Structural Integrity in Severe Thermal Environments

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<ul><li><p>Structural Integrity in Severe Thermal Environments A. G . EVANS and E. A. CHARLES </p><p>Science Center, Rockwell International, Thousand Oaks, California 91360 </p><p>The theory of structural stability of ceramics in severe thermal environments was explored via the use of precracking. It was shown that structural stability should be possible when an array of precracks is used, but that the precracks required for this purpose are relatively large. The approach is thus limited to systems which are not required to sustain a significant mechan- ical stress. Preliminary quenching experiments on short cylin- ders confirmed that stabilization can be achieved by pre- </p><p>cracking. </p><p>I. Introduction </p><p>N MANY important applications, ceramic materials are subjected I to severe* thermal stresses which inevitably cause crack propa- gation and eventual failure. Typical examples are UO, nuclear fuel pellets and A1,0, liners for coal gasification reactors. In such applications it is desirable (and often imperative) that crack exten- sion be minimized and be amenable to effective prediction so that the useful life of the ceramic can be both optimized and predeter- mined. </p><p>The first semiquantitative approach to thermal fracture was pro- posed in Hasselmansi unified theory which correlates the onset and arrest conditions for fracture. The onset criterion has since been essentially confirmed for small preexisting flaws and has been quantified by showing how the effects of slow crack growth can be incorporate8 and how the pertinent heat-transfer coefficients can be measured acc~rately.~ Hence, the onset conditions for thermal fracture that pertain to small inherent flaws can now be effectively predetermined. However, for the applications considered herein, this onset condition is invariably exceeded, and the propagation and arrest behavior in the macrocrack regime are of principal impor- tance. </p><p>A rapidlyt propagating crack is thought to arrest either when all of the strain energy is converted into fracture surface energy or when the (statically defined) stress intensity factor, K, equals the critical stress-intensity factor, K,.4 The latter applies when the crack is small compared with the component dimensions (whereupon K is uniquely related to v (the velocity of crack pr~pagation)~); but a recent study6 has indicated that when the cracks are relatively large, neither condition accurately represents crack arrest. Generally, only a proportion, @, of the strain energy is converted into fracture surface energy, and the crack extends farther than predicted by equating K to K,. The proportion @ depends on the component geometry, the fracture onset condition, and the test ambience. The dependence of @ on these variables has not been quantified and, because of the complexity of the problem, it is doubtful that satisfac- tory crack-arrest predictions will be possible for some time. </p><p>However, if stable crack propagation, i.e. a monotonic decrease in K with an increase in crack length, can be induced by introducing precracks, this uncertainty could be averted and the crack exten- sion might become quite predictable. Also, as an additional benefit, since K = K , during stable propagation, the arrest crack length should be less (for a specific thermal environment) than when rapid propagation is allowed. In the present paper, the behavior of relatively large precracks in thermal stress fields will be explored in detail, as needed to determine the utility of precracking for assuring </p><p>Received June 16, 1976; revised copy received September 13, 1976. Supported by the Rockwell International Independent Research and Development </p><p>Severe is used to define the situation wherein there are no reasonable fabrication modifications that would allow the s u e of the preexisting flaws to be sufficiently reduced that fracture from such flaws could be prevented. </p><p>tRapid is defined in this context as crack propagation at a stress intensity factor, K , that exceeds the critical stress-intensity factor, K,. </p><p>R~pTUll. Member, the American Ceramic Society. </p><p>fracture predictability in severe thermal environments. Initially, the crack propagation condition is analyzed as a function of the pre- crack length and thermal variables, and the implications for pre- cracking are discussed. Then the experimental studies suggested by the analysis are performed to determine the practical feasibility of the precrack approach to fracture predictability. </p><p>11. Large Cracks in Thermal Stress Fields </p><p>(1) Analysis The behavior of a crack in a thermal stress field can be deduced </p><p>from the dependence of the crack length on K. This is a complex problem which, in general, can be solved only for given specimen geometries, using numerical techniques. However, a physical ap- preciation of the problem and an understanding of the general trends in K can be obtained by using solutions from the literature for problems which approximate various phases of the thermal stress cycle. This approach allows the constraints on the practicality of precracking to be ascertained quickly, then specific issues can be addressed in detail by using numerical K calculations or by empiri- cal methods. </p><p>There are 2 primary sources of stress in a cracked body subjected to a temperature gradient. First, when the direction of heat flow is at some finite inclination to the crack plane, the heat flow is impeded and shear stresses are induced. These stresses generate a mode-I1 K . 