Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mechanical Engineering & Applied Mechanics

First Advisor

Celia Reina


Elastic/Acoustic metamaterials have exhibited a rapid increase of interest due to their surprising dynamic properties, such as subwavelength bandgaps and negative effective elastic constant and/or density. The design and optimization of these novel architectures call for an enhanced understanding of the microstructure-properties relations including micro-inertia effects. However, similarly to regular composites, their direct numerical simulation is often prohibitively time consuming, and multiscale strategies that can consider the effect of the local microstructure under dynamic conditions and, at the same time, save computational time, are in need.

We start by revisiting some fundamental theories in homogenization, such as Hill's averaging relations and Hill-Mandel condition, initially derived under static loading. We first prove that these fundamental relations hold exactly under finite element discretization, (i.e.~no numerical error is introduced by the finite element discretization during the scale transition in the multiscale analyses) and further discuss their extension to the dynamic setting. Then, we present a variational coarse-graining framework for heterogeneous media under dynamic loading conditions, which is applicable to general material behavior as well as to discrete or continuous representations of the material and its deformation, e.g., finite element discretization or atomistic system. The proposed theoretical framework can be used to perform multiscale numerical simulations in the spirit of multilevel finite element method (FE^2), which has been implemented with the help of an open-source finite element library deal.II. Various time integration algorithms, i.e. explicit and implicit Newmark methods, have been employed, and implementation in series and in parallel have been developed. The resulting multiscale strategy has been tested for different applications, namely layered materials and locally resonant structures with space/time modulation. In all cases, comparisons with single scale finite element simulations showcase the efficiency and accuracy of the method.

Finally, we also report a material architecture design based on locally resonant hierarchical structures to deliver metamaterial structures with enhanced bandwidth. The theoretical analyses based on lattice systems are then certified using continuum models and finite element simulations.

The text and results in this thesis closely follow the articles by the author Liu & Reina (2016, 2017, 2018a) and Liu & Reina (2018b).

Files over 3MB may be slow to open. For best results, right-click and select "save as..."