Differentiation of energy functionals in the problem on a curvilinear crack with possible contact between the shores

V. Z. Parton and E. M. Morozov, Mechanics of Elastoplastic Fracture (Nauka, Moscow, 1974) [in Russian].

G. P. Cherepanov, Mechanics of Brittle Fracture (Nauka, Moscow, 1974; McGraw-Hill, New York, 1979).

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT-Press, Southampton-Boston, 2000).

R. V. Goldstein and A. A. Spektor, “Variational Estimates of Solutions of Some Mixed Spatial Problems of Elasticity with Unknown Boundary,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 2, 82–94 (1978) [Mech. Solids (Engl. Transl.)].

R. V. Goldstein and Yu. V. Zhitnikov, “Equilibrium of Cavities and Cracks-Cuts with Domains of Overlapping and Opening in an Elastic Medium,” Prikl. Mat. Mekh. 50(5), 826–834 (1986) [J. Appl. Math. Mech. (Engl. Transl.)].

V. A. Kovtunenko, “Shape Sensitivity of a Plane Crack Front,” Math. Meth. Appl. Sci. 26(5), 359–374 (2003).

K. Ohtsuka, “Mathematics of Brittle Fracture,” in Theoretical Studies on Fracture Mechanics in Japan (Hiroshima-Denki Inst. Technol., Hiroshima, 1997).

V. G. Maz’ya and S. A. Nazarov, “Asymptotics of Energy Integrals under Small Perturbations of the Boundary Close to Angular and Conic Points,” Trudy Moskov. Mat. Obsch. 50, 79–129 (1987) [Trans. Mosc. Math. Soc. (Engl. Transl.)].

N. V. Banichuk, “Determination of the Shape of a Curved Crack by the Small-Parameter Method,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 2, 130–137 (1970) [Mech. Solids (Engl. Transl.)].

R. V. Goldstein and R. L. Salganik, “The Plane Problem on Curved Cracks in Elastic Body,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 3, 69–82 (1970) [Mech. Solids (Engl. Transl.)].

R. V. Goldstein and R. L. Salganik, “Brittle Fracture of Bodies with Arbitrary Cracks,” in Progress in Mechanics of Strains Media (Nauka, Moscow, 1975) [in Russian].

B. Cotterell and J. R. Rice, “Slightly Curved or Kinked Cracks,” Intern. J. Fract. 16(2), 155–169 (1980).

M. Amestoy and J. B. Leblond, “Crack Paths in Plane Situations — II: Detailed Form of the Expansion of the Stress Intensity Factors,” Intern. J. Solids and Structures 29(4), 465–501 (1992).

J. B. Leblond, “Crack Paths in Three-Dimensional Elastic Solids. I: Two-Term Expansion of the Stress Intensity Factors — Application to Cracks Path Stability in Hydraulic Fracturing,” Intern. J. Solids and Structures 36(1), 79–103 (1999).

D. Leguillon, “Asymptotic and Numerical Analysis of a Crack Branching in Non-Isotropic Materials,” Eur. J. Mech. A / Solids 12(1), 33–51 (1993).

H. Gao and C.-H. Chiu, “Slightly Curved or Kinked Cracks in Anisotropic Elastic Solids,” Intern. J. Solids and Structures 29(8), 947–972 (1992).

P. A. Martin, “Perturbed Cracks in Two-Dimensions: An Integral-Equation Approach,” Intern. J. Fract. 104(4), 317–327 (2000).

A. M. Khludnev and J. Sokolowski, “The Griffith Formula and the Rice-Cherepanov Integral for Crack Problems with Unilateral Conditions in Nonsmooth Domains,” Euro. J. Appl. Math. 10(4), 379–394 (1999).

V. A. Kovtunenko, “Invariant Energy Integrals for the Non-Linear Crack Problem with Possible Contact of the Crack Surfaces,” Prikl. Mat. Mekh. 67(1), 109–123 (2003) [J. Appl. Math. Mech. (Engl. Transl.) 67 (1), 99–110 (2003)].

J. Sokolowski and A. M. Khludnev, “On the Differentiation of Energy Functionals in Crack Theory with Possible Contact of the Crack Faces,” Dokl. Ross. Akad. Nauk 374(6), 776–779 (2000) [Russian Acad. Sci. Dokl. Math. (Engl. Transl.)].

E. M. Rudoi, “The Griffith Formula for a Plate with a Crack,” Sibirsk. Zh. Industr. Mat. 5(3), 155–161 (2002).

V. A. Kovtunenko, “Shape Sensitivity of Curvilinear Cracks on Interface to Non-Linear Perturbations,” ZAMP 54(4), 410–423 (2003).

V. A. Kovtunenko, “Sensitivity of Interfacial Cracks to Non-Linear Crack Front Perturbation,” ZAMM 82(6), 387–398 (2002).

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(2), 99–109 (2002).

E. M. Rudoi, “Invariant Integrals for the Equilibrium Problem for a Plate with a Crack,” Sibirsk. Mat. Zh. 45(2), 466–477 (2004) [Siberian Math. J. (Engl. Transl.) 45 (2), 388–397 (2004)].

E. M. Rudoi, “Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks,” Zh. Prikl. Mekh. Tekhn. Fiz. 45(6), 83–94 (2004) [J. Appl. Mech. Tech. Phys. 45 (6), 843–852 (2004)].

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions (Boca Raton, Ann Arbor, and London, 1992; Nauchnaya Kniga, Novosibirsk, 2002).

G. Duvaut and J.-L. Lions, Les Inequations en Mecanique et Physique (Dunod, Paris, 1972; Nauka, Moscow, 1980).

J. K. Knowles and E. Sternberg, “On a Class of Conservation Laws in Linearized and Finite Elastoplastic,” Arch. Ration. Mech. and Analysis 44(3), 187–211 (1972).