Response functions in linear viscoelastic constitutive equations and related fractional operators
This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing t...
Saved in:
| Published in | Mathematical modelling of natural phenomena Vol. 14; no. 3; p. 305 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Les Ulis
EDP Sciences
01.01.2019
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators. |
|---|---|
| AbstractList | This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators. |
| Author | Hristov, Jordan |
| Author_xml | – sequence: 1 givenname: Jordan surname: Hristov fullname: Hristov, Jordan email: jordan.hristov@mail.bg organization: Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 8 Kl. Ohridsky Blvd., 1756 Sofia, Bulgaria |
| BookMark | eNp9kE9r3DAQxUVJoZt0b_0AglzjZiTZkvYYlmzSNrRQWnoUY1kCbby2V5KX5NvH-4ccAs1cZuD93szwzslZ13eOkC8MvjKo2PVm0w3XHJgGqT6QGVMSCsmAnZEZLJQoKlHqT2Se0hqmEqwUADOCv10a-i456sfO5jCNNHS0DZ3DSHch2d61mHKw1E5aDnnMYeeo2454pLFraJyY7BrqIx52YEv7wUXMfUyfyUePbXLzU78gf1e3f5b3xcOvu2_Lm4fCCr7IBaq6snUltQcU3C_A6lrWTVM32jNfee81VugYlFrXCwtlzV0twSNvGoFSigtyedw7xH47upTNuh_j9EoynGvBNQNQ71NSlcArtaeujpSNfUrReTPEsMH4bBiYfdpmn7Y5pT3h_A1uQz7EkyOG9n-m4mgKKbun1wMYH82kqspo-Gfu1I9V-fP7ytyLFyVYlu0 |
| CitedBy_id | crossref_primary_10_1007_s10973_020_09835_0 crossref_primary_10_1016_j_cam_2022_114840 crossref_primary_10_1016_j_chaos_2021_111078 crossref_primary_10_1177_14644207211062497 crossref_primary_10_3390_fractalfract3020014 crossref_primary_10_1080_09720502_2020_1800225 crossref_primary_10_1155_2021_5572823 crossref_primary_10_1186_s13662_020_02671_4 crossref_primary_10_3390_fractalfract3020016 crossref_primary_10_3390_fractalfract6080434 crossref_primary_10_1007_s40819_023_01564_w crossref_primary_10_1016_j_polymer_2023_126565 crossref_primary_10_1016_j_joes_2020_10_004 crossref_primary_10_29105_ingenierias23_88_3 crossref_primary_10_1016_j_matlet_2021_130238 crossref_primary_10_3390_fractalfract3010012 crossref_primary_10_1155_2021_8817794 crossref_primary_10_1016_j_cnsns_2020_105490 crossref_primary_10_1016_j_ijheatmasstransfer_2019_04_094 crossref_primary_10_3390_math12020357 crossref_primary_10_1051_mmnp_2021030 crossref_primary_10_3390_sym11070879 crossref_primary_10_1002_mma_5949 crossref_primary_10_3389_fphy_2019_00189 crossref_primary_10_3390_app13053065 crossref_primary_10_1155_2023_6401067 crossref_primary_10_3390_fractalfract3020027 crossref_primary_10_1007_s11253_022_02000_w crossref_primary_10_1080_25765299_2024_2423467 crossref_primary_10_3389_fphy_2018_00135 crossref_primary_10_1016_j_cnsns_2020_105186 crossref_primary_10_2298_TSCI221219223H crossref_primary_10_1186_s13662_019_2082_8 crossref_primary_10_3390_axioms13020099 crossref_primary_10_2478_jamsi_2021_0003 crossref_primary_10_3390_sym12061034 crossref_primary_10_1002_mma_6343 crossref_primary_10_3390_math7090766 crossref_primary_10_1515_geo_2022_0446 crossref_primary_10_1155_2020_3842946 crossref_primary_10_1140_epjp_i2019_12697_7 crossref_primary_10_1002_num_22476 crossref_primary_10_37863_umzh_v73i9_805 crossref_primary_10_1016_j_chaos_2019_109454 crossref_primary_10_1111_jfpe_14060 crossref_primary_10_1142_S0217984924500866 crossref_primary_10_1016_j_chaos_2019_109530 crossref_primary_10_1155_2020_1727358 |
| Cites_doi | 10.1016/S0020-7683(96)00178-3 10.1016/j.mechmat.2008.11.008 10.1186/s13662-018-1680-1 10.1016/j.matdes.2015.03.063 10.1016/0021-9290(93)90088-V 10.1016/0377-0257(93)85052-C 10.1016/0022-5096(95)00004-3 10.18576/pfda/030402 10.1016/j.tecto.2017.10.021 10.1016/j.biomaterials.2005.10.034 10.1016/j.mechmat.2017.11.003 10.1142/p614 10.3390/math6020015 10.1002/pen.760320206 10.1122/1.4892114 10.1016/j.mechmat.2010.07.007 10.1007/BF01993026 10.1016/j.cpc.2018.04.015 10.1002/(SICI)1097-0207(19961130)39:22<3903::AID-NME34>3.0.CO;2-C 10.1016/j.ijsolstr.2015.04.018 10.1016/S0093-6413(01)00185-9 10.1007/978-3-642-88210-4 10.1007/BF00411984 10.1016/j.amc.2015.10.021 10.1103/RevModPhys.61.41 10.1007/s11043-011-9162-9 10.1016/j.mechmat.2018.07.012 10.3390/e17064439 10.1016/j.triboint.2016.06.003 10.1007/978-3-642-73602-5 10.1088/0266-5611/8/6/005 