{"id":"https://openalex.org/W7204534965","doi":"https://doi.org/10.1016/j.rineng.2026.112684","title":"Analytical modeling of thermoelastic\u2013diffusion in a microstructured rod subjected to gradually increasing internal heating","display_name":"Analytical modeling of thermoelastic\u2013diffusion in a microstructured rod subjected to gradually increasing internal heating","publication_year":2026,"publication_date":"2026-08-28","ids":{"openalex":"https://openalex.org/W7204534965","doi":"https://doi.org/10.1016/j.rineng.2026.112684"},"language":"en","primary_location":{"id":"doi:10.1016/j.rineng.2026.112684","doi":"https://doi.org/10.1016/j.rineng.2026.112684","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rineng.2026.112684","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2590123026036996/pdf","source":{"id":"https://openalex.org/S4210226253","display_name":"Results in Engineering","issn_l":"2590-1230","issn":["2590-1230"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Engineering","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.sciencedirect.com/science/article/pii/S2590123026036996/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5114190268","display_name":"M. Adel","orcid":"https://orcid.org/0000-0001-7069-697X","observed_orcids":["https://orcid.org/0000-0001-7069-697X"]},"institutions":[{"id":"https://openalex.org/I4210151290","display_name":"Islamic University of Madinah","ror":"https://ror.org/03rcp1y74","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210151290"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"M. Adel","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-7069-697X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia","institution_ids":["https://openalex.org/I4210151290"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008773817","display_name":"M. A. El-Naggar","orcid":"https://orcid.org/0000-0002-9840-3187","observed_orcids":["https://orcid.org/0000-0002-9840-3187"]},"institutions":[{"id":"https://openalex.org/I4210165529","display_name":"University of Business and Technology","ror":"https://ror.org/05tcr1n44","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210165529"]},{"id":"https://openalex.org/I84524832","display_name":"Alexandria University","ror":"https://ror.org/00mzz1w90","country_code":"EG","type":"education","lineage":["https://openalex.org/I84524832"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Mohamed Abbas El-Naggar","raw_affiliation_strings":["Chemical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt","Department of General Subjects, University of Business and Technology, Jeddah 21361, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chemical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt","institution_ids":["https://openalex.org/I84524832"]},{"raw_affiliation_string":"Department of General Subjects, University of Business and Technology, Jeddah 21361, Saudi Arabia","institution_ids":["https://openalex.org/I4210165529"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5088104924","display_name":"Eslam S. Elidy","orcid":"https://orcid.org/0000-0003-2989-0716","observed_orcids":["https://orcid.org/0000-0003-2989-0716"]},"institutions":[{"id":"https://openalex.org/I192398990","display_name":"Zagazig University","ror":"https://ror.org/053g6we49","country_code":"EG","type":"education","lineage":["https://openalex.org/I192398990"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Eslam S. Elidy","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt"],"raw_orcid":"https://orcid.org/0000-0003-2989-0716","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt","institution_ids":["https://openalex.org/I192398990"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091376113","display_name":"Kh. Lotfy","orcid":"https://orcid.org/0000-0001-9383-1361","observed_orcids":["https://orcid.org/0000-0001-9383-1361"]},"institutions":[{"id":"https://openalex.org/I192398990","display_name":"Zagazig University","ror":"https://ror.org/053g6we49","country_code":"EG","type":"education","lineage":["https://openalex.org/I192398990"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"Khaled Lotfy","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt"],"raw_orcid":"https://orcid.org/0000-0001-9383-1361","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt","institution_ids":["https://openalex.org/I192398990"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079073624","display_name":"Praveen Ailawalia","orcid":"https://orcid.org/0000-0003-4381-6299","observed_orcids":["https://orcid.org/0000-0003-4381-6299"]},"institutions":[{"id":"https://openalex.org/I101407740","display_name":"Chandigarh