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EFG and XFEM Cohesive Fracture Analysis EFG and XFEM Cohesive Fracture Analysis
Methods in LSMethods in LS--DYNADYNA
Yong Guo*, Cheng-Tang Wu
Livermore Software Technology Corporation
LS-DYNA Seminar
Stuttgart, Germany
November 24, 2010
LS-DYNA Seminar 2
1. Overview on Failure/Crack Simulations
2. EFG and XFEM Cohesive Fracture
Methods
3. Numerical Examples
4. Conclusions
OutlineOutline
LS-DYNA Seminar 3
1. Overview on Failure/Crack 1. Overview on Failure/Crack
SimulationsSimulations
Tie-break Interface
Force/stress-based failure + spring element, rigid rods, or other constraints
Suitable for delamination, debonding, known weak areas
Element Erosion
Stress/strain-based failure + contact force
Loss of conservation, strong mesh dependence and inadequate accuracy
Cohesive Interface Element
Cohesive zone model + interface element + contact force
Crack along interfaces: Mesh dependence
EFG
Cohesive zone model + moving least-square + EFG visibility
XFEM
Cohesive zone model + level sets + extended finite element
LS-DYNA Seminar 4
2. EFG and XFEM Cohesive Failure 2. EFG and XFEM Cohesive Failure
MethodsMethods
EFG and XFEM Failure Analysis
Both are discrete approaches (strong discontinuity).
Crack initiation and propagation are governed by cohesive law
(Energy release rate).
Crack propagates cell-by-cell in current implementation.
EFG is defined by visibility; XFEM is defined by Level Set.
Minimized mesh sensitivity and orientation effects in cracks.
Applied to quasibrittle materials and some ductile materials.
EFG for Solid with 4-noded integration cells.
XFEM for 2D plain strain and shells.
LS-DYNA Seminar 5
2.1 EFG Cohesive Failure Method2.1 EFG Cohesive Failure Method
+Ω0
−Ω0
( )ηX
Meshfree Method: MLS + Visibility Criterion (Belytschko et al. 1996)
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ),hW
,hWΦ
Φ
IJ
T
J
J
II
T
I
I
I
I
h
XXXPXPXA
XXXPXAXP
uXXu
−=
−=
=
∑
∑−
=
1
1
( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( ) ( )η
ηΨΨ
η
ηΦ
η
η
ηΨηΨηΦη
ΩJ ΩJ
JJ
JJ
FEM
I
I
I
ΩJ ΩJ
JJJJI
I
FEM
I
∂∂
∂∂
⊗+∂
∂⊗+
∂∂
⊗=∂
∂
++=
∑ ∑∑
∑ ∑∑
+ −
+ −
∈ ∈=
∈ ∈=
X
X
Xu
X
XuX
x
uXuXXx
0 0
0 0
2
1
2
1
2
1
2
1
Mid-plane fracture surface
The domain of influence of particles on one side of the crack cannot go
through the crack surface and the particles on one side of the crack cannot
interact with the particles on other side of the crack
Moving Least Square
Visibility Criterion Intrinsic (implicit) crack: no additional unknowns
LS-DYNA Seminar 6
Minimization of Mesh Size Effect Minimization of Mesh Size Effect
in Modein Mode--I Failure TestI Failure Test
Coarse elements Fine elements
Failure is
limited in this
area
01.0== crD λ005.0== crD λ
Tn
D=0.01 D=0.005
LS-DYNA Seminar 7
IDIM EQ. 1: Local boundary condition method
EQ. 2: Two-points Guass integration (default)
EQ.-1: Stabilized EFG method (applied to 8-noded, 6-noded and combination of them)
EQ.-2: Fractured EFG method (applied to 4-noded & SMP only)
Variable DX DY DZ ISPLINE IDILA IEBT IDIM TOLDEF
Type F F F I I I I F
Default 1.01 1.01 1.01 0 0 -1 2 0.01
Card 2
Variable IGL STIME IKEN SF CMID IBR DS ECUT
Type I F I F I I F F
Default 0 1.e+20 0 0.0 1 1.01 0.1
*SECTION_SOLID_EFG
5, 41
1.1, 1.1, 1.1, , ,4, -2,
, , , , 100, 1, 2.0, 0.2
*MAT_COHESIVE_TH
100,1.0e-07, ,1, 330.0, 0.0001,
Card 3
Input Format for EFG Failure Analysis
