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@PHDTHESIS{Kikis:846871,
author = {Kikis, Georgia},
othercontributors = {Klinkel, Sven and De Lorenzis, Laura and Dornisch,
Wolfgang},
title = {{L}ocking and brittle fracture in isogeometric
{R}eissner-{M}indlin plate and shell analysis},
volume = {15 (2022)},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {Rheinisch-Westfälische Technische Hochschule Aachen,
Fakultät für Bauingenieurwesen, Lehrstuhl für Baustatik
und Baudynamik},
reportid = {RWTH-2022-04922},
isbn = {978-3-946090-14-4},
series = {Schriftenreihe des Lehrstuhls für Baustatik und Baudynamik
der RWTH Aachen},
pages = {1 Online-Ressource : Illustrationen, Diagramme},
year = {2022},
note = {Druckausgabe: 2022. - Auch veröffentlicht auf dem
Publikationsserver der RWTH Aachen University; 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.},
cin = {311810},
ddc = {624},
cid = {$I:(DE-82)311810_20140620$},
pnm = {Angepasste Approximationsräume und Mehrfeldfunktionale zur
Eliminierung von Versteifungseffekten für isogeometrische
Schalenelemente (266714483) / TRR 280:
Konstruktionsstrategien für materialminimierte
Carbonbetonstrukturen – Grundlagen für eine neue Art zu
bauen (417002380)},
pid = {G:(GEPRIS)266714483 / G:(GEPRIS)417002380},
typ = {PUB:(DE-HGF)11 / PUB:(DE-HGF)3},
doi = {10.18154/RWTH-2022-04922},
url = {https://publications.rwth-aachen.de/record/846871},
}
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