# Locking and brittle fracture in isogeometric Reissner-Mindlin plate and shell analysis

Kikis, Georgia; Klinkel, Sven (Thesis advisor); De Lorenzis, Laura (Thesis advisor); Dornisch, Wolfgang (Thesis advisor)

*Aachen : Rheinisch-Westfälische Technische Hochschule Aachen, Fakultät für Bauingenieurwesen, Lehrstuhl für Baustatik und Baudynamik (2022)*

Book, Dissertation / PhD Thesis

In: Schriftenreihe des Lehrstuhls für Baustatik und Baudynamik der RWTH Aachen 15 (2022)*Page(s)/Article-Nr.: 1 Online-Ressource : Illustrationen, Diagramme*

Dissertation, RWTH Aachen University, 2022

Abstract

The present work focuses on two main topics, the treatment of locking effects in the framework of an isogeometric Reissner-Mindlin shell formulation and the correct description of brittle fracture in Reissner-Mindlin plates and shells using a phase-field model. In both cases the geometry is described by the mid-surface of the structure with Non-Uniform Rational B-Spline (NURBS) basis functions that are common in CAD tools and a director vector field is used for the description of the thickness direction. Since only small deformations are considered, the director vector is updated using a difference vector formulation. In addition to the three displacements, two rotational degrees of freedom that account for the transverse shear effects are defined.Regarding the first objective of treating locking in the framework of isogeometric analysis, the focus lies on the two main locking effects that occur in the present Reissner-Mindlin shell formulation, namely, transverse shear locking and membrane locking. These undesirable effects lead to an artificial stiffening of the system, an underestimation of the deformation and oscillations in the stress resultants. They are intensified with a decreasing thickness, i.e. in the Kirchoff limit. In a first step, a method to eliminate transverse shear locking in plates and shells is introduced. The method is based on the fact that transverse shear locking occurs due to a mismatch of the approximation spaces of the displacements and rotations in the strain formulation. Thus, adjusted approximation spaces are defined for the two rotations, namely, their basis functions are in the relevant direction one order lower than the ones of the displacements. The three different control meshes are created using the same starting geometry and applying different degrees of refinement. This way, the isogeometric concept still holds. The meshes have the same number of elements and together they form the global mesh which is used in the weak formulation. The efficiency and accuracy of the method is assessed with the help of numerical examples. The results highlight the superior behavior of the method compared to the standard Reissner-Mindlin shell formulation without any anti-locking measures. Oscillations in the stress resultants are eliminated and the method is shown to be competitive with other methods used in isogeometric analysis against locking. It is generally applicable for any polynomial degree and leads to less degrees of freedom in the system of equations compared to the standard shell formulation. In a second step, a displacement-stress mixed method based on the Hellinger-Reissner variational principle is proposed in order to alleviate both membrane and transverse shear locking in plates and shells. The stress resultants that are related to these locking effects are considered to be additional unknowns and have to be interpolated with carefully chosen shape functions. Namely, in the relevant direction, one order lower splines are chosen for the stress resultants than for the displacements and rotations. The additional unknowns that are used in mixed formulations are in general eliminated from the resulting system of equations using static condensation. In contrast to the classical finite element method where C0-continuous shape functions are used and static condensation is performed on the element level, in isogeometric analysis the high continuity of splines does not allow that anymore. Static condensation has to be performed on the patch level, which includes the inversion of a matrix on the patch level and leads to a fully populated stiffness matrix. This on the other hand increases the computational cost and thus, two local approaches are proposed that enable static condensation on the element level. The first one includes stress resultants that are defined discontinuously across the element boundaries and leads to a sparse matrix that has the same bandwidth as the standard displacement-based shell formulation. It is shown that this method improves the results for low polynomial degrees and is attractive due to its low computational cost. However, because of the discontinuity of the stress resultant fields locking is not completely eliminated and the results are not greatly improved for higher polynomial degrees. In the second local approach, a reconstruction algorithm is used and the local control variables are weighted in order to compute blended global variables. In the numerical examples it is shown that this method has almost the same accuracy as the global approach on the patch level, however, in contrast to that it leads to a banded stiffness matrix and is computed partly on the element level, thus, reducing the overall computational cost. The mixed continuous approach on the patch level and the mixed reconstructed approach are competitive compared to other methods used against locking.The second main objective of this work is the development of a phase-field model for the description of brittle fracture in isogeometric Reissner-Mindlin plates and shells. A continuous crack phase-field which is defined on the shell mid-surface and interpolated with NURBS basis functions is used to describe the transition between cracked and uncracked material. Since Reissner-Mindlin formulations are used for both thin and thick structures, fracture due to transverse shear deformations is possible. Thus, a special focus lies on the incorporation of the transverse shear strains in the phase-field model. The spectral decomposition for the tension-compression split is applied on the total strain tensor, which varies through the thickness, in order to avoid unphysical fracture in compressive areas. The plane stress condition cannot be applied by a simple elimination of the thickness normal strain and thickness normal stress from the constitutive law but has to be enforced numerically. In each integration point through the thickness, the thickness normal strain is computed using a local algorithm with quadratic convergence in order to achieve a zero thickness normal stress. The ability of the phase-field model of brittle fracture to correctly describe crack initiation, propagation and merging in plates and shells is assessed with the help of various numerical examples. A comparison to two existing formulations, namely, a 3D solid and a Kirchhoff-Love shell is carried out. It is shown that in the cases of thin plates and shells a good agreement between the three different element types is observed. However, in cases where shearing plays a crucial role, the results of the Kirchhoff-Love shell differ from the other two since it does not consider transverse shear deformations.

### Identifier

- ISBN: 978-3-946090-14-4
- DOI: 10.18154/RWTH-2022-04922
- RWTH PUBLICATIONS: RWTH-2022-04922