{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T18:29:16Z","timestamp":1790360956995,"version":"4.1.0"},"reference-count":68,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T00:00:00Z","timestamp":1708905600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2022R1A2C2007429"],"award-info":[{"award-number":["2022R1A2C2007429"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Damage Mechanics"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:p>\n                    This work introduces a theoretical framework for continuum damage and healing mechanics by extending stress decomposition to account for tensile, compressive, and shear stresses. In addition to the spectral stress decomposition into tensile and compressive components, we extend the existing stress decomposition method to address shear stresses. The extraction of shear stresses employs two hypotheses, considering both same-signed and opposite-signed principal stresses. This stress decomposition approach yields three damage variables\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mo>(<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi>\u03c6<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\u00b1<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and three healing variables\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mo>(<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\u00b1<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . The damage formulation is discussed in terms of equivalent strain and conjugate force, while the healing formulation is based on the initial damage state and healing time. We explore the influence of material parameters and healing time on damage and healing evolution. Furthermore, we analyze the relationship between nominal stress-to-effective stress ratio, damage variables, and healing time. Lastly, we present a thermodynamically consistent formulation for damage-healing processes, acknowledging that this work establishes a theoretical formulation. The proposed method is validated by analyzing the performance of an L-shaped concrete specimen using three damage variables and one healing variable. These results illustrate the model's ability to effectively capture the damage and healing phenomena. The practical implementation of the proposed formulation will be pursued numerically using innovative healing techniques and a pseudo-damage healing approach, which will be detailed in future work.\n                  <\/jats:p>","DOI":"10.1177\/10567895241233833","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T06:22:26Z","timestamp":1708928546000},"page":"855-877","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Multi-stress damage and healing mechanics in quasi-brittle materials: Theoretical overview"],"prefix":"10.1177","volume":"34","author":[{"given":"Bilal","family":"Ahmed","sequence":"first","affiliation":[{"name":"Civil and Urban Engineering, New York University Abu Dhabi, Abu Dhabi, UAE"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7840-3959","authenticated-orcid":false,"given":"Taehyo","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Hanyang University, Seoul, Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6657-7265","authenticated-orcid":false,"given":"Jong-Su","family":"Jeon","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Hanyang University, Seoul, Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2024,2,26]]},"reference":[{"key":"e_1_3_2_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijengsci.2010.09.016"},{"key":"e_1_3_2_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2010.04.007"},{"key":"e_1_3_2_4_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789508097541"},{"key":"e_1_3_2_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2019.11.023"},{"key":"e_1_3_2_6_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789521998728"},{"key":"e_1_3_2_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-020-01939-6"},{"key":"e_1_3_2_8_1","doi-asserted-by":"publisher","DOI":"10.1186\/s40323-018-0106-7"},{"key":"e_1_3_2_9_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789516635729"},{"key":"e_1_3_2_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2015.02.009"},{"key":"e_1_3_2_11_1","first-page":"20","article-title":"Characterization of self-healing fiber-reinforced polymer-matrix composite with distributed damage","volume":"39","author":"Barbero E","year":"2007","unstructured":"Barbero E, Ford K (2007) Characterization of self-healing fiber-reinforced polymer-matrix composite with distributed damage. Journal of Advanced Materials 39: 20\u201327.","journal-title":"Journal of Advanced Materials"},{"key":"e_1_3_2_12_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789505045928"},{"key":"e_1_3_2_13_1","doi-asserted-by":"crossref","unstructured":"Barbero EJ Lonetti P (2003) Application of continuum damage healing mechanics to Self-Healing composites [paper presentation]. ASME 2003 international mechanical engineering congress and exposition pp. 515\u2013519.","DOI":"10.1115\/IMECE2003-43738"},{"key":"e_1_3_2_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10853-006-0512-y"},{"key":"e_1_3_2_15_1","doi-asserted-by":"publisher","DOI":"10.1088\/0964-1726\/3\/2\/006"},{"key":"e_1_3_2_16_1","doi-asserted-by":"crossref","unstructured":"Carolyn D Sottos NR (1993) Passive Smart Self-Repair in Polymer Matrix Composite Materials Smart Structures and Materials 1993: Smart Materials. North American Conference on Smart Structures and Materials. Albuquerque NM United States: SPIE pp. 