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The Flamant Problem

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Elasticity for Geotechnicians

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 204))

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Abstract

In 1892, the French mechanist Alfred-Aimé Flamant (1839–1914) posed and solved the equilibrium problem of a linearly elastic, isotropic and homogeneous body occupying a half-space acted upon by a perpendicular line load of constant magnitude per unit length and infinitely long support. In this chapter, we solve the Flamant Problem by a method different from his.

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Notes

  1. 1.

    See Sect. A.3.1 for an exposition of the classical Airy method to construct balanced and compatible plane stress fields.

  2. 2.

    This circumference has the Cartesian equation

    $$\begin{aligned} c^2=(x_1-c)^2+x_2^2=(\rho \cos \vartheta -c)^2+(\rho \sin \vartheta )^2= \rho ^2(1-(2c)\rho ^{-1}\cos \vartheta +c^2/\rho ^2). \end{aligned}$$

    In the geotechnical literature, this locus is called the pressure bulb.

  3. 3.

    A. Musesti, private communication, June 2004.

  4. 4.

    Here \(\widehat{{{\varvec{f}}}}(\mathcal{A},\mathcal{B})\) denotes the total contact force exerted by part \(\mathcal B\) over part \(\mathcal A\) along their common boundary.

  5. 5.

    Roughly speaking, the reduced boundary of a set—a measure-theoretic notion carefully introduced, e.g., in [1], p. 154—is the subset of all points of the topological boundary where a (inner) normal is well defined.

  6. 6.

    Private communication, August 2004.

  7. 7.

    In view of (2.54), the bounds (2.44)\(_2\) on \(\nu \), that descend from the positivity requirement for the density of elastic energy, translate into the following equivalent bounds for \(\nu _0\):

    $$\begin{aligned} -\frac{1}{2}<\nu _0<1. \end{aligned}$$
  8. 8.

    To find this result, (i) differentiate (4.32), and get:

    $$\begin{aligned} \frac{2(1-\nu _0)f}{\pi E_0}\cos \vartheta +\widehat{v}^{\prime \prime }(\vartheta )+\widehat{v}(\vartheta )=0; \end{aligned}$$

    (ii) recall that the well-known homogeneous equation associated with this second-order ODE admits a family of even solutions:

    $$\begin{aligned} \widehat{v}_h(\vartheta )=v_0\cos \vartheta ; \end{aligned}$$

    (iii) confirm that function

    $$\begin{aligned} \widehat{v}_p(\vartheta )=-\frac{(1-\nu _0)f}{\pi E_0}\vartheta \sin \vartheta \end{aligned}$$

    is a particular integral of the complete equation.

  9. 9.

    As a rule, when a boundary-value problem is formulated over an unbounded domain, the consequent lack of boundary conditions is compensated by posing on the solution a convenient set of conditions at infinity. This is not doable for the Flamant Problem, where the behavior at infinity of the elastic state is not tunable.

References

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Authors and Affiliations

  1. Department of Civil Engineering and Information Engineering, University of Rome Tor Vergata, via del Politecnico 1, 00173, Rome, Italy

    P. Podio-Guidugli

  2. Institute of Continuum Mechanics and Material Mechanics, Technical University of Hamburg, Eißendorfer Straße 42, 21073, Hamburg, Germany

    A. Favata

Authors
  1. P. Podio-Guidugli
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  2. A. Favata
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Correspondence to P. Podio-Guidugli .

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Podio-Guidugli, P., Favata, A. (2014). The Flamant Problem. In: Elasticity for Geotechnicians. Solid Mechanics and Its Applications, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-01258-2_4

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  • DOI: https://doi.org/10.1007/978-3-319-01258-2_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-01257-5

  • Online ISBN: 978-3-319-01258-2

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