Abstract
In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain
Funding statement: This material is based on work supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INDAM) under the Project “OptiFrac: fratture e problemi a discontinuità libera”.
Acknowledgements
The author wishes to thank Gianni Dal Maso, Antonio DeSimone, Alessandro Lucantonio, Giovanni Noselli, and Rodica Toader for many helpful discussions.
References
[1] S. Almi, Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening, ESAIM Control Optim. Calc. Var. (2016), 10.1051/cocv/2016014. 10.1051/cocv/2016014Search in Google Scholar
[2] S. Almi, G. Dal Maso and R. Toader, Quasi-static crack growth in hydraulic fracture, Nonlinear Anal. 109 (2014), 301–318. 10.1016/j.na.2014.07.009Search in Google Scholar
[3] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167 (2003), no. 3, 211–233. 10.1007/s00205-002-0240-7Search in Google Scholar
[4] P. G. Ciarlet, Mathematical Elasticity. Vol. I, Stud. Math. Appl. 20, North-Holland, Amsterdam, 1988. Search in Google Scholar
[5] G. Dal Maso and M. Morandotti, A model for the quasistatic growth of cracks with fractional dimension, Nonlinear Anal. (2016), 10.1016/j.na.2016.03.007. 10.1016/j.na.2016.03.007Search in Google Scholar
[6] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002), no. 2, 101–135. 10.1007/s002050100187Search in Google Scholar
[7] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: Existence and regularity results, Arch. Ration. Mech. Anal. 186 (2007), no. 3, 477–537. 10.1007/s00205-007-0082-4Search in Google Scholar
[8] G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), no. 10, 1465–1500. 10.1002/cpa.3039Search in Google Scholar
[9] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. 10.1016/S0022-5096(98)00034-9Search in Google Scholar
[10] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel, 1993. Search in Google Scholar
[11] M. Giaquinta and S. Hildebrandt, Calculus of Variations. I, Grundlehren Math. Wiss. 310, Springer, Berlin, 1996. Search in Google Scholar
[12] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. Ser. A 221 (1921), no. 582–593, 163–198. 10.1098/rsta.1921.0006Search in Google Scholar
[13] D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci. 31 (2008), no. 5, 501–528. 10.1002/mma.922Search in Google Scholar
[14] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci. 18 (2008), no. 9, 1529–1569. 10.1142/S0218202508003121Search in Google Scholar
[15] S. G. Krantz and H. R. Parks, The Implicit Function Theorem, Birkhäuser, Boston, 2002. 10.1007/978-1-4612-0059-8Search in Google Scholar
[16] A. Lucantonio, G. Noselli, X. Trepat, A. DeSimone and M. Arroyo, Hydraulic fracture and toughening of a brittle layer bonded to a hydrogel, Phys. Rev. Lett. 115 (2015), Article ID 188105. 10.1103/PhysRevLett.115.188105Search in Google Scholar
[17] A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), 461–559. 10.1016/S1874-5717(06)80009-5Search in Google Scholar
[18] M. Negri and R. Toader, Scaling in fracture mechanics by Bažant law: From finite to linearized elasticity, Math. Models Methods Appl. Sci. 25 (2015), no. 7, 1389–1420. 10.1142/S0218202515500360Search in Google Scholar
[19] C. A. Rogers, Hausdorff Measures, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1998. Search in Google Scholar
[20] I. N. Sneddon and M. Lowengrub, Crack Problems in the Classical Theory of Elasticity, John Wiley & Sons, New York, 1969. Search in Google Scholar
[21] R. Temam, Problèmes mathématiques en plasticité, Méthodes Math. Inform. 12, Gauthier-Villars, Montrouge, 1983. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Sobolev homeomorphisms with gradients of low rank via laminates
- Stationary Kirchhoff equations with powers
- Quasi-static hydraulic crack growth driven by Darcy’s law
- A note on transport equation in quasiconformally invariant spaces
- On the regularity of solutions of one-dimensional variational obstacle problems
Articles in the same Issue
- Frontmatter
- Sobolev homeomorphisms with gradients of low rank via laminates
- Stationary Kirchhoff equations with powers
- Quasi-static hydraulic crack growth driven by Darcy’s law
- A note on transport equation in quasiconformally invariant spaces
- On the regularity of solutions of one-dimensional variational obstacle problems