MSP 33-37 -New problems arising in Mathematical Modeling of smart and biological materials I -II -III -IV -V

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The Mini-symposium is devoted to most recent results concerning mathematical models, with particular interest in smart and biological materials.

Notably, challenging analytical problems are originated from applications; indeed, new materials are more and more widely studied since they are used in a variety of different environments. Specifically, in recent years there has been a growing interest, on one hand, in materials with memory, and smart material in general, and, on the other hand, in biological materials. Indeed, all these materials exhibit crucial physical properties; in the case of materials with memory their behaviour depends on time not only through the present time but also through their past history. This peculiarity leads to the use of such materials in various different applications, like in biomedical tools and, in general, in sensors and actuators. Also biological materials require new analytical instruments to be modeled. More generally, other problems originating from different areas of biology as well as of medicine are of interest. The Minisymposium aims to bring together researchers who are investigating mathematical problems arising in modeling of such materials.

Organizers:

Boundary controllability and source reconstruction in a viscoelastic string under external traction

Luciano Pandolfi

Dipartimento di Scienze Matematiche, Politecnico di Torino Corso Duca degli Abruzzi 24—10129 Torino, Italy

luciano.pandolfi@polito.it

Treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions. In fact, usually a “system” is not an isolated body but it is part of a chain of mechanisms which disturb the “system” for example due to the periodic rotation of shafts. Ths kind of problem has been rarely studied in control theory. In the specific case we shall study, the case of a viscoelastic string, the effect of such external action is on the horizontal component of the traction, and so it affects the coefficients of the corresponding wave type equation, which will will be time dependent. The usual methods used in controllability are not naturally adapted to this case. For example at first sight it might seem that moment methods can only be used in case of coefficients which are constant in time. Instead, we shall see that moment methods can be extended to study controllability of a viscoelastic string subject to external traction and in particular we shall study a controllability problem which is encountered in the solution of the inverse problem consisting in the identification of a distributed disturbance source.

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Stabilization for nonlinear integro-differential equations with weakly singular kernels.

Daniela Sforza

Dipartimento SBAI Sezione di Matematica, Sapienza Universit`a di Roma via Antonio Scarpa, 16 00161 -Roma Italy

sforza@dmmm.uniroma1.it

In this talk we will show stabilization results for nonlinear abstract evolution equations in the presence of a convolution integral term.

For concrete problems, e.g. the motion of some viscoelastic materials, the effects of the past time cannot be neglected: in the partial integro-differential equations describing the motion, the integral term just represents the memory of the past.

We will give decay estimates for the solutions when the convolution kernels are allowed to be weakly singular functions (integrable functions, singular at ”t=0”) and satisfy sign conditions consistent with thermodynamical restrictions.

Our main tools are coercive estimates, properties of positive definite kernels and the multipliers method: we succeeded in finding multipliers which work even in the presence of integral terms.

Joint work with Piermarco Cannarsa.

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Discrete Observability

Paola Loreti

Dipartimento SBAI Sezione di Matematica, Sapienza Universit`a di Roma via Antonio Scarpa, 16 00161 -Roma Italy

loreti@dmmm.uniroma1.it

In the framework of a classical theorem due to Ingham, we will discuss some aspects of observability problems when the observation is given at a finite number of points of the given time.

The talk is based on joint works with Vilmos Komornik.

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Some existence and uniqueness results in viscoelasticity: connections with free energies

Sandra Carillo

Dipartimento di Scienze di Base e Applicate per l’Ingegneria – Sez. Matematica Sapienza Universit`a di Roma, 16, Via Antonio Scarpa, 00161 Rome, Italy sandra.carillo@sbai.uniroma1.it or sandra.carillo@uniroma1.it

Integro-differential model problems arise in viscoelasticity. Here, some existence, and, possibly uniqueness, results recently obtained in joint work with V. Valente and G. Vergara Caffarelli, in the case, in turn, of a magneto-viscoelasticity problem [3], [4], or of a singular viscoelasticity problem [5] are considered. The connection between analytic results and the choice of suitable free energies is pointed out. Furthermore, the problem of the choice of the free energy, studied in various different physical models, in joint work with G. Amendola and A. Manes [1,2], is also considered. In particular, free energies in viscoelastic solids and fluids, [2], are also mentioned emphasizing their analytical implications as far as asymptotic behaviour of solutions is concerned.

