Calculus of Variations and Geometric Measure Theory

R. Llerena - P. Piovano

Solutions for a free-boundary problem modeling multilayer films with coherent and incoherent interfaces

created by llerena on 25 Jan 2024
modified by piovano on 26 Jan 2024


Submitted Paper

Inserted: 25 jan 2024
Last Updated: 26 jan 2024

Pages: 27
Year: 2024


In this paper, we move forward from the results of "R. Llerena, P. Piovano (2023)" by introducing a variational model for the study of multilayer films that allows for the treatment of both coherent and incoherent interfaces between layers. The model is designed in the framework of the theory of Stress Driven Rearrangement Instabilities, which are characterized by the competition between elastic and surface energy effects. The surface of each film layer is assumed to satisfy the "exterior graph condition'' already introduced in "R. Llerena, P. Piovano (2023)", for which bulk cracks are allowed to be of non-graph type. By applying the direct method of calculus of variations under a constraint on the number of connected components of the cracks not connected to the surface of the film layers the existence of energy minimizers is established in dimension 2. As a byproduct of the analysis the state of art on the variational modeling of single-layered films deposited on a fixed substrate is advanced by letting the substrate surface free, by addressing the presence of multiple layers of various materials, and by including the possibility of delamination between the various film layers.

Keywords: free boundary problem, surface energy, delamination, elastic energy, deformable layers, multiphase morphology, multilayer films