Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Cover of the book

2016 | OriginalPaper | Chapter

1. Basic Equations of Continuum Mechanics

Authors : Jože Korelc, Peter Wriggers

Published in: Automation of Finite Element Methods

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter contains a summary of the continuum mechanics background that is needed for the finite element formulation of solid mechanics and structural problems. In detail the kinematical relations and the balance laws with their weak forms are described in this chapter.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
next chapter Automation of Research in Computational Modeling
Footnotes
1
It is not necessary that the body assumes the reference configuration at any time. Since the reference configuration can be chosen arbitrarily it is often assumed for practical purposes that the configuration of body B at the beginning of the deformation (initial configuration) is equivalent to the reference configuration. However there are applications like isoparametric interpolation functions within finite element formulations for which reference configurations will be defined which are purely fictitious.
 
2
In the following we will use indices \(a,b,c,\ldots \) to refer in index notation to the referential description while indices \(i,j,k,\ldots \) refer to the current or spatial configuration.
 
3
The Green–Lagrange strain measure is often used in nonlinear structural engineering applications. Mostly this strain measure is applied for problems with large displacements but small strains (e.g. within beam or shell theory) since it can describe arbitrary rigid body motions correctly.
 
4
For the Green–Lagrange strain tensor \(\varvec{E} = \frac{1}{2}\,(\,\varvec{F}^T \varvec{F} - \varvec{1}\,)\) this leads with \( \varvec{F}^T\varvec{F} = (\varvec{G}^i \otimes \varvec{g}_i) (\varvec{g}_k \otimes \varvec{G}^k)\) to
where \(g_{ik}\) is the so called metric tensor that defines the deformation of the solid. This result is equivalent to the pull back operation in (1.26).
 
5
The following formulation can also be applied to inelastic constitutive equations when the elastic stresses are computed from the strain energy function under the constraints of the inelastic evolution equations.
 
Literature
Chadwick, P. 1999. Continuum mechanics, concise theory and problems. Mineola: Dover Publications.
Ciarlet, P.G. 1988. Mathematical elasticity I: three-dimensional elasticity. Amsterdam: North-Holland.MATH
Flory, P. 1961. Thermodynamic relations for high elastic materials. Transactions of the Faraday Society 57: 829–838.MathSciNetCrossRef
Hencky, H. 1933. The elastic behaviour of vulcanized rubber. Journal of Applied Mechanics 1: 45–53.
Holzapfel, G.A. 2000. Nonlinear solid mechanics. Chichester: Wiley.MATH
Hudobivnik, B., and J. Korelc. 2016. Closed-form representation of matrix functions in the formulation of nonlinear material models. Finite Elements in Analysis and Design 111: 19–32.MathSciNetCrossRef
Johnson, C. 1987. Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press.MATH
Malvern, L.E. 1969. Introduction to the mechanics of a continuous medium. Englewood Cliffs: Prentice-Hall.MATH
Marsden, J.E., and T.J.R. Hughes. 1983. Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall.MATH
Ogden, R.W. 1984. Non-linear elastic deformations. Chichester: Ellis Horwood and John Wiley.MATH
Simo, J.C., R.L. Taylor, and K.S. Pister. 1985. Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Computer Methods in Applied Mechanics and Engineering 51: 177–208.MathSciNetCrossRefMATH
Truesdell, C., and W. Noll. 1965. In The nonlinear field theories of mechanics, ed. Flügge, S., Handbuch der Physik III/3 Berlin: Springer.
Truesdell, C., and R. Toupin. 1960. The classical field theories., Handbuch der Physik III/1 Berlin: Springer.CrossRef
Washizu, K. 1975. Variational methods in elasticity and plasticity, 2nd ed. Oxford: Pergamon Press.MATH
Metadata
Title
Basic Equations of Continuum Mechanics
Authors
Jože Korelc
Peter Wriggers
Copyright Year
2016
Book
Automation of Finite Element Methods

Print ISBN: 978-3-319-39003-1

Electronic ISBN: 978-3-319-39005-5

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-39005-5_1

Premium Partners

    Image Credits