---
_id: '60124'
article_number: '113465'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  last_name: Mahnken
citation:
  ama: Westermann H, Mahnken R. Thermodynamically consistent phase-field modeling
    for polycrystalline multi-phase continua. <i>International Journal of Solids and
    Structures</i>. Published online 2025. doi:<a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">10.1016/j.ijsolstr.2025.113465</a>
  apa: Westermann, H., &#38; Mahnken, R. (2025). Thermodynamically consistent phase-field
    modeling for polycrystalline multi-phase continua. <i>International Journal of
    Solids and Structures</i>, Article 113465. <a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">https://doi.org/10.1016/j.ijsolstr.2025.113465</a>
  bibtex: '@article{Westermann_Mahnken_2025, title={Thermodynamically consistent phase-field
    modeling for polycrystalline multi-phase continua}, DOI={<a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">10.1016/j.ijsolstr.2025.113465</a>},
    number={113465}, journal={International Journal of Solids and Structures}, publisher={Elsevier
    BV}, author={Westermann, Hendrik and Mahnken, Rolf}, year={2025} }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “Thermodynamically Consistent Phase-Field
    Modeling for Polycrystalline Multi-Phase Continua.” <i>International Journal of
    Solids and Structures</i>, 2025. <a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">https://doi.org/10.1016/j.ijsolstr.2025.113465</a>.
  ieee: 'H. Westermann and R. Mahnken, “Thermodynamically consistent phase-field modeling
    for polycrystalline multi-phase continua,” <i>International Journal of Solids
    and Structures</i>, Art. no. 113465, 2025, doi: <a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">10.1016/j.ijsolstr.2025.113465</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “Thermodynamically Consistent Phase-Field
    Modeling for Polycrystalline Multi-Phase Continua.” <i>International Journal of
    Solids and Structures</i>, 113465, Elsevier BV, 2025, doi:<a href="https://doi.org/10.1016/j.ijsolstr.2025.113465">10.1016/j.ijsolstr.2025.113465</a>.
  short: H. Westermann, R. Mahnken, International Journal of Solids and Structures
    (2025).
date_created: 2025-06-03T19:48:12Z
date_updated: 2025-06-06T14:24:50Z
doi: 10.1016/j.ijsolstr.2025.113465
language:
- iso: eng
publication: International Journal of Solids and Structures
publication_identifier:
  issn:
  - 0020-7683
publication_status: published
publisher: Elsevier BV
status: public
title: Thermodynamically consistent phase-field modeling for polycrystalline multi-phase
  continua
type: journal_article
user_id: '60816'
year: '2025'
...
---
_id: '52233'
abstract:
- lang: eng
  text: ELDIRK methods are defined to have an <jats:italic>Explicit Last</jats:italic>
    stage in the general Butcher array of <jats:italic>Diagonal Implicit Runge-Kutta</jats:italic>
    methods, with the consequence, that no additional system of equations must be
    solved, compared to the embedded RK method. Two general formulations for second-
    and third-order ELDIRK methods have been obtained recently in Mahnken [21] with
    specific schemes,  e.g. for the embedded implicit Euler method, the embedded trapezoidal-rule
    and the embedded Ellsiepen method. In the first part of this paper, we investigate
    some general stability characteristics of ELDIRK methods, and it will be shown
    that the above specific RK schemes are not A-stable. Therefore, in the second
    part, the above-mentioned general formulations are used for further stability
    investigations, with the aim to construct new second- and third-order ELDIRK methods
    which simultaneously are A-stable. Two numerical examples are concerned with the
    curing for a thermosetting material and phase-field RVE modeling for crystallinity
    and orientation. The numerical results confirm the theoretical results on convergence
    order and stability.
author:
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
citation:
  ama: Mahnken R, Westermann H. Construction of A-stable explicit last-stage diagonal
    implicit Runge–Kutta (ELDIRK) methods. <i>Computational Mechanics</i>. Published
    online 2024. doi:<a href="https://doi.org/10.1007/s00466-024-02442-y">10.1007/s00466-024-02442-y</a>
  apa: Mahnken, R., &#38; Westermann, H. (2024). Construction of A-stable explicit
    last-stage diagonal implicit Runge–Kutta (ELDIRK) methods. <i>Computational Mechanics</i>.
