{"id":"https://openalex.org/W2048965806","doi":"https://doi.org/10.1137/070700206","title":"Bounds for the Effective Stress of Classical and Strain Gradient Plastic Composites","display_name":"Bounds for the Effective Stress of Classical and Strain Gradient Plastic Composites","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2048965806","doi":"https://doi.org/10.1137/070700206","mag":"2048965806"},"language":"en","primary_location":{"id":"doi:10.1137/070700206","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070700206","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046387242","display_name":"Vi\u00eat H\u00e0 H\u00f2ang","orcid":"https://orcid.org/0000-0001-9990-5106"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Viet Ha Hoang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5046387242"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.12011484,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"69","issue":"6","first_page":"1598","last_page":"1617"},"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/T11558","display_name":"Composite Material Mechanics","score":0.9941999912261963,"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":0.9850999712944031,"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/plasticity","display_name":"Plasticity","score":0.6760827898979187},{"id":"https://openalex.org/keywords/stress-space","display_name":"Stress space","score":0.6716223955154419},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5977452397346497},{"id":"https://openalex.org/keywords/deformation","display_name":"Deformation (meteorology)","score":0.5571505427360535},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.5267614126205444},{"id":"https://openalex.org/keywords/stress","display_name":"Stress (linguistics)","score":0.5259623527526855},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49250444769859314},{"id":"https://openalex.org/keywords/length-scale","display_name":"Length scale","score":0.4696919918060303},{"id":"https://openalex.org/keywords/scale","display_name":"Scale (ratio)","score":0.4465722441673279},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.44028759002685547},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.37359392642974854},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.33878475427627563},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3009682893753052},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.29546409845352173},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.2743039131164551},{"id":"https://openalex.org/keywords/composite-material","display_name":"Composite material","score":0.18223556876182556},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10583126544952393},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.10555046796798706},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.08796948194503784},{"id":"https://openalex.org/keywords/constitutive-equation","display_name":"Constitutive equation","score":0.08537057042121887}],"concepts":[{"id":"https://openalex.org/C79186407","wikidata":"https://www.wikidata.org/wiki/Q472074","display_name":"Plasticity","level":2,"score":0.6760827898979187},{"id":"https://openalex.org/C2781314585","wikidata":"https://www.wikidata.org/wiki/Q7623452","display_name":"Stress space","level":4,"score":0.6716223955154419},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5977452397346497},{"id":"https://openalex.org/C204366326","wikidata":"https://www.wikidata.org/wiki/Q3027650","display_name":"Deformation (meteorology)","level":2,"score":0.5571505427360535},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.5267614126205444},{"id":"https://openalex.org/C21036866","wikidata":"https://www.wikidata.org/wiki/Q181767","display_name":"Stress (linguistics)","level":2,"score":0.5259623527526855},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49250444769859314},{"id":"https://openalex.org/C11135259","wikidata":"https://www.wikidata.org/wiki/Q2477027","display_name":"Length scale","level":2,"score":0.4696919918060303},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.4465722441673279},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.44028759002685547},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.37359392642974854},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.33878475427627563},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3009682893753052},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.29546409845352173},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.2743039131164551},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.18223556876182556},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10583126544952393},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.10555046796798706},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.08796948194503784},{"id":"https://openalex.org/C202973686","wikidata":"https://www.wikidata.org/wiki/Q1937401","display_name":"Constitutive equation","level":3,"score":0.08537057042121887},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/070700206","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070700206","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Affordable and clean energy","score":0.6100000143051147,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1977882974","https://openalex.org/W1991152985","https://openalex.org/W1999447349","https://openalex.org/W2004066071","https://openalex.org/W2010265814","https://openalex.org/W2010672249","https://openalex.org/W2025309403","https://openalex.org/W2027417473","https://openalex.org/W2033890098","https://openalex.org/W2045978947","https://openalex.org/W2049225037","https://openalex.org/W2064743347","https://openalex.org/W2086082521","https://openalex.org/W2093861381","https://openalex.org/W2098048629","https://openalex.org/W2098345730","https://openalex.org/W2106769821","https://openalex.org/W2139661959","https://openalex.org/W3156190131"],"related_works":["https://openalex.org/W2132137594","https://openalex.org/W2069783012","https://openalex.org/W2085967955","https://openalex.org/W4256335355","https://openalex.org/W3098612666","https://openalex.org/W2076324502","https://openalex.org/W1499116563","https://openalex.org/W2138867060","https://openalex.org/W2023859141","https://openalex.org/W2589886142"],"abstract_inverted_index":{"Given":[0],"an":[1,33],"average":[2],"strain,":[3],"rigorous":[4],"bounds":[5,83,128],"are":[6],"established":[7],"for":[8,75,97,154],"the":[9,43,58,88,93,98,104,107,112,115,118,121,131,139,155],"stress":[10,82,116,122,133],"in":[11,120,129],"a":[12,15,21],"deformation":[13,25,45],"of":[14,27,47,111,114],"plastic":[16],"composite":[17],"material,":[18],"which":[19,31,84,130],"follows":[20],"power":[22],"law.":[23],"The":[24,55,136],"theory":[26,46],"strain":[28],"gradient":[29],"plasticity,":[30],"introduces":[32],"internal":[34,108],"material":[35],"length":[36,51,109],"scale,":[37],"is":[38],"used.":[39],"It":[40],"falls":[41],"into":[42],"classical":[44],"elasto-plasticity":[48],"when":[49],"this":[50],"scale":[52,110],"equals":[53],"zero.":[54],"method":[56],"employs":[57],"idea":[59],"by":[60,125,142,159],"Milton":[61],"and":[62,72,92,117,144],"Serkov":[63],"[J.":[64,146],"Mech.":[65,147],"Phys.":[66,148],"Solids,":[67,149],"48":[68],"(2000),":[69],"pp.":[70,152],"1259\u20131324]":[71],"other":[73],"techniques":[74],"bounding":[76],"effective":[77],"energy.":[78,99],"We":[79,100],"derive":[80],"two":[81,127],"closely":[85],"relate":[86],"to":[87],"Reuss":[89],"lower":[90],"bound":[91,96],"Hashin\u2013Shtrikman":[94],"upper":[95],"then":[101],"study":[102],"numerically":[103],"dependence":[105],"on":[106],"magnitude":[113],"region":[119],"space":[123],"determined":[124],"these":[126],"macro":[132],"must":[134],"lie.":[135],"results":[137],"confirm":[138],"prediction":[140],"made":[141],"Fleck":[143],"Willis":[145],"52":[150],"(2004),":[151],"1855\u20131888]":[153],"macroscopic":[156],"uniaxial":[157],"response":[158],"differentiating":[160],"their":[161],"energy":[162],"bounds.":[163]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}