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

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632 Paper received: 17.06.2010DOI:10.5545/sv-jme.2010.138 Paper accepted: 20.06.2011Computation of Stress Intensity Factorin Functionally Graded Plates under Thermal ShockNazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.Mohammad Bagher Nazari 1,* ‒ Mahmoud Shariati 1 ‒ Mohammad Reza Eslami 2 ‒ Behrooz Hassani 11 Shahrood University of Technology, Iran2 Amir-Kabir University of Technology, IranThis paper addresses the implementation of the element-free Galerkin method, which is enrichedintrinsically for fracture analysis of functionally graded materials under mode I steady-state and transientthermal loading. The stress intensity factors are evaluated by means of both equivalent domain integraland displacement correlation technique. Continuum functions and the micromechanical model are usedto describe the distribution of material properties. For thermal shock analysis, the modal decompositionmethod, which is a semi-discretization approach, is implemented to obtain the transient temperature field.Also, few parametric analyses are performed to study the effect of material gradation on the stress intensityfactors. The results imply that the magnitude of the stress intensity factor reaches its peak a short whileafter the thermal shock, indicating its significant role in the fracture failure.©2011 Journal of Mechanical Engineering. All rights reserved.Keywords: functionally graded materials, element-free Galerkin method, equivalent domainintegral, displacement correlation technique, thermal stresses6220 INTRODUCTIONFunctionally graded materials (FGMs)are a new type of advanced composites thatare introduced for use in high temperatureenvironments. The composition, microstructureand/or crystal structure of the FGMs changegradually, forming a non-homogeneous materialwith continuously varying thermomechanicalproperties. In recent years, FGMs have been usedwidely in other applications. According to theexperimental studies of Kawasaki and Watanabe[1], when sudden cooling is applied to ceramic/metal FGMs, some edge cracks are created onthe ceramic surface. Therefore, examining thesurface crack problem in FGMs under thermalloading,especially thermal shock, is important infailure analysis of these materials.Jin and Noda [2] derived the general formof the thermoelastic crack-tip fields in FGMs.They assumed that the material properties arecontinuous and piecewise differentiable functionof spatial position and some of them are notzero at the crack-tip. According to their study,the variation of material properties does notaffect the order of singularity of thermoelasticcrack-tip fields. Kishimoto et al. [3] showedthat in the presence of thermal loading, the pathindependency of original J-integral is lost. Theypresented a path-independent form of J-integralincluded extra term to regard the thermal effect.Analytical approaches including the perturbationmethod and singular integral equations have beenused to consider thermal fracture of FGMs [4] and[5]. It is important to know that using analyticalapproaches is limited to some simple problems orespecial conditions. For example, Noda and Guo[5] have studied the edge crack problem in FGMsunder thermal shock using the perturbation method.For the sake of simplification, they assumed thatthe Poisson’s ratio is constant. Yildirim [6] andDag [7] developed an equivalent domain integralto compute the mode-I stress intensity factor (SIF)under steady-state and transient thermal loadingin isotropic and orthotropic FGMs, respectively.These analyses were performed by using veryfine meshes of regular elements in HEAT2Dand FRAC2D software. KC and Kim [8] and[9] used the interaction integral to evaluate themixed-mode SIFs under steady-state thermalloading. Chen [10] used the interaction integralin conjunction with element-free Galerkin (EFG)method to compute SIFs for an interface crackin orthotropic functionally graded coating understeady-state thermal loading. These results wereobtained by using first-order polynomial basisfunctions, which lead to a fine node arrangement.Also, Chen reported the value of J-integral was*Corr. Author’s Address: Shahrood University of Technology,7 tir Square, 3619995161 Shahrood, Iran, mbnazari@yahoo.com

