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A novel computational strategy is presented to calculate from first principles the coefficient of thermal expansion and the elastic constants of a material over meaningful intervals of temperature and pressure. This strategy combines a novel implementation of the quasiharmonic approximation to calculate the isothermal-isochoric linear and nonlinear elastic constants of a material, with elementary equations of nonlinear continuum mechanics. Our implementation of the quasiharmonic approximation relies on finite deformations, the use of nonprimitive supercells to describe a material, a recently proposed technique to calculate generalized mode Grüneisen parameters, and the numerical differentiation of the stress tensor to calculate both second- and third-order elastic constants. The combination of this method with nonlinear continuum mechanics is shown to yield accurate predictions of lattice parameters and linear elastic constants of a material over finite intervals of temperature and pressure, at the cost of calculating isothermal second- and third-order elastic constants for a single reference state. Here, the validity and limits of our novel methods are assessed by carrying out calculations of MgO based on classical interatomic potentials. To demonstrate potential, our methods are then used in conjunction with a density functional theory approach to calculate thermal expansion and elastic properties of silicon, lithium hydrate, and graphite.