{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,11]],"date-time":"2026-01-11T11:24:34Z","timestamp":1768130674087,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:00:00Z","timestamp":1767916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004421","name":"Warsaw University of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004421","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>The magnetoelastic effect in grain-oriented electrical steels arises from interactions between magnetocrystalline anisotropy, domain wall confinement, and applied mechanical stress. This presents a comprehensive model based on the minimization of total magnetic energy in a two-domain system separated by a 180\u00b0 Bloch wall. The model uniquely permits independent variation in the magnetization angle and external field direction, allowing accurate representation of energy competition among magnetostatic coupling, inter-domain interactions, and multi-component anisotropic confinement. The effective anisotropic wall energy incorporates isotropic, uniaxial, and six-fold crystallographic anisotropies modified by stress-induced terms. The Bloch wall position and the actual direction of magnetization are the variables that minimize the energy. Transformation to dimensionless variables enables efficient parameter identification via tri-division search. Experimental validation on M120-27s grain-oriented steel demonstrates that the model quantitatively reproduces stress-dependent 2D permeability tensors across arbitrary cutting orientations with very good quality, confirmed by determination coefficient R-squared exceeding 98%, which verifies the physical validity of the proposed model. This satisfactory agreement, together with the concept of anisotropic domain wall effective energy, represents a genuine novelty in the analysis of low-field magnetic permeability in grain-oriented electrical steels.<\/jats:p>","DOI":"10.3390\/ma19020274","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T10:35:52Z","timestamp":1767954952000},"page":"274","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modelling Stress-Dependent Magnetic Permeability Using Two-Domain Approach with an Effective Anisotropic Wall Energy in Grain-Oriented Electrical Steel"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0430-4805","authenticated-orcid":false,"given":"Tadeusz","family":"Szumiata","sequence":"first","affiliation":[{"name":"Faculty of Mechanical Engineering, Department of Physics, Kazimierz Pulaski Radom University, Stasieckiego 54, 26-600 Radom, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1214-1009","authenticated-orcid":false,"given":"Roman","family":"Szewczyk","sequence":"additional","affiliation":[{"name":"Faculty of Mechatronics, Warsaw University of Technology, sw. Andrzeja Boboli 8, 02-525 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3398-5796","authenticated-orcid":false,"given":"Pawe\u0142","family":"R\u0119kas","sequence":"additional","affiliation":[{"name":"Faculty of Mechatronics, Warsaw University of Technology, sw. Andrzeja Boboli 8, 02-525 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2513-952X","authenticated-orcid":false,"given":"Micha\u0142","family":"Nowicki","sequence":"additional","affiliation":[{"name":"Faculty of Mechatronics, Warsaw University of Technology, sw. Andrzeja Boboli 8, 02-525 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1080\/03019233.2016.1211122","article-title":"Grain-oriented electrical steels for power and distribution transformers","volume":"43","author":"Price","year":"2016","journal-title":"Ironmak. Steelmak."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"470","DOI":"10.2478\/jee-2018-0078","article-title":"Prediction of anisotropic properties of grain-oriented steels based on magnetic measurements","volume":"69","author":"Zurek","year":"2018","journal-title":"J. Electr. Eng."},{"key":"ref_3","unstructured":"(2025, September 02). 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