    <a href="https://doi.org/10.1007/s00466-024-02442-y">https://doi.org/10.1007/s00466-024-02442-y</a>
  bibtex: '@article{Mahnken_Westermann_2024, title={Construction of A-stable explicit
    last-stage diagonal implicit Runge–Kutta (ELDIRK) methods}, DOI={<a href="https://doi.org/10.1007/s00466-024-02442-y">10.1007/s00466-024-02442-y</a>},
    journal={Computational Mechanics}, publisher={Springer Science and Business Media
    LLC}, author={Mahnken, Rolf and Westermann, Hendrik}, year={2024} }'
  chicago: Mahnken, Rolf, and Hendrik Westermann. “Construction of A-Stable Explicit
    Last-Stage Diagonal Implicit Runge–Kutta (ELDIRK) Methods.” <i>Computational Mechanics</i>,
    2024. <a href="https://doi.org/10.1007/s00466-024-02442-y">https://doi.org/10.1007/s00466-024-02442-y</a>.
  ieee: 'R. Mahnken and H. Westermann, “Construction of A-stable explicit last-stage
    diagonal implicit Runge–Kutta (ELDIRK) methods,” <i>Computational Mechanics</i>,
    2024, doi: <a href="https://doi.org/10.1007/s00466-024-02442-y">10.1007/s00466-024-02442-y</a>.'
  mla: Mahnken, Rolf, and Hendrik Westermann. “Construction of A-Stable Explicit Last-Stage
    Diagonal Implicit Runge–Kutta (ELDIRK) Methods.” <i>Computational Mechanics</i>,
    Springer Science and Business Media LLC, 2024, doi:<a href="https://doi.org/10.1007/s00466-024-02442-y">10.1007/s00466-024-02442-y</a>.
  short: R. Mahnken, H. Westermann, Computational Mechanics (2024).
date_created: 2024-03-03T13:23:28Z
date_updated: 2024-03-19T12:14:07Z
department:
- _id: '154'
- _id: '321'
doi: 10.1007/s00466-024-02442-y
keyword:
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics
- Mechanical Engineering
- Ocean Engineering
- Computational Mechanics
language:
- iso: eng
publication: Computational Mechanics
publication_identifier:
  issn:
  - 0178-7675
  - 1432-0924
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta
  (ELDIRK) methods
type: journal_article
user_id: '335'
year: '2024'
...
---
_id: '48464'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>Initial value problems can be solved
    efficiently by means of Runge–Kutta algorithms with adaptive step size control.
    Diagonally implicit Runge–Kutta (DIRK) methods are the most popular class among
    the diverse family of Runge–Kutta algorithms. In this paper, the novel class of
    low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods
    are explored, which combine implicit schemes with an additional explicit evaluation
    as an explicit last stage. ELDIRK Butcher tableaus are used to control embedded
    RK methods to obtain solutions of different orders. The lower‐order solution is
    obtained by classical implicit RK stages and the higher‐order solution is obtained
    by additional explicit evaluation. As a result, a significant reduction in computational
    cost is achieved by skipping the iterative solution of nonlinear systems for the
    additional step. The examination of the heat problem and the use of the innovative
    Butcher tableau in the finite‐element method are the main contributions of this
    work. Thus, it is possible to establish adaptive step size control for the new
    low‐order embedded methods based on an empirical method for error estimation.
    Two‐dimensional simulations are used to show an appropriate algorithm for the
    ELDIRK schemes. The new Runge–Kutta schemes' predictions of higher‐order convergence
    are confirmed, and their successful outcomes are illustrated.</jats:p>
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  ama: Westermann H, Mahnken R. Numerical investigations of new low‐order explicit
    last stage diagonal implicit Runge–Kutta schemes with the finite‐element method.
