1,720,977 research outputs found
Non-local approximation of the Griffith functional
Full text linkAn approximation, in the sense of Γ-convergence and in any dimension d ≥ 1, of Griffith-type functionals, with p-growth (p > 1) in
the symmetrized gradient, is provided by means of a sequence of nonlocal integral functionals depending on the average of the symmetrized gradients on small balls
Functionals Defined on Piecewise Rigid Functions: Integral Representation and Γ -Convergence
No full textWe analyse integral representation and Γ-convergence properties of functionals defined on piecewise rigid functions, that is, functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for Γ-convergence and a careful adaption of the global method for relaxation (Bouchitté et al. in Arch Ration Mech Anal 165:187–242, 2002; Bouchitté et al. in Arch Ration Mech Anal 145:51–98, 1998), to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation
Mean-field analysis of multipopulation dynamics with label switching
Full text linkThe mean-dield analysis of a multipopulation agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markov-type jump process on the probability that each agent has of belonging to a given population. A general functional analytic framework for the well-posedness of the problem is established, and some concrete applications are presented, both in the cases of a discrete and continuous set of labels. In the particular case of a leader-follower dynamics, the existence and approximation results recently obtained in [G. Albi et al., Math. Models Methods Appl. Sci., 29 (2019), pp. 633{679] are recovered and generalized as a byproduct of the abstract approach proposed
A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension
Full text linkIn this paper we study the vanishing inertia and viscosity limit of a second order system set in an Euclidean space, driven by a possibly nonconvex time-dependent potential satisfying very general assumptions. By means of a variational approach, we show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation
Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD
No full textIn this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV-ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBDp functions
A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games
Full text linkA multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided
Radial solutions for a dynamic debonding model in dimension two
No full textIn this paper we deal with a debonding model for a thin film in dimension two, where the wave equation on a time-dependent domain is coupled with a flow rule (Griffith's principle) for the evolution of the domain. We propose a general definition of energy release rate, which is central in the formulation of Griffith's criterion. Next, by means of an existence result, we show that such definition is well posed in the special case of radial solutions, which allows us to employ representation formulas typical of one-dimensional models
Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case
No full textWe study the spatially uniform case of the quasistatic evolution in Cam-Clay plasticity, a relevant example of small strain nonassociative elasto- plasticity. Introducing a viscous approximation, the problem reduces to de- termine the limit behavior of the solutions of a singularly perturbed system of ODE’s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity; the slow dynamics, when the stress evolves smoothly on the yield surface and plas- tic flow is produced; the fast dynamics, which may happen only in the softening regime, when viscous solutions exhibit a jump determined by the heteroclinic orbit of an auxiliary system. We give an iterative procedure to construct a viscous solution
Integral representation for energies in linear elasticity with surface discontinuities
Full text linkIn this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation (GSBD p GSBDp) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51-98] and a recent Korn-type inequality in GSBD p GSBDp, cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré-Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBD pSBDp, obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p SBDp in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337-1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation (GSBV pGSBVp)
Lower semicontinuity and relaxation for free discontinuity functionals with non-standard growth
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