1. Amendola G., Carillo S. and Manes A. 2010 Classical free energies of a heat conductor with memory and the minimum free energy for its discrete spectrum model, Boll. U. M.l., sect. B, 421 – 446 (3) n.ro 3, ISSN: 1972-6724.
2. Amendola, G., Carillo S. and Manes, A. 2012 Free energies for viscoelastic fluids. Differential problems and asymptotic behaviour. preprint;
3.  Carillo S., Valente V. and Vergara Caffarelli G. 2011 A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity, Applicable Analisys: An International Journal, 1791–1802, (90) n.ro 12, ISSN: 0003-6811, doi: 10.1080/00036811003735832
4. Carillo S., Valente V. and Vergara Caffarelli G. 2012 An existence theorem for the magneto-viscoelastic problem Discrete and Continuous Dynamical Systems Series S. , 435 – 447 (5) n.ro 3. doi:10.3934/dcdss.2012.5.435;
5. Carillo S., Valente V. and Vergara Caffarelli G. 2012 A singular viscoelastic problem: existence and uniqueness results, submitted.
   

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A phase transitions model for shape memory polymers

Elena Bonetti

Universit`a degli Studi di Pavia via Ferrata 1, 27100 -Pavia Italy

elena.bonetti@unipv.it

We introduce a new predictive model for shape memory effects occurring in polymers. In spite of the great importance such materials have in new technologies and applications, a very few mathematical literature have been developed. Shape memory polymers may be deformed at some temperature and then recover their original shape just by thermal actions. This is not so far from shape memory alloys. However, the process on the basis of this effect is fairly different (mainly it is based on some order structure of the polymers chains), so that a completely new model has been introduced. Starting from experimental data we have introduced energy and dissipation functionals to get constitutive relations fitting with experiments. hence, by a generalized version of the principle of virtual powers we have recover an evolution PDE system for which we investigate existence of solutions. This research is developed in a collaboration with the Research Center on polymers of Polimeri Europa (ENI).

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Free Energies and Phase Transitions in Materials with Hysteresis

Alessia Berti

Universit`a e-Campus 22060 -Novedrate (CO), Italy

alessia.berti@ing.unibs.it

Claudio Giorgi

Universit`a degli Studi di Brescia via Valotti, 9 25133 -Brescia Italy

claudio.giorgi@ing.unibs.it

Elena Vuk

Universit`a degli Studi di Brescia via Valotti, 9 25133 -Brescia Italy

elena.vuk@ing.unibs.it

A lot of mathematical models for magnetic hysteresis can be found in the literature, many of which are exhaustive and fit for applications (see, for instance, [1] and [2]). Unfortunately, none of them is able to describe the temperature-induced phase transition between the paramagnetic and the ferromagnetic regime. Recently, some efforts have been made in order to apply the Ginzburg-Landau theory in this direction [3]. In the talk we present a new approach to paramagnetic-ferromagnetic transition which involves a suitable order parameter related to the remnant magnetization. Starting from the skeleton curve description and exploiting the minimum (Gibbs) free energy representation, we are able to highlight the role of the Ginzburg-Landau equation when phase transitions in materials with hysteresis are involved. In particular, applications to ferrofluids and magnetic shape-memory alloys are expected.

1. Atherton D. L., Jiles D. C. 1986 Theory of ferromagnetic hysteresis. J. Magn. Magn. Mater. 61, 48–60.
2. Coleman B. D., Hodgdon M. L. 1986 A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials. Internat. J. Eng. Sci. 24, 897–919.
3. Fabrizio M., Giorgi C., Morro A. 2009 Phase transition in ferromagnetism. Internat.

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On a modified viscous Cahn-Hilliard type system

Gianni Gilardi

Universit`a degli Studi di Pavia Strada Nuova 65, 27100 -Pavia Italy

gianni.gilardi@unipv.it

A system modelling phase segregation that is similar to the viscous Cahn-Hilliard system is studied from the mathematical point of view. The talk regards a number of results in several directions (well-posedness, long-time behavior, asymptotic analysis with respect to a small parameter, optimal control) that the author recently obtained in collaboration with P. Colli, J. Sprekels e P. Podio-Guidugli.

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Modelling of active particles in a mixture of fluids

Giacomo Caviglia

Universit`a degli Studi di Genova, DIMA

via Dodecaneso 35, 16146 -Genova Italy

caviglia@dima.unige.it

Angelo Morro

Universit`a degli Studi di Genova, DIBRIS via Opera Pia 11a, 16145 -Genova Italy

angelo.morro@unige.it

The motion of a cell population within an extracellular fluid is modelled by viewing cells, fluid and chemical attractant (or chemical factor) as constituents of a mixture. The cells are regarded as active (or self-propelled) particles. Their geometric structure is represented by a vector p directed along a privileged direction of the cells. The evolution of p is affected by the stretching of the fluid and the gradient of the attractant. The pertinent equations are the continuity equations for all constituents and the equations of motion for the fluid and the attractant. Because of the active character, the velocity of the cells, relative to the fluid, is modelled by a function of p and of the gradient of the attractant. Relations with some classical mathematical models of related biological systems [1-3] are investigated.

