The XFEM using Crack Tip Enrichment with Large Support for ... · XFEM with Large Support for Curved Cracks M. Baydoun T.P. Fries Motivation New Alternative Other Alternatives Studies

XFEM with LargeSupport for

Curved Cracks

M. BaydounT.P. Fries

Motivation

New Alternative

OtherAlternatives

StudiesSigned Distance

Derivatives of LevelSets

Radius Enrichment

Stress IntensityFactors

Conclusions

The XFEM using Crack Tip Enrichment withLarge Support for Curved Cracks

Malak Baydoun and Thomas Peter Fries

AACHEN INSTITUTE FOR ADVANCED STUDYIN COMPUTATIONAL ENGINEERING SCIENCE

ECCM 2010, Paris

1


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Outline

1 Motivation

2 New Alternative

3 Other Alternatives

4 Studies

5 Conclusions

2


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

eXtended Finite Element Formulation

u(x,y) = ∑i∈I

Ni(x,y)ui︸︷︷︸Continuous

+ ∑j∈I?1

N?j (x,y) ·H(x,y)aj + ∑

k∈I?2

N?k (x,y) ·

(4

∑m=1

Bmbmk

)︸︷︷︸

Discontinuous

!

t

!

t

!

t

3


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


u(x,y) = ∑i∈I


+ ∑j∈I?1

N?j (x,y) ·H(x,y)aj + ∑

k∈I?2

N?k (x,y) ·

(4

∑m=1

Bmbmk

)︸︷︷︸

Discontinuous

!

t

!

t

!

t

!

t

!

t

!

t

3


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


u(x,y) = ∑i∈I


+ ∑j∈I?1

N?j (x,y) ·H(x,y)aj + ∑

k∈I?2

N?k (x,y) ·

(4

∑m=1

Bmbmk

)︸︷︷︸

Discontinuous

!

t

!

t

!

t

!

t

!

t

!

t

• In XFEM, Optimal Convergence Rates with Fixed Radius forBranch Enrichments are achieved. [LABORDE ET AL.]

3


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Coordinate SystemsFor Curved Cracks/Crack Propagation, Different CoordinateSystems to evaluate the SIFs and/or Enrichments exist:

!"#$%#&$!$

!

"

!"

#"

Alternative 0• Easy to evaluate by Crack Tip

Information only.• Discontinuity follows a Straight Path.• Not Suitable for “Large” Radius

Enrichment.

4


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Coordinate SystemsFor Curved Cracks/Crack Propagation, Different CoordinateSystems to evaluate the SIFs and/or Enrichments exist:

!"#$%#&$!$

!

"

!"

#"

Alternative 0• Easy to evaluate by Crack Tip

Information only.• Discontinuity follows a Straight Path.• Not Suitable for “Large” Radius

Enrichment.

!

"

!"

!#

Alternative 1• Discontinuity follows Curved Path.

• Bases e1 and e2.

• Drawbacks.

4


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.

• ∀p ∈Ω: Two Signed Values and Bases.

• Bases are no longer Orthogonal.

• Inconvenient values away from the Tip.

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!"#

!"$

!"%

!"&

!"'

!"(

!")

!"*

!!""#!!"+$

!!"+%

!!"+&

!!"+'

!!+ (

!!"+)

!!+ *

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!"!"#!"!"$!"!"%&

#!"$#!"'

#!"&

#!"#

#!"(

#!")

#!"*

#!"+

#!""$#!"%'

#!"%&

#!"%#

#!"%(

#!% )

#!"%*

#!% +

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!!"#!!"$

!!"%

!!"&

!!"'

!!"(

!!")

!!"*

"#$%$&'$($!$%$&')

"#$%$*&'$($

!$%$$*&')!!""#!!"+$

!!"+%

!!"+&

!!"+'

!!+ (

!!"+)

!!+ *

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!!"#!!"$

!!"%

!!"&

!!"'

!!"(

!!")

!!"*

"#$%$&'($)$

!$%$*+'(,

!!""#!!"+$

!!"+%

!!"+&

!!"+'

!!+ (

!!"+)

!!+ *

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!!"#!!"$

!!"%

!!"&

!!"'

!!"(

!!")