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

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632where:andwhereS = ⎡ S⎣ ⎢10KKthB = ⎡ ∂ φi∂ x1i ⎢∂φi∂x2ijγij⎣Ti⎤⎥ , (7e)⎦= ∫DB B Ω, (8a)ΩTij d= γ∫ ϕ S ϕ d Γ , (8b)ΓuF = bϕdΩ+tϕdΓ, (8c)i∫FΩγiTi∫jΓtTi= γ∫ Suϕ dΓ, (8d)Γui0 ⎤⎥ = ⎧ 1 if uigivenon Γu, Si⎨ , (8e)S2⎦ ⎩ 0 if uinot givenon Γu⎡∂φi∂x10 ⎤B i =⎢⎢0 ∂φi∂x⎥2 ⎥,⎣⎢∂φi∂x∂φi∂x⎦⎥2 1(8f)iϕ i = ⎡ ϕ 0⎣ ⎢ ⎤⎥. 0 ϕi⎦(8g)In the enriched EFG method, thesingularity problems due to the presence of acrack is alleviated by enrichment functions. In theintrinsic enrichment, the standard basis (usuallypolynomials) vector is enriched by including thenear-tip asymptotic displacement field [12]:⎧θ θ ⎫1, x1, x2, r cos , r sin ,T ⎪p ( x2 2 ⎪) = ⎨⎬, (9)⎪ θθr sin sin θ, r cos sinθ⎪⎩⎪2 2 ⎭⎪where r and θ are the usual crack-tip polarcoordinates.3 EQUIVALENT DOMAIN INTEGRAL FORTHERMAL FRACTUREThe J-integral is an energy-based methodwhich is widely used to calculate SIFs. TheJ-integral originally was derived in the form of acontour integral [13]:J =∫ ( Wδ1j −σijui,1 ) njdΓA , (10)Γ Awhere Γ A is an arbitrary contour enclosing thecrack-tip and n j is the j th component of theoutward unit vector normal to Γ A . For the sakeof simplyfying the calculation, it is suitable toconvert this contour form into an equivalentdomain integral. Defining a smooth weightfunction q and applying the divergence theorem,the equivalent domain form of the J-integral isderived as [7]:∫J = ( σ u − Wδ) q dA + ( W ) expl qdA,(11)Aij i, 1 1j , j, 1Awhere A is the area inside the contour Γ A . The firstintegral contains W ,1 , i.e., the partial derivativesof W with respect to x 1 . It should be noted thatin FGMs the temperature field and materialproperties are dependent on spatial coordinates.In linear elastic fracture mechanics, J-integral isequal to the energy release rate and the relationshipbetween the energy release rate and the mode I SIFis given by:*∫J = K 2 E* , (12)Iwhere Etip= Etipfor plane stress and E tip ( 1−νtip )for plane strain. E tip and ν tip are Young's modulusand Poisson's ratio, respectively, evaluated at thecrack-tip.4 TRANSIENT HEAT CONDUCTIONPROBLEMTo obtain the temperature field, the firstordermatrix differential equation (6a) shouldbe solved. Among many methods, the modaldecomposition technique [14] was chosen.Modal decomposition is an analytical approachto solve systems of ordinary differentialequations (ODEs) without the introductionof additional approximations. Based on themodal decomposition procedure, a coupledsystem of ODEs is turned into uncoupledequations by using eigenvectors. The completesolution of Eq. (6a) can be expressed as a linearcombination of all eigenvectors of the systemT(t) = [T 1 , T 2 , ..., T N ]ψ(t) = Mψ(t), where M isan N×N square matrix whose columns are theeigenvectors. Substituting the above definitioninto Eq. (6a) and premultiplying it by M T , theuncoupled system of equation is obtained,th* th*T th thC ψ+ K ψ = M ( F + F γ ), (13)tip2624 Nazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632whereK th* = M T · K th · M,C th* = M T · C th · M.(14a)(14b)The system of Eq. (13) contains N uncoupledequations,Λψii + siψi= ,( i = 12 , ,..., N),(15)*C iith* th*where s i = K ii / Ciiand Λ = M T (F th + F γth).The initial condition ψ (0) can be obtained fromT(0) = M · ψ(0). Depending on the complexity ofthe right-hand side of Eq. (15), it is solved eitheranalytically or numerically.5 NUMERICAL RESULTS ANDDISSCUSSIONIn this section, the calculation of the mode ISIF for an edge crack in functionally graded plate(FGP) under thermal stresses is considered. Inaddition, a few parametric analyses are performedto study the effect of the gradation of materialproperties on the stress intensity factor. Thedistribution of material properties is determinedby means of continuum functions e.g., exponentialfunction or micromechanics models e.g., selfconsistentmodel. Examples are presented here:• An edge cracked plate: exponential gradation.