*SECTION_SOLID_EFG
SF: Failure strain
CMID: Cohesive material ID
IBR: Branching indicator
DS: Normalized support for displacement jump
ECUT: Minimum edge cut
LS-DYNA Seminar 8
2.2 XFEM Cohesive Failure Method2.2 XFEM Cohesive Failure Method
Extended FEM: Level Set + Local PU (Belytschko et al. 2000)
( ) ( ) ( )
( ) ( ) ( )( ) ( )( )[ ]( ) ( )( ) ( )( )[ ]
−
−=
+= ∑∑∈=
tip crack contain
element cutfully
I
*FEM
I
I
FEM
I
I
I
wI
II
I
FEM
I
h
fHfHξΦ
fHfHξΦΨ
ΨξΦ
XX
XXX
qXuXu1
*ξ
0=f
Level Set
Discontinuity defined by two implicit functions: )( and )( XX gf
( )[ ]XXnXXXX
−⋅−=αΓ∈
signf min)(
00 >=Γ∈ α ),( and )( if tgf XXX0
Signed distance function
Discontinuity
Local Partition of Unity
Define implicit functions locally
∑=I
IINff )()( XX
npropagatio crack eelementwis forindex by replaced ),( tg X
LS-DYNA Seminar 9
- Song, Areias and Belytschko (2006)Approximation of crack in element
( ) ( )[ ] ∑ −+=I
IIII
h fHfHttNt )()()()()(),( XXquXXu
Phantom Nodes and Phantom ElementsPhantom Nodes and Phantom Elements
Rewrite into superposition of two phantom elements
( ) ( )∑∑∈∈
+−=2 21 1
21
SIt
II
SIt
II
h fHNtfHNtt )()()()()()(),(
),(),(
XXuXXuXu
XuXu
4342143421
where
>
<+=
>−
<=
0
0
0
0
2
1
)( if
)( if
)( if
)( if
II
III
I
III
II
I
f
f
f
f
Xu
Xquu
Xqu
Xuu
0<)(xf
0>)(xf
11
22
3 3
0=)(xf
0>)(xf
0<)(xf
1 2
34 44
1
2
= +
Real node
Phantom node
LS-DYNA Seminar 10
Easier in implementation
Domain Integration SchemesDomain Integration Schemes
Integration in phantom elements and assembly according to area ratios
[ ]
[ ]
∫∫
∫∫
∫∫
∫
∫
ΓΓ
ΓΩ
ΓΩ
Ω
Ω
Γ=Γ−=
Γ+Ω=
Γ−+Ω=
Ω=
Ω=
ec
2ec
1
et
e
2
2
et
e
1
1
e
1
1
1e
1
1
e
c
ccoh
e
e
c
ccoh
e
e
t
eeext
e
e
t
eeext
e
ee(eint
e(e
e(e
ee(ekin
e(e
dd
dfHdA
A
dfHdA
A
dA
A
dA
A
,0,0
,00
,00
0
2
2
20
2
2
,00,00
,0
0
00
0
,0
0
00
0
0
0
)/
)/
)/00
0
)/
)/
)(
)(
nτNfnτNf
XtNbNf
XtNbNf
PBf
uNNf
TT
TT
TT
T
T
ρ
ρ
ρ &&
0<)(xf
0>)(xf
11
22
3 344
1
2
1e 2e
- Song, Areias and Belytschko (2006)Phantom Element Integration
Sub-domain integration
Integration conducted in two sub-domains cut by cracks
More accurate results
Difficulties in implementation: Varied integration schemes,
Different data structure, Transfer of state variables
LS-DYNA Seminar 11
Variable CMID IOPBASE IDIM INITC
Type I I I I
Default 13,16 0 1
*SECTION_SHELL
5, 53
0.1, 0.1, 0.1, 0.1
100, 16, 0, 1
*MAT_COHESIVE_TH
100,1.0e-07, ,1, 330.0, 0.0001,
Card 3
Input Format for XFEM Failure Analysis
*SECTION_SHELL_XFEM
Variable SECID ELFORM SHRF NIP PROPT QR/IRID ICOMP SETYP
Type I I F I F F I I
Card 1
ELFORM EQ. 52: Plane strain (x-y plane) XFEM
EQ. 54: Shell XFEM
CMID: Cohesive material ID
IOPBASE: Base element type
Type 13 for plain strain XFEM
Type 16 for shell XFEM
IDIM: Domain integration method
0 for phantom element integration
1 for subdomain integration
INITC: Criterion for crack initiation
1 for maximum tensile stress
LS-DYNA Seminar 12
Crack is consisted of mathematical crack (cohesive zone) and
physical crack.
Cohesive zone crack initiates when maximum stress reached.
Physical crack occurs when critical COD reached.
Cohesive work = critical energy release rate
Constitutive cohesive law relates the traction forces to
displacement jumps through a potential:
( )δ
qδT
∂Φ∂
=,
2.3 Cohesive Fracture Model2.3 Cohesive Fracture Model
Different Cohesive Laws Displacement jumps can have various components
due to different crack modes.