10.9734/ajr2p/2018/v1i324617 10.1007/s11012-003-6437-5 10.1016/S0020-7683(98)00194-2 10.1016/j.jbiomech.2015.02.001 10.1016/j.ijmecsci.2017.06.032 10.1016/j.ijrmms.2010.11.004 10.1090/qam/295683 10.1016/j.physa.2017.10.002 10.1016/j.jbiomech.2011.12.023 10.1088/0305-4470/28/23/012 10.1088/0953-8984/21/34/345802 10.1016/j.jmbbm.2015.03.011 10.1016/j.cnsns.2018.02.019 10.3389/fphy.2018.00135 10.1002/(SICI)1099-0488(199706)35:8<1225::AID-POLB8>3.0.CO;2-S 10.1007/s11071-018-4289-8 10.1016/j.mechmat.2016.04.002 10.1016/j.taml.2015.11.002 10.1016/j.actbio.2011.07.035 10.1016/j.ultras.2015.09.003 10.1016/j.foodhyd.2016.05.020 10.1016/j.cnsns.2014.03.014 10.1111/j.1365-246X.1967.tb02303.x 10.1016/j.dental.2007.08.008 10.1007/BF00281562 10.1016/j.cnsns.2017.12.001 10.1016/0045-7949(90)90096-K 10.1115/1.2812256 10.1122/1.549887 10.1007/BF00366673 10.1007/s11012-016-0456-5 10.3389/fphy.2017.00052 10.1007/BF00366960 10.1061/(ASCE)0733-9399(1999)125:8(914) 10.1016/S0016-0032(21)90171-6 10.1016/j.physa.2016.10.066 10.1007/s00397-005-0443-6 10.1016/j.jfoodeng.2014.05.019 10.1007/s00033-017-0808-6 10.1016/0016-0032(46)90636-9 10.1016/0021-9290(94)00130-V 10.1007/s11043-005-3442-1 10.18576/pfda/020101 10.1007/s11012-003-9400-6 10.1007/s11012-016-0376-4 10.1016/S0041-624X(00)00024-X 10.1371/journal.pone.0143090 10.1002/app.31093 10.1016/j.jpowsour.2010.08.075 10.1007/BF02124750 10.1016/j.amc.2017.08.048 10.1016/j.tsf.2016.03.064 10.5772/64251 10.1016/j.tws.2017.06.030 10.1061/(ASCE)ST.1943-541X.0000822 10.1016/S0895-7177(01)00137-6 10.1007/978-1-4615-9970-8 10.1016/0013-7944(93)90041-P 10.1016/j.cnsns.2009.05.004 10.1016/j.jmbbm.2014.05.031 10.1137/1.9781611971484 10.1016/j.eswa.2011.12.033 10.3390/fractalfract2030020 10.1016/j.chaos.2018.07.033 10.1016/j.cjph.2018.02.007 10.1016/0032-3861(95)90680-Z 10.1016/j.actamat.2008.07.030 10.2298/TSCI160229115H 10.1007/BF01515713 10.1007/BF01976435 10.1016/j.polymer.2013.08.062 10.1111/j.1475-1305.2008.00502.x 10.1016/j.cnsns.2017.08.002 10.1002/andp.19314030608 10.1177/058310249903100201 10.1016/S0030-4018(96)00384-7 10.1016/j.oceaneng.2015.06.023 10.1142/2905 10.1016/j.jmbbm.2015.10.008 10.1016/j.matpr.2016.03.037 10.2298/TSCI160111018A 10.1007/978-0-387-73861-1 10.1007/s11043-007-9035-4 10.1016/j.physa.2018.04.047 10.1016/j.conbuildmat.2015.09.016 10.1016/j.cnsns.2017.12.003 10.1002/cta.2564 10.1103/RevModPhys.33.239 10.1016/j.compstruc.2007.01.024 10.1016/0377-0257(92)80043-W 10.1016/j.camwa.2016.05.002 10.3233/BIR-1997-34603 10.1016/j.chaos.2018.07.022 10.1063/1.1700076 10.1016/0022-247X(78)90234-2 10.1016/j.engstruct.2016.04.024 10.1063/1.470346 |
| ContentType | Journal Article |
| Copyright | 2019. Notwithstanding the ProQuest Terms and conditions, you may use this content in accordance with the associated terms available at https://www.mmnp-journal.org/articles/mmnp/abs/2019/03/mmnp180121/mmnp180121.html . |
| Copyright_xml | – notice: 2019. Notwithstanding the ProQuest Terms and conditions, you may use this content in accordance with the associated terms available at https://www.mmnp-journal.org/articles/mmnp/abs/2019/03/mmnp180121/mmnp180121.html . |
| DBID | BSCLL AAYXX CITATION 7QF 7QO 7QQ 7SC 7SE 7SP 7SR 7TA 7TB 7TK 7U5 8BQ 8FD F28 FR3 H8D H8G JG9 JQ2 KR7 L7M L~C L~D P64 |
| DOI | 10.1051/mmnp/2018067 |
| DatabaseName | Istex CrossRef Aluminium Industry Abstracts Biotechnology Research Abstracts Ceramic Abstracts Computer and Information Systems Abstracts Corrosion Abstracts Electronics & Communications Abstracts Engineered Materials Abstracts Materials Business File Mechanical & Transportation Engineering Abstracts Neurosciences Abstracts Solid State and Superconductivity Abstracts METADEX Technology Research Database ANTE: Abstracts in New Technology & Engineering Engineering Research Database Aerospace Database Copper Technical Reference Library Materials Research Database ProQuest Computer Science Collection Civil Engineering Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Biotechnology and BioEngineering Abstracts |