1. Bellomo, N., De Angelis, E. and Preziosi, L. 2003 Multiscale modeling and mathematical problems related to tumor evolution and medical therapy. J. Theor. Med. 5, 111–136.

2. Hill, N. A. and Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 1–20.

3. Vald´es-Parada, F. J., Porter, M. L., Narayanaswamy, K., Ford, R. M. and Wood B.

D. 2009 Upscaling microbial chemotaxis in porous media. Adv. Water Resour. 32, 1413–1428.

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Modeling magnetostriction with phase change

Ulisse Stefanelli

IMATI – CNR via Ferrata 1, I-27100 Pavia, Italy

ulisse.stefanelli@imati.cnr.it

I shall report on some variational modeling of magnetostrictive effects in solids [1], especially in shape memory crystals. The idea is that of flagging the different phases (that is the different crystallographic variants) via the corresponding easy axes of magnetization. Different modeling regimes will be addressed and the corresponding existence, approximability, and control analysis will be outlined [2,3].

This is joint work with F. Auricchio, A. Reali, A.-L. Bessoud, and C. Zanini.

1. DeSimone, A. and James, R. D. 2002 A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50, 283–320.

2. Bessoud, A.-L. and Stefanelli, U. 2011 Magnetic Shape Memory Alloys: three-dimensional modeling and analysis. Math. Models Meth. Appl. Sci. 21, 1043–1069.

3. Stefanelli, U. 2012 Magnetic control of magnetic shape-memory crystals. Phys. B 407, 1316–1321.

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A Ginzburg-Landau model in superfluidity

Alessia Berti

Universit`a e-Campus via Isimbardi, 10 22060 -Novedrate (CO) Italy

alessia.berti@ing.unibs.it

Valeria Berti

Universit`a degli Studi di Bologna P.zza di Porta S.Donato, 5 40127 -Bologna Italy

berti@dm.unibo.it

We propose a thermodynamically consistent model to study the superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. According to the Ginzburg-Landau theory, the (second order) phase transition is described by means of an order parameter, describing the concentration of the superfluid phase and emphasizing the analogies between superfluidity and superconductivity. We decompose the velocity of the fluid as the sum of a normal and a superfluid component and we assume that the normal component is compressible and the superfluid component satisfies an evolution equation similar to the differential equation governing the motion of the superconducting electrons inside a superconductor. With these assumptions, the usual phase diagram of liquid helium is recovered and the continuity equation leads to a dependence between density and temperature in agreement with the experimental data.

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Asymptotic dynamics of nonlinear coupled suspension bridge equations

Ivana Bochicchio

Universit`a degli Studi di Salerno via Ponte Don Melillo, 84084 -Fisciano (SA) Italy

ibochicchio@unisa.it

Claudio Giorgi

Universit`a degli Studi di Brescia via D. Valotti, 9 25133 -Brescia Italy

claudio.giorgi@ing.unibs.it

Elena Vuk

Universit`a degli Studi di Brescia via D. Valotti, 9 25133 -Brescia Italy

elena.vuk@ing.unibs.it

The long-term dynamics of a doubly nonlinear abstract system is analyzed. In particular, for a special choice of the nonlinear terms, the system describes the motion of a suspension bridge where the road bed and the main cable are modeled as a nonlinear beam and a vibrating string, respectively, and their coupling is carried out by nonlinear springs. The set of stationary solutions turns out to be nonempty and bounded in the energy norm. As the external loads vanish, the null solution of the system is proved to be exponentially stable provided that the axial load does not exceed some critical value. Finally, the existence of a bounded global attractor of optimal regularity is proved in connection with quite general nonlinear terms, exploiting a particular decomposition of the associated semigroup and bootstrap arguments (see [1], [2]).

1. Conti, M. and Pata, V. 2005 Weakly dissipative semilinear equations of viscoelasticity. Commun. Pure Appl. Anal. 4, 705–720.

2. Giorgi, C., Pata, V. and Vuk, E. 2008 On the extensible viscoelastic beam. Nonlinearity 21, 713–733.

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Some qualitative results in the mixtures of thermoelastic solids

Maria Grazia Naso

Universit`a degli Studi di Brescia via Valotti, 9 25133 -Brescia Italy

naso@ing.unibs.it

In this talk we present a model governing the thermomechanical deformations for mixtures of thermoelastic solids. In particular we discuss the asymptotic behavior of the associated solutions under suitable assumptions on the constitutive constants related to the mixtures.

