In this paper, mixed formulations are presented in the framework of isogeometric Reissner–Mindlin plates and shells with the aim of alleviating membrane and shear locking. The formulations are based on the Hellinger-Reissner functional and use the stress resultants as additional unknowns, which have to be interpolated in appropriate approximation spaces. The additional unknowns can be eliminated by static condensation. In the framework of isogeometric analysis static condensation is performed globally on the patch level, which leads to a high computational cost. Thus, two additional local approaches to the existing continuous method are presented, an approach with discontinuous stress resultant fields at the element boundaries and a reconstructed approach which is blending the local control variables by using weights in order to compute the global ones. Both approaches allow for a static condensation on the element level instead of the patch level. Various numerical examples are investigated in order to verify the accuracy and effectiveness of the different approaches and a comparison to existing elements that include mechanisms against locking is carried out.
One of the biggest challenges in fracture modeling is the correct physical description of the fracture processes. Over the past decade, the well‐established phase‐field modeling framework has gained a lot of attention, due to its ability to predict the fracture phenomena adequately. It has shown very promising results for a three dimensional solid formulation and has been extended to an isogeometric Kirchhoff‐Love formulation in order to describe brittle fracture in plates and shells, see [1]. However, the Kirchhoff‐Love theory only describes shells in the thin regime. For thick shells, the deformation behavior and subsequently the fracture behavior additionally depends on the transverse shear strains which are not present in the Kirchhoff‐Love approach. In this work, the phase‐field fracture framework is extended to an isogeometric Reissner‐Mindlin shell formulation [2]. In this approach, the shell is described using the midsurface and a director vector field for the thickness direction. Subsequently, the phase‐field is also defined only on the midsurface. In order to distinguish the cracking behavior in tension and compression, as proposed in [1], the spectral decomposition of the strain for the tension‐compression split is done on the total strain, which varies through the thickness. The proposed method is tested for several numerical examples and compared to a three dimensional solid formulation and the Kirchhoff‐Love shell formulation in order to confirm its accuracy and efficiency.
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...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.