February  2013, 6(1): 147-165. doi: 10.3934/dcdss.2013.6.147

Crack propagation by a regularization of the principle of local symmetry

1. 

Dipartimento di Matematica, Università degli Studi di Pavia, Via A. Ferrata 1 - 27100 Pavia, Italy

Received  April 2011 Revised  December 2011 Published  October 2012

For planar mixed mode crack propagation in brittle materials many similar criteria have been proposed. In this work the Principle of Local Symmetry together with Griffith Criterion will be the governing equations for the evolution. The Stress Intensity Factors, a crucial ingredient in the theory, will be employed in a 'non-local' (regularized) fashion. We prove existence of a Lipschitz path that satisfies the Principle of Local Symmetry (for the approximated stress intensity factors) and then existence of a $BV$-parametrization that satisfies Griffith Criterion (again for the approximated stress intensity factors).
Citation: Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 147-165. doi: 10.3934/dcdss.2013.6.147
References:
[1]

M. Amestoy and J. B. Leblond, Crack paths in plane situations. II. Detailed form of the expansion of the stress intensity factors,, Internat. J. Solids Structures, 29 (1992), 465. doi: 10.1016/0020-7683(92)90210-K. Google Scholar

[2]

A. Chambolle, G. A. Francfort and J.-J. Marigo, Revisiting energy release rates in brittle fracture,, J. Nonlinear Sci., 20 (2010), 395. doi: 10.1007/s00332-010-9061-2. Google Scholar

[3]

A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in elastic bodies,, Arch. Ration. Mech. Anal., 188 (2008), 309. doi: 10.1007/s00205-007-0080-6. Google Scholar

[4]

B. Cotterell, On brittle fracture paths,, Internat. J. Fracture, 1 (1965), 96. doi: 10.1007/BF00186747. Google Scholar

[5]

B. Cotterell and J. R. Rice, Slightly curved or kinked cracks,, Int. J. Fracture, 16 (1980), 155. doi: 10.1007/BF00012619. Google Scholar

[6]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar

[7]

A. Friedman and Y. Liu, Propagation of cracks in elastic media,, Arch. Rational Mech. Anal., 136 (1996), 235. doi: 10.1007/BF02206556. Google Scholar

[8]

R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks,, Internat. J. Fracture, 10 (1974), 507. Google Scholar

[9]

M. Gosz, J. Dolbow and B. Moran, Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks,, Int. J. Solids Struct., 35 (1998), 1763. doi: 10.1016/S0020-7683(97)00132-7. Google Scholar

[10]

P. Grisvard, Singularités en elasticité,, Arch. Rational Mech. Anal., 107 (1989), 157. doi: 10.1007/BF00286498. Google Scholar

[11]

A. M. Khludnev, V. A. Kovtunenko and A. Tani., On the topological derivative due to kink of a crack with non-penetration. Anti-plane model,, J. Math. Pures Appl., 94 (2010), 571. doi: 10.1016/j.matpur.2010.06.002. Google Scholar

[12]

G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity,, J. Math. Pures Appl., 95 (2011), 565. doi: 10.1016/j.matpur.2011.01.001. Google Scholar

[13]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189. Google Scholar

[14]

M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation,, Adv. Calc. Var., 3 (2010), 149. doi: 10.1515/acv.2010.008. Google Scholar

[15]

G. C. Sih and F. Erdogan, On the crack extension in plates under plane loading and transverse shear,, J. Basic Engineering, 85 (1963), 519. doi: 10.1115/1.3656897. Google Scholar

[16]

G. J. Williams and P. D. Ewing, Fracture under complex stress - the angled crack problem,, Int. J. Fracture, 8 (1972), 441. doi: 10.1007/BF00191106. Google Scholar

[17]

M. L. Williams, On the stress distribution at the base of a stationary crack,, J. Appl. Mech., 24 (1957), 109. Google Scholar

show all references

References:
[1]

M. Amestoy and J. B. Leblond, Crack paths in plane situations. II. Detailed form of the expansion of the stress intensity factors,, Internat. J. Solids Structures, 29 (1992), 465. doi: 10.1016/0020-7683(92)90210-K. Google Scholar

[2]

A. Chambolle, G. A. Francfort and J.-J. Marigo, Revisiting energy release rates in brittle fracture,, J. Nonlinear Sci., 20 (2010), 395. doi: 10.1007/s00332-010-9061-2. Google Scholar

[3]

A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in elastic bodies,, Arch. Ration. Mech. Anal., 188 (2008), 309. doi: 10.1007/s00205-007-0080-6. Google Scholar

[4]

B. Cotterell, On brittle fracture paths,, Internat. J. Fracture, 1 (1965), 96. doi: 10.1007/BF00186747. Google Scholar

[5]

B. Cotterell and J. R. Rice, Slightly curved or kinked cracks,, Int. J. Fracture, 16 (1980), 155. doi: 10.1007/BF00012619. Google Scholar

[6]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar

[7]

A. Friedman and Y. Liu, Propagation of cracks in elastic media,, Arch. Rational Mech. Anal., 136 (1996), 235. doi: 10.1007/BF02206556. Google Scholar

[8]

R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks,, Internat. J. Fracture, 10 (1974), 507. Google Scholar

[9]

M. Gosz, J. Dolbow and B. Moran, Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks,, Int. J. Solids Struct., 35 (1998), 1763. doi: 10.1016/S0020-7683(97)00132-7. Google Scholar

[10]

P. Grisvard, Singularités en elasticité,, Arch. Rational Mech. Anal., 107 (1989), 157. doi: 10.1007/BF00286498. Google Scholar

[11]

A. M. Khludnev, V. A. Kovtunenko and A. Tani., On the topological derivative due to kink of a crack with non-penetration. Anti-plane model,, J. Math. Pures Appl., 94 (2010), 571. doi: 10.1016/j.matpur.2010.06.002. Google Scholar

[12]

G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity,, J. Math. Pures Appl., 95 (2011), 565. doi: 10.1016/j.matpur.2011.01.001. Google Scholar

[13]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189. Google Scholar

[14]

M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation,, Adv. Calc. Var., 3 (2010), 149. doi: 10.1515/acv.2010.008. Google Scholar

[15]

G. C. Sih and F. Erdogan, On the crack extension in plates under plane loading and transverse shear,, J. Basic Engineering, 85 (1963), 519. doi: 10.1115/1.3656897. Google Scholar

[16]

G. J. Williams and P. D. Ewing, Fracture under complex stress - the angled crack problem,, Int. J. Fracture, 8 (1972), 441. doi: 10.1007/BF00191106. Google Scholar

[17]

M. L. Williams, On the stress distribution at the base of a stationary crack,, J. Appl. Mech., 24 (1957), 109. Google Scholar

[1]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[2]

Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i

[3]

T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215

[4]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257

[5]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649

[6]

Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585

[7]

Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070

[8]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145

[9]

Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i

[10]

Christopher J. Larsen. Local minimality and crack prediction in quasi-static Griffith fracture evolution. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 121-129. doi: 10.3934/dcdss.2013.6.121

[11]

Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256

[12]

Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257

[13]

Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591

[14]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411

[15]

Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167

[16]

Dorothee Knees, Andreas Schröder. Computational aspects of quasi-static crack propagation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 63-99. doi: 10.3934/dcdss.2013.6.63

[17]

Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625

[18]

G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705

[19]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495

[20]

Goro Akagi, Jun Kobayashi, Mitsuharu Ôtani. Principle of symmetric criticality and evolution equations. Conference Publications, 2003, 2003 (Special) : 1-10. doi: 10.3934/proc.2003.2003.1

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]