    <i>PAMM</i>. 2023;23(2). doi:<a href="https://doi.org/10.1002/pamm.202300071">10.1002/pamm.202300071</a>
  apa: Westermann, H., &#38; Mahnken, R. (2023). Numerical investigations of new low‐order
    explicit last stage diagonal implicit Runge–Kutta schemes with the finite‐element
    method. <i>PAMM</i>, <i>23</i>(2). <a href="https://doi.org/10.1002/pamm.202300071">https://doi.org/10.1002/pamm.202300071</a>
  bibtex: '@article{Westermann_Mahnken_2023, title={Numerical investigations of new
    low‐order explicit last stage diagonal implicit Runge–Kutta schemes with the finite‐element
    method}, volume={23}, DOI={<a href="https://doi.org/10.1002/pamm.202300071">10.1002/pamm.202300071</a>},
    number={2}, journal={PAMM}, publisher={Wiley}, author={Westermann, Hendrik and
    Mahnken, Rolf}, year={2023} }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “Numerical Investigations of New
    Low‐order Explicit Last Stage Diagonal Implicit Runge–Kutta Schemes with the Finite‐element
    Method.” <i>PAMM</i> 23, no. 2 (2023). <a href="https://doi.org/10.1002/pamm.202300071">https://doi.org/10.1002/pamm.202300071</a>.
  ieee: 'H. Westermann and R. Mahnken, “Numerical investigations of new low‐order
    explicit last stage diagonal implicit Runge–Kutta schemes with the finite‐element
    method,” <i>PAMM</i>, vol. 23, no. 2, 2023, doi: <a href="https://doi.org/10.1002/pamm.202300071">10.1002/pamm.202300071</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “Numerical Investigations of New Low‐order
    Explicit Last Stage Diagonal Implicit Runge–Kutta Schemes with the Finite‐element
    Method.” <i>PAMM</i>, vol. 23, no. 2, Wiley, 2023, doi:<a href="https://doi.org/10.1002/pamm.202300071">10.1002/pamm.202300071</a>.
  short: H. Westermann, R. Mahnken, PAMM 23 (2023).
date_created: 2023-10-25T10:46:57Z
date_updated: 2023-11-07T14:34:44Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202300071
intvolume: '        23'
issue: '2'
keyword:
- Electrical and Electronic Engineering
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
publication: PAMM
publication_identifier:
  issn:
  - 1617-7061
  - 1617-7061
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Numerical investigations of new low‐order explicit last stage diagonal implicit
  Runge–Kutta schemes with the finite‐element method
type: journal_article
user_id: '335'
volume: 23
year: '2023'
...
---
_id: '48465'
article_number: '116545'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  ama: Westermann H, Mahnken R. On the accuracy, stability and computational efficiency
    of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK) for the
    adaptive solution of phase-field problems. <i>Computer Methods in Applied Mechanics
    and Engineering</i>. 2023;418. doi:<a href="https://doi.org/10.1016/j.cma.2023.116545">10.1016/j.cma.2023.116545</a>
  apa: Westermann, H., &#38; Mahnken, R. (2023). On the accuracy, stability and computational
    efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK)
    for the adaptive solution of phase-field problems. <i>Computer Methods in Applied
    Mechanics and Engineering</i>, <i>418</i>, Article 116545. <a href="https://doi.org/10.1016/j.cma.2023.116545">https://doi.org/10.1016/j.cma.2023.116545</a>
  bibtex: '@article{Westermann_Mahnken_2023, title={On the accuracy, stability and
    computational efficiency of explicit last-stage diagonally implicit Runge–Kutta
    methods (ELDIRK) for the adaptive solution of phase-field problems}, volume={418},
    DOI={<a href="https://doi.org/10.1016/j.cma.2023.116545">10.1016/j.cma.2023.116545</a>},
    number={116545}, journal={Computer Methods in Applied Mechanics and Engineering},
    publisher={Elsevier BV}, author={Westermann, Hendrik and Mahnken, Rolf}, year={2023}
    }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “On the Accuracy, Stability and
    Computational Efficiency of Explicit Last-Stage Diagonally Implicit Runge–Kutta
    Methods (ELDIRK) for the Adaptive Solution of Phase-Field Problems.” <i>Computer
    Methods in Applied Mechanics and Engineering</i> 418 (2023). <a href="https://doi.org/10.1016/j.cma.2023.116545">https://doi.org/10.1016/j.cma.2023.116545</a>.
