![Page 1: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/1.jpg)
A tutorial on multiple crack growth and intersections with XFEM
Danas Sutula
Prof. Stéphane Bordas
Dr. Pierre Kerfriden
24/03/2015
![Page 2: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/2.jpg)
Content
1. Problem statement
2. Crack growth
3. XFEM discretization
4. Results/verification
5. Summary
6. Appendix: Review of crack intersection management
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
2
![Page 3: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/3.jpg)
• Consider a cracked linear-elastic isotropic solid subject to an external load whose quasistatic behavior can be described by the following total Lagrangian form:
Problem statement
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
3
• Consider a cracked linear-elastic isotropic solid subjected to an external load whose quasistatic behavior can be described by the following total Lagrangian form:
• The solutions for and are obtained by satisfying the stationarity of during the evolution of , subject to :
Danas Sutula (MMAM)
![Page 4: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/4.jpg)
• Consider a cracked linear-elastic isotropic solid subject to an external load whose quasistatic behavior can be described by the following total Lagrangian form:
• The solution for u(a) and a(t) are obtained by satisfying the stationarity of L(u,a) during the evolution of t, subject to Δai ≥ 0:
Problem statement
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
4
• Consider a cracked linear-elastic isotropic solid subjected to an external load whose quasistatic behavior can be described by the following total Lagrangian form:
• The solutions for and are obtained by satisfying the stationarity of during the evolution of , subject to :
Danas Sutula (MMAM)
![Page 5: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/5.jpg)
Crack growth maximum hoop stress
• Post processing of solution to evaluate SIF
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
5 Danas Sutula (MMAM)
![Page 6: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/6.jpg)
Crack growth maximum hoop stress
• Post processing of solution to evaluate SIF
• Crack growth direction
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
6 Danas Sutula (MMAM)
![Page 7: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/7.jpg)
Crack growth maximum hoop stress
• Post processing of solution to evaluate SIF
• Crack growth direction
• Crack growth criterion
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
7 Danas Sutula (MMAM)
![Page 8: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/8.jpg)
• Energy release rate w.r.t. crack increment direction, θi:
Crack growth energy minimization
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
8 Danas Sutula (MMAM)
![Page 9: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/9.jpg)
• Energy release rate w.r.t. crack increment direction, θi:
Crack growth energy minimization
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
9 Danas Sutula (MMAM)
![Page 10: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/10.jpg)
• Energy release rate w.r.t. crack increment direction, θi:
• The rates of energy release rate:
• Updated directions:
Crack growth energy minimization
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
11 Danas Sutula (MMAM)
![Page 11: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/11.jpg)
Discretization The eXtended Finite Element Method (XFEM)
• Approximation function (single crack)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
12
singular tip enrichment discontinuous enrichment standard part
Danas Sutula (MMAM)
FEM
XFEM
![Page 12: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/12.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
13 Danas Sutula (MMAM)
Discretization The eXtended Finite Element Method (XFEM)
![Page 13: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/13.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
14
Danas Sutula (MMAM)
Discretization The eXtended Finite Element Method (XFEM)
![Page 14: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/14.jpg)
Results pressure driven growth of 10 random cracks
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
15 Danas Sutula (MMAM)
![Page 15: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/15.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
16 Danas Sutula (MMAM)
Results pressure driven growth of 10 random cracks
![Page 16: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/16.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
17 Danas Sutula (MMAM)
Results pressure driven growth of 10 random cracks
![Page 17: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/17.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
18 Danas Sutula (MMAM)
Results pressure driven growth of 10 random cracks
![Page 18: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/18.jpg)
Results tension splitting of plate containing 10 cracks
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
19
![Page 19: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/19.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
20 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 20: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/20.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
21 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 21: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/21.jpg)
Results 10 crack problem
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
22 Danas Sutula (MMAM)
![Page 22: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/22.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
23 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 23: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/23.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
24 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 24: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/24.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
25 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 25: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/25.jpg)
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
26 Danas Sutula (MMAM)
Results tension splitting of plate containing 10 cracks
![Page 26: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/26.jpg)
1. Robust approach to determining multiple crack growth directions based on the principle of minimum global energy
2. The criteria converge to fracture paths that are, in the global sense, in close agreement. Consequence of local-symmetry.
3. The criteria can be used to estimate the upper/lower bound of the true fracture path for smooth crack growth problems
Summary
Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
27 Danas Sutula (MMAM)
Theoretical Limitations…
![Page 27: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/27.jpg)
Managing crack intersections overview
• Elements that feature crack intersections need multiple jump enrichments to capture the kinematics of crack opening correctly (Daux et al. 2000, Budyn et al. 2004)
• Multiple jump enrichments are superposed on an element by extending the element’s approximation space for each crack that the element contains whilst averting linear dependence between approximations.
