On junction problem for elastic Timoshenko inclusion and semi-rigid inclusion

  • Khludnev Alexander M., khlud@hydro.nsc.ru Lavrentiev Institute of Hydrodynamics, 15 Lavrentiev Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk, 630090, Russia
  • Popova Tatyana S., ptsokt@mail.ru M. K. Ammosov North-Eastern Federal University, 58 Belinsky Street, Yakutsk 677000, Russia
Keywords: Timoshenko inclusion, semi-rigid inclusion, elastic body, crack, nonlinear boundary conditions

Abstract

An equilibrium problem for elastic bodies with a thin elastic inclusion and a thin semi-rigid inclusion is investigated. The inclusions are assumed to be delaminated from the elastic bodies, forming therefore a crack between the inclusions and the elastic matrix. Nonlinear boundary conditions are considered at the crack faces to prevent mutual penetration between the crack faces. The inclusions have a joint point. We present both differential formulation in the form of a boundary value problem and a variational formulation in the form of a minimization problem for an energy functional on a convex set of admissible displacements. The unique solvability of the problem is substantiated. Equivalence of differential and variational statements is shown. Passage to the limit is investigated as the rigidity parameter of the elastic inclusion goes to infinity. The limit model is analyzed. Junction boundary conditions are found at the joint point for the considered problem as well as for the limit problem.