• FGP with an edge crack: power law gradation.• Edge crack in an FGP: micromechanicsmodel.The FGP of length W and height H witha crack of length a, as depicted in Fig. 1a, isconsidered. The thickness (in the x 3 direction) ofthe plate is assumed to be quite thin for plane stressanalysis and large enough for plane strain analysis.The crack is aligned parallel to the direction ofmaterial property gradation. Initially, the FGP is ata uniform stress-free temperature T 0 . Temperaturesof x 1 = 0 and x 1 = W faces are decreased to constanttemperatures T 1 and T 2 , respectively. All otherfaces, including the crack surfaces, are assumed tobe insulated, which results in a dimensional heatconduction problem in the x 1 direction. In all cases,the calculated SIFs will be normalized by dividingto:K = E( 0) α( 0) T πa( 1−ν ( 0)).(16)0 05.1. An Edge Cracked Plate with ExponentiallyGradationFig. 1a shows an unconstrained FGP withan edge crack of length a, Fig. 1b presents thecomplete node arrangement of the FGP whichconsists of 1695 regular nodes and 40 crack-tipnodes, with a total of 1735. Fig. 1c shows thecrack-tip node arrangement. In this case, twodifferent types of functionally graded materialsare considered with exponentially varyingthermomechanical properties (E, ν, α, k, ρc), in thex 1 direction, e.g.:Ex ( ) = E( 0)exp( P x ), (17)1 E 1where the nonhomogeniety parameters are definede.g., as:EWPE = 1 ⎛ ( ) ⎞ln ⎜ ⎟. (18)W ⎝ E( 0)⎠Fig. 1. An FGM plate with an edge crack;a) geometry, b) complete node arrangement, c)crack-tip node arrangementThe values of the nonhomogeneityparameters for the first material are selectedarbitrarily (academic materials) as they followto provide conditions for which the referencessolutions are available.E(0) = k(0) = α(0) = ρc(0) = 1.0, ν(0) = 0.3.For the second case, the ceramic/metalFGM ZrO 2 /Ti-6Al-4V material with properties ofTable 1 is assumed.Computation of Stress Intensity Factor in Functionally Graded Plates under Thermal Shock625

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632For the sake of comparison, two differentcases of the thermal boundary conditions areconsidered in the steady-state analysis. In the thirdcase, a transient analysis is also carried out fordifferent temperatures at the left and right sides ofthe plate.In order to verify the implementation ofDCT and EDI approaches in the framework ofEFG method, comparisons of the calculated SIFsand the available reference solutions are firstpresented. In this case, the temperature of x 1 = 0and x 1 = W faces are decreased from T 0 to T 1 andT 2 , respectively. Table 2 compares the normalizedSIFs with the results provided by Erdogan andWu [4], KC and Kim [8] and Yildirim [6]. Theobtained solutions are in good agreement with thereferences. It is interesting to note that our modelis comprised of 1735 nodes, while the 2D meshdiscretization in KC and Kim [8] consists of 966elements and 2937 nodes in the framework of thefinite element method.Since the surface crack is usually createdduring cooling, the FGP problem subjected toa cooling shock is considered here. To considerthe thermal shock, it is assumed that the FGP isinitially at a uniform stress-free temperature T 0 andsuddenly cooled down to constant temperaturesT 1 = 0.2 T 0 and T 2 = 0.5 T 0 at the left and righthand side faces, respectively. The obtained resultsfor the transient temperature distribution in theZrO 2 /Ti-6Al-4V FGM versus normalized time τ,as defined in Eq. (19), is depicted in Fig. 2.ρτ = k ( 0) ( 0) c ( 0) t .