---
_id: '64187'
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>
article_number: '10812865261420809'
author:
- first_name: Yingjie
  full_name: Zhan, Yingjie
  id: '93591'
  last_name: Zhan
- first_name: Ismail
  full_name: Caylak, Ismail
  last_name: Caylak
- first_name: Richard
  full_name: Ostwald, Richard
  id: '106876'
  last_name: Ostwald
  orcid: 0000-0003-2147-8444
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Enrico
  full_name: Barth, Enrico
  last_name: Barth
- first_name: Eckart
  full_name: Uhlmann, Eckart
  last_name: Uhlmann
citation:
  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>
  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>
  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} }'
  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>.
  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>.'
  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>.
  short: Y. Zhan, I. Caylak, R. Ostwald, R. Mahnken, E. Barth, E. Uhlmann, Mathematics
    and Mechanics of Solids (2026).
date_created: 2026-02-17T11:21:00Z
date_updated: 2026-02-17T11:22:49Z
department:
- _id: '9'
- _id: '952'
- _id: '321'
doi: 10.1177/10812865261420809
language:
- iso: eng
publication: Mathematics and Mechanics of Solids
publication_identifier:
  issn:
  - 1081-2865
  - 1741-3028
publication_status: published
publisher: SAGE Publications
quality_controlled: '1'
status: public
title: A fully implicit mean-field damage formulation with consistent linearization
  at large deformations
type: journal_article
user_id: '85414'
year: '2026'
...
---
_id: '39412'
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>
article_number: '108128652211080'
article_type: original
author:
- first_name: Jacky
  full_name: Cresson, Jacky
  last_name: Cresson
- first_name: Khaled
  full_name: Hariz-Belgacem, Khaled
  last_name: Hariz-Belgacem
citation:
  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>
  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>
  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}
    }'
  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>.
  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>.'
  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>.
  short: J. Cresson, K. Hariz-Belgacem, Mathematics and Mechanics of Solids (2022).
date_created: 2023-01-24T10:28:32Z
date_updated: 2023-07-27T16:07:04Z
doi: 10.1177/10812865221108094
keyword:
- Mechanics of Materials
- General Materials Science
- General Mathematics
language:
- iso: eng
publication: Mathematics and Mechanics of Solids
publication_identifier:
  issn:
  - 1081-2865
  - 1741-3028
publication_status: published
publisher: SAGE Publications
status: public
title: About the structure of the discrete and continuous Eringen’s nonlocal elastica
type: journal_article
user_id: '98857'
year: '2022'
...
---
_id: '39400'
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>
article_number: '108128652211080'
author:
- first_name: Jacky
  full_name: Cresson, Jacky
  last_name: Cresson
- first_name: Khaled
  full_name: Hariz Belgacem, Khaled
  id: '98857'
  last_name: Hariz Belgacem
citation:
  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>
  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>
  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}
    }'
  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>.
  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>.'
  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>.
  short: J. Cresson, K. Hariz Belgacem, Mathematics and Mechanics of Solids (2022).
date_created: 2023-01-24T10:18:34Z
date_updated: 2023-08-01T11:52:17Z
doi: 10.1177/10812865221108094
keyword:
- Mechanics of Materials
- General Materials Science
- General Mathematics
language:
- iso: eng
publication: Mathematics and Mechanics of Solids
publication_identifier:
  issn:
  - 1081-2865
  - 1741-3028
publication_status: published
publisher: SAGE Publications
status: public
title: About the structure of the discrete and continuous Eringen’s nonlocal elastica
type: journal_article
user_id: '98857'
year: '2022'
...