{"id":"https://openalex.org/W3020455564","doi":"https://doi.org/10.3390/sym12040673","title":"Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness","display_name":"Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness","publication_year":2020,"publication_date":"2020-04-23","ids":{"openalex":"https://openalex.org/W3020455564","doi":"https://doi.org/10.3390/sym12040673","mag":"3020455564"},"language":"en","primary_location":{"id":"doi:10.3390/sym12040673","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12040673","pdf_url":"https://www.mdpi.com/2073-8994/12/4/673/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/4/673/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000224565","display_name":"Gioacchino Alotta","orcid":"https://orcid.org/0000-0002-9091-5655"},"institutions":[{"id":"https://openalex.org/I59725666","display_name":"University of Reggio Calabria","ror":"https://ror.org/041sz8d87","country_code":"IT","type":"education","lineage":["https://openalex.org/I59725666"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Gioacchino Alotta","raw_affiliation_strings":["Dipartimento di Ingegneria Civile, dell\u2019Energia, dell\u2019Ambiente e dei Materiali, Universit\u00e0 degli Studi \u201cMediterranea\u201d di Reggio Calabria, 89124 Reggio Calabria, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria Civile, dell\u2019Energia, dell\u2019Ambiente e dei Materiali, Universit\u00e0 degli Studi \u201cMediterranea\u201d di Reggio Calabria, 89124 Reggio Calabria, Italy","institution_ids":["https://openalex.org/I59725666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032646291","display_name":"E Bologna","orcid":"https://orcid.org/0000-0002-6209-1599"},"institutions":[{"id":"https://openalex.org/I900890020","display_name":"University of Palermo","ror":"https://ror.org/044k9ta02","country_code":"IT","type":"education","lineage":["https://openalex.org/I900890020"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Emanuela Bologna","raw_affiliation_strings":["Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy","institution_ids":["https://openalex.org/I900890020"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039529942","display_name":"Massimiliano Zingales","orcid":"https://orcid.org/0000-0001-9093-9529"},"institutions":[{"id":"https://openalex.org/I900890020","display_name":"University of Palermo","ror":"https://ror.org/044k9ta02","country_code":"IT","type":"education","lineage":["https://openalex.org/I900890020"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Massimiliano Zingales","raw_affiliation_strings":["Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria, Viale delle scienze eq.8, 90128 Palermo, Italy","institution_ids":["https://openalex.org/I900890020"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5039529942"],"corresponding_institution_ids":["https://openalex.org/I900890020"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.1052,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.48939047,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"12","issue":"4","first_page":"673","last_page":"673"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/superposition-principle","display_name":"Superposition principle","score":0.664089024066925},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6244509816169739},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.4956163167953491},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.45285043120384216},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.4517136216163635},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4368334114551544},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.42206886410713196},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3611829876899719},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.34076064825057983},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.08772227168083191}],"concepts":[{"id":"https://openalex.org/C27753989","wikidata":"https://www.wikidata.org/wiki/Q284885","display_name":"Superposition principle","level":2,"score":0.664089024066925},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6244509816169739},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.4956163167953491},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.45285043120384216},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.4517136216163635},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4368334114551544},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.42206886410713196},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3611829876899719},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.34076064825057983},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.08772227168083191},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym12040673","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12040673","pdf_url":"https://www.mdpi.com/2073-8994/12/4/673/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:0b6d7639ceb54fa19a93c92b622bbbab","is_oa":true,"landing_page_url":"https://doaj.org/article/0b6d7639ceb54fa19a93c92b622bbbab","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 12, Iss 4, p 673 (2020)","raw_type":"article"},{"id":"pmh:oai:iris.unipa.it:10447/593677","is_oa":true,"landing_page_url":"https://hdl.handle.net/10447/593677","pdf_url":null,"source":{"id":"https://openalex.org/S4306401065","display_name":"Nova Science Publishers (Nova Science Publishers, Inc.)