On the equilibrium problem for a two-layer structure with the upper layer covering a defect tip

I. V. Frankina

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 141-160.

Résumé

The equilibrium problem of a two-layer elastic structure is investigated. In the lower layer there is a rectilinear defect. The upper layer covers one of the defect tips and is glued to the lower layer along its edge. Nonlinear boundary conditions are used to model the defect. Using the variational approach, the existence of a solution of the problem is established. Passages to the limit in the problem with respect to a parameter characterizing the elasticity of the upper layer, as well as to the defect damage parameter are carried out. The optimal control problem is considered, in which the cost functional is the derivative of the energy functional with respect to the defect length, and two parameters mentioned above act as control functions. The solvability of the optimal control problem is proved.

Détail

Keywords: two-layer structure, nonpenetration condition, defect, damage parameter, variational inequality, optimal control problem.

Bibliographie

[1] A.M. Khludnev, “On modeling elastic bodies with defects”, Sib. Electron. Math. Izv., 15 (2018), 153–166 | MR | Zbl

[2] A.M. Khludnev, “On thin inclusions in elastic bodies with defects”, Z. Angew. Math. Phys., 70:45 (2019) | MR | Zbl

[3] A.M. Khludnev, “On thin Timoshenko inclusions in elastic bodies with defects”, Arch. Appl. Mech., 89:8 (2019), 1691–1704 | DOI | MR

[4] A.M. Khludnev, Elasticity Problems in Nonsmooth Domains, Fizmatlit, M., 2010

[5] T.S. Popova, “Contact problem for two viscoelastic plates”, Mat. Notes YaGU, 12:2 (2005), 60–92

[6] N.V. Neustroeva, “Rigid inclusion in the contact problem for elastic plates”, J. Appl. Ind. Math., 4:4 (2010), 526–538 | DOI | MR | Zbl

[7] T.A. Rotanova, “On formulation and solvability of problems on the contact of two plates with rigid inclusions”, Sib. Zh. Ind. Mat., 15:2 (2012), 107–118 | MR | Zbl

[8] D. Knees, A. Schroder, “Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints”, Math. Methods Appl. Sci., 35:15 (2012), 1859–1884 | DOI | MR | Zbl

[9] L. Freddi, T. Roubicek, C. Zanini, “Quasistatic delamination of sandwich-like Kirchhoff-Love plates”, J. Elasticity, 113:2 (2013), 219–250 | DOI | MR | Zbl

[10] V.A. Kovtunenko, G. Leugering, “A shape-topological control problem for nonlinear crack-defect interaction: the antiplane variational model”, SIAM J. Control Optim., 54:3 (2016), 1329–1351 | DOI | MR | Zbl

[11] S. Almi, “Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening”, ESAIM, Control Optim. Calc. Var., 23:3 (2017), 791–826 | DOI | MR | Zbl

[12] A. Chambolle, S. Conti, G. Francfort, “Approximation of a brittle fracture energy with a constraint of non-interpenetration”, Arch. Ration. Mech. Anal., 228:3 (2018), 867–889 | DOI | MR | Zbl

[13] A.M. Khludnev, “On crack problem with overlapping domain”, Z. Angew. Math. Mech., 88:8 (2008), 650–660 | DOI | MR | Zbl

[14] A.M. Khludnev, A. Tani, “Overlapping domain problems in the crack theory with possible contact between crack faces”, Q. Appl. Math., 66:3 (2008), 423–435 | DOI | MR | Zbl

[15] M.P. Savruk, V.S. Kravets, “Influence of reinforcement pads on distribution stress in cracked plates”, Prikl. Mech., 29:3 (1993), 48–55

[16] A. Yu. Zemlyanova, V.V. Sil'vestrov, “A problem on reinforcement of a plate with a cut by a two-dimensional covering strap”, J. Appl. Math. Mech., 71:1 (2007), 40–51 | DOI | MR | Zbl

[17] Yu.O. Vasil'eva, V.V. Sil'vestrov, “The problem of an interface crack with a rigid patch plate on part of its edge”, J. Appl. Math. Mech., 75:6 (2011), 716–730 | DOI | MR | Zbl

[18] A. Gaudiello, A.M. Khludnev, “Crack on the boundary of two overlapping domains”, Z. Angew. Math. Phys., 61:2 (2010), 341–356 | DOI | MR | Zbl

[19] A.M. Khludnev, G.R. Leugering, “Optimal control of cracks in elastic bodies with thin rigid inclusion”, Z. Angew. Math. Mech., 91:2 (2011), 125–137 | DOI | MR | Zbl

[20] A.M. Khludnev, “On the equilibrium of a two-layer elastic body with a crack”, J. Appl. Ind. Math., 7:3 (2013), 370–379 | DOI | MR | Zbl

[21] E.V. Pyatkina, “Optimal control of the layer size in the problem of equilibrium of elastic bodies with overlapping domains”, J. Appl. Ind. Math., 10:3 (2016), 435–443 | DOI | MR | Zbl

[22] E.V. Pyatkina, “On control problem for two-layers elastic body with a crack”, J. Math. Sci., 230:1 (2018), 159–166 | DOI | MR

[23] E.M. Rudoy, N.A. Kazarinov, V. Yu. Slesarenko, “Numerical simulation of the equilibrium of an elastic two-layer structure with a through crack”, Numer. Anal. Appl., 10:1 (2017), 63–73 | DOI | MR

[24] G.P. Cherepanov, Mechanics of Brittle Fracture, Nauka, M., 1974 | MR

[25] V.Z. Parton, E.M. Morozov, Mechanics of elastic-plastic fracture, Hemisphere Publishing Corp., Washington, DC, 1989 | MR | Zbl

[26] E. M. Rudoy, “Differentiation of energy functionals in two-dimensional elasticity theory for solids with curvilinear cracks”, J. Appl. Mech. Tech. Phys., 45:6 (2004), 843–852 | DOI | MR | Zbl

[27] N.P. Lazarev, “The Griffith formula for a Timoshenko-type plate with a curvilinear track”, Sib. Zh. Ind. Mat., 16:2 (2013), 98–108 | MR | Zbl

[28] V. Shcherbakov, “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions”, Z. Angew. Math. Phys., 68:26 (2017) | MR | Zbl

[29] N. P. Lazarev, S. Das, M. P. Grigoryev, “Optimal control of a thin rigid stiffener for a model describing equilibrium of a Timoshenko plate with a crack”, Sib. Elektron. Mat. Izv., 15 (2018), 1485–1497 | DOI | MR | Zbl

[30] V.P. Mikhailov, Partial Differential Equations, Nauka, M., 1976 | MR

[31] R. Temam, Mathematical Problems of Plasticity Theory, Nauka, M., 1991 | MR | Zbl