×

Analysis of a dynamic peeling test with speed-dependent toughness. (English) Zbl 1420.35400

The authors analyse a one-dimensional model of dynamic debonding for a thin film, where the local toughness of the glue between the film and the substrate also depends on the debonding speed. The wave equation on the debonded region is strongly coupled with Griffith’s criterion for the evolution of the debonding front. They provide an existence and uniqueness result and find explicitly the solution in some concrete examples. They study the limit of solutions as inertia tends to zero, observing phases of unstable propagation, as well as time discontinuities, even though the toughness diverges at a limiting debonding speed.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35L05 Wave equation
35R35 Free boundary problems for PDEs
74H20 Existence of solutions of dynamical problems in solid mechanics
35B40 Asymptotic behavior of solutions to PDEs
74K35 Thin films
74R99 Fracture and damage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. Bourdin, G. A. Francfort, and J.-J. Marigo, {\it The variational approach to fracture}, J. Elasticity, 91 (2008), pp. 5-148. · Zbl 1176.74018
[2] A. Chambolle, {\it A density result in two-dimensional linearized elasticity, and applications}, Arch. Ration. Mech. Anal., 167 (2003), pp. 211-233. · Zbl 1030.74007
[3] G. Dal Maso, G. A. Francfort, and R. Toader, {\it Quasistatic crack growth in nonlinear elasticity}, Arch. Ration. Mech. Anal., 176 (2005), pp. 165-225. · Zbl 1064.74150
[4] G. Dal Maso and C. J. Larsen, {\it Existence for wave equations on domains with arbitrary growing cracks}, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 22 (2011), pp. 387-408. · Zbl 1239.35086
[5] G. Dal Maso, C. J. Larsen, and R. Toader, {\it Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition}, J. Mech. Phys. Solids, 95 (2016), pp. 697-707. · Zbl 1482.74017
[6] G. Dal Maso and G. Lazzaroni, {\it Quasistatic crack growth in finite elasticity with non-interpenetration}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), pp. 257-290. · Zbl 1188.35205
[7] G. Dal Maso, G. Lazzaroni, and L. Nardini, {\it Existence and uniqueness of dynamic evolutions for a peeling test in dimension one}, J. Differential Equations, 261 (2016), pp. 4897-4923. · Zbl 1347.35143
[8] G. Dal Maso and I. Lucardesi, {\it The wave equation on domains with cracks growing on a prescribed path: Existence, uniqueness, and continuous dependence on the data}, Appl. Math. Res. Express, 1 (2017), pp. 184-241. · Zbl 1456.35123
[9] G. Dal Maso and R. Toader, {\it A model for the quasi-static growth of brittle fractures: Existence and approximation results}, Arch. Ration. Mech. Anal., 162 (2002), pp. 101-135. · Zbl 1042.74002
[10] P.-E. Dumouchel, J.-J. Marigo, and M. Charlotte, {\it Dynamic fracture: An example of convergence towards a discontinuous quasistatic solution}, Contin. Mech. Thermodyn., 20 (2008), pp. 1-19. · Zbl 1160.74401
[11] G. A. Francfort and C. J. Larsen, {\it Existence and convergence for quasi-static evolution in brittle fracture}, Commun. Pure Appl. Math., 56 (2003), pp. 1465-1500. · Zbl 1068.74056
[12] G. A. Francfort and J.-J. Marigo, {\it Revisiting brittle fracture as an energy minimization problem}, J. Mech. Phys. Solids, 46 (1998), pp. 1319-1342. · Zbl 0966.74060
[13] L. B. Freund, {\it Dynamic Fracture Mechanics}, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 1990. · Zbl 0712.73072
[14] T. Goldman, A. Livne, and J. Fineberg, {\it Acquisition of inertia by a moving crack}, Phys. Rev. Lett., 104 (2010), article number 114301.
