{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T07:31:37Z","timestamp":1777707097771,"version":"3.51.4"},"reference-count":41,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2013,7,4]],"date-time":"2013-07-04T00:00:00Z","timestamp":1372896000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Damage Mechanics"],"published-print":{"date-parts":[[2014,3]]},"abstract":"<jats:p>Over the past years, the non-local method has established itself as an effective remedy to the well-known pathological mesh dependency that inherently affects softening media. The non-local method incorporates an intrinsic length into the traditional continuum theory and therefore the size of the localising zone is resolved, attenuating the unwanted effects of spurious mesh dependency. However, despite many contributions that have successfully employed the non-local theory, it is still not clear how exactly should non-locality be formulated in the general sense. Aiming to answer the question of which non-local formulations effectively lead to mesh-insensitive results, we select in this article several constitutive variables to be non-local quantities by taking both Lemaitre and Gurson\u2013Tvergaard\u2013Needleman models as the base for the non-local enhancement. The resulting non-local constitutive models are employed in the numerical simulation of various specimens which are subjected to different values of stress triaxiality and third invariant at the fracture zone. The goal is to find which models present the best performance in the task of providing mesh-insensitive solutions for different stress states. The results show that strain-softening mesh dependency is stronger in plane strain than in the axisymmetric case. It is also found that the variables that regularise the solution in the axisymmetric case do not necessarily eliminate mesh sensitivity in the other cases. Furthermore, the results indicate that damage should be the preferred non-local variable in the case of implicit damage models. This result is in sharp contrast with the case of explicit damage models, for which it has already been shown in the literature that damage is a bad candidate for non-local variable.<\/jats:p>","DOI":"10.1177\/1056789513493103","type":"journal-article","created":{"date-parts":[[2013,7,4]],"date-time":"2013-07-04T21:56:31Z","timestamp":1372974991000},"page":"261-296","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":37,"title":["Assessment and comparison of non-local integral models for ductile damage"],"prefix":"10.1177","volume":"23","author":[{"given":"FXC","family":"Andrade","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering \u2013 DEMec, Faculty of Engineering, University of Porto, Porto, Portugal"}]},{"given":"JMA C\u00e9sar","family":"de S\u00e1","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering \u2013 DEMec, Faculty of Engineering, University of Porto, Porto, Portugal"}]},{"given":"FM Andrade","family":"Pires","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering \u2013 DEMec, Faculty of Engineering, University of Porto, Porto, Portugal"}]}],"member":"179","published-online":{"date-parts":[[2013,7,4]]},"reference":[{"key":"bibr1-1056789513493103","unstructured":"Andrade FXC (2012)\n                      Non-local modelling of ductile damage: Formulation and numerical issues\n                      . PhD Thesis, Faculty of Engineering, University of Porto."},{"key":"bibr2-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1177\/1056789510386850"},{"key":"bibr3-1056789513493103","unstructured":"Bai Y (2008)\n                      Effect of loading history on necking and fracture\n                      . PhD Thesis, Department of Ocean Engineering, Massachusetts Institute Technology."},{"key":"bibr4-1056789513493103","unstructured":"Bao Y (2003)\n                      Prediction of ductile crack formation in uncracked bodies\n                      . PhD Thesis, Department of Ocean Engineering, Massachusetts Institute Technology."},{"key":"bibr5-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1115\/1.1755244"},{"key":"bibr6-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2004.02.006"},{"key":"bibr7-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2006.09.031"},{"key":"bibr8-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2007.01.010"},{"key":"bibr9-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1984)110:10(1441)"},{"key":"bibr10-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1002\/nme.856"},{"key":"bibr11-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2007.12.001"},{"key":"bibr12-1056789513493103","doi-asserted-by":"crossref","unstructured":"C\u00e9sar de S\u00e1 J, Andrade Pires F and Andrade F (2010) Local and nonlocal modeling of ductile damage. In: Vaz M Jr, De Souza Neto E and Mu\u00f1oz-Rojas P (eds)\n                      Advanced Computational Materials Modelling: From Classical to Multi-Scale Techniques\n                      , Chap. 2. Weinheim: Wiley-VCH, pp.23\u201372.","DOI":"10.1002\/9783527632312.ch2"},{"key":"bibr13-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.02.037"},{"key":"bibr14-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620350307"},{"key":"bibr15-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.511"},{"key":"bibr16-1056789513493103","doi-asserted-by":"crossref","unstructured":"De Souza Neto, E, Peri\u0107, D and Owen, D (2008)\n                      Computational Methods for Plasticity: Theory and Applications\n                      . Chichester, UK: John Wiley and Sons, Ltd.","DOI":"10.1002\/9780470694626"},{"key":"bibr17-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1007\/BF00251544"},{"key":"bibr18-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1007\/BF00251543"},{"key":"bibr19-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/S0749-6419(01)00042-0"},{"key":"bibr20-1056789513493103","doi-asserted-by":"crossref","unstructured":"Gao, X, Zhang, G and Roe, C (2010) A study on the effect of the stress state on ductile fracture.\n                      International Journal of Damage Mechanics\n                      19: 75\u201394.","DOI":"10.1177\/1056789509101917"},{"key":"bibr21-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1002\/nme.654"},{"key":"bibr22-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1115\/1.3443401"},{"key":"bibr23-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5096(83)90017-0"},{"key":"bibr24-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7683(97)00306-5"},{"key":"bibr25-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7225(03)00027-2"},{"key":"bibr26-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0013-7944(85)90052-9"},{"key":"bibr27-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1115\/1.3225775"},{"key":"bibr28-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(85)90026-X"},{"key":"bibr29-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0013-7944(77)90062-5"},{"key":"bibr30-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1115\/1.3601204"},{"key":"bibr31-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.10.001"},{"key":"bibr32-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2007.08.002"},{"key":"bibr33-1056789513493103","doi-asserted-by":"crossref","unstructured":"Nguyen, G and Einav, I (2010) A stress-return algorithm for nonlocal constitutive models of softening materials.\n                      International Journal for Numerical Methods in Engineering\n                      82: 637\u2013670.","DOI":"10.1002\/nme.2790"},{"key":"bibr34-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D"},{"key":"bibr35-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)0733-9399(1987)113:10(1512)"},{"key":"bibr36-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2008.09.009"},{"key":"bibr37-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5096(69)90033-7"},{"key":"bibr38-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(96)00997-8"},{"key":"bibr39-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/0001-6160(84)90213-X"},{"key":"bibr40-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijplas.2010.01.015"},{"key":"bibr41-1056789513493103","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2005.03.003"}],"container-title":["International Journal of Damage Mechanics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1056789513493103","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/1056789513493103","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1056789513493103","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:48:48Z","timestamp":1777456128000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/1056789513493103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,4]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["10.1177\/1056789513493103"],"URL":"https:\/\/doi.org\/10.1177\/1056789513493103","relation":{},"ISSN":["1056-7895","1530-7921"],"issn-type":[{"value":"1056-7895","type":"print"},{"value":"1530-7921","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,7,4]]}}}