ZAOUI, Yassine (2024) Asymptotic analysis of small emerging cracks PRE - Research Project, ENSTA.

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Abstract

The goal of this research internship was to perform an asymptotic analysis of small emerging cracks, a subject of critical importance in material science and structural engineering. The study focused on understanding the perturbations caused by small cracks in a material, specifically within the context of the conductivity equation. We explored both conductive and insulating cracks, with a primary emphasis on the conductive case. To achieve this, we developed a mathematical model derived from energy minimization principles, which laid the groundwork for analyzing the perturbations in the elliptic solutions caused by small emerging cracks. Our approach involved both theoretical and computational methods. The theoretical aspect focused on deriving convergence results and a representation formula for the perturbed solutions using advanced mathematical tools such as capacities and Green functions. On the computational side, we implemented numerical approximations of the capacity using MATLAB. These tools allowed us to validate our theoretical findings by approximating the perturbed solutions and confirming their asymptotic behavior. The results of this internship provided valuable insights into the asymptotic behavior of small emerging cracks, with potential implications for the development of more accurate nucleation crack models. The study also contributed to a deeper understanding of the capacity notion in both theoretical and numerical contexts.

Item Type:Thesis (PRE - Research Project)
Uncontrolled Keywords:Asymptotic analysis, small cracks, elliptic equations, capacity, Green functions, numerical approximation
Subjects:Mathematics and Applications
ID Code:10077
Deposited By:Yassine ZAOUI
Deposited On:28 août 2024 19:18
Dernière modification:28 août 2024 19:18

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