8 For most ceramic fracture problems of practical interest, this is the least important source of stress intensification and will not be considered in detail. The more important stress intensification de- rives from the thermal stresses associated with the nonlinear tem- perature distribution in the body. These stresses can be described completely for convective heat-transfer conditions by the nor- malized parametersg: </p><p>u ( 1 - u ) a= ~ E a A T </p><p>where E is Youngs modulus, a is the linear thermal-expansion coefficient, u is Poissons ratio, AT is the initial (or imposed) temperature differential between the body and the environment, t is the time after exposure to AT, k is thermal conductivity, p is density, c is the specific heat, h is the heat transfer coefficient, ro is the pertinent semidimension of the specimen (e.g. the radius for a cylinder) andp is referred to as the Biot modulus. Figure 1 shows a typical example of the relation between time and surface stress and the corresponding stress distributions. </p><p>The K relations that pertain to the precracking problem are affected by certain consequences of precracking . Thermal stresses are usually predicated by the compatibility of displacements, both between elements and at external boundaries.1 Hence, because a crack affects the compliance of a body, the introduction of a crack not only modifies the local stress distribution but also reduces the stress level throughout the body., This result is relatively unimportant when the crack is small but has a vitally important effect on K when the crack depth is a significant fraction of the component thickness. Under severe thermal conditions, reduction of the stress levelcaused by a single crack is generally not sufficient to prevent the activation and uncontrolled propagation of inherent flaws at some distance from the precrack. The propagation of these </p><p>22 </p></li><li><p>Jan.-Feb. 1977 Structural Integrity in Severe Thermal Environments 23 0. </p><p>51 I I 1 I I </p><p>L o . 0 0 1 </p><p>Fig. 1. ( A ) Schematic of time dependence of peak surface stress in a transient thermal field under convective heat-transfer conditions. ( B ) Stress gradient corresponding to ( A ) . </p><p>flaws would negate the effects of precracking on fracture predicta- bility. However, because the reduction in the stress level is approx- imately proportional to the number of precracks (in the regime where the crack length and density substantially reduce the effective modulus of the material), the activation of inherent flaws can be eliminated by incorporating crack arrays containing the appropriate precrack density. The stress intensity factors that pertain to the present precrack problems are, thus, the multicrack stress-intensity factors. </p><p>( A ) Short Cracks: When the cracks are small enough that the component compliance is unaffected by the cracks and the interac- tion with the opposite boundary is negligible, it is possible to use the stress u (x) that exists in the absence of the crack (Fig. l(B)), to obtain K.,I3 One method uses the concentrated force result (for a crack in a semiinfinite plane) as a Greens function to obtain14 </p><p>where </p><p>F ( x / u ) = ( I -~ /~) [0 .2945-0 .3912(~/~)2 + 0 . 7 6 8 5 ( ~ / ~ ) ~ - 0.9942(~/a)~ +0.5094(x/a)s] (2a ) </p><p>Substituting u from Eq. (1) and normalizing the length parameters then gives </p><p>Another method uses the Greens function for the (uniform) stress acting over a small segment of the crack and obtains K by a mapping techNque.13 Both methods yield similar results, and some typical values obtained for the solid cylinders are plotted in Fig. 2. The parameter of principal importance for fracture prediction is the peak </p><p>l 0 . 0 0 5 </p><p>I I I I I 0.1 0.2 </p><p>alr, 3 </p><p>Fig. 2. Crack length dependence of normalized stress-intensity factor, K , for several values of normalized time, 0 (indicated opposite each curve), for Biot moduli of ( A ) m and ( B ) 15, as determined analytically at the short crack extreme. </p><p>I I I I 1 1 </p><p>01 I I I 1 I 0 0.2 0.4 0.6 0.8 1 .o </p><p>ah, </p><p>Fig. 3. Crack length dependence of normalized 2 obtained analytically for the short crack extreme. Also shown are data points for large cracks determined by a finite element technique (Ref. 19). </p><p>value of the dimensionless stress-intensity factor, i , and its depen- dence on crack length. Values for 2 obtained for two /3 values are plottedinFig. 