Sergio Conti, Matteo Focardi and Flaviana Iurlano Existence of strong minimizers for the Griffith static fracture model in dimension two Ann. Inst. Henri Poincaré C, Anal. Non Linéaire, 36: 455-474 2019 10.1016/j.anihpc.2018.06.003
Sergio Conti, Martin Lenz, Nora Lüthen, Martin Rumpf and Barbara Zwicknagl Geometry of martensite needles in shape memory alloys 2019 https://arxiv.org/abs/1912.02274
2018
P. Ariza, S. Conti, A. Garroni and M. Ortiz Variational modeling of dislocations in crystals in the line-tension limit In V. Mehrmann and M. Skutella, editor, European Congress of Mathematics, Berlin, 2016
page 583-598.
Publisher: EMS
2018 10.4171/176-1/27
Antonin Chambolle, Sergio Conti and Gilles A. Francfort Approximation of a britte fracture energy with the constraint of non-interpenetration Arch. Ration. Mech. Anal., 228: 867-889 2018 10.1007/s00205-017-1207-z
Sergio Conti, Martin Rumpf, Rüdiger Schultz and Sascha Tölkes Stochastic Dominance Constraints in Elastic Shape Optimization SIAM J. Control Optim., 56: 3021-3034 2018 10.1137/16M108313X
Abstract: This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The underlying cost functional is of compliance type plus a perimeter term, in the implementation shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations.
S. Conti and G. Dolzmann An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers J. Mech. Phys. Solids, 113: 126-143 2018 10.1016/j.jmps.2018.02.001
Sergio Conti, Matteo Focardi and Flaviana Iurlano Which special functions of bounded deformation have bounded variation Proc. Roy. Soc. Edinb. A, 148: 33-50 2018 10.1017/S030821051700004X
Abstract: Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions which are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that $SBD^p$ functions are approximately continuous $H^n-1$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in BV and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.
Sergio Conti, Stefan Müller and Michael Ortiz Data-driven problems in elasticity Arch. Ration. Mech. Anal., 229: 79-123 2018 10.1007/s00205-017-1214-0
Sergio Conti, Martin Lenz, Matthäus Pawelczyk and Martin Rumpf Homogenization in magnetic-shape-memory polymer composites In Volker Schulz and Diaraf Seck, editor, Shape Optimization, Homogenization and Optimal Control, Volume 169 of International Series of Numerical Mathematics
page 1-17.
Publisher: Birkhäuser, Cham
2018 10.1007/978-3-319-90469-6_1
S. Conti, M. Goldman, F. Otto and S. Serfaty A branched transport limit of the Ginzburg-Landau functional Journal de l'École polytechnique -- Mathématiques, 5: 317-375 2018 10.5802/jep.72
Sergio Conti, Benedict Geihe, Martin Lenz and Martin Rumpf A posteriori modeling error estimates in the optimization of two-scale elastic composite materials ESAIM: Mathematical Modelling and Numerical Analysis, 52: 1457-1476 2018 10.1051/m2an/2017004
Celia Reina, Landry Fokoua Djodom, Michael Ortiz and Sergio Conti Kinematics of elasto-plasticity: Validity and limits of applicability of $F=F_eF_p$ for general three-dimensional deformations Journal of the Mechanics and Physics of Solids, 121: 99--113 2018 10.1016/j.jmps.2018.07.006
W. Schill, S. Heyden, S. Conti and M. Ortiz The anomalous yield behavior of fused silica glass Journal of the Mechanics and Physics of Solids, 113: 105 - 125 2018 10.1016/j.jmps.2018.01.004
2017
David 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.
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
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.
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.
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
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.
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