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Some results on the mathematical analysis of crack problems with forces applied on the fracture lips
Full text linkThis thesis is devoted to the study of some models of fracture growth in elastic materials, characterized by the presence of forces acting on the crack lips. Working in the general framework of rate-independent processes, we first discuss a variational formulation of the problem of quasi-static crack evolution in hydraulic fracture. Then, we investigate the crack growth process in a cohesive fracture model, showing the existence of an evolution satisfying a weak Griffith's criterion. Finally, in the last chapter of this work we investigate, in the static case, the interaction between the energy spent in order to create a new fracture and the energy spent by the applied surface forces. This leads us to study the lower semicontinuity properties of a free discontinuity functional F(u) that can be written as the sum of a crack term, depending on the jump set of u, and of a boundary term, depending on the trace of u
A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations
No full textWe derive a strain-gradient theory for plasticity as the Gamma-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido et al. (Adv Calc Var 17:1039–1055, 2024), we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order 1-alpha. As alpha goes to 0, we show that suitably rescaled energies Gamma-converge to the macroscopic strain-gradient model of Garroni et la. (J Eur Math Soc (JEMS) 12:1231–1266, 2010)
Mean-Field Selective Optimal Control via Transient Leadership
Full text linkA mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. In this way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via [Formula: see text] -convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments
A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity
Full text linkIn this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the Γ-limit of the functional as the thickness tends to zero.
The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements.
Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation
A Pontryagin maximum principle for agent-based models with convex state space
Full text linkWe derive a first order optimality condition for a class of agent-based systems, as well as for their mean-field counterpart. A relevant difficulty of our analysis is that the state equation is formulated on possibly infinite-dimensional convex subsets of Banach spaces. This is a typical feature of many problems in multi-population dynamics, where a convex set of probability measures may account for the population, the degree of influence or the strategy attached to each agent. Due to the lack of a linear structure and of local compactness, the usual tools of needle variations and linearisation procedures used to derive Pontryagin type conditions have to be generalised to the setting at hand. This is done by considering suitable notions of differentials and by a careful inspection of the underlying functional structures
Quasistatic crack growth in hydraulic fracture
No full textWe present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption
Geometric rigidity on Sobolev spaces with variable exponent and applications
No full textWe present extensions of rigidity estimates and of Korn’s inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-Hölder continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result à la Nitsche in Sobolev spaces with variable exponents. As an application, by means of
Gamma-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well
A lower semicontinuity result for a free discontinuity functional with a boundary term
No full textWe study the lower semicontinuity in GSBV^{p}(\Om;\R^{m}) of a free discontinuity functional~\F(u) that can be written as the sum of a crack term, depending only on the jump set~, and of a boundary term, depending on the trace of~ on~\partial\Om. We give sufficient conditions on the integrands for the lower semicontinuity of~\F. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of~\F can be represented by the sum of two integrals on~ and~\partial\Om, respectively
Convergence of discrete and continuous unilateral flows for Ambrosio-Tortorelli energies and application to mechanics
No full textWe study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovered a posteriori by a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateral -gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments
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