{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T19:50:28Z","timestamp":1790711428712,"version":"4.1.0"},"reference-count":67,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T00:00:00Z","timestamp":1605484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["19-15326S"],"award-info":[{"award-number":["19-15326S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["CZ.02.1.01\/0.0\/0.0\/16_019\/0000778"],"award-info":[{"award-number":["CZ.02.1.01\/0.0\/0.0\/16_019\/0000778"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-\/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin\u2013Francfort\u2013Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one\/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.<\/jats:p>","DOI":"10.3390\/ma13225153","type":"journal-article","created":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T21:48:52Z","timestamp":1605563332000},"page":"5153","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1174-2822","authenticated-orcid":false,"given":"Jaroslav","family":"Schmidt","sequence":"first","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 166 29 Prague 6 \u2013 Dejvice, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7613-2099","authenticated-orcid":false,"given":"Alena","family":"Zemanov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 166 29 Prague 6 \u2013 Dejvice, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2503-8120","authenticated-orcid":false,"given":"Jan","family":"Zeman","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 166 29 Prague 6 \u2013 Dejvice, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michal","family":"\u0160ejnoha","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 166 29 Prague 6 \u2013 Dejvice, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haldimann, M., Luible, A., and Overend, M. 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