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Giuseppe Tomassetti

Publications and source records attributed to Giuseppe Tomassetti.

At least 19 recordsLinked to original sources
arXiv2026

On the single field formulation in magnetostatics

We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field on the one hand and the single field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, despite the fact that the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.

math-ph↗
arXiv2025

Using stochastic thermodynamics with internal variables to capture orientational spreading in cell populations undergoing cyclic stretch

We revisit the modeling framework introduced in [N. Loy and L. Preziosi: Bull. Math. Bio., 85, 2023] to describe the dynamics of cell orientation under cyclic stretch. We propose a reformulation based on the principles of Stochastic Thermodynamics with Internal Variables introduced in [T. Leadbetter, P. Purohit, and C. Reina: PNAS Nexus, 2, 2023]. This approach allows us to describe not only the evolution of the orientation distribution, but also the observed spreading phenomenon. The insight provided by our model reveals an interesting phenomenon, which we call two-stage reorientation: when cells begin aligned with an energy maximum, their orientations spread before concentrating at the energy minimum. This theoretical prediction suggests a new experiment to test this modeling framework.

cond-mat.stat-mech↗
arXiv2025

Homogenization of magnetoelastic materials with rigid magnetic inclusions at small strains

We investigate a homogenization problem for a linearly elastic magnetic material that incorporates elastically rigid magnetic inclusions firmly bonded to the matrix. By considering a periodic arrangement of this material, we identify an effective magnetoelastic energy, obtained by homogenization when the period approaches zero. For comparison, we also briefly discuss alternative, essentially equivalent magnetic models naturally linked by a Legendre-Fenchel transform of magnetic energy density where the elastic deformation enters as a parameter.

math.AP↗
arXiv2024

Driving forces in cell migration and pattern formation in a soft tissue

We give a description of cell diffusion in a soft tissue, paying special attention to the coupling of force, matter, and microforce balance laws through a suitable dissipation principle. To this end, we cast our framework into a multi-level schematics, comprising both kinematics and kinetics, based on a characterization of the free energy. We lay down first a force balance law, where force and stress fields are defined as power conjugate quantities to velocity fields and their gradients, then we give a species molar balance law, with chemical potential test fields, as power conjugate quantities to the rate of change of the species concentration, and finally a microforce balance law. The main feature of this framework is the constitutive expression for the chemical potential which is split into a term derived from the homogeneous convex part of the free energy and an active external chemical potential giving rise to the spinodal decomposition. The active part of the chemical potential is given an expression depending on the cell concentration and resembling the one defined in [Oster, Murray and Harris, J. Embryol. Exp. Morph. 78 (1983)], where it is meant to characterize an upward cell diffusion induced by cell motility. Further we show how an external vector field, entering the microforce balance law as a power conjugate quantity to the rate of change of the concentration gradient, can guide the diffusion process to a different limit stationary pattern. This vector field could possibly model any directional cue or bias characterizing the interaction of the migrating cells and the surrounding tissue.

cond-mat.soft↗
arXiv2024

A coordinate-free guide to the mechanics of thin shells

In this tutorial, we provide a coordinate-free derivation of the system of equations that govern equilibrium of a thin shell that can undergo shear. This system involves tensorial fields representing the internal force and couple per unit length that adjacent parts of the shell exchange at their common boundary. By an appropriate decomposition of those quantities, we obtain a representation of the internal power in terms of time derivatives of suitable strain measures. Subsequently, we propose constitutive equations that employ these strain measures as independent variables. After specializing the theory to the case of unshearable shells, we linearize the resulting equations. As an application, we study the free vibrations of a pressurized spherical shell, showcasing the advantages of a coordinate-free perspective, which simplifies both the deduction and the solution of the final governing equations.

physics.class-ph↗
arXiv2024

Simulation and design of isostatic thick origami structures

Thick origami structures are considered here as assemblies of polygonal panels hinged to each other along their edges according to a corresponding origami crease pattern. The determination of the internal actions caused by external loads in such structures is not an easy task, owing to their high degree of static indeterminacy, and the likelihood of unwanted self-balanced internal actions induced by manufacturing imperfections. Here we present a method for reducing the degree of static indeterminacy which can be applied to several thick origami structures to make them isostatic. The method utilizes sliding hinges, which permit also the relative translation along the hinge axis, to replace conventional hinges. After giving the analytical description of both types of hinges and describing a rigid folding simulation procedure based on the integration of the exponential map, we present the static analysis of a series of noteworthy examples based on the Miura-ori pattern, the Yoshimura pattern, and the Kresling pattern. The method can be applied for the design and realization of thick origami structures with adequate strength to resist external actions.

cond-mat.soft↗
arXiv2024

An existence result for accretive growth in elastic solids

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

math.AP↗
arXiv2023

A discrete model for layered growth

In this work we present a discrete model that captures the fundamental properties of additively manufactured solids in a minimal setting. The model is based on simplified kinematics and allows for the onset of incompatible deformations between discrete layers of an additively manufactured stack. Thanks to the discrete nature of the model, we obtain an averaged formulation of mechanical equilibrium for the growing stack, leading to closed-form solutions that are both analytically simple and physically transparent. In particular, we are able to explain the origin of residual stresses by the accumulation of incompatible deformations between adjacent layers. At the same time, we are able to formulate the technologically relevant inverse problem that provides the deposition protocol required to produce a desired state of internal stress in the manufactured stack. Another important aspect analyzed in the work is the role played by an ideal ``glue'' between the layers, whose presence is fundamental to prevent their sliding and whose mechanical behavior can quantitatively influence the final stress distribution in the stack. Although the model is an elementary approximation of additive manufacturing, its simplicity makes it possible to highlight how the controls exerted during deposition will have qualitative or quantitative effects on the final stress state of the stack. This understanding is crucial in shedding light on the complex mechanical behavior of additive manufactured solids.

