Virtual Crack Closure Technique Based on Meshless Shepard Interpolation Method (MSIM )

1. Virtual Crack Closure Technique Based on Meshless Shepard Interpolation Method (MSIM) Ph.D. Feng Su1,2, Jie Wu1,2,*, Prof. Yongchang Cai1,2 1State Key Laboratory for disaster reduction in Civil Engineering 2Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education Tongji University June 17, 2013 Presented to: 13th International Conference on Fracture *Corresponding author:Jie.voo@gmail.com

2. Motivations • Fractures widely exist in materials such as Rock mass and concrete. • The accurate calculation of the stress intensity factor is critical important and is hence of great interest to researchers. • An simple and efficient numerical calculation method for crack is desirable to the practical modeling of the complex geotechnical engineering.

3. Outline • General review of representative methods modeling fractures • Meshless Shepard interpolation method (MSIM) • Virtual crack closure technique & J integral • Numerical investigation • conclusions

4. How to numerically simulate the discontinuities? General review of representative methods modeling fractures

5. Finite element method (FEM): • ‘joint element’ or ‘interface element’ • mesh coincides with the fractures, meshing complicated; • remeshing, simulation tedious, time-consuming • Modifications to the FEM • XFEM: incorporate enrichment functions to represent discontinuities; • GFEM: incorporate high-order terms or handbook functions to tackle multiple corners, voids, cracks, etc. • Boundary element method(BEM): • not efficient in dealing with material heterogeneity and non-linearity ;

6. Numerical manifold method The numerical manifold method was first proposed by Shi (1991) • A regular mesh is adopted throughout the calculation • The discontinuity can be treated in a straightforward manner • The generation of the finite cover system is complex which is hard to be applied to 3D analysis

7. Meshless method • Advantage • h-adaptivityis simpler to incorporate in MMs than in mesh-based methods, • Problems with moving boundaries can be treated with ease • Large deformation can be handled more robustly • Higher-order continuous shape functions • Disadvantage • Higher computational cost compared with FEM • Difficulties with the treatment of essential boundary

8. Meshless Shepard interpolation method Interpolations/weight functions & discrete equations

9. 100 i(r) 0 r 0 1 PU-based interpolation Cover Ci not on the boundary Cover Cj on the boundary Shepard function Singular at Xi Subdomain:

10. Cover interpolation forNode i Similar procedure is implemented in y-direction where: It means that the cover function of the node is interpolated in terms of nodal displacements .

11. Cover interpolation for Node j Minimization , where: This approximation does not fit the nodal displacement values A modification is made like this:

12. Treatment of discontinuity Concepts of the mathematical and physical cover in NMM is employed to express the discontinuity Crack 1 Crack 1 Crack 2 Gauss point Crack 1 Crack 2

13. Virtual crack closure technique & J integral Differences are investigated by a numerical example

14. y x J integral • Developed by Cherepanov,1967 & by Jim Rice,1968,independently • Accurate and widely used in the calculation of stress intensity factor • Path-independent • However, the broken crack cannot be properly simulated by this method • Decreasing the radius of the contour path and enrich function will often be employed. Contour path crack

15. y x Virtual crack closure technique (VCCT) • Assumption: • The energy released when crack extended from i to j is identical to the energy required to close the crack between i and j • The two displacementsare approximately the same. j m i • Only the node displacement and force are required • Fracture mode separation is determined explicitly • The calculation results are free of the affect of crack length • Always incorporated in FEM, and the remeshing cannot be avoided a ∆a ∆a

16. Implementation VCCT in MSIM • Calculate the constructed MSIM model; • Set the assistant mesh near the crack tip • Acquire the nodal displacement U in the assistant mesh based on the MSIM calculation result. • Construct the global stiffness matrix K of the assistant mesh • Get the nodal force near the crack tip by F=K*U • Calculate the SERR • Transform SERR into SIF by

17. Numerical investigations Cracked plate under remote tension & A star-shaped crack in a square plate under bi-axial tension

18. Cracked plate under remote tension =1kPa =1*107 Pa, u=0.3 h=5m, W=5m Vary a from 0.1W to 0.8W J integral V.S. VCCT Collocation method J integral (linear basis) VCCT (linear basis) VCCT (enrich basis) J integral (linear basis) VCCT (linear basis) VCCT (enrich basis) The relative error of F1 Remark: • Accurate results can be get from both of the methods • The relative error get from J integral seen a big change with the increase of the crack length, while that get from VCCT is more stable SIF F1 Relative error of F1 Geometric and MSIM model

19. A star-shaped crack in a square plate under bi-axial tension =1kPa =1*107 Pa, u=0.3 W=2m, =60 Normalized SIF: Geometric and MSIM model The relative error of FIA The relative error of FIA VCCT (linear basis) VCCT (Enriched basis) Relative error of FIA

20. Star-shaped crack (cont.) VCCT (linear basis) The relative error of FIB The relative error of FIB VCCT (Enriched basis) Relative error of FIB VCCT (Enriched basis) The relative error of FIIB The relative error of FIIB VCCT (linear basis) Relative error of FIIB

21. Conclusions The essential boundary could be easily imposed in MSIM due to delta property The discontinuity problem could be well addressed in the MSIM Only the node displacement and force are required in VCCT, and fracture mode separation is determined explicitly The relative error get from J integral seen a big change with the increase of the crack length, while that get from VCCT is more stable Further investigation results show that the VCCT in the framework of MSIM is prominent in modeling complex crack problem.

22. Thanks you very much The authors gratefully acknowledge the support of: Program for New Century Excellent Talents (NCET-12-0415) National Science and Technology Support Program (2011BAB08B01) & Fundamental Research Funds for the Central Universities.