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We obtain the minimal energy solution of the Plateau problem with elastic boundary as a variational limit of the minima of the Kirchhoff-Plateau problems with a rod boundary when the cross-section of the rod vanishes. The limit boundary is a framed curve that can sustain bending and twisting.

We present a continuum hyperelastic model which describes the mechanical response of a skeletal muscle tissue when its strength and mass are reduced by aging. Such a reduction is typical of a geriatric syndrome called sarcopenia. The passive behavior of the material is described by a hyperelastic, polyconvex, transversely isotropic strain energy function, and the activation of the muscle is modeled by the so called active strain approach. The loss of ability of activating of an elder muscle is then obtained by lowering of some percentage the active part of the stress, while the loss of mass is modeled through a multiplicative decomposition of the deformation gradient. The obtained stress-strain relations are graphically represented and discussed in order to study some of the effects of sarcopenia.

We study a simplified version of a class of constitutive relations used to describe large deformations of soft tissues, where the elastic energy density involves an exponential term. The class was originally introduced by Y.C. Fung as a model of many biological soft tissues in a series of papers during the Seventies. We prove existence and uniqueness of the equilibrium solution for a general measure-valued external load, under quite general boundary conditions, and study the validity of the associated Euler–Lagrange equation in the sense of distributions.

Lake Garda, the largest lake in Italy at the southern front of the central Alps, provides a unique opportunity to reconstruct Quaternary geological processes related to the interplay between deglaciation driven sedimentation and tectonic activity. In fact, the topographic depression presently occupied by the lake was site of a major glacial tongue during the LGM, and is located close to the epicentral areas of two of the largest magnitude historical earthquakes recorded in the Po Plain, the Verona (Jan. 13, 1117) and Brescia, (Dec. 25, 1222) seismic events, characterized by Io IX-X MCS, respectively. Despite this peculiarity, geophysical surveys of the lake are scant. We present results of a waterborne geophysical survey of Lake Garda, leading to the acquisition of a densely-spaced grid of high-resolution seismic reflection profiles which image the lake floor and uppermost sedimentary sequence. These data enabled us to compile thematic maps which provide insights on active geologic processes in different sectors of the basin. We focused our work on a prominent seismic reflector (H1), correlated over the entire lake surface, to estimate the sediment deposition rate at the scale of the last 10 ka, and to evaluate the variability of seismic facies and geological processes in different sectors of the basin. Based on available stratigraphic data, we discuss whether H1 could mark the diachronic glacial/fluvio-glacial transition after the Last Glacial Maximum, when the ice tongues south of the Alps started to melt and collapse. We observed that sediments of the southern Lake Garda are affected by incipient tectonic deformation, and two major seismoturbidite-homogenite beds might represent over 50% of the Holocene sedimentary record in the lake depocenter, making the study of the Lake Garda stratigraphy an interesting tool for assessing earthquake hazards at the southern front of the Alpine chain.

We study the equilibrium problem of a system consisting of several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.

We introduce a model of fluid which has four main features: it readily emerges by a general continuum mechanical framework; it is a generalization maintaining most of the physical features of incompressible Newtonian fluids; it can model adherence interactions with one-dimensional structures surrounded by the fluid; the associated initial boundary-value problem is well-posed on three-dimensional domains. © 2010 Springer-Verlag.

The ageing of biological tissues can be accelerated by many factors, mainly of physiological and nutritional nature. In the case of skeletal muscle tissue, one of the main consequences of ageing is a progressive loss of muscle mass and a worsening of the quality of muscle tissue, termed "sarcopenia". The correlation between the deterioration of muscle tissue and what we usually refer to as the "lifestyle", although being the subject of several studies, up to now has been considered only from a clinical and a statistical viewpoint. However, the construction of a sound mathematical model of the muscle tissue, accounting for the changes due to ageing, can provide a more refined quantitative tool. Such a tool could determine in an improved way the variations of some measurable physiological parameters, such as the mass and the electrical impedance of the tissue, caused by the variation of other controllable factors, such as diet, physical activity, pharmacological treatments, air pollution exposure. A specific mathematical model, once implemented on a computer, makes it possible to perform "virtual" experiments, facilitating the search for a suitable treatment of sarcopenia. Moreover, test situations can be studied which would not be reproducible in vivo, such as drug overdoses, extreme nutritional deficiencies, environmental overexposure to harmful substances, and so on.

We give a general formulation of the Principle of virtual powers in Continuum Mechanics from a distributional point of view, and study some of its relevant consequences in the field of balance equations. Keywords: Virtual Powers, Contact Interactions, Balance Equations Mathematics Subject Classification (2000): 74A10, 74G70, 74A30, 74A60

By means of balanced virtual powers, an axiomatic approach is developed, in the spirit of Noll, to second-gradient continua. The measure-theoretical formulation allows a considerable simplification since the existence of an edge stress density is regarded as a special case of a surface stress which is a singular measure with respect to the area. To prove our results, we introduce a particular class of subbodies, namely the sets with curvature measure.

