Computer Research and Modeling
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Computer Research and Modeling:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Computer Research and Modeling, 2021, Volume 13, Issue 3, Pages 541–555
DOI: https://doi.org/10.20537/2076-7633-2021-13-3-541-555
(Mi crm900)
 

This article is cited in 1 scientific paper (total in 1 paper)

MODELS IN PHYSICS AND TECHNOLOGY

Variational principle for shape memory solids under variable external forces and temperatures

V. A. Grachev, Yu. S. Nayshtut

Academy of Building and Architecture, Samara State Technical University, 194 Molodogvardeiskaya st., Samara, 443001, Russia
Full-text PDF (413 kB) Citations (1)
References:
Abstract: The quasistatic deformation problem for shape memory alloys is reviewed within the phenomenological mechanics of solids without microphysics analysis. The phenomenological approach is based on comparison of two material deformation diagrams. The first diagram corresponds to the active proportional loading when the alloy behaves as an ideal elastoplastic material; the residual strain is observed after unloading. The second diagram is relevant to the case when the deformed sample is heated to a certain temperature for each alloy. The initial shape is restored: the reverse distortion matches deformations on the first diagram, except for the sign. Because the first step of distortion can be described with the variational principle, for which the existence of the generalized solutions is proved under arbitrary loading, it becomes clear how to explain the reverse distortion within the slightly modified theory of plasticity. The simply connected surface of loading needs to be replaced with the doubly connected one, and the variational principle needs to be updated with two laws of thermodynamics and the principle of orthogonality for thermodynamic forces and streams. In this case it is not difficult to prove the existence of solutions either. The successful application of the theory of plasticity under the constant temperature causes the need to obtain a similar result for a more general case of variable external forces and temperatures. The paper studies the ideal elastoplastic von Mises model at linear strain rates. Taking into account hardening and arbitrary loading surface does not cause any additional difficulties.
The extended variational principle of the Reissner type is defined. Together with the laws of thermal plasticity it enables to prove the existence of the generalized solutions for three-dimensional bodies made of shape memory materials. The main issue to resolve is a challenge to choose a functional space for the rates and deformations of the continuum points. The space of bounded deformation, which is the main instrument of the mathematical theory of plasticity, serves this purpose in the paper. The proving process shows that the choice of the functional spaces used in the paper is not the only one. The study of other possible problem settings for the extended variational principle and search for regularity of generalized solutions seem an interesting challenge for future research.
Keywords: solids, variational principle, shape memory materials, thermal plasticity, space of bounded deformation, generalized solutions.
Received: 01.03.2021
Accepted: 27.04.2021
Document Type: Article
UDC: 519.6; 539.3
Language: Russian
Citation: V. A. Grachev, Yu. S. Nayshtut, “Variational principle for shape memory solids under variable external forces and temperatures”, Computer Research and Modeling, 13:3 (2021), 541–555
Citation in format AMSBIB
\Bibitem{GraNay21}
\by V.~A.~Grachev, Yu.~S.~Nayshtut
\paper Variational principle for shape memory solids under variable external forces and temperatures
\jour Computer Research and Modeling
\yr 2021
\vol 13
\issue 3
\pages 541--555
\mathnet{http://mi.mathnet.ru/crm900}
\crossref{https://doi.org/10.20537/2076-7633-2021-13-3-541-555}
Linking options:
  • https://www.mathnet.ru/eng/crm900
  • https://www.mathnet.ru/eng/crm/v13/i3/p541
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Computer Research and Modeling
    Statistics & downloads:
    Abstract page:82
    Full-text PDF :21
    References:23
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024