Abstract
In this paper, we extend the asymptotic analysis in (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013) performed, in the framework of small strains, on a structure consisting of two linearly elastic bodies connected by a thin soft nonlinear Kelvin–Voigt viscoelastic adhesive layer to the case in which the total mass of the layer remains strictly positive as its thickness tends to zero.
We obtain convergence results by means of a nonlinear version of Trotter’s theory of approximation of semigroups acting on variable Hilbert spaces. Differently from the limit models derived in (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013), in the present analysis the dynamic effects on the surface to which the layer shrinks do not disappear. Thus, the limiting behavior of the remaining bodies is described not only in terms of their displacements on the contact surface, but also by an additional variable that keeps track of the dynamics in the adhesive layer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonetti, E., Bonfanti, G., Rossi, R.: Global existence for a contact problem with adhesion. Math. Methods Appl. Sci. 31, 1029–1064 (2008)
Bonetti, E., Bonfanti, G., Lebon, F., Rizzoni, R.: A model of imperfect interface with damage. Meccanica 52(8), 1911–1922 (2017)
Bonetti, E., Bonfanti, G., Lebon, F.: Derivation of imperfect interface models coupling damage and temperature. Comput. Math. Appl. 11, 2906–2916 (2019)
Brezis, H.: Opérateurs Maximaux-Monotones et Semi-Groupes de Contraction dans les Espaces de Hilbert. North-Holland, Amsterdam (1972)
Dumont, S., Lebon, F., Rizzoni, R.: An asymptotic approach to the adhesion of thin stiff films. Mech. Res. Commun. 58, 24–35 (2014)
Freddi, L., Paroni, R., Roubìček, T., Zanini, C.: Quasistatic delamination models for Kirchhoff-Love plates. Z. Angew. Math. Mech. 91, 845–865 (2011)
Freddi, L., Roubìček, T., Zanini, C.: Quasistatic delamination of sandwich-like Kirchhoff-Love plates. J. Elast. 113, 219–250 (2013)
Iosifescu, O., Licht, C., Michaille, G.: Nonlinear boundary conditions in Kirchhoff-Love plate theory. J. Elast. 96, 57–79 (2009)
Klarbring, A.: Derivation of the adhesively bonded joints by the asymptotic expansion method. Int. J. Eng. Sci. 29, 493–512 (1991)
Kočvara, M., Mielke, A., Roubíček, T.: A rate-independent approach to the delamination problem. Math. Mech. Solids 11, 423–447 (2006)
Lebon, F., Rizzoni, R.: Asymptotic behavior of a hard thin linear elastic interphase: an energy approach. Int. J. Solids Struct. 48, 441–449 (2011)
Licht, C.: Comportement asymptotique d’une bande dissipative mince de faible rigidité. C. R. Acad. Sci. Paris, Sér. I Math. 317, 429–433 (1993)
Licht, C., Michaille, G.: A modelling of elastic bonded joints. Adv. Math. Sci. Appl. 7, 711–740 (1997)
Licht, C., Weller, T.: Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete Contin. Dyn. Syst., Ser. S 12, 1709–1741 (2019)
Licht, C., Léger, A., Orankitjaroen, S., Ould Khaoua, A.: Dynamics of elastic bodies connected by a thin soft viscoelastic layer. J. Math. Pures Appl. 99, 685–703 (2013)
Mielke, A., Roubíček, T., Thomas, M.: From damage to delamination in nonlinear elastic materials at small strains. J. Elast. 109, 235–273 (2012)
Roubíček, T., Scardia, L., Zanini, C.: Quasistatic delamination problem. Contin. Mech. Thermodyn. 21, 223–235 (2009)
Trotter, H.F.: Approximation of semi-groups of operators. Pac. J. Math. 8, 887–919 (1958)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was partially carried out during a visit of C.L. at the Sezione di Matematica of DICATAM (University of Brescia), also supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Rights and permissions
About this article
Cite this article
Bonetti, E., Bonfanti, G., Licht, C. et al. Dynamics of Two Linearly Elastic Bodies Connected by a Heavy Thin Soft Viscoelastic Layer. J Elast 141, 75–107 (2020). https://doi.org/10.1007/s10659-020-09776-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-020-09776-7
Keywords
- Dimension reduction
- Kelvin-Voigt viscoelasticity
- Trotter’s theory of approximation of semigroups
- Variational convergence