## Multiscale Modeling, with Applications in Contact Mechanics

Bearbeitung: | Prof.Dr.-Ing. U. Nackenhorst, Dr.-Ing W. Shan |

Förderung durch: | Deutsche Forschungsgemeinschaft (DFG), Graduiertenkolleg 614 (GRK615) |

Contact problems are typical problems with more than one characteristic length scales which motivate the development of multiscale methods. Far away from the contacting surface, material can be usually approximated as solid continuum and simulated by the finite element (FE) method. However, for the region near the contacting surface, the continuum approximation is not good enough when we are interested in the material behavior at the microscale. Rough surface profile at microscale results in high local contact pressure and makes studying what really happens at the contacting surface at microscale almost impossible by using traditional FE methods based on continuum mechanics, Fig. 1. On the other hand, the constitutive contact laws used in the continuum mechanics are not good enough for describing the contact interaction at microscale. Molecular dynamics (MD) which explicitly models each atom gives much more accurate description of material behavior than the continuum mechanics does. However, for our current computer technology, it is too expensive to be commonly applied for regular simulations. Therefore, it is better to apply them only to where they are needed and leave the rest to the continuum mechanics.

Fig. 1: Contact problem with realistic surface profile at microscale.

For the contact problem, the place of interest where MD model should be applied is the region near the contacting surface, where the surface profile at microscale can be realisticly modeled and the atomic potentials can be used as the contact interaction, Fig. 2.

Fig. 2: Indentaion model (left) and the FE/MD coupled model (right) where the MD model is applied to the region near the contact surface.

To replace the constitutive contact laws by the atomic potential, a contact region is defined to include all the surface atoms that are likely to be in contact and then the interaction neighbor-list is built for evaluating the atomic interaction in an efficient way (O(N)), Fig. 3.

Fig. 3: Predefined contact region (left, shadowed) and the atoms in contact (right, connect by red lines).

The FE/MD coupled model is formulated within a modified Quasicontinuum (QC) framework, where the material properties of the FE model are obtained from an underlying lattice structure which is consistent with the MD model, by applying the Cauchy-Born rule which maps the distance vector between atoms into deformation gradient when the lattice is assumed to be perfect and under uniform deformation, Fig. 4.

Fig. 4: Demonstration of applying the Cauchy-Born rule where (a) is the representative atom, (b) is the reference lattice used to obtain the material properties and (c) is the entire lattice approximated by solid continuum.

The FE and MD models are coupled by the dummy-atoms created within the elements of the FE model at the coupling region, where the position of the dummy-atoms are interpolated from the nodal positions and the dummy-atoms interact with the real atoms in the MD model via atomic potential, Fig. 5.

Fig. 5: Coupling of FE and MD models: direct nodal coupling (left, used in the original QC method) and the indirect coupling via dummy-atoms (right, used in our work).

The simulated indentation process is shown in Fig. 6, where the displacement in the z-direction u_{3} and the adaptive propagation of the MD model are demonstrated. The displacement field translates smoothly from the FE model to the MD model and the conversion of the FE model to the MD model takes place in the region right below the indentation tip, which is expected to be the critical region in the model. As the indentation depth increases, the MD model propagates further into the main body. Moreover, due to the indirect coupling technique, the mesh condition of the FE model is not significantly disturbed during the refinement.

Fig: 6: Indentation process simulated by the coupled FE/MD model: displacement field (top) and the adaptive propagation of the MD model (bottom).

The load-displacement curve, Fig. 7, indicates a close match between the solution obtained from the coupled FE/MD model and the lattice solution obtained from the fully atomic model which consists of 36610 atoms, while the coupled FE/MD model contains no more than 4052 nodes dureing the entire process. The total computational time for the coupled FE/MD model is about 45 minutes, impelemented in MATLAB, while the fully atomic model takes about 15 hours. Due to the application of the neighbor-list algorithm, the computational cost of the FE/MD model increases linearly with the size of the model, Fig. 8.

Fig. 7: Load-displacement curves for the indentation test.

Fig. 8: Computation time per iteration step for the adaptive, coupled FE/MD model for the indentation example.