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An Extended Bridging Domain Method for Modeling Dynamic Fracture

Hossein Talebi. An Extended Bridging Domain Method for Modeling Dynamic Fracture. Outline. Introduction Multiscale Modeling of Fracture The Bridging Domain Method Governing Equations Implementation Aspects Numerical Example Future Challenges. Multiscale Modeling of Fracture.

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An Extended Bridging Domain Method for Modeling Dynamic Fracture

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  1. Hossein Talebi An Extended Bridging Domain Method for Modeling Dynamic Fracture

  2. Outline • Introduction • Multiscale Modeling of Fracture • The Bridging Domain Method • Governing Equations • Implementation Aspects • Numerical Example • Future Challenges

  3. Multiscale Modeling of Fracture

  4. Multiscale Modeling of Fracture • The global response of the system is often governed by the behavior at the smaller length scales(eg. shear bands). • A more fundamental understanding on the phenomenon ‘material failure’. • Subscale behavior must be computed accurately for good predictions of the full scale behavior. • The most accurate and versatile method of modeling material failure is with Molecular dynamics. • Often, with the current computer capacity, one can model a very tiny fraction of the material and that comes with high costs. • Therefore it makes sense to model only the hotspots like crack tip areas and the rest with continuum models.

  5. The bridging domain Method

  6. Governing Equations • With FE approximation and in the continuum domain we have: • The Hamiltonian of the system will be: • The Hamiltonian of the continuum domain will be: and p is linear momentum and W is the internal energy(strain energy).

  7. Governing Equations • In the Molecular dynamics region, the motion of particles is computed via classical MD equation of motion and a potential e.g. the Lennard-Jones potential: • The hamiltonian of the MD domain is: where is dirac delta function, M is mass of the atom and W is the potential of the bond joining atoms i and j.

  8. The bridging Domain The key concept here is that the total Hamiltonian is a varying combination of the two Hamiltonians in the overlapping subdomain.

  9. Governing Equations • To enforce the compatibilty between the two domains Lagrange multipliers are used. • The total Hamiltonian of the system is then: • Where lambda is the Lagrange multiplier (called interaction energy)

  10. Governing Equations • The Lagrangian of the system is then: • The equation of motion can be obtained by: where q=[d u], ie all displacement degrees of freedom.

  11. Semi-discrete equations

  12. Semi-discrete equations • And the corrector forces are: • P is the nominal stress and it is obtained from the Cauchy-Bond rule. For the LJ potential it is: • The Cauchy-Born rule is valid only in small deformation.

  13. Time integration • We use the Verlet Method:

  14. the lagrange multipliers

  15. Implementation • We need: • Continuum FEM/XFEM in 3D • MD implementation which can handle more than 1 potential (LJ and EAM minimum) • MD implementation should not be slow and naive(possibly parallel) • A proper post-processing (XFEM-MD) • Future Extensions are possible for coarsening and refinement.

  16. Implementation Aspects • Molecular Dynamics: • Q: Implement or use a library? LAMMPS? A: Library • Q: Which Molecular Dynamics library to use? A: Warp(Fortran 90) • Q: How easy is the implementation, changes, communication? :Modify Warp(Fortran2003)

  17. Implementation Aspects • Continuum: • Q: Can we use a commercial product? Eg. Abaqus A: No(limitations, commercial results!) • Q: How to do Preprocessing XFEM and finding Level-sets? A: Use Abaqus INP files • Q: How to visualize XFEM? A: Implement yourself in Tecplot

  18. Full MD results • Potential: Aluminum(3.986) EAM • Full Region: 398.6 x398.6x398.6 • Uncoupled full Atomistic:4020000 • Atoms with high Centro-symmetry is shown

  19. The Example

  20. Example Specifications • Dimensions of the whole domain are: 1000x1000x150 angestroms • Crack length is 500 through the whole domain • The Full atomistic domain is 365x365x150 • The Lennard-Jones potential is used with sigma=2.29,epsilon=.467 and cut-off redaius of 4.0 • Atomic mass is 65 g/mol • 1368575 active atoms, 231890 bridging atoms and 308067 ghost atoms

  21. Crack and Dislocation Propagation • Atoms with high centro-symmetry value are shown. • Note, atoms in the bridging region are not shown

  22. Crack and Dislocation Propagation

  23. Atomic Stress Plot

  24. Atomic Stress Plot

  25. Atomic Stress Plot

  26. Atomic Stress Plot

  27. Future challenges • Adaptive refinement of the MD region • Detection of cracks and dislocations in the MD domain • Coarse Graining of the detected cracks and dislocations to the continuum domain • Parallelization of the code to run sizes close to macroscopic scale.

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