References

[1]Khludnev A. M., Negri M., “Crack on the boundary of a thin elastic inclusion inside an elastic body”, Z. Angew. Math. Mech., 92:5 (2012), 341–354  crossref  mathscinet  zmath  elib  scopus
[2]Itou H., Khludnev A. M., “On delaminated thin Timoshenko inclusions inside elastic bodies”, Math. Meth. Appl. Sci., 39:17 (2016), 4980–4993  crossref  mathscinet  zmath  scopus
[3]Khludnev A. M., Leugering G. R., “Delaminated thin elastic inclusion inside elastic bodies”, Math. Mech. Complex Systems, 2:1 (2014), 1–21  crossref  mathscinet  zmath  elib  scopus
[4]Khludnev A. M., Leugering G. R., “On Timoshenko thin elastic inclusions inside elastic bodies”, Mathematics and Mechanics of Solids, 20:5 (2015), 495–511  crossref  mathscinet  zmath  scopus
[5]Khludnev A. M., “Optimal control of inclusions in an elastic body crossing the external boundary”, Sib. Zh. Ind. Mat., 18:4 (2015), 75–87  mathnet  zmath  elib
[6]Morozov N. F., Mathematical Questions of the Crack Theory, Nauka, Moscow, 1984
[7]Khludnev A. M., “Singular invariant integrals for elastic body with delaminated thin elastic inclusion”, Quart. Appl. Math., 72:4 (2014), 719–730  crossref  mathscinet  zmath  elib  scopus
[8]Itou H., Khludnev A. M., Rudoy E. M., Tani A., “Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity”, Z. Angew. Math. Mech., 92:9 (2012), 716–730  crossref  mathscinet  zmath  elib  scopus
[9]Khludnev A. M., Kovtunenko V. A., Analysis of cracks in solids, WIT Press, Southampton; Boston, 2000
[10]Khludnev A. M., Problems of Elasticity Theory in Nonsmooth Domains, Fizmatlit, Moscow, 2010
[11]Kovtunenko V. A., “Invariant energy integrals for the nonlinear crack problem with possible contact of the crack surfaces”, J. Appl. Math. Mech., 67:1 (2003), 99–110  crossref  mathscinet  zmath  scopus
[12]Kovtunenko V. A., “Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration”, IMA J. Appl. Math., 71:5 (2006), 635–657  crossref  mathscinet  zmath  elib  scopus
[13]Knees D., Schroder A., “Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints”, Math. Methods Appl. Sci., 35:15 (2012), 1859–1884  crossref  mathscinet  zmath  scopus
[14]Rudoy E. M., “The Griffith formula and Cherepanov–Rice integral for a plate with a rigid inclusion and a crack”, J. Math. Sci., 186:3 (2012), 511–529  mathnet  crossref  mathscinet  elib  scopus
[15]Rudoy E. M., “Asymptotic behavior of the energy functional for a three-dimensional body with a rigid inclusion and a crack”, J. Appl. Math. Mech., 75:6 (2011), 731–738  crossref  scopus
[16]Khludnev A. M., “The problem for a crack on the border of a rigid inclusion in a elastic plate”, Izv. Akad. Nauk, Mekh. Tvyord. Tela, 2010, no. 5, 98–110
[17]Lazarev N. P., “The equilibrium problem for a Timoshenko-type shallow shell containing a through crack”, J. Appl. Ind. Math., 7:1 (2013), 78–88  mathnet  crossref  mathscinet  zmath  elib  scopus
[18]Khludnev A. M., “On the equilibrium of a two-layer elastic body with a crack”, J. Appl. Ind. Math., 7:3 (2013), 370–379  mathnet  crossref  mathscinet  zmath  elib  scopus
[19]Shcherbakov V. V., “On an optimal control problem for the shape of thin inclusions in elastic bodies”, J. Appl. Ind. Math., 7:3 (2013), 435–443  mathnet  crossref  mathscinet  zmath  elib  scopus
[20]Shcherbakov V. V., “Existence of an optimal shape of the thin rigid inclusions in the Kirchhoff–Love plate”, J. Appl. Ind. Math., 8:1 (2014), 97–105  mathnet  crossref  mathscinet  elib  scopus
[21]Lazarev N. P., “Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion”, Z. Angew. Math. Phys., 66:4 (2015), 2025–2040  crossref  mathscinet  zmath  elib  scopus
[22]Lazarev N. P., Rudoy E. M., “Shape sensitivity analysis of Timoshenko's plate with a crack under the nonpenetration condition”, Z. Angew. Math. Mech., 94 (2014), 730–739  crossref  mathscinet  zmath  elib  scopus
[23]Rudoy E. M., Khludnev A. M., “Unilateral contact of a plate with a thin elastic obstacle”, Sib. Zh. Ind. Mat., 12:2 (2009), 120–130  mathnet  mathscinet  zmath
[24]Shcherbakov V. V., “The Griffith formula and J-integral for elastic bodies with Timoshenko inclusions”, Z. Angew. Math. Mech., 96:11 (2016), 1306–1317  crossref  mathscinet  elib  scopus
[25]Bessoud A.-L., Krasucki F., Serpilli M., “Plate-like and shell-like inclusions with high rigidity”, Compt. Rend. Math., 346 (2008), 697–702  crossref  mathscinet  zmath  scopus
[26]Bessoud A.-L., Krasucki F., Michaille G. Multi-materials with strong interface: Variational modelings, Asymptotic Analysis, 61:1 (2009), 1–19  mathscinet  zmath
[27]Pasternak I. M., “Plane problem of elasticity theory for anisotropic bodies with thin elastic inclusions”, J. Math. Sci., 186:1 (2012), 31–47  crossref  mathscinet  elib  scopus
[28]Vynnytska L., Savula Y., “Mathematical modeling and numerical analysis of elastic body with thin inclusion”, Comput. Mech., 50:5 (2004), 533–542  crossref  mathscinet  scopus
[29]Kozlov V. A., Maz'ya V. G., Movchan A. B., Asymptotic analysis of fields in a multi-structure, Oxford Math. Monogr., Oxford Univ. Press, New York, 1999  mathscinet
[30]Le Dret H., “Modeling of the junction between two rods”, J. Math. Pure Appl., 68 (1989), 365–397  mathscinet  zmath
[31]Titeux I., Sanchez-Palencia E., “Junction of thin plate”, Europ. J. Mech. A/Solids, 19:3 (2000), 377–400  crossref  mathscinet  zmath  adsnasa  scopus
[32]Gaudiello A., Zappale E., “Junction in a thin multidomain for a forth order problem”, Math. Models Methods Appl. Sci., 16:12 (2006), 1887–1918  crossref  mathscinet  zmath  scopus
[33]Gaudiello A., Zappale E., “A model of joined beams as limit of a 2D plate”, J. Elasticity, 103:2 (2011), 205–233  crossref  mathscinet  zmath  elib  scopus
[34]Ciarlet P. G., Le Dret H., Nzengwa R., “Junctions between three dimensional and two dimensional linearly elastic structures”, J. Math. Pures Appl., 6 (1989), 261–295  mathscinet
[35]Faella L., Khludnev A. M., Popova T. S., “Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies”, Mathematics and Mechanics of Solids, 22:4 (2017), 737– 750  crossref  mathscinet  zmath  scopus
[36]Khludnev A. M., Popova T. S., “On the mechanical interplay between Timoshenko and semirigid inclusions embedded in elastic bodies”, Z. Angew. Math. Mech., 97:11 (2017), 1406– 1417  mathscinet  elib
[37]Khludnev A. M., Popova T. S., “Junction problem for rigid and semi-rigid inclusions in elastic bodies”, Arch. Appl. Mech., 86:9 (2016), 1565–1577  crossref  mathscinet  elib  scopus
[38]Khludnev A. M., Popova T. S., “Junction problem for Euler-Bernoulli and Timoshenko elastic inclusions in elastic bodies”, Quart. Appl. Math., 74:4 (2016), 705–718  crossref  mathscinet  zmath  elib  scopus
[39]Grigolyuk E. I., Selezov I. T., Mekhanika Tvyordyh Deformiruemyh Tel. Neklassicheskie Teorii Kolebaniy Sterzhney, Plastin i Obolochek, v. 5, Nauka, Moscow, 1973
How to Cite
Khludnev, A. and Popova, T. ( ) “On junction problem for elastic Timoshenko inclusion and semi-rigid inclusion”, Mathematical notes of NEFU, 25(1), pp. 73-89. doi: https://doi.org/10.25587/SVFU.2018.1.12770.
Section
Mathematics