Rod models for cardiovascular mechanics - Davide Ambrosi
Cardiovascular mechanics is an extremely fertile field of application of continuum mechanics, and specifically finite elasticity. In this talk I will illustrate some modelling issues that arise in cardiovascular modelling that can be at some extent addressed with bare-bone mathematical models. In particular, I will show as a simple mechanical representation in terms of rods can provide insight in diverse applications, as the mitral valve configuration under blood pressure at systole and the embryonic morphogenesis of the mitral annulus. In all these examples the application of classical models to a specific issue naturally generates interesting mathematical questions about contact conditions, comparison of shapes and stability, respectively, that will be discussed in the talk.
Elasto-Plasticity Theory for Crystalline Solids - Paolo Biscari
We discuss a unified elasto-plastic theory for crystalline hyperelastic solids. We provide the theoretical foundation necessary for the analytical construction of an explicit hyperelastic strain potential, enabling the study of key thermo-mechanical features in crystalline materials. Our approach encompasses the investigation of elasticity, plasticity, and martensitic transitions.
Additionally, we discuss the results of numerical simulations, highlighting the emergence of typical experimental features such as intermittency, and the creation/annihilation of dislocations.
This comprehensive framework contributes to a more profound understanding of the intricate behavior exhibited by crystalline materials across a range of mechanical and thermal loading.
Supramolecular Polymers for Turbulent Drag Reduction - Carlo Casciola
It took decades from Tom’s discovery before DNS demonstrated viscoelasticity as crucial in polymer-induced drag reduction (DR) – Sureshkumar (1997), Casciola (2020). The polymer chain was modeled as a dumbbell with two massless beads connected by a nonlinear spring stretched by the solvent (FENE model). To make the model tractable, the Peterlin approximation was adopted (FENE-P model). Recently, see the review by Ching (2024), such approximation was removed by following every single dumbbell – Serafini (2022) – and the analysis was extended to multiple beads, Serafini (2024). Covalently bonded polymers suffer from mechanical degradation, eventually rendering them futile. Here, we intend to move forward, addressing supramolecular polymers that can assemble and de-assemble, making them particularly interesting for DR. They also present anti-mist properties useful in preventing explosions, e.g., in the event of aircraft crashes. In the talk, besides presenting state-of-the-art and new numerical results for the more usual covalently bonded chains, a statistical mechanics model for supramolecular polymers will be illustrated, discussing preliminary results concerning their behavior in turbulence.
R. Sureshkumar, A.N. Beris, and R.A. Handler. Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids, 9(3), 743-755, 1997.
E. De Angelis, C.M. Casciola, and R. Piva. DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Computers & fluids, 31(4-7), 495-509, 2002.
F. Serafini, F. Battista, P. Gualtieri, and C.M. Casciola. Drag reduction in turbulent wall- bounded flows of realistic polymer solutions, Phys. Rev. Lett., 129(10),104502, 2022.
F. Serafini, F. Battista, P. Gualtieri, C.M. Casciola. Polymers in turbulence: any better than dumbbells?, JFM, 987, R1, 2024.
E.S.C Ching. Less is more: modelling polymers in turbulent flows, JFM, 994, F1, 2024.
Hydrodynamic simulations and Hydrodynamic limit of the Scattering Function via Time-dependent Non-equilibrium Molecular Dynamics - Giovanni Ciccotti
The derivation of the hydrodynamic limit of the Van Hove function from a rigorous probabilistic approach is presented. Then, after recalling the extension of the standard stationary state (time averages!) Molecular Dynamics to time dependent nonequilibrium situations, we show how the same idea can be used to compute hydrodynamic relaxation processes. The procedure, which we have called Dynamical Non-Equilibrium Molecular Dynamics (D-NEMD), to distinguish it from standard (stationary) NEMD, is based on a generalization of linear response theory. The idea has been implicitly formulated by Onsager in the thirties in metaphysical language; has been given a solid statistical mechanical foundation in the fifties by Green and Kubo (in the linear and nonlinear regime); then has been proven useful in molecular simulation by the present author in collaboration with G.Jacucci and I.R.Mac Donald in the seventies. It has been called the nonlinear Kubo-Onsager relation. It permits to connect dynamical nonequilibrium averages or dynamical relaxations to the final stationary state (the chosen initial distribution, suitably sampled, is the key ingredient of the game). To show the power of the method we apply it to get the hydrodynamic relaxation of an interface between two immiscible liquids.
