Papers by Giuseppe Tomassetti
We derive a large-strain plate model that allows to describe transient, coupled processes involvi... more We derive a large-strain plate model that allows to describe transient, coupled processes involving elasticity and solvent migration, by performing a dimensional reduction of a three-dimensional poroe-lastic theory. We apply the model to polymer gel plates, for which a specific kinematic constraint and constitutive relations hold. Finally, we assess the accuracy of the plate model with respect to the parent three-dimensional model through two numerical benchmarks, solved by means of the finite element method. Our results show that the theory offers an efficient computational framework for the study of swelling-induced morphing of composite gel plates.
Taking the cue from experiments on actin growth on spherical beads, we formulate and solve a mode... more Taking the cue from experiments on actin growth on spherical beads, we formulate and solve a model problem describing the accretion of an incompressible elastic solid on a rigid sphere due to attachment of diffusing free particles. One of the peculiar characteristics of this problem is that accretion takes place on the interior surface that separates the body from its support rather than on its exterior surface, and hence is responsible for stress accumulation. Simultaneously, ablation takes place at the outer surface where material is removed from the body. As the body grows, mechanical effects associated with the build-up of stress and strain energy slow down accretion and promote ablation. Eventually, the system reaches a point where internal accretion is balanced by external ablation. The present study is concerned with this stationary regime called " treadmilling ". The principal ingredients of our model are: a nonstandard choice of the reference configuration, which allows us to cope with the continually evolving material structure; and a driving force and a kinetic law for accretion/ablation that involves the difference in chemical potential, strain energy and the radial stress. By combining these ingredients we arrive at an algebraic system which governs the stationary treadmilling state. We establish the conditions under which this system has a solution and we show that this solution is unique. Moreover, by an asymptotic analysis we show that for small beads the thickness of the solid is proportional to the radius of the support and is strongly affected by the stiffness of the solid, whereas for large beads the stiffness of the solid is essentially irrelevant , the thickness being proportional to a characteristic length that depends on the parameters that govern diffusion and accretion kinetics.
We analyze dissipative scale effects within a one-dimensional theory, developed in [L. Anand et a... more We analyze dissipative scale effects within a one-dimensional theory, developed in [L. Anand et al., J. Mech. Phys. Solids, 53 (2005), pp. 1789–1826], which describes plastic flow in a thin strip undergoing simple shear. We give a variational characterization of the yield (shear) stress—the threshold for the onset of plastic flow—and we use this characterization, together with results from [M. Amar et al., J. Math. Anal. Appl., 397 (2011), pp. 381–401], to obtain an explicit relation between the yield stress and the height of the strip. The relation we obtain confirms that thinner specimens are stronger, in the sense that they display higher yield stress.
Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and to... more Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and torsion. We deduce a system of equations that rule accretion and remodeling in a morphoelastic rod by combining balance laws involving non-standard forces with constitutive prescriptions filtered by a dissipation principle that takes into account both standard and non-standard working. We find that, as in the theory of three-dimensional bulk growth proposed [DiCarlo, A and Quiligotti, S. Mech Res Commun 2002; 29: 449–456], it is possible to identify a universal coupling mechanism between stress and growth, conveyed by an Eshelbian driving force.
We derive the evolution equation for a sharp concentric interface in a two-phase elastic solid of... more We derive the evolution equation for a sharp concentric interface in a two-phase elastic solid of spherical shape. The solid is immersed in a reservoir of interstitial species whose diffusion triggers phase transformation. We find that mismatch strain accelerates phase–transformation processes that initiate at the center of the specimen, and slows down those that begin at the boundary.
We consider a two-phase elastic solid subject to diffusion-induced phase transformation by inters... more We consider a two-phase elastic solid subject to diffusion-induced phase transformation by interstitial hydrogen. We derive a simple analytical model to quantify the effect of misfit strain on the kinetic of phase transformation and to calculate the amplitude of the well-know hysteresis cycle observed when a sequence of forward and reverse phase transformations takes place.
We propose a thermodynamically consistent general-purpose model describing diffusion of a solute ... more We propose a thermodynamically consistent general-purpose model describing diffusion of a solute or a fluid in a solid undergoing possible phase transformations and damage, beside possible visco-inelastic processes. Also heat genera-tion/consumption/transfer is considered. Damage is modelled as rate-independent. The applications include metal-hydrogen systems with metal/hydride phase transformation, poroelastic rocks, structural and ferro/para-magnetic phase transformation , water and heat transport in concrete, and if diffusion is neglected, plasticity with damage and viscoelasticity, etc. For the ensuing system of partial differential equations and inclusions, we prove existence of solutions by a carefully devised semi-implicit approximation scheme of the fractional-step type. Mathematics Subject Classification. 35K55 · 35Q74 · 74A15 · 74R20 · 74N10 · 74F10 · 76S99 · 80A17 · 80A20.
We consider a residually stressed plate-like body having the shape of a cylinder of cross-section... more We consider a residually stressed plate-like body having the shape of a cylinder of cross-section ω and thickness hε, subjected to a system of loads proportional to a positive multiplier λ. We look for the smallest value of the multiplier such that the plate buckles, the so-called critical multiplier. The critical multiplier is computed by minimizing a functional whose domain of definition is a collection of vector fields defined in the three-dimensional region ε = ω × (−εh/2, +εh/2). We let ε → 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with ε in a suitable manner, then the critical multiplier converges to a limit that can be computed by minimizing a functional whose domain is a collection of scalar fields defined on the two-dimensional region ω. In the special case of null residual stress, the Euler–Lagrange equations associated to this functional coincide with the classical equations governing plate buckling.