cond-mat.soft↗
arXiv2023

Inhomogeneous finitely-strained thermoplasticity with hardening by an Eulerian approach

A standard elasto-plasto-dynamic model at finite strains based on the Lie-Liu-Kröner multiplicative decomposition, formulated in rates, is here enhanced to cope with spatially inhomogeneous materials by using the reference (called also return) mapping. Also an isotropic hardening can be involved. Consistent thermodynamics is formulated, allowing for both the free and the dissipation energies temperature dependent. The model complies with the energy balance and entropy inequality. A multipolar Stokes-like viscosity and plastic rate gradient are used to allow for a rigorous analysis towards existence of weak solutions by a semi-Galerkin approximation.

math.AP↗
arXiv2021

A theory of magneto-elastic nanorods obtained through rigorous dimension reduction

Starting from a two-dimensional theory of magneto-elasticity for fiber-reinforced magnetic elastomers we carry out a rigorous dimension reduction to derive a rod model that describes a thin magneto-elastic strip undergoing planar deformations. The main features of the theory are the following: a magneto-elastic interaction energy that manifests itself through a distributed torque; a penalization term that prevent local interpenetration of matter; a regularization that depends on the second gradient of the deformation and models microstructure-induced size effects. As an application, we study a problem involving magnetically-induced buckling and we show that the intensity of the field at the onset of the instability increases if the length of the rod is decreased. Finally, we assess the accuracy of the deduced model by performing numerical simulations where we compare the two-dimensional and the one-dimensional theory in some special cases and we observe excellent agreement.

cond-mat.soft↗
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↗
arXiv2021

A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling

Devised towards geophysical applications for various processes in the lithosphere or the crust, a model of poro-elastodynamics with inelastic strains and other internal variables like damage (aging) and porosity as well as with diffusion of water is formulated fully in the Eulerian setting. Concepts of gradient of the total strain rate as well as the additive splitting of the total strain rate are used while eliminating the displacement from the formulation. It relies on that the elastic strain is small while only the inelastic and the total strains can be large. The energetics behind this model is derived and used for analysis as far as the existence of global weak energy-conserving solutions concerns. By this way, the model in [V. Lyakhovsky et al., Pure Appl. Geophys., 171:3099-3123, 2014] and [V. Lyakhovsky et al., Izvestiya, Physics of the Solid Earth, 43:13-23, 2007] is completed to make it mechanically consistent and amenable for analysis.

math.AP↗
arXiv2021

Shape programming of a magnetic elastica

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.

math.AP↗
arXiv2021

Equilibrium of Kirchhoff's rods subject to a distribution of magnetic couples

The equilibrium of magneto-elastic rods, formed of an elastic matrix containing a uniform distribution of paramagnetic particles, that are subject to terminal loads and are immersed in a uniform magnetic field, is studied. The deduced nonlinear equilibrium equations are fully consistent with Kirchhoff's theory in the sense that they hold at the same order of magnitude. Exact solutions of those equations in terms of Weierstrass elliptic functions are presented with reference to magneto-elastic cantilevers that undergo planar deformations under the action of a terminal force and a magnetic field whose directions are either parallel or orthogonal. The exact solutions are applied to the study of a problem of remotely controlled deformation of a rod and to a bifurcation problem in which the end force and the magnetic field act as an imperfection parameter and a bifurcation parameter, respectively.

cond-mat.soft↗
arXiv2020

An interpretation of Temam's extra force in the quasi-incompressible Navier-Stokes system

We discuss the role of the extra force $$ \mathbf f_{\rm e}=-\frac 1 2(\nabla\cdot\mathbf v)\mathbf v $$ in the system of partial differential equations \begin{equation*} \left\{ \begin{aligned} &\frac{\partial\mathbf v}{\partial \rm t}+(\mathbf v\cdot\nabla)\mathbf v+\nabla \mathrm p-\frac 1 {\rm Re}Δ\mathbf v=\mathbf f+\mathbf f_{\rm e},\\ &\frac 1 {\mathrm K}\frac{\partial \mathrm p}{\partial \rm t}+\nabla\cdot\mathbf v=0,\qquad \mathrm K>>1, \end{aligned}\right. \end{equation*} whose weak solution has been proved in [Arch. Rat. Mech. Analysis, 32:135-153] to approximate, in the limit $\mathrm K\to\infty$, the weak solution of the incompressible Navier-Stokes system. Taking the cue from [Ann. Mat. Pura Appl. 172:103-124], we provide a physical interpretation of the extra force $\mathbf f_{\rm e}$, showing that it is a manifestation of inertia.

math.AP↗
arXiv2020

Treadmilling stability of a one-dimensional actin growth model

Actin growth is a fundamental biophysical process and it is, at the same time, a prototypical example of diffusion-mediated surface growth. We formulate a coupled chemo-mechanical, one-dimensional growth model encompassing both material accretion and ablation. A solid bar composed of bound actin monomers is fixed at one end and connected to an elastic device at the other. This spring-like device could, for example, be the cantilever tip of an atomic force microscope. The compressive force applied by the spring on the bar increases as the solid grows and affects the rate of growth. The mechanical behaviour of the bar, the diffusion of free actin monomers in a surrounding solvent and the kinetic growth laws at the accreting/ablating ends are accounted for. The constitutive response of actin is modeled by a convex but otherwise arbitrary elastic strain energy density function. Treadmilling solutions, characterized by a constant length of the continuously evolving body, are investigated. Existence and stability results are condensed in the form of simple formulas and their physical implications are discussed.

cond-mat.soft↗
arXiv2019

Bounded solutions and their asymptotics for a doubly nonlinear Cahn-Hilliard system

In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscosity and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.

math.AP↗