Citation: Stefano Almi, Giuliano Lazzaroni, Ilaria Lucardesi. Crack growth by vanishing viscosity in planar elasticity[J]. Mathematics in Engineering, 2020, 2(1): 141-173. doi: 10.3934/mine.2020008
[1] |
Almi S (2017) Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening. ESAIM Control Optim Calc Var 23: 791-826. doi: 10.1051/cocv/2016014
![]() |
[2] |
Almi S (2018) Quasi-static hydraulic crack growth driven by Darcy's law. Adv Calc Var 11: 161-191. doi: 10.1515/acv-2016-0029
![]() |
[3] |
Almi S, Lucardesi I (2018) Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks. Nonlinear Differ Equ Appl 25: 43. doi: 10.1007/s00030-018-0536-4
![]() |
[4] | Argatov II, Nazarov SA (2002) Energy release in the kinking of a crack in a plane anisotropic body. Prikl Mat Mekh 66: 502-514. |
[5] |
Babadjian JF, Chambolle A, Lemenant A (2015) Energy release rate for non-smooth cracks in planar elasticity. J Éc polytech Math 2: 117-152. doi: 10.5802/jep.19
![]() |
[6] |
Brokate M, Khludnev A (2004) On crack propagation shapes in elastic bodies. Z Angew Math Phys 55: 318-329. doi: 10.1007/s00033-003-3026-3
![]() |
[7] |
Chambolle A (2003) A density result in two-dimensional linearized elasticity, and applications. Arch Ration Mech Anal 167: 211-233. doi: 10.1007/s00205-002-0240-7
![]() |
[8] |
Chambolle A, Francfort GA, Marigo JJ (2010) Revisiting energy release rates in brittle fracture. J Nonlinear Sci 20: 395-424. doi: 10.1007/s00332-010-9061-2
![]() |
[9] |
Chambolle A, Giacomini A, Ponsiglione M (2008) Crack initiation in brittle materials. Arch Ration Mech Anal 188: 309-349. doi: 10.1007/s00205-007-0080-6
![]() |
[10] |
Chambolle A, Lemenant A (2013) The stress intensity factor for non-smooth fractures in antiplane elasticity. Calc Var Partial Dif 47: 589-610. doi: 10.1007/s00526-012-0529-9
![]() |
[11] |
Costabel M, Dauge M (2002) Crack singularities for general elliptic systems. Math Nachr 235: 29-49. doi: 10.1002/1522-2616(200202)235:1<29::AID-MANA29>3.0.CO;2-6
![]() |
[12] |
Crismale V, Lazzaroni G (2017) Quasistatic crack growth based on viscous approximation: A model with branching and kinking. Nonlinear Differ Equ Appl 24: 7. doi: 10.1007/s00030-016-0426-6
![]() |
[13] |
Dal Maso G, DeSimone A, Solombrino F (2011) Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling. Calc Var Partial Dif 40: 125-181. doi: 10.1007/s00526-010-0336-0
![]() |
[14] |
Dal Maso G, DeSimone A, Solombrino F (2012) Quasistatic evolution for Cam-Clay plasticity: Properties of the viscosity solution. Calc Var Partial Dif 44: 495-541. doi: 10.1007/s00526-011-0443-6
![]() |
[15] |
Dal Maso G, Francfort GA, Toader R (2005) Quasistatic crack growth in nonlinear elasticity. Arch Ration Mech Anal 176: 165-225. doi: 10.1007/s00205-004-0351-4
![]() |
[16] |
Dal Maso G, Lazzaroni G (2010) Quasistatic crack growth in finite elasticity with noninterpenetration. Ann Inst H Poincaré Anal Non Linéaire 27: 257-290. doi: 10.1016/j.anihpc.2009.09.006
![]() |
[17] |
Dal Maso G, Morandotti M (2017) A model for the quasistatic growth of cracks with fractional dimension. Nonlinear Anal 154: 43-58. doi: 10.1016/j.na.2016.03.007
![]() |
[18] |
Dal Maso G, Orlando G, Toader R (2015) Laplace equation in a domain with a rectilinear crack: Higher order derivatives of the energy with respect to the crack length. Nonlinear Differ Equ Appl 22: 449-476. doi: 10.1007/s00030-014-0291-0
![]() |
[19] |
Dal Maso G, Toader R (2002) A model for the quasi-static growth of brittle fractures: Existence and approximation results. Arch Ration Mech Anal 162: 101-135. doi: 10.1007/s002050100187
![]() |
[20] | Dauge M (1988) Smoothness and asymptotics of solutions, In: Elliptic Boundary Value Problems on Corner Domains, Berlin: Springer-Verlag. |
[21] |
Destuynder P, Djaoua M (1981) Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile. Math Method Appl Sci 3: 70-87. doi: 10.1002/mma.1670030106
![]() |
[22] | Efendiev MA, Mielke A (2006) On the rate-independent limit of systems with dry friction and small viscosity. J Convex Anal 13: 151-167. |
[23] |
Fonseca I, Fusco N, Leoni G, et al. (2007) Equilibrium configurations of epitaxially strained crystalline films: Existence and regularity results. Arch Ration Mech Anal 186: 477-537. doi: 10.1007/s00205-007-0082-4
