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
2016
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
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.
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.
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
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
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.
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
Sergio Conti, Georg Dolzmann and Stefan Müller Korn's second inequality and geometric rigidity with mixed growth conditions Calc. Var., 50: 437-454 2014 http://dx.doi.org/10.1007/s00526-013-0641-5
Abstract: Geometric rigidity states that a gradient field which is \( L^p\) -close to the set of proper rotations is necessarily \( L^p\) -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in \( L^p+L^q\) and in \( L^p,q\) interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces.
Sergio Conti and Geog Dolzmann Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions Math. Models. Metods App. Sci., 24(14): 2929-2942 2014 http://dx.doi.org/10.1142/S0218202514500419
Abstract: We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.