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.

%B Annali di Matematica Pura ed Applicata (1923 -) %8 Jul %G eng %U https://doi.org/10.1007/s10231-019-00887-0 %R 10.1007/s10231-019-00887-0 %0 Journal Article %J ESAIM: COCV %D 2017 %T Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish %A Riccardo Scala %XWe 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.

%B ESAIM: COCV %V 23 %P 593-625 %G eng %U https://doi.org/10.1051/cocv/2016006 %R 10.1051/cocv/2016006 %0 Journal Article %D 2016 %T Confinement of dislocations inside a crystal with a prescribed external strain %A Ilaria Lucardesi %A Marco Morandotti %A Riccardo Scala %A Davide Zucco %X We 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. %G en %U http://urania.sissa.it/xmlui/handle/1963/35247 %1 35558 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2016-10-20T11:44:46Z No. of bitstreams: 1 LMSZ-preprint.pdf: 547676 bytes, checksum: 5c7add921deefef560aac54da6a584cc (MD5) %0 Journal Article %J Methods and Applications of Analysis %D 2016 %T Currents and dislocations at the continuum scale %A Riccardo Scala %A Nicolas Van Goethem %XA 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.

%B Methods and Applications of Analysis %I International Press of Boston %V 23 %P 1–34 %G eng %R 10.4310/MAA.2016.v23.n1.a1 %0 Journal Article %J Mathematical Methods in the Applied Sciences %D 2015 %T A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity %A Maggiani, Giovanni Battista %A Riccardo Scala %A Nicolas Van Goethem %K 35J58 %K 35Q74 %K compatibility conditions %K elasticity %K Korn inequality %K strain decomposition %K subclass74B05 %XIn 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.

%B Mathematical Methods in the Applied Sciences %V 38 %P 5217-5230 %G eng %U https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.3450 %R 10.1002/mma.3450 %0 Report %D 2014 %T Dislocations at the continuum scale: functional setting and variational properties %A Riccardo Scala %A Nicolas Van Goethem %G eng %U http://cvgmt.sns.it/paper/2294/ %0 Journal Article %J Journal of Dynamics and Differential Equations %D 2014 %T Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes %A Gianni Dal Maso %A Riccardo Scala %XWe 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.

%B Journal of Dynamics and Differential Equations %V 26 %P 915–954 %8 Dec %G eng %U https://doi.org/10.1007/s10884-014-9409-7 %R 10.1007/s10884-014-9409-7 %0 Thesis %D 2014 %T A variational approach to statics and dynamics of elasto-plastic systems %A Riccardo Scala %K delamination %X We 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. %I SISSA %G en_US %U http://urania.sissa.it/xmlui/handle/1963/7471 %1 7583 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by Riccardo Scala (rscala@sissa.it) on 2014-10-29T12:04:30Z No. of bitstreams: 1 tesi.pdf: 1322338 bytes, checksum: 5fcf0395d13a0dc16eac4edaef440913 (MD5)