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Journal of Applied and Industrial Mathematics

, Volume 13, Issue 2, pp 208–218 | Cite as

A Contact Problem for a Plate and a Beam in Presence of Adhesion

  • A. I. FurtsevEmail author
Article
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Abstract

Under consideration is the problem of contact between a plate and a beam. It is assumed that no mutual penetration between the plate and the beam can occur, and so an appropriate nonpenetration condition is used. On the other hand, the adhesion of the bodies is taken into account which is characterized by a numerical adhesion parameter. We study the existence and uniqueness of a solution for the contact problem as well as the passage to the limit with respect to the adhesion parameter. The accompanying optimal control problem is investigated in which the adhesion parameter acts as a control parameter.

Keywords

contact plate beam thin obstacle nonpenetration condition defect adhesion minimization problem variational inequality optimal control 

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References

  1. 1.
    L. A. Caffarelli, “Further Regularity for the Signorini Problem,” Comm. Partial Differential Equations 4 (9), 1067–1075 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. Athanasopoulos and L. A. Caffarelli, “Optimal Regularity of Lower-Dimensional Obstacle Problems,” J. Math. Sci. 132 (3), 274–284 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Petrosyan, H. Shahgholian, and N. N. Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems (Amer. Math. Soc., Providence, 2012).CrossRefzbMATHGoogle Scholar
  4. 4.
    L. A. Caffarelli and A. Friedman, “The Obstacle Problem for the Biharmonic Operator,” Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6 (1), 151–184 (1979).MathSciNetzbMATHGoogle Scholar
  5. 5.
    L. A. Caffarelli, A. Friedman, and A. Torelli, “The Two-Obstacle Problem for the Biharmonic Operator,” Pacific J. Math. 103 (3), 325–335 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    B. Schild, “On the Coincidence Set in Biharmonic Variational Inequalities with Thin Obstacles,” Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV.Ser. 13 (4), 559–616 (1986).MathSciNetzbMATHGoogle Scholar
  7. 7.
    G. Dal Maso and G. Paderni, “Variational Inequalities for the Biharmonic Operator with Varying Obstacles,” Ann. Mat. Pura Appl. 153 (1), 203–227 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics (Birkhauser, Basel, 1997).zbMATHGoogle Scholar
  9. 9.
    A. M. Khludnev, “On Unilateral Contact of Two Plates Aligned at Angle to Each Other,” J. Appl. Mech. Tech. Phys. 49 (4), 553–567 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. M. Khludnev and G. Leugering, “Unilateral Contact Problems for Two Perpendicular Elastic Structures,” Z. Anal. Anwend. 27 (2), 157–177 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. V. Neustroeva, “A Rigid Inclusion in the Contact Problem for Elastic Plates,” Sibir. Zh. Industr. Mat. 12 (4), 92–105 (2009) [J. Appl. Indust. Math. 4 (4), 526–538 (2010)].MathSciNetzbMATHGoogle Scholar
  12. 12.
    T. A. Rotanova, “Unilateral Contact Problem for Two Plates with a Rigid Inclusion in the Lower Plate,” J. Math. Sci. 188 (4), 452–462 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  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)].zbMATHGoogle Scholar
  14. 14.
    A. M. Khludnev and A. Tani, “Unilateral Contact Problem for Two Inclined Elastic Bodies,” European J. Mech. A: Solids 27 (3), 365–377 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. I. Furtsev, “About the Contact of a Thin Obstacle and a Plate Containing a Thin Inclusion,” Sibir. Zh. Chist. Prikl. Mat. 17 (4), 94–111 (2017).MathSciNetGoogle Scholar
  16. 16.
    A. I. Furtsev, “Differentiation of Energy Functional with Respect to Delamination Length in Problem of Contact between Plate and Beam,” Sibir. Elektr. Mat. Izv. 15, 935–949 (2018); http://semr.math.nsc.ru.MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. M. Khludnev, “On Modeling Thin Inclusions in Elastic Bodies with a Damage Parameter,” Math. Mech. Solids. 2018; doi  https://doi.org/10.1177/1081286518796472.
  18. 18.
    A. M. Khludnev, “On Modeling Elastic Bodies with Defects,” Sibir. Electr. Mat. Izv. 15, 153–166 (2018); URL: http://semr.math.nsc.ru.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lavrent’ev Institute of HydrodynamicsNovosibirskRussia

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