We consider an ε-parametrized collection of cylinders of cross section εω, where ω ⊂ R 2 , and of... more We consider an ε-parametrized collection of cylinders of cross section εω, where ω ⊂ R 2 , and of fixed length. By Korn's inequality, there exists a positive constant K ε such that ε |sym∇u| 2 d 3 x ≥ K ε ε |∇u| 2 d 3 x provided that u ∈ H 1 (; R 3) satisfies a condition that rules out infinitesimal rotations. We show that K ε /ε 2 converges to a strictly positive limit, and we characterize this limit in terms of certain parameters that depend on the geometry of ω and on .
A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is pr... more A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized " gas " is outlined, too.
We derive a thermodynamically consistent general continuum-mechanical model describing mutually c... more We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magneto-mechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regular-ized scheme.
We consider a cylinder Ω ε having fixed length and small cross-section εω with ω ⊂ R 2. Let 1/K ε... more We consider a cylinder Ω ε having fixed length and small cross-section εω with ω ⊂ R 2. Let 1/K ε be the Korn constant of Ω ε. We show that, as ε tends to zero, K ε /ε 2 converges to a positive constant. We provide a characterization of this constant in terms of certain parameters that depend on ω.
We study elasto-plastic torsion in a thin wire within the framework of the strain-gradient plasti... more We study elasto-plastic torsion in a thin wire within the framework of the strain-gradient plasticity theory elaborated by Gurtin and Anand in 2005. The theory in question envisages two material scales: an energetic length-scale, which takes into account the so-called " geometrically-necessary dislocations " through a dependence of the free energy on the Burgers tensor, and a dissipa-tive length-scale. For the rate-independent case with null dissipative length-scale, we construct and characterize a special class of solutions to the evolution problem. With the aid of such characterization , we estimate the dependence on the energetic scale of the ratio between the torque and the twist. Our analysis confirms that the energetic scale is responsible for size-dependent strain-hardening, with thinner wires being stronger. We also detect, and quantify in terms of the energetic length-scale, both a critical twist, after which the wire becomes fully plastified, and two boundary layers near the external boundary of the wire and near the boundary of the plastified region, respectively.
We propose a new derivation of the evolution equation of a sharp, coherent interface in a two-pha... more We propose a new derivation of the evolution equation of a sharp, coherent interface in a two-phase body having elongated shape, a body which we regard as a one-dimensional micropolar continuum. To this aim, we introduce a system of forces acting at the interface, and we apply the method of virtual powers to derive a balance law involving these forces. By exploiting the dissipation inequality, we manage to write this balance law in terms of a scalar field whose form is reminiscent of a well-known expression for the configurational stress in three dimensional micropolar continua.
We consider a initially stressed hyperelastic body in equilibrium in its undeformed configuration... more We consider a initially stressed hyperelastic body in equilibrium in its undeformed configuration under a system of dead loads. We give sufficient conditions on the stored energy which guarantee that when the loads undergo a small perturbation, the energy functional converges, after some re-scaling, to the energy functional of linear elasticity with initial stress. We also show, under stronger conditions, that quasi-minimizers of the non-linear problem converge to a minimizer of the incremental problem.
Permanent magnet arrays are often employed in a broad range of applications: actuators, sensors, ... more Permanent magnet arrays are often employed in a broad range of applications: actuators, sensors, drug targeting and delivery systems, fabrication of self-assembled particles, just to name a few. An estimate of the magnetic forces in play between arrays is required to control devices and fabrication procedures. Here, we introduce analytical expressions for calculating the attraction force between two arrays of cylindrical permanent magnets and compare the predictions with experimental data obtained from force measurements with NdFeB magnets. We show that the difference between predicted and measured force values is less than 10%.
Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities ... more Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic ¯eld, and electric ¯eld in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.
We consider a system of partial differential equations which describes anti-plane shear in the co... more We consider a system of partial differential equations which describes anti-plane shear in the context of a strain-gradient theory of plasticity proposed by Gurtin in [6]. The problem couples a fully nonlinear degenerate parabolic system and an elliptic equation. It features two types of degenera-cies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field. Furthermore, the elliptic equation depends on the divergence of such vector field – which is not controlled by twice the curl – and the boundary conditions suggested in [6] are of mixed type. To overcome the latter complications we use a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress. To handle the nonlinearities, by a suitable reformulation of the problem we transform the original system into one satisfying a monotonicity property which is more " robust " than the gradient flow structure inherited as an intrinsic feature of the mechanical model. These two insights make it possible to prove existence and uniqueness of a solution to the original system.
We propose a continuum theory describing the evolution of magnetization and temperature in a rigi... more We propose a continuum theory describing the evolution of magnetization and temperature in a rigid magnetic body. The theory is based on a microforce balance, an energy balance, and an entropy imbalance. We advance the choice of a class of constitutive equations, consistent with the entropy imbalance, that appear appropriate to describe the phase transition taking place in a ferromagnet at the Curie point. By combining these constitutive equations with the balance laws, we formulate an initial-boundary value problem for the magnetization and temperature fields, and we prove existence of weak solutions.
Phase transformation in shape-memory alloys is known to cause electric resistivity variation that... more Phase transformation in shape-memory alloys is known to cause electric resistivity variation that, under electric current, may conversely influence Joule heat production and thus eventually the martensitic transformation itself. A ther-modynamically consistent general continuum-mechanical model at large strains is presented. In special cases, a proof of the existence of a weak solution is outlined, using a semidiscretization in time. Mathematics Subject Classification (2000). 35K55 · 74A15 · 74N10 · 80A17.
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Papers by Giuseppe Tomassetti