  ieee: 'H. Westermann and R. Mahnken, “On the accuracy, stability and computational
    efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK)
    for the adaptive solution of phase-field problems,” <i>Computer Methods in Applied
    Mechanics and Engineering</i>, vol. 418, Art. no. 116545, 2023, doi: <a href="https://doi.org/10.1016/j.cma.2023.116545">10.1016/j.cma.2023.116545</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “On the Accuracy, Stability and Computational
    Efficiency of Explicit Last-Stage Diagonally Implicit Runge–Kutta Methods (ELDIRK)
    for the Adaptive Solution of Phase-Field Problems.” <i>Computer Methods in Applied
    Mechanics and Engineering</i>, vol. 418, 116545, Elsevier BV, 2023, doi:<a href="https://doi.org/10.1016/j.cma.2023.116545">10.1016/j.cma.2023.116545</a>.
  short: H. Westermann, R. Mahnken, Computer Methods in Applied Mechanics and Engineering
    418 (2023).
date_created: 2023-10-25T10:47:23Z
date_updated: 2023-11-07T14:34:56Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1016/j.cma.2023.116545
intvolume: '       418'
keyword:
- Computer Science Applications
- General Physics and Astronomy
- Mechanical Engineering
- Mechanics of Materials
- Computational Mechanics
language:
- iso: eng
publication: Computer Methods in Applied Mechanics and Engineering
publication_identifier:
  issn:
  - 0045-7825
publication_status: published
publisher: Elsevier BV
quality_controlled: '1'
status: public
title: On the accuracy, stability and computational efficiency of explicit last-stage
  diagonally implicit Runge–Kutta methods (ELDIRK) for the adaptive solution of phase-field
  problems
type: journal_article
user_id: '335'
volume: 418
year: '2023'
...
---
_id: '44891'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  ama: Westermann H, Mahnken R. A thermodynamic framework for the phase‐field approach
    considering carbide precipitation during phase transformations. <i>PAMM</i>. 2023;22(1).
    doi:<a href="https://doi.org/10.1002/pamm.202200080">10.1002/pamm.202200080</a>
  apa: Westermann, H., &#38; Mahnken, R. (2023). A thermodynamic framework for the
    phase‐field approach considering carbide precipitation during phase transformations.
    <i>PAMM</i>, <i>22</i>(1). <a href="https://doi.org/10.1002/pamm.202200080">https://doi.org/10.1002/pamm.202200080</a>
  bibtex: '@article{Westermann_Mahnken_2023, title={A thermodynamic framework for
    the phase‐field approach considering carbide precipitation during phase transformations},
    volume={22}, DOI={<a href="https://doi.org/10.1002/pamm.202200080">10.1002/pamm.202200080</a>},
    number={1}, journal={PAMM}, publisher={Wiley}, author={Westermann, Hendrik and
    Mahnken, Rolf}, year={2023} }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “A Thermodynamic Framework for the
    Phase‐field Approach Considering Carbide Precipitation during Phase Transformations.”
