A rigorous formulation is presented for the frictionless axisymmetric interaction of a rigid disk with a two-layered inhomogeneous medium. The materials are considered to be linearly elastic transversely isotropic materials and the exponential variation of properties along the depth of each layer is assumed in order to model the effect of inhomogeneity. The disk is considered to be at the top of the coating layer or embedded at the interface of the two media. By satisfying the boundary conditions, the problem leads to a dual integral equation and is reducible to a Fredholm integral equation of the second kind which is solved numerically. The numerical solutions survey the effect of material inhomogeneity through two special cases: a functionally graded coating on a homogeneous transversely isotropic half-space and a homogeneous transversely isotropic coating on an inhomogeneous half-space. The accuracy of the numerical solutions is verified by comparison with the corresponding solution for homogeneous material in the literature.
ReddyJN.Thermomechanical behavior of functionally graded materials. Final report for Air Force Office of Scientific Research (grant F49620-95-1-0342), Arlington, VA, 1998.
2.
BirmanVByrdL.Modeling and analysis of functionally graded materials and structures. Appl Mech Rev2007; 60: 195–216.
3.
WangXPanERoyAK.A functionally graded plane with a circular inclusion under uniform antiplane eigenstrain. J Appl Mech2008; 75: 014501.
4.
StapletonSWaasAMArnoldSM.Functionally graded adhesives for composite joints. Int J Adhes Adhes2012; 35: 36–49.
5.
SelvaduraiA.Elastic contact between a flexible circular plate and a transversely isotropic elastic halfspace. Int J Solid Struct1980; 16: 167–176.
6.
SelvaduraiA.The settlement of a rigid circular foundation resting on a halfspace exhibiting a near surface elastic non-homogeneity. Int J Numer Anal Meth1996; 20: 251–364.
7.
SelvaduraiALanQ.A disc inclusion at a non-homogeneous elastic interface. Mech Res Commun1997; 24: 473–488.
8.
SelvaduraiALanQ.Axisymmetric mixed boundary value problems for an elastic halfspace with a periodic nonhomogeneity. Int J Solid Struct1998; 35: 1813–1826.
9.
KatebiAKhojastehARahimianM. Axisymmetric interaction of a rigid disc with a transversely isotropic elastic half-space. Int J Numer Anal Meth2009; 34(12): 1211–1236.
10.
KatebiASelvaduraiA.A frictionless contact problem for a flexible circular plate and an incompressible nonhomogeneous elastic halfspace. Int J Mech Sci2015; 90: 239–245.
11.
SelvaduraiAKatebiA.An adhesive contact problem for an incompressible non-homogeneous elastic halfspace. Acta Mech, 2015; 226(2): 249–265. DOI: 10.1007/s00707-014-1171-8.
12.
RahmanMNewazG.Elastostatic surface displacements of a half-space reinforced by a thin film due to an axial ring load. Int J Eng Sci1997; 35(6): 603–611.
13.
RahmanMNewazG.Boussinesq type solution for a transversely isotropic half-space coated with a thin film. Int J Eng Sci2000; 38: 807–822.
14.
ShodjaHEskandariM.Axisymmetric time-harmonic response of a transversely isotropic substrate—coating system. Int J Eng Sci2007; 45: 272–287.
15.
KhojastehARahimianMPakR. Asymmetric dynamic Green’s functions in a two-layered transversely isotropic half-space. J Eng Mech (ASCE)2008; 134(9): 777–787.
16.
AizikovichSAlexandrovVKalkerJ. Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties. Int J Solid Struct2002; 39(10): 2745–2772.
GulerMAErdoganF.The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings. Int J Mech Sci2007; 49(2): 161–182.
19.
AizikovichSKrenevLTrubchikI.The deformation of a half-space with a gradient elastic coating under arbitrary axisymmetric loading. J Appl Math Mech-USS2008; 72(4): 461–467.
20.
AizikovichSAlexandrovVMTrubchikI. Introduction of a spherical indenter into a half-space with a functionally graded elastic coating. Dokl Phys2008; 53(1): 29–33.
LiuTJWangYS.Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus. Acta Mech2008; 199: 151–165.
23.
ChoiHJ.On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch. J Mech Sci Tech2009; 23: 2703–2713.
