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

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 DomainThe 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

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

ImplementationWe 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 AspectsMolecular 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 AspectsContinuum: 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: 398.6 x398.6x398.6 Uncoupled full Atomistic:4020000 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 1368575 active atoms, 231890 bridging atoms and

308067 ghost atoms

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

Crack and Dislocation Propagation

Crack and Dislocation Propagation

Atomic Stress Plot

Atomic Stress Plot

Atomic Stress Plot

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.