Title: Extended finite element and meshfree methods: 11' EXtended finite elements Timon Rabczuk Prof' Wolfg
1Extended finite element and meshfree methods11.
EXtended finite elements Timon RabczukProf.
Wolfgang Wall
2Outline
- Tracking the crack path
- Embedded Elements
- Interface elements
- Extended Meshfree Methods
- Introductions into cohesive zone models
- Crack initiation/propagation
- Problems
3Tracking the crack path
- Global methods
- Local methods
- The level set method
4Embedded elements
- Statical optimal symmetric elements traction
- continuity is enforced but wrong kinematics
(rigid - motion of two element parts not guaranteed)
- Kinematical optimal symmetric elements correct
- kinematic but no traction continuity
- Kinematicstatic optimal non-symmetric elements
- correct kinematicenforcement of traction
continuity - but non symmetric
SOS shows stress locking-gt mixed formulation
5Embedded vs. XFEM
Embedded
XFEM
6Embedded elements
Elemental enrichment
S
7Embedded elements
Stiffness matrices
8Embedded elements
9Interface separation models
Advantages
Drawbacks
- Sensitive with respect to element size, form and
arrangement - Meshdependent results
Ortiz et al., Needleman et al.
10Discontinuities in meshfree methods
XFEM (Belytschko et al. 1999)
11Discontinuities in meshfree methods
Two solutions to model a meshfree crack
without the branch enrichment
Domain decrease method
Lagrange multiplier method
12Discontinuities in meshfree methods
- Tracing the crack paths
- Evolution law for the level set
- Difficulties
- Crack branching/intersections
- Crack propagation
13Discontinuities in meshfree methods
An alternative approach to treat
discontinuities with meshfree methods
14The cracking particle approach
Methods that do not enforce crack path continuity
Continuous crack
model
15The cracking particle approach
Decomposition of the displacements
W
Discretization
- Difference to methods before
- crack goes directly through a particle
- crack is modeled by a set of
- cracked particles
- no representation of the crack path is necessary
crack
16Drawback of FE
Continuous crack path
Crack segments
17Application to shear bands
Decomposition of the displacements
W
Discretization
Shear band
18Cracks in thin shells
- Meshfree thin shell
- Combination of KL shell (kinematical assumption)
- and continuum based shell
Kinematics
Extension to cracks
Discretization
19Cohesive zone model
COHESIVE ZONE MODELS A BRIEF INTRODUCTION
20Introduction into cohesive cracks
load
Strain softening
displacement
- Problem IBVP is ill-posed, mesh-dependent
results (localization - in a set of measure zero,
zero-energy dissipation with mesh - refinement)
21Regularization
Regularization techniques for unstable materials
- Gradient models
- Integral or nonlocal models
- Viscous (rate dependent) models
22Outline
Gradient model
Integral model
Viscous model
23Outline
24Cohesive cracks
- Important
- Traction continuity
- Complementary problem
25Cohesive cracks
wmax
w
Prozesszone
ft
Rissoberflaeche
tf(w)
26Cohesive cracks
Starke Form
27Cohesive cracks
Variational formulation
28Smeared cracks
- Fixed single cracks
- Fixed orthorgonal cracks
- Fixed multiaxial cracks
- Rotating cracks
29Smeared cracks
Advantages
Drawbacks
- Wrong crack kinematics
- Stress locking
- diffuses crack pattern
- mesh-dependent results (mesh alignment
sensitivity) - Problems, when process-zone larger than element
size - Fixed cracks too stiff
- Rotating cracks too soft
30Crack initiation/propagation
Crack initiation
- When is a crack initiated
- Crack orientation
- Crack length
Crack propagation
- Basically same as initiation but simpler
31Crack onset
Stress based Rankine
Strain based
Others Damage
Loss of material stability (ellipticity/hyperbolic
ity)
Acoustic tensor
- Difficulty
- 2 angles obtained in the
- localization analysis
32Discontinuous bifurcation analysis
33Crack orientation
Mode I
Mode II
h
h
n
Mixed Mode
34Crack orientation/length
- Crack orientation
- Rankine perpendicular to principal tensile
stress - Strain based perpendicular to principal tensile
strain - Others ???
- Loss of Material Stability obtained from
bifurcation analysis
- Crack length
- through the entire element
- Control crack length
- e0
35Problems
- Cohesive models
- Unloading/reloading
- fmax cannot be exceeded
- Convergence difficulties with rigid cohesve
models (unloading) - Dynamic fracture energy?
- Bifurcation analysis
- Bifurcation analysis for problems with
constraints (shells) (shear bands) - Ambiguous normal
- PDE does not loose ellipticity for large time
steps
36The end
Thank you very much for your attention