(19)2WAccording to these results, the temperaturegradient near the plate edges is considerably largeat the early times after imposing the thermalshock.Figs. 3 and 4 present normalized SIFs in theZrO 2 /Ti-6Al-4V plate resulting from the transienttemperature field versus the normalized time τ andthe normalized crack length a/W for plane strainand plane stress cases, respectively. As shown inthese figures, the SIF quickly increases to a peakvalue that is drastically larger than the steady valueand then decreases rapidly to the correspondingsteady value for all crack lengths. In addition, themagnitude of SIF decreases as the normalizedcrack length a/W becomes larger in both transientand steady states that are in agreement with theresults that have recently been reported by Nodaand Guo [5].Table 1. Material properties of ZrO 2 and Ti-6Al-4VMaterialsYoung'smodulus[GPa]Poisson'sratioCoefficientof thermalexpansion[10 -6 /K]Thermalconductivity[W/(m K)]Mass density[kg/m 3 ]Specific heat[J/(kg K)]ZrO 2 151.0 0.33 10.0 2.09 5331 456.7Ti-6Al-4V 116.7 0.33 9.5 7.5 4420 537.0Table 2. Normalized mode I SIF in FGP under steady-state thermal loadingNormalized SIFMaterialAnalysisLoadPresent Erdogan KC and YildirimparameterstypeEDI DCT and Wu [4] Kim [8] [6]T 1 = 0.5 T 0 Plane strain 0.0124 0.0126 0.0125 0.0128 0.0128WP E =ln(5) T 2 = 0.5 T 0 Plane stress 0.0090 0.0088 - 0.0090 0.0090WP α =ln(2) T 1 = 0.05 T 0Plane strain 0.0246 0.0240 0.0245 0.244 -T 2 = 0.05 T 0TWP E =ln(5) 1 = 0.2 T 0 Plane strain 0.0334 0.0343 0.0335 0.0334 0.034TWP α =ln(2) 2 = 0.5 T 0 Plane stress 0.0234 0.0239 - 0.0235 0.024TWP k =ln(10) 1 = 0.05 T 0Plane strain 0.0405 0.0411 0.0410 0.0406 -T 2 = 0.5 T 0626 Nazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632Fig. 2. Transient temperature distribution in theFGP (ZrO 2 /Ti-6Al-4V) for various normalizedtimes with T 1 / T 0 = 0.2 and T 2 / T 0 = 0.5The exponent p is a positive constantused as an adjusting parameter to obtain certaindistribution for material properties. As theexponent p can be chosen independently from thecomprised materials, this function is significantlyflexible and hence widely used in practice forthe analysis of the FGMs. In the proportionalmaterial properties, the exponent p is assumedthe same value for all material properties while itcan be selected differently for non-proportionalmaterials.Fig. 3. Normalized K I in the ZrO 2 /Ti-6Al-4Vplate versus normalized time and different cracklengths in plane strain conditionAs the final point, the magnitude of SIF forthe plane strain is larger than plane stress. Noda etal. [14] have derived thermal stresses analyticallyfor a homogeneous isotropic strip under onedimensionaltransient temperature distribution.These results indicate that the thermal stresses forthe plane strain case are equal to those of the planestress multiplied by a factor of 1/(1-ν). Regardingthe fact 0 < ν < 0.5, this factor is greater than one,which implies a larger SIF for the plane strain incomparison to the plane stress problem, which canbe noticed from Figs. 2 and 3.5.2. FGP with an Edge Crack with Power LawGradationA Ni/TiC plate with the configuration ofthe first example is considered here and a powerlawfunction is assumed to describe the materialproperties in the x 1 -direction e.g., as follows.Ex ( ) = E( 0) + ( E( W) − E( 0))( x / W) p . (20)1 1Fig. 4. Normalized K I in the ZrO 2 /Ti-6Al-4V plateversus normalized time for different crack lengthsin plane stress conditionMoreover, here different thermal boundaryconditions are imposed on the uncracked face ofFGP. To apply a thermal shock, the cracked face isassumed to be quenched to a constant temperatureof T 1 = 0 while having the free convection atthe other face with a convection coefficient ofh=10 W/(m 2 K) and the ambient temperature isassumed T 0 . The transient temperature distributionin the Ni/TiC plate is presented in Fig. 5 for theproportional case with p = 5. The effect of theconvection boundary condition at the x 1 = Wface on the temperature distribution is moreapparent at the steady-state. Figs. 6 and 7 showthe transient thermal SIF versus crack lengthsfor the proportional case with p = 5 and p = 0.2,respectively. As can be seen, the variation of thethermal SIF is completely different for these cases.In the ceramic-riched case (p = 5), at the beginningof the thermal shock the SIF increases to a peakvalue and declines to its minimum quickly andthen increase gradually to a steady-state value.However, in the metal-riched case (p = 0.2)the SIF increases quickly to a peak value and thendecreases rapidly until the crack is closed. TheComputation of Stress Intensity Factor in Functionally Graded Plates under Thermal Shock627

