Claudio Giorgi

On the extensible viscoelastic beam

This work is focused on the equation describing the motion of an extensible viscoelastic beam. Under suitable boundary conditions, the related dynamical system in the history space framework is shown to possess a global attractor of optimal regularity. The result is obtained by exploiting an appropriate decomposition of the solution semigroup, together with the existence of a Lyapunov functional.

Internal dissipation, relaxation property, and free energy in materials with fading memory

This paper examines some features of the standard theory of materials with fading memory and emphasizes that the commonly-accepted notion of dissipation yields unexpected consequences. First, application of the Clausius-Duhem inequality to linear viscoelasticity shows that there is a free energy functional such that the so-called internal dissipation vanishes in spite of the dissipative character of the model. Second, upon the choice of a suitable function norm, the relaxation property is proved not to hold for viscoelastic solids. Finally, the particular case is considered when the relaxation function is a superposition of exponentials. Different descriptions of state are then possible which prove to be inequivalent as far as the free energy is concerned.

STABILITY OF ABSTRACT LINEAR THERMOELASTIC SYSTEMS WITH MEMORY

An abstract linear thermoelastic system with memory is here considered. Existence, uniqueness, and continuous dependence results are given. In presence of regular and convex memory kernels, the system is shown to be exponentially stable. An application to the Kirchhoff plate equation is given.

Long-Term Damped Dynamics of the Extensible Suspension Bridge

This work is focused on the doubly nonlinear equation 𝜕 𝑡 𝑡 𝑢 + 𝜕 𝑥 𝑥 𝑥 𝑥 𝑢 + ( 𝑝 − ‖ 𝜕 𝑥 𝑢 ‖ 2 𝐿 2 ( 0 , 1 ) ) 𝜕 𝑥 𝑥 𝑢 + 𝜕 𝑡 𝑢 + 𝑘 2 𝑢 + = 𝑓 , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness 𝑘 2 . When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load 𝑝 and stiffness 𝑘 2 . For a general external source 𝑓 , we prove the existence of bounded absorbing sets. When 𝑓 is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

Abstract. In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an integro-differential equation describing the heat flow in a rigid heat conductor with memory. This model arises matching the energy balance, in presence of a nonlinear time-dependent heat source, with a linearized heat flux law of the Gurtin-Pipkin type. Existence and uniqueness of solutions for the corresponding semilinear system (subject to initial history and Dirichlet boundary conditions) is provided. Moreover, under proper assumptions on the heat flux memory kernel and the magnitude of nonlinearity, the existence of a uniform absorbing set is achieved.

Minimum principles, convexity, and thermodynamics in viscoelasticity

A linear viscoelastic solid is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is shown that such reslrictions imply the validity of a dissipativity condition, so far regarded as unrelated to the second law. Next it is remarked that the thermodynamic restrictions imply the convexity of a commonly used bilinear functional and the stationarity only if the class of displacement field is appropriate. Then it is proved that the Laplace transform of the solution to the mixed problem gives the strict minimum of a bilinear functional and vice versa. Finally, a bilinear functional with a weight function is considered and it is shown that the solution to the mixed problem gives the strict minimum and vice versa.

Mathematical Models of Reissner–Mindlin Thermoviscoelastic Plates

In this paper we investigate mathematical models describing deformations and thermal variations of a thin homogeneous thermoviscoelastic plate. A hereditary non-Fourier constitutive law for the heat flux and some heat power constitutive equation with linear memory are considered. The resulting models are derived in the framework of the well-established theory, due to Gurtin and Pipkin, and according to the standard approximation procedure for the Reissner–Mindlin plate model.

Long-term dynamics of the coupled suspension bridge system

In this paper we study the long-term dynamics of a nonlinear suspension bridge system. 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 one-sided springs. First, we scrutinize the set of stationary solutions, which turns out to be nontrivial when the axial load exceeds some critical value. Then, we prove the existence of a bounded global attractor of optimal regularity and we give its characterization in terms of the steady states of the problem.

Modeling and steady state analysis of the extensible thermoelastic beam

In this work we formulate a nonlinear mathematical model for the thermoelastic beam assuming the Fourier heat conduction law. Boundary conditions for the temperature are imposed on the ending cross sections of the beam. A careful analysis of the resulting steady states is addressed and the dependence of the Euler buckling load on the beam mean temperature, besides the applied axial load, is also discussed. Finally, under some simplifying assumptions, we deduce the model for the bending of an extensible thermoelastic beam with fixed ends. The behavior of the resulting dissipative system accounts for both the elongation of the beam and the Fourier heat conduction. The nonlinear term enters the motion equation, only, while the dissipation is entirely contributed by the heat equation, ruling the thermal evolution.

ENERGY DECAY OF ELECTROMAGNETIC SYSTEMS WITH MEMORY

We consider a linear evolution problem with memory arising in the theory of hereditary electromagnetism. Under general assumptions on the memory kernel, all single trajectories are proved to decay to zero, but the decay rate is not uniform in dependence of the initial data, so that the system is not exponentially stable. Nonetheless, if the kernel is rapidly fading and close to the Dirac mass at zero, then the solutions are close to exponentially stable trajectories.