Summary
A group analysis approach is developed for the model describing a non linear elastic rod of variable cross-section. Thus, some sets of exact invariant solutions to the system of governing equations are obtained.
Meanwhile, via the group theoretic methods considered herein, possible functional forms of the stress-strain laws as well as the cross-section area are characterized.
Riassunto
In questo lavoro si considera, nell'ambito della teoria dei gruppi di trasformazioni infinitesime, il modello che descrive una corda elastica non lineare di sezione variabile. In tal∘modo è possibile ottenere delle classi di soluzioni esatte per il sistema differenziale in esame. Inoltre la richiesta di invarianza del modello considerato rispetto a gruppi di transformazione consente di caratterizzare delle classi di legami costitutivi per lo stress e l'area.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. F. Ames,Nonlinear partial differential equations in engineering, Vol. II. Academic Press, New York 1972.
I. V. Ovsiannikov,Group analysis of differential equations, Russian ed. Nauka (1978). English ed., Academic Press, New York 1982.
G. W. Bluman and J. D. Cole,Similarity methods for differential equations. Springer, Berlin 1974.
R. Chand, D. T. Davy and W. F. Ames,On the similarity solutions of wave propagation for a general class of non-linear dissipative materials. Int. J. Non-linear Mech.11, 191–205 (1976).
W. F. Ames and I. Suliciu,Some exact solutions for wave propagation in viscoelastic, viscoplastic and electrical transmission lines. Int. J. Non-linear Mech.17, 223–230 (1982).
D. Fusco,Group analysis and constitutive laws for fluid filled elastic tubes. Int. J. Non-linear Mech.19, 565–574 (1984).
A. Donato and D. Fusco,Wave features and infinitesimal group analysis for a second order quasilinear equation in conservative form, to appear.
W. F. Ames and R. J. Lohner,Group properties of u tt=[f(u)u x]x. Int. J. Non-linear Mech.16, 439–447 (1981).
M. Torrisi and A. Valenti,Group properties and invariant solutions for infinitesimal transformations of a nonlinear wave equation. Int. J. Non-linear Mech.20, 135–144 (1985).
A. Jeffrey,Acceleration wave propagation in hyperelastic rods of variable cross-section. Wave Motion4, 173–180 (1982).
H. M. Cekirge and E. Varley,Large amplitude waves in bounded media: Reflection and transmission of large amplitude shockless pulses at interface. Phil. Trans. Roy. Soc. London Ser. A273, 261–313 (1973).
J. F. Bell,The physics of large deformations of crystalline solids. Springer Verlag, Berlin 1968.
C. Rogers and W. F. Shadwick,Bäcklund transformations and their applications. Academic Press, New York 1982.
N. Cristescu,Dynamic plasticity. North Holland, Amsterdam 1967.
G. B. Whitham,Linear and nonlinear waves. John Wiley, New York 1974.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Currò, C., Fusco, D. Invariant solutions and constitutive laws for a nonlinear elastic rod of variable cross-section. Z. angew. Math. Phys. 37, 244–255 (1986). https://doi.org/10.1007/BF00945085
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00945085