Abstract
In this paper, we contribute to studying the issue of quasistatic limit in the context of Griffith’s theory by investigating a one-dimensional debonding model. It describes the evolution of a thin film partially glued to a rigid substrate and subjected to a vertical loading. Taking viscosity into account and under suitable assumptions on the toughness of the glue, we prove that, in contrast to what happens in the undamped case, dynamic solutions converge to the quasistatic one when inertia and viscosity go to zero, except for a possible discontinuity at the initial time. We then characterise the size of the jump by means of an asymptotic analysis of the debonding front.
Similar content being viewed by others
References
Agostiniani, V.: Second order approximations of quasistatic evolution problems in finite dimension. Discrete Contin. Dyn. Syst. 32, 1125–1167 (2012)
Almi, S., Dal Maso, G., Toader, R.: Quasi-static crack growth in hydraulic fracture. J. Nonlinear Anal. 109, 301–318 (2014)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon Press, Oxford (2000)
Bourdin, B., Francfort, G.A., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)
Burridge, R., Keller, J.B.: Peeling, slipping and cracking: some one-dimensional free boundary problems in mechanics. SIAM Rev. 20, 31–61 (1978)
Conti, M., Danese, V., Giorgi, C., Pata, V.: A model of viscoelasticity with time-dependent memory kernels. Am. J. Math. 140, 349–389 (2018)
Dal Maso, G., Scala, R.: Quasistatic evolution in perfect plasticity as limit of dynamic processes. J. Differ. Equ. 26, 915–954 (2014)
Dal Maso, G., Lazzaroni, G., Nardini, L.: Existence and uniqueness of dynamic evolutions for a peeling test in dimension one. J. Differ. Equ. 261, 4897–4923 (2016)
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 1, Physical Origins and Classical Methods. Springer, Berlin (1992)
Dumouchel, P.-E., Marigo, J.-J., Charlotte, M.: Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution. Contin. Mech. Thermodyn. 20, 1–19 (2008)
Freund, L.B.: Dynamic Fracture Mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1990)
Hellan, K.: Debond dynamics of an elastic strip-I. Timoshenko beam properties and steady motion. Int. J. Fract. 14, 91–100 (1978a)
Hellan, K.: Debond dynamics of an elastic strip-II. Simple transient motion. Int. J. Fract. 14, 173–184 (1978b)
Hellan, K.: Introduction to Fracture Mechanics. McGraw-Hill, New York (1984)
Lazzaroni, G., Nardini, L.: On the 1d wave equation in time-dependent domains and the problem of debond initiation. ESAIM: COCV. (2017). https://doi.org/10.1051/cocv/2019006
Lazzaroni, G., Nardini, L.: Analysis of a dynamic peeling test with speed-dependent toughness. SIAM J. Appl. Math. 78, 1206–1227 (2018a)
Lazzaroni, G., Nardini, L.: On the quasistatic limit of dynamic evolutions for a peeling test in dimension one. J. Nonlinear Sci. 28, 269–304 (2018b)
Lazzaroni, G., Bargellini, R., Dumouchel, P.-E., Marigo, J.-J.: On the role of kinetic energy during unstable propagation in a heterogeneous peeling test. Int. J. Fract. 175, 127–150 (2012)
Lazzaroni, G., Rossi, R., Thomas, M., Toader, R.: Rate-independent damage in thermo-viscoelastic materials with inertia. J. Dyn. Differ. Equ. 30, 1311–1364 (2018)
Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, vol. 193. Springer, New York (2015)
Misra, S., Gorain, G.C.: Stability of an inhomogeneous damped vibrating string. Appl. Appl. Math. 9, 435–448 (2014)
Nardini, L.: A note on the convergence of singularly perturbed second order potential-type equations. J. Dyn. Differ. Equ. 29, 783–797 (2017)
Riva, F.: A continuous dependence result for a dynamic debonding model in dimension one. Milan J. Math. (2019). https://doi.org/10.1007/s00032-019-00303-5
Riva, F., Nardini, L.: Existence and uniqueness of dynamic evolutions for a one-dimensional debonding model with damping. (2018). arXiv:1810.12006v2
Roubíček, T.: Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45, 101–126 (2013)
Scilla, G., Solombrino, F.: Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach. Nonlinearity 31, 5036–5074 (2018)
Scilla, G., Solombrino, F.: A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension. J. Differ. Equ. 267, 6216–6264 (2019)
Slepyan, L.I.: Models and Phenomena in Fracture Mechanics. Springer, New York (2002)
Zanini, C.: Singular perturbations of finite dimensional gradient flows. Discrete Contin. Dyn. Syst. Ser. A 18, 657–675 (2007)
Acknowledgements
The author wishes to thank Prof. Gianni Dal Maso for many helpful discussions on the topic. The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dr. Anthony Bloch.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Riva, F. On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity. J Nonlinear Sci 30, 903–951 (2020). https://doi.org/10.1007/s00332-019-09595-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-019-09595-8
Keywords
- Thin films
- Dynamic debonding
- Quasistatic debonding
- Griffith’s criterion
- Quasistatic limit
- Vanishing inertia and viscosity limit