[{"author":[{"id":"93591","full_name":"Zhan, Yingjie","last_name":"Zhan","first_name":"Yingjie"},{"first_name":"Ismail","last_name":"Caylak","full_name":"Caylak, Ismail"},{"first_name":"Richard","last_name":"Ostwald","orcid":"0000-0003-2147-8444","id":"106876","full_name":"Ostwald, Richard"},{"last_name":"Mahnken","full_name":"Mahnken, Rolf","id":"335","first_name":"Rolf"},{"last_name":"Barth","full_name":"Barth, Enrico","first_name":"Enrico"},{"last_name":"Uhlmann","full_name":"Uhlmann, Eckart","first_name":"Eckart"}],"date_created":"2026-02-17T11:21:00Z","publisher":"SAGE Publications","date_updated":"2026-02-17T11:22:49Z","doi":"10.1177/10812865261420809","title":"A fully implicit mean-field damage formulation with consistent linearization at large deformations","publication_identifier":{"issn":["1081-2865","1741-3028"]},"quality_controlled":"1","publication_status":"published","citation":{"short":"Y. Zhan, I. Caylak, R. Ostwald, R. Mahnken, E. Barth, E. Uhlmann, Mathematics and Mechanics of Solids (2026).","mla":"Zhan, Yingjie, et al. “A Fully Implicit Mean-Field Damage Formulation with Consistent Linearization at Large Deformations.” <i>Mathematics and Mechanics of Solids</i>, 10812865261420808, SAGE Publications, 2026, doi:<a href=\"https://doi.org/10.1177/10812865261420809\">10.1177/10812865261420809</a>.","bibtex":"@article{Zhan_Caylak_Ostwald_Mahnken_Barth_Uhlmann_2026, title={A fully implicit mean-field damage formulation with consistent linearization at large deformations}, DOI={<a href=\"https://doi.org/10.1177/10812865261420809\">10.1177/10812865261420809</a>}, number={10812865261420808}, journal={Mathematics and Mechanics of Solids}, publisher={SAGE Publications}, author={Zhan, Yingjie and Caylak, Ismail and Ostwald, Richard and Mahnken, Rolf and Barth, Enrico and Uhlmann, Eckart}, year={2026} }","apa":"Zhan, Y., Caylak, I., Ostwald, R., Mahnken, R., Barth, E., &#38; Uhlmann, E. (2026). A fully implicit mean-field damage formulation with consistent linearization at large deformations. <i>Mathematics and Mechanics of Solids</i>, Article 10812865261420808. <a href=\"https://doi.org/10.1177/10812865261420809\">https://doi.org/10.1177/10812865261420809</a>","ieee":"Y. Zhan, I. Caylak, R. Ostwald, R. Mahnken, E. Barth, and E. Uhlmann, “A fully implicit mean-field damage formulation with consistent linearization at large deformations,” <i>Mathematics and Mechanics of Solids</i>, Art. no. 10812865261420808, 2026, doi: <a href=\"https://doi.org/10.1177/10812865261420809\">10.1177/10812865261420809</a>.","chicago":"Zhan, Yingjie, Ismail Caylak, Richard Ostwald, Rolf Mahnken, Enrico Barth, and Eckart Uhlmann. “A Fully Implicit Mean-Field Damage Formulation with Consistent Linearization at Large Deformations.” <i>Mathematics and Mechanics of Solids</i>, 2026. <a href=\"https://doi.org/10.1177/10812865261420809\">https://doi.org/10.1177/10812865261420809</a>.","ama":"Zhan Y, Caylak I, Ostwald R, Mahnken R, Barth E, Uhlmann E. A fully implicit mean-field damage formulation with consistent linearization at large deformations. <i>Mathematics and Mechanics of Solids</i>. Published online 2026. doi:<a href=\"https://doi.org/10.1177/10812865261420809\">10.1177/10812865261420809</a>"},"year":"2026","department":[{"_id":"9"},{"_id":"952"},{"_id":"321"}],"user_id":"85414","_id":"64187","language":[{"iso":"eng"}],"article_number":"10812865261420809","publication":"Mathematics and Mechanics of Solids","type":"journal_article","status":"public","abstract":[{"lang":"eng","text":"<jats:p>Carbon fiber-reinforced plastics (CFRPs) have become increasingly significant in recent decades due to their remarkable mechanical properties and lightweight nature. This study aims to advance the understanding and simulation of CFRP behavior through the development of a hyperelastic-plastic-damage homogenization method combined with mean-field theory. The material responses of both the fiber and matrix are modeled using strain energy functions that account for damage evolution, while a complete linearization of the homogenization process is derived to ensure the consistent implementation of the Newton–Raphson iteration scheme in large deformation simulations. The innovative aspect of this work lies in the constitutive linearization for the hyperelastic-plastic-damage formulation within a mean-field homogenization framework, providing an efficient Newton algorithm for modeling the nonlinear behavior of CFRP. A failure criterion for the hyperelastic model of fibers is introduced, along with a damage saturation variable in rate form for the matrix, effectively capturing damage evolution. Through discrete formulations for the homogenization, the proposed model’s capability is demonstrated via three numerical examples and validated against experimental investigations, proving its effectiveness and reliability in simulating CFRP damage.