Abstract
The well-known Lame problem, posed in 1852, involves solving the static equilibrium of a parallelepiped with free side surfaces subjected to action of opposite end forces. In this article, the same problem for a more complicated case of impacts of end forces is considered.
An exact analytical solution of this problem is found.
Emphasizing the particular difficulty of solving this problem, Lamé, in his book “Leçons sur la thorie mathematique de Ielasticite des corps solides” (Paris, 1852), wrote: “C’est une sorte d’engine aussi digne d’exercer la sagasite des analystes que le fameux problem des trios corps de la Mécanique celeste”,—“This is a kind of drive, as worthy of training the clairvoyance of analysts as the famous three-body problem of celestial mechanics.” At that time, this problem was the subject of a prize from the Paris Academy of Sciences, that was intended for the one who solved the Lamé problem. Despite this, to date, no solution has been found even for a static case of this problem, not to mention the complicated version of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. A. Medick, “Extensional waves in elastic bars of rectangular cross sections,” J. Acoust. Soc. Am. 43 (1), 152–161 (1968). https://doi.org/10.1121/1.1910744
A. E. Vovk, V. V. Gudkov, T. V. Levtchenkova, and V. V. Tyutekin, “Modes of solid rectangular waveguide,” Akust. Zh. 26 (3), 356–363 (1980).
W. B. Fraser, “Stress wave propagation in rectangular bars,” Int. J. Solids Struct. 5 (4), 379–397 (1969). https://doi.org/10.1016/0020-7683(69)90020-1
E. Volterra and M. Asce, “Dispersion of longitudinal waves,” J. Eng. Mech. 83 (3), 13–22 (1957). https://doi.org/10.1061/JMCEA3.0000032
P. Hertelendy, “An approximate theory governing simmetric motions of elastic rods of rectangular or square cross section,” J. Appl. Mech. 35 (2), 333–341 (1968). https://doi.org/10.1115/1.3601200
K. Tanaka and Y. Iwahashi, “Dispersion relations of elastic wave in bars of rectangular cross sections,” Bull. JSME 20 (146), 922–929 (1977). https://doi.org/10.1299/jsme1958.20.922
K. Tanaka and Y. Iwahashi, “Longitudinal impact of a semi-infinite rectangular bar,” Bull. JSME 21 (156), 980–985 (1978). https://doi.org/10.1299/jsme1958.21.980
N. B. Rasulova, “Propagation of waves in a prismatic beam subjected to axial loads,” Mech. Solids 32 (6), 149–152 (1997).
N. B. Rassoulova, “On dynamic of bar of rectangular cross section,” J. Appl. Mech. 68 (4), 662–666 (2001). https://doi.org/10.1115/1.1352063
N. B. Rasulova and G. R. Shamilova, “Stress wave propagation in a rectangular bar,” Mech. Solids 51, 494–500 (2016). https://doi.org/10.3103/S0025654416040117
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by A. Borimova
About this article
Cite this article
Rasulova, N.B., Mahmudzade, T.M. Solution of the Dynamic Lame Problem. Mech. Solids 58, 1545–1550 (2023). https://doi.org/10.3103/S0025654423600137
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654423600137