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.