LS-DYNA Seminar 13
Constitutive Cohesive LawConstitutive Cohesive Law
maxˆ
ˆTMTTTT tttnefs ≥
αβ
+
αβ
+
αβ
+≡ 2
1
2
2
2
2
2
22
1
2
1
12
Equivalent fracture stress
2
2
2
2
22
2
2
1
12
2
1
θ∆θ∆
β+
δβ+
δβ+
δ=λ
max
ˆ
t
t
t
t
n
n uuu
Effective displacement jump
Initially Rigid Cohesive Law
0 1
1
λ
maxTT
maxλ
max
max
1
max2
2
2
2
2
max1
1
1
1
1
max
ˆ1
1
1
1
TM
Tu
uT
Tu
uT
Tu
uT
t
t
t
t
t
t
t
t
t
n
n
n
n
αθθ
λλ
θ
αδλ
λ
αδλ
λ
δλλ
∆∆−
=∆∂Φ∂
=
−=
∂Φ∂
=
−=
∂Φ∂
=
−=
∂Φ∂
=
Tractions
max
ˆˆ , ,θ∆δ
β=α
δδ
β=α
δδ
β=α n
t
n
t
n 2
2
2
22
1
2
11
ncI
cIII
cI
cII
ncI
t
G
G
G
GTG
δθ∆
=ββ=β=δ=62
121
maxmax
ˆ , , ,
- Zavattieri (2001, 2005)
( )maxmaxmax
max
if ,0
,max
λλλλλ
λλλ
>==
=
LS-DYNA Seminar 14
2.4 Computation Procedures2.4 Computation Procedures
1. Representation of Cracks
2. Cohesive Law
Crack initiation/propagation
3. Branching/Multiple cracks
4. State Variables Transfer
5. Numerical Integration
( )
[ ][ ] c
ccoh
t
ext
kin
cohextkin
dW
ddW
dW
dW
uXuWWWW
c
t
Γ⋅−=
Γ⋅+Ω⋅=
Ω∂∂
=
Ω⋅=
∈∀+−=
∫
∫∫
∫
∫
Γ
ΓΩ
Ω
Ω
τu
tubu
PX
u
uu
δδ
δρδδ
δδ
ρδδ
δδδδδ
00
00
0
int
00
0
int
00
0
0
:
&&
∫
∫∫
∫
∫
Γ
ΓΩ
Ω
Ω
Γ−=
Γ−+Ω−=
Ω−=
Ω−=
+−=
et
et
e
e
e
e
t
cecoh
e
e
t
eeeext
e
eeint
e
eekin
e
cohextkin
d
dfHdfH
dfH
dfH
,0
,00
0
0
,00
,000
0
00
int
)1(
))()1(())()1((
))()1((
))()1((
nNf
XtNXbNf
XBf
uXNNf
ffff
T
TT
T
T
τ
ρ
σ
ρ &&
Cohesive tractions treated as external forces
LS-DYNA Seminar 15
75mm
25mm
100mm
100mm
50mm
100mm
0vx
y
m/s5.16
m10245.5 ,N/m10213.1
300 ,GPa190 ,kg/m8000
0
5
max
4
3
=
×=×=
===−
v
δ
νρ
IcG
.E
3.1 Kalthoff Plate Crack Propagation3.1 Kalthoff Plate Crack Propagation
LS-DYNA Seminar 16
EFG 3D
Maximum Principle Stress Contour
XFEM Plain Strain
Failure Indicator
Kalthoff Plate Crack PropagationKalthoff Plate Crack Propagation
o0.69
o0.70
Average Crack Angle: Average Crack Angle:
Average Crack Angle from Experiment:
o5.62
LS-DYNA Seminar 17
3.2 EFG 3D Edge3.2 EFG 3D Edge--cracked Plate cracked Plate
under Threeunder Three--point Bendingpoint Bending
101 x 31 x 6 nodes
Elastic
EFG Fracture
Linear Cohesive Law
Explicit analysis
front
back
Resultant Displacement Contour
Failure Contour
d
LS-DYNA Seminar 18
3.3 Rigid Ball Impact on EFG 3.3 Rigid Ball Impact on EFG
Concrete PlateConcrete Plate
Progressive Crack Propagation
Elastic
EFG Fracture
Linear Cohesive Law
Explicit analysis
Time-velocity of the rigid ball
LS-DYNA Seminar 19
3.4 Steel Ball Impact on Steel Plate3.4 Steel Ball Impact on Steel Plate
image01
Elastic_plastic
EFG Fracture
Linear Cohesive Law
Explicit analysis
LS-DYNA Seminar 20
Time-velocity of the metal ball
Steel Ball Impact on Steel PlateSteel Ball Impact on Steel Plate
LS-DYNA Seminar 21
3.5 EFG Glass under Impact3.5 EFG Glass under Impact
101 x 101 x 4 nodes
Elastic + Rubber
EFG Fracture
Linear Cohesive Law
Explicit analysis
front
back
Failure Contour
LS-DYNA Seminar 22
Rigid ball Metal ball
3.5 EFG Glass under Impact3.5 EFG Glass under Impact
LS-DYNA Seminar 23
3.6 Thin Cylinder Pulling3.6 Thin Cylinder Pulling
m/s5
mm5 mm,200 mm,50
MPa200 ,k500
MPa200100 30 ,70 ,7830 3
=
===
==
ε+=σ=ν==ρ
v
hLR
TG
.E
Ic
p
y
maxN/m
)(,GPakg/m
1860 elements
Clamped edge
v
Rigid diaphragms
Pre-crack
LS-DYNA Seminar 24
4. Conclusions4. Conclusions
EFG and XFEM cohesive failure methods are successfully
applied to brittle and semi-brittle materials.
EFG failure analysis with visibility criterion is more robust and
capable of handling crack branching and interaction.
XFEM cohesive failure analysis is more suitable for crack
analysis with pre-cracks and without crack branching or interaction.
Further research is needed for ductile fracture analysis.