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632Table 3. Material properties of Ni and TiCMaterialsYoung’smodulus[GPa]Poisson’sratioCoefficientof thermalexpansion[10 -6 /K]Thermalconductivity[W/(m K)]Mass density[kg/m 3 ]Specific heat[J/(kg K)]TiC 320 0.195 7.4 25.1 4940 134Ni 206 0.312 13.3 90.5 8890 439.5corresponding time of the crack closure increasesas the crack length is increased. In this example,the crack closure occurred in steady-state for allcrack lengths.condition. According to these results, whilethe value of the SIF is independent of the typeof the thermal boundary condition applied onthe uncracked face, the steady-state value iscompletely dependent. Moreover, a greater valuefor the steady-state SIF is obtained for the case ofconstant temperature at both faces.Fig. 5. Transient temperature distribution in theFGP (Ni/TiC) for various normalized times withT 1 = 0 and h = 10Fig. 7. Normalized K I in the Ni/TiC plate versusnormalized time for different crack lengths inplane strain condition and p = 0.2Fig. 6. Normalized K I in the Ni/TiC plate versusnormalized time and different crack lengths inplane strain condition and p = 5The effect of the thermal boundarycondition applied on the uncracked face for thelinear proportional material i.e., p = 1, is illustratedin the Fig. 8. Here, the h = 0 corresponds to theinsulated thermal boundary condition and h = ∞corresponds to a known temperature boundaryFig. 8. The effect of thermal boundary conditionat x 1 = W on the variation of normalized K INow, the effect of the material gradationis studied and some parametric analyses arecarried out to assess their effect on the SIFs. Inall cases, it is assumed that the exponent p getsdifferent values for the special property and628 Nazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632p = 0.2 for other material properties. Figs. 9 and10 present the effect of variation in FGP elasticproperties, i.e. Young’s modulus and Poisson’sratio, on the SIF for the crack length a/W = 0.3and the plane strain problem. According to Fig.9, the magnitude of SIF, especially its peak value,increases significantly as the parameter p E isincreased. These results indicate that for all valuesof p E , the peak and the crack closure time occurroughly simultaneously. This can be explained bythe fact that the transient temperature distributionis independent of the variations of the parameterp E . We believe that the effect of Young’s modulusis responsible for the slight difference between thepeak time and the steady time.The influence of Poisson’s ratio gradationon the SIF is shown in Fig. 10. It can be seen that,by a decrease in p ν , i.e. for metal-riched whosegreater Poisson’s ratio, the magnitude of SIF andthe crack closure time increase.The analytical solutions for thermal stressdistribution in an uncracked FGP under onedimensionaltemperature distribution for the planestrain and plane stress cases are given as [4]:σthxx 2 2andσEx ( 1)= ( Cx C α x ν T x t2 1 1+ 2 − ( 1)( 1+) ∆ ( 1, ), ) (21a)1 − ν( )thxx 2 2 1 1 1 2 1 1= Ex ( ) Cx + C −α( x ) ∆ T( x , t),(21b)The effects of the gradation of the thermalproperties on the SIF during the shock period areshown in Figs. 11, 12 and 13. Fig. 11 depicts thenormalized SIF versus normalized time for variousvalues of the exponent p for the thermal expansioncoefficient, i.e., p α . According to this figure, thepeak value of SIF for the case p = 0.2 is significantlygreater than others. Also, depending on the p αvalue, the trend of SIF might be completelydifferent. For example, the crack is closed for thep α = 0.2 and p α = 1 cases, while the SIF for p α = 5increases gradually to a steady-state value after itpeaks and reduces to a local minimum.Fig. 10. Normalized K I versus normalized timefor different p ν ; plane strain with a/W = 0.3,p = 0.2 for other material propertiesrespectively, where C 1 and C 2 are unknowncoefficients determined from the force and momentboundary conditions in the x 2 direction. From Eq.