## Journal of the Mathematical Society of Japan

### Evolution of a crack with kink and non-penetration

#### Abstract

The nonlinear evolution problem for a crack with a kink in elastic body is considered. This nonlinear formulation accounts the condition of mutual non-penetration between the crack faces. The kinking crack is presented with the help of two unknown shape parameters of the kink angle and of the crack length, which minimize an energy due to the Griffith hypothesis. Based on the obtained results of the shape sensitivity analysis, solvability of the evolutionary minimization problem is proved, and the necessary conditions for the optimal crack are derived.

#### Article information

Source
J. Math. Soc. Japan, Volume 60, Number 4 (2008), 1219-1253.

Dates
First available in Project Euclid: 5 November 2008

https://projecteuclid.org/euclid.jmsj/1225894039

Digital Object Identifier
doi:10.2969/jmsj/06041219

Mathematical Reviews number (MathSciNet)
MR2467876

Zentralblatt MATH identifier
1153.49040

#### Citation

KHLUDNEV, Alexander M.; KOVTUNENKO, Victor A.; TANI, Atusi. Evolution of a crack with kink and non-penetration. J. Math. Soc. Japan 60 (2008), no. 4, 1219--1253. doi:10.2969/jmsj/06041219. https://projecteuclid.org/euclid.jmsj/1225894039

#### References

• M. Amestoy and J.-B. Leblond, Crack paths in plane situations, II: Detailed form of the expansion of the stress intensity factors, Int. J. Solids Struct., 29 (1992), 465–501.
• I. I. Argatov and S. A. Nazarov, Energy release caused by the kinking of a crack in a plane anisotropic solid, J. Appl. Math. Mech., 66 (2002), 491–503.
• A. Ben Abda, H. Ben Ameur and M. Jaoua, Identification of 2D cracks by elastic boundary measurements, Inverse Problems, 15 (1999), 67–77.
• M. Brokate and A. M. Khludnev, On crack propagation shapes in elastic bodies, Z. Angew. Math. Phys., 55 (2004), 318–329.
• B. Cotterell and J. R. Rice, Slightly curved or kinked cracks, Int. J. Fracture, 16 (1980), 155–169.
• G. Dal Maso and R. Toader, A model for the quasistatic growth of brittle fractures: Existence and approximation results, Arch. Rational Mech. Anal., 162 (2002), 101–135.
• M. C. Delfour and J.-P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, 2001, p.,482.
• P. Destuynder and M. Djaoua, Équivalence de l'intégrale de Rice et du taux de restitution de l'énergie en mécanique de la rupture fragile, C. R. Acad. Sci. Paris Sér. A–B, 290 (1980), A347–A350.
• G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319–1342.
• A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J., 38 (1989), 527–556.
• M. Hintermüller, V. A. Kovtunenko and K. Kunisch, An optimization approach for the delamination of a composite material with non-penetration. In: Free and Moving Boundaries: Analysis, Simulation and Control, (eds. R. Glowinski and J.-P. Zolesio), Lecture Notes in Pure and Applied Mathematics, 252, Chapman & Hall/CRC, 2007, 331–348.
• H. Itou and A. Tani, Existence of a weak solution in an infinite viscoelastic strip with a semi-infinite crack, Math. Models Methods Appl. Sci., 14 (2004), 975–986.
• T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1966, p.,592.
• A. M. Khludnev, A. N. Leontiev and J. Herskovits, Nonsmooth domain optimization for elliptic equations with unilateral conditions, J. Math. Pures Appl., 82 (2003), 197–212.
• A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000, p.,408.
• A. M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel, 1997, p.,384.
• A. M. Khludnev, K. Ohtsuka and J. Sokolowski, On derivative of energy functional for elastic bodies with cracks and unilateral conditions, Quart. Appl. Math., 60 (2002), 99–109.
• V. A. Kovtunenko, Invariant energy integrals for the non-linear crack problem with possible contact of the crack surfaces, J. Appl. Maths. Mechs., 67 (2003), 99–110.
• V. A. Kovtunenko, Primal-dual methods of shape sensitivity analysis for curvilinear cracks with non-penetration, IMA J. Appl. Math., 71 (2006), 635–657.
• V. A. Kovtunenko, Interface cracks in composite orthotropic materials and their delamination via global shape optimization, Optim. Eng., 7 (2006), 173–199.
• V. A. Kovtunenko and K. Kunisch, Problem of crack perturbation based on level sets and velocities, Z. Angew. Math. Mech., 87 (2007), 809–830.
• V. A. Kovtunenko and I. V. Sukhorukov, Optimization formulation of the evolutionary problem of crack propagation under quasibrittle fracture, Appl. Mech. Tech. Phys., 47 (2006), 704–713.
• V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, AMS, Providence, RI, 1997, p.,414.
• J.-B. Leblond and J. Frelat, Crack kinking from an interface crack with initial contact between the crack lips, European J. Mech. A Solids, 20 (2001), 937–951.
• D. Leguillon, Asymptotic and numerical analysis of a crack branching in nonisotropic materials, European J. Mech. A Solids, 12 (1993), 33–51.
• K. Ohtsuka, Generalized $J$-integrals and three dimensional fracture mechanics I, Hiroshima Math. J., 11 (1981), 21–52.
• E. M. Rudoy, Differentiation of energy functionals in two-dimensional elasticity theory for solids with curvilinear cracks, J. Appl. Mech. Tech. Phys., 45 (2004), 843–852.