Experiments and simulations of the transmission of mechanical forces from the cell to the nucleus and chromatin - Fabrizio Cleri
The mechanical impact of the nucleus on cellular function becomes evident during cell migration in 3-D environments. With its large volume and relative rigidity, governed by the nuclear envelope proteins and internal chromatin organization, the nucleus acts as physical barrier, particularly relevant to immune cells and invading cancer cells. Such cells must move through tissue pores and clefts, often smaller than the size of the nucleus, which induce substantial compressive and shearing stresses, and may also lead to the temporary rupture of the nuclear envelope. Recent studies amply demonstrated that extreme nuclear deformations during confined migration, can lead to DNA damage and increased genomic instability in cancer cells. We recently started a multi-faceted, experimental-theoretical study of the molecular processes of stress transfer and relaxation from the cell body to the nucleus, down to the scale of the individual chromatin units, the nucleosomes. I will outline our innovative experimental techniques that provide high sensitivity to examine different biophysical properties (size, stiffness, viscosity, etc.), and high-throughput; and show molecular- and single-cell-scale simulations of force-induced deformation of nuclear elements, under ideally controlled conditions. The concerted actions of mechanical deformation and remodeler enzymes open the way for a new paradigm, to understand the microscopic control of chromatin organization by mechanical forces, and the downstream effects on gene expression and transcription factor activity.
Dynamics and dissipation in 2D fluid foams: the foamy continuum model - Cesare Davini
Fluid foams are bi-phase systems consisting of small bubbles of gas separated by thin liquid walls in such a proportion that the volume fraction is small. Their physics is dominated by the interfacial forces that act at the walls’ level; in spite of the simplicity of
the first cause, though, the phenomenology of the foams is very rich. Important phenomena that occur at the microscopic level may be of importance for the gross behavior: drainage, coarsening, topological changes such as the T1 transitions. The mathematical modeling, in this case more than in other two-scale modeling problems, puts the ancient question to establish what is macroscopic and what is microscopic, and how to take account of the latter in a continuum description.
Following the pioneering work by Princen (1983, 1985), and previous papers by Davini (2010, 2024), with Podio-Guidugli we focussed on the modeling of a 2D fluid foam under a simple shearing motion. We adopted the standard measure of representing the foam as a hexagonal tessellation of gas-filled tiles with liquid contour and gave a macroscopic account of the microscopic order by means of an ad hoc Cauchy-Born rule (CBR). The use of CBR and the kinematical splitting between macroscopic deformation and microscopic structure lead to the introduction of the notion of a fictitious continuous body equipped with a vectorial order-parameter field, a so-called foamy continuum, whose motion mimics the foam shearing motion closely. The approach provides a general theory of equilibrium. It turns out that the targeted shearing motion consists of a succession of slow and fast phases: the former (quasi static) consisting of a sequence of intermediate equilibrium states; the latter dissipative and modeled as an instantaneous T1-transition that takes place for stability reasons.
In this lecture, I sketch how to get the governing equations of the dynamics of a foamy continuum. This special continuum is modeled as a dissipative ordered fluid, following the lead by Sonnet&Virga [17]; attention to both regular and topological transformations is taken.
Diffeomorphism invariant minimization of functionals with nonuniform coercivity - Marco Degiovanni
We consider the minimization of a functional of the Calculus of Variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the Calculus of Variations can be applied in a suitable space, which is in turn diffeomorphism invariant.
