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

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