Hellinger–Reissner principle based stress–displacement formulation for three-dimensional isogeometric analysis in linear elasticity

Abstract

In case of Lagrangian finite element formulation, three-dimensional (3-D) stress-based hybrid solid elements have shown excellent coarse mesh accuracy for a wide range of applications. However, to the best of our knowledge, there is no work available towards the development of 3-D stress-based hybrid solid elements for isogeometric analysis (IGA). In this work, we propose stress-based hybrid solid elements to alleviate the issue of locking arising in non-uniform rational B-spline (NURBS)-based IGA. The present work primarily focuses on the linear elasticity, though the formulation can be extended to the non-linear regime. We believe that the excellent coarse mesh accuracy provided by the proposed elements will further enhance the IGA in various applications, especially involving structures with high aspect ratios and nearly incompressible materials. The proposed elements are constructed based on a two-field Hellinger–Reissner variational statement, where stress and displacement fields are interpolated separately. The stress interpolation functions have been derived systemically for various orders of NURBS displacement interpolation functions. Further, we show mathematically that the choice of the stress interpolation functions is free from spurious zero-energy modes. Lastly, the results of numerous 3-D linear-elastic benchmark problems are presented to demonstrate the efficacy and robustness of the proposed elements. The results confirm the superior coarse mesh accuracy for structures with high aspect ratios and almost incompressible materials.