{"id":"https://openalex.org/W4205954652","doi":"https://doi.org/10.3390/sym14020188","title":"On Aspects of Gradient Elasticity: Green\u2019s Functions and Concentrated Forces","display_name":"On Aspects of Gradient Elasticity: Green\u2019s Functions and Concentrated Forces","publication_year":2022,"publication_date":"2022-01-19","ids":{"openalex":"https://openalex.org/W4205954652","doi":"https://doi.org/10.3390/sym14020188"},"language":"en","primary_location":{"id":"doi:10.3390/sym14020188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020188","pdf_url":"https://www.mdpi.com/2073-8994/14/2/188/pdf?version=1642557481","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/2073-8994/14/2/188/pdf?version=1642557481","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014166874","display_name":"Igor V. Andrianov","orcid":"https://orcid.org/0000-0002-2762-8565"},"institutions":[{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Igor V. Andrianov","raw_affiliation_strings":["Institute of General Mechanics, RWTH Aachen University, 52062 Aachen, Germany"],"affiliations":[{"raw_affiliation_string":"Institute of General Mechanics, RWTH Aachen University, 52062 Aachen, Germany","institution_ids":["https://openalex.org/I887968799"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026764868","display_name":"Steve Koblik","orcid":"https://orcid.org/0000-0001-9046-9766"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Steve G. Koblik","raw_affiliation_strings":["Independent Researcher, 8110 Birchfield Dr., Indianapolis, IN 46268, USA"],"affiliations":[{"raw_affiliation_string":"Independent Researcher, 8110 Birchfield Dr., Indianapolis, IN 46268, USA","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027242808","display_name":"Galina A. Starushenko","orcid":"https://orcid.org/0000-0003-4331-4723"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Galina A. Starushenko","raw_affiliation_strings":["Dnipropetrovs\u2019k Regional Institute of Public Administration, National Academy of Public Administration under the President of Ukraine, 49631 Dnipro, Ukraine","Dnipropetrovs'k Regional Institute of Public Administration, National Academy of Public Administration under the President of Ukraine, 49631 Dnipro, Ukraine"],"affiliations":[{"raw_affiliation_string":"Dnipropetrovs\u2019k Regional Institute of Public Administration, National Academy of Public Administration under the President of Ukraine, 49631 Dnipro, Ukraine","institution_ids":[]},{"raw_affiliation_string":"Dnipropetrovs'k Regional Institute of Public Administration, National Academy of Public Administration under the President of Ukraine, 49631 Dnipro, Ukraine","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5086294916","display_name":"Askat K. Kudaibergenov","orcid":"https://orcid.org/0000-0003-4773-0580"},"institutions":[{"id":"https://openalex.org/I185571130","display_name":"Al-Farabi Kazakh National University","ror":"https://ror.org/03q0vrn42","country_code":"KZ","type":"education","lineage":["https://openalex.org/I185571130"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Askat K. Kudaibergenov","raw_affiliation_strings":["Department of Mathematical and Computer Modelling, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Computer Modelling, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan","institution_ids":["https://openalex.org/I185571130"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5014166874"],"corresponding_institution_ids":["https://openalex.org/I887968799"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.075,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.28054149,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"14","issue":"2","first_page":"188","last_page":"188"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":1.0,"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":0.9991000294685364,"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"}},{"id":"https://openalex.org/T10221","display_name":"Composite Structure Analysis and Optimization","score":0.9948999881744385,"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/elasticity","display_name":"Elasticity (physics)","score":0.8276228904724121},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.7864314317703247},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6750441193580627},{"id":"https://openalex.org/keywords/dirac-delta-function","display_name":"Dirac delta function","score":0.6644816398620605},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49181678891181946},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4465482234954834},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40240657329559326},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14849629998207092}],"concepts":[{"id":"https://openalex.org/C121854251","wikidata":"https://www.wikidata.org/wiki/Q62932","display_name":"Elasticity (physics)","level":2,"score":0.8276228904724121},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.7864314317703247},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6750441193580627},{"id":"https://openalex.org/C54486999","wikidata":"https://www.wikidata.org/wiki/Q209675","display_name":"Dirac delta function","level":2,"score":0.6644816398620605},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49181678891181946},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4465482234954834},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40240657329559326},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14849629998207092},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14020188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020188","pdf_url":"https://www.mdpi.com/2073-8994/14/2/188/pdf?version=1642557481","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:cd8e6aa556224a03a202cce1cc4f5daa","is_oa":true,"landing_page_url":"https://doaj.org/article/cd8e6aa556224a03a202cce1cc4f5daa","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 