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Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture

Final Report Summary - QUADYNEVOPRO (Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture)

This research project deals with nonlinear evolution problems that arise in the study of the inelastic behaviour of solids, in particular in plasticity and fracture. The project focuses on selected problems, grouped into three main topics, namely:
• Plasticity,
• Quasistatic crack growth,
• Dynamic fracture mechanics.

The analysis of the models of these mechanical problems leads to deep mathematical questions originated by two common features: the energies are not convex and the solutions exhibit discontinuities both with respect to space and time. In addition, plasticity problems often lead to concentration of the strains, whose mathematical description requires singular measures. Most of these problems have a variational structure and are governed by partial differential equations. Therefore, the construction of consistent models and their analysis need advanced mathematical tools from the calculus of variations, from (geometric) measure theory, and from the theory of nonlinear partial differential equations.

The main achievements are the original mathematical results on the topics of the project. They are contained in 66 research papers published or accepted in high level scientific journals with a wide international diffusion. Further results, obtained in the last months, are described in 9 preprints, submitted to international journals.

The purpose of these results is to give a sound mathematical justification of some theoretical models in fracture mechanics or in the theory of plasticity and dislocations. They also provide a theoretical background for some algorithms used for numerical simulations in these fields. The mathematical tools developed for the main topics of the project have a wide range of applicability in neighbouring fields.

A group of papers deals with quasistatic evolution problems in elasto-plasticity (including the hard mathematical problems connected with heterogeneous materials), approximation of quasistatic and dynamic perfect plasticity, dimension reduction problems which lead to a rigorous theory of elasto-plastic plates, and some nonassociative models in elasto-plasticity.

Some papers study mathematical models of dislocations and develop a rigorous discrete-to-continuum analysis to derive macroscopic models from atomic interactions.

Another group of papers develops new theoretical tools to solve the difficult mathematical problems that arise in the variational formulation of linearly elastic fracture mechanics (LEFM), when the crack path or the shape of the crack are not prescribed. These tools allow us to solve, in any space dimension, the weak formulation of the incremental minimum problems that are used in the discrete time approximation of the quasistatic evolution in LEFM. The corresponding strong formulation has been solved in dimension two. So far a rigorous existence result for the evolution problem in LEFM has been proved only in dimension two.

Some papers are devoted to homogenisation problems in fracture mechanics and to different mathematical models of crack growth, including cohesive zone models and models obtained by a vanishing viscosity approach. Others deal with quasistatic evolution problems for damaged elastic or elasto-plastic materials, and highlight their close connection to fracture problems.

Some papers concern the approximation of the fracture process by suitable variants of the phase-field model introduced by Ambrosio and Tortorelli and widely used in numerical simulations. In particular they contain the first complete proof of this approximation in the case of linear elasticity in dimension three. Other results provide a rigorous derivation of linearly elastic fracture mechanics from fracture models in finite elasticity, using a scaling argument.

The papers on dynamic fracture contain some promising results for a simplified problem in dimension one and some preliminary results for the fracture problem in 2d with a prescribed crack path.