arXiv subjects

Roberto Paroni

Publications and source records attributed to Roberto Paroni.

At least 19 recordsLinked to original sources

arXiv2026

When Volumetric Growth Selects Surface Growth

We investigate the relationship between volumetric and surface growth within a recently proposed optimization-driven framework for linearly elastic solids. In this approach, growth is not prescribed through an evolution law; instead, the growth distribution is determined as the solution of a constrained optimization problem. Focusing on processes driven by the minimization of the work performed by external loads in one-dimensional and axisymmetric settings, we derive explicit analytical solutions for the resulting growth distributions. Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, we show that the optimal growth distributions are singular and concentrate on boundaries or internal interfaces. These results provide a variational mechanism through which, under certain conditions, surface growth is selected as the optimal realization of volumetric growth.

math-ph↗

arXiv2026

Surface Growth Driven by an Optimality Criterion

We propose a variational framework for accretive surface growth driven by an optimality principle. Rather than prescribing a kinetic law, the configuration at each time step is obtained, within a time-discrete setting, as the solution of a constrained minimization problem. Growth is modeled as an irreversible surface deposition process subject to a global mass constraint, while the driving mechanism is encoded in an objective functional, here taken to be the structural mean compliance. The approach is illustrated on a linearly elastic cantilever beam whose cross-sectional height evolves through layered accretion, possibly involving prestrain and precurvature. Growth-induced residual stresses can alter the convexity of the compliance functional, leading to nonuniqueness and localization phenomena. We explore the possibility of adding a regularization term penalizing deviations from the previous-step configuration. Finally, through a formal limiting procedure, we derive from the time-discrete formulation a time-continuous limit in the form of a constrained gradient flow.

math-ph↗

arXiv2026

Volumetric Growth in Linear Elasticity Driven by an Optimality Criterion

Using linearized elasticity as a convenient mechanical framework, we show that volumetric growth can be formulated as an optimization-driven process in which the growth tensor is determined implicitly by constrained optimization rather than prescribed through phenomenological evolution laws. At each incremental step, the displacement and growth fields satisfy equilibrium, mass-balance constraints, and an irreversibility condition enforcing accretive growth, while an objective functional encodes the driving mechanism of the process. Finite element discretization leads to a finite-dimensional constrained minimization problem in the growth variables alone and makes explicit the interpretation of the evolution as a projected gradient flow. Numerical examples illustrate the proposed framework.

math-ph↗

arXiv2024

A beam that can only bend on the Cantor set

In this work we address the following question: is it possible for a one-dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by $n$, each one only able to bend on the set associated with the $n$-th step in the construction of the Cantor set and compute the $\Gamma$-limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension $\ln 2/\ln 3$, which is the dimension of the Cantor set.

math.AP↗

arXiv2024

Sharp-interface limits for brittle fracture via the inverse-deformation formulation

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Gamma-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks; crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied load.

math.AP↗

arXiv2022

On a notion of nonlocal curvature tensor

In the literature various notions of nonlocal curvature can be found. Here we propose a notion of nonlocal curvature tensor. This we do by generalizing an appropriate representation of the classical curvature tensor and by exploiting some analogies with certain fractional differential operators.

math.DG↗

arXiv2021

Stability of boundary conditions for the Sadowsky functional

It has been proved by the authors that the (extended) Sadowsky functional can be deduced as the Gamma-limit of the Kirchhoff energy on a rectangular strip of height $\varepsilon$, as $\varepsilon$ tends to 0. In this paper we show that this Gamma-convergence result is stable when affine boundary conditions are prescribed on the short sides of the strip. These boundary conditions include those corresponding to a M\"obius band.

math.AP↗

arXiv2021

Linear Models of a Stiffened Plate via $Γ$-convergence

We consider a family of three-dimensional stiffened plates whose dimensions are scaled through different powers of a small parameter $\varepsilon$. The plate and the stiffener are assumed to be linearly elastic, isotropic, and homogeneous. By means of $Γ$-convergence, we study the asymptotic behavior of the three-dimensional problems as the parameter $\varepsilon$ tends to zero. For different relative values of the powers of the parameter $\varepsilon$, we show how the interplay between the plate and the stiffener affects the limit energy. We derive twenty-three limit problems.

math-ph↗

arXiv2021

Competition between epithelial tissue elasticity and surface tension in cancer morphogenesis

We derive a continuum mechanical model to capture the morphological changes occurring at the pretumoral stage of epithelial tissues. The proposed model aims to investigate the competition between the bulk elasticity of the epithelium and the surface tensions of the apical and basal sides. According to this model, when the apico-basal tension imbalance reaches a critical value, a subcritical bifurcation is triggered and the epithelium attains its physiological folded shape. Based on data available in the literature, our model predicts that pretumoral cells are softer than healthy cells.

