In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to ``fill the hole'' in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

1 aBellettini, Giovanni1 aElshorbagy, Alaa1 aPaolini, Maurizio1 aScala, Riccardo uhttps://doi.org/10.1007/s10231-019-00887-001179nas a2200121 4500008004100000245009100041210006900132300001200201490000700213520077600220100002000996856004101016 2017 eng d00aLimit of viscous dynamic processes in delamination as the viscosity and inertia vanish0 aLimit of viscous dynamic processes in delamination as the viscos a593-6250 v233 aWe introduce a model of dynamic evolution of a delaminated visco-elastic body with viscous adhesive. We prove the existence of solutions of the corresponding system of PDEs and then study the behavior of such solutions when the data of the problem vary slowly. We prove that a rescaled version of the dynamic evolutions converge to a “local” quasistatic evolution, which is an evolution satisfying an energy inequality and a momentum balance at all times. In the one-dimensional case we give a more detailed description of the limit evolution and we show that it behaves in a very similar way to the limit of the solutions of the dynamic model in [T. Roubicek, SIAM J. Math. Anal. 45 (2013) 101–126], where no viscosity in the adhesive is taken into account.

1 aScala, Riccardo uhttps://doi.org/10.1051/cocv/201600601304nas a2200133 4500008004100000245008300041210006900124520084400193100002201037700002201059700002001081700001801101856005101119 2016 en d00aConfinement of dislocations inside a crystal with a prescribed external strain0 aConfinement of dislocations inside a crystal with a prescribed e3 aWe study screw dislocations in an isotropic crystal undergoing antiplane shear. In the framework of linear elasticity, by fixing a suitable boundary condition for the strain (prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Ƭ-convergence theory, in the framework of the so-called core radius approach.1 aLucardesi, Ilaria1 aMorandotti, Marco1 aScala, Riccardo1 aZucco, Davide uhttp://urania.sissa.it/xmlui/handle/1963/3524701884nas a2200145 4500008004100000245005300041210005300094260003400147300001100181490000700192520140800199100002001607700002501627856008601652 2016 eng d00aCurrents and dislocations at the continuum scale0 aCurrents and dislocations at the continuum scale bInternational Press of Boston a1–340 v233 aA striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in $L^p$ with $1\leq p<2$, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

1 aScala, Riccardo1 aVan Goethem, Nicolas uhttps://www.math.sissa.it/publication/currents-and-dislocations-continuum-scale-001802nas a2200229 4500008004100000245010400041210006900145300001400214490000700228520107100235653001001306653001001316653002901326653001501355653002001370653002501390653001801415100003301433700002001466700002501486856006101511 2015 eng d00aA compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity0 acompatibleincompatible decomposition of symmetric tensors in Lp a5217-52300 v383 aIn this paper, we prove the Saint-Venant compatibility conditions in $L^p$ for $p\in(1,∞)$, in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in $L^p$ to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where $1 < p < 2$. This justifies the need to generalize and prove these rather classical results in the Hilbertian case ($p = 2$), to the full range $p\in(1,∞)$. Copyright © 2015 John Wiley & Sons, Ltd.

10a35J5810a35Q7410acompatibility conditions10aelasticity10aKorn inequality10astrain decomposition10asubclass74B051 aMaggiani, Giovanni, Battista1 aScala, Riccardo1 aVan Goethem, Nicolas uhttps://onlinelibrary.wiley.com/doi/abs/10.1002/mma.345000376nas a2200097 4500008004100000245008700041210006900128100002000197700002500217856003600242 2014 eng d00aDislocations at the continuum scale: functional setting and variational properties0 aDislocations at the continuum scale functional setting and varia1 aScala, Riccardo1 aVan Goethem, Nicolas uhttp://cvgmt.sns.it/paper/2294/00819nas a2200157 4500008004100000022001400041245007800055210006900133260000800202300001400210490000700224520034300231100002100574700002000595856004600615 2014 eng d a1572-922200aQuasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes0 aQuasistatic Evolution in Perfect Plasticity as Limit of Dynamic cDec a915–9540 v263 aWe introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

1 aDal Maso, Gianni1 aScala, Riccardo uhttps://doi.org/10.1007/s10884-014-9409-700850nas a2200121 4500008004300000245007700043210006900120260001000189520044200199653001700641100002000658856005000678 2014 en_Ud 00aA variational approach to statics and dynamics of elasto-plastic systems0 avariational approach to statics and dynamics of elastoplastic sy bSISSA3 aWe prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions. We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers.10adelamination1 aScala, Riccardo uhttp://urania.sissa.it/xmlui/handle/1963/7471