{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T10:00:44Z","timestamp":1783504844268,"version":"3.55.0"},"reference-count":161,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T00:00:00Z","timestamp":1689292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education","award":["01.0.13.00\/1.02.001\/SUBB.MKKM.23.002"],"award-info":[{"award-number":["01.0.13.00\/1.02.001\/SUBB.MKKM.23.002"]}]},{"name":"Ministry of Science and Higher Education","award":["02.0.20.00\/1.02.001, SUBB. BKTK. 23.002"],"award-info":[{"award-number":["02.0.20.00\/1.02.001, SUBB. BKTK. 23.002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>The article is a continuation of two previous review papers on the fracture mechanism of structural metals through the nucleation, growth and coalescence of voids. In the present paper, the literature on the numerical modelling of void nucleation and development has been reviewed. The scope of the work does not include porous material models and their numerical implementation. As part of the discussion on void initiation, nucleation around second phase particles and nucleation as an effect of the discontinuity of the crystal structure were discussed separately. The basic void cell models, finite element method (FEM) models of periodically distributed particles\/voids and models based on the results of the observations of the actual microstructure of materials have been characterised. Basic issues related to the application of the cohesive approach in void nucleation modelling have been considered. A separate issue is the characteristics of atomistic simulations and peridynamic modelling, which have been developed in recent years. Numerical approaches to modelling the growth and coalescence of voids are described, with particular emphasis on the influence of the stress state and strain localisation. Basic conclusions from the simulation are presented, pointing to the contribution of FEM modelling to the understanding of microstructural phenomena leading to ductile fracture.<\/jats:p>","DOI":"10.3390\/ma16144998","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T08:40:06Z","timestamp":1689324006000},"page":"4998","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Voids Development in Metals: Numerical Modelling"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6586-5019","authenticated-orcid":false,"given":"Wiktor","family":"Wci\u015blik","sequence":"first","affiliation":[{"name":"Faculty of Civil Engineering and Architecture, Kielce University of Technology, 25-314 Kielce, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4430-3989","authenticated-orcid":false,"given":"Sebastian","family":"Lipiec","sequence":"additional","affiliation":[{"name":"Faculty of Mechatronics and Mechanical Engineering, Kielce University of Technology, 25-314 Kielce, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1016\/j.actamat.2007.12.058","article-title":"High-Density Three-Dimensional Mapping of Internal Strain by Tracking Microstructural Features","volume":"56","author":"Kobayashi","year":"2008","journal-title":"Acta Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1177\/0309324711408975","article-title":"Three-Dimensional Strain Mapping Using in Situ X-ray Synchrotron Microtomography","volume":"46","author":"Toda","year":"2011","journal-title":"J. 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