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Encyclopedia of Continuum Mechanics pp 1–7Cite as

Direct Method of Calculus of Variations in Elasticity

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Definition

Theorems on the existence of minimizers for functionals defined on Banach spaces, and related approximation methods, applied to minimization problems arising in elasticity theory.

Introduction

Variational methods are a powerful tool in elasticity and in fact the only known approach able to guarantee sufficient generality in the treatment of problems arising in hyperelasticity (see the corresponding entry), in the asymptotic derivation of two-dimensional elastic models (Ciarlet, 1988, 1997), and in many other cases in continuum mechanics (Pedregal, 2000; Fonseca, 1987; dell’Isola and Placidi, 2011). Variational problems take usually the form of a minimization problem that can be described as follows: we search the minimum of a functional F(u) defined on a subset S of a Banach space B (i.e., a normed, complete vector space) and taking values in [−∞, +∞]. Direct method provides sufficient conditions on S, B, and F for the existence of a minimizer \(\tilde {u}\in S\). The main...

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Notes

  1. 1.

    We recall that compactness is a topological invariant.

  2. 2.

    A set is said to be compact if each of its open covers admits a finite subcover.

  3. 3.

    In French literature (e.g., in Ciarlet 1988), it is sometimes used the more logical name infimizing sequence.

  4. 4.

    Here B∗ denotes the dual space of B, i.e., the vector space of continuous linear functionals on B, and 〈a, b∗〉 denotes the action of the continuous linear functional b∗∈ B∗ on a ∈ B. Note that in a Hilbert space H (e.g., the Sobolev spaces Hk := Wk, 2), 〈a, b∗〉 can be expressed by means of an inner product 〈a, b〉 with b ∈ H.

  5. 5.

    By this we mean the convergence induced by the usual functional norm \(\vert \vert v^* \vert \vert _{B^*}:=\sup \langle u,v^*\rangle :\vert \vert u \vert \vert =1\).

  6. 6.

    In fact, the same can be said for generic integral functionals.

  7. 7.

    A real function f defined on a topological space X is called lower semicontinuous if the set {x ∈ X : f(x) > α} is an open real set for every \(\alpha \in \mathbb {R}\). A well-known example of a functional which is lower semicontinuous but not continuous is the arc-length functional, i.e., \(L(f):=\int _a^b \sqrt {1+(f'(x))^2} dx\). It is geometrically evident that, in the C0 norm, L is not continuous, as we can find functions arbitrarily close to a given rectifiable function whose graph has arbitrarily large length. On the other hand, it is easy to see that it is lower semicontinuous. This simple example shows how natural the concept of lower semicontinuity is. Clearly a similar result holds true for the existence of maxima of upper semicontinuous functions.

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Author information

Authors and Affiliations

  1. DIMA, University of Rome La Sapienza, Rome, Italy

    Alessandro Della Corte

  2. International Research Center M&MoCS, University of L’Aquila, L’Aquila, Italy

    Alessandro Della Corte & Francesco dell’Isola

  3. DISG, University of Rome La Sapienza, Rome, Italy

    Francesco dell’Isola

Authors
  1. Alessandro Della Corte
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  2. Francesco dell’Isola
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Corresponding author

Correspondence to Alessandro Della Corte .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. DISG, University of Rome La Sapienza, Rome, Italy

    Francesco dell’Isola

  2. International Research Center M&MoCS, L’Aquila, Italy

    Francesco dell’Isola

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Corte, A.D., dell’Isola, F. (2018). Direct Method of Calculus of Variations in Elasticity. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_174-1

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  • DOI: https://doi.org/10.1007/978-3-662-53605-6_174-1

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