A weak formulation of the stress boundary conditions in Continuum Mechanics is pro- posed. This condition has the form of a balance law, allows also singular measure data and is consistent with the regular case. An application to the Flamant solution in linear elasticity is shown.

Some nonlinear evolution problems arising in the theory of elastic beams with linear memory are considered. Under proper assumptions on the memory kernel the existence of uniform absorbing sets and of a global attractor is achieved.

An approach to weak balance laws in Continuum Mechanics is presented, involving densities with only divergence measure, which relies on the balance of power. An equivalence theorem between Cauchy powers and Cauchy fluxes is proved. As an application of this method, the construction of the stress tensor when the body is an orientable differential manifold is achieved under very general assumptions.

Balance laws of the type of entropy are treated in the framework of geometric measure theory, and a weak version, although conceptually simple, of the Second Law of Thermodynamics is introduced, allowing extensions to measure-valued entropy productions and to sets of finite perimeter as subbodies.

In this paper we study the long time behaviour of dynamical systems acting in topological spaces. We give suitable definitions of attractivity and prove some properties of limit sets, as well as theorems on existence of attractors in the class of regular spaces.

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory. Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In case of quasiperiodic dependence of time of the external heat supply, the above attractor is shown to have finite Hausdorff dimension.

We consider the free fall of slender rigid bodies in a viscous incompressible fluid. We show that the dimensional reduction (DR), performed by substituting the slender bodies with one-dimensional rigid objects, together with a hyperviscous regularization (HR) of the Navier--Stokes equation for the three-dimensional fluid lead to a well-posed fluid-structure interaction problem. In contrast to what can be achieved within a classical framework, the hyperviscous term permits a sound definition of the viscous force acting on the one-dimensional immersed body. Those results show that the DR/HR procedure can be effectively employed for the mathematical modeling of the free fall problem in the slender-body limit., Comment: arXiv admin note: substantial text overlap with arXiv:1305.0707

Asymptotic behavior in time for a model of nonlinear plane air waves when a viscosity term is present is investigated. The existence of a uniform bound of several norms is established, and in particular the continuity of the solution, which implies the nonexistence of shock waves.

We model the interaction of one-dimensional moving structures with a surrounding three-dimensional fluid, physically close to a Newtonian liquid. The interaction is the adherence of the fluid to the immersed structures, which drag it while moving as rigid bodies. To get solutions of the dynamical problem, we need a model of viscous fluid slightly more general than the Newtonian one, in which the Cauchy stress tensor depends upon higher-order derivatives of the velocity field. Assuming reasonable hypotheses on the motion of the one-dimensional rigid bodies, existence and uniqueness of the solution for the dynamical problem can be proved. © 2010 Elsevier Ltd. All rights reserved.

A “saddle point” (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving the Fourier transform. The “minimax” property is shown to hold as a direct consequence of thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed.

Consider the nonlinear parabolic equation in a cylinder with homogeneous Dirichlet boundary conditions on ▽ω. We show that the L 2-norm of u is equivalent to C t −1/(p−2) where C is a constant which can be related to an “eigenvalue” of the p-laplacian. This results is also generalized in various directions.

This paper contains some results on asymptotic behaviour of solutions to strongly damped abstract nonlinear wave equations. After reviewing sufficient hypotheses for existence and uniqueness, uniform time estimates are given and a global attractor for the trajectories of the associated dynamical system is constructed. Finally, applications are made to nonlinear wave equations such as Sine–Gordon equation, proving the finite dimensionality of the corresponding attractors.

1 Dipartimento di Matematica e Fisica, Universita Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy m.degiovanni@dmf.unicatt.it 2 Dipartimento di Matematica, Universita di Brescia, Via Valotti 9, I-25133 Brescia, Italy alfredo.marzocchi@ing.unibs.it 3 Dipartimento di Matematica e Fisica, Universita Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy a.musesti@dmf.unicatt.it

In this paper we show the existence, uniqueness and regularity of the solutions to the thermoelastic transmission problem. Moreover, when the solutions are symmetrical we show that the energy decays exponentially as time goes to infinity, no matter how small is the size of the thermoelastic part.

This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.

Cauchy interactions between subbodies of a continuous body are introduced in the framework of Measure Theory, extending the class of previously admissible ones. A decomposition theorem into a volume and a surface interaction is proved, as well as characterizations of the single components. Finally, an extension result and a generalized balance law are given.

Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized. The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved.