high gradient enrichment functions for crack propagation ... · high gradient enrichment functions...

High gradient enrichment functions for crackpropagation in cohesive and cohesion-less cracks

Safdar AbbasThomas-Peter Fries

AICES, RWTH Aachen University, Aachen, Germany

IV European Conference on Computational Mechanics,Paris, France, May 17, 2010

1

Outline

Outline

Motivation

XFEM in fracture mechanics

Numerical examples (cohesionless cracks)

Numerical examples (cohesive cracks)

Conclusions

Future outlook

2

Motivation

Outline

Motivation




Conclusions

Future outlook

3

Motivation Types of fracture

Types of fracture

Strain

Ductile Fracture

Brittle FractureStress Brittel Ductile

From: Fracture mechanics: fundamentals andapplications by Ted L. Anderson

4


Types of fractureσ

Fracture Behavior

Linear ElasticFracture Mechanics

Fracture Process ZoneCohesive

Plastic Collapse

5


Cohesionless fracture

• Negligible Plastic Zone.• Infinite Stresses at Crack Tip.• Different Criteria for Crack Growth.

6



• Negligible Plastic Zone.

• Infinite Stresses at Crack Tip.• Different Criteria for Crack Growth.

6



• Negligible Plastic Zone.• Infinite Stresses at Crack Tip.

• Different Criteria for Crack Growth.

6



• Negligible Plastic Zone.• Infinite Stresses at Crack Tip.• Different Criteria for Crack Growth.

6


Cohesive fracture

• Presence of ”Damaged” zone:1. Plastic Zone for metals.2. Fracture Process Zone for cementitious materials and ceramics.

• Finite Stresses at Crack Tip.• Different size of the Plastic or Process zone.• Different criteria for crack growth.

7


Cohesive fracture


• Finite Stresses at Crack Tip.

• Different size of the Plastic or Process zone.• Different criteria for crack growth.

7


Cohesive fracture


• Finite Stresses at Crack Tip.• Different size of the Plastic or Process zone.

• Different criteria for crack growth.

7


Cohesive fracture


• Finite Stresses at Crack Tip.• Different size of the Plastic or Process zone.• Different criteria for crack growth.

7


Linear elastic fractureu

x

x

σ

Cohesive fractureu

x

x

σ

8

Motivation High-gradient enrichment functions

High-gradient enrichment functions

ψ = r x ;dψ

dx= xr x−1

=>dψ

dx=

∞ x < 1,0 x > 1 & x < 2,

0 x > 2.

9



ψ = r x ;dψ

dx= xr x−1 =>

dψ

dx=

∞ x < 1,

0 x > 1 & x < 2,

0 x > 2.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

r

d! /

dr

9



ψ = r x ;dψ

dx= xr x−1 =>

dψ

dx=

∞ x < 1,0 x > 1 & x < 2,

0 x > 2.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

r

d! /

dr

9


• Interpolated function f• Interpolation functions Ψ = [ψ1, ψ2, ψ3]• Find

∫ωuh =

∫ωf , for ω ∈ Ψ,

where uh =∑ψiui = Ψ

Tu

10


• Interpolated function f

• Interpolation functions Ψ = [ψ1, ψ2, ψ3]• Find

∫ωuh =



Tu

10


• Interpolated function f• Interpolation functions Ψ = [ψ1, ψ2, ψ3]

• Find∫ωuh =



Tu

10


• Interpolated function f• Interpolation functions Ψ = [ψ1, ψ2, ψ3]• Find

∫ωuh =



Tu

10


Optimal set of functions

Ψ = {r1.1, r1.3, r1.5, r1.7}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

r

d! /

dr

11



Ψ = {r1.1, r1.3, r1.5, r1.7}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

r

d! /

dr

11



Ψ = {r1.1, r1.3, r1.5, r1.7}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

r

d! /

dr

11


Outline

Motivation




Conclusions

Future outlook

12


XFEM Formulation in Fracture Mechanics

uh(x) =∑i∈I

Ni (x)ui︸︷︷︸Continuous

+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.• Set of crack tip nodes.• Partition-of-unity functions.• Enrichment functions.• Additional degrees of freedom.

13



uh(x) =∑i∈I


+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.

• Set of crack tip nodes.• Partition-of-unity functions.• Enrichment functions.• Additional degrees of freedom.

135 140 145 150 15592

94

96

98

100

102

104

106

108

13



uh(x) =∑i∈I


+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.• Set of crack tip nodes.