Programmed morphing in soft and biological matter - Antonio De Simone
In recent years we have studied problems related to shape control, motivated by our interest in biological systems and their motility. The aim of this research is to distil lessons useful for the design of bio-inspired robotic and bio-medical devices. This has required the use of tools from the calculus of variations and differential geometry, from theoretical and computational mechanics of solids and fluids, of physical experiments and observations at the microscope in the case of unicellular swimmers, and manufacturing of prototypes.
We will review some of the instructive lessons that have emerged from this research line, with special emphasis on unicellular swimmers.
On the concept of self-interpenetration for tubular elastic bodies - Alfredo Marzocchi
After a brief review of the most famous conditions for non-interpenetration of continuum bodies, a proposal coming from the theory of knots is presented, together with some analytical results.
Some issues on activation in hyperelastic materials - Alessandro Musesti
Nearly all biological tissues are "active," meaning they have the ability to grow, adapt, deform, and more. Following the significant example of skeletal muscle tissue, I will explore some issues related to modeling active deformations in hyperelastic materials. Additionally, I will present a new model, based on a coercive and polyconvex elastic energy density, where activation is described using the "mixture active strain" approach. This approach effectively captures the essential aspects of activation observed in experiments on skeletal muscles.
Computational Topology, Boolean Algebras, and Solid Modeling - Alberto Paoluzzi (e Giorgio Scorzelli)
I will present a computational theory providing an original and very unusual solution to the problem of Boolean operations of solids. Our approach reduces any (finite) formula of solid algebra in 2D and 3D --- Constructive Solid Geometry (CSG) --- to the resolution of binary Boolean forms between atoms of set algebras. This approach, firmly based on chain complexes and computational topology, has been ported to the scientific language Julia, together with the functional design language Plasm, a geometric extension of FL (at Function Level) by Backus’ group at IBM Research (1993). The talk will present Plasm.jl and discuss its Boolean Algebras.
Towards a 1d model for tape spring devices - Roberto Paroni
Tape springs, which are slender metallic strips featuring a pre-curved cross-section, serve as an appealing structural option and hinge mechanism for deployable structures due to their lightweight design, affordability, and overall simplicity. For these reasons, the study of these devices has been particularly prolific in recent years. In this talk, we model tape spring devices using shell theory and analyze their asymptotic behavior as the width of the cross-section approaches zero.
The talk is based on ongoing work with Marco Picchi-Scardaoni.
Dissipate, dissipate, qualcosa resterà - Paolo Podio-Guidugli
La nozione di dissipazione della termodinamica dei continui viene spezzata additivamente in due parti, una meccanica e una termica, entrambe per definizione non-negative. Si propone di chiamare corpi a dissipazione scissa quei corpi continui la cui risposta termomeccanica è coerente con questa nozione rinforzata, se ne caratterizza la classe dei processi ammissibili e si congettura che corpi siffatti potrebbero avere comportamenti simili a quelli di certi sistemi fisici microstrutturati per i quali l'usuale formulazione di problemi ai valori iniziali e di bordo è impossibile, perché essi occupano regioni che non si può pensare consistano di punti materiali.
Multi-level biomechanical models for cell migration in dense fibrous environments - Luigi Preziosi
Cell-extracellular matrix interaction and the mechanical properties of the cell nucleus have been demonstrated to play a fundamental role in cell movement across fibrous networks, micro-channels and therefore the microstructure of porous scaffolds. So, their study is important to
understand both motion and growth in confined environments and also the spread of cancer metastases.
This talk will merge the results of some continuum mechanics models and individual cell-based models that take into account of cell adhesion mechanics and nucleus mechanical properties to finally deduce a macroscopic model able describe the motion and growth in dense fibrous
environments.