(21), it is observed that the thermal stresses are anincreasing function of Young’s modulus.Fig. 11. Normalized K I versus normalized timefor different p α values; plane strain witha/W = 0.3, p = 0.2 for other material propertiesFig. 9. Normalized K I versus normalized time fordifferent p E ; plane strain with a/W = 0.3, p = 0.2for other material propertiesFig. 12 illustrates the SIF variation interms of time for various values of conductivityparameter p k . It can be seen that an increase of theparameter p k causes a delay in the occurrence ofthe peak value of SIF and steady-state. This delayis not surprising since the diffusivity k/ρc is anincreasing function of the conductivity and thep k = 0.2 correspond to metal-riched compositionComputation of Stress Intensity Factor in Functionally Graded Plates under Thermal Shock629

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632with greater conductivity, Moreover, for ceramicrichedcase (p k = 5) the peak value of SIF and thecrack closure time is greater than the metal-richedcase. The variation of SIF with the normalizedtime and the nonhomogeniety parameter of ρc ispresented in Fig. 13. These results indicate that thepeak value of SIF is almost independent from p ρc .The peak time and the crack closure time increasefor the ceramic-riched case (p ρc = 5).Fig. 12. Normalized K I versus normalized timefor different p k . Plane strain with a/W = 0.3,p = 0.2 for other material propertiesFig. 13. Normalized K I versus normalized timefor different p ρc values; plane strain witha/W = 0.3, p = 0.2 for other material propertiesa benefit for relate optimal property distributions.Moreover, in this method the properties aredetermined independently of the phase of theinclusion and the matrix. This is significantfor FGMs in which the volume fraction of theconstituent phases varies in a wide range. For twophaseFGMs, the volume fraction of the ceramicis assumed in the form of a power function, i.e.V c = 1‒(x 1 /L) p , in which L is the material gradationlength and the exponent p is known as the gradientindex. Here x 1 = 0 corresponds to pure matrixphase (ceramic) and x 1 = L to pure inclusionmaterial (metal). For a two-phase composite, theeffective materials are determined from [16],1 VcVm= + , (22a)κ + 4µ / 3 κ + 4µ / 3 κ + 4µ/ 3c⎛ Vcκc Vmκm⎞⎜ + ⎟ +⎝ κ c + 4µ/ 3 κ m + 4µ/ 3⎠⎛ Vcµm Vmµc⎞+ 5⎜+ ⎟ + 2=0, (22b)⎝ µ − µ m µ − µ c ⎠( αc −αm)( 1/ κ −1/ κm)α = αm+,( 1/ κ −1/ κ )cmm(22c)Vm( km− k) Vc( kc− k)+ = 0 . (22d)k + 2kk + 2kmWe consider an edge crack in anunconstrained FGP of length W and height H = 8W. To consider the thermal shock, it is assumedthat only the cracked face of the FGP is suddenlycooled down to constant temperature T 1 = 0 fromthe stress-free temperature T 0 . Fig. 14 presentsthe transient temperature distribution in the FGP.Here, it is assumed that ΔT = T(x 1 ,t) ‒ T 0 .c5.3 Edge Crack in an FGP with MicromechanicsModelPrediction of the effective macroscopicproperties is one of the basic issues in compositematerial theory. For FGMs, as the gradedcomposites, some micromechanics models ofcomposites have been developed. Among manymicromechanics models extended for FGMs, selfconsistentmethod (SCM) was used. Zuiker [16]has pointed out that the SCM provides a simpleand initial estimate of effective properties which isFig. 14. Transient temperature distribution in theFGP for various normalized times with T 1 =0630 Nazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-6324. Comparison of the obtained numerical resultswith the reference solutions indicates thatboth energy-based EDI and direct approachDCT methods, in the framework of enrichedEFG, are efficient tools to analyze the thermalfracture of FGMs.7 REFERENCESFig. 15. Normalized mode I stress intensity factorin the FGP versus normalized time and differentcrack lengths in plane strain conditionFig. 