The formulation of the Lamé–Gadolin problem on the equilibrium of a hollow sphere consisting of two parts, one of which is preliminarily deformed, is given for finite deformations. An analytical solution of the problem for the elastic potential of a special form is obtained. A numerical solution of the problem for an elastoplastic material is obtained. When modeling plastic deformations, the Drucker–Prager plasticity condition is used, and the plastic flow is described using an unassociated law.
Similar content being viewed by others
References
A. V. Gadolin, Artilleriiskii Zh., No. 12, 1033 (1861).
L. I. Sedov, A Course in Continuum Mechanics, Vol. 1, Wolters-Noordhoff, Groningen (1972).
V. I. Levitas, Int. J. Plast., 140, 102914 (2021); https://doi.org/10.1016/j.ijplas.2020.102914.
V. I. Levitas, Nat. Commun., 13, 6291 (2022); https://doi.org/10.1038/s41467-022-33802-y.
L. Tajčmanová, Y. Podladchikov, R. Powell, et al., J. Metamorph. Geol., 32, 195 (2014); https://doi.org/10.1111/jmg.12066.
X. Zhong, E. Moulas, and L. Tajcmanová, Solid Earth, 11, 223 (2020); 10.5194/se-11-223-2020.
K. M. Zingerman and V. A. Levin, J. Appl. Math. Mech., 77, 235 (2013); https://doi.org/10.1016/j.jappmathmech.2013.07.016.
P. J. Blatz and W. L. Ko, Trans. Soc. Rheol., 6, 223 (1962).
F. D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A., 30, 244 (1944).
T. J. B. Holland and R. Powell, J. Metamorph. Geol., 16, 309 (1998); https://doi.org/10.1111/j.1525-1314.1998.00140.x.
E. Moulas, Y. Podladchikov, K. Zingerman, et al., Am. J. Sci., 323, 2 (2023); https://doi.org/10.2475/001c.68195.
V. A. Levin, Y. Y. Podladchikov, and K. M. Zingerman, Eur. J. Mech. A/Solids, 90, 104345 (2021); https://doi.org/10.1016/j.euromechsol.2021.104345.
M. M. Carroll, J. Elast., 20, 65 (1988); https://doi.org/10.1007/bf00042141.
D. C. Drucker and W. Prager, Q. Appl. Math., 10, 157 (1952).
J. C. Simo, Comput. Meth. Appl. Mech. Eng., 62, 169 (1987); https://doi.org/10.1016/0045-7825(87)90022-3.
A. Golovanov and L. Sultanov, Int. Appl. Mech., 41, 614 (2005); https://doi.org/10.1007/s10778-005-0129-x.
D. Komatitsch and J. Tromp, Geophys. J. Int., 139, 806 (1999); https://doi.org/10.1046/j.1365-246x.1999.00967.x.
A. Vershinin, Contin. Mech. Thermodyn. 35, 1245 (2023); https://doi.org/10.1007/s00161-022-01117-4.
L. Tajčmanová, Y. Podladchikov, E. Moulas, et al., Sci. Rep., 11, 18740 (2021); https://doi.org/10.1038/s41598-021-97568-x.
V. I. Levitas, V. F. Nesterenko, and M. A. Meyers, Acta Mater., 46, 5947 (1998); https://doi.org/10.1016/S1359-6454(98)00214-6.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Levin, V.A., Zingerman, K.M. & Vershinin, A.V. Approaches to the Solution of the Lamé–Gadolin Problem for a Composite Hollow Ball Made of Nonlinear Elastic and Elasto-Plastic Materials Under Superimposed Finite Deformations. Russ Phys J 66, 1060–1068 (2023). https://doi.org/10.1007/s11182-023-03044-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11182-023-03044-6