Computational Homogenization: An Approach to Computational Multi-scale Modeling of Mechanical Metamaterials

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Insinööritieteiden korkeakoulu | Bachelor's thesis
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Computational Engineering
Degree programme
Aalto Bachelor’s Programme in Science and Technology
Metamaterials have played a crucial part in the development of materials science in recent years due to their unnatural novel properties. However, studying these properties using traditional materials modeling is both time-consuming and computationally expensive, and the accuracy is, most of the time, dependent on the scale of the simulation. Computational homogenization is a multi-scale method which aims at addressing this shortcoming: the properties of the metamaterials can be determined by analyzing the known artificial structures of the material on the micro-scale. When applying to elasticity problems of first order, computational homogenization consists of two main parts: solving the localization problem and determining the effective properties. In the first part, a localization problem is defined to determine the deformation of a single unit cell inside the material. Here, periodic boundary condition is implemented in order to preserve the periodicity of the structure and the homogeneity of the deformation. The deformation map at micro-scale is then transferred back to macro-scale by utilizing the averaging quantities: one of which is the Hill-Mandel lemma stating that the average energy at micro-scale is equivalent to that at the macro-scale. Finally, the deformation map at macro-scale can be evaluated to determine the macroscopic properties of the material. In this thesis, an analysis of triangular truss and frame lattices is conducted. In the triangular truss lattice where the bar members are subjected to axial loading, the local strain energy is determined by the direct stiffness method, and the elastic tensor is derived using Hill-Mandel lemma. The Young's modulus of this structure is dependent on the relative density of the cell, whereas the Poisson's value is invariant. In the triangular frame lattice where the beam members are subjected to additional shear and bending effects, the direct stiffness method also takes into account the slope-deflection equations for these flexural effects. Therefore, the elastic tensor of frame lattice comprises an extra term which is dependent on both the relative density of the unit cell and the geometry of the beam members, and this term describes the flexural properties of the structure. Both truss and frame lattices are isotropic, and the null angular displacement shows that these structures are stretching-dominated.
St-Pierre, Luc
Thesis advisor
Niiranen, Jarkko
computational homogenization, mechanical metamaterials, multi-scale modeling, linear elasticity, computational mechanics
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