cond-mat.soft↗

arXiv2019

An atomistic-based Föppl-von Kármán model for graphene

We deduce a non-linear continuum model of graphene for the case of finite out-of-plane displacements and small in-plane deformations. On assuming that the lattice interactions are governed by the Brenner's REBO potential of 2nd generation and that self-stress is present, we introduce discrete strain measures accounting for up-to-the-third neighbor interactions. The continuum limit turns out to depend on an average (macroscopic) displacement field and a relative shift displacement of the two Bravais lattices that give rise to the hexagonal periodicity. On minimizing the energy with respect to the shift variable, we formally determine a continuum model of Föppl-von Kármá type, whose constitutive coefficients are given in terms of the atomistic interactions.

cond-mat.mes-hall↗

arXiv2017

On the Nonlocal Curvatures of Surfaces with or without Boundary

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

math.DG↗

arXiv2017

A 2D metamaterial with auxetic out-of-plane behavior and non-auxetic in-plane behavior

Customarily, in-plane auxeticity and synclastic bending behavior (i.e. out-of-plane auxeticity) are not independent, being the latter a manifestation of the former. Basically, this is a feature of three-dimensional bodies. At variance, two-dimensional bodies have more freedom to deform than three-dimensional ones. Here, we exploit this peculiarity and propose a two-dimensional honeycomb structure with out-of-plane auxetic behavior opposite to the in-plane one. With a suitable choice of the lattice constitutive parameters, in its continuum description such a structure can achieve the whole range of values for the bending Poisson coefficient, while retaining a membranal Poisson coefficient equal to 1. In particular, this structure can reach the extreme values, $-1$ and $+1$, of the bending Poisson coefficient. Analytical calculations are supported by numerical simulations, showing the accuracy of the continuum formulas in predicting the response of the discrete structure.

physics.app-ph↗

arXiv2017

A new material property of graphene: the bending Poisson coefficient

The in-plane infinitesimal deformations of graphene are well understood: they can be computed by solving the equilibrium problem for a sheet of isotropic elastic material with suitable stretching stiffness and Poisson coefficient $ν^{(m)}$. Here, we pose the following question: does the Poisson coefficient $ν^{(m)}$ affect the response to bending of graphene? Despite what happens if one adopts classical structural models, it does not. In this letter we show that a new material property, conceptually and quantitatively different from $ν^{(m)}$, has to be introduced. We term this new parameter bending Poisson coefficient; we propose for it a physical interpretation in terms of the atomic interactions and produce a quantitative evaluation.

cond-mat.mtrl-sci↗

arXiv2017

A REBO-potential-based model for graphene bending by $Γ$-convergence

An atomistic to continuum model for a graphene sheet undergoing bending is presented. Under the assumption that the atomic interactions are governed by a harmonic approximation of the 2nd-generation Brenner REBO (reactive empirical bond-order) potential, involving first, second and third nearest neighbors of any given atom, we determine the variational limit of the energy functionals. It turns out that the $Γ$-limit depends on the linearized mean and Gaussian curvatures. If some specific contributions in the atomic interaction are neglected, the variational limit is non-local.

math-ph↗

arXiv2017

The Gaussian Stiffness of Graphene deduced from a Continuum Model based on Molecular Dynamics Potentials

We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations.

cond-mat.mtrl-sci↗

arXiv2017

Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height epsilon. We then study, by means of Gamma-convergence, the asymptotic behavior as epsilon goes to zero of the sequence of complementary energies. The limit functional is then identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields.

math-ph↗

arXiv2017

Linear models for thin plates of polymer gels

Within the linearized three-dimensional theory of polymer gels, we consider a sequence of problems formulated on a family of cylindrical domains whose height tends to zero. We assume that the fluid pressure is controlled at the top and bottom faces of the cylinder, and we consider two different scaling regimes for the diffusivity tensor. Through asymptotic-analysis techniques we obtain two plate models where the transverse displacement is governed by a plate equation with an extra contribution from the fluid pressure. In the limit obtained within the first scaling regime the fluid pressure is affine across the thickness and hence it is determined by its instantaneous trace on the top and bottom faces. In the second model, instead, the value of the fluid pressure is governed by a three-dimensional diffusion equation.

cond-mat.soft↗

arXiv2017

One-dimensional von Kármán models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármán plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the "linearized" von Kármán energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Kármán energy is a non-linear model that comprises stretching, bendings, and twisting. The "constrained" von Kármán energy, instead, leads to a new Sadowsky type of model.

math.AP↗