• Partition-of-unity functions.• Enrichment functions.• Additional degrees of freedom.

135 140 145 150 15592

94

96

98

100

102

104

106

108

13



uh(x) =∑i∈I


+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.• Set of crack tip nodes.• Partition-of-unity functions.

• Enrichment functions.• Additional degrees of freedom.

135 140 145 150 15592

94

96

98

100

102

104

106

108

13



uh(x) =∑i∈I


+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.• Set of crack tip nodes.• Partition-of-unity functions.• Enrichment functions.

• Additional degrees of freedom.

135 140 145 150 15592

94

96

98

100

102

104

106

108

13



uh(x) =∑i∈I


+∑j∈I?1

N?j (x) · H(x)aj +∑

k∈I?2N?k (x) ·

(4∑

m=1

Bm(x)bmk

)︸︷︷︸

Discontinuous

• Set of nodes whose support is cut by the interface.• Set of crack tip nodes.• Partition-of-unity functions.• Enrichment functions.• Additional degrees of freedom.

135 140 145 150 15592

94

96

98

100

102

104

106

108

13


Crack-tip polar coordinate system

From: Aspects of Energy Minimization in Solid Mechanics: Evolution of Inelastic Microstructures and CrackPropagation by Ercan Gürses

• Modes I & II.• r is the length of a small vector extending forward from the

crack tip.• θ is the angle from rectangular to polar coordinates.

14


Asymptotic Enrichment functions

B(x) = {√

r sinθ

2,√

r sinθ

2sin θ,

√r cos

θ

2,√

r cosθ

2sin θ}

From: T.P. Fries, T. Belytschko, 2010 accepted. 15



B(x) = {r1.1θ, r1.3θ, r1.5θ, r1.7θ, r1.1 sin θ2

sin θ}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

r

d! /

dr

16


Outline

Motivation




Conclusions

Future outlook

17


Cohesionless fracture: Problem statement

F

nΓc

Γt Ω

Γu

f int = f ext∫Ω

�(uh) : C : �(v)dΩ =

∫Γt

F · v

u ∈ U = {v ∈ V : v = 0 on Γu}

18


Pure Mode-I edge crack

H/2

H

W

a = W/2

KI = 1, KII = 0E = 10000 MPaν = 0.3

19


Pure Mode-I edge crack: L2-Norm

10−2 10−110−3

10−2

Element Size (h)

L2 N

orm

Branch Enr. Func.High−Grad. Enr. Func.

20


Mixed Mode I and II edge crackqx

L

W

a

W = 7, L = 16E = 10000 MPaν = 0.3, qx = 1

21


Mixed Mode I and II edge crack: Convergence of KI

0 1 2 3 4 5 6x 104

30

30.5

31

31.5

32

32.5

33

33.5

34

Degrees of Freedom

K I


22


Mixed Mode I and II edge crack: Convergence of KII

0 1 2 3 4 5 6x 104

4.3

4.35

4.4

4.45

4.5

Degrees of Freedom

K II


23


Single edge notched beam (Areias & Belytschko [2005])

E = 3× 107 psiν = 0.3, q = −10000 lbs

q

262.5 75

50

300

150

37.5

75

24


(Loading Movies/Movie3.avi)

25

Movie3.aviMedia File (video/avi)


Single edge notched beam

330 335 340 345 350 355 360 3650

50

100

150

Crackpath for High Grad. Enr.Crackpath for the Branch Enr.

Crackpath for the Branch enrichments and High-gradient enrichments

26


Single edge notched beam (Areias & Belytschko [2005])

E = 3× 107 psi,ν = 0.3, q = −10000 lbs,

37.5

262.5 75

50

300

15075

q

37.5

27


(Loading Movies/Movie4.avi)

28

Movie4.aviMedia File (video/avi)


Single edge notched beam

300 320 340 360 380 400 420 4400

50

100

150

Crackpath for High Grad. Enr.Crackpath for the Branch Enr.