[15] J. K. Hale, {\it Ordinary Differential Equations}, 2nd ed. Robert E. Krieger Publishing, Huntington, NY, 1980. · Zbl 0433.34003
[16] D. Knees, A. Mielke, and C. Zanini, {\it On the inviscid limit of a model for crack propagation}, Math. Models Methods Appl. Sci. 18 (2008), pp. 1529-1569. · Zbl 1151.49014
[17] C. M. Landis, T. Pardoen, and J. W. Hutchinson, {\it Crack velocity dependent toughness in rate dependent materials}, Mech. Mater., 32 (2000), pp. 663-678.
[18] C. J. Larsen, C. Ortner, and E. Süli, {\it Existence of solutions to a regularized model of dynamic fracture}, Math. Models Methods Appl. Sci., 20 (2010), pp. 1021-1048. · Zbl 1425.74418
[19] G. Lazzaroni and L. Nardini, {\it On the quasistatic limit of dynamic evolutions for a peeling test in dimension one}, J. Nonlinear Sci., 28 (2018), pp. 269-304. · Zbl 1516.35556
[20] G. Lazzaroni, R. Bargellini, P.-E. Dumouchel, and J.-J. Marigo, {\it On the role of kinetic energy during unstable propagation in a heterogeneous peeling test}, Int. J. Fract., 175 (2012), pp. 127-150.
[21] G. Lazzaroni, R. Rossi, M. Thomas, and R. Toader, {\it Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia}, preprint, SISSA 52/2014/MATE. · Zbl 1412.35325
[22] G. Lazzaroni and R. Toader, {\it A model for crack propagation based on viscous approximation}, Math. Models Methods Appl. Sci., 21 (2011), pp. 2019-2047. · Zbl 1277.74066
[23] Y. Lee and V. Prakash, {\it Dynamic fracture toughness versus crack tip speed relationship at lower than room temperature for high strength} 4340{\it VAR structural steel}, J. Mech. Phys. Solids, 46 (1998), pp. 1943-1967. · Zbl 0945.74685
[24] M. Marder, {\it New dynamical equation for cracks}, Phys. Rev. Lett., 66 (1991), pp. 2484-2487. · Zbl 1050.74657
[25] D. P. Miannay, {\it Time-Dependent Fracture Mechanics}, Mechanical Engineering Series, Springer-Verlag, New York, 2001. · Zbl 0977.74001
[26] A. Mielke and T. Roubíček, {\it Rate-Independent Systems: Theory and Application}, Applied Mathematical Sciences 193, Springer, New York, 2015. · Zbl 1339.35006
[27] L. Nardini, {\it A note on the convergence of singularly perturbed second order potential-type equations}, J. Dynam. Differential Equations, 29 (2017), pp. 783-797. · Zbl 1376.34055
[28] L. Nardini, {\it A Mathematical Analysis of a One-Dimensional Model for Dynamic Debonding}, Ph.D. Thesis, SISSA, Trieste, 2017.
[29] S. Nicaise and A.-M. Sändig, {\it Dynamic crack propagation in a} 2{\it D elastic body: The out-of-plane case}, J. Math. Anal. Appl., 329 (2007), pp. 1-30. · Zbl 1342.74143
[30] R. Rossi and T. Roubíček, {\it Thermodynamics and analysis of rate-independent adhesive contact at small strains}, Nonlinear Anal., 74 (2011), pp. 3159-3190. · Zbl 1217.35108
[31] R. Rossi and M. Thomas, {\it From adhesive to brittle delamination in visco-elastodynamics}, Math. Models Methods Appl. Sci., 27 (2017), pp. 1489-1546. · Zbl 1375.49019
[32] T. Roubíček, {\it Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity}, SIAM J. Math. Anal., 45 (2013), pp. 101-126. · Zbl 1264.35131
[33] T. W. Webb and E. C. Aifantis, {\it Oscillatory fracture in polymeric materials}, Int. J. Solids Struct., 32 (1995), pp. 2725-2743. · Zbl 0869.73059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.