Sezione F7 Meccanica del continuo

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Power and incremental work for a thermoelastic body: their consequences on control of a rest state. M. Padula, Dept. Mathematics, University of Ferrara, Italy Abstract Let C k be the reference configuration of a thermo-elastic body B, and let C ∗ be a rest state configuration of B. We reach three objectives: i) control in time of perturbations to C ∗ in terms of the data; ii) asymptotic decay to zero, as t → ∞, of perturbations to C ∗, in case of dissipative systems; iii) nonlinear instability in case of nodissipative mechanical systems. To not obscure the lines of reasoning, mechanical systems are considered first. The concept of power is used to construct a Lyapunov functional V correct to furnish stability in the mean when the total energy E has a minimum at C ∗, then introducing the new concept of incremental work another functional F is constructed which provides at once asymptotic stability if E has a minimum at C ∗ and instability if E has a maximum at C ∗. Paper ends with study of thermo-elastic systems, where a different energy H is introduced, it remembers the Helmohltz free energy. Stability in the mean is proven, for nondissipative systems, when the basic temperature is uniform, furthermore, asymptotic stability is proven in case the basic temperature gradient is not to large. Stability in the mean for hyperelastic, or thermo-elastic continua, is proved under the assumtpion that the basic configuration is a local minimum for a suitable energy E(S). To this end, we employ the difference between the balance equations of power, valid for the motion C t, and for the rest C ∗. In this way, the Lyapunov functional is the difference between the energy E in the states C t and C ∗, that is V = E(C t) − E(C ∗), the viscous dissipative term can be zero. Next, we analize the same continua when the viscous dissipative term is not zero. In this case, the time derivative of E(C t) − E(C ∗) is strictly negative, and a decay of the motion to the rest is expected in the general theory. Multiplying the momentum equation times the displacement vector joining x ∈ C t to x ∗ ∈ C ∗, and then integrating over C t, we obtain a fictious energy V 1 that must be added to V. The stability in the mean may be achieved for general boundary conditions: displacement and traction problem. The asymptotic result is achieved for displacement problem. We confine ourselves to displacement problem at boundary. 1
A LINEAR MAGNETIC BÉNARD PROBLEM WITH HALL AND ION-SLIP EFFECTS L. Palese University Campus, Mathematics Department, Bari, Italia. A.Georgescu University of Pitesti, Faculty of Mathematics and Computer Science, Pitesti, Romania. For normal mode perturbations, in the hypothesis that the principle of exchange of stabilities holds, the eigenvalue problems defining the neutral curves of the linear stability for some magnetic electroanisotropic Bénard problem are solved by Budiansky-DiPrima method. The unknown functions are taken as Fourier series on some total sets of separable Hilbert spaces and the expansion functions satisfied only part of the boundary conditions of the problem. This introduces some constraints to be satisfied by the Fourier coefficients. In order to keep the number of these constraints as low as possible we are lead to use total sets for the even velocity and temperature fields different from the case when velocity and temperature are odd.The splitting of the unknown functions into even and odd parts leads to two problems of the same order as the given one each of which containing even as well as odd order parts of these functions. The secular equations involve series which are truncated to one and two terms, the last situation corresponding to best results. A closed form of the neutral curve is obtained. The presence of the Hall and ion-slip currents is proved to be destabilizing. We recover the destabilizing effect of the Hall current and, in the case when the Hall effect is absent, we prove that the singularities ot the secular equation are eigenvalues of the given boundary value problem. In other words, the smallest singular value of the secular equation defines the neutral curve of the linear instability. 1
Page 1: Sezione F7 Meccanica del continuo D
Page 4 and 5: XVII Congresso U.M.I. Milano, 8-13
Page 20 and 21: MATERIAL FORCES INHOMOGENEOUS FERRO
Page 22: XVII Congresso U.M.I. Milano, 8-13

Sezione F7 Meccanica del continuo

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