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Another basic feature of our model is that there is conditional compatibility, entering the model as kinematic constraints and depending on the magnitude of an internal force associated with dislocations. Moreover, due to the kinematic constraints, the virtual velocities depend on the solutions of the problem. As a consequence, the variational formulation of the problem and the related mathematical analysis are neither standard nor straightforward. We adopt the strategy to invert the kinematic constraints through Green propagators, obtaining a system of integro-differential coupled equations. As a first mathematical step, we develop the analysis of the model in a simplified setting, i.e., considering the quasi-stationary version of the full system where we neglect inertia. In this context, we prove the existence of a global in time strong solution in three space dimensions for the system, employing techniques from PDEs and convex analysis, thus obtaining a novel breakthrough in the field of three-dimensional finite visco-elasticity described in terms of the stretch and rotation variables. We also study a limit problem, letting the magnitude of the internal force associated with dislocations tend to zero, in which case the deformation becomes incompatible and the equations take the form of a coupled system of PDEs. For the limit problem, we obtain global existence, uniqueness and continuous dependence from data in three space dimensions.<\/jats:p>","DOI":"10.1007\/s00332-025-10168-1","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T16:24:18Z","timestamp":1747844658000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Large Deformations in Terms of Stretch and Rotation and Global Solution to the Quasi-stationary Problem"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5706-3772","authenticated-orcid":false,"given":"A.","family":"Agosti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7921-5041","authenticated-orcid":false,"given":"P.","family":"Colli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-9739-1614","authenticated-orcid":false,"given":"M.","family":"Fr\u00e9mond","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"issue":"2","key":"10168_CR1","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1023\/A:1007653400249","volume":"56","author":"A Acharya","year":"1999","unstructured":"Acharya, A.: On compatibility conditions for the left Cauchy-Green deformation field in three dimensions. 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