University","ror":"https://ror.org/05t4pvx35","country_code":"IN","type":"education","lineage":["https://openalex.org/I101407740"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Praveen Ailawalia","raw_affiliation_strings":["Department of Mathematics, University Institute of Sciences, Chandigarh University, Gharuan, Mohali, 140413, Punjab, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University Institute of Sciences, Chandigarh University, Gharuan, Mohali, 140413, Punjab, India","institution_ids":["https://openalex.org/I101407740"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5091376113"],"corresponding_institution_ids":["https://openalex.org/I192398990"],"apc_list":{"value":1940,"currency":"USD","value_usd":1940},"apc_paid":{"value":1940,"currency":"USD","value_usd":1940},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"32","issue":null,"first_page":"112684","last_page":"112684"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.9994999766349792,"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/T11646","display_name":"Nonlocal and gradient elasticity in micro/nano structures","score":0.00019999999494757503,"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":0.0,"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/thermoelastic-diffusion","display_name":"thermoelastic diffusion","score":0.9300000071525574},{"id":"https://openalex.org/keywords/microconcentrations","display_name":"microconcentrations","score":0.8399999737739563},{"id":"https://openalex.org/keywords/internal-heat-generation","display_name":"internal heat generation","score":0.7250000238418579},{"id":"https://openalex.org/keywords/thermal-diffusion","display_name":"thermal diffusion","score":0.8090000152587891},{"id":"https://openalex.org/keywords/mass-transfer","display_name":"mass transfer","score":0.6159999966621399},{"id":"https://openalex.org/keywords/stress-distribution","display_name":"stress distribution","score":0.38999998569488525},{"id":"https://openalex.org/keywords/temperature-and-concentration-profiles","display_name":"temperature and concentration profiles","score":0.5109999775886536},{"id":"https://openalex.org/keywords/inverse-laplace-transform","display_name":"inverse Laplace transform","score":0.6000000238418579},{"id":"https://openalex.org/keywords/finite-rod","display_name":"finite rod","score":0.3089999854564667}],"concepts":[{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.6093000173568726},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.42500001192092896},{"id":"https://openalex.org/C204366326","wikidata":"https://www.wikidata.org/wiki/Q3027650","display_name":"Deformation (meteorology)","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.29429998993873596},{"id":"https://openalex.org/C19474535","wikidata":"https://www.wikidata.org/wiki/Q131172","display_name":"Welding","level":2,"score":0.2847000062465668},{"id":"https://openalex.org/C107551265","wikidata":"https://www.wikidata.org/wiki/Q1458245","display_name":"Displacement (psychology)","level":2,"score":0.2637999951839447},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.26019999384880066},{"id":"https://openalex.org/C50517652","wikidata":"https://www.wikidata.org/wiki/Q179635","display_name":"Heat transfer","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C180016635","wikidata":"https://www.wikidata.org/wiki/Q2712821","display_name":"Compression (physics)","level":2,"score":0.2578999996185303},{"id":"https://openalex.org/C162184086","wikidata":"https://www.wikidata.org/wiki/Q505544","display_name":"Rod","level":3,"score":0.2517000138759613}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/j.rineng.2026.112684","doi":"https://doi.org/10.1016/j.rineng.2026.112684","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rineng.2026.112684","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2590123026036996/pdf","source":{"id":"https://openalex.org/S4210226253","display_name":"Results in Engineering","issn_l":"2590-1230","issn":["2590-1230"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Engineering","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1016/j.rineng.2026.112684","doi":"https://doi.org/10.1016/j.rineng.2026.112684","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rineng.2026.112684","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2590123026036996/pdf","source":{"id":"https://openalex.org/S4210226253","display_name":"Results in Engineering","issn_l":"2590-1230","issn":["2590-1230"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Engineering","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320331150","display_name":"Islamic University of Madinah","ror":"https://ror.org/03rcp1y74"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204534965.pdf","grobid_xml":"https://content.openalex.org/works/W7204534965.