Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.
Mots-clés : Wave equation in time-dependent domains, singularities, dynamic debonding, dynamic energy release rate, Griffith’s criterion, dynamic fracture, crack initiation, thin films
@article{COCV_2019__25__A80_0,
author = {Lazzaroni, Giuliano and Nardini, Lorenzo},
title = {On the 1d wave equation in time-dependent domains and the problem of debond initiation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2019006},
zbl = {1437.35445},
mrnumber = {4040713},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2019006/}
}
TY - JOUR AU - Lazzaroni, Giuliano AU - Nardini, Lorenzo TI - On the 1d wave equation in time-dependent domains and the problem of debond initiation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2019006/ DO - 10.1051/cocv/2019006 LA - en ID - COCV_2019__25__A80_0 ER -
%0 Journal Article %A Lazzaroni, Giuliano %A Nardini, Lorenzo %T On the 1d wave equation in time-dependent domains and the problem of debond initiation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2019006/ %R 10.1051/cocv/2019006 %G en %F COCV_2019__25__A80_0
Lazzaroni, Giuliano; Nardini, Lorenzo. On the 1d wave equation in time-dependent domains and the problem of debond initiation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 80. doi : 10.1051/cocv/2019006. http://archive.numdam.org/articles/10.1051/cocv/2019006/
[1] , and , On some abstract variable domain hyperbolic differential equations. Ann. Mat. Pura Appl. 174 (1998) 209–239. | DOI | MR | Zbl
[2] and , Sur un problème de Mlle Z. Szmydt relatif à l’équation ∂2z∕∂xy = f(x, y, z, ∂z∕∂x, ∂z∕∂y). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958) 321–325. | MR | Zbl
[3] , and , The variational approach to fracture. J. Elastic. 91 (2008) 5–148. | DOI | MR | Zbl
[4] , Linear hyperbolic systems in domains with growing cracks. Milan J. Math. 85 (2017) 149–185. | DOI | MR | Zbl
[5] , and , Crack initiation in brittle materials. Arch. Ration. Mech. Anal. 188 (2008) 309–349. | DOI | MR | Zbl
[6] , Local decay of solutions of the wave equation in the exterior of a moving body. J. Math. Anal. Appl. 49 (1975) 130–153. | DOI | MR | Zbl
[7] and , A nonlinear wave equation in a time dependent domain. J. Math. Anal. Appl. 42 (1973) 29–60. | DOI | MR | Zbl
[8] and , Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 22 (2011) 387–408. | DOI | MR | Zbl
[9] and , The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness and continuous dependence on the data. Appl. Math. Res. Express 2017 (2016) 184–241. | MR | Zbl
[10] , and , Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition. J. Mech. Phys. Solids 95 (2016) 697–707. | DOI | MR | Zbl
[11] , and , Existence and uniqueness of dynamic evolutions for a peeling test in dimension one. J. Differ. Equ. 261 (2016) 4897–4923. | DOI | MR | Zbl
[12] , and , Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition. J. Math. Pures Appl. 127 (2019) 160–191. | DOI | MR | Zbl
[13] and , Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8 of Évolution: semi-groupe, variationnel. Masson, Paris (1988). | MR | Zbl
[14] , and , Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution. Contin. Mech. Thermodyn. 20 (2008) 1–19. | DOI | MR | Zbl
[15] , Dynamic fracture mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1990). | MR | Zbl
[16] , Sur un problème relatif à la théorie des équations aux dérivées partielles du second ordre (second mémoire). Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 6 (1904) 117–144. | JFM | Numdam | MR
[17] , Sur un procédé alterné. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 1 (1909) 129–143. | JFM | Numdam | MR
[18] , Generalized Goursat problem. Pacific J. Math. 12 (1962) 207–224. | DOI | MR | Zbl
[19] , Sur cmu + u3 = f dans un domaine non cylindrique. C. R. Acad. Sci. Paris Sér. A-B 275 (1972) A659–A662. | MR | Zbl
[20] , , and , An assessment of the phase field formulation for crack growth. Comput. Methods Appl. Mech. Eng. 294 (2015) 313–330. | DOI | MR | Zbl
[21] and , On the solvability of boundary value problems for the wave equation with nonlinear dissipation in noncylindrical domains. Sibirsk. Mat. Zh. 6 (2001) 1278–1299. | MR | Zbl
[22] , and , Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010) 1021–1048. | DOI | MR | Zbl
[23] , , and , On the role of kinetic energy during unstable propagation in a heterogeneous peeling test. Int. J. Fract. 175 (2012) 127–150. | DOI
[24] and , On the quasistatic limit of dynamic evolutions for a peeling test in dimension one. J. Nonlinear Sci. 28 (2018) 269–304. | DOI | MR | Zbl
[25] and , Analysis of a dynamic peeling test with speed-dependent toughness. SIAM J. Appl. Math. 78 (2018) 1206–1227. | DOI | MR | Zbl
[26] , , and , Rate-independent damage in thermo-viscoelastic materials with inertia. J. Dyn. Differ. Equ. 30 (2018) 1311–1364. | DOI | MR | Zbl
[27] , Une remarque sur les problèmes d’évolution non linéaires dans des domaines non cylindriques. Rev. Roumaine Math. Pures Appl. 9 (1964) 11–18. | MR | Zbl
[28] , and , Dynamics of wave equations with moving boundary. J. Differ. Equ. 262 (2017) 3317–3342. | DOI | MR | Zbl
[29] and , Dynamic crack propagation in a 2D elastic body: the out-of-plane case. J. Math. Anal. Appl. 329 (2007) 1–30. | DOI | MR | Zbl
[30] , A mixed problem for the wave equation in a time dependent domain. Arch. Ratl. Mech. Anal. 22 (1966) 24–36. | DOI | MR | Zbl
[31] and , Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlin. Anal. 74 (2011) 3159–3190. | DOI | MR | Zbl
[32] and , From adhesive to brittle delamination in visco-elastodynamics. Math. Models Methods Appl. Sci. 27 (2017) 1489–1546. | DOI | MR | Zbl
[33] , Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45 (2013) 101–126. | DOI | MR | Zbl
[34] , A linear wave equation in a time-dependent domain. J. Math. Anal. Appl. 153 (1990) 533–548. | DOI | MR | Zbl
[35] and , On a one-dimensional parabolic problem arising in the study of some problems with free boundaries. Zap. Nauchn. Sem. S.- Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 269 (2000) 322–338. | MR | Zbl
[36] , Sur l’existence de solutions de certains problèmes aux limites relatifs à un système d’équations différentielles hyperboliques. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958) 31–36. | MR | Zbl
[37] and , Stabilization of wave dynamics by moving boundary. Nonlinear Anal. Real World Appl. 39 (2018) 213–232. | DOI | MR | Zbl
Cité par Sources :
