An Extended Bridging Domain Method for Modeling Dynamic Fracture Hossein Talebi

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

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.

The bridging domain Method

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).

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.

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

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)

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.

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.

Time integration  We use the Verlet Method:

the lagrange multipliers

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.

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)

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

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

The Example

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 active atoms, bridging atoms and ghost atoms

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

Atomic Stress Plot

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.