Almost all problems in science and engineering are multiscale in nature. Things are made up of electrons and atoms at the atomic scale, while at the same time they are characterized by their own geometric dimensions that are usually several orders of magnitude larger. Therefore, to model the plasticity of a material, several length scales are involved. We need to find the scale of (i) the crystal lattice, (ii) the dislocation core, (iii) the mean distance between dislocations, (iv) the grain size and (v) the dimension of the structure. The first two scales can be studied by either first principle calculations or molecular statics/dynamics calculations.
However, when molecular statics/ dynamics are considered, the interaction forces between neighboring atoms are calculated based on a semi-empirical potential. As first principle calculations are limited to a few hundred atoms, which give too small a representative volume element to study the plasticity, the semi-empirical potential constitutes the first bridge of a hierarchical multiscale framework. Depending on the accuracy of the potential, it can be used to model the mechanical behavior of a system containing several million atoms. However, as the characteristic length that is accessible using molecular statics/dynamics modeling is smaller than the mean free path of the dislocations, the yield stress and hardening are not controlled by the dislocation interactions but by the nucleation of dislocations, and therefore a strong size effect is usually observed. To model the mechanical behavior of a material at a length scale, where the hardening is controlled by the dislocation interactions, dislocations are modeled by lines of singularity in an elastic continuum and their dynamics are solved using a discrete dislocation framework. Although other mechanisms, such as twinning deformation or grain boundary sliding, can accommodate the deformation, only dislocation motion and interaction are taken into account into the discrete dislocation framework. Due to the large amount of degree of freedom involved in a discrete dislocations simulations, this method is limited to volume of dimension 10 microm. Therefore, to model polycrystalline materials with grain size in the range of approximately 10-100 microm, the number of degree of freedom needs to be decrease, and in that case, the mechanical behavior can be predicted using a polycrystalline plasticity model.
- Dislocation Dynamics: Numerical methods to model the hardening of a materials.
- Density Functional Theory: Numerical tools to model the collective behavior of atoms by solving the Schrodinger equations.
- Molecular Dynamics: Numerical methods to model the collective behavior of atoms using semi-empirical potentials to describe the interaction between atoms.
- Finite Element Method: Numerical method to find approximate solution of a system of partial differential equations.