Abstract
Nonlinear viscoelastic media are investigated within the framework of the group analysis. By requiring the governing system of equations to be invariant under the dilatation group, some classes of material response functions are characterized. Furthermore several sets of self-similar solutions to the model of interest are determined. Finally for the Maxwell media wave propagation into a non constant state described by a similarity solution is studied and the occurrence of a shock wave is considered.
Riassunto
Si considera un modello per i mezzi viscoelastici non lineari nell'ambito dell'analisi gruppale. Richiedendo che il sistema di equazioni di interesse sia invariante rispetto al gruppo di dilatazione, si caratterizzano delle classi di funzioni di risposta per il materiale. Inoltre si determinano diverse soluzioni di similarità. Successivamente nel caso dei mezzi di Maxwell si studia la propagazione di onde di discontinuità in uno stato non costante descritto da una soluzione di similarità. Infine si discutono le condizioni per l'esistenza di un tempo critico a cui un'onda d'urto può formarsi.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Lebon, C. Perez-Garcia and J. Casas-Vazquez,On a new thermodynamic description of viscoelastic materials, Physica137A, 531 (1986).
G. Lebon, C. Perez-Garcia and J. Casas-Vazquez,On the thermodynamic foundations of viscoelasticity, preprint.
W. F. Ames,Nonlinear partial differential equations in engineering, Vol. II, Academic Press, New York 1972.
G. W. Bluman and J. D. Cole,Similarity methods for differential equations, Springer, Berlin 1974.
L. V. Osiannikov,Group analysis of differential equations, English ed., Academic Press, New York 1982.
D. Fusco,Group analysis and constitutive laws for fluid filled elastic tubes, Int. J. Non-linear Mech.19, 565–574 (1984).
C. Curró and D. Fusco,Invariant solutions and constitutive laws for a nonlinear elastic rod of variable cross-section, J. Appl. Math. Phys. (ZAMP)37, 244–255 (1987).
A. Donato and D. Fusco,Wave features and infinitesimal group analysis for a second order quasilinear equation in conservative form, Int. J. Nonlinear Mech.22, 37–46 (1987).
A. Donato,Similarity analysis and non-linear wave propagation, Int. J. Nonlinear Mech.22, 307–314 (1987).
W. F. Ames and A. Donato,On the evolution of weak discontinuities in a state characterized by similarity solutions, Int. J. Nonlinear Mech.23, 167–174 (1988).
A. Jeffrey,Quasilinear hyperbolic systems and waves, Pitman Publ., London 1976.
G. Boillat,La propagation des ondes, Gauthier-Villars, Paris 1965.
G. Boillat and T. Ruggeri,On the evolution law of weak discontinuities for hyperbolic quasi-linear systems, Wave Motion1, 149–152 (1979).
H. M. Cekirge and E. Varley,Large amplitude waves in bounded media: I. Reflection and transmission of large amplitude shockless pulses at an interface, Phil. Trans. R. Soc. London Ser. A,273, 261–313 (1973).
J. Y. Kazakia and E. Varley,Large amplitude waves in bounded media: II. The deformation of an impulsively loaded slab: the first reflection, Phil. Trans. R. Soc. London Ser. A,277, 191–237 (1974).
B. D. Coleman, M. E. Gurtin, R. I. Herrera and C. Truesdell,Wave propagation in dissipative materials. A reprint of five memoirs, Springer-Verlag Inc., New York 1965.
M. Renardy, Hrusa and Nohel,Mathematical problems in viscoelasticity, Longman, Harlow 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fusco, D., Palumbo, A. Similarity solutions and model constitutive laws to viscoelasticity with application to nonlinear wave propagation. Z. angew. Math. Phys. 40, 78–92 (1989). https://doi.org/10.1007/BF00945311
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00945311