Advertisement

Compatibility of the Infinitesimal Deformation Tensor

  • Ciprian D. ComanEmail author
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 238)

Abstract

Our next main task will be to investigate how the displacement field can be recovered if the infinitesimal stress or strain tensors are known. Although the scope of this question turns out to be rather broad, the gist of what we are about to do is contained in the following basic scenario. Assume that \(D\subset \mathbb {E}^2\) is an open set and \(g_j\in C^1(D)\) are given scalar fields (\(j=1,2\)); we are interested in finding a new scalar field \(\phi \in C^2(D)\) such that \(\phi _{\,,\,1} = g_1\) and \(\phi _{\,,\,2}=g_2\) in D. Of course, if such a field exists then \(\phi _{\,,\,12}=\phi _{\,,\,21}\) or \(g_{1,\,2}=g_{2,\,1}\). This latter condition is necessary for the existence of a \(\phi \) having the foregoing stated properties. To put it differently, the obtained condition ensures the compatibility (or consistency) of the two equations satisfied by \(\phi \). It is fairly straightforward to show that under certain circumstances the condition is also sufficient. We note in passing that if \(\varvec{g}:=(g_1,\,g_2,\,0)\), the compatibility condition in this basic case can be cast as \(\varvec{\nabla }\wedge \varvec{g}= \varvec{0}\). The extension of these naive calculations to vector and tensor fields (as explained in the next sections) leads naturally to a discussion of the Beltrami–Michell equations and the concept of Weingarten-Volterra dislocation in multiply connected linearly elastic bodies.

Bibliography

  1. 1.
    Adams RA (1999) Calculus: a complete course. Addison-Wesley, Longman Ltd., OntarioGoogle Scholar
  2. 2.
    Salas SL, Hille E (1995) Calculus: one and several variables. Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Chou PC, Pagano NJ (1967) Elasticity: tensor, dyadic, and engineering approaches. D. Van Nostrand Company Inc, Princeton, New JerseyGoogle Scholar
  4. 4.
    Fraeijs de Veubeke BM (1979) A course in elasticity. Springer, New YorkGoogle Scholar
  5. 5.
    Gurtin ME (1972) Theory linear theory of elasticity. In: Truesdell C (ed) Handbuch der Physik. Springer, Berlin, pp 1–273Google Scholar
  6. 6.
    Love AEH (2011) A treatise of the mathematical theory of elasticity. Dover Publications, Mineola, New YorkGoogle Scholar
  7. 7.
    Malvern LE (1969) Introduction to the mechanics of a continuum medium. Prentice-Hall Inc, Englewood Cliffs, New JerseyGoogle Scholar
  8. 8.
    Nadeau G (1964) Introduction to elasticity. Holt, Rinehart and Winston Inc, New YorkGoogle Scholar
  9. 9.
    Teodosiu C (1982) Elastic models of crystal defects. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.School of Computing and EngineeringUniversity of HuddersfieldHuddersfieldUK

Personalised recommendations