24.
KashtalyanMMenshykovaM.Effect of a functionally graded interlayer on three-dimensional elastic deformation of coated plates subjected to transverse loading. Compos Struct2009; 89(2): 167–176.
25.
LeeDBarberJThoulessM.Indentation of an elastic half space with material properties varying with depth. Int J Eng Sci2009; 47: 1274–1283.
26.
KeLLWangYS.Fretting contact of two dissimilar elastic bodies with functionally graded coatings. Mech Adv Mater Struct2010; 17: 433–447.
27.
LiuT-jWangY-sXingY-m.Fretting contact of two elastic solids with graded coatings under torsion. Int J Solid Struct2012; 49: 1283–1293.
28.
ChenPChenS.Partial slip contact between a rigid punch with an arbitrary tip-shape and an elastic graded solid with a finite thickness. Mech Mater2013; 59: 24–35.
29.
VolkovSAizikovichSWangY. Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer. Acta Mech Sin2013; 29(2): 196–201.
30.
Kulchytsky-ZhyhailoRBajkowskiAS.Stresses in coating with gradient interlayer caused by contact loading. Acta Mech Autom2014; 8(1): 33–37.
31.
LiuTJXingYM.Analysis of graded coatings for resistance to contact deformation and damage based on a new multi-layer model. Int J Mech Sci2014; 81(7): 158–164.
32.
LiuTJXingYMWangYS.The axisymmetric contact problem of a coating/substrate system with a graded interfacial layer under a rigid spherical punch. Math Mech Solid. Epub online ahead of print 20October2014. DOI: 10.1177/1081286514555563.
33.
MaJKeLLWangYS.Sliding frictional contact of functionally graded magneto-electro-elastic materials under a conducting flat punch. J Appl Mech2015; 82(1): 011009.
34.
LiuTJWangYS.Reissner–Sagoci problem for functionally graded materials with arbitrary spatial variation of material properties. Mech Res Commun2009; 36: 322–329.
35.
MatysiakSJKulchytsky-ZhyhailoRPerkowskiDM.Reissner–Sagoci problem for a homogeneous coating on a functionally graded half-space. Mech Res Commun2011; 38: 320–325.
36.
LeeDTriantafyllidisNBarberJ. Surface instability of an elastic half space with material properties varying with depth. J Mech Phys Solid2008; 56(3): 858–868.
37.
WuZMengJLiuY. A state space method for surface instability of elastic layers with material properties varying in thickness direction. J Appl Mech2014; 81: 171–188.
38.
MaoJJKeLLWangYS.Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane. Int J Solid Struct2014; 51(23–24): 3962–3972.
39.
MaoJJKeLLWangYS.Thermoelastic instability of a functionally graded layer interacting with a homogeneous layer. Int J Mech Sci2015; 99: 218–227.
40.
ShulCWLeeKY.A subsurface eccentric crack in a functionally graded coating layer on the layered half-space under an anti-plane shear impact load. Compos Struct2002; 39(7): 2019–2029.
41.
GuoLCWuLZZengT. The dynamic fracture behavior of a functionally graded coating–substrate system. Compos Struct2004; 64(3–4): 433–441.
42.
GuoLCWuLZ. The interface crack problem under a concentrated load for a functionally graded coating–substrate composite system. Compos Struct2004; 63(3–4): 397–406.
43.
WangCLinYJengY. Wave propagation in an inhomogeneous cross-anisotropic medium. Int J Numer Anal Meth2010; 34: 711–732.
44.
SneddonIN.The use of integral transforms. New York, NY: McGraw–Hill, 1972.
45.
KhojastehARahimianMEskandariM. Asymmetric wave propagation in a transversely isotropic half-space in displacement potentials. Int J Eng Sci2008; 46: 690–710.
46.
SneddonIN.Mixed boundary value problems in potential theory. New York, NY: North-Holland, 1966.
47.
NobleB.The solution of Bessel function dual integral equations by a multiplying-factor method. Math Proc Cambr Philos Soc1963; 59: 351–362.
48.
EskandariMShodjaH.Green’s functions of an exponentially graded transversely isotropic half-space. Int J Solid Struct2010; 47: 1537–1545.