A periodic beam-like structure modelled as a planar third gradient 1D continuum - Nicola Rizzi
A planar, periodic, beam-like structure made up of straight bars linked by (perfect) hinges, is considered. A periodic unit cell and a straight middle line are identified in the reference configuration. By assuming the bars to be rigid bodies, the structure has one degree of freedom (not considering the overall rigid motion). It is proved that, during the mechanism's motion, the middle line changes its shape into a circumference. By adding elastic joints or considering some bars as elastic beams, the middle line can assume a shape whose local curvature is different from the former. The scalar parameter accounting for this difference is taken as a deformation measure. It is proved that when the length of the periodic unit cell tends to zero, the structure can be modelled as a 1D third gradient continuum.
Generalised invariants and pseudo-universal relationships for hyperelastic materials - Giuseppe Saccomandi
Constitutive modelling of nonlinear isotropic elastic materials requires a general formulation of the strain–energy function in terms of invariants, or equivalently in terms of the principal stretches. Yet, when choosing a particular form of a model, the representation in terms of either the principal invariants or stretches becomes important, since a judicious choice between one or the other can lead to a better encapsulation and interpretation of much of the behaviour of a given material. The question is then: what is the choice of invariants that best captures the behaviour of a subject material? Here, we provide a systematic method to obtain invariants that generalise the classical ones and that can be tuned to match pseudo-universal relationships in simple deformations, independently of the specific form of the strain–energy function.
Integrating Cell Dynamics into Generalized Continuum Mechanics: Towards a Mechanobiological Model of Bone Remodeling - Vittorio Sansalone
In the early 2000s, Antonio Di Carlo's pioneering contributions paved the way for the development of a new mechanical modeling approach to describe the adaptive behavior of living tissues within the framework of generalized continuum mechanics. This approach has been successfully applied to the growth and remodeling of both soft and hard tissues. One of the most delicate aspects lies in the constitutive theory, specifically in how biological phenomena are incorporated into the mechanical continuum model. In this work, we address the main modeling challenges involved in coupling mechanics and biology. As an application, we present a model of bone remodeling that integrates bone cell population dynamics into a generalized continuum mechanics framework. By coupling mechanical and biochemical phenomena at the cellular scale, the model bridges microscale processes governing cell activity with the macroscale mechanical responses of bone tissue. This multiscale approach not only deepens our understanding of bone tissue adaptation, but also embodies Di Carlo's ideas of enriching continuum mechanics with biological complexity through a mechanobiological perspective.
Driving forces in cell migration and pattern formation in a soft tissue - Amabile Tatone
I 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. I lay down a force balance law, where force and stress fields are defined as power conjugate quantities to velocity fields and their gradients, then 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 which stems out of the gradient energy leading to a Cahn-Hilliard equation. 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 chemical potential, giving rise to the spinodal decomposition which is meant to characterize an upward cell diffusion induced by cell motility. Further I 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. This vector field could possibly model any directional cue or bias characterizing the interaction of the migrating cells and the surrounding tissue. Some numerical simulations illustrate how, in a phase separation and aggregation process, an initial uniform concentration evolving to a non uniform stationary pattern is characterized by moving interfaces possibly guided by the above vector field.
Bifurcation analysis of pressure-induced detachment of a rod adhered to a plate - Stefano Turzi
We study the lift of an elastica adhering to a flat rigid surface induced by a pressure difference. Adhesion is
modelled by a cohesive force that decreases linearly with separation. Using a nonlinear local analysis, we determine the bifurcation diagram that governs the peeling process under quasi-static conditions. We show that the delamination emerges through a discontinuous transition: a normal form of the bifurcation diagram allows us to draw in a simple way the main physical mechanism, elucidating the local validity of the theory at the transition. We predict that the pressure, as a function of the detachment length, undergoes an initial drop followed by an approximately constant behaviour, while the detachment length at the transition is always finite and is roughly proportional to the elasto-adhesion length. This analysis can be the starting point to understand more complex-related problems which arise in fracture mechanics or in biology, such as testing of adhesives in a flowfield and the arterial dissection.