We prove the existence of a solution to a problem modeling the stationary heat distribution in an inhomogeneous plane with a crack and derive explicit representations of singular terms of an asymptotic expansion of the heat flow in the vicinity of the crack tips.
G.Anlas, J.Lambros and M.H.Santare, Dominance of asymptotic crack tip fields in elastic functionally graded materials, Int. J. Fract.115 (2002), 193–204.
2.
V.Birman and L.W.Byrd, Modeling and analysis of functionally graded materials and structures, ASME Appl. Mech. Rev.60 (2007), 195–216.
3.
Y.-S.Chan, L.J.Gray, T.Kaplan and G.H.Paulino, Green’s function for a two-dimensional exponentially graded elastic medium, Proc. Royal Soc. Lond. A460 (2004), 1689–1706.
4.
T.-C.Chiu, S.-W.Tsai and C.-H.Chue, Heat conduction in a functionally graded medium with an arbitrarily oriented crack, Int. J. Heat Mass Transfer67 (2013), 514–522.
5.
O.Chkadua, S.E.Mikhailov and D.Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics, J. Integral Equations and Applications22(1) (2010), 19–37.
6.
A.V.Glushko, Acoustic oscillations and intrusion in viscous compressible stratified fluid, Doklady Akademii Nauk365(1) (1999), 26–30.
7.
A.V.Glushko, Acoustic vibrations and intrusion in a viscous compressible stratified fluid, Doklady Mathematics59(2) (1999), 307–311.
8.
A.V.Glushko and E.A.Loginova, Asymptotic properties of the solution to the problem of the stationary heat distribution in an inhomogeneous plane with a crack, Vestnik VGU. Ser.: Physics and Mathematics2 (2010), 47–50(in Russian).
9.
A.V.Glushko and A.S.Ryabenko, Localization principle and an estimate of the rate of damping of oscillations in a viscous compressible stratified liquid, Mathematical Notes85(3–4) (2009), 558–565.
10.
A.V.Glushko and S.O.Rybakov, Localization theorem for the problem of rotary viscous compressible fluid dynamics, Siberian Mathematical Journal33(1) (1992), 24–33.
11.
N.Jain, R.Chona and A.Shukla, Asymptotic stress fields for thermomechanically loaded cracks in FGMs, Journal of ASTM International3(7) (2006), 88–90.
12.
Z.-H.Jin and N.Noda, Crack-tip singular fields in nonhomogeneous materials, J. Appl. Mech.61 (1994), 738–740.
13.
D.V.Kubair and B.Bhanu-Chandar, Stress concentration factor due to a circular hole in functionally graded panels under uniaxial tension, Int. J. Mech. Sci.50 (2008), 732–742.
14.
P.A.Martin, J.D.Richardson, L.J.Gray and J.Berger, Green’s functions for an exponentially graded elastic material, Proc. Royal Soc.458 (2002), 1931–1948.
15.
V.P.Mikhailov, Partial Differential Equations, Mir Publishers, Moscow, 1978.
16.
V.Petrova and S.Schmauder, Thermal fracture of a functionally graded/homogeneous bimaterial with a system of cracks, Theor. Appl. Fract. Mech.55 (2011), 148–157.
17.
V.Petrova and S.Schmauder, Mathematical modelling and thermal stress intensity factors evaluation for an interface crack in the presence of a system of cracks in functionally graded/homogeneous bimaterials, Comp. Mater. Sci.52 (2012), 171–177.
18.
A.Shukla, N.Jain and R.Chona, A review of dynamic fracture studies in functionally graded materials, Strain43 (2007), 76–95.
19.
G.C.Sih, Heat conduction in the infinite medium with lines of discontinuities, J. Heat Transfer87(2) (1965), 293–298.
20.
J.Sladek, V.Sladek and C.Zhan, An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials, Comp. Mater. Sci.32 (2005), 532–543.
21.
V.S.Vladimirov, Equations of Mathematical Physics, Marcel Dekker, New York, 1971.
22.
V.S.Vladimirov, A Collection of Problems on the Equations of Mathematical Physics, Mir Publishers, Moscow, 1986(in Russian).
23.
G.N.Watson, A Treatise on the Theory of Bessel Functions, The University Press, Cambridge, 1922.
