In this paper, we analyze equilibrium problems for 2D elastic bodies with two thin inclusions in the presence of damage. A delamination of the inclusions from the matrix is assumed, thus forming a crack between the elastic body and the inclusions. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. Weak and strong formulations of the problems are discussed. The paper provides an asymptotic analysis with respect to the damage parameter. An optimal control problem is analyzed with a cost functional to be equal to the derivative of the energy functional with respect to the crack length, and the damage parameter being a control function.
KikuchiNOdenJT.Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Philadelphia: SIAM, 1988.
2.
HanWSofoneaM.Quasistatic contact problems in viscoelasticity and viscoplasticity. In: Studies in Advanced Mathematics. Providence, RI:American Mathematical Society/Somerville, MA: International Press, 2002.
3.
KhludnevAMKovtunenkoVA.Analysis of Cracks in Solids. Southampton: WIT Press, 2000.
4.
KhludnevAM.Elasticity Problems in Non-smooth Domains. Moscow: Fizmatlit, 2010.
5.
ItouHKovtunenkoVARajagopalKR.Nonlinear elasticity with limiting small strain for cracks subject to non-penetration. Math Mech Solids2017; 22(6): 1334–1346.
6.
ItouHKovtunenkoVARajagopalKR.On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body. Math Mech Solids2018; 23(3): 433–444.
7.
LazarevNPRudoyEM.Shape sensitivity analysis of Timoshenko plate with a crack under the nonpenetration condition. Z Angew Math Mech2014; 94(9): 730–739.
8.
KhludnevAM.Singular invariant integrals for elastic body with delaminated thin elastic inclusion. Quart Appl Math2014; 72(4): 719–730.
KhludnevAMShcherbakovVV.Singular path-independent energy integrals for elastic bodies with Euler–Bernoulli inclusions. Math Mech Solids2017; 22(11): 2180–2195.
12.
LazarevNP.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 Phys2015; 66: 2025–2040.
13.
KhludnevAMNegriM.Crack on the boundary of a thin elastic inclusion inside an elastic body. Z Angew Math Mech2012; 92(5): 341–354.
14.
RudoiYM.An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem. J Appl Math Mech2011; 75(6): 731–738.
15.
MallickPK.Fiber-reinforced composites. In: Materials, Manufacturing, and Design. New York: Marcel Dekker, 1993.
16.
NasserMHassenA.Embedded beam under equivalent load induced from a surface moving load. Acta Mech1987; 67: 237–247.
17.
SaccomandiGBeattyMF.Universal relations for fiber-reinforced elastic materials. Math Mech Solids2002; 7: 99–110.
18.
ShcherbakovVV.Optimal control of rigidity parameter of thin inclusions in elastic bodies with curvilinear cracks. J Math Sci2014; 203(4): 591–604.
19.
ShcherbakovVV.On an optimal control problem for the shape of thin inclusions in elastic bodies. J Appl Indust Math2013; 7(3): 435–443.
20.
ShcherbakovVV.Choosing an optimal shape of thin rigid inclusions in elastic bodies. J Appl Mech Tech Phys2015; 56(2): 321–329.
21.
KhludnevAM.Shape control of thin rigid inclusions and cracks in elastic bodies. Arch Appl Mech2013; 83(10): 1493–1509.
22.
KhludnevAM.Rigidity parameter identification for thin inclusions located inside elastic bodies. J Opt Theory Appl2017; 172(1): 281–297.
23.
LazarevNP.Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack. Z Angew Math Mech2016; 96(4): 509–518.
24.
GaudielloAZappaleE.A model of joined beams as limit of a 2D plate. J Elasticity2011; 103: 205–233.
25.
GaudielloASiliA.Homogenization of highly oscillating boundaries with strongly contrasting diffusivity. SIAM J Math Anal2015; 47(3): 1671–1692.
26.
GaudielloAZappaleE.Junction in a thin multidomain for a forth order problem. Math Mod Meth Appl Sci2006; 16(12): 1887–1918.
27.
TiteuxISanchez-PalenciaE.Junction of thin plates. Eur J Mech A - Solid2000; 19(3): 377–400.
28.
CiarletPGLe DretHNzengwaR.Junctions between three-dimensional and two-dimensional linearly elastic structures. J Math Pures Appl1989; 68: 261–295.
29.
FaellaLKhludnevAMPopovaTS.Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies. Math Mech Solids2017; 22(4): 737–750.
30.
FaellaLKhludnevAM.Junction problem for elastic and rigid inclusions in elastic bodies. Math Meth Appl Sci2016; 39 (12): 3381–3390.
31.
KhludnevAMPopovaTS.Junction problem for rigid and semi-rigid inclusions in elastic bodies. Arch Appl Mech2016; 86(9): 1565–1577.
