{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:39:39Z","timestamp":1760243979771,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2009,3,27]],"date-time":"2009-03-27T00:00:00Z","timestamp":1238112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Within the framework of multifield continua, we move from the model of elastic microcracked body introduced in (Mariano, P.M. and Stazi, F.L., Strain localization in elastic microcracked bodies, Comp. Methods Appl. Mech. Engrg. 2001, 190, 5657\u20135677) and propose a few novel variational formulations of mixed type along with relevant mixed FEM discretizations. To this goal, suitably extended Hellinger-Reissner principles of primal and dual type are derived. A few numerical studies are presented that include an investigation on the interaction between a single cohesive macrocrack and diffuse microcracks (Mariano, P.M. and Stazi, F.L., Strain localization due to crack\u2013microcrack interactions: X\u2013FEM for a multifield approach, Comp. Methods Appl. Mech. Engrg. 2004, 193, 5035\u20135062).<\/jats:p>","DOI":"10.3390\/a2010606","type":"journal-article","created":{"date-parts":[[2009,3,30]],"date-time":"2009-03-30T13:11:40Z","timestamp":1238418700000},"page":"606-622","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework"],"prefix":"10.3390","volume":"2","author":[{"given":"Matteo","family":"Bruggi","sequence":"first","affiliation":[{"name":"Department of Structural Mechanics, University of Pavia, via Ferrata 1, Pavia, Italy"}]},{"given":"Paolo","family":"Venini","sequence":"additional","affiliation":[{"name":"Department of Structural Mechanics, University of Pavia, via Ferrata 1, Pavia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2009,3,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF03167064","article-title":"PEERS: a new mixed finite element for plane elasticity","volume":"1","author":"Arnold","year":"1984","journal-title":"Japan Journal of Applied Mathematics"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Brezzi, F., and Fortin, M. 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Random heterogeneous materials, Springer Verlag.","DOI":"10.1007\/978-1-4757-6355-3"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/2\/1\/606\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:10:07Z","timestamp":1760220607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/2\/1\/606"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3,27]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,3]]}},"alternative-id":["a2010606"],"URL":"https:\/\/doi.org\/10.3390\/a2010606","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2009,3,27]]}}}