{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T04:59:13Z","timestamp":1777093153279,"version":"3.51.4"},"reference-count":50,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2021,8,30]],"date-time":"2021-08-30T00:00:00Z","timestamp":1630281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>In this work, a small-strain phase-field model is presented, which is able to predict crack propagation in systems with anisotropic brittle and ductile constituents. To model the anisotropic brittle crack propagation, an anisotropic critical energy release rate is used. The brittle constituents behave linear-elastically in a transversely isotropic manner. Ductile crack growth is realised by a special crack degradation function, depending on the accumulated plastic strain, which is calculated by following the J2-plasticity theory. The mechanical jump conditions are applied in solid-solid phase transition regions. The influence of the relevant model parameters on a crack propagating through a planar brittle-ductile interface, and furthermore a crack developing in a domain with a single anisotropic brittle ellipsoid, embedded in a ductile matrix, is investigated. We demonstrate that important properties concerning the mechanical behaviour of grey cast iron, such as the favoured growth of cracks along the graphite lamellae and the tension\u2013compression load asymmetry of the stress\u2013strain response, are covered by the model. The behaviour is analysed on the basis of a simulation domain consisting of three differently oriented elliptical inclusions, embedded in a ductile matrix, which is subjected to tensile and compressive load. The material parameters used correspond to graphite lamellae and pearlite.<\/jats:p>","DOI":"10.3390\/ma14174956","type":"journal-article","created":{"date-parts":[[2021,8,30]],"date-time":"2021-08-30T22:49:51Z","timestamp":1630363791000},"page":"4956","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Phase-Field Model for the Simulation of Brittle-Anisotropic and Ductile Crack Propagation in Composite Materials"],"prefix":"10.3390","volume":"14","author":[{"given":"Christoph","family":"Herrmann","sequence":"first","affiliation":[{"name":"Institute of Applied Materials (IAM-CMS), Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, D-76131 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9250-2918","authenticated-orcid":false,"given":"Daniel","family":"Schneider","sequence":"additional","affiliation":[{"name":"Institute of Applied Materials (IAM-CMS), Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, D-76131 Karlsruhe, Germany"},{"name":"Institute of Digital Materials Science (IDM), Karlsruhe University of Applied Sciences, Moltkestrasse 30, D-76133 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6821-7263","authenticated-orcid":false,"given":"Ephraim","family":"Schoof","sequence":"additional","affiliation":[{"name":"Institute of Applied Materials (IAM-CMS), Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, D-76131 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0401-0105","authenticated-orcid":false,"given":"Felix","family":"Schwab","sequence":"additional","affiliation":[{"name":"Groupe de Physique des Mat\u00e9riaux, University of Rouen, CNRS, Avenue de l\u2019Universit\u00e9, F-76800 Saint Etienne du Rouvray, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Britta","family":"Nestler","sequence":"additional","affiliation":[{"name":"Institute of Applied Materials (IAM-CMS), Karlsruhe Institute of Technology (KIT), Kaiserstrasse 12, D-76131 Karlsruhe, Germany"},{"name":"Institute of Digital Materials Science (IDM), Karlsruhe University of Applied Sciences, Moltkestrasse 30, D-76133 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0013-7944(01)00083-2","article-title":"The cohesive zone model: Advantages, limitations and challenges","volume":"69","author":"Elices","year":"2002","journal-title":"Eng. 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