Green’s Functions method for modeling of materials
Principal Investigator: Vinod Tewary- Materials Reliability Division, NIST, Boulder.
Workshops:
Green's function gives response to a probe and is also called response function. Our interest is in the elastic response of a solid to a mechanical force. For time dependent forces, we use dynamic or time-dependent Green's function. The static Green's functions are used for time independent problems. In case of dynamic Green's functions, our interest is only in the causal Green's functions.
At the atomistic level, the response of a crystal lattice is given by the lattice Green's function that we calculate using the Born-von Karman model of a crystal lattice. A defect or a discontinuity in the lattice may exert the force that causes lattice distortion. The lattice statics Green's function method is used to calculate the lattice distortion. The physical measurements that are sensitive to lattice distortions are, for example, neutron or X-ray scattering cross-sections. Lattice static Green's function depends upon interatomic potentials and detailed lattice structure.
The bulk properties of solids, such as elastic constants, can be modeled by assuming the solid to be a continuum of matter. In this case, we use the continuum model. The continuum model Green's function or the elastic Green's function is determined by the elastic constants of the solid. The elastostatic Green's functions are used to calculate the stress distribution in a solid, fracture properties etc. The elastodynamic Green's function is used for calculating elastic wave forms in solids and non-destructive characterization of solids.
The continuum or the elastostatic Green's function is the asymptotic limit of the lattice statics Green's function. Thus, the Green's function method provides a convenient framework for an integrated model of a solid or a unified formulation that can model a solid in different physical regions and over different length scales.
A major computational advantage of the Green’s function is that it is a characteristic of the material and its geometry, and is independent of the probe. The Green’s function can be calculated in steps of increasing geometrical or structural complexities using the previous value as input. It is, therefore, possible to calculate and store the Green’s functions for basic geometrical shapes and structures and different material parameters, for use in further calculations. Thus it is useful to set up a library of Green's functions. Lattice static Green's functions for several fcc and bcc solids will be made available at this site in near future.
In the continuum case, the strategy is that we calculate the Green’s function analytically for simple geometrical shapes and use the boundary element method for application to complicated geometrical shapes for engineering applications. We are in the process of setting up a library of pre-computed elastic Green's functions using boundary element method for materials of industrial interest on this site. The work on this project was done in collaboration with Frank Rizzo of Iowa State University, who proposed the idea of a library of Green's function for the continuum case, and Prof. John Berger of Colorado School of Mines.