On unilateral contact problems with friction for anelastic body with cracks
Hiromichi Itou

Department of Mathematics, Tokyo University of Science
We discuss about frictional contact problems in two dimensional elasticity, which are
arising from a wide variety of phenomena in mechanical engineering and concerning withsome inverse problems and control problems. Contact conditions for cracks are so-called
non-penetration conditions deﬁned as unilateral conditions on the displacements of bodies toexclude nonphysical phenomenon such as mutual penetration of crack faces, see [1] for thedetails. In this talk, we introduce mathematical results obtained in [5] and [6]. First, in [5]Coulomb friction in the static case is considered and the existence of the solution is shown by
using penalization method. Moreover, we derive asymptotic expansions of the solution near
the crack tip. Second, an interfacial crack of two elastic bodies is considered in [6]. On thecrack the non-penetration condition and Tresca friction condition which is an approximationof Coulomb friction (e.g. [2]) are imposed. On the other part of the interface, both ofadhesion forces and friction force are taken into account. Then we derive a formula for thederivative of the energy functional with respect to the crack length, which can be represented
as a path-independent integral (J -integral) and also some numerical results are introduced.
Finally, we mention about dynamic unilateral contact problems. In my knowledge, there area few mathematical results dealing with the existence of a solution to such kinds of problems(cf. [3, 4, 7]). Then, we overview the related known results and clarify the main diﬃculty.

AcknowledgmentsH. Itou was partially supported by Grant-in-Aid for Scientiﬁc Research (C)(No. 18K03380)
and (B)(No. 17H02857) of Japan Society for the Promotion of Science and JSPS and RFBR
under the Japan - Russia Research Cooperative Program (project No. J19-721). I would liketo thank V. A. Kovtunenko (University of Graz), A. Tani (Keio University), A. I. Furtsev andE. M. Rudoy (Lavrentyev Institute of Hydrodynamics of the Russian Academy of Sciences)
for the cooperation and fruitful discussions.

References[1] Khludnev A. M. and Kovtunenko V. A., Analysis of Cracks in Solids. WIT-Press,
Southampton, 2000.
[2] Kikuchi N. and Oden J. T., Contact problems in elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, 1988.
[3] Duvalt G. and Lions J-L., Inequalities in Mechanics and Physics. Springer, 1976.
[4] Eck C., Jarušek J. and Krbec M., Unilateral Contact Problems. Variational Methods andExistence Theorems. Chapman & Hall / CRC, 2005.
[5] Itou H., Kovtunenko V. A. and Tani A., The interface crack with Coulomb friction
between two bonded dissimilar elastic media. Appl. Math., 56(2011), pp. 69–97.
[6] Furtsev A., Itou H. and Rudoy E., Modeling of bonded elastic structures by a variational
method: Theoretical analysis and numerical simulation. Internat. J. Solids Structures,
182–183(2020), pp. 100-111.
[7] Tani A., Dynamic unilateral contact problem with averaged friction for a viscoelastic
body with cracks. Mathematical Analysis of Continuum Mechanics and Industrial Applications III -Proceedings of the International Conference CoMFoS18- (Eds. Itou, H. etal.), Springer, to appear.