    <i>PAMM</i> 22, no. 1 (2023). <a href="https://doi.org/10.1002/pamm.202200080">https://doi.org/10.1002/pamm.202200080</a>.
  ieee: 'H. Westermann and R. Mahnken, “A thermodynamic framework for the phase‐field
    approach considering carbide precipitation during phase transformations,” <i>PAMM</i>,
    vol. 22, no. 1, 2023, doi: <a href="https://doi.org/10.1002/pamm.202200080">10.1002/pamm.202200080</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “A Thermodynamic Framework for the Phase‐field
    Approach Considering Carbide Precipitation during Phase Transformations.” <i>PAMM</i>,
    vol. 22, no. 1, Wiley, 2023, doi:<a href="https://doi.org/10.1002/pamm.202200080">10.1002/pamm.202200080</a>.
  short: H. Westermann, R. Mahnken, PAMM 22 (2023).
date_created: 2023-05-16T12:20:19Z
date_updated: 2023-05-16T12:21:15Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202200080
intvolume: '        22'
issue: '1'
keyword:
- Electrical and Electronic Engineering
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
publication: PAMM
publication_identifier:
  issn:
  - 1617-7061
  - 1617-7061
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: A thermodynamic framework for the phase‐field approach considering carbide
  precipitation during phase transformations
type: journal_article
user_id: '335'
volume: 22
year: '2023'
...
---
_id: '23794'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Alexander
  full_name: Reitz, Alexander
  id: '24803'
  last_name: Reitz
  orcid: 0000-0001-9047-467X
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Mirko
  full_name: Schaper, Mirko
  id: '43720'
  last_name: Schaper
- first_name: Olexandr
  full_name: Grydin, Olexandr
  id: '43822'
  last_name: Grydin
citation:
  ama: Westermann H, Reitz A, Mahnken R, Schaper M, Grydin O. Microstructure transformations
    in a press hardening steel during tailored thermo‐mechanical processing. <i>steel
    research international</i>. Published online 2022. doi:<a href="https://doi.org/10.1002/srin.202100346">10.1002/srin.202100346</a>
  apa: Westermann, H., Reitz, A., Mahnken, R., Schaper, M., &#38; Grydin, O. (2022).
    Microstructure transformations in a press hardening steel during tailored thermo‐mechanical
    processing. <i>Steel Research International</i>. <a href="https://doi.org/10.1002/srin.202100346">https://doi.org/10.1002/srin.202100346</a>
  bibtex: '@article{Westermann_Reitz_Mahnken_Schaper_Grydin_2022, title={Microstructure
    transformations in a press hardening steel during tailored thermo‐mechanical processing},
    DOI={<a href="https://doi.org/10.1002/srin.202100346">10.1002/srin.202100346</a>},
    journal={steel research international}, author={Westermann, Hendrik and Reitz,
    Alexander and Mahnken, Rolf and Schaper, Mirko and Grydin, Olexandr}, year={2022}
    }'
  chicago: Westermann, Hendrik, Alexander Reitz, Rolf Mahnken, Mirko Schaper, and
    Olexandr Grydin. “Microstructure Transformations in a Press Hardening Steel during
    Tailored Thermo‐mechanical Processing.” <i>Steel Research International</i>, 2022.
    <a href="https://doi.org/10.1002/srin.202100346">https://doi.org/10.1002/srin.202100346</a>.
  ieee: 'H. Westermann, A. Reitz, R. Mahnken, M. Schaper, and O. Grydin, “Microstructure
    transformations in a press hardening steel during tailored thermo‐mechanical processing,”
    <i>steel research international</i>, 2022, doi: <a href="https://doi.org/10.1002/srin.202100346">10.1002/srin.202100346</a>.'
  mla: Westermann, Hendrik, et al. “Microstructure Transformations in a Press Hardening
    Steel during Tailored Thermo‐mechanical Processing.” <i>Steel Research International</i>,
    2022, doi:<a href="https://doi.org/10.1002/srin.202100346">10.1002/srin.202100346</a>.
  short: H. Westermann, A. Reitz, R. Mahnken, M. Schaper, O. Grydin, Steel Research
    International (2022).
date_created: 2021-09-06T12:00:55Z
date_updated: 2023-04-27T16:39:38Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
- _id: '158'
doi: 10.1002/srin.202100346
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: 'https://doi.org/10.1002/srin.202100346 [Titel anhand dieser DOI in Citavi-Projekt
    übernehmen] '
oa: '1'
publication: steel research international
publication_identifier:
  issn:
  - 1611-3683
  - 1869-344X
publication_status: published
quality_controlled: '1'
status: public
title: Microstructure transformations in a press hardening steel during tailored thermo‐mechanical
  processing
type: journal_article
user_id: '43720'
year: '2022'
...
