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

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