Separate control meshes for displacements and rotations for a shear locking free isogeometric Reissner‐Mindlin plate

Abstract

In the analysis of plate and shell bending problems using an isogeometric Reissner-Mindlin approach, transversal shear locking effects may occur especially for thin structures. One possibility to overcome locking effects is to increase the polynomial order of the NURBS basis functions. However, there are certain examples where this method shows some deficiencies, like oscillations. For low polynomial degrees, there exist only a few effective concepts for the elimination of locking effects. One is the enhanced assumed strain (EAS) method which is used in finite element formulations and which is sensitive to distorted element geometries. Beirão da Veiga et al.[1] introduced a new approach for a Reissner-Mindlin plate formulation where the displacements and rotations of the mesh are approximated using different control meshes. The physical space of the structure always remains the same. Hence, the method is in accordance to the isoparametric paradigm. However, the shape functions for the approximation of the displacements and the rotations may have different polynomial degrees and number of control points. In this way, the compatibility requirement for pure bending is fulfilled and shear locking is avoided. The method is tested for an isogeometric Reissner-Mindlin plate formulation, which is based on a degenerated shell formulation [2]. Basic examples are chosen and the results are compared to the unaltered isogeometric Reissner-Mindlin plate and the finite element Method using MITC elements. The results show that the method has similar accuracy and efficiency as the MITC element and is also applicable for skew element geometries. The used plate element is derived from continuum theory and is only described by its midsurface. The thickness direction is defined by the director vector [2]. Since the considered examples are only linear elastic problems, the update of the director vector is given as d = D + b, where b T = β 1 β 2 0 . Using the linear strain tensor we get the following strains for the plate,where the first three entries denote the change of curvature and the last two the transversal shear. Finally, using Hooke's law for the definition of the stress resultants σ, the following weak form of equilibrium occurswhere the solution variables u T = w β 1 β 2 include the vertical displacement and the two rotations. Adjusted approximation spaces for the treatment of transversal shear locking effectsIn pure bending problems of thin structures, two compatibility requirements must holdSince the equations include the derivative of w in combination with the rotations β i , the use of the same shape functions for w and β i would lead to not conforming interpolations. This mismatch is the reason for the transversal shear locking effects. A very simple way to overcome this difficulty would be to use adjusted approximation spaces for the displacement and the rotations in order to fullfill the compatibility requirements and to avoid the coupling of shear strains and curvature [1]. The different control meshes are construct...