Computational Continua for Heterogeneous Solids
The computational continua (C2) framework, which is the focus of the present thesis, is a coarse-scale continuum description coupled with an underlying fine-scale description of material heterogeneity of finite size. It is intended to account for a variation of the coarse-scale stresses (strains) over a unit cell (UC) domain. It was originally developed to overcome the theoretical and computational limitations of higher order homogenization models and generalized continuum theories, namely the need for higher order finite element continuity, additional degrees-of-freedom, and non-classical boundary conditions. The key feature of the C2 is so-called nonlocal quadrature scheme (NLQS) defined over a computational continua domain consisting of a disjoint union of so-called computational unit cells (CUC). The CUCs, which are merely computational entities, have a shape and size of the physical periodic microstructure, but their positions depend on the size of the unit cell domain and is determined to reproduce the weak form of the governing equations on the fine scale. In the original C2 formulation the unit cell domains when mapped onto the parent element domains preserved their original shape. Thus, the nonlocal quadrature scheme was limited to structured meshes or meshes with slightly distorted elements. In the present thesis, it is accounted for that the CUCs when mapped onto the parent element domain, may no longer preserve their initial shape. Towards this end, an exact nonlocal quadrature scheme for distorted elements, which matches the two-dimensional monomials of the element, and an approximate tensor-product based nonlocal quadrature that eliminates the need for costly evaluation of the quadrature points for each element were developed. The performance of both nonlocal quadrature schemes is demonstrated in two-dimensional linear elasticity problems on several meshes and microstructures and compared with the classical first-order (O(1)) homogenization theory and the direct numerical simulation (DNS). The error in the overall behavior (total strain energy stored and L2 norm error in von Mises stress) of the C2 formulations offers a 10-20 (%) improvement over the O(1) theory. More substantial is the gain of the C2 formulations over the O(1) theory in the accuracy of the local stresses in critical locations. Finally, the performance of the tensor-product based approximate quadrature is comparable to that of the computationally costly exact nonlocal quadrature in terms of both the global and local error measures making it more attractive. In the wave propagation regime, the computational continua formulation showed strikingly accurate dispersion curves. Unlike classical dispersive methods pioneered more than a half a century ago where the unit cell is quasi-static and provides effective mechanical and dispersive properties to the coarse-scale problem, the dispersive C2 gives rise to transient problems at all scales and for all microphases involved. An efficient block time-integration scheme is proposed that takes advantage of the fact that the transient unit cell problems are not coupled to each other, but rather to a single coarse-scale finite element they are positioned in. It is showed that the computational cost of the method is comparable to the classical dispersive methods for short load durations. The scheme is proved to be stable. Finally, accuracy analysis on a wave propagation model problem demonstrates that the proposed scheme is substantially more accurate when compared with a O(1) homogenization scheme with microinertia effects.