Crackpath for the Branch enrichments and High-gradient enrichments

29


Outline

Motivation




Conclusions

Future outlook

30


Cohesive fracture: Problem statement

F

nΓc

ΓCoh

Γt Ω

Γu

Fc

ft

Str

ess

Crack opening displacementwc

Gf

f int =λf ext + f coh

f int =

∫Ω

�(uh) : C : �(v)dΩ = K · uh

f ext =

∫Γt

F · vdΓ

f coh =−∫

ΓCoh

Fcn · vdΓ

Fc =ft − kw(uh)

u ∈ U = {v ∈ V : v = 0 on Γu}

31


Cohesive fracture: Problem statement

F

nΓc

ΓCoh

Γt Ω

Γu

Fc

r =

[K · uh − λf ext − f coh

ft − Suh]

A =

K− ∂f coh∂u −f ext−S 0

S = MTCB; M = n ⊗ n{4uh4λ

}i= −(Ai−1)−1·ri−1

Goangseup Zi and Ted Belytschko: New crack-tip elements for XFEM and applications to cohesive cracks, Internat.

J. Numer. Methods Engrg., 2003; 57:2221–2240

32


Double cantilever beam (straight crack)

ft

Str

ess

Crack opening displacementwc

Gf

h

P

P

0.3L

L

L = 400 mm, E = 36500 MPa, ν = 0.18, ft = 3.19 MPa, Gf = 50N/m

33


Double cantilever beam (straight crack)

0 100 200 300 400−0.5

0

0.5

1

1.5

Distance [mm]

Dim

ensi

onle

ss s

tress

0 0.2 0.4 0.6 0.80

10

20

30

40

50

60

Deflection [mm]

Load

[KN]

High gradient Enr. Str.Zi and Belytschko (2003) Str.

34


Double cantilever beam (curved crack)

0 50 100 150 200 250 300 350 4000

50

100

150

200

0 0.2 0.4 0.6 0.80

10

20

30

40

50

60

Deflection [mm]

Load

[KN]

High gradient Enr. Str.High gradient Enr. Cur.Zi and Belytschko (2003) Str.Zi and Belytschko (2003) Cur.

35


Three point bending test

a

2b 2b

b

E = 36500 MPa, ν = 0.1, ft = 3.19 MPa, b = 150, a = 0

36



0 0.2 0.4 0.6 0.8 1x 10−3

0

0.05

0.1

0.15

0.2

0.25

Deflection/b

Load/f tb2

High gradient EnrCarpinteri and Colombo (1989)

Load-point displacement curvefor Gf = 50 N/m

0 1 2 3 4x 10−4

0

0.05

0.1

0.15

0.2

0.25

Deflection/b

Load/f tb2


37



0 0.2 0.4 0.6 0.8 1x 10−3

0

0.05

0.1

0.15

0.2

0.25

Deflection/b

Load/f tb2

High gradient EnrCarpinteri and Colombo (1989)


0 1 2 3 4x 10−4

0

0.05

0.1

0.15

0.2

0.25

Deflection/bLoad/f tb2


37

Conclusions

Outline

Motivation




Conclusions

Future outlook

38

Conclusions

Conclusions

• An enrichment scheme for the XFEM has been proposedwhich enables highly accurate approximations of high gradientstresses/strains near the crack-tip.

• The scheme is independent of the fracture model.• Accuracy in the case of linear elastic fracture mechanics is

better than that achieved through classical branchenrichments.

39

Future outlook

Outline

Motivation




Conclusions

Future outlook

40

Future outlook

Future outlook

• Application of proposed enrichment functions to more generalmaterial and geometric non-linearities.

• Considering for the shear traction condition.• General interest in model-independent methods for problems

involving high gradients.

A. Alizada, T.P. Fries: Cracks and crack propagation with xfemand hanging nodes in 2d (17.05.2010 @ 17:20 )

41

Acknowledgements

Financial support from the DeutscheForschungsgemeinschaft (German ResearchAssociation) through grant GSC 111 andthe Emmy-Noether program is gratefullyacknowledged.

42

Acknowledgements

Thanks for your attention

43

Outline

0.0: 0.1: 0.2: 0.3: 0.4: 0.5: 0.6: 0.7: 0.8: 0.9: 0.10: 0.11: anm0: 1.0: 1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 1.8: 1.9: 1.10: 1.11: anm1: 2.0: 2.1: 2.2: 2.3: 2.4: 2.5: 2.6: 2.7: 2.8: 2.9: 2.10: anm2: 3.0: 3.1: 3.2: 3.3: 3.4: 3.5: 3.6: 3.7: 3.8: 3.9: 3.10: anm3: 4.0: 4.1: 4.2: 4.3: 4.4: 4.5: 4.6: 4.7: 4.8: 4.9: 4.10: anm4:

high gradient enrichment functions for crack propagation ... · high gradient enrichment functions...

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