We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.
BellieudMBouchitteG.Homogenization of a soft elastic material reinforced by fibers. Asymptotic Anal2002; 32: 153–183.
2.
El JarroudiM. Homogenization of an elastic material reinforced with thin rigid von Karman ribbons. Math Mech Solids2019; 24(7):1965–1991.
3.
KhludnevAM.Elasticity problems in non-smooth domains. Moscow: Fizmatlit, 2010.
4.
RudoyEMLazarevNP.Domain deecomposition technique for a model of an elastic body reinforced by a Timoshenko’s beam. J Comput Appl Math2018; 334: 18–26.
5.
FankinaIVFurtsevAIRudoyEM, et al. Asymptotic modeling of curvilinear narrow inclusions with rough boundaries in elastic bodies: case of a soft inclusion. Siber Electr Math Reports2022; 19(2): 935–948.
6.
FankinaIVFurtsevAIRudoyEM, et al. A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion. Math Mech Solids2024; 29(4): 796–817.
7.
ShcherbakovVV.Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks. Nonlinear Anal Real World Appl2022; 65: 103505.
8.
RudoyE.Asymptotic justification of models of plates containing inside hard thin inclusions. Technologies2020; 8(4): 59.
9.
FurtsevARudoyE.Variational approach to modeling soft and stiff interfaces in the Kirchhoff-Love theory of plates. Int J Solids Struct2020; 202: 562–574.
10.
KhludnevAM.On modeling thin inclusions in elastic bodies with a damage parameter. Math Mech Solids2019; 24(9): 2742–2753.
11.
FurtsevARudoyEItouH.Modeling of bonded elastic structures by a variational method: theoretical and numerical simulation. Int J Solids Structures2020; 182–183: 100–111.
12.
LazarevNPItouH.Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff-Love plates with a crack. Math Mech Solids2019; 24(12): 3743–3752.
13.
MorassiARossetEVessellaS.Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-Love plate. SIAM J Math Anal2019; 51: 731–747.
14.
KovtunenkoVALeugeringG.A shape-topological control problem for nonlinear crack-defect interaction: the anti-plane variational model. SIAM J Control Optim2016; 54: 1329–1351.
15.
GaudielloAZappaleE.A model of joined beams as limit of a 2D plate. J Elasticity2011; 103: 205–233.
16.
GaudielloAGómezDPérez-MartnezME.Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure. J Math Pures Appl2020; 134: 299–327.
17.
GaudielloASiliA.Limit models for thin heterogeneous structures with high contrast. J Diff Equat2021; 302: 37–63.
18.
KozlovVAMazyaVGMovchanAB.Asymptotic analysis of fields in a multi-structure (Oxford mathematical monographs). New York: Oxford University Press, 1999.
19.
PanasenkoG.Multi-scale modelling for structures and composites. New York: Springer, 2005.
20.
MallickPK.Fiber-reinforced composites. Materials, manufacturing, and design. New York: Marcel Dekker, 1993.
21.
AttouchHButtazzoGMichailleG.Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. 2nd ed.Philadelphia, PA: SIAM, 2014.
22.
GoelevenD.Noncoercive hemivariational inequality and its applications in nonconvex unilateral mechanics. Appl Math1996; 41(3): 203–229.
23.
KohnJJNirenbergL.Non-coercive boundary value problems. Comm Pure Appl Math1965; 18: 443–492.
24.
KhludnevAM.Non-coercive problems for Kirchhoff-Love plates with thin rigid inclusion. Z Angew Math Phys2022; 73:54.
25.
KhludnevAM.Asymptotics of solutions for two elastic plates with thin junction. Siber Electr Math Reports2022; 19(2): 484–501.
26.
KhludnevAMRodionovAA.Elastic body with thin nonhomogeneous inclusion in non-coercive case. Math Mech Solids2023; 28(10): 2141–2154.
27.
KhludnevAMRodionovAA.Elasticity tensor identification in elastic body with thin inclusions: non-coercive case. J Opt Theory Appl2023; 197(3): 993–1010.
