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

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Strojniški vestnik - Journal <strong>of</strong> Mechanical Eng<strong>in</strong>eer<strong>in</strong>g 57(2011)7-8, 622-632Fig. 2. Transient temperature distribution <strong>in</strong> 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 adjust<strong>in</strong>g parameter to obta<strong>in</strong> certa<strong>in</strong>distribution for material properties. As theexponent p can be chosen <strong>in</strong>dependently from thecomprised materials, this function is significantlyflexible and hence widely used <strong>in</strong> practice forthe analysis <strong>of</strong> 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 <strong>in</strong> the ZrO 2 /Ti-6Al-4Vplate versus normalized time and different cracklengths <strong>in</strong> plane stra<strong>in</strong> conditionAs the f<strong>in</strong>al po<strong>in</strong>t, the magnitude <strong>of</strong> SIF forthe plane stra<strong>in</strong> is larger than plane stress. Noda etal. [14] have derived thermal stresses analyticallyfor a homogeneous isotropic strip under onedimensionaltransient temperature distribution.These results <strong>in</strong>dicate that the thermal stresses forthe plane stra<strong>in</strong> case are equal to those <strong>of</strong> the planestress multiplied by a factor <strong>of</strong> 1/(1-ν). Regard<strong>in</strong>gthe fact 0 < ν < 0.5, this factor is greater than one,which implies a larger SIF for the plane stra<strong>in</strong> <strong>in</strong>comparison 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 <strong>of</strong>the first example is considered here and a powerlawfunction is assumed to describe the materialproperties <strong>in</strong> 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 <strong>in</strong> the ZrO 2 /Ti-6Al-4V plateversus normalized time for different crack lengths<strong>in</strong> plane stress conditionMoreover, here different thermal boundaryconditions are imposed on the uncracked face <strong>of</strong>FGP. To apply a thermal shock, the cracked face isassumed to be quenched to a constant temperature<strong>of</strong> T 1 = 0 while hav<strong>in</strong>g the free convection atthe other face with a convection coefficient <strong>of</strong>h=10 W/(m 2 K) and the ambient temperature isassumed T 0 . The transient temperature distribution<strong>in</strong> the Ni/TiC plate is presented <strong>in</strong> Fig. 5 for theproportional case with p = 5. The effect <strong>of</strong> 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 <strong>of</strong> thethermal SIF is completely different for these cases.In the ceramic-riched case (p = 5), at the beg<strong>in</strong>n<strong>in</strong>g<strong>of</strong> the thermal shock the SIF <strong>in</strong>creases to a peakvalue and decl<strong>in</strong>es to its m<strong>in</strong>imum quickly andthen <strong>in</strong>crease gradually to a steady-state value.However, <strong>in</strong> the metal-riched case (p = 0.2)the SIF <strong>in</strong>creases quickly to a peak value and thendecreases rapidly until the crack is closed. The<strong>Computation</strong> <strong>of</strong> <strong>Stress</strong> <strong>Intensity</strong> <strong>Factor</strong> <strong>in</strong> <strong>Functionally</strong> <strong>Graded</strong> <strong>Plates</strong> under Thermal Shock627

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