---
_id: '29089'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Alexander
  full_name: Reitz, Alexander
  id: '24803'
  last_name: Reitz
  orcid: 0000-0001-9047-467X
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Olexandr
  full_name: Grydin, Olexandr
  id: '43822'
  last_name: Grydin
- first_name: Mirko
  full_name: Schaper, Mirko
  id: '43720'
  last_name: Schaper
citation:
  ama: Westermann H, Reitz A, Mahnken R, Grydin O, Schaper M. Constitutive modeling
    of viscoplasticity including phase transformations for graded thermo‐mechanical
    processing. <i>PAMM</i>. Published online 2021. doi:<a href="https://doi.org/10.1002/pamm.202100041">10.1002/pamm.202100041</a>
  apa: Westermann, H., Reitz, A., Mahnken, R., Grydin, O., &#38; Schaper, M. (2021).
    Constitutive modeling of viscoplasticity including phase transformations for graded
    thermo‐mechanical processing. <i>PAMM</i>. <a href="https://doi.org/10.1002/pamm.202100041">https://doi.org/10.1002/pamm.202100041</a>
  bibtex: '@article{Westermann_Reitz_Mahnken_Grydin_Schaper_2021, title={Constitutive
    modeling of viscoplasticity including phase transformations for graded thermo‐mechanical
    processing}, DOI={<a href="https://doi.org/10.1002/pamm.202100041">10.1002/pamm.202100041</a>},
    journal={PAMM}, author={Westermann, Hendrik and Reitz, Alexander and Mahnken,
    Rolf and Grydin, Olexandr and Schaper, Mirko}, year={2021} }'
  chicago: Westermann, Hendrik, Alexander Reitz, Rolf Mahnken, Olexandr Grydin, and
    Mirko Schaper. “Constitutive Modeling of Viscoplasticity Including Phase Transformations
    for Graded Thermo‐mechanical Processing.” <i>PAMM</i>, 2021. <a href="https://doi.org/10.1002/pamm.202100041">https://doi.org/10.1002/pamm.202100041</a>.
  ieee: 'H. Westermann, A. Reitz, R. Mahnken, O. Grydin, and M. Schaper, “Constitutive
    modeling of viscoplasticity including phase transformations for graded thermo‐mechanical
    processing,” <i>PAMM</i>, 2021, doi: <a href="https://doi.org/10.1002/pamm.202100041">10.1002/pamm.202100041</a>.'
  mla: Westermann, Hendrik, et al. “Constitutive Modeling of Viscoplasticity Including
    Phase Transformations for Graded Thermo‐mechanical Processing.” <i>PAMM</i>, 2021,
    doi:<a href="https://doi.org/10.1002/pamm.202100041">10.1002/pamm.202100041</a>.
  short: H. Westermann, A. Reitz, R. Mahnken, O. Grydin, M. Schaper, PAMM (2021).
date_created: 2021-12-22T12:41:34Z
date_updated: 2023-01-24T12:54:26Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202100041
language:
- iso: eng
publication: PAMM
publication_identifier:
  issn:
  - 1617-7061
  - 1617-7061
publication_status: published
status: public
title: Constitutive modeling of viscoplasticity including phase transformations for
  graded thermo‐mechanical processing
type: journal_article
user_id: '335'
year: '2021'
...
