From elastic shallow shells to beams with elastic hinges by \(\Gamma\)-convergence (Q6592255)

The article explores a complex and theoretically rich topic -- the Γ-limit of the family of energies of elastic sloping shells with a width decreasing to zero. The work is motivated by observable mechanical phenomena such as bending and localized deformations in curved ribbon structures. Particular attention is paid to the definition and substantiation of the existence of the so-called elastic hinges formed when the strip width is reduced, which is an innovative approach that for the first time simulates such elastic effects in this structure. Although the introduction is related to practical examples, such as tape measures and aerospace applications, the main focus is on previous theoretical research, which limits discussion of broader engineering or materials science implications. Despite the presence of practical engineering motivation, the direct applicability of the results of the work remains unclear. For example, it is not clear how these models can be used for the design of materials or their practical applications. \N\NIn this work, modern variational methods are used to determine the Γ-limit of energy functionals and to study their compactness. Modeling of elastic joints based on the use of Γ-convergence to analyze the reduction of the model dimensionality, including jump discontinuities, was carried out mathematically rigorously and consistently. The main findings show that Γ-convergence captures localized curvature effects and elastic hinge formation in beam models, however, validation of the results remains purely theoretical, with no computational or experimental confirmation. For example, the use of the equations of a shallow shell seems doubtful, since when the width decreases, the shell model is not valid, the basic assumptions of the shell theory are violated, and, therefore, the ultimate transition to a new model should be based on the general model of a three-dimensional elastic body with large deformations. Nevertheless, the work is a valuable theoretical study, it offers significant innovations in the modeling of elastic joints in a rigorous mathematical formulation. However, there is no convincing evidence of its practical significance. The study is likely to be highly appreciated by specialists with deep mathematical knowledge, but its impact outside of a narrow academic circle may be limited. A more detailed discussion of the engineering implications of elastic joints and curvature could make the paper more accessible to an interdisciplinary audience, and a simplified presentation of technical details could be understood by non-specialists.