Numerical Constitutive Modelling for Continuum Mechanics Simulation
Three investigations are described in this dissertation, on the common theme of obtaining constitutive laws describing bulk properties of crystalline materials. We use ‘first-principles’ techniques where possible, an approach offering results which are predictive, with applicability to a wide class of materials, and with a systematic way to apply the techniques to any particular material of interest. We apply these techniques to silicon.
The first investigation aims at developing an equation of state model for temperature dependent, anisotropic non-linear hyperelasticity. A method is presented for finding deformed states of a material on the same isentrope as a given starting configuration. The energies and stresses of a number of elastic deformations are sampled from dft molecular dynamics using this method, over a given range of the seven-dimensional space of deformation and potential temperature. The complete energy surface within this range can then be reliably reconstructed using the technique of Gaussian process regression. This is a machine learning technique that has particular merit here due to its ability to reconstruct a smooth surface without over-fitting. An equation of state model is then constructed for dft silicon, and demonstrated within a finite-volume continuum elasticity simulation for several problems of interest involving shock waves.
The second investigation is concerned with the computation of properties of shock waves. We describe a simple annealing procedure to obtain the Hugoniot locus (states accessible by a shock wave) for a given material in a computationally efficient manner, particularly suited to first-principles calculations. We apply this method to determine the Hugoniot locus in bulk silicon from ab initio molecular dynamics with forces from density-functional theory, up to 70 GPa. In addition, we perform direct non-equilibrium molecular dynamics simulations of shock waves using empirical interatomic potentials and compare with our indirect method. We also present a direct ab initio molecular dynamics simulation of an elastic shock-wave in silicon, the first performed, to our knowledge.
The third and final investigation is into the computation of thermal conductivity from atomistic simulations. We produce a number of model interatomic potentials for silicon, using the non-parametric, Bayesian approach of Gaussian Approximation Potentials, which are improved systematically through a database of training configurations. We compute the thermal conductivity from these at the level of the phonon-Boltzmann transport equation. The best of these potentials reproduces the dft value of phonon-Boltzmann conductivity to within a few percent, which is itself in good agreement with experiment. We consider several issues relating to computing thermal conductivity from molecular dynamics simulations.