5 research outputs found
An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching
An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
We present an adaptive space-time phase field formulation for dynamic
fracture of brittle shells. Their deformation is characterized by the
Kirchhoff-Love thin shell theory using a curvilinear surface description. All
kinematical objects are defined on the shell's mid-plane. The evolution
equation for the phase field is determined by the minimization of an energy
functional based on Griffith's theory of brittle fracture. Membrane and bending
contributions to the fracture process are modeled separately and a thickness
integration is established for the latter. The coupled system consists of two
nonlinear fourth-order PDEs and all quantities are defined on an evolving
two-dimensional manifold. Since the weak form requires -continuity,
isogeometric shape functions are used. The mesh is adaptively refined based on
the phase field using Locally Refinable (LR) NURBS. Time is discretized based
on a generalized- method using adaptive time-stepping, and the
discretized coupled system is solved with a monolithic Newton-Raphson scheme.
The interaction between surface deformation and crack evolution is demonstrated
by several numerical examples showing dynamic crack propagation and branching.Comment: In this version, typos were fixed, Fig. 16 is added, the literature
review is extended and clarifying explanations and remarks are added at
several places. Supplementary movies are available at
https://av.tib.eu/series/641/supplemental+videos+of+the+paper+an+adaptive+space+time+phase+field+formulation+for+dynamic+fracture+of+brittle+shells+based+on+lr+nurb
-convergence for high order phase field fracture: continuum and isogeometric formulations
We consider second order phase field functionals, in the continuum setting,
and their discretization with isogeometric tensor product B-splines. We prove
that these functionals, continuum and discrete, -converge to a brittle
fracture energy, defined in the space . In particular, in the
isogeometric setting, since the projection operator is not Lagrangian (i.e.,
interpolatory) a special construction is needed in order to guarantee that
recovery sequences take values in ; convergence holds, as expected, if
, being the size of the physical mesh and
the internal length in the phase field energy
Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models
This work presents numerical techniques to enforce continuity constraints on
multi-patch surfaces for three distinct problem classes. The first involves
structural analysis of thin shells that are described by general Kirchhoff-Love
kinematics. Their governing equation is a vector-valued, fourth-order,
nonlinear, partial differential equation (PDE) that requires at least
-continuity within a displacement-based finite element formulation. The
second class are surface phase separations modeled by a phase field. Their
governing equation is the Cahn-Hilliard equation - a scalar, fourth-order,
nonlinear PDE - that can be coupled to the thin shell PDE. The third class are
brittle fracture processes modeled by a phase field approach. In this work,
these are described by a scalar, fourth-order, nonlinear PDE that is similar to
the Cahn-Hilliard equation and is also coupled to the thin shell PDE. Using a
direct finite element discretization, the two phase field equations also
require at least a -continuous formulation. Isogeometric surface
discretizations - often composed of multiple patches - thus require constraints
that enforce the -continuity of displacement and phase field. For this,
two numerical strategies are presented: For this, two numerical strategies are
presented: A Lagrange multiplier formulation and a penalty method. The
curvilinear shell model including the geometrical constraints is taken from
Duong et al. (2017) and it is extended to model the coupled phase field
problems on thin shells of Zimmermann et al. (2019) and Paul et al. (2020) on
multi-patches. Their accuracy and convergence are illustrated by several
numerical examples considering deforming shells, phase separations on evolving
surfaces, and dynamic brittle fracture of thin shells.Comment: In this version, typos were fixed, Chapter 6.4 is added, Table 1 is
updated, and clarifying explanations and remarks are added at several place