Unique solvability of crack problem with time-dependent friction condition in linearized elastodynamic body

  • Itou Hiromichi, h-itou@rs.tus.ac.jp Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
  • Kashiwabara Takahito, tkashiwa@ms.u-tokyo.ac.jp Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
Keywords: crack, friction, linearized elastodynamics


This study considers a crack problem with a time-dependent friction condition in a linearized elastodynamic body. We suppose that the crack is fixed and the frictional force acting on the crack is given and depends on the time as well as space variables. The problem is then reduced to a variational inequality of the hyperbolic type. The unique existence of a solution is proved by using Galerkin’s method and deriving a priori estimates for the penalized problem.


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

Bach M., Khludnev A. M., and Kovtunenko V. A., “Derivatives of the energy functional for 2D-problems with a crack under Signorini and friction conditions,” Math. Methods Appl. Sci., 23, No. 6, 515–534 (2000).

Duvalt G. and Lions J-L., Inequalities in Mechanics and Physics, Springer-Verl., Berlin (1976).

Eck C., Jaruˇsek J., and Krbec M., Unilateral Contact Problems, Variational Methods and Existence Theorems, Chapman & Hall/CRC, Boca Raton, FL (2005).

Furtsev A., Itou H., Kovtunenko V. A., Rudoy E., and Tani A., “On unilateral contact problems with friction for an elastic body with cracks,” RIMS Kˆokyˆuroku, Kyoto Univ., 2174, 43–58 (2021).

Furtsev A., Itou H., and Rudoy E., “Modeling of bonded elastic structures by a variational method: theoretical analysis and numerical simulation,” Int. J. Solids Structures, 182-183, 100–111 (2020).

Kikuchi N. and Oden J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, PA (1988).

Kovtunenko V. A., “Crack in a solid under Coulomb friction law,” Appl. Math., 45, No. 4, 265–290 (2000).

Neˇcas J., Jaruˇsek J., and Haslinger J., “On the solution of the variational inequality to the Signorini problem with a small friction,” Bol. Unione Mat. Ital., Ser. VII, B, 17, No. 2, 796–811 (1980).

Itou H., Kovtunenko V. A., and Tani A., “The interface crack with Coulomb friction between two bonded dissimilar elastic media,” Appl. Math., 56, No. 1, 69–97 (2011).

Tani A., “Dynamic unilateral contact problem with averaged friction for a viscoelastic body with cracks,” in: Mathematical Analysis of Continuum Mechanics and Industrial Applications, III, Proc. Int. Conf. CoMFoS18, pp. 3–21, Springer-Verl., Singapore (2020).

Hirano S. and Itou H., “Parameter interdependence of dynamic self-similar crack with distance-weakening friction,” Geophys. J. Int., 223, No. 3, 1584–1596 (2020).

Kashiwabara T., “On a strong solution of the non-stationary Navier–Stokes equations under slip or leak boundary conditions of friction type,” J. Differ. Equ., 254, No. 2, 756–778 (2013).

Lions J.-L. and Magenes E., Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer-Verl., Berlin; Heidelberg (1972).

Evans L. C., Partial Differential Equations, Amer. Math. Soc., Providence, RI (2010).

Simon J., “Compact Sets in the space Lp(0, T;B),” Ann. Mat. Pura Appl., 146, 65–96 (1986).
How to Cite
Itou, H. and Kashiwabara, T. (2021) “Unique solvability of crack problem with time-dependent friction condition in linearized elastodynamic body”, Mathematical notes of NEFU, 28(3), pp. 121-134. doi: https://doi.org/10.25587/SVFU.2021.38.33.008.