marine, and civil engineering. Several structural theories have been developed to analyze this type of structure, such as the Kirchhoff–Love and Reissner–Mindlin models. However, these formulations typically involve kinematic assumptions that include rotational degrees of freedom, which makes their development more complex for problems involving large rotations. In addition, shell theories often neglect transverse normal strain, which limits the use of full 3D constitutive models. To overcome these problems, this work adopts a solid-shell approach, in which continuum elements are modeled from the mid-surface of the shell and do not require rotational degrees of freedom in the kinematic description, as in solid elements, but with a lower computational cost. This formulation considers the complete 3D stress state, enabling the use of a full three-dimensional constitutive law. However, this approach can lead to Poisson locking, since an imbalance arises between the transverse normal stress and strain terms due to the assumed displacement field. To avoid this problem, two alternatives are adopted: one that simplifies the material constitutive matrix, and another that enriches the displacement field. The numerical solution is obtained using a NURBS-based isogeometric formulation, which employs the same basis functions used to define the geometry in CAD models to approximate the displacement field. This ensures an exact representation of the geometry and simplifies both h and p refinement. Furthermore, shells have characteristics that can lead to shear and membrane locking. Therefore, the influence of different numerical integration schemes on displacement convergence is investigated, as well as the effect of degree elevation in minimizing these locking phenomena. The accuracy of the proposed isogeometric formulations is evaluated through a series of linear and nonlinear examples involving plates and shells. The linear numerical examples show that the proposed formulations achieve good agreement with results reported in the literature and that the techniques used to avoid locking are effective. The nonlinear problems indicate that these formulations present good agreement with literature results up to high displacement levels, although they cannot describe the complete load-displacement curve in all cases.