Abstract
Well-posedness in the sense of Hadamard is examined for the Cauchy problem of a one-dimensional hyperbolic system of partial differential equations describing the longitudinal vibrations of a Maxwell-type viscoelastic rod with constant cross section. Some properties of the system and its solutions, including the conservation of modified “energy,” the finite propagation speed, and the dispersion and dissipation of solutions, are discussed.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. In 10 Vols., Vol. 7: Theory of Elasticity (Fizmatlit, Moscow, 2003) [in Russian].
A. R. Rzhanitsyn, Cree** Theory (Stroiizdat, Moscow, 1968) [in Russian].
A. R. Rzhanitsyn, Some Mechanical Problems on Time Deformable Systems (Gos. Izd. Tekhn.-Tekh. Lit., Moscow, 1949) [in Russian].
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed. (Soc. for Industrial and Application Mathematics, Philadelphia, 2004).
L. C. Evans, Partial Differential Equations, 2nd ed. (Am. Math. Soc., Providence, RI, 2010).
V. I. Korzyuk and J. V. Rudzko, “Classical solution of the initial-value problem for a one-dimensional quasilinear wave equation,” Dokl. Nats. Akad. Nauk Belarusi 67 (1), 14–19 (2023).
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, Boca Raton, FL, 2002).
V. I. Korzyuk and J. V. Rudzko, “A particular solution of a problem for a system of equations from mechanics with nonsmooth Cauchy conditions,” Izv. Nats. Akad. Nauk Belarusi, Ser. Fiz.-Mat. Nauk 58 (3), 300–311 (2022).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct thisparticular research were obtained.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by M. Shmatikov
Publisher’s Note.
Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Korzyuk, V.I., Rudzko, J.V. & Kolyachko, V.V. Problem of Longitudinal Vibrations of a Maxwell-Type Viscoelastic Rod. Mech. Solids 58, 2631–2639 (2023). https://doi.org/10.3103/S0025654423070129
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654423070129