---
_id: '24384'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  ama: Westermann H, Mahnken R. Constitutive modeling of dynamic recrystallization
    coupled to viscoplasticity. <i>PAMM</i>. Published online 2021. doi:<a href="https://doi.org/10.1002/pamm.202000186">10.1002/pamm.202000186</a>
  apa: Westermann, H., &#38; Mahnken, R. (2021). Constitutive modeling of dynamic
    recrystallization coupled to viscoplasticity. <i>PAMM</i>. <a href="https://doi.org/10.1002/pamm.202000186">https://doi.org/10.1002/pamm.202000186</a>
  bibtex: '@article{Westermann_Mahnken_2021, title={Constitutive modeling of dynamic
    recrystallization coupled to viscoplasticity}, DOI={<a href="https://doi.org/10.1002/pamm.202000186">10.1002/pamm.202000186</a>},
    journal={PAMM}, author={Westermann, Hendrik and Mahnken, Rolf}, year={2021} }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “Constitutive Modeling of Dynamic
    Recrystallization Coupled to Viscoplasticity.” <i>PAMM</i>, 2021. <a href="https://doi.org/10.1002/pamm.202000186">https://doi.org/10.1002/pamm.202000186</a>.
  ieee: 'H. Westermann and R. Mahnken, “Constitutive modeling of dynamic recrystallization
    coupled to viscoplasticity,” <i>PAMM</i>, 2021, doi: <a href="https://doi.org/10.1002/pamm.202000186">10.1002/pamm.202000186</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “Constitutive Modeling of Dynamic Recrystallization
    Coupled to Viscoplasticity.” <i>PAMM</i>, 2021, doi:<a href="https://doi.org/10.1002/pamm.202000186">10.1002/pamm.202000186</a>.
  short: H. Westermann, R. Mahnken, PAMM (2021).
date_created: 2021-09-14T11:44:19Z
date_updated: 2023-01-24T12:53:55Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1002/pamm.202000186
language:
- iso: eng
publication: PAMM
publication_identifier:
  issn:
  - 1617-7061
  - 1617-7061
publication_status: published
status: public
title: Constitutive modeling of dynamic recrystallization coupled to viscoplasticity
type: journal_article
user_id: '335'
year: '2021'
...
---
_id: '24390'
article_number: '102988'
author:
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
citation:
  ama: Mahnken R, Westermann H. A non-equilibrium thermodynamic framework for viscoplasticity
    incorporating dynamic recrystallization at large strains. <i>International Journal
    of Plasticity</i>. Published online 2021. doi:<a href="https://doi.org/10.1016/j.ijplas.2021.102988">10.1016/j.ijplas.2021.102988</a>
  apa: Mahnken, R., &#38; Westermann, H. (2021). A non-equilibrium thermodynamic framework
    for viscoplasticity incorporating dynamic recrystallization at large strains.
    <i>International Journal of Plasticity</i>, Article 102988. <a href="https://doi.org/10.1016/j.ijplas.2021.102988">https://doi.org/10.1016/j.ijplas.2021.102988</a>
  bibtex: '@article{Mahnken_Westermann_2021, title={A non-equilibrium thermodynamic
    framework for viscoplasticity incorporating dynamic recrystallization at large
    strains}, DOI={<a href="https://doi.org/10.1016/j.ijplas.2021.102988">10.1016/j.ijplas.2021.102988</a>},
    number={102988}, journal={International Journal of Plasticity}, author={Mahnken,
    Rolf and Westermann, Hendrik}, year={2021} }'
  chicago: Mahnken, Rolf, and Hendrik Westermann. “A Non-Equilibrium Thermodynamic
    Framework for Viscoplasticity Incorporating Dynamic Recrystallization at Large
    Strains.” <i>International Journal of Plasticity</i>, 2021. <a href="https://doi.org/10.1016/j.ijplas.2021.102988">https://doi.org/10.1016/j.ijplas.2021.102988</a>.
  ieee: 'R. Mahnken and H. Westermann, “A non-equilibrium thermodynamic framework
    for viscoplasticity incorporating dynamic recrystallization at large strains,”
    <i>International Journal of Plasticity</i>, Art. no. 102988, 2021, doi: <a href="https://doi.org/10.1016/j.ijplas.2021.102988">10.1016/j.ijplas.2021.102988</a>.'
  mla: Mahnken, Rolf, and Hendrik Westermann. “A Non-Equilibrium Thermodynamic Framework
    for Viscoplasticity Incorporating Dynamic Recrystallization at Large Strains.”
