{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:45:49Z","timestamp":1760233549174,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,16]],"date-time":"2021-01-16T00:00:00Z","timestamp":1610755200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["19-51-52003"],"award-info":[{"award-number":["19-51-52003"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["108-2923-E-110-002-MY3"],"award-info":[{"award-number":["108-2923-E-110-002-MY3"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["AAAA-A20-120011690136-2"],"award-info":[{"award-number":["AAAA-A20-120011690136-2"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution\u2019s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill\u2019s quadratic yield criterion is adopted. Both elastic\/plastic and rigid\/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic\/plastic and rigid\/plastic solutions is negligible, except at the very beginning of the process. However, the rigid\/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.<\/jats:p>","DOI":"10.3390\/sym13010145","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"145","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7289-0856","authenticated-orcid":false,"given":"Sergei","family":"Alexandrov","sequence":"first","affiliation":[{"name":"Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elena","family":"Lyamina","sequence":"additional","affiliation":[{"name":"Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yeong-Maw","family":"Hwang","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Emamian, S., Awang, M., and Yusof, F. 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