Computation of Stress Intensity Factor in Functionally Graded Plates ...

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632not completely path-independent and results wereunreliable for small integral domain size.The EFG method provides an efficient androbust framework of analyzing fracture mechanicsproblems. This method has been implementedfor fracture analysis of cracks in FGMs undermechanical loading e.g. [11] or steady-statethermal stresses [10]. In this paper, the EFGmethod is applied in both steady-state andtransient thermal fracture of FGMs. The transientthermal loading is imposed in the form of thermalshock.This paper is organized as follows. Section1 presents the thermoelastic governing equations.Section 2 provides the EFG discretization formof governing equations. Section 3 explains theuse of the equivalent domain integral for thermalfracture of FGMs. Section 4 describes the modaldecomposition technique to obtain the transienttemperature field. Section 5 presents the obtainednumerical results of thermal SIF as well asparametric analyses and the relevant aspects ofthe results are discussed. Finally, in Section 6conclusions are drawn.1 GOVERNING EQUATIONSA body occupying a space Ω surroundedby a surface Γ under external actions, body forcesand prescribed thermal boundary conditions hasbeen considered. The governing equations forstatic linear thermoelasticity in the domain Ω are:Fourier law:where,∇⋅ σ + b = 0, (1a)∂−∇ q + Q = c T ∂t(1b)Also, the heat flux is obtained based on theq=−kI∇T. (2)The constitutive equation is defined as:σ = C :( ε −εth ), (3)ε =∇ s u,(4a)ε th = α( T −T0 ) I .(4b)Here, the material properties are the forthorderHooke tensor C , isotropic conductivity k,expansion coefficient α, density ρ and specific heatc. The field variables are displacement u, straintensor ε, stress tensor σ, and thermal strain ε th andthe imposed values are heat source Q and bodyforce b. Moreover, I is the identity second-ordertensor and ∇ s is the symmetric gradient operatoron a vector field. The boundary conditions are asfollows:T = T on Γ T , (5a)kI∇T⋅ n= q on Γ q , (5b)kI∇T⋅ n+ hT ( − T∞) = q on Γ c , (5c)u= u on Γ u , (5d)σ ⋅ n=t on Γ t , (5e)where h is the convection coefficient and n is theoutward unit vector which is normal to Γ.2 ELEMENT-FREE GALERKIN METHOD INTHERMOELASTICITYWe implement the EFG method to solvegoverning partial differential equations (PDEs)of 2D thermoelastic problems. This methodneeds only a set of nodes to construct thediscretized model. In EFG, using moving leastsquare (MLS) approximation leads stability infunction approximation and applying the Galerkinprocedure leads to a stable and well-behavedsystem of discretized equations. Here, the EFGdiscretization in the space dimension only is usedand the Kantorovitch semi-discretization processis followed. According to the EFG method, thefinal discrete equations can be obtained as:( ) = +( K + K ) U = F + FthC T +thK +th th thKγ T F Fγ , (6a)γ γ , (6b)where the dot ( . ) denotes differentiation withrespect to time and:F iththijthC ij=∫ ρφφ c d Ω , (7a)ΩthT thi j∫i jΩ ∫Γc∫ ∫ ∫ ∞iK = kB B dΩ+hφφ dΓ, (7b)= Qφ dΩ+ qφ dΓ+hθ φ dΓ, (7c)Ωi iΓqΓcithF γ i = γ∫ θφi ΓΓTj, (7d)Computation of Stress Intensity Factor in Functionally Graded Plates under Thermal Shock623