Shelleman_pt2_Cring_Oring_JTEVA91 (2)

Shelleman_pt2_Cring_ - D L Shelleman,1 O M Jadacm,2 J C Conway Jr,3 and J Mecholsky Jr.4 Prediction of the Strength of Ceramic Tubular Components

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Unformatted text preview: D. L. Shelleman,1 O. M. Jadacm,2 J. C. Conway, Jr.,3 and]. J. Mecholsky, Jr.4 Prediction of the Strength of Ceramic Tubular Components: Part |I—'—Experimental Verification REFERENCE: Shelleman, D. L., Jadaan, O. M., Conway, J. 0, Jr., and Mecholsky, J. J ., Jr., “Prediction of the Strength of Ceramic Tubular Components: Part II—Experimental Verification,” Journal of Testing and Evaluation, JTEVA, Vol. 19, No. 3, May 1991, pp. 192—200. ABSTRACT: The strength distribution of reaction-bonded silicon car- bide tubes that failed by internal pressurization was predicted from strength distributions obtained from simple laboratory test specimens at room temperature. The strength distributions of flexure bars, C-rings tested in tension, C-rings tested in compression, diametrally compressed O-rings, and internally pressurized short tubes were com— pared with the strength distribution of internally pressurized long tubes. The methodology involved application of Weibull statistics using elasticity theory to define the stress distributions in the simple specimens. The flexural specimens did not yield acceptable results, since they were ground before testing, thereby altering their flaw population in comparison with the processing-induced flaw popula« tions of the tubular specimens. However, the short tube internal pressure test, the C—ring tested in tension, and the diametrally com- pressed O—ring test configurations yielded accurate strength predic- tions of full—scale tubular components, since these specimens more accurately represent the strength~limiting flaw population in the long tubes. KEY WORDS: failure probability, failure statistics, Weibull statistics, Weibull modulus, C—ring, O-ring Because of their better strength and corrosion resistance as compared to metals, ceramics such as silicon carbide and silicon nitride are being contemplated for structural applications at high temperatures in natural gas environments. Examples may include recuperative heat exchangers and radiant tube heaters. Due to their low strain tolerance, low fracture toughness, and large vari- ations in strength that are attributed to the variation in inherent microscopic flaws or defects that may be introduced during pro- cessing, handling, or service, ceramics must be designed using statistics and reliability analysis for failure predictions [1—6]. The objective of this paper is to define and develop test meth- odologies applicable to the use and design of tubular compo— nents. A companion paper [7] presented the stress distributions Manuscript received 3/23/90; accepted for publication 11/6/90. ‘Graduate Assistant, Center for Advanced Materials, The Pennsyl- vania State University, University Park, PA 16802. 2Assistant Professor, University of Wisconsin-Platteville, Platteville', WI 53818. 3Professor, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802. 4Professor, Department of Materials Science and Engineering, Uni« versity of Florida, Gainesville, FL 32611. 192 ' - ' t’. and the relationships between the fracture stress and the failure probability for the specimen configurations used in this study, The selection of specimen configurations (C-rings, O-rings, and flexure bars) was based on the following criteria: simple geom. etry, easy machining, easy loading, and unaltered flaw popula. tion in comparison with the tubular sections from which the specimens were cut. The exception, flexure bars, were machined and ground from plates of the same material before testing. The flexure bar specimen was selected because of its widespread use and to check for the effects of grinding on the strength prediction of tubular components. I The most widely accepted statistical method for characterizing the strength behavior of brittle materials is Weibull analysis, which is based on the weakest link theory [8—11]. However, other statistical methods, which also incorporate shear stresses and/or multiaxial stress states, have been proposed [2, 1218]. Due to its wide acceptability and previous successes in characterizing strength distributions, the scaling ability of Weibull statistical methods for predicting the strengths of large industrial tubular components from strength distributions obtained from laboratory size specimens will be assessed. Although previous investigators have attempted to predict the mechanical behavior of one simple specimen configuration from another using Weibull statistics, no comprehensive reports were found in the literature discussing the strength prediction of full-scale tubular components from simple specimens. The reader is referred to the companion paper [7] for a dis- cussion of Weibull statistical analysis and for the derivation of the stress analysis of the specimen configurations used in this study. Reference [7] used computer-generated data to demon- strate how the strength of largetubular components can be pre- dicted from the O-ring strength distribution. This paper will apply the analysis described in Part I to scale Weibull strength distri— butions obtained from simple specimen configurations (e.g., C- ring, O-ring‘, and flexure bar) to predict the strength distribution of large tubular specimens. The remainder of this paper will , discuss the experimental procedure, the strength results, and the applicability of the analysis presented in Ref 7 for predicting the strength of large tubular components. Experimental Procedure A siliconized silicon carbide material (SCRB210)5 was used in this study. SCRB210 has a bimodal SiC grain size distribution 5Reaction-bonded silicon carbide, Coors Ceramics Company, Golden, Colorado. © 1991 by the American Society for Testing and Materials » , 4 1 rigfizkm ., .rmgzr a ,,v,_._.. .—m_ where exhibl‘ dark P urathI figural C—ring (4) dj