{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,4]],"date-time":"2026-11-04T21:18:35Z","timestamp":1793827115878,"version":"4.6.1"},"reference-count":67,"publisher":"The Royal Society","issue":"2203","license":[{"start":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T00:00:00Z","timestamp":1624233600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/royalsociety.org\/journals\/ethics-policies\/data-sharing-mining\/"}],"funder":[{"name":"Danish Hydrocarbon Research and Technology Centre","award":["DHRTC"],"award-info":[{"award-number":["DHRTC"]}]},{"DOI":"10.13039\/501100000700","name":"Royal Commission for the Exhibition of 1851","doi-asserted-by":"crossref","award":["RF496\/2018"],"award-info":[{"award-number":["RF496\/2018"]}],"id":[{"id":"10.13039\/501100000700","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/R010161\/1"],"award-info":[{"award-number":["EP\/R010161\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/R017727\/1"],"award-info":[{"award-number":["EP\/R017727\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["royalsocietypublishing.org"],"crossmark-restriction":true},"short-container-title":["Phil. Trans. R. Soc. A."],"published-print":{"date-parts":[[2021,8,9]]},"abstract":"<jats:p>The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith\u2019s energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness- for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimization\u2014without the need for ad hoc criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms.<\/jats:p>\n                  <jats:p>This article is part of a discussion meeting issue \u2018A cracking approach to inventing new tough materials: fracture stranger than friction\u2019.<\/jats:p>","DOI":"10.1098\/rsta.2021.0021","type":"journal-article","created":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T03:05:16Z","timestamp":1624244716000},"page":"20210021","update-policy":"https:\/\/doi.org\/10.1098\/crossmark-policy","source":"Crossref","is-referenced-by-count":156,"title":["An assessment of phase field fracture: crack initiation and growth"],"prefix":"10.1098","volume":"379","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0178-9985","authenticated-orcid":true,"given":"Philip K.","family":"Kristensen","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian F.","family":"Niordson","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1562-097X","authenticated-orcid":true,"given":"Emilio","family":"Mart\u00ednez-Pa\u00f1eda","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2021,6,21]]},"reference":[{"key":"e_1_3_5_2_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1921.0006"},{"key":"e_1_3_5_3_2","doi-asserted-by":"crossref","unstructured":"Irwin GR. 1956 Onset of fast crack propagation in high strength steel and aluminum alloys. In Sagamore Research Conference Proceedings vol. 2 pp. 289\u2013305. New York NY: Springer.","DOI":"10.21236\/AD0099305"},{"key":"e_1_3_5_4_2","doi-asserted-by":"publisher","DOI":"10.1201\/9781420058215"},{"key":"e_1_3_5_5_2","volume-title":"Crack control: using fracture theory to create tough new materials","author":"Kendall K","year":"2021","unstructured":"Kendall K. 2021 Crack control: using fracture theory to create tough new materials. 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