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W1973038067","https://openalex.org/W1978796508","https://openalex.org/W1979042256","https://openalex.org/W1982499709","https://openalex.org/W1987145007","https://openalex.org/W2002001991","https://openalex.org/W2005831147","https://openalex.org/W2008118527","https://openalex.org/W2032023503","https://openalex.org/W2034402995","https://openalex.org/W2038000836","https://openalex.org/W2038940027","https://openalex.org/W2050110207","https://openalex.org/W2070972422","https://openalex.org/W2073161947","https://openalex.org/W2077234159","https://openalex.org/W2104344979","https://openalex.org/W2121105689","https://openalex.org/W2129700287","https://openalex.org/W2350746007","https://openalex.org/W2401188399","https://openalex.org/W2549047809","https://openalex.org/W2624769052","https://openalex.org/W2898386350","https://openalex.org/W3033257955","https://openalex.org/W3151540233","https://openalex.org/W3155151422","https://openalex.org/W3183334332","https://openalex.org/W3197019436","https://openalex.org/W3198621701","https://openalex.org/W4213147903","https://openalex.org/W4230822328","https://openalex.org/W4381890150","https://openalex.org/W4392502012","https://openalex.org/W4400700869","https://openalex.org/W4404498123","https://openalex.org/W4405357514","https://openalex.org/W4411540420","https://openalex.org/W4413738912","https://openalex.org/W4414058755","https://openalex.org/W4414695981","https://openalex.org/W4415389342","https://openalex.org/W4415602766","https://openalex.org/W4416021852","https://openalex.org/W7154950290"],"related_works":["https://openalex.org/W2918477135","https://openalex.org/W4253064868","https://openalex.org/W4206198225","https://openalex.org/W7201970185","https://openalex.org/W2089382104","https://openalex.org/W3200449893","https://openalex.org/W4230070482","https://openalex.org/W2968378470","https://openalex.org/W2464274450","https://openalex.org/W151439462"],"abstract_inverted_index":{"This":[0],"study":[1],"investigates":[2],"the":[3,60,75,142,146,168],"transient":[4],"interaction":[5,28],"of":[6,145,164],"heat,":[7],"deformation,":[8],"and":[9,40,46,56,74,125,157,178,185],"mass":[10,97,199],"transport":[11,42,98,200],"in":[12,35,86,99,201],"a":[13,88,100],"homogeneous":[14],"isotropic":[15],"rod":[16],"with":[17,94],"microconcentrations":[18],"subjected":[19],"to":[20,69,109,117,123,130,139,154,182],"gradually":[21],"increasing":[22,90],"internal":[23],"heat":[24,92],"generation.":[25],"Understanding":[26],"this":[27],"is":[29,54],"important":[30],"for":[31,196],"thermally":[32],"active":[33],"components":[34],"which":[36],"temperature-driven":[37],"species":[38],"migration":[39],"microscale":[41],"can":[43],"alter":[44],"stress":[45],"structural":[47],"reliability.":[48],"A":[49],"one-dimensional":[50],"generalized":[51],"thermoelastic-diffusion":[52],"model":[53],"nondimensionalized":[55],"solved":[57],"analytically":[58],"using":[59],"Laplace":[61],"transform.":[62],"The":[63,83],"coupled":[64],"transformed":[65],"equations":[66],"are":[67,78,194],"reduced":[68],"an":[70],"eighth-order":[71],"differential":[72],"equation,":[73],"time-domain":[76],"fields":[77],"recovered":[79],"through":[80],"numerical":[81],"inversion.":[82],"novelty":[84],"lies":[85],"combining":[87],"progressively":[89],"volumetric":[91],"source":[93],"microconcentration-dependent":[95],"higher-order":[96,198],"finite":[101],"rod.":[102],"As":[103],"dimensionless":[104],"time":[105],"increases":[106,135],"from":[107,114,121,128,136,151],"0.1":[108],"5.0,":[110],"peak":[111,119,126,144],"displacement":[112],"rises":[113],"approximately":[115,152],"0.21":[116],"1.14,":[118],"temperature":[120],"0.16":[122],"0.87,":[124],"concentration":[127,158],"0.18":[129],"1.02.":[131],"Maximum":[132],"microconcentration":[133,169,192],"simultaneously":[134],"about":[137],"0.015":[138],"0.078,":[140],"while":[141],"positive":[143],"mass-diffusion":[147],"flux":[148],"moment":[149],"grows":[150],"0.22":[153],"1.22.":[155],"Temperature":[156],"exhibit":[159],"related":[160],"interior":[161],"profiles":[162],"because":[163],"thermodiffusion":[165],"coupling,":[166],"whereas":[167],"variables":[170],"show":[171],"distinct":[172],"spatial":[173],"behavior.":[174],"Stress":[175],"develops":[176],"tensile":[177],"compressive":[179],"regions":[180],"owing":[181],"constrained":[183],"thermal":[184],"diffusion":[186],"expansion.":[187],"These":[188],"results":[189],"demonstrate":[190],"that":[191],"effects":[193],"essential":[195],"predicting":[197],"internally":[202],"heated":[203],"rods.":[204]},"counts_by_year":[],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2026-08-29T00:00:00"}