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In the case of a large load step, the solution of the local nonlinear system to compute the plastic multiplier may not be attained. To overcome this problem, a sub-stepping scheme is generally used to improve the convergence of the local nonlinear system solution strategy. Nevertheless, the complexity of the tangent operator of the sub-stepping scheme is high. This complicates the use of Newton\u2013Raphson algorithm to obtain global quadratic convergence. In this paper, a formulation for consistent tangent operator is developed for implicit sub-stepping integration for the modified Cam-Clay model and unified Clay and Sand model. This formulation is highly efficient and can be used with problem involving arbitrary large load step, such as tunnel simulation.<\/jats:p>","DOI":"10.1007\/s00366-026-02309-1","type":"journal-article","created":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T02:58:10Z","timestamp":1775012290000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Implicit sub-stepping scheme for critical state soil models"],"prefix":"10.1007","volume":"42","author":[{"given":"Hoang-Giang","family":"Bui","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jelena","family":"Nin\u00edc","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G\u00fcnther","family":"Meschke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,1]]},"reference":[{"key":"2309_CR1","doi-asserted-by":"publisher","first-page":"1737","DOI":"10.1002\/(SICI)1097-0207(19960530)39:10<1737::AID-NME927>3.0.CO;2-5","volume":"39","author":"AJ Abbo","year":"1996","unstructured":"Abbo AJ, Sloan SW (1996) An automatic load stepping algorithm with error control. 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