</jats:p>"}]},{"status":"public","abstract":[{"lang":"eng","text":"<jats:p> The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica. </jats:p>"}],"publication":"Mathematics and Mechanics of Solids","type":"journal_article","language":[{"iso":"eng"}],"keyword":["Mechanics of Materials","General Materials Science","General Mathematics"],"article_type":"original","article_number":"108128652211080","user_id":"98857","_id":"39412","citation":{"ieee":"J. Cresson and K. Hariz-Belgacem, “About the structure of the discrete and continuous Eringen’s nonlocal elastica,” <i>Mathematics and Mechanics of Solids</i>, Art. no. 108128652211080, 2022, doi: <a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>.","chicago":"Cresson, Jacky, and Khaled Hariz-Belgacem. “About the Structure of the Discrete and Continuous Eringen’s Nonlocal Elastica.” <i>Mathematics and Mechanics of Solids</i>, 2022. <a href=\"https://doi.org/10.1177/10812865221108094\">https://doi.org/10.1177/10812865221108094</a>.","ama":"Cresson J, Hariz-Belgacem K. About the structure of the discrete and continuous Eringen’s nonlocal elastica. <i>Mathematics and Mechanics of Solids</i>. Published online 2022. doi:<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>","mla":"Cresson, Jacky, and Khaled Hariz-Belgacem. “About the Structure of the Discrete and Continuous Eringen’s Nonlocal Elastica.” <i>Mathematics and Mechanics of Solids</i>, 108128652211080, SAGE Publications, 2022, doi:<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>.","bibtex":"@article{Cresson_Hariz-Belgacem_2022, title={About the structure of the discrete and continuous Eringen’s nonlocal elastica}, DOI={<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>}, number={108128652211080}, journal={Mathematics and Mechanics of Solids}, publisher={SAGE Publications}, author={Cresson, Jacky and Hariz-Belgacem, Khaled}, year={2022} }","short":"J. Cresson, K. Hariz-Belgacem, Mathematics and Mechanics of Solids (2022).","apa":"Cresson, J., &#38; Hariz-Belgacem, K. (2022). About the structure of the discrete and continuous Eringen’s nonlocal elastica. <i>Mathematics and Mechanics of Solids</i>, Article 108128652211080. <a href=\"https://doi.org/10.1177/10812865221108094\">https://doi.org/10.1177/10812865221108094</a>"},"year":"2022","publication_identifier":{"issn":["1081-2865","1741-3028"]},"publication_status":"published","doi":"10.1177/10812865221108094","title":"About the structure of the discrete and continuous Eringen’s nonlocal elastica","author":[{"full_name":"Cresson, Jacky","last_name":"Cresson","first_name":"Jacky"},{"first_name":"Khaled","last_name":"Hariz-Belgacem","full_name":"Hariz-Belgacem, Khaled"}],"date_created":"2023-01-24T10:28:32Z","publisher":"SAGE Publications","date_updated":"2023-07-27T16:07:04Z"},{"publication":"Mathematics and Mechanics of Solids","type":"journal_article","abstract":[{"lang":"eng","text":"<jats:p> The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica. </jats:p>"}],"status":"public","_id":"39400","user_id":"98857","keyword":["Mechanics of Materials","General Materials Science","General Mathematics"],"article_number":"108128652211080","language":[{"iso":"eng"}],"publication_identifier":{"issn":["1081-2865","1741-3028"]},"publication_status":"published","year":"2022","citation":{"short":"J. Cresson, K. Hariz Belgacem, Mathematics and Mechanics of Solids (2022).","mla":"Cresson, Jacky, and Khaled Hariz Belgacem. “About the Structure of the Discrete and Continuous Eringen’s Nonlocal Elastica.” <i>Mathematics and Mechanics of Solids</i>, 108128652211080, SAGE Publications, 2022, doi:<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>.","bibtex":"@article{Cresson_Hariz Belgacem_2022, title={About the structure of the discrete and continuous Eringen’s nonlocal elastica}, DOI={<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>}, number={108128652211080}, journal={Mathematics and Mechanics of Solids}, publisher={SAGE Publications}, author={Cresson, Jacky and Hariz Belgacem, Khaled}, year={2022} }","apa":"Cresson, J., &#38; Hariz Belgacem, K. (2022). About the structure of the discrete and continuous Eringen’s nonlocal elastica. <i>Mathematics and Mechanics of Solids</i>, Article 108128652211080. <a href=\"https://doi.org/10.1177/10812865221108094\">https://doi.org/10.1177/10812865221108094</a>","ama":"Cresson J, Hariz Belgacem K. About the structure of the discrete and continuous Eringen’s nonlocal elastica. <i>Mathematics and Mechanics of Solids</i>. Published online 2022. doi:<a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>","ieee":"J. Cresson and K. Hariz Belgacem, “About the structure of the discrete and continuous Eringen’s nonlocal elastica,” <i>Mathematics and Mechanics of Solids</i>, Art. no. 108128652211080, 2022, doi: <a href=\"https://doi.org/10.1177/10812865221108094\">10.1177/10812865221108094</a>.","chicago":"Cresson, Jacky, and Khaled Hariz Belgacem. “About the Structure of the Discrete and Continuous Eringen’s Nonlocal Elastica.” <i>Mathematics and Mechanics of Solids</i>, 2022. <a href=\"https://doi.org/10.1177/10812865221108094\">https://doi.org/10.1177/10812865221108094</a>."},"publisher":"SAGE Publications","date_updated":"2023-08-01T11:52:17Z","author":[{"last_name":"Cresson","full_name":"Cresson, Jacky","first_name":"Jacky"},{"first_name":"Khaled","full_name":"Hariz Belgacem, Khaled","id":"98857","last_name":"Hariz Belgacem"}],"date_created":"2023-01-24T10:18:34Z","title":"About the structure of the discrete and continuous Eringen’s nonlocal elastica","doi":"10.1177/10812865221108094"}]