A 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.

}, doi = {10.4310/MAA.2016.v23.n1.a1}, author = {Riccardo Scala and Nicolas Van Goethem} } @article {doi:10.1002/mma.3450, title = {A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity}, journal = {Mathematical Methods in the Applied Sciences}, volume = {38}, number = {18}, year = {2015}, pages = {5217-5230}, abstract = {In this paper, we prove the Saint-Venant compatibility conditions in $L^p$ for $p\in(1,$\infty$)$, 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 $p\in(1,$\infty$)$. Copyright {\textcopyright} 2015\ John Wiley \& Sons, Ltd.

}, keywords = {35J58, 35Q74, compatibility conditions, elasticity, Korn inequality, strain decomposition, subclass74B05}, doi = {10.1002/mma.3450}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.3450}, author = {Maggiani, Giovanni Battista and Riccardo Scala and Nicolas Van Goethem} } @article {scala2014dislocations, title = {Dislocations at the continuum scale: functional setting and variational properties}, year = {2014}, url = {http://cvgmt.sns.it/paper/2294/}, author = {Riccardo Scala and Nicolas Van Goethem} } @article {2013, title = {Fields of bounded deformation for mesoscopic dislocations}, year = {2013}, publisher = {SISSA}, abstract = {In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.}, url = {http://hdl.handle.net/1963/6378}, author = {Nicolas Van Goethem} } @article {2013, title = {Minimal partitions and image classification using a gradient-free perimeter approximation}, year = {2013}, institution = {SISSA}, abstract = {In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.}, keywords = {Image classification, deblurring, optimal partitions, perimeter approximation}, url = {http://hdl.handle.net/1963/6976}, author = {Samuel Amstutz and Nicolas Van Goethem and Antonio Andr{\'e} Novotny} } @article {2012, title = {Topological sensitivity analysis for high order elliptic operators}, number = {SISSA;36/2012/M}, year = {2012}, institution = {SISSA}, abstract = {The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.}, keywords = {Topological derivative, Elliptic operators, Polarization tensor}, url = {http://hdl.handle.net/1963/6343}, author = {Samuel Amstutz and Antonio Andr{\'e} Novotny and Nicolas Van Goethem} }