!!"*

"#$%$&'($)$

!$%$*+'(,

"#$%$-'+$)$

!$%$*.'(,

!!""#!!"+$

!!"+%

!!"+&

!!"+'

!!+ (

!!"+)

!!+ *

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions





!!""#!!"$%

!!"$&

!!"$'

!!"$(

!!"$)

!!"$*

!!"$+

"#$%$&'($)$

!$%$*+'(,

"#$%$-'+$)$

!$%$*.'(,

"#$%$+'/$)$$!$%$*.'+,

5


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Alternative 2• Geometrical Reconstruction.

• Split the Domain into Triangles.

!"!"#!"!"$!"!"%&

#!"$#!"'

#!"&

#!"#

#!"(

#!")

#!"*

#!"+

#!""$#!"%'

#!"%&

#!"%#

#!"%(

#!% )

#!"%*

#!% +

6


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!"% '(

6


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(

6


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(

6


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(

6


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


!"!"#!"!"$!"!"%&

#!""$

#!% '(#!% &'(#!% #'(

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


!!""#

!!$ %&

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


!!""#

!!$ %&

!!"$ %&'(

!!"$ %&')

!!"$*%&'(

!!"$ %&'+

!!$ ,%&'(

!!"$-%&'(

!!$ .%&'(

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


!!""#

!!$ %&

!!"$ %&'(

!!"$ %&')

!!"$*%&'(

!!"$ %&'+

!!$ ,%&'(

!!"$-%&'(

!!$ .%&'(

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions


!!""#

!!$ %&

!!"$ %&'(

!!"$ %&')

!!"$*%&'(

!!"$ %&'+

!!$ ,%&'(

!!"$-%&'(

!!$ .%&'(

"#!"*"#!"#

7


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Alternative 2• Convenient Values γ.

• Bases are almost Orthogonal.

!"!"#!"!"$!"!"%&

#!""$#!"%'()*

#!"% '()+

#!"%#'()*

#!"% '()&

#!% ,'()*

#!"%-'()*

#!% .'()*

#!% '(#!"% #'(

#!"% &'(

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!!""#

"#!"$"#!"#"#!"%&

!!""#!!"%'

!!"%&

!!"%$

!!"%(

!!% )

!!"%*

!!% +

$,-. /-.0)1

!!""#!!"%-.02

!!"% -.0(

!!"%$-.02

!!"% -.0&

!!% )-.02

!!"%*-.02

!!% +-.02

!!% -.$,-. /&-.0)1

$,-. /$-.0)1$,-. /(-.0)1

!!% &-.!!"% $-.

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#'() *()+,-

$!""$$!"% ()+.

$!"% ()+/

$!"%#()+.

$!"% ()+&

$!% ,()+.

$!"%0()+.

$!% 1()+.

$!"% ()#'() *&()+,-

#'() *#()+,-#'() */()+,-

$!"% &()#'&()"*()+,-

#'&()"*&()+,-#'&()"*#()+,-

#'&()"*/()+,-

$!"% #()

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!% '(#!% &'(#!% #'(

$!"%"&'(")"

#"%*'+'(,

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!% '(#!% &'(#!% #'(

$!"%"&'(")"

#"%*+'(,'(-

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions



!"!"#!"!"$!"!"%&

#!""$

#!% '(#!% &'(#!% #'(

$!"%"&'(")"

#"%*&'+,'(-

8


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(

Alternative 2– a• Applicable if Radius includes

more than One Increment.

9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$!"!"%&

#!""$

#!"% '(

#!"% &'(#!"% #'(

Alternative 2– b• Fits Straight Cracks and Small

Angle Increments: Quadrilateral.

9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$!"!"%&

#!""$

#!"% '(

#!"% &'(#!"% #'(



9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$

#!""$

#!"% &'

#!"% #&'

!"!"%(

#!"% (&'



9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$

#!""$

#!"% &'

#!"% (&'#!"% #&'

!"!"%(



9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$

#!""$

#!"% &'

#!"% (&'#!"% #&'

!"!"%(



!"!"#!"!"$

#!""$

#!"% &'

#!"% (&'#!"% #&'

!"!"%(

Alternative 2– c• Mid Angle of Orthogonal Limiter

Level Sets.

