{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T19:53:33Z","timestamp":1781121213712,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,8]],"date-time":"2018-03-08T00:00:00Z","timestamp":1520467200000},"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>We consider material bodies exhibiting a response function for free energy, which depends on both the strain and its gradient. Toupin\u2013Mindlin\u2019s gradient elasticity is characterized by Cauchy stress tensors, which are given by space-like Euler\u2013Lagrange derivative of the free energy with respect to the strain. The present paper aims at developing a first version of gradient elasticity of non-Toupin\u2013Mindlin\u2019s type, i.e., a theory employing Cauchy stress tensors, which are not necessarily expressed as Euler\u2013Lagrange derivatives. This is accomplished in the framework of non-conventional thermodynamics. A one-dimensional boundary value problem is solved in detail in order to illustrate the differences of the present theory with Toupin\u2013Mindlin\u2019s gradient elasticity theory.<\/jats:p>","DOI":"10.3390\/e20030179","type":"journal-article","created":{"date-parts":[[2018,3,8]],"date-time":"2018-03-08T12:07:33Z","timestamp":1520510853000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Non-Conventional Thermodynamics and Models of Gradient Elasticity"],"prefix":"10.3390","volume":"20","author":[{"given":"Hans-Dieter","family":"Alber","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Technische Universit\u00e4t Darmstadt, Schlossgartenstra\u00dfe 7, D-64289 Darmstadt, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carsten","family":"Broese","sequence":"additional","affiliation":[{"name":"Department of Continuum Mechanics, Faculty of Civil Engineering, Technische Universit\u00e4t Darmstadt, Franziska-Braun-Str. 7, D-64287 Darmstadt, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Charalampos","family":"Tsakmakis","sequence":"additional","affiliation":[{"name":"Department of Continuum Mechanics, Faculty of Civil Engineering, Technische Universit\u00e4t Darmstadt, Franziska-Braun-Str. 7, D-64287 Darmstadt, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dimitri","family":"Beskos","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, University of Patras, 26500 Patras, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/BF00253945","article-title":"Elastic materials with couple-stresses","volume":"11","author":"Toupin","year":"1962","journal-title":"Arch. 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