The main focus of this thesis is on investigating the minimum size of the Representative Volume Element (RVE) and finite-size scaling of properties of random linear and nonlinear elastic composites. The RVE is a material volume which accurately describes the overall behavior of a heterogeneous solid, and is the core assumption of continuum mechanics theory. If the composite microstructure admits the assumption of spatial homogeneity and ergodicity, the RVE can be attained within a specific accuracy on a finite length-scale. Determining this scale is the key objective of this thesis. In order to theoretically analyze the scale-dependence of the apparent response of random microstructures, essential and natural boundary conditions which satisfy Hill's averaging theorem in finite deformation elasticity are first considered. It is shown that the application of the partitioning method and variational principles in nonlinear elasticity and thermoelasticity, under the two above-mentioned boundary conditions, leads to the hierarchy of mesoscale bounds on the effective strain- and free-energy functions, respectively. These theoretical derivations lay the ground for the quantitative estimation of the scale-dependence of nonlinear composite responses and their RVE size. The hierarchies were computed for planar matrix-inclusion composites with the microstructure modeled by a homogeneous Poisson point field. Various nonlinear composites with Ogden-type strain-energy function are considered. The obtained results are compared with those where both matrix and inclusions are described by a neo-Hookean strain-energy function as well as with the results obtained from the linear elasticity theory. The trends toward the RVE are also computed for nonlinear elastic composites subjected to non-isothermal loading. The accuracy of the RVE size estimation is calculated in terms of the discrepancy between responses under essential and natural boundary conditions. Overall, the results show that the trends toward the RVE as well as its minimum size are functions of the deformation, deformation mode, temperature, and the mismatch between material properties of the phases. The last part of the thesis presents an investigation of the size effect on thermoelastic damping of a micro-/nanobeam resonator. It does not follow the framework described above. The main concern here is the size and the vibration frequency, at which the classical Fourier law of heat conduction is no longer valid, and the finite speed of heat propagation has to be taken into account.

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