{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:10:28Z","timestamp":1788801028726,"version":"build-2803163510"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A thermodynamically consistent model of shape memory alloys is developed for a body in a uniaxial setting under a tensile stress. The evolution properties are described using the temperature, the martensite fraction, and the stress as independent variables. The innovative approach is based on a general form of the Clausius\u2013Duhem inequality (really, an equality) where the entropy flux and the entropy production rate are given by constitutive functions. Thermodynamic restrictions and a suitable splitting of the entropy and deformation functions transform the Clausius\u2013Duhem inequality into an evolutionary partial differential equation. As a result, both temperature-induced and stress-induced phase transitions and their related hysteretic loops are carefully modelled by properly choosing the free energy, dynamic functions, and the entropy production rate. Furthermore, a region of equilibrium states follows from a stationary condition on the free energy. Next, a generalization is given by letting the constitutive function depend on appropriate gradients within a Lagrangian and an Eulerian formulation. Both formulations are allowed by the occurrence of the extra-entropy flux that turns out to be proportional to the pertinent rates of temperature, stress, and mass fraction.<\/jats:p>","DOI":"10.3390\/e28090998","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:06:58Z","timestamp":1788800818000},"page":"998","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Role of Entropy Flux and Entropy Production in the Modeling of Shape Memory Alloys"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2587-1668","authenticated-orcid":false,"given":"Claudio","family":"Giorgi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria Civile Ambiente Territorio Architettura e Matematica, Universit\u00e0 di Brescia, Via D. Valotti 9, 25133 Brescia, Italy"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5766-1107","authenticated-orcid":false,"given":"Angelo","family":"Morro","sequence":"additional","affiliation":[{"name":"Dipartimento di Informatica, Bioingegneria, Robotica e Ingegneria dei Sistemi, Universit\u00e0 di Genova, Via All\u2019Opera Pia 13, 16145 Genova, Italy"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Truesdell, C. 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