Free energies and pseudo-elastic transitions for shape memory alloys
Alessia Berti Claudio Giorgi Elena Vuk
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 293-316 doi: 10.3934/dcdss.2013.6.293
A one-dimensional model for a shape memory alloy is proposed. It provides a simplified description of the pseudo-elastic regime, where stress-induced transitions from austenitic to oriented martensitic phases occurs. The stress-strain evolution is ruled by a bilinear rate-independent o.d.e. which also accounts for the fine structure of minor hysteresis loops and applies to the case of single crystals only. The temperature enters the model as a parameter through the yield limit $y$.Above the critical temperature $\theta_A^*$, the austenite-martensite phase transformations are described by a Ginzburg-Landau theory involving an order parameter $φ$, which is related to the anelastic deformation. As usual, the basic ingredient is the Gibbs free energy, $\zeta$, which is a function of the order parameter, the stress and the temperature. Unlike other approaches, the expression of this thermodynamic potential is derived rather then assumed, here. The explicit expressions of the minimum and maximum free energies are obtained by exploiting the Clausius-Duhem inequality, which ensures the compatibility with thermodynamics, and the complete controllability of the system. This allows us to highlight the role of the Ginzburg-Landau equation when phase transitions in materials with hysteresis are involved.
keywords: hysteresis loops Shape memory alloys stress-rate materials austenite-martensite transition Ginzburg-Landau theory. pseudo-elasticity free energy potentials
On the viscoelastic coupled suspension bridge
Ivana Bochicchio Claudio Giorgi Elena Vuk
Evolution Equations & Control Theory 2014, 3(3): 373-397 doi: 10.3934/eect.2014.3.373
In this paper we discuss the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. The main cable is modeled by a viscoelastic string and is connected to the road-bed by a distributed system of one-sided elastic springs. A constant axial force $p$ is applied at one end of the deck, and time-independent vertical loads are allowed to act both on the road-bed and on the suspension cable. For this general model we obtain original results, including the existence of a regular global attractor for all $p\in\mathbb{R}$.
keywords: viscoelastic beam Suspension bridge system viscoelastic string nonlinear oscillations global attractor.

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