    <i>International Journal of Plasticity</i>, 102988, 2021, doi:<a href="https://doi.org/10.1016/j.ijplas.2021.102988">10.1016/j.ijplas.2021.102988</a>.
  short: R. Mahnken, H. Westermann, International Journal of Plasticity (2021).
date_created: 2021-09-14T12:02:07Z
date_updated: 2023-01-24T13:05:23Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.1016/j.ijplas.2021.102988
language:
- iso: eng
publication: International Journal of Plasticity
publication_identifier:
  issn:
  - 0749-6419
publication_status: published
quality_controlled: '1'
status: public
title: A non-equilibrium thermodynamic framework for viscoplasticity incorporating
  dynamic recrystallization at large strains
type: journal_article
user_id: '335'
year: '2021'
...
---
_id: '24388'
author:
- first_name: Hendrik
  full_name: Westermann, Hendrik
  id: '60816'
  last_name: Westermann
  orcid: 0000-0002-5034-9708
- first_name: Rolf
  full_name: Mahnken, Rolf
  id: '335'
  last_name: Mahnken
citation:
  ama: 'Westermann H, Mahnken R. On the Thermodynamics of Dynamic Recrystallization
    for Viscoplasticity at Large Strains. In: <i>14th WCCM-ECCOMAS Congress</i>. ;
    2021. doi:<a href="https://doi.org/10.23967/wccm-eccomas.2020.261">10.23967/wccm-eccomas.2020.261</a>'
  apa: Westermann, H., &#38; Mahnken, R. (2021). On the Thermodynamics of Dynamic
    Recrystallization for Viscoplasticity at Large Strains. <i>14th WCCM-ECCOMAS Congress</i>.
    <a href="https://doi.org/10.23967/wccm-eccomas.2020.261">https://doi.org/10.23967/wccm-eccomas.2020.261</a>
  bibtex: '@inproceedings{Westermann_Mahnken_2021, title={On the Thermodynamics of
    Dynamic Recrystallization for Viscoplasticity at Large Strains}, DOI={<a href="https://doi.org/10.23967/wccm-eccomas.2020.261">10.23967/wccm-eccomas.2020.261</a>},
    booktitle={14th WCCM-ECCOMAS Congress}, author={Westermann, Hendrik and Mahnken,
    Rolf}, year={2021} }'
  chicago: Westermann, Hendrik, and Rolf Mahnken. “On the Thermodynamics of Dynamic
    Recrystallization for Viscoplasticity at Large Strains.” In <i>14th WCCM-ECCOMAS
    Congress</i>, 2021. <a href="https://doi.org/10.23967/wccm-eccomas.2020.261">https://doi.org/10.23967/wccm-eccomas.2020.261</a>.
  ieee: 'H. Westermann and R. Mahnken, “On the Thermodynamics of Dynamic Recrystallization
    for Viscoplasticity at Large Strains,” 2021, doi: <a href="https://doi.org/10.23967/wccm-eccomas.2020.261">10.23967/wccm-eccomas.2020.261</a>.'
  mla: Westermann, Hendrik, and Rolf Mahnken. “On the Thermodynamics of Dynamic Recrystallization
    for Viscoplasticity at Large Strains.” <i>14th WCCM-ECCOMAS Congress</i>, 2021,
    doi:<a href="https://doi.org/10.23967/wccm-eccomas.2020.261">10.23967/wccm-eccomas.2020.261</a>.
  short: 'H. Westermann, R. Mahnken, in: 14th WCCM-ECCOMAS Congress, 2021.'
date_created: 2021-09-14T11:54:38Z
date_updated: 2023-01-24T13:06:01Z
department:
- _id: '9'
- _id: '154'
- _id: '321'
doi: 10.23967/wccm-eccomas.2020.261
language:
- iso: eng
publication: 14th WCCM-ECCOMAS Congress
publication_status: published
quality_controlled: '1'
status: public
title: On the Thermodynamics of Dynamic Recrystallization for Viscoplasticity at Large
  Strains
type: conference
user_id: '335'
year: '2021'
...