438\u2013444.","DOI":"10.1117\/12.148501"},{"key":"e_1_3_2_17_1","doi-asserted-by":"publisher","DOI":"10.3390\/ma10040433"},{"key":"e_1_3_2_18_1","doi-asserted-by":"publisher","DOI":"10.3390\/ma10040433"},{"key":"e_1_3_2_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2017.11.017"},{"key":"e_1_3_2_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2007.03.006"},{"key":"e_1_3_2_21_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1711937"},{"key":"e_1_3_2_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2011.10.017"},{"key":"e_1_3_2_23_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2012.02.003"},{"key":"e_1_3_2_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(97)00119-4"},{"key":"e_1_3_2_25_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.conbuildmat.2015.10.192"},{"key":"e_1_3_2_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2017.10.019"},{"key":"e_1_3_2_27_1","doi-asserted-by":"crossref","unstructured":"Giannaros P Kanellopoulos A Al-Tabbaa A (2016) Research data supporting. Sealing of Cracks in Cement Using Microencapsulated Sodium Silicate. Smart Materials and Structures 25: 084005.","DOI":"10.1088\/0964-1726\/25\/8\/084005"},{"key":"e_1_3_2_28_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.matdes.2014.11.012"},{"key":"e_1_3_2_29_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2011.08.012"},{"key":"e_1_3_2_30_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2013.07.008"},{"key":"e_1_3_2_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.conbuildmat.2017.04.111"},{"key":"e_1_3_2_32_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2007.03.010"},{"key":"e_1_3_2_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cemconres.2014.04.003"},{"key":"e_1_3_2_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.04.017"},{"key":"e_1_3_2_35_1","first-page":"26","article-title":"Time of the rupture process under creep conditions","volume":"8","author":"Kachanov LM","year":"1958","unstructured":"Kachanov LM (1958) Time of the rupture process under creep conditions. Izvestiia Akademii Nauk SSSR, Otdelenie Teckhnicheskikh Nauk 8: 26\u201331.","journal-title":"Izvestiia Akademii Nauk SSSR, Otdelenie Teckhnicheskikh Nauk"},{"key":"e_1_3_2_36_1","doi-asserted-by":"publisher","DOI":"10.3390\/ma7043136"},{"key":"e_1_3_2_37_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cemconres.2010.12.002"},{"key":"e_1_3_2_38_1","doi-asserted-by":"publisher","DOI":"10.1155\/2016\/3231092"},{"key":"e_1_3_2_39_1","doi-asserted-by":"publisher","DOI":"10.1016\/S1359-8368(98)00034-1"},{"key":"e_1_3_2_40_1","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1995)121:10(1122)"},{"key":"e_1_3_2_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10853-014-8657-6"},{"key":"e_1_3_2_42_1","doi-asserted-by":"publisher","DOI":"10.3390\/ma10010049"},{"key":"e_1_3_2_43_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-81598-0_28"},{"key":"e_1_3_2_44_1","unstructured":"Oliver J Cervera M Oller S et\u00a0al. (1990) Isotropic Damage Models and Smeared Crack Analysis of Concrete. Conference: II International Conference on Computer Aided Analysis and Design of Concrete Structures. Zell am See Austria: SCI-C'90At."},{"key":"e_1_3_2_45_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6636(85)90007-9"},{"key":"e_1_3_2_46_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tafmec.2018.04.010"},{"key":"e_1_3_2_47_1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(87)90130-7"},{"key":"e_1_3_2_48_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2010.02.010"},{"key":"e_1_3_2_49_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2019.04.001"},{"key":"e_1_3_2_50_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-012-0769-8"},{"key":"e_1_3_2_51_1","doi-asserted-by":"publisher","DOI":"10.3390\/app8061004"},{"key":"e_1_3_2_52_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789515616447"},{"key":"e_1_3_2_53_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.12.027"},{"key":"e_1_3_2_54_1","doi-asserted-by":"crossref","unstructured":"Voyiadjis GZ Kattan PI (1999) Advances in Damage Mechanics Metals and Metal Matrix Composites. Elsevier pp. 1\u20139. https:\/\/doi.org\/10.1016\/B978-0-08-043601-2.50006-4.","DOI":"10.1016\/B978-0-08-043601-2.50006-4"},{"key":"e_1_3_2_55_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789513491773"},{"key":"e_1_3_2_56_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2015.02.006"},{"key":"e_1_3_2_57_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789520905347"},{"key":"e_1_3_2_58_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2010.11.002"},{"key":"e_1_3_2_59_1","doi-asserted-by":"publisher","DOI":"10.1177\/1056789510397069"},{"key":"e_1_3_2_60_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2008.04.002"},{"key":"e_1_3_2_61_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cemconres.2013.11.009"},{"key":"e_1_3_2_62_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2018.02.004"},{"key":"e_1_3_2_63_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2012.11.012"},{"key":"e_1_3_2_64_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2005.05.038"},{"key":"e_1_3_2_65_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.conbuildmat.2011.08.086"},{"key":"e_1_3_2_66_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cemconres.2009.01.013"},{"key":"e_1_3_2_67_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2016.04.038"},{"key":"e_1_3_2_68_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2017.10.004"},{"key":"e_1_3_2_69_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2013.07.008"}],"container-title":["International Journal of Damage Mechanics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10567895241233833","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/10567895241233833","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10567895241233833","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:50:59Z","timestamp":1777456259000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/10567895241233833"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,26]]},"references-count":68,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["10.1177\/10567895241233833"],"URL":"https:\/\/doi.org\/10.1177\/10567895241233833","relation":{},"ISSN":["1056-7895","1530-7921"],"issn-type":[{"value":"1056-7895","type":"print"},{"value":"1530-7921","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,26]]}}}