2, p 188 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/2/188/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14020188","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/sym14020188","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020188","pdf_url":"https://www.mdpi.com/2073-8994/14/2/188/pdf?version=1642557481","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":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4205954652.pdf","grobid_xml":"https://content.openalex.org/works/W4205954652.grobid-xml"},"referenced_works_count":76,"referenced_works":["https://openalex.org/W655548073","https://openalex.org/W1543511296","https://openalex.org/W1603735942","https://openalex.org/W1912109891","https://openalex.org/W1965135771","https://openalex.org/W1965790903","https://openalex.org/W1988183318","https://openalex.org/W1989009426","https://openalex.org/W1989745566","https://openalex.org/W1999827612","https://openalex.org/W2006763798","https://openalex.org/W2009484082","https://openalex.org/W2011036834","https://openalex.org/W2013758901","https://openalex.org/W2026828076","https://openalex.org/W2034338538","https://openalex.org/W2042682843","https://openalex.org/W2049376337","https://openalex.org/W2049738325","https://openalex.org/W2054046240","https://openalex.org/W2056922185","https://openalex.org/W2060435198","https://openalex.org/W2061631288","https://openalex.org/W2069169719","https://openalex.org/W2072100735","https://openalex.org/W2076111901","https://openalex.org/W2078908559","https://openalex.org/W2080797205","https://openalex.org/W2082346015","https://openalex.org/W2084113519","https://openalex.org/W2092324331","https://openalex.org/W2094250534","https://openalex.org/W2106974008","https://openalex.org/W2131298010","https://openalex.org/W2148926175","https://openalex.org/W2164264278","https://openalex.org/W2166873416","https://openalex.org/W2168362335","https://openalex.org/W2258021863","https://openalex.org/W2288562070","https://openalex.org/W2318172824","https://openalex.org/W2342705689","https://openalex.org/W2467702777","https://openalex.org/W2485527793","https://openalex.org/W2495417478","https://openalex.org/W2509557402","https://openalex.org/W2551592702","https://openalex.org/W2761390097","https://openalex.org/W2790681150","https://openalex.org/W2799788277","https://openalex.org/W2885315243","https://openalex.org/W2895368257","https://openalex.org/W2912597783","https://openalex.org/W2916263707","https://openalex.org/W2917236682","https://openalex.org/W2956105178","https://openalex.org/W2962798321","https://openalex.org/W2963472892","https://openalex.org/W2990564406","https://openalex.org/W3014272419","https://openalex.org/W3030272700","https://openalex.org/W3080424106","https://openalex.org/W3120576735","https://openalex.org/W3162614091","https://openalex.org/W3163245979","https://openalex.org/W3169609532","https://openalex.org/W4230695952","https://openalex.org/W4232916866","https://openalex.org/W4233153886","https://openalex.org/W4246867916","https://openalex.org/W4252639978","https://openalex.org/W4252713891","https://openalex.org/W4396757266","https://openalex.org/W6647311167","https://openalex.org/W6665617932","https://openalex.org/W6778421945"],"related_works":["https://openalex.org/W4245687327","https://openalex.org/W2905483137","https://openalex.org/W2072289333","https://openalex.org/W2112549839","https://openalex.org/W4254637135","https://openalex.org/W4223602307","https://openalex.org/W2952168696","https://openalex.org/W581800798","https://openalex.org/W2116637251","https://openalex.org/W4205954652"],"abstract_inverted_index":{"In":[0,42],"the":[1,10,14,18,28,35,43,47,49,55,75,85,91,98,108,111,118,136,140,157],"first":[2],"part":[3,45],"of":[4,12,17,30,46,51,58,71,78,87,110,117,160],"our":[5],"review":[6],"paper,":[7,48],"we":[8],"consider":[9],"problem":[11,50,120],"approximating":[13],"Green\u2019s":[15],"function":[16,94],"Lagrange":[19],"chain":[20],"by":[21,97,156],"continuous":[22,31],"analogs.":[23],"It":[24],"is":[25,64,81,95,113,121],"shown":[26],"that":[27,106],"use":[29],"equations":[32],"based":[33],"on":[34],"two-point":[36],"Pad\u00e9":[37],"approximants":[38],"gives":[39],"good":[40],"results.":[41],"second":[44],"singularities":[52,132],"arising":[53],"in":[54],"classical":[56,137],"theory":[57,77,159],"elasticity":[59],"with":[60],"affecting":[61],"concentrated":[62,88],"loadings":[63],"considered.":[65],"To":[66],"overcome":[67],"this":[68],"problem,":[69],"instead":[70],"a":[72,149],"transition":[73],"to":[74,83,135,153],"gradient":[76,158],"elasticity,":[79],"it":[80],"proposed":[82],"change":[84],"concept":[86],"effort.":[89],"Namely,":[90],"Dirac":[92],"delta":[93],"replaced":[96],"Whittaker\u2013Shannon\u2013Kotel\u2019nikov":[99],"interpolating":[100],"function.":[101],"The":[102,126,145],"only":[103],"additional":[104],"parameter":[105,142],"characterizes":[107],"microheterogeneity":[109,141],"medium":[112],"used.":[114],"An":[115],"analog":[116],"Flamant":[119],"considered":[122],"as":[123],"an":[124],"example.":[125],"found":[127],"solution":[128],"does":[129],"not":[130],"contain":[131],"and":[133],"tends":[134],"one":[138],"when":[139],"approaches":[143],"zero.":[144],"derived":[146],"formulas":[147],"have":[148],"simpler":[150],"form":[151],"compared":[152],"those":[154],"obtained":[155],"elasticity.":[161]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}