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:mdpi.com:/2073-8994/12/4/673/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym12040673","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym12040673","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12040673","pdf_url":"https://www.mdpi.com/2073-8994/12/4/673/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Life in Land","score":0.6800000071525574,"id":"https://metadata.un.org/sdg/15"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3020455564.pdf","grobid_xml":"https://content.openalex.org/works/W3020455564.grobid-xml"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W27573504","https://openalex.org/W92553431","https://openalex.org/W775420623","https://openalex.org/W1481508052","https://openalex.org/W1896434679","https://openalex.org/W1947246103","https://openalex.org/W1967212620","https://openalex.org/W1971103981","https://openalex.org/W1977688072","https://openalex.org/W1977698160","https://openalex.org/W1979016111","https://openalex.org/W1981378325","https://openalex.org/W1988988010","https://openalex.org/W1992138695","https://openalex.org/W1998956322","https://openalex.org/W1999295163","https://openalex.org/W2000009742","https://openalex.org/W2009995486","https://openalex.org/W2010310037","https://openalex.org/W2015260916","https://openalex.org/W2019630492","https://openalex.org/W2024561775","https://openalex.org/W2034559234","https://openalex.org/W2035568585","https://openalex.org/W2037306545","https://openalex.org/W2048827596","https://openalex.org/W2054444450","https://openalex.org/W2055701161","https://openalex.org/W2058077104","https://openalex.org/W2064626973","https://openalex.org/W2067078126","https://openalex.org/W2080267628","https://openalex.org/W2085679833","https://openalex.org/W2087945961","https://openalex.org/W2090593300","https://openalex.org/W2094076403","https://openalex.org/W2123753708","https://openalex.org/W2130781647","https://openalex.org/W2136912222","https://openalex.org/W2151797558","https://openalex.org/W2152152827","https://openalex.org/W2187932833","https://openalex.org/W2211280137","https://openalex.org/W2469039151","https://openalex.org/W2479963550","https://openalex.org/W2488108814","https://openalex.org/W2556784265","https://openalex.org/W2767075836","https://openalex.org/W2801123864","https://openalex.org/W2887863142","https://openalex.org/W2898911173","https://openalex.org/W2914828727","https://openalex.org/W3005063071","https://openalex.org/W3014313712","https://openalex.org/W4211052487","https://openalex.org/W4213355710","https://openalex.org/W6686907548","https://openalex.org/W6730176618","https://openalex.org/W6755545742"],"related_works":["https://openalex.org/W4308749380","https://openalex.org/W2037841693","https://openalex.org/W2372481764","https://openalex.org/W1977264433","https://openalex.org/W2746368218","https://openalex.org/W4308448008","https://openalex.org/W2347371661","https://openalex.org/W2012943989","https://openalex.org/W3005471417","https://openalex.org/W1584385814"],"abstract_inverted_index":{"Non-local":[0],"time":[1,24],"evolution":[2],"of":[3,35,42,49,61,70,87,90,112,124,137],"material":[4,11,126],"stress/strain":[5],"is":[6,30,45,99],"often":[7],"referred":[8],"to":[9,21,133],"as":[10,84],"hereditariness.":[12],"In":[13],"this":[14],"paper,":[15],"the":[16,33,43,62,68,85,88,91,102,125,135,138],"widely":[17],"used":[18],"non-linear":[19,40,59,103],"approach":[20,29],"single":[22,51],"integral":[23,52],"non-local":[25],"mechanics":[26],"named":[27],"quasi-linear":[28],"proposed":[31,139],"in":[32,47,110],"context":[34],"fractional":[36],"differential":[37],"calculus.":[38],"The":[39,73],"model":[41,74],"springpot":[44,104],"defined":[46],"terms":[48,111],"a":[50,58,76,82,113],"with":[53,57,121],"separable":[54],"kernel":[55],"endowed":[56],"transform":[60],"state":[63],"variable":[64],"that":[65,79,101],"allows":[66,80],"for":[67,81],"use":[69],"Boltzmann":[71],"superposition.":[72],"represents":[75],"self-similar":[77],"hierarchy":[78,109],"time-invariance":[83],"result":[86],"application":[89],"conservation":[92],"laws":[93],"at":[94],"any":[95],"resolution":[96],"scale.":[97],"It":[98],"shown":[100],"possess":[105],"an":[106],"equivalent":[107],"mechanical":[108],"functionally-graded":[114],"elastic":[115],"column":[116],"resting":[117],"on":[118],"viscous":[119],"dashpots":[120],"power-law":[122],"decay":[123],"properties.":[127],"Some":[128],"numerical":[129],"applications":[130],"are":[131],"reported":[132],"show":[134],"capabilities":[136],"model.":[140]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}