Sergio Conti, Matteo Focardi and Flaviana Iurlano Phase field approximation of cohesive fracture models Annales de l'Institut Henri Poincar{\'e} / Analyse non lin{\'e}aire, 33: 1033-1067 2016 10.1016/j.anihpc.2015.02.001
Abstract: We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.
Sergio Conti, Martin Lenz and Martin Rumpf Hysteresis in Magnetic Shape Memory Composites: Modeling and Simulation 2016 10.1016/j.jmps.2015.12.010
Abstract: Magnetic shape memory alloys are characterized by the coupling between a structural phase transition and magnetic one. This permits to control the shape change via an external magnetic field, at least in single crystals. Composite materials with single-crystalline particles embedded in a softer matrix have been proposed as a way to overcome the blocking of the transformation at grain boundaries. We investigate hysteresis phenomena for small NiMnGa single crystals embedded in a polymer matrix for slowly varying magnetic fields. The evolution of the microstructure is studied within the rate-independent variational framework proposed by Mielke and Theil (1999). The underlying variational model incorporates linearized elasticity, micromagnetism, stray field and a dissipation term proportional to the volume swept by the phase boundary. The time discretization is based on an incremental minimization of the sum of energy and dissipation. A backtracking approach is employed to approximately ensure the global minimality condition. We illustrate and discuss the influence of the particle geometry (volume fraction, shape, arrangement) and the polymer elastic parameters on the observed hysteresis and compare with recent experimental results.
Sergio Conti, Matteo Focardi and Flaviana Iurlano Existence of minimizers for the 2d stationary Griffith fracture model C. R. Math. Acad. Sci. Paris, 354(11): 1055--1059 2016 10.1016/j.crma.2016.09.003
Sergio Conti, Matteo Focardi and Flaviana Iurlano Some recent results on the convergence of damage to fracture Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27(1): 51--60 2016 10.4171/RLM/722
Celia Reina, Anja Schlömerkemper and Sergio Conti Derivation of $\bf F=\bf F^\rm e\bf F^\rm p$ as the continuum limit of crystalline slip J. Mech. Phys. Solids, 89: 231--254 2016 10.1016/j.jmps.2015.12.022
2015
Mathias Beiglböck, Alexander M. G. Cox, Martin Huesmann, Nicolas Perkowski and David J. Prömel Pathwise super-replication via Vovk's outer measure ArXiv e-prints 2015 http://arxiv.org/abs/1504.03644
Sunil Chhita and Patrik L. Ferrari A combinatorial identity for the speed of growth in an anisotropic KPZ model arXiv e-prints 2015 http://arxiv.org/abs/1508.01665
Abstract: The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [arXiv:1503.05339] and for this generalization the stationary measure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.
Sergio Conti, Janusz Ginster and Martin Rumpf A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery ESAIM: Mathematical Modelling and Numerical Analysis, 49(5): 1463-1487 2015 http://dx.doi.org/10.1051/m2an/2015036
Abstract: The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.
Sergio Conti, Adriana Garroni and Michael Ortiz The line-tension approximation as the dilute limit of linear-elastic dislocations Arch. Ration. Mech. Anal., 218(2): 699-755 2015 http://dx.doi.org/10.1007/s00205-015-0869-7
Abstract: We prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the $\Gamma$-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length $\psi(b,t)$, which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length $\psi_0(b,t)$ obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation the classical energy down to its $\calH^1$-elliptic envelope.
Abstract: Dislocations in crystals can be studied by a Peierls-Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing, and for dislocations restricted to planes, we show that it reduces to a line-tension model, with an energy depending on the local orientation and Burgers vector of the dislocation. This model predicts, for specific geometries, spontaneous formation of microstructure, in the sense that straight dislocations are unstable towards a zig-zag pattern. Coupling between dislocations in different planes can lead to microstructures over several length scales.
Sergio Conti, Adriana Garroni and Annalisa Massaccesi Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity Calc. Var. PDE, 54(2): 1847-1874 2015 http://dx.doi.org/10.1007/s00526-015-0846-x
Abstract: In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.
Sergio Conti and Georg Dolzmann On the theory of relaxation in nonlinear elasticity with constraints on the determinant Arch. Rat. Mech. Anal., 217(2): 413-437 2015 http://dx.doi.org/10.1007/s00205-014-0835-9
Abstract: We consider vectorial variational problems in nonlinear elasticity of the form I[u]=∫W(Du)dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional ∫Wqc(Du)dx is an upper bound on the relaxation of I, and coincides with the relaxation if the quasiconvex envelope W qc of W is polyconvex and has p-growth from below with p≧n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.
Carlota M. Cuesta, Maria Calle and Juan J. L. Velázquez Interfaces determined by capillarity and gravity in a two-dimensional porous medium 2015 http://arxiv.org/abs/1505.03676
Stefanie Heyden, Bo Li, Kerstin Weinberg, Sergio Conti and Michael Ortiz A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity J. Mech. Phys. Solids, 74: 175-195 2015 http://dx.doi.org/10.1016/j.jmps.2014.08.005
Abstract: We derive and numerically verify scaling laws for the macroscopic fracture energy of poly- mers undergoing crazing from a micromechanical model of damage. The model posits a local energy density that generalizes the classical network theory of polymers so as to account for chain failure and a nonlocal regularization based on strain-gradient elasticity. We specifically consider periodic deformations of a slab subject to prescribed opening dis- placements on its surfaces. Based on the growth properties of the energy densities, scaling relations for the local and nonlocal energies and for the specific fracture energy are derived. We present finite-element calculations that bear out the heuristic scaling relations.
Brandon Runnels, Irene Beyerlein, Sergio Conti and Michael Ortiz A relaxation method for the energy and morphology of grain boundaries and interfaces J. Mech. Phys. Solids 2015 http://dx.doi.org/10.1016/j.jmps.2015.11.007
2014
Mathias Beiglböck, Alexander M.G. Cox and Martin Huesmann Optimal Transport and Skorokhod Embedding ArXiv eprints 2014 http://arxiv.org/abs/1307.3656
Abstract: The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.
Alexei Borodin, Ivan Corwin, Patrik L. Ferrari and Balint Vető Height fluctuations for the stationary KPZ equation Math. Phys. Anal. Geom., 18(1, Art. 20): 1-95 2014 http://arxiv.org/abs/1407.6977
Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.
Fabio Cavalletti and Martin Huesmann Self-intersection of optimal geodesics Bulletin of the London Mathematical Society, 46(3): 653-656 2014 http://dx.doi.org/10.1112/blms/bdu026
Abstract: Let (X,d,m)(X,d,m) be a geodesic metric measure space. Consider a geodesic μtμt in the L2L2-Wasserstein space. Then as ss goes to tt, the support of μsμs and the support of μtμt have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X,d,m)(X,d,m) and t∈[0,1]t∈[0,1], we consider the set of times for which this geodesic belongs to the support of μtμt. We prove that tt is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K,∞)CD(K,∞). The non-branching property is not needed.
Abstract: Let (X,d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X,d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.
We also prove a stability property for Assumption 1: If (X,d,m) satisfies Assumption 1 and View the MathML source, for some continuous function g>0, then also View the MathML source verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.