Multiscale computational homogenization is an efficient method to upscale the microstructural behavior of micro-heterogeneous materials. In this method, a representative volume element (RVE) is assigned to a macroscale material point and the constitutive law for the macroscopic model at that point is obtained by solving a boundary value problem on the RVE. Among the conventional boundary conditions, the so-called strong periodic boundary conditions tend to converge faster towards the actual microstructural response. Nonetheless, applying strong periodic boundary conditions to a batch of 48 fiber-matrix RVEs under uniaxial load with varying orientations introduces a dependency between the average ultimate principal stress (σ₁) and the orientation angle (θ).

This treatise investigates the effects on this dependency by applying so-called weak periodic boundary conditions instead. These boundary conditions soften the strong requirement of periodicity of displacements at the RVE boundary by coarsening the traction mesh and requiring periodicity to hold only in an average sense over this coarser mesh. Three main questions are asked: Do weak periodic boundary conditions alleviate the dependency between σ₁ and θ? Is this dependency the result of RVEs prone to localization under shear? Do smaller coarsening factors widen the range of θ over which localization with a single shear band is permissible? Overall, it is concluded that only the weakest form of weak periodic boundary conditions reduces the dependency between σ₁ and θ, which is indeed caused by RVEs that are prone to strain localization under shear, particularly towards θ = 45^o. Increasing coarsening factors quickly introduce stricter periodicity requirements, thus limiting the possibility of strain localization with a single shear band at angles other than θ = 45^o. Recommendations are provided to alleviate the dependency between σ₁ and θ as well as how to more realistically model the behavior of RVEs used in this research.

Shells tend to be thin because their curvature enables them to carry distributed loads as membrane forces. The property of thinness stems from shells’ capacity to store membrane strain energy without much deformation. As a result, buckling failures often govern the design of shell structures. These buckling failures usually start locally, at a location where a combination of curvature and membrane forces is met. Moreover, shells tend to be imperfection-sensitive structures, that is, real-life shells (with initial geometric imperfections) usually cannot resist loads as high as the theoretically predicted critical buckling load. Advanced finite element analyses can accurately predict these so-called nonlinear buckling loads but require significant time and computation effort. On the other hand, current design equations are simple yet highly inaccurate and often penalize strength significantly. This treatise caters a Python script that executes nonlinear finite element analyses (using ANSYS Mechanical APDL) to generate a database of the nonlinear buckling loads of shell portions with varying membrane forces, curvatures and magnitudes of initial geometric imperfections. The aim, beyond the scope of this treatise, is to perform a parametric regression on said database to device design equation(s) that accurately predict the nonlinear buckling load of linear-elastic shell structures with initial geometric imperfections based merely on the linear elastic results of a geometrically perfect shell model.