Remarks on Saint Venant’s principle and application to the matching problem.

*(English)*Zbl 0738.73013
Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Vol. IX, Paris/Fr. 1985-86, Pitman Res. Notes Math. Ser. 181, 362-375 (1988).

From the introduction: As the statement of the principle was not very exact (and it was unproven for long time),other interpretations of it are possible, in particular for non-slender bodies acted upon by forces with small support. In this case, the principle is wrong. To my knowledge, the first to notice this was R. von Mises [Bull. Am. Math. Soc. 51, 555–562 (1945; Zbl 0063.04019)]. Next, E. Sternberg [J. Appl. Math. 11, 393–402 (1954; Zbl 0057.40004)] and H. B. Keller [Q. Appl. Math. 22, 293–304 (1965; Zbl 0151.36003)] gave new statements and partial proofs of cases where the principle holds under additional symmetry hypotheses.

In this lecture I give an asymptotic expansion of the solution in the case of nonslender bodies when the forces are applied in a support tending to zero in two cases: body forces in interior points of \(\Omega\) and surface forces on \(\partial\Omega\). The solution is obtained on the grounds of a moment expansion [see E. Sanchez-Palencia, Res. Notes Math. 70, 309–325 (1982; Zbl 0505.35020) for other problems handled by this method]. It turns out that the Saint Venant principle holds in this case when considering the complete moment matrix \(M_{ij}\) instead of the mechanical moment \(\underline{M}\), and this allows us to deal with general data without the symmetry hypotheses of Keller above mentioned (*). Moreover, the resultant and the first order moments appear at different orders of the asymptotic expansion, and any number of terms may be obtained by considering higher order moments. An estimate of the error is obtained in Sobolev spaces \(H^{-s}(\Omega)\) with negative order [J. L. Lions and E. Magenes, Problèmes aux limites nonhomogènes et applications. Vols. 1, 2. Paris: Dunod (1968; Zbl 0165.10801)].

Sect. 4 contains some remarks on the problem of the matching of a slender cylinder and a “flat body”. This problem is far from solved here, but some indications are given on the use of the inner and outer expansion method (see W. Eckhaus [Asymptotic analysis of singular perturbations. Amsterdam etc.: North-Holland (1979; Zbl 0421.34057)] for instance); both forms of the Saint Venant’s principle are necessary and we show that if the slender body is submitted to torsion, the action on the flat body is highly dependent on the local geometric form of the junction region.

[For the entire collection see Zbl 0653.00012.]

In this lecture I give an asymptotic expansion of the solution in the case of nonslender bodies when the forces are applied in a support tending to zero in two cases: body forces in interior points of \(\Omega\) and surface forces on \(\partial\Omega\). The solution is obtained on the grounds of a moment expansion [see E. Sanchez-Palencia, Res. Notes Math. 70, 309–325 (1982; Zbl 0505.35020) for other problems handled by this method]. It turns out that the Saint Venant principle holds in this case when considering the complete moment matrix \(M_{ij}\) instead of the mechanical moment \(\underline{M}\), and this allows us to deal with general data without the symmetry hypotheses of Keller above mentioned (*). Moreover, the resultant and the first order moments appear at different orders of the asymptotic expansion, and any number of terms may be obtained by considering higher order moments. An estimate of the error is obtained in Sobolev spaces \(H^{-s}(\Omega)\) with negative order [J. L. Lions and E. Magenes, Problèmes aux limites nonhomogènes et applications. Vols. 1, 2. Paris: Dunod (1968; Zbl 0165.10801)].

Sect. 4 contains some remarks on the problem of the matching of a slender cylinder and a “flat body”. This problem is far from solved here, but some indications are given on the use of the inner and outer expansion method (see W. Eckhaus [Asymptotic analysis of singular perturbations. Amsterdam etc.: North-Holland (1979; Zbl 0421.34057)] for instance); both forms of the Saint Venant’s principle are necessary and we show that if the slender body is submitted to torsion, the action on the flat body is highly dependent on the local geometric form of the junction region.

[For the entire collection see Zbl 0653.00012.]