{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:41:17Z","timestamp":1760366477419,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,23]],"date-time":"2020-04-23T00:00:00Z","timestamp":1587600000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Non-local time evolution of material stress\/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus. The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear springpot possess an equivalent mechanical hierarchy in terms of a functionally-graded elastic column resting on viscous dashpots with power-law decay of the material properties. Some numerical applications are reported to show the capabilities of the proposed model.<\/jats:p>","DOI":"10.3390\/sym12040673","type":"journal-article","created":{"date-parts":[[2020,4,23]],"date-time":"2020-04-23T10:46:22Z","timestamp":1587638782000},"page":"673","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9091-5655","authenticated-orcid":false,"given":"Gioacchino","family":"Alotta","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria Civile, dell\u2019Energia, dell\u2019Ambiente e dei Materiali, Universit\u00e0 degli Studi \u201cMediterranea\u201d di Reggio Calabria, 89124 Reggio Calabria, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6209-1599","authenticated-orcid":false,"given":"Emanuela","family":"Bologna","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9093-9529","authenticated-orcid":false,"given":"Massimiliano","family":"Zingales","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,23]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Mainardi, F. 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