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

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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.
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