3. Finally, notethat, exceptforlargeP values ( ~ 2 5 ) , 2 values near and beyond the maximum can be obtained with reasonable accuracy simply by taking the best linear fit to the </p></li><li><p>24 Journal of The American Ceramic Society-Evans and Charles Vol. 60, No. 1-2 </p><p>Fig. 4. ( A ) Schematic of unconstrained contraction of an element sub- jected to a thermal gradient. ( B ) Rotation andextensionneeded to restore the element in ( A ) to equilibrium with surrounding elements. </p><p>1 .a </p><p>lL </p><p>near-surface stress distribution (Fig. 1(B)) and applying the solu- tion for a linear stress gradient,15 </p><p>a lb </p><p>Fig. 5. for several values of element size and ( B ) end extension function F2. </p><p>Plots of crack length dependence of ( A ) end rotation function F I </p><p>K = R m -[ 1.12-0 .68 r* - "I ( 4 ) where R,,, is the value of SZ at the surface and r* is the r position where the linear fit to R ( r ) passes through 0 = 0. This approach is especially useful for complex geometries where the stress fields are not given by simple analytic expressions. </p><p>(B) Intermediate Cracks: The contraction of an uncon- strained transverse element in a body subjected to a transient heat loss at the surface has the form shown in Fig. 4 ( A ) . Thermal stress- es in this element are generated when the contraction is resisted by adjacent elements." In fact, subjecting this element to rotations that restore its original shape and (if necessary) applying a uniform extension to restore the boundary to a position dictated by the aver- age temperature, Taw, (Fig. 4 ( B ) ) allows the thermal stresses to be evaluated.'" Hence, superimposing the results obtained for a sheet subjected to end rotationsI6 and uniform extension1' and setting the element thickness, 2b, (Fig. 4 ( B ) ) equal to the crack spacing yields a K solution that is useful for gaging the relative ef- fects of crack density changes on K . The result is most pertinent in the intermediate range where.the cracks are still small compared with the sample dimensions (a /r050.3) , whereupon boundary in- teraction effects are negligible, but large enough that the sample compliance is significantly reduced. </p><p>The stress intensity factor, K , , for a sheet subjected to end rota- tions id6 </p><p>where 4 is the angle of rotation and F1 is the function plotted in Fig. 5 ( A ) . The rotation is determined by the relative displacements, Ab, across the element and is given by </p><p>(6) </p><p>where To is the temperature at the center of the body and TI is the surface temperature (Fig. 4(A)). Substituting for $ in Eq. (5) and rearranging gives </p><p>$= -= A b b N T o - T i ) r0 TO </p><p>The stress intensity factor K e for a thin infinite strip containing an edge crack subjected to an extension, A x , is given by" </p><p>where F 2 is the function plotted in Fig. 5 ( B ) and Ax is given by </p><p>A x =ba [ T,, -( v) ] Substituting for Ax in Eq. (8) and rearranging gives </p><p>7 </p><p>(9) </p><p>(10) </p><p>The utility of this method for evaluating crack density effects is demonstrated in Fig. 6, where K for 9 cracks in a cylinder under a steady-state temperature gradient is compared with a finite element solution.Is The correlation is quite good, but it should be noted that the steady-state condition should provide the closest correlation with the above analogy because the steady-state displacement gra- dient is nearly linear.I8 </p><p>(C) Large Cracks: There are no analytic solutions or analogs for large cracks. The K information can only be obtained numeri- cally. Some numerical data are available for single cracks in cylin- denig at relative crack lengths of 0.5 and 1 .O*; these are included in Fig. 3. Also, since the thermal stress tends to zero as the crack approaches the opposite boundary, R should approach zero as a -+2ro . (For a cylindrical or spherical geometry and &gt; 1 crack, the cracks tend to meet at the sample center, and 2+0 as a+ro.) (2) zmpricutions </p><p>The trends in the normalized 2 values suggested by the foregoing analysis are summarized in Fig. 7. The rapid reduction in ri w i b an increase in aho in the stable regime suggests the possibility that </p><p>*There are 2 e m s in the figures in Ref. 1 9 The decimals have been omitted from the normalizedK values (these should be 0.400, etc.) and the (1 -v) term should be in the numerator (A. F. Emery; private communication). </p></li><li><p>Jan.-Feb. 1977 Structural Integrity in Severe Thermal Environments </p><p>0'5 r------ hFINITE ELEMENT (Trantina </p><p>and Roberts) 0.4 - </p><p>0.3 </p><p>Y </p><p>0.2 </p><p>APPROXIMATE ANALYTK: </p><p>0.1 METHOD </p><p>I I I I 0 0.1 0.2 0.3 0.4 </p><p>alr, </p><p>Fig. 6. with equivalent finite element result. </p><p>Comparison of K predictions of approximate analytical method </p><p>precracking might prevent crack propagation (even though the propagation of the inherent flaws in uncracked samples may be inevitable). The potential for fracture prevention is determined by R and the pertinent material properties (E, a, v, K , , k , p, and c), thermal environment ( A T , h ) , and specimen size(ro). The criterion for the prevention of crack propagation (see Fig. 7) is </p><p>( 1 1 ) R &lt; ; = Kc(1-v) C - E a A T V T o </p><p>or, when slow crack growth occurs, </p><p>(12) ;</p></li><li><p>26 </p><p>0.E </p><p>0.: </p><p>3 i m </p><p>0.1 </p><p>0. A , </p><p>Journal of The America...</p></li></ul>

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