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Reconstruction of a convolution kernel in a parabolic problem with a memory term in the boundary conditions

Davide Guidetti

Dipartimento di Matematica, Universit`a degli Studi di Bologna Piazza di Porta S. Donato, 5 40126-Bologna Italy

davide.guidetti@unibo.it

We consider a parabolic problem of the form

Here A is a linear strongly elliptic operator of the second order in the open subset Ω, B is a linear operator of the first order, h is a convolution kernel depending only on t, f, q and u0 are known data. M is a nonlinear memory operator, that is, an operator that, at time t, depends only on the restriction of u to (0,t) × Ω. We suppose that h is unknown, together with u and want to reconstruct them. To this aim, we assume that, for every t the datum

is given. In applications, a problem of this type is a model of an automatic control problem, based on a feedback device. We may also think of h as regulator that we have at disposal, in order to obtain the prescribed Φ(u(t)).

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An irreversible phase transition problem with a nonlinear heat flux law

Giovanna Bonfanti

Universita` degli Studi di Brescia via Valotti, 9 25133 -Brescia Italy

bonfanti@ing.unibs.it

We deal with a phase transition problem characterized by an irreversible evolution and ruled by a nonlinear heat flux law. It is described by the following system

where Ω is a smooth bounded domain in R³ with boundary ∂Ω, T> 0 is a given final time, ϑ stands for the absolute temperature, ϑc for the critical transition temperature, and χ for a phase variable. Moreover ∂I[0,+∞[(·), denoting the subdifferential of the indicator function I[0,+∞[(·) (= 0 if the argument is nonnegative, = +∞ otherwise), forces ∂tχ to be nonnegative and it renders the irreversible character of the evolution. Finally, the positive function k, representing the heat conductivity of the process, satisfies suitable growth conditions and can model the case of high temperature phenomena.

The derivation of (15)-(18) comes from the theory proposed by M. Frémond [2]. The main analytical difficulty connected with this problem is due to the presence of the quadratic terms in (15) which can prevent to deduce global-in-time existence (and uniqueness) results. Also exploiting the proper features of the heat flux law (k is assumed to be positive and to behave like  ϑ^p ,p ≥ 2 for large values of ϑ), we canwork ina functional setting which is regular enough to deal with the nonlinearities of the system. We obtain a global-in-time well-posedness result [1].

1. Bonfanti, G. and Luterotti, F. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete Contin. Dyn. Syst. Ser. S, in print.

2. Frémond, M. 2002 Non-smooth Thermomechanics. Springer-Verlag, Berlin.

These results have been obtained in collaboration with: Fabio Luterotti Dipartimento di Matematica, Facolt`a di Ingegneria, Universit`a di Brescia Via Valotti 9, 25133 Brescia, Italy. luterott@ing.unibs.it

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A phase-field model for shape memory alloys at macroscopic scale: uni-axial deformation tests under different control conditions

Mirko Maraldi

DICAM, Universit`a di Bologna Viale Risorgimento, 2 -40136 Bologna

mirko.maraldi@unibo.it

Luisa Molari

DICAM, Universit`a di Bologna Viale Risorgimento, 2 -40136 Bologna

luisa.molari@unibo.it

Diego Grandi

Universit`a di Brescia Via Valotti, 9 25133 -Brescia Italy

diego.grandi@unibo.it

We propose a model for the thermomechanical behaviour of a shape memory alloy based on a phase-field approach to the underlying martensitic phase transition [1]. The model is applied at the macroscopic scale and it is formulated to capture essentially the one-dimensional mechanical behaviour; in fact, the effect of the martensitic phase transition has been included in terms of a two-variants ruled uni-axial deformation along a fixed direction. In addition to the usual momentum balance equation, a time-dependent Ginzburg-Landau equation is introduced to describe the evolution of the phase; a central constitutive relation couples the phase to the deformation. Finally, the heat equation accounts for the non isothermal effects due to the phase transformation in a thermodynamically consistent way. The model has been implemented within a finite-element framework and the predictions of the mechanical response under different control conditions are investigated in a number of numerical tests; the results obtained are analysed and compared with the experimental evidences available in literature.

1. D. Grandi, M. Maraldi, L. Molari, A macroscale phase-field model for shape memory alloys with non-isothermal effects: Influence of strain rate and environmental conditions on the mechanical response, Acta Materialia 60 (2012) 179–191.