9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Other Alternatives

!"!"#!"!"$!"!"%&

#!""$

#!"% '(#!"% &'(#!"% #'(



!"!"#!"!"$

#!""$

#!"% &'

#!"% (&'#!"% #&'

!"!"%(



!"!"#!"!"$

#!""$

#!"% &'#!"% (&'#!"% #&'

!"!"%(

Alternative 2– c• Mid Angle of Orthogonal Limiter

Level Sets.

9


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Signed Distance

0

2

4

6

01

23

45

67

−4

−3

−2

−1

0

1

2

3

4

Alternative 1• Signed Distance is Tangent to the

Crack at the Tip.

10


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Signed Distance

0

2

4

6

01

23

45

67

−4

−3

−2

−1

0

1

2

3

4


Crack at the Tip.

0

2

4

6

02

46

−4

−3

−2

−1

0

1

2

3

4

5

6

Alternative 2–b• Signed Distance is Tangent to the

Crack Path.

10


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Signed Distance

0

2

4

6

01

23

45

67

−4

−3

−2

−1

0

1

2

3

4


Crack at the Tip.

0

2

4

6

01234567−4

−3

−2

−1

0

1

2

3

4

5

6

Alternative 2–b• Signed Distance is Tangent to the

Crack Path.

10


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Derivatives of Level SetsDerivatives of Level Sets at the Integration points are Required.

• Option 1:1 Find the Signed Distance at the Integration Points.

2 Not easy to evaluate the Derivatives.

• Option 2:1 Find the Signed Distance at the Nodes.

2 Evaluate the Derivatives by using the Shape Functions.

• Comparing both Options for ∇xφ :

11


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions







11


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions







11


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Radius Enrichment

0 1 2 3 4 5 6

−2

−1

0

1

2

Alternative 0• Enrichment does not conform to

the Discontinuity.

12


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Radius Enrichment

0 1 2 3 4 5 6

−2

−1

0

1

2


the Discontinuity.

0 1 2 3 4 5 6

−1.5

−1

−0.5

0

0.5

1

1.5

2

Alternative 1• Enrichment conforms to the

Discontinuity under someRestrictions away from the Tip.

12


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Radius Enrichment

0 1 2 3 4 5 6

−2

−1

0

1

2


the Discontinuity.

0 1 2 3 4 5 6

−1.5

−1

−0.5

0

0.5

1

1.5

2

Alternative 1• Enrichment conforms to the

Discontinuity under someRestrictions away from the Tip.

0 1 2 3 4 5 6

−2

−1

0

1

2

Alternative 2–b• Enrichment conforms to the

Discontinuity.

12


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Stress Intensity Factors• Inclined Center Crack: β = 40.

around the crack tip, where 9! 9 Gauss quadrature wasused. Quadratic XFEM seems to perform as well as the EFGmethod. While the EFG method tends to overestimate thestress intensity factors, quadratic XFEM tends to under-estimate the result.

6.3Edge crack under shear stressA plate is clamped on the bottom and loaded by a sheartraction s " 1:0 psi over the top edge. The material pa-rameters are 3! 107 psi for Young’s modulus and 0.25 forPoisson’s ratio. The reference mixed mode stress intensityfactors as given in [35] and [31] are:

KI " 34:0 psi!!!!!

inp

KII " 4:55 psi!!!!!

inp

The equivalent stress intensity factor Keq obtained fromthe J-integral for plain strain problem is:

J " 1# m2

EK2eq $46%

The equivalent stress intensity factor is compared to!!!!!!!!!!!!!!!!!!!!!

$K2I & K2

II%p

using an elliptic criterion described in Bazant[1]. In Fig. 17, we can see that the quadratic elementconverges slightly faster that the linear element and ismore accurate. In Fig. 18, KI and KII is seen to exhibit thesame behavior.

6.4Mixed mode crack in infinite bodyThe problem of an angled center crack in a body wasconsidered as shown in Fig. 19. The plate is subjected to afar-field state of stress r equal to unity. The crack is oflength 2a and is oriented with an angle b with respect tothe x-axis. The material parameters are 3! 107 psi forYoung’s modulus and 0.25 for Poisson’s ratio. The stressintensity factors KI and KII are given in terms of the angleb by Yau et al. [35] and Dolbow et al. [11].

KI " r!!!!!!!!!

$pa%p

cos2$b% $47%

KII " r!!!!!!!!!

