RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zh. Vychisl. Mat. Mat. Fiz.:
Year:
Volume:
Issue:
Page:
Find






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


Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 9, Pages 1640–1668 (Mi zvmmf4938)  

This article is cited in 2 scientific papers (total in 2 papers)

Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity

V. A. Garanzha

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia

Abstract: It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that BallТs approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.

Key words: elasticity equations, polyconvexity, entropy solutions, quasi-isometric mappings.

Full text: PDF file (362 kB)
References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2010, 50:9, 1561–1587

Bibliographic databases:

UDC: 519.634
Received: 28.12.2009
Revised: 27.04.2010

Citation: V. A. Garanzha, “Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity”, Zh. Vychisl. Mat. Mat. Fiz., 50:9 (2010), 1640–1668; Comput. Math. Math. Phys., 50:9 (2010), 1561–1587

Citation in format AMSBIB
\Bibitem{Gar10}
\by V.~A.~Garanzha
\paper Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2010
\vol 50
\issue 9
\pages 1640--1668
\mathnet{http://mi.mathnet.ru/zvmmf4938}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2760642}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010CMMPh..50.1561G}
\elib{https://elibrary.ru/item.asp?id=15241673}
\transl
\jour Comput. Math. Math. Phys.
\yr 2010
\vol 50
\issue 9
\pages 1561--1587
\crossref{https://doi.org/10.1134/S0965542510090095}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000282212600009}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77957106382}


Linking options:
  • http://mi.mathnet.ru/eng/zvmmf4938
  • http://mi.mathnet.ru/eng/zvmmf/v50/i9/p1640

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Garanzha V.A. Kudryavtseva L.N. Utyuzhnikov S.V., “Variational Method For Untangling and Optimization of Spatial Meshes”, J. Comput. Appl. Math., 269 (2014), 24–41  crossref  mathscinet  zmath  isi  elib  scopus
    2. Dobrev V., Knup P., Kolev T., Mittal K., Tomov V., “The Target-Matrix Optimization Paradigm For High-Order Meshes”, SIAM J. Sci. Comput., 41:1 (2019), B50–B68  crossref  mathscinet  zmath  isi  scopus
  • ∆урнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
    Number of views:
    This page:478
    Full text:102
    References:39
    First page:7

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020