| DatabaseTitle | CrossRef Materials Research Database Civil Engineering Abstracts Aluminium Industry Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Mechanical & Transportation Engineering Abstracts Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Ceramic Abstracts Neurosciences Abstracts Materials Business File METADEX Biotechnology and BioEngineering Abstracts Computer and Information Systems Abstracts Professional Aerospace Database Copper Technical Reference Library Engineered Materials Abstracts Biotechnology Research Abstracts Solid State and Superconductivity Abstracts Engineering Research Database Corrosion Abstracts Advanced Technologies Database with Aerospace ANTE: Abstracts in New Technology & Engineering |
| DatabaseTitleList | Materials Research Database Materials Research Database CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Sciences (General) |
| EISSN | 1760-6101 |
| ExternalDocumentID | 10_1051_mmnp_2018067 ark_67375_80W_G7KF4NJF_H |
| GroupedDBID | -E. .FH 0E1 169 3V. 4.4 5GY 5VS 7~V 8FE 8FG 8FH AADXX AAFWJ AAOTM ABDBF ABJCF ABJNI ABKKG ABUBZ ABZDU ACACO ACGFS ACIMK ACIWK ACPRK ACQPF ADBBV AEMTW AENEX AFAYI AFHSK AFKRA AFRAH AFUTZ AJPFC ALMA_UNASSIGNED_HOLDINGS ARABE ARAPS AZPVJ BENPR BHPHI BPHCQ BSCLL C0O CS3 DC4 EBS EJD ESX F5P FRP GFF GI~ HCIFZ HG- HST HZ~ I-F I.6 IL9 I~P J36 J38 J9A K6V K7- L6V LK8 LO0 M-V M0N M7P M7S O9- OK1 P2P P62 PQQKQ PROAC RCA RED RR0 S6- TUS WQ3 WXU WXY ~8M AAOGA AAYXX ABGDZ ABNSH ACRPL ACUHS ADNMO AGQPQ AMVHM CITATION 7QF 7QO 7QQ 7SC 7SE 7SP 7SR 7TA 7TB 7TK 7U5 8BQ 8FD F28 FR3 H8D H8G JG9 JQ2 KR7 L7M L~C L~D P64 |
| ID | FETCH-LOGICAL-c329t-a7b5cb568f0a32f90c8b6bddbd8f1f5fff8a5ae10488b9c04b2eb60fa2dd3a663 |
| ISSN | 0973-5348 |
| IngestDate | Mon Jun 30 10:18:56 EDT 2025 Mon Jun 30 10:10:41 EDT 2025 Tue Jul 01 01:22:53 EDT 2025 Thu Apr 24 23:05:15 EDT 2025 Wed Oct 30 09:24:39 EDT 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Language | English |
| License | https://www.edpsciences.org/en/authors/copyright-and-licensing |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c329t-a7b5cb568f0a32f90c8b6bddbd8f1f5fff8a5ae10488b9c04b2eb60fa2dd3a663 |
| Notes | href:https://www.mmnp-journal.org/articles/mmnp/abs/2019/03/mmnp180121/mmnp180121.html ark:/67375/80W-G7KF4NJF-H publisher-ID:mmnp180121 istex:4A53D21D7E070F2E07BC98191F2DE70B5B28B02C ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 2267402577 |
| PQPubID | 626347 |
| ParticipantIDs | proquest_journals_2283281007 proquest_journals_2267402577 crossref_primary_10_1051_mmnp_2018067 crossref_citationtrail_10_1051_mmnp_2018067 istex_primary_ark_67375_80W_G7KF4NJF_H |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2019-01-01 |
| PublicationDateYYYYMMDD | 2019-01-01 |
| PublicationDate_xml | – month: 01 year: 2019 text: 2019-01-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationPlace | Les Ulis |
| PublicationPlace_xml | – name: Les Ulis |
| PublicationTitle | Mathematical modelling of natural phenomena |
| PublicationYear | 2019 |
| Publisher | EDP Sciences |
| Publisher_xml | – name: EDP Sciences |
| References | Schneider (R129) 2016; 122 Boltzmann (R23) 1874; 70 de Prony (R120) 1975; 1 Guisti (R64) 2018; 56 O’Down (R112) 1995; 43 Enderlein (R50) 1997; 134 Hristov (R72) 2017; 21 Holzapfel (R70) 1996; 39 Caputo (R26) 1967; 13 Vandenberghe (R149) 2014; 142 Berry (R19) 1997; 36 Phillips (R116) 2006; 27 Zhou (R155) 2016; 48 Atangana (R12) 2018; 114 R22 R24 R29 Shou (R134) 2018; 117 Sliker (R137) 2016; 2012 R1 Czyz (R44) 1990; 37 Ciambella (R34) 2014; 37 Andrews (R5) 2015; 48 Ravikumar (R122) 2015; 47 Lin (R92) 2001; 28 Huang (R76) 2016; 618 Wintner (R150) 1959; IX Ikeno (R77) 2018; 230 R31 Adolfsson (R2) 2005; 9 Caputo (R28) 2016; 2 Prieto-Munoz (R119) 2013; 140 Ortigueira (R113) 2018; 59 Valerio (R148) 2014; 19 Hanyga (R68) 2005; 44 Singh (R135) 2018; 316 Jaloha (R79) 2015; 67–68 R102 Schiessel (R128) 1995; 28 DeHoff (R46) 2008; 24 Babaei (R14) 2016; 55 R41 Scott-Blair (R130) 1951; A2 Ciambella (R33) 2010; 42 Coleman (R35) 1961; 33 Colombaro (R38) 2017; 68 Londono (R95) 2016; 98 Long (R96) 2018; 127 R48 Glockle (R60) 1994; 33 Kneser (R86) 1931; 6 Mauro (R101) 2018; 506 Johnson (R80) 1999; 31 Baumgaertel (R17) 