15 depicts the transient thermal SIFversus normalized crack lengths a/W for planestrain case. Although the steady value of SIF isgreater for longer cracks, the peak value of SIF issignificantly large for short cracks.6 CONCLUSIONIn this paper, the domain form of J-integral(EDI) and displacement correlation technique(DCT) in conjunction with element-free Galerkinmethod are implemented to evaluate the mode Istress intensity factor in FGMs under steady-stateand transient temperature fields. The present studypoints out that:1. In the enriched EFG framework a relativelycoarse mesh in compared with FEM andcommon XFEM is sufficient for analysis ofcracks in FGMs under thermal loading.2. A short while after the thermal shock, SIFincreases to a large peak value, which issignificantly greater than the correspondingsteady value and then decreases rapidly to asteady value. Moreover, although the crackis closed at steady state for some cases, thevalue of SIF might reach to a large positivevalue during the thermal shock period. Theseimply that in thermal fracture analysis ofFGMs, the SIF at the beginning of thermalloading might be the main factor in fracturefailure analysis.3. Parametric analyses indicate that thevariation in the thermomechanical properties,especially thermal characteristics, hasa significantly influence on the fracturebehaviour of FGMs.[1] Kawasaki, A., Watanabe, R. (2002).Thermal fracture behavior of metal/ceramicfunctionally graded materials. EngineeringFracture Mechanics, vol. 69, p. 1713-1728.[2] Jin, Z-H., Noda, N. (1994). Crack-tip singularfields in nonhomogeneous materials. Journalof Applied Mechanics, Transactions ASME,vol. 61, p. 738-740.[3] Kishimoto, K., Aoki, S., Sakata, M.(1980). On the path independent J-integral.Engineering Fracture Mechanics, vol. 13, p.841-850.[4] Erdogan, F., Wu, B.H. (1996). Crackproblems in FGM layers under thermalstresses. Journal of Thermal Stresses, vol. 19,p. 237-265.[5] Noda, N., Guo, L.C. (2008). Thermal shockanalysis for a functionally graded plate witha surface crack. Acta Mechanica, vol. 195, p.157-166.[6] Yildirim, B. (2006). An equivalent domainintegral method for fracture analysis offunctionally graded materials under thermalstresses. Journal of Thermal Stresses, vol. 29,p. 371-397.[7] Dag, S. (2006). Thermal fracture analysisof orthotropic functionally graded materialsusing an equivalent domain integral approach.Engineering Fracture Mechanics, vol. 73, p.2802-2828.[8] KC, A., Kim, J.H. (2008). Interaction integralsfor thermal fracture of functionally gradedmaterials. Engineering Fracture Mechanics,vol. 75, p. 2542-2565.[9] Kim, J.H., KC, A. (2008). A Generalizedinteraction integral method for the evaluationof the T-stress in orthotropic functionallygraded materials under thermal loading.Journal of Applied Mechanics, vol. 75, p.1-11.Computation of Stress Intensity Factor in Functionally Graded Plates under Thermal Shock631

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)7-8, 622-632[10] Chen, J. (2005). Determination of thermalstress intensity factors for an interface crackin a graded orthotropic coating-substratestructure. International Journal of Fracture,vol. 133, p. 303-328.[11] Rao, B.N., Rahman, S. (2003). Mesh-freeanalysis of cracks in isotropic functionallygraded materials. Engineering FractureMechanics, vol. 70, p. 1-27.[12] Fleming, M., Chu, Y.A., Moran, B.,Belytschko, T. (1997). Enriched elementfreegalerkin methods for crack tip fields.International Journal for Numerical Methodsin Engineering, vol. 40, p. 1483-1504.[13] Rice, J.R. (1968). A path independent integraland the approximate analysis of strainconcentration by notches and cracks. Journalof Applied Mechanics, vol. 35, p. 379-386.[14] Zienkiewics, O.C., Taylor, R.L. (2000).The Finite Element Method. Butterworth-Heinemann, Oxford.[15] Noda, N., Hetnarski, R.B., Tanigawa, Y.(2003). Thermal Stresses. Taylor and Francis,New York.[16] Zuiker, J.R. (1995). Functionally gradedmaterials: choice of micromechanics modeland limitations in property variation.Composites Engineering, vol. 5, p. 807-819.632 Nazari, M.B. ‒ Shariati, M. ‒ Eslami, M.R. ‒ Hassani, B.