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Diffusion and current-voltage curves in potassium channels

Daniele Andreucci, Dario Bellaveglia, Emilio N.M. Cirillo, Silvia Marconi

Sapienza Universit`a di Roma Dip. SBAI, via A.Scarpa, 16 00161 -Roma Italy

daniele.andreucci@sbai.uniroma1.it

We consider the outflux of ions through a ion channel in a cell membrane. It is well known that such channels exhibit, among other peculiar features, the capability of selecting ions of preferred chemical species, e.g., K⁺ over Na⁺ .

The existence of an external voltage difference across the cell membrane has important effects on the total current through the channel. We examine the possibility to predict the behavior of the current-voltage curves on the basis of a model similar to the one in [1].

There we introduced a random-walk model where the channel is lumped to a two state stochastic point system, and gating of the channel (i.e., switching between closed and open phases) is achieved by means of changes in affinity for the (preferred) ions, following an idea in [2]. Differently from other models we take into account also the influence of diffusion in the cell. A mild modification is needed to take into account the effect of the trans-membrane voltage difference.

We shall compare the current-voltage behavior predicted by our model with experimental results.

1. Andreucci, D., Bellaveglia, D., Cirillo, E.N.M. and Marconi, S. 2011 Monte Carlo study of gating and selection in potassium channels. Physical Review E 84, 021920.

2. VanDongen, A.M.J. 2004 K channel gating by an affinity-switching selectivity filter. PNAS 101, 3248-3252.

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Two-phase mass transfer in drug delivery systems: an application to the drug-eluting stent

Giuseppe Pontrelli, Andrea Di Mascio

IAC-CNR via dei Taurini 19, 00185 -Roma, Italy

g.pontrelli, a.dimascio@iac.cnr.it

Filippo de Monte

DIMEG, University of L’Aquila via G. Gronchi 18, 67100 -L’Aquila, Italy

filippo.demonte@univaq.it

Biodegradable polymeric coatings on cardiovascular stents are commonly used for local delivery of therapeutic drug to diseased coronary arteries after stenting procedures. The stent-based drug delivery is not completely understood and the elution mechanism may be influenced by different concurrent processes. One of them is the structure of the arterial wall that is recognized constituted by a sequence of adjacent layers. The multilayered wall accounts for a relatively detailed structure for the macromolecular transport inside the biological tissues and provides a better description when compared with a homogeneous monolayered model in diffusion processes. Another aspect is the porous coating structure where the drug, initially present in solid phase, passes into the liquid phase and, as such, is released in the tissue. We define a characteristic transfer time and we solve the transient drug dynamics problem in the adjoining wall layers faced with the coating. The endothelium, intima, internal elastic lamina and media are all treated as homogeneous porous media and the drug release from the coating through them is modelled by a set of coupled partial differential equations. The local non-equilibrium mass transport in the coating is described by adding a solid phase equation containing a solid-fluid transfer term. A spectral decomposition method for a multi-layer configuration is used. Drug concentration level and mass profiles in each layer at various times are given and discussed. As long as built on a realistic set up, the current simulation is able to estimate the local concentration, offers an easy tool for computing useful indicators such as the residence time of a drug and can be used as a guideline for designing better delivery systems.

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Nonlinear heat transport in nanosystems

Vito Antonio Cimmelli

Department of Mathematics and Computer Science, University of Basilicata, Campus Macchia Romana, 85100, Potenza, Italy

vito.cimmelli@unibas.it

David Jou

Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Catalonia, Spain

david.jou@uab.es

Antonio Sellitto

Department of Mathematics and Computer Science, University of Basilicata, Campus Macchia Romana, 85100, Potenza, Italy

ant.sellitto@gmail.com

The aim of this presentation is to analyze some nonlinear effects arising in heat transport at the nanoscale, which are especially relevant because, due to the small size of the heat conductor, small temperature differences could lead to high values of temperature gradient.

Such effects are discussed on the basis of a weakly nonlocal and nonlinear heat transport equation and a dynamical nonequilibrium temperature [1,2].

Our study will concern not only the nonlinear effects arising by the dependence of the thermal conductivity, or of the specific heat, on the temperature, but also the ones due to terms in the heat transport equation which are nonlinear in the heat flux itself.

The consequences of nonlinearity are illustrated in different situations of practical interest, involving nanowires, carbon nanotubes, and graphene sheets. The thermodynamic aspects are considered as well.

1. V. A. Cimmelli, A. Sellitto and D. Jou, 2010 Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations, Phys. Rev. B 81, 054301 (9 pages).

2. V. A. Cimmelli, A. Sellitto and D. Jou, 2010 Nonlinear evolution and stability of the heat flow in nanosystems: Beyond linear phonon hydrodynamics, Phys. Rev. B 82, 184302 (9 pages).