Journal of Applied and Industrial Mathematics

, Volume 4, Issue 4, pp 526–538

# A rigid inclusion in the contact problem for elastic plates

• N. V. Neustroeva
Article

## Abstract

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.

## Keywords

the Kirchhoff-Love model contact problem rigid inclusion

## References

1. 1.
A. M. Khludnev, “On the Unilateral Contact of Two Plates Situated Under an Angle to Each Other,” Prikl. Mekh. i Tekhn. Fiz. 49(4), 42–58 (2008).
2. 2.
G. Fichera, Existence Theorems in Elasticity. Handbuch der Physic, Vol. 6a/2 (Springer, Berlin, 1972; Nauka, Moscow, 1974).Google Scholar
3. 3.
L. A. Caffarelli and A. Friedman, “The Obstacle Problem for the Biharmonic Operator,” Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 6(1), 151–184 (1979).
4. 4.
L. A. Caffarelli, A. Friedman, and A. Torelli, “The Two-Obstacle Problem for the Biharmonic Operator,” Pacific J. Math. 103(2), 325–335 (1982).
5. 5.
B. Schild, “On the Coincidence Set in Biharmonic Variational Inequalities with Thin Obstacles,” Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 13(4), 559–616 (1986).
6. 6.
G. Dal Maso and G. Paderni, “Variational Inequalities for the Biharmonic Operator with Varying Obstacles,” Ann. Mat. Pura Appl. 153, 203–227 (1988).
7. 7.
A. M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics (Birkhauser, Basel, 1997).
8. 8.
I. Hlavachek, J. Haslinger, J. Nechas, and J. Lovishek, Solution of Variational Inequalities in Mechanics (Alfa, Bratislava, 1982; Springer, New York, 1988; Mir, Moscow, 1986).Google Scholar
9. 9.
A. M. Khludnev, The Problem of a Crack on the Boundary of a Rigid Inclusion in an Elastic Plate Preprint No. 1-2009 (Inst. of Hydrodynamics, Novosibirsk, 2009).Google Scholar
10. 10.
A. M. Khludnev and V. A. Kovtunenko Analysis of Cracks in Solids (WIT-Press, Boston, 2000).Google Scholar
11. 11.
A. M. Khludnev and G. Leugering, “Unilateral Contact Problems for Two Perpendicular Elastic Structures,” J. Anal. Appl. 27(2), 157–177 (2008).
12. 12.
A. M. Khludnev and A. Tani, “Unilateral Contact Problem for Two Inclined Elastic Bodies,” European J. Mech. A Solids 27(3), 365–377 (2008).
13. 13.
A. M. Khludnev, K.-H. Hoffmann, and N. D. Botkin, “The Variational Contact Problem for Elastic Objects of Different Dimensions,” Sibir. Mat. Zh. 47(3), 707–717 (2006) [Siberian Math. J. 47 (3), 584–593 (2006)].
14. 14.
N. V. Neustroeva, “The Contact Problem for Elastic Bodies of Different Dimensions,” Vestnik Novosibirsk. Gos. Univ. Ser. Mat. Mekh. Inform. 8(4), 60–75 (2008).Google Scholar