$pa%p

sin$b% cos$b% $48%where a is the half crack length. For the computations bwas chosen to be 41:9872'.

Figure 20 shows the convergence for KI and KII .Good accuracy is obtained for a reasonable number ofnodes. The stress intensity factors are computed by an

Table 1. Stress intensity factors computed by quadratic XFEMcompared to EFG

Cracklength

KI XFEM(linear)

KI XFEM(quadratic)

KI EFG byBelytschkoet al. [7]

KI exact

0.21 1.0616 1.1243 1.1401 1.13410.22 1.1000 1.1691 1.1779 1.18160.23 1.1321 1.2187 1.2487 1.23030.24 1.1558 1.2707 1.2807 1.27880.28 1.3783 1.4760 1.5036 1.49350.50 3.1299 3.5064 3.5512 3.5423

Fig. 17. Convergence for edge crack under shear. K is computedfrom the J-integral

Fig. 18. Convergence for edgecrack under shear. KI and KII

computed by the interactionintegral

Fig. 19. Discretization used for angled crack in a plate underuniaxial tension

45

Alternative 2–b• SIFs Match.

13


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Stress Intensity Factors• Inclined Center Crack: β = 40.

around the crack tip, where 9! 9 Gauss quadrature wasused. Quadratic XFEM seems to perform as well as the EFGmethod. While the EFG method tends to overestimate thestress intensity factors, quadratic XFEM tends to under-estimate the result.

6.3Edge crack under shear stressA plate is clamped on the bottom and loaded by a sheartraction s " 1:0 psi over the top edge. The material pa-rameters are 3! 107 psi for Young’s modulus and 0.25 forPoisson’s ratio. The reference mixed mode stress intensityfactors as given in [35] and [31] are:

KI " 34:0 psi!!!!!

inp

KII " 4:55 psi!!!!!

inp

The equivalent stress intensity factor Keq obtained fromthe J-integral for plain strain problem is:

J " 1# m2

EK2eq $46%

The equivalent stress intensity factor is compared to!!!!!!!!!!!!!!!!!!!!!

$K2I & K2

II%p

using an elliptic criterion described in Bazant[1]. In Fig. 17, we can see that the quadratic elementconverges slightly faster that the linear element and ismore accurate. In Fig. 18, KI and KII is seen to exhibit thesame behavior.

6.4Mixed mode crack in infinite bodyThe problem of an angled center crack in a body wasconsidered as shown in Fig. 19. The plate is subjected to afar-field state of stress r equal to unity. The crack is oflength 2a and is oriented with an angle b with respect tothe x-axis. The material parameters are 3! 107 psi forYoung’s modulus and 0.25 for Poisson’s ratio. The stressintensity factors KI and KII are given in terms of the angleb by Yau et al. [35] and Dolbow et al. [11].

KI " r!!!!!!!!!

$pa%p

cos2$b% $47%

KII " r!!!!!!!!!

$pa%p

sin$b% cos$b% $48%where a is the half crack length. For the computations bwas chosen to be 41:9872'.

Figure 20 shows the convergence for KI and KII .Good accuracy is obtained for a reasonable number ofnodes. The stress intensity factors are computed by an

Table 1. Stress intensity factors computed by quadratic XFEMcompared to EFG

Cracklength

KI XFEM(linear)

KI XFEM(quadratic)

KI EFG byBelytschkoet al. [7]

KI exact

0.21 1.0616 1.1243 1.1401 1.13410.22 1.1000 1.1691 1.1779 1.18160.23 1.1321 1.2187 1.2487 1.23030.24 1.1558 1.2707 1.2807 1.27880.28 1.3783 1.4760 1.5036 1.49350.50 3.1299 3.5064 3.5512 3.5423

Fig. 17. Convergence for edge crack under shear. K is computedfrom the J-integral

Fig. 18. Convergence for edgecrack under shear. KI and KII

computed by the interactionintegral

Fig. 19. Discretization used for angled crack in a plate underuniaxial tension

45

Alternative 2–b• SIFs Match.

13


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Stress Intensity Factors• Curved Center Crack: β = 30

interaction integral as described in Sect. 5. For coarsediscretization, the values for KI are more accurate thanthe values for KII . This is probably due to the fact thatthe computations were made in a finite body. If a largermodel relative to the crack length were used this differ-ence would have been less noticeable. The results givenin Table 2 also show very good symmetry in the behaviorat the two tips.

