Effets de gradient dans les réseaux élastiques discrets capturés par homogénéisation asymptotique

Architected structures have been more and more popular in a wide range of engineering applications due to their exotic mechanical properties, such as lightweight, high-strength, wave transmission, and shock-absorbing features. Their unique mechanical behavior can be finely tuned by altering their topology. Recent advancements in additive manufacturing have made it possible to fabricate truss-based architected structures with precisely designed and customizable microstructures. A straightforward approach to predicting the mechanical behavior of these lattice structures is to carry out full discrete simulations. However, this approach is restricted by the scale and complexity of the structure, making it impractical to study large or intricate designs. This limitation highlights the necessity of adopting a homogenized model.Recent work shows Cauchy-continuum obtained by the classical homogenization method is limited, for example, it cannot capture the size effect and ignores the gradient effect. This Ph.D. explores a general energy approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The homogenized energy achieves asymptotic accuracy two orders beyond the results of classical homogenization. The model is validated through a series of examples, including lattice with various topologies, and varying geometric or material properties, with or without prestrain/prestress, either in 2D or 3D. For each example, we utilize the shoal library, an open-source homogenization engine designed for the homogenization of discrete structures, to homogenize the target lattice and we compare the homogenization results with those extracted from full, discrete simulations to evaluate the accuracy of our homogenized model. Our findings show that the asymptotic homogenized model remains highly accurate even when the strain gradient effects are significant.We further extend the linear homogenization method to analyze the Kagome lattice with springs under finite deformation. The Kagome with springs has mechanisms due to its topology and recent work shows that Cauchy's continuum is not enough. By introducing an enriched kinematic variable and applying a leading-order two-scale expansion, we obtain an asymptotic, enrichment homogenized model for the Kagome lattice with finite deformation. We present a 1D and a 2D example to demonstrate that the model can effectively predict deformation under applied loads by solving the corresponding boundary value problems. Our effective model is capable of capturing some interesting phenomena, such as the (left-right) asymmetry in the deformation of the discrete structure under symmetric boundary conditions. Moreover, it predicts that this asymmetry progressively decreases when the microscopic length scale is refined, ultimately achieving perfect symmetry as the microscopic length approaches zero.The homogenization approach used in this Ph.D. can provide a versatile and efficient tool to analyze the mechanical performance of discrete structures and is powerful for designing and optimizing microstructures.