![]() |
[24] |
Francfort GA, Larsen CJ (2003) Existence and convergence for quasi-static evolution in brittle fracture. Commun Pur Appl Math 56: 1465-1500. doi: 10.1002/cpa.3039
![]() |
[25] |
Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46: 1319-1342. doi: 10.1016/S0022-5096(98)00034-9
![]() |
[26] |
Friedrich M, Solombrino F (2018) Quasistatic crack growth in 2d-linearized elasticity. Ann Inst H Poincaré Anal Non Linéaire 35: 27-64. doi: 10.1016/j.anihpc.2017.03.002
![]() |
[27] |
Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc London A 221: 163-198. doi: 10.1098/rsta.1921.0006
![]() |
[28] | Grisvard P (1985) Monographs and Studies in Mathematics, In: Elliptic Problems in Consmooth Domains, Boston: Pitman. |
[29] |
Khludnev AM, Shcherbakov VV (2018) A note on crack propagation paths inside elastic bodies. Appl Math Lett 79: 80-84. doi: 10.1016/j.aml.2017.11.023
![]() |
[30] |
Khludnev AM, Sokolowski J (2000) Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. Eur J Mech A Solids 19: 105-119. doi: 10.1016/S0997-7538(00)00138-8
![]() |
[31] | Knees D (2011) A short survey on energy release rates, In: Dal Maso G, Larsen CJ, Ortner C, Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Oberwolfach Reports, 8: 1216-1219. |
[32] |
Knees D, Mielke A (2008) Energy release rate for cracks in finite-strain elasticity. Math Method Appl Sci 31: 501-528. doi: 10.1002/mma.922
![]() |
[33] |
Knees D, Mielke A, Zanini C (2008) On the inviscid limit of a model for crack propagation. Math Mod Meth Appl Sci 18: 1529-1569. doi: 10.1142/S0218202508003121
![]() |
[34] |
Knees D, Rossi R, Zanini C (2013) A vanishing viscosity approach to a rate-independent damage model. Math Mod Meth Appl Sci 23: 565-616. doi: 10.1142/S021820251250056X
![]() |
[35] |
Knees D, Rossi R, Zanini C (2015) A quasilinear differential inclusion for viscous and rateindependent damage systems in non-smooth domains. Nonlinear Anal Real 24: 126-162. doi: 10.1016/j.nonrwa.2015.02.001
![]() |
[36] |
Knees D, Zanini C, Mielke A (2010) Crack growth in polyconvex materials. Physica D 239: 1470-1484. doi: 10.1016/j.physd.2009.02.008
![]() |
[37] | Krantz SG, Parks HR (2002) The Implicit Function Theorem: History, theory, and applications, Boston: Birkhäuser Boston Inc. |
[38] |
Lazzaroni G, Toader R (2011) A model for crack propagation based on viscous approximation. Math Mod Meth Appl Sci 21: 2019-2047. doi: 10.1142/S0218202511005647
![]() |
[39] |
Lazzaroni G, Toader R (2011) Energy release rate and stress intensity factor in antiplane elasticity. J Math Pure Appl 95: 565-584. doi: 10.1016/j.matpur.2011.01.001
![]() |
[40] | Lazzaroni G, Toader R (2013) Some remarks on the viscous approximation of crack growth. Discrete Contin Dyn Syst Ser S 6: 131-146. |
[41] |
Mielke A, Rossi R, Savaré G (2009) Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin Dyn Syst 25: 585-615. doi: 10.3934/dcds.2009.25.585
![]() |
[42] |
Mielke A, Rossi R, Savaré G (2012) BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim Calc Var 18: 36-80. doi: 10.1051/cocv/2010054
![]() |
[43] |
Mielke A, Rossi R, Savaré G (2016) Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J Eur Math Soc 18: 2107-2165. doi: 10.4171/JEMS/639
![]() |
[44] | Mielke A, Roubíček T (2015) Rate-Independent Systems, New York: Springer. |
[45] |
Nazarov SA, Specovius-Neugebauer M, Steigemann M (2014) Crack propagation in anisotropic composite structures. Asymptot Anal 86: 123-153. doi: 10.3233/ASY-131187
![]() |
[46] |
Negri M (2011) Energy release rate along a kinked path. Math Method Appl Sci 34: 384-396. doi: 10.1002/mma.1362
![]() |
[47] |
Negri M (2014) Quasi-static rate-independent evolutions: Characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim Calc Var 20: 983-1008. doi: 10.1051/cocv/2014004
![]() |
[48] |
Negri M, Ortner C (2008) Quasi-static crack propagation by Griffith's criterion. Math Mod Meth Appl Sci 18: 1895-1925. doi: 10.1142/S0218202508003236
![]() |
[49] |
Negri M, Toader R (2015) Scaling in fracture mechanics by Bažant law: From finite to linearized elasticity. Math Mod Meth Appl Sci 25: 1389-1420. doi: 10.1142/S0218202515500360
![]() |