First-Order Phase Transitions and Hysteresis
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First-Order Phase Transitions and Hysteresis Alessia Berti · Claudio Giorgi · Elena Vuk
Received: 4 February 2014 / Accepted: 24 March 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A simple phase-field model for first-order phase transitions with hysteresis is proposed. It describes both temperature- and stress-induced transitions between austenitic and (oriented) martensitic regimes in a shape memory alloy (SMA). Finally, numerical simulations of local paths of the system are performed in the (ε, σ ) and (ε, θ ) planes, respectively, when either stress or temperature cyclic processes are considered and phase diffusion is neglected. Keywords Shape memory alloys · Austenite-martensite transition · Pseudo-elasticity · Hysteresis loops · Stress-rate materials · Free energy potentials · Ginzburg-Landau theory Mathematics Subject Classification 74N30 · 74C05 · 74F05 · 80A22 1 Introduction Many real-world materials exhibit hysteresis (elasto-plastic solids, shape memory alloys, ferromagnetic and ferroelectric materials, . . .) and many mathematical models for hysteresis phenomena have been proposed. Unfortunately, the most part of these models are isothermal (see, for instance, [6, 8, 15] and references therein. More recently, some efforts have been spent to apply the phase-field machinery and the Ginzburg-Landau theory to model phase transitions with hysteresis. In particular, we refer to some papers on SMAs [2, 3, 5] and on ferromagnetic materials [4, 9–11].
This work was produced under the auspices of GNFM–INDAM. A. Berti Facoltà di Ingegneria, Università e-Campus, Novedrate (CO), Italy e-mail: [email protected]
B
C. Giorgi ( ) · E. Vuk DICATAM, Università degli studi di Brescia, Brescia, Italy e-mail: [email protected] E. Vuk e-mail: [email protected]
A. Berti et al. Fig. 1 Major (gray) and minor (solid) hysteresis loops
The focus of this paper is the relation between hysteresis and phase-transition, with special emphasis on processes when the temperature field θ is varying. In particular, we discuss a simple one-dimensional model of the pseudo-elastic behavior in SMAs. It consists in a Ginzburg-Landau phase-field model describing phase transitions from austenitic (A) to oriented martensitic (M− , M+ ) phases. The martensitic transformation can be driven either by temperature or by stress and produces hysteresis loops both in the (ε, σ ) and in the (ε, θ ) planes. The pseudo-elastic regime occurs at temperatures which exceed a threshold value, say θA (see [7, 16]). In order to describe the (mechanical) pseudo-elastic behavior, a simple Duhem’s rateindependent model is considered. More precisely, we assume that the stress σ and the (logarithmic) strain ε are related by ⎧ α ⎪ ⎪ ⎪ ⎨
dσ = F (σ, ε, sgn ε˙ ) = ⎪ dε ⎪ ⎪ ⎩ 0
if (ε, σ ) ∈ Σ1 ∪ Σ2 or (ε, σ ) ∈ Ξ1 and sgn ε˙ = 1 or (ε, σ ) ∈ Ξ2 and sgn ε˙ = −1 otherwise,
(1)
where the sets Σ1 , Σ2 , Ξ1 , Ξ2 are as in Fig. 1. The states (ε, σ ) ∈ OA rely in the austenitic regime A, those belonging to CB are in the martensitic regime M+ .
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