Measure-valued solutions in elasticity and plasticity

This project investigates how materials deform under stress, focusing on both elastic and plastic behaviors, particularly in situations involving large deformations and temperature changes. These questions are central to applied mathematics and mechanics, with wide-ranging applications in engineering, materials science, and biomechanics. The first part of the project focuses on elastic materials, hence materials that return to their original shape after being deformed. While the static behavior of such materials is well understood, much less is known about how they evolve dynamically, especially under thermal effects. We aim to study these systems using variational methods, which describe material behavior in terms of energy principles. Our goal is to construct measure-valued solutions, a flexible mathematical framework that can capture complex effects such as oscillations or microstructures. These solutions will conserve total energy and allow for entropy dissipation, reflecting key thermodynamic laws. The second part of the project examines plasticity, permanent deformation that remains even after the stress is removed. We focus on non-associative finite plasticity, a challenging class of models that better represent many real-world materials but are harder to treat mathematically. These models do not follow standard energy-based structures, making them less accessible to known techniques. We aim to design new variational schemes that can describe how these materials evolve over time under quasistatic conditions, and to prove that these schemes produce stable, energy-consistent solutions. We also plan to extend this framework to handle the effects of temperature. A major difficulty in both parts of the project comes from the fact that the underlying energies are nonconvex, which complicates the mathematical analysis of this problem. We address this by developing new approaches based on generalized convexity assumptions and Young measures, which are powerful tools for handling complex material responses. The main goals of the project are: (A) To find an appropriate thermodynamic potential for one-dimensional thermoelastic systems and show that it can produce entropy-dissipating, energy-conserving solutions using step-by- step minimization schemes; (B) To develop generalized solutions for dynamic thermoelasticity in higher dimensions; (C) To establish the existence of reliable solutions for non-associative finite plasticity and extend these methods to include thermal effects.