Crack propagation by a regularization of the principle of local symmetry

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

Crack propagation by a regularization of the principle of local symmetry

Matteo Negri ^1,

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

Abstract
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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).

Keywords: rate-independent evolution., principle of local symmetry, Crack propagation.

Mathematics Subject Classification: Primary: 74R10; Secondary: 74A4.

Citation: Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete and 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), 465-501. doi: 10.1016/0020-7683(92)90210-K.
[2]	A. Chambolle, G. A. Francfort and J.-J. Marigo, Revisiting energy release rates in brittle fracture, J. Nonlinear Sci., 20 (2010), 395-424. doi: 10.1007/s00332-010-9061-2.
[3]	A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in elastic bodies, Arch. Ration. Mech. Anal., 188 (2008), 309-349. doi: 10.1007/s00205-007-0080-6.
[4]	B. Cotterell, On brittle fracture paths, Internat. J. Fracture, 1 (1965), 96-103. doi: 10.1007/BF00186747.
[5]	B. Cotterell and J. R. Rice, Slightly curved or kinked cracks, Int. J. Fracture, 16 (1980), 155-169. doi: 10.1007/BF00012619.
[6]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.
[7]	A. Friedman and Y. Liu, Propagation of cracks in elastic media, Arch. Rational Mech. Anal., 136 (1996), 235-290. doi: 10.1007/BF02206556.
[8]	R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks, Internat. J. Fracture, 10 (1974), 507-523.
[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), 1763-1783. doi: 10.1016/S0020-7683(97)00132-7.
[10]	P. Grisvard, Singularités en elasticité, Arch. Rational Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498.
[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), 571-596. doi: 10.1016/j.matpur.2010.06.002.
[12]	G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity, J. Math. Pures Appl., 95 (2011), 565-584. doi: 10.1016/j.matpur.2011.01.001.
[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), 189-206.
[14]	M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation, Adv. Calc. Var., 3 (2010), 149-212. doi: 10.1515/acv.2010.008.
[15]	G. C. Sih and F. Erdogan, On the crack extension in plates under plane loading and transverse shear, J. Basic Engineering, 85 (1963), 519-527. doi: 10.1115/1.3656897.
[16]	G. J. Williams and P. D. Ewing, Fracture under complex stress - the angled crack problem, Int. J. Fracture, 8 (1972), 441-446. doi: 10.1007/BF00191106.
[17]	M. L. Williams, On the stress distribution at the base of a stationary crack, J. Appl. Mech., 24 (1957), 109-114.

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), 465-501. doi: 10.1016/0020-7683(92)90210-K.
[2]	A. Chambolle, G. A. Francfort and J.-J. Marigo, Revisiting energy release rates in brittle fracture, J. Nonlinear Sci., 20 (2010), 395-424. doi: 10.1007/s00332-010-9061-2.
[3]	A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in elastic bodies, Arch. Ration. Mech. Anal., 188 (2008), 309-349. doi: 10.1007/s00205-007-0080-6.
[4]	B. Cotterell, On brittle fracture paths, Internat. J. Fracture, 1 (1965), 96-103. doi: 10.1007/BF00186747.
[5]	B. Cotterell and J. R. Rice, Slightly curved or kinked cracks, Int. J. Fracture, 16 (1980), 155-169. doi: 10.1007/BF00012619.
[6]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.
[7]	A. Friedman and Y. Liu, Propagation of cracks in elastic media, Arch. Rational Mech. Anal., 136 (1996), 235-290. doi: 10.1007/BF02206556.
[8]	R. V. Goldstein and R. L. Salganik, Brittle fracture of solids with arbitrary cracks, Internat. J. Fracture, 10 (1974), 507-523.
[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), 1763-1783. doi: 10.1016/S0020-7683(97)00132-7.
[10]	P. Grisvard, Singularités en elasticité, Arch. Rational Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498.
[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), 571-596. doi: 10.1016/j.matpur.2010.06.002.
[12]	G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity, J. Math. Pures Appl., 95 (2011), 565-584. doi: 10.1016/j.matpur.2011.01.001.
[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), 189-206.
[14]	M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation, Adv. Calc. Var., 3 (2010), 149-212. doi: 10.1515/acv.2010.008.
[15]	G. C. Sih and F. Erdogan, On the crack extension in plates under plane loading and transverse shear, J. Basic Engineering, 85 (1963), 519-527. doi: 10.1115/1.3656897.
[16]	G. J. Williams and P. D. Ewing, Fracture under complex stress - the angled crack problem, Int. J. Fracture, 8 (1972), 441-446. doi: 10.1007/BF00191106.
[17]	M. L. Williams, On the stress distribution at the base of a stationary crack, J. Appl. Mech., 24 (1957), 109-114.

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