| Contemporary heterogeneous materials possess intricate microstructures not easily treated with analytical and existing computational techniques. Particularly for the non-linear regime, conditions found in realistic manufacturing and in situ applications warrant methods that integrate micromechanical principles with conventional continuum field theories. A step in this direction is the advent of the so-called asymptotic expansion homogenization approach. The central premise is to approximate the primary variables, such as displacements, flow velocity, or temperature, with a perturbative asymptotic series over a length scale parameter, 3 . Based only on this approximation, new field equations can be derived to embrace the rapidly oscillating material properties using causal global and local boundary value problems over multi-scales. Both linear and non-linear problems are tractable. The length scales under consideration are presently assumed to adhere to the continuum approximation. Definitive but coupled sets of boundary value problems are obtained that ensure mathematical rigor in the formulations as well as mathematically consistent schemes, hence the resolution of multi-scales.;The study first describes the computational approach in great detail for linear elasticity. Then four Finite Element approaches are presented that encompass multi-scale time variant linear and non-linear, thermal and structural problems. These are viscoelasticity, non-linear thermal beat conduction, thermo-viscoelasticity, and transient elasto-plasticity.;The findings are validated with experimental measurements when permitted by existing literature. Verifications with analytical and numerical solutions are also shown. The findings indicate that the present developments are degenerative in the sense that they return the conventional homogeneous material solution. Present studies of simple shapes and geometries also agree with experimental results and existing approximation formulae in the literature. The contributions encompass computational developments for multi-scale heterogeneous problems. Although some of the mathematical developments exist to an extent, the computational and implementational aspects for engineering problems that help elucidate the physics are noteworthy. Furthermore, to demonstrate the utility of the approaches, illustrative examples and practical applications are presented for complicated geometries where existing methods are inapplicable.;The study concludes with discussions of the results, a summary of the proposed contributions stemming from the work, limitations of the computational developments, and proposed additional investigations for multi-scale computational mechanics. |