1992; 44 Bagley (R15) 1986; 30 Cui (R43) 2011; 196 Guisti (R63) 2018; 93 Trzmiel (R146) 2009; 21 Caputo (R27) 2015; 1 Choi (R32) 2008; 56 R56 Renardy (R123) 1982; 21 R58 Ahmed (R3) 2018; 56 Bhattarcharjee (R20) 2012; 16 Enelund (R51) 1999; 36 Friedrich (R57) 1993; 46 Knaus (R85) 2007; 11 Mitra (R106) 2012; 39 Storm (R139) 1951; 22 R127 Drozdov (R49) 1997; 34 Tayeb (R144) 2017; 130 R124 Bilston (R21) 1997; 34 Atangana (R8) 2016; 20 Colombaro (R39) 2018; 3 R69 Fernandez (R55) 2011; 47 Lai (R89) 1995; 36 Gurtin (R65) 1968; 28 Hansen (R66) 1992; 8 Zanzotto (R153) 1997; 35 Hristov (R75) 2018; 6 R118 Tarasov (R142) 2018; 62 Johnson (R81) 1997; 119 R117 Atangan (R7) 2016; 273 Sasaki (R125) 1993; 26 Guisti (R62) 2016; 51 Nutting (R111) 1946; 242 Joseph (R82) 1989; 61 Gomez-Aguilar (R61) 2005; 17 Wyatt (R151) 2010; 114 Atangana (R10) 2015; 7 Scott-Blair (R131) 1965; 6 Troyer (R145) 2012; 8 Colombaro (R37) 2017; 52 Zhang (R154) 2016; 64 Atangana (R11) 2015; 114 R73 Fabrzio (R53) 2004; 39 Hristov (R71) 2016; 20 Kim (R84) 2013; 54 Liu (R94) 2012; 5 Morales Delgado (R107) 2018; 46 Mun (R108) 2009; 41 R121 Allison (R4) 2018; 733 Hanyga (R67) 2001; 34 Carre (R30) 1999; 125 dos Santos (R47) 2018; 2 Fajman (R54) 2007; 85 Kosa (R87) 2016; 3 Laksari (R90) 2012; 45 Tateishi (R143) 2017; 5 R147 Nutting (R110) 1921; 191 Araujo (R6) 2000; 24 Singh (R136) 2018; 2018 Canestrati (R25) 2015; 76 Crook (R42) 1993; 44 Kumar (R88) 2018; 492 Corless (R40) 1996; 5 Markovitz (R100) 1977; 21 Fabrzio (R52) 2004; 39 Jaishankar (R78) 2014; 58 Jozwiak (R83) 2015; 10 Scott-Blair (R132) 1972; 11 Sheng (R133) 2015; 105 Papagiannnopoulos (R114) 2016; 61 Nunciato (R109) 1971; 29 Sun (R141) 2017; 468 Das (R45) 2018; 1 Garbarski (R59) 1992; 32 Baleanu (R16) 2018; 59 Sasaki (R126) 1995; 28 Atangana (R13) 2015; 7 Xu (R152) 2017; 73 Song (R138) 2006; 18 R91 Pelayo (R115) 2017; 119 Meral (R103) 2010; 15 Miller (R105) 1978; 66 Colombaro (R36) 2018; 6 R98 Sun (R140) 2015; 99 R99 Atangana (R9) 2016; 17 R18 Hristov (R74) 2017; 3 Losada (R97) 2015; 1 Linde (R93) 2000; 38 Metzler (R104) 1995; 103 |
| References_xml | – volume: 34 start-page: 2685 year: 1997 ident: R49 publication-title: Int. J. Solids Struct doi: 10.1016/S0020-7683(96)00178-3 – volume: 41 start-page: 339 year: 2009 ident: R108 publication-title: Mech. Mater doi: 10.1016/j.mechmat.2008.11.008 – volume: 2018 start-page: 231 issue: 2018 year: 2018 ident: R136 publication-title: Adv. Differ. Equ doi: 10.1186/s13662-018-1680-1 – volume: 76 start-page: 141 year: 2015 ident: R25 publication-title: Mater. Des doi: 10.1016/j.matdes.2015.03.063 – volume: 26 start-page: 1369 year: 1993 ident: R125 publication-title: J. Biomech doi: 10.1016/0021-9290(93)90088-V – volume: 1 start-page: 73 year: 2015 ident: R27 publication-title: Progr. Fract. Differ. Appl – volume: 46 start-page: 307 year: 1993 ident: R57 publication-title: J. Non-Newton. Fluid Mech. doi: 10.1016/0377-0257(93)85052-C – ident: R127 – volume: 43 start-page: 771 year: 1995 ident: R112 publication-title: J. Mech. Phys. Solids doi: 10.1016/0022-5096(95)00004-3 – volume: 3 start-page: 255 year: 2017 ident: R74 publication-title: Progr. Fract. Differ. Appl doi: 10.18576/pfda/030402 – volume: 733 start-page: 232 year: 2018 ident: R4 publication-title: Tectonophysics doi: 10.1016/j.tecto.2017.10.021 – volume: 27 start-page: 2162 year: 2006 ident: R116 publication-title: Biomaterials doi: 10.1016/j.biomaterials.2005.10.034 – volume: 117 start-page: 116 year: 2018 ident: R134 publication-title: Mech. Mater doi: 10.1016/j.mechmat.2017.11.003 – ident: R98 doi: 10.1142/p614 – ident: R56 – volume: 6 start-page: 15 year: 2018 ident: R36 publication-title: Mathematics doi: 10.3390/math6020015 – volume: 32 start-page: 107 year: 1992 ident: R59 publication-title: Polym. Eng. Sci doi: 10.1002/pen.760320206 – volume: 58 start-page: 1751 year: 2014 ident: R78 publication-title: J. Rheol doi: 10.1122/1.4892114 – ident: R1 – volume: 42 start-page: 932 year: 2010 ident: R33 publication-title: Mech. Mater doi: 10.1016/j.mechmat.2010.07.007 – volume: 11 start-page: 237 year: 1972 ident: R132 publication-title: Rheol. Acta doi: 10.1007/BF01993026 – volume: 230 start-page: 135 year: 2018 ident: R77 publication-title: Comput. Phys. Commun doi: 10.1016/j.cpc.2018.04.015 – volume: 39 start-page: 3903 year: 1996 ident: R70 publication-title: Int. J. Numer. Methods Eng doi: 10.1002/(SICI)1097-0207(19961130)39:22<3903::AID-NME34>3.0.CO;2-C – volume: 67–68 start-page: 169 year: 2015 ident: R79 publication-title: Int. J. Solids Struct doi: 10.1016/j.ijsolstr.2015.04.018 – volume: 28 start-page: 365 year: 2001 ident: R92 publication-title: Mech. Res. Commun doi: 10.1016/S0093-6413(01)00185-9 – ident: R91 doi: 10.1007/978-3-642-88210-4 – volume: A2 start-page: 225 year: 1951 ident: R130 publication-title: Appl. Sci. Res doi: 10.1007/BF00411984 – volume: 273 start-page: 948 year: 2016 ident: R7 publication-title: Appl. Math. Comput doi: 10.1016/j.amc.2015.10.021 – volume: 61 start-page: 41 year: 1989 ident: R82 publication-title: Rev. Mod. Phys doi: 10.1103/RevModPhys.61.41 – volume: 16 start-page: 287 year: 2012 ident: R20 publication-title: Mech. Time-Depend. Mater doi: 10.1007/s11043-011-9162-9 – volume: 127 start-page: 55 year: 2018 ident: R96 publication-title: Mech. Mater doi: 10.1016/j.mechmat.2018.07.012 – volume: 17 start-page: 4439 year: 2016 ident: R9 publication-title: Entropy doi: 10.3390/e17064439 – volume: 2012 start-page: 472 year: 2016 ident: R137 publication-title: Tribol. Int doi: 10.1016/j.triboint.2016.06.003 – ident: R147 doi: 10.1007/978-3-642-73602-5 – volume: 8 start-page: 849 year: 1992 ident: R66 publication-title: Inverse Probl doi: 10.1088/0266-5611/8/6/005 – ident: R99 – volume: 1 start-page: 1 year: 2018 ident: R45 publication-title: Asian J. Res. Rev. Phys doi: 10.9734/ajr2p/2018/v1i324617 – volume: 24 start-page: 91 year: 2000 ident: R6 publication-title: J. Comput. Math – volume: 39 start-page: 531 year: 2004 ident: R52 publication-title: Meccanica doi: 10.1007/s11012-003-6437-5 – volume: 36 start-page: 4447 year: 1999 ident: R51 publication-title: Int. J. Solids Struct doi: 10.1016/S0020-7683(98)00194-2 – volume: 48 start-page: 1485 year: 2015 ident: R5 publication-title: J. Biomech doi: 10.1016/j.jbiomech.2015.02.001 – volume: 130 start-page: 437 year: 2017 ident: R144 publication-title: Int. J. Mech. Sci doi: 10.1016/j.ijmecsci.2017.06.032 – volume: 1 start-page: 87 year: 2015 ident: R97 publication-title: Progr. Fract. Differ. Appl – volume: 48 start-page: 116 year: 2016 ident: R155 publication-title: Int. J. Rock Mech. Min. Sci doi: 10.1016/j.ijrmms.2010.11.004 – volume: 29 start-page: 187 year: 1971 ident: R109 publication-title: Q. Appl. Math doi: 10.1090/qam/295683 – volume: 492 start-page: 155 year: 2018 ident: R88 publication-title: Physica A doi: 10.1016/j.physa.2017.10.002 – volume: 45 start-page: 642 year: 2012 ident: R90 publication-title: J. Biomech doi: 10.1016/j.jbiomech.2011.12.023 – volume: 28 start-page: 6567 year: 1995 ident: R128 publication-title: J. Phys. A doi: 10.1088/0305-4470/28/23/012 – volume: 21 start-page: 345802 year: 2009 ident: R146 publication-title: J. Phys. Condens. Matter doi: 10.1088/0953-8984/21/34/345802 – volume: 47 start-page: 87 year: 2015 ident: R122 publication-title: J. Mech. Behav. Biomed. Mater doi: 10.1016/j.jmbbm.2015.03.011 – volume: 62 start-page: 157 year: 2018 ident: R142 publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2018.02.019 – volume: 6 start-page: 135 year: 2018 ident: R75 publication-title: Front. Phys doi: 10.3389/fphy.2018.00135 – volume: 35 start-page: 1225 year: 1997 ident: R153 publication-title: J. Polym. Sci. B doi: 10.1002/(SICI)1099-0488(199706)35:8<1225::AID-POLB8>3.0.CO;2-S – volume: 93 start-page: 1757 year: 2018 ident: R63 publication-title: Nonlinear Dyn doi: 10.1007/s11071-018-4289-8 – volume: 98 start-page: 81 year: 2016 ident: R95 publication-title: Mech. Mater doi: 10.1016/j.mechmat.2016.04.002 – ident: R73 – volume: 5 start-page: 222 year: 2012 ident: R94 publication-title: Theoret. Appl. Mech. Lett doi: 10.1016/j.taml.2015.11.002 – volume: 8 start-page: 240 year: 2012 ident: R145 publication-title: Acta Biomater doi: 10.1016/j.actbio.2011.07.035 – volume: 64 start-page: 170 year: 2016 ident: R154 publication-title: Ultrasonics doi: 10.1016/j.ultras.2015.09.003 – volume: 20 start-page: 765 year: 2016 ident: R71 publication-title: Therm. Sci – volume: 61 start-page: 201 year: 2016 ident: R114 publication-title: Food Hydrocolloids doi: 10.1016/j.foodhyd.2016.05.020 – volume: 19 start-page: 3419 year: 2014 ident: R148 publication-title: Commun. Nonlinear Sci. Numer. Simul doi: 10.1016/j.cnsns.2014.03.014 – volume: 13 start-page: 529 year: 1967 ident: R26 publication-title: Geophys. J. R. Astron. Soc doi: 10.1111/j.1365-246X.1967.tb02303.x – ident: R31 – volume: 24 start-page: 744 year: 2008 ident: R46 publication-title: Dent. Mater doi: 10.1016/j.dental.2007.08.008 – volume: 28 start-page: 40 year: 1968 ident: R65 publication-title: Arch. Ration. Mech. Anal doi: 10.1007/BF00281562 – volume: 59 start-page: 608 year: 2018 ident: R113 publication-title: Commun. Nonlinear Sci. Numer. Simul doi: 10.1016/j.cnsns.2017.12.001 – volume: 37 start-page: 637 year: 1990 ident: R44 publication-title: Comput. Struct doi: 10.1016/0045-7949(90)90096-K – volume: 119 start-page: 273 year: 1997 ident: R81 publication-title: ASME J. Eng. Mater. Technol doi: 10.1115/1.2812256 – volume: 30 start-page: 133 year: 1986 ident: R15 publication-title: J. Rheol doi: 10.1122/1.549887 – volume: 36 start-page: 320 year: 1997 ident: R19 publication-title: Rheol. Acta doi: 10.1007/BF00366673 – volume: 52 start-page: 825 year: 2017 ident: R37 publication-title: Meccanica doi: 10.1007/s11012-016-0456-5 – volume: 5 start-page: 52 year: 2017 ident: R143 publication-title: Front. Phys doi: 10.3389/fphy.2017.00052 – volume: 33 start-page: 337 year: 1994 ident: R60 publication-title: Rheol. Acta doi: 10.1007/BF00366960 – volume: 125 start-page: 914 year: 1999 ident: R30 publication-title: J. Eng. Mech doi: 10.1061/(ASCE)0733-9399(1999)125:8(914) – volume: 191 start-page: 679 year: 1921 ident: R110 publication-title: J. Franklin Inst. doi: 10.1016/S0016-0032(21)90171-6 – volume: 468 start-page: 590 year: 2017 ident: R141 publication-title: Physica A doi: 10.1016/j.physa.2016.10.066 – volume: 44 start-page: 614 year: 2005 ident: R68 publication-title: Rheol. Acta doi: 10.1007/s00397-005-0443-6 – volume: 18 start-page: 18 year: 2006 ident: R138 publication-title: Korea-Aust. Rheol. J – volume: 7 start-page: 1 year: 2015 ident: R10 publication-title: Adv. Mech. Eng – volume: 142 start-page: 31 year: 2014 ident: R149 publication-title: J. Food. Eng doi: 10.1016/j.jfoodeng.2014.05.019 – volume: 68 start-page: 62 year: 2017 ident: R38 publication-title: Z. Angew. Math. Phys doi: 10.1007/s00033-017-0808-6 – volume: 242 start-page: 449 year: 1946 ident: R111 publication-title: J Franklin Inst doi: 10.1016/0016-0032(46)90636-9 – volume: 28 start-page: 809 year: 1995 ident: R126 publication-title: J. Biomech doi: 10.1016/0021-9290(94)00130-V – volume: 9 start-page: 15 year: 2005 ident: R2 publication-title: Mech. Time-Depend Mater doi: 10.1007/s11043-005-3442-1 – volume: 2 start-page: 1 year: 2016 ident: R28 publication-title: Progr. Fract. Differ. Appl doi: 10.18576/pfda/020101 – volume: 39 start-page: 547 year: 2004 ident: R53 publication-title: Meccanica doi: 10.1007/s11012-003-9400-6 – volume: 51 start-page: 2321 year: 2016 ident: R62 publication-title: Meccanica doi: 10.1007/s11012-016-0376-4 – volume: 38 start-page: 945 year: 2000 ident: R93 publication-title: Ultrasonics doi: 10.1016/S0041-624X(00)00024-X – volume: 10 start-page: e0143090 year: 2015 ident: R83 publication-title: PloS One doi: 10.1371/journal.pone.0143090 – volume: 114 start-page: 4076 year: 2010 ident: R151 publication-title: J. Appl. Polym. Sci doi: 10.1002/app.31093 – volume: 196 start-page: 1216 year: 2011 ident: R43 publication-title: J. Power Sources doi: 10.1016/j.jpowsour.2010.08.075 – ident: R69 – volume: 5 start-page: 329 year: 1996 ident: R40 publication-title: Adv. Comput. Math doi: 10.1007/BF02124750 – volume: 316 start-page: 504 year: 2018 ident: R135 publication-title: Appl. Math. Comput doi: 10.1016/j.amc.2017.08.048 – volume: 618 start-page: 2 year: 