• A dedicated book-keeping practice is required to efficiently manage multi-layer enrichments (e.g. as pertains to intersections) in a topologically consistent manner with regard to each crack occurrence.
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
28
![Page 28: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/28.jpg)
Managing crack intersections merging cracks in 2D
• Minimum distance criterion:
– intersection of crack A onto crack B is forced once the distance between A’s tip and B’s surface becomes less than a prescribed tolerance (e.g. typically, the size of A’s tip enrichment radius)
– crack A is extended normal to B’s surface and deflected along it
• As intersection happens:
– intersection between crack A and crack B is registered once it is detected that A’s tip increment crosses B thereby forming an ‘X’-type intersection
– crack A is pulled back until it’s tip lies on B; A is then extended along B
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
29
![Page 29: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/29.jpg)
Managing crack intersections blending crack intersections
• The deflected fracture extent of A is called the blending region of the fracture junction. The size of it needs to be large enough to smoothly merge A onto B
• Crack merging is accommodated by blending elements that lie on the blending region a short distance from the junction.
• Blending elements are jump-enriched elements that are cut by the blending region and that have at least one node whose support is cut by the part of A that can be thought as the relative complement of A with respect to B.
• In a 3D context, where cracks are polygonal surfaces, the approach to crack intersections is analogous to 2D in that the surface of crack A needs to be deflected along surface B followed by an effective blending procedure.
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
30
![Page 30: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/30.jpg)
Managing crack intersections multi-jump enrichment
• The discontinuous part of the displacement approximation over an element cut by several cracks can be expressed (using shifted enrichment) as follows:
• If the element serves to blend a particular crack (e.g. A) onto another crack (e.g. B), some of the enriched DOFs pertaining to A will have to be set to zero.
• This is to prevent linear dependence between the enriched shapes relating to A with those relating to B, because they are identical by virtue of A perfectly overlying B in the blending region.
• The blending zero-degrees of freedom are determined as those DOFs whose corresponding nodal support is not cut by the main branch of crack A, yet the DOF belongs to at least one element that has at least one of its nodal supports cut by the main branch of A
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
31
![Page 31: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/31.jpg)
Managing crack intersections system updating
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
32
• Evaluate the fracture growth criterion to determine which crack grow
A snap-shot during crack evolution
(fig. shows enriched elements)
![Page 32: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/32.jpg)
Managing crack intersections system updating
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
33
• Evaluate the fracture growth criterion to determine which crack grow
• Advance cracks
A snap-shot during crack evolution
(fig. shows enriched elements)
![Page 33: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/33.jpg)
Managing crack intersections system updating
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
34
A snap-shot during crack evolution
(fig. shows enriched elements)
• Evaluate the fracture growth criterion to determine which crack grow
• Advance cracks
• Use an intersection criterion to merge cracks (e.g. min. dist.)
![Page 34: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/34.jpg)
Managing crack intersections system updating
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
35
A snap-shot during crack evolution
(fig. shows enriched elements)
• Evaluate the fracture growth criterion to determine which crack grow
• Advance cracks
• Use an intersection criterion to merge cracks (e.g. min. dist.)
• Update enrichment topology
![Page 35: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/35.jpg)
Managing crack intersections system updating
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
36
A snap-shot during crack evolution
(fig. shows enriched elements)
• Evaluate the fracture growth criterion to determine which crack grow
• Advance cracks
• Use an intersection criterion to merge cracks (e.g. min. dist.)
• Update enrichment topology
close-up
![Page 36: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/36.jpg)
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
37
crack (B) crack (A)
Managing crack intersections system updating (close-up view)
![Page 37: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/37.jpg)
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
38
crack-A is deflected along crack-B to from a blending region
crack (B) crack (A)
Managing crack intersections system updating (close-up view)
![Page 38: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/38.jpg)
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
39
crack (B) crack (A)
Managing crack intersections system updating (close-up view)
Doubly jump-enriched elements for crack-A,B. Full element support for both cracks
Doubly jump-enriched elements for crack-A,B. Partial element support for crack A (a blending element)
![Page 39: A tutorial on multiple crack growth and intersections with ... tutorial on... · A tutorial on multiple crack growth and intersections with XFEM Danas Sutula Prof. Stéphane Bordas](https://reader036.vdocument.in/reader036/viewer/2022062413/5ad9c06e7f8b9a53618bad32/html5/thumbnails/39.jpg)
Danas Sutula (MMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM
40
Heaviside blending elements (mc): at least one node has support for the minor crack
crack (B) crack (A)
Enriched nodes for crack-A: element nodal support is cut by the main branch of A
Blending nodes for crack-A: nodal support is entirely within blending region (set DOFs to zero)
Managing crack intersections system updating (close-up view)