6.5Center crack in a finite plateThe problem of a finite plate with a center crack wasstudied [10]. The geometry of the plate is described inFig. 21. The analytical solution to this problem is given inSuo and Combescure [10]. The stress intensity factor isgiven by:

KI ! r

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

pa secpa2w

" #" #

r

"49#

where a is the half crack-length and w ! W=2 is the halfwidth of the plate, and r is the tensile load applied at thetop of the plate.

Figure 22 shows the improved accuracy and conver-gence of the quadratic element over the linear element.Again, we can observe symmetric behavior at the twocrack tips.

Fig. 20. Stess intensity factorserror for the angled crack ininfinite plate. KI (left) and KII

(right) computed by the inter-action integral; ‘‘tip 1’’ and ‘‘tip2’’ refer to the two crack tips

Table 2. Stress intensity factors for angled center crack byquadratic elements

NumNodes

KIKanaI

tip 1 KIKanaI

tip 2 KIIKanaII

tip 1 KIIKanaII

tip 2 Mesh

1661 0.6619 0.6647 0.2207 0.2205 11 · 212377 0.6916 0.6940 0.7279 0.7278 13 · 253449 1.0464 1.0491 1.1171 1.1172 15 · 314193 1.0260 1.0288 1.0670 1.0673 17 · 335293 1.0224 1.0251 1.0512 1.0515 19 · 37

Fig. 21. Finite plate containing a centered crack

Fig. 22. Stress intensity factor error for a centered crack in afinite plate

Fig. 23. Curved crack in an infinite plate

46

• SIFs Convergence.

14


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Stress Intensity Factors• Curved Center Crack: β = 30

interaction integral as described in Sect. 5. For coarsediscretization, the values for KI are more accurate thanthe values for KII . This is probably due to the fact thatthe computations were made in a finite body. If a largermodel relative to the crack length were used this differ-ence would have been less noticeable. The results givenin Table 2 also show very good symmetry in the behaviorat the two tips.

6.5Center crack in a finite plateThe problem of a finite plate with a center crack wasstudied [10]. The geometry of the plate is described inFig. 21. The analytical solution to this problem is given inSuo and Combescure [10]. The stress intensity factor isgiven by:

KI ! r

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

pa secpa2w

" #" #

r

"49#

where a is the half crack-length and w ! W=2 is the halfwidth of the plate, and r is the tensile load applied at thetop of the plate.

Figure 22 shows the improved accuracy and conver-gence of the quadratic element over the linear element.Again, we can observe symmetric behavior at the twocrack tips.

Fig. 20. Stess intensity factorserror for the angled crack ininfinite plate. KI (left) and KII

(right) computed by the inter-action integral; ‘‘tip 1’’ and ‘‘tip2’’ refer to the two crack tips

Table 2. Stress intensity factors for angled center crack byquadratic elements

NumNodes

KIKanaI

tip 1 KIKanaI

tip 2 KIIKanaII

tip 1 KIIKanaII

tip 2 Mesh

1661 0.6619 0.6647 0.2207 0.2205 11 · 212377 0.6916 0.6940 0.7279 0.7278 13 · 253449 1.0464 1.0491 1.1171 1.1172 15 · 314193 1.0260 1.0288 1.0670 1.0673 17 · 335293 1.0224 1.0251 1.0512 1.0515 19 · 37

Fig. 21. Finite plate containing a centered crack

Fig. 22. Stress intensity factor error for a centered crack in afinite plate

Fig. 23. Curved crack in an infinite plate

46

Alternative 2–b Alternative 2–c

• SIFs Convergence.

14


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

Conclusions

• New Bases System (φ ,γ): Triangles and a Quadrilateral.

• Improved Definition of γ.

• Improved Stress Intensity Factors Convergence Rates.

• Improved Evaluation of Enrichment Functions.

• Flexibility and Multiplicity of Alternatives.

15


Curved Cracks


Motivation

New Alternative

OtherAlternatives



Radius Enrichment


Conclusions

THANK YOU FOR YOUR ATTENTION !

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The XFEM using Crack Tip Enrichment with Large Support for ... · XFEM with Large Support for Curved Cracks M. Baydoun T.P. Fries Motivation New Alternative Other Alternatives Studies