arXiv subjects

Sergio Conti

Publications and source records attributed to Sergio Conti.

At least 55 records · Page 3Linked to original sources

arXiv2016

Stochastic Dominance Constraints in Elastic Shape Optimization

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.

math.NA↗

arXiv2016

Existence of minimizers for the $2$d stationary Griffith fracture model

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $SBD^2$ and for which existence is well-known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford-Shah problem.

math.AP↗

arXiv2015

Energy Bounds for a Compressed Elastic Film on a Substrate

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.

math.AP↗

arXiv2015

Deformation concentration for martensitic microstructures in the limit of low volume fraction

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.

math.AP↗

arXiv2015

Symmetry breaking in indented elastic cones

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.

math.AP↗

arXiv2015

Folding patterns in partially delaminated thin films

Michael Ortiz and Gustavo Gioia showed in the 90s that the complex patterns arising in compressed elastic films can be analyzed within the context of the calculus of variations. Their initial work focused on films partially debonded from the substrate, subject to isotropic compression arising from the difference in thermal expansion coefficients between film and substrate. In the following two decades different geometries have been studied, as for example anisotropic compression. We review recent mathematical progress in this area, focusing on the rich phase diagram of partially debonded films with a lateral boundary condition.

math.AP↗

arXiv2015

Dislocation microstructures and strain-gradient plasticity with one active slip plane

We study dislocation networks in the plane using the vectorial phase-field model introduced by Ortiz and coworkers, in the limit of small lattice spacing. We show that, in a scaling regime where the total length of the dislocations is large, the phase field model reduces to a simpler model of the strain-gradient type. The limiting model contains a term describing the three-dimensional elastic energy and a strain-gradient term describing the energy of the geometrically necessary dislocations, characterized by the tangential gradient of the slip. The energy density appearing in the strain-gradient term is determined by the solution of a cell problem, which depends on the line tension energy of dislocations. In the case of cubic crystals with isotropic elasticity our model shows that complex microstructures may form, in which dislocations with different Burgers vector and orientation react with each other to reduce the total self energy.

math.AP↗

arXiv2015

Integral representation for functionals defined on $SBD^p$ in dimension two

We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(Ω)$, for $Ω\subset\mathbb{R}^2$. The space $SBD^p$ of functions whose distributional strain is the sum of an $L^p$ part and a bounded measure supported on a set of finite $\mathcal{H}^{1}$-dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by $W^{1,p}$ functions. We also obtain a generalization of Korn's inequality in the $SBD^p$ setting.

math.AP↗

arXiv2015

Low volume-fraction microstructures in martensites and crystal plasticity

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.

math.AP↗

arXiv2015

Branched Microstructures in the Ginzburg-Landau Model of Type-I Superconductors

We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We determine the optimal scaling law of the minimal energy in terms of the parameters of the problem, when the applied magnetic field is sufficiently small and the sample sufficiently thick. This optimal scaling law is proven via ansatz-free lower bounds and an explicit branching construction which refines further and further as one approaches the surface of the sample. Two different regimes appear, with different scaling exponents. In the first regime, the branching leads to an almost uniform magnetic field pattern on the boundary; in the second one the inhomogeneity survives up to the boundary.

math.AP↗

arXiv2015

Derivation of $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$ as the continuum limit of crystalline slip

In this paper we provide a proof of the multiplicative kinematic description of crystal elastoplasticity in the setting of large deformations, i.e. $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$, for a two dimensional single crystal. The proof starts by considering a general configuration at the mesoscopic scale, where the dislocations are discrete line defects (points in the two-dimensional description used here) and the displacement field can be considered continuous everywhere in the domain except at the slip surfaces, over which there is a displacement jump. At such scale, as previously shown by two of the authors, there exists unique physically-based definitions of the total deformation tensor $\mathbf{F}$ and the elastic and plastic tensors $\mathbf{F}^e$ and $\mathbf{F}^p$ that do not require the consideration of any non-realizable intermediate configuration and do not assume any a priori relation between them of the form $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$. This mesoscopic description is then passed to the continuum limit via homogenization i.e., by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. We show for two-dimensional deformations of initially perfect single crystals that the classical continuum formulation is recovered in the limit with $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$, $\det \mathbf{F}^p= 1$ and $\mathbf{G}=\text{Curl}\ \mathbf{F}^p$ the dislocation density tensor.

cond-mat.mtrl-sci↗

arXiv2015

A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery

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.

math.AP↗

arXiv2015

Which special functions of bounded deformation have bounded variation?

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 ${\mathcal{H}}^{n-1}$-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.

math.AP↗

arXiv2015

Hysteresis in Magnetic Shape Memory Composites: Modeling and Simulation

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.

math.NA↗

arXiv2014

Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance

Domain branching near the boundary appears in many singularly-perturbed models for microstructure in materials and was first demonstrated mathematically by Kohn and Müller for a scalar problem modeling the elastic behavior of shape-memory alloys. We study here a model for shape-memory alloys based on the full vectorial problem of nonlinear elasticity, including invariance under rotations, in the case of two wells in two dimensions. We show that, for two wells with two rank-one connections, the energy scales proportional to the power $2/3$ of the surface energy, in agreement with the scalar model. In a case where only one rank-one connection is present, we show that the energy exhibits a different behavior, proportional to the power $4/5$ of the surface energy. This lower energy is achieved by a suitable interaction of the two components of the deformations and hence cannot be reproduced by the scalar model. Both scalings are proven by explicit constructions and matching lower bounds.

math.AP↗

arXiv2014

Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

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.

math.AP↗

arXiv2014

Phase field approximation of cohesive fracture models

We obtain a cohesive fracture model as a $Γ$-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.

math.AP↗

arXiv2014

On the theory of relaxation in nonlinear elasticity with constraints on the determinant

We consider vectorial variational problems in nonlinear elasticity of the form $I[u]=\int W(Du)dx$, where $W$ is continuous on matrices with positive determinant and diverges to infinity long sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional $\int W^{qc}(Du)dx$ 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 $p\ge n$. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.

math.AP↗