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Elisa Davoli

Publications and source records attributed to Elisa Davoli.

At least 55 records · Page 3Linked to original sources
arXiv2020

Two-well linearization for solid-solid phase transitions

In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus not being detected by the limiting deformations. By Gamma-convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [32] for a model with anisotropic second-order perturbation.

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arXiv2020

Magnetoelastic thin films at large strains

Starting from the three-dimensional setting, we derive a limit model of a thin magnetoelastic film by means of Γ-convergence techniques. As magnetization vectors are defined on the elastically deformed configuration, our model features both Lagrangian and Eulerian terms. This calls for qualifying admissible three-dimensional deformations of planar domains in terms of injectivity. In addition, a careful treatment of the Maxwell system in the deformed film is required.

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arXiv2019

Two-well rigidity and multidimensional sharp-interface limits for solid-solid phase transitions

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $Γ$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.

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arXiv2019

Local asymptotics for nonlocal convective Cahn-Hilliard equations with W^{1,1} kernel and singular potential

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.

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arXiv2019

Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms

We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

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arXiv2019

Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

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arXiv2019

Dynamic perfect plasticity and damage in viscoelastic solids

In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto-plasticity, and unidirectional rate-dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic-strain measures and bounded-deformation displacements, on sophisticated time-regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate-independent processes for the energy conservation in the dynamical-plastic part, and on the proof of the strong convergence of the elastic strains. Existence of a suitably defined weak solutions is proved rather constructively by using a staggered two-step time discretization scheme.

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arXiv2019

Homogenization in $BV$ of a model for layered composites in finite crystal plasticity

In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space $BV$ of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.

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arXiv2018

Analytical validation of the Yound-Dupré law for epitaxially-strained thin films

We present here an analysis of the regularity of minimizers of a variational model for epitaxially strained thin-films identified by the authors in the companion paper [Davoli E., Piovano P., Derivation of a heteroepitaxial thin-film model. Submitted 2018]. The regularity of energetically-optimal film profiles is studied by extending previous methods and by developing new ideas based on transmission problems. The achieved regularity results relate to both the Stranski-Krastanow and the Volmer-Weber modes, the possibility of different elastic properties between the film and the substrate, and the presence of the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. Finally, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact point of films with the substrate. In particular, the Young-Dupré law is shown to hold, yielding what appears to be the first analytical validation of such law for a thin-film model in the context of Continuum Mechanics.

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arXiv2018

Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films

An advective Cahn-Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn-Hilliard equation.

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arXiv2018

Derivation of a heteroepitaxial thin-film model

A variational model for epitaxially-strained thin films on rigid substrates is derived both by Γ-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles.

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arXiv2017

One dimensional fractional order $TGV$: Gamma-convergence and bilevel training scheme

New fractional $r$-order seminorms, $TGV^r$, $r\in \mathbb R$, $r\geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k\in\mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $Γ$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.

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arXiv2017

Analytical validation of a 2+1 dimensional continuum model for epitaxial growth with elastic substrate

An analytical validation is obtained for the evolution equation $$h_t=Δ[ \mathcal{F}^{-1}(-aE \mathcal{F}(h)) - r/h^2 -Δh ],$$ introduced in {\cite{TS}} by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, $h$ denotes the surface height of the film, $\mathcal{F}$ is the Fourier transform, and $a$, $E$, $r$ are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum.

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arXiv2016

Homogenization for A-quasiconvexity with variable coefficients

A homogenization result for a family of oscillating integral energies is presented, where the fields under consideration are subjected to first order linear differential constraints depending on the space variable x. The work is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of A-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

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arXiv2015

Multiscale homogenization in Kirchhoff's nonlinear plate theory

The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff's nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter $h$ and the two homogenization scales $\ep$ and $\ep^2$.

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