Fleur Hendriks

Mechanical metamaterials are engineered materials whose mechanical behavior is governed not only by the material they are made from, but also by the geometry of their internal microstructure. By carefully designing this microstructure, these materials can exhibit unusual and useful properties, such as tunable stiffness, negative Poisson’s ratio, programmable shape changes, and controllable wave propagation. A particularly powerful mechanism behind these functionalities is buckling. Under loading, a structure may suddenly switch from one deformation pattern to another, giving rise to large changes in the overall mechanical response. This makes buckling useful for design, but also difficult to simulate and predict. The main difficulty addressed in this thesis is that the mechanical response of buckling metamaterials is nonlinear, strongly geometry-dependent, and governed by symmetry. In addition, the post-buckling solution space can be multimodal: a single microstructure under a fixed loading condition may admit several physically valid deformation patterns. This is challenging not only for machine-learning models, but also for conventional finite-element simulations, which can accurately compute individual deformation paths, but do not automatically reveal the full set of possible post-buckling solutions. The solution that is obtained may depend on imperfections, perturbations, or numerical choices. Exploring multiple admissible responses therefore requires repeated simulations or post-processing. This can become computationally prohibitive when many simulations are needed for design, optimization, or uncertainty analysis. Machine-learning surrogate models offer a promising route to faster prediction, but standard deterministic models often ignore important physical structure. They may fail to respect geometric symmetries, generalize poorly to unseen microstructures, or collapse several valid post-buckling solutions into one averaged, nonphysical prediction. This thesis investigates how machine learning can be adapted to this setting by embedding physical and geometric structure directly into the learning problem. The work is organized around four research questions. 1. How can neural-network surrogate models be designed so that they respect the symmetries and physical constraints of mechanical metamaterials? To incorporate these symmetries, this thesis develops a similarity-equivariant graph neural network, SimEGNN. The model represents the microstructure as a graph and predicts both effective macroscopic quantities, such as energy, stress, and stiffness, and local deformation fields. Translation, rotation, reflection, scaling, and periodicity are treated as inductive biases that guide the model towards physically consistent predictions. 2. How can a dataset be constructed to study and learn mechanical responses across a broad range of loading conditions and metamaterial designs, including all periodic planar symmetry groups? This thesis introduces a dataset of two-dimensional mechanical metamaterials based on the 17 wallpaper groups. The dataset contains 1,020 microstructures together with their simulated mechanical responses under multiple loading paths in the finite-strain regime, including cases where buckling occurs. 3. How can machine-learning models represent symmetry breaking, where one symmetric input may lead to multiple valid outputs? This challenge is addressed using equivariant flow matching. Flow matching is a generative modeling approach that learns how to transform simple random samples into samples from a target distribution. The learned distribution respects the symmetries of the physical system, while individual samples are still allowed to break those symmetries. This thesis demonstrates that equivariant generative models can represent such symmetry-breaking behavior without collapsing multiple solutions into a nonphysical average. 4. How can equivariant generative models be scaled to realistic buckling trajectories in mechanical metamaterials? The thesis combines equivariant graph neural networks, flow matching, temporal modeling, and a hierarchical architecture into one probabilistic framework. The graph representation captures the geometry, the equivariant architecture preserves symmetries, the generative formulation represents multiple solutions, and the hierarchical structure helps coordinate local and global deformation patterns. Together, the four parts of this thesis develop a coherent approach to learning the nonlinear and symmetry-governed behavior of buckling mechanical metamaterials, including cases with multiple admissible post-buckling solutions. The main outcome is an equivariant data-driven framework that enables faster simulation, improved generalization, and a systematic treatment of multiple possible post-buckling responses.