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

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... 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 i [...]