| 2017David Bourne, Sergio Conti and Stefan Müller
Energy bounds for a compressed elastic film on a substrate
J. Nonlinear Science, 27: 453-494 2017
10.1007/s00332-016-9339-0
Abstract: We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimization into a free-boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $σ$, and a measure of the bonding strength between the film and substrate, $γ$. We prove upper bounds on the minimum energy of the form $σ^a γ^b$ and find that there are four different parameter regimes corresponding to different values of $a$ and $b$ and to different folding patterns of the film. In some cases the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds. |
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| Anton Bovier, Loren Coquille and Rebecca Neukirch
The recovery of a recessive allele in a Mendelian dipoloid model
2017
https://arxiv.org/abs/1703.02459
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| Andrea Braides, Sergio Conti and Adriana Garroni
Density of polyhedral partitions
Calc. Var. Partial Differential Equations, 56(2): Art. 28, 10 2017
10.1007/s00526-017-1108-x
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| Emanuel Carneiro, Diogo Oliveira e Silva and Mateus Sousa
Extremizers for Fourier restriction on hyperboloids
arXiv e-prints: arXiv:1708.03826 2017
https://ui.adsabs.harvard.edu/abs/2017arXiv170803826C
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| S. Chhita, P.L. Ferrari and F.L. Toninelli
Speed and fluctuations for some driven dimer models
preprint: arXiv:1705.07641 2017
https://arxiv.org/abs/1705.07641
Abstract: We consider driven dimer models on the square and honeycomb graphs, starting from a stationary Gibbs measure. Each model can be thought of as a two dimensional stochastic growth model of an interface, belonging to the anisotropic KPZ universality class. We use a combinatorial approach to determine the speed of growth and show logarithmic growth in time of the variance of the height function fluctuations. |
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| Sergio Conti, Johannes Diermeier and Barbara Zwicknagl
Deformation concentration for martensitic microstructures in the limit of low volume fraction
Calc. Var. PDE, 56: 16 2017
10.1007/s00526-016-1097-1
Abstract: We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $Γ$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation. |
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| Sergio Conti, Matteo Focardi and Flaviana Iurlano
Integral representation for functionals defined on $SBD^p$ in dimension two
Arch. Ration. Mech. Anal., 223(3): 1337--1374 2017
10.1007/s00205-016-1059-y
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| Sergio Conti, Heiner Olbermann and Ian Tobasco
Symmetry breaking in indented elastic cones
Mathematical Models and Methods in Applied Sciences, 27: 291-321 2017
10.1142/S0218202517500026
Abstract: Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von-Kármán plate model. Specifically, we find that three different regimes arise with increasing indentation $δ$: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger $δ$ a localized inversion takes place in the central region. Then we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations. |
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| S. Conti, M. Klar and B. Zwicknagl
Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity
Proc. Roy. Soc. A, 473(2203) 2017
http://rspa.royalsocietypublishing.org/content/473/2203/20170235
Abstract: We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small. |
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| Sergio Conti, Adriana Garroni and Stefan Müller
Homogenization of vector-valued partition problems and dislocation cell structures in the plane
Boll. Unione Mat. Ital., 10(1): 3--17 2017
10.1007/s40574-016-0083-z
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| Carlota M. Cuesta, Hans Knüpfer and J.J. L. Velázquez
Self-similar lifting and persistent touch-down points in the thin film equation
2017
https://arxiv.org/abs/1708.00243
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| Celia Reina and Sergio Conti
Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition $F=F^eF^i$
J. Mech. Phys. Solids, 107: 322--342 2017
10.1016/j.jmps.2017.07.004
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| 2016Beatrice Acciaio, Alexander M. G. Cox and Martin Huesmann
Model-independent pricing with insider information: A Skorokhod embedding approach
arxiv e-print 1610.09124 2016
https://arxiv.org/abs/1610.09124
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| V. Beffara, S. Chhita and K. Johansson
Airy point process at the liquid-gas boundary
arXiv:1606.08653 2016
http://arxiv.org/abs/1606.08653
Abstract: {Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.} |
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| A. Borodin, I. Corwin and P.L. Ferrari
Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes
arXiv:1612.00321 2016
https://arxiv.org/abs/1612.00321
Abstract: We consider a discrete model for anisotropic (2+1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit to the (2+1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE. |
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| S. Chhita, P.L. Ferrari and H. Spohn
Limit distributions for KPZ growth models with spatially homogeneous random initial conditions
preprint, arXiv:1611.06690 2016
http://arxiv.org/abs/1611.06690
Abstract: For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the roughness of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at is conical point. |
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| Sergio Conti, Adriana Garroni and Stefan Müller
Dislocation microstructures and strain-gradient plasticity with one active slip plane
J. Mech. Phys. Solids, 93: 240-251 2016
10.1016/j.jmps.2015.12.008
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| Sergio Conti and Michael Ortiz
Optimal Scaling in Solids Undergoing Ductile Fracture by Crazing
Arch. Rat. Mech. Anal., 219(2): 607-636 2016
http://dx.doi.org/10.1007/s00205-015-0901-y
Abstract: We derive optimal scaling laws for the macroscopic fracture energy of polymers failing by crazing. We assume that the effective deformation-theoretical free-energy density is additive in the first and fractional deformation-gradients, with zero growth in the former and linear growth in the latter. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. For this particular geometry, we derive optimal scaling laws for the dependence of the specific fracture energy on cross-sectional area, micromechanical parameters, opening displacement and intrinsic length of the material. In particular, the upper bound is obtained by means of a construction of the crazing type. |
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| Sergio Conti, Felix Otto and Sylvia Serfaty
Branched Microstructures in the Ginzburg-Landau Model of Type-I Superconductors
SIAM J. Math. Anal., 48: 2994-3034 2016
10.1137/15M1028960
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| Sergio Conti and Barbara Zwicknagl
Low volume-fraction microstructures in martensites and crystal plasticity
Math. Models Methods App. Sci.: 1319-1355 2016
10.1142/S0218202516500317
Abstract: We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small. |
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