2016 ident: R76 publication-title: Thin Solid Films doi: 10.1016/j.tsf.2016.03.064 – ident: R29 doi: 10.5772/64251 – volume: 119 start-page: 324 year: 2017 ident: R115 publication-title: Thin-Walled Struct doi: 10.1016/j.tws.2017.06.030 – volume: IX start-page: 165 issue: 38 year: 1959 ident: R150 publication-title: Math. Pure Appl – volume: 21 start-page: 384 year: 1977 ident: R100 publication-title: Trans. Soc. Rheol – volume: 140 start-page: 04013053 year: 2013 ident: R119 publication-title: J. Struct. Eng doi: 10.1061/(ASCE)ST.1943-541X.0000822 – volume: 34 start-page: 1399 year: 2001 ident: R67 publication-title: Math. Comput. Model doi: 10.1016/S0895-7177(01)00137-6 – ident: R117 doi: 10.1007/978-1-4615-9970-8 – volume: 44 start-page: 167 year: 1993 ident: R42 publication-title: Eng. Fract. Mech doi: 10.1016/0013-7944(93)90041-P – volume: 15 start-page: 939 year: 2010 ident: R103 publication-title: Commun. Nonlinear Sci. Numer. Simul doi: 10.1016/j.cnsns.2009.05.004 – volume: 37 start-page: 286 year: 2014 ident: R34 publication-title: J. Mech. Behav. Biomed. Mater doi: 10.1016/j.jmbbm.2014.05.031 – volume: 3 start-page: 26 year: 2018 ident: R39 publication-title: Int. J. Theor. Appl. Mech – ident: R22 doi: 10.1137/1.9781611971484 – volume: 7 start-page: 1 year: 2015 ident: R13 publication-title: Adv. Mech. Eng – volume: 39 start-page: 6370 year: 2012 ident: R106 publication-title: Expert Syst. Appl doi: 10.1016/j.eswa.2011.12.033 – volume: 2 start-page: 20 year: 2018 ident: R47 publication-title: Fractal Fract doi: 10.3390/fractalfract2030020 – volume: 114 start-page: 516 year: 2018 ident: R12 publication-title: Chaos Soliton. Fract doi: 10.1016/j.chaos.2018.07.033 – volume: 56 start-page: 630 year: 2018 ident: R3 publication-title: Chin. J. Phys doi: 10.1016/j.cjph.2018.02.007 – volume: 36 start-page: 93 year: 1995 ident: R89 publication-title: Polymer doi: 10.1016/0032-3861(95)90680-Z – volume: 56 start-page: 5377 year: 2008 ident: R32 publication-title: Acta Mater doi: 10.1016/j.actamat.2008.07.030 – volume: 21 start-page: 827 year: 2017 ident: R72 publication-title: Therm. Sci doi: 10.2298/TSCI160229115H – volume: 21 start-page: 251 year: 1982 ident: R123 publication-title: Rheol. Acta doi: 10.1007/BF01515713 – volume: 6 start-page: 201 year: 1965 ident: R131 publication-title: Rheol. Acta doi: 10.1007/BF01976435 – volume: 54 start-page: 5993 year: 2013 ident: R84 publication-title: Polymer doi: 10.1016/j.polymer.2013.08.062 – ident: R121 – volume: 47 start-page: 188 year: 2011 ident: R55 publication-title: Strain doi: 10.1111/j.1475-1305.2008.00502.x – volume: 56 start-page: 138 year: 2018 ident: R64 publication-title: Commun. Nonlinear Sci Numer. Simul doi: 10.1016/j.cnsns.2017.08.002 – volume: 6 start-page: 761 year: 1931 ident: R86 publication-title: Ann. Phys doi: 10.1002/andp.19314030608 – volume: 31 start-page: 91 year: 1999 ident: R80 publication-title: Shock Vib. Digest doi: 10.1177/058310249903100201 – volume: 134 start-page: 371 year: 1997 ident: R50 publication-title: Opt. Commun doi: 10.1016/S0030-4018(96)00384-7 – volume: 105 start-page: 43 year: 2015 ident: R133 publication-title: Ocean Eng doi: 10.1016/j.oceaneng.2015.06.023 – ident: R48 doi: 10.1142/2905 – volume: 55 start-page: 32 year: 2016 ident: R14 publication-title: J. Mech. Behav. Biomed. Mater doi: 10.1016/j.jmbbm.2015.10.008 – volume: 3 start-page: 1003 year: 2016 ident: R87 publication-title: Mater. Today Proc doi: 10.1016/j.matpr.2016.03.037 – volume: 20 start-page: 763 year: 2016 ident: R8 publication-title: Therm. Sci doi: 10.2298/TSCI160111018A – ident: R24 doi: 10.1007/978-0-387-73861-1 – volume: 70 start-page: 275 year: 1874 ident: R23 publication-title: Akad. Wiss. Wien. Mathem.- Naturwiss – ident: R58 – ident: R102 – ident: R18 – volume: 11 start-page: 199 year: 2007 ident: R85 publication-title: Mech. Time-Depend. Mater doi: 10.1007/s11043-007-9035-4 – volume: 506 start-page: 75 year: 2018 ident: R101 publication-title: Physica A doi: 10.1016/j.physa.2018.04.047 – ident: R124 – ident: R118 – volume: 99 start-page: 226 year: 2015 ident: R140 publication-title: Constr. Build. Mater doi: 10.1016/j.conbuildmat.2015.09.016 – volume: 59 start-page: 444 year: 2018 ident: R16 publication-title: Commun. Nonlinear Sci. Numer. Simul doi: 10.1016/j.cnsns.2017.12.003 – volume: 1 start-page: 24 year: 1975 ident: R120 publication-title: J. Ecole Polytech – volume: 46 start-page: 1 year: 2018 ident: R107 publication-title: Int. J. Circ. Theor. Appl doi: 10.1002/cta.2564 – ident: R41 – volume: 33 start-page: 239 year: 1961 ident: R35 publication-title: Rev. Mod. Phys doi: 10.1103/RevModPhys.33.239 – volume: 17 start-page: 6239 year: 2005 ident: R61 publication-title: Entropy – volume: 85 start-page: 1514 year: 2007 ident: R54 publication-title: Comput. Struct doi: 10.1016/j.compstruc.2007.01.024 – volume: 44 start-page: 15 year: 1992 ident: R17 publication-title: J. Non-Newton. Fluid Mech doi: 10.1016/0377-0257(92)80043-W – volume: 73 start-page: 1377 year: 2017 ident: R152 publication-title: Comput. Math. Appl doi: 10.1016/j.camwa.2016.05.002 – volume: 34 start-page: 377 year: 1997 ident: R21 publication-title: Bioreology doi: 10.3233/BIR-1997-34603 – volume: 114 start-page: 347 year: 2015 ident: R11 publication-title: Chaos Soliton. Fract doi: 10.1016/j.chaos.2018.07.022 – volume: 22 start-page: 940 year: 1951 ident: R139 publication-title: J. Appl. Phys doi: 10.1063/1.1700076 – volume: 66 start-page: 313 year: 1978 ident: R105 publication-title: J. Math. Anal. Appl doi: 10.1016/0022-247X(78)90234-2 – volume: 122 start-page: 42 year: 2016 ident: R129 publication-title: Eng. Struct doi: 10.1016/j.engstruct.2016.04.024 – volume: 103 start-page: 7180 year: 1995 ident: R104 publication-title: J. Chem. Phys doi: 10.1063/1.470346 |
| SSID | ssj0000314300 |
| Score | 2.3641598 |
| SecondaryResourceType | review_article |
| Snippet | This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate... |
| SourceID | proquest crossref istex |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 305 |
| SubjectTerms | 26A33 35K05 40C10 Caputo–Fabrizio operator Constitutive equations Constitutive relationships Fractional calculus Integrals Kernels Mathematical models Mittag–Leffler relaxation kernel Modelling non-singular fading memory nonlinear relaxation Operators (mathematics) Polymers Power law Prony series Response functions Solid polymer rheology Strain relaxation Stress relaxation Viscoelasticity |
| Title | Response functions in linear viscoelastic constitutive equations and related fractional operators |
| URI | https://api.istex.fr/ark:/67375/80W-G7KF4NJF-H/fulltext.pdf https://www.proquest.com/docview/2267402577 https://www.proquest.com/docview/2283281007 |
| Volume | 14 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1db9MwFLXY9sILYnyIjjH5ARCoypYPO04eEbQrYxSEWtE3K3ZsMbGlpe0mxAO_nXsdJ23FhoCXqEodp_Kxr6_dc44JeZowHZpYZ4FSURowESsYUiIK8oQpXuZpYbUjyA7TwZidTPhkdXChU5cs1aH-ca2u5H9QhXuAK6pk_wHZtlK4AZ8BX7gCwnD9K4w_1QRX08XZqSWFY-JYzLtXZws9NZAcoyWrnnpSABKFzLdLT4Bz5HKkw0Haaee1xgGz05lx_74v1lPX963BKwpO8ACdc8-YduagqOj6Yio0dGgj_cAZF1zVu_Tz0ndEv8eAsqaNPYbem49NqNnYQRRJwJPaKPPQ1PFTpLga9c82AZatdaTk2rgNoQEa--KimqFCBU3F6kM6Ng2yhx9kf3x6Kke9yWiL7MSwMnD67rc_2201dONPauFR8_O83AFecYQvOPLVbyQiOzimvv82H7skY3SX3PGrA_qqhnqX3DLVPbLbNAp94U3CX94nRYM9bbGnZxWtsafr2NN17GmLPQXsqceerrCnLfYPyLjfG70eBP64jEAncb4MCqG4VjzNbFgksc1DnalUlaUqMxtZbq3NCl6YCGO2ynXIVGxUGtoiLsukgMzzIdmuppV5RKhiIsmzohSWp8xapqASy2zO4xINGEWHdJu2k9p7yeORJufScRp4JLGlpW_pDnnWlp7VHio3lHvuYGgLFfOvyDsUXGbhZ3ks3vXZ8KQvBx2y3-Ak_XBcSFhHCAYZvBA3fA1zV4acoL0_P_2Y3F4NgX2yvZxfmieQeC7VAdnK-scHrsP9AqvKjKo |
| linkProvider | EDP |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Response+functions+in+linear+viscoelastic+constitutive+equations+and+related+fractional+operators&rft.jtitle=Mathematical+modelling+of+natural+phenomena&rft.au=Hristov%2C+Jordan&rft.date=2019-01-01&rft.pub=EDP+Sciences&rft.issn=0973-5348&rft.eissn=1760-6101&rft.volume=14&rft.issue=3&rft_id=info:doi/10.1051%2Fmmnp%2F2018067&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0973-5348&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0973-5348&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0973-5348&client=summon |