title: Thermoelastic and Fracture Responses of Periodic Materials: Theory and Applications to Laminates and Triply Periodic Minimal Surfaces
Creators: name: Teixeria Préve, Deison 
Publisher: name: IMT Alti Studi Lucca
Year: 2022

description: In order to provide better quality on a variety of equipment, services and new technologies to the community, periodic materials as laminates and periodic structures as foams are constantly gaining more attention world-wide, due to the fact that these structures present suitable mechanical behaviours, enhanced physical properties and are yet low-cost. There- fore, it is crucial to understand how these structures respond for different physical and mechanical problems. The present thesis exploits how cer-tain periodic structures behave and respond under thermo-mechanical loading and fracture phenomena. In the first part of the thesis, a multi-scale variational-asymptotic ho-mogenization method for periodic microstructured materials for ther-moelastic problems with one relaxation time is exploited. The asymp-totic expansions of the micro-displacement and the micro-temperature fields are rewritten on the transformed Laplace space and expressed as power series of the microstructural length scale, leading to a set of re-cursive differential problems over the periodic unit cell.The solution of such cell problems leads to the perturbation functions. Up-scaling and down-scaling relations are then defined, and the latter allow expressing the microscopic fields in terms of the macroscopic ones and their gra-dients. The variational-asymptotic scheme to establish an equivalence between the equations at macro-scale and micro-scale is developed. Av- erage field equations of infinite order are also derived. The efficiency of the proposed technique was tested in relation to a bi-dimensional or-thotropic layered bodies with orthotropy axis parallel to the direction of the layers, where the mechanical and temperature constitutive prop-erties were well established. The dispersion curves of the homogenized medium, truncated at the first order are compared with the dispersion curves of the heterogeneous continuum obtained by the Floquet-Bloch theory. The results obtained with the two different approaches show a very good agreement. The second part of the thesis is focused on assessing the occurrence of fracture in Triply Periodic Minimal Surfaces (TPMS) foams subjected to compressive loading. TPMS, described by the mathematics commu-nity, may be exploited as a backbone for developing a new class of foams with open porosity for a wide range of engineering and biomedical appli-cations. Therefore, a comprehensive analysis of their fracture response is fundamental and is herein attempted. To this aim, a 3D phase field model is outlined and applied to TPMS foam structures under com-pression, with the goal to predict critical points for crack nucleation, potential crack paths, and the stiffness and maximum compressive stress of the unit cell, which can be related to the apparent Young’s modulus and apparent strength of a macro-scale composite made of such TPMS unit cells. A careful mesh sensitivity analysis was conducted on the specimens, to provide guidelines on how to identify the optimal finite el- ement discretization consistent with the internal length scale parameter of the phase field approach to fracture. The major predicted mechanical properties for five different TPMS open foams, and for different levels of porosity, are summarized in Ashby plots. The predicted trends are in agreement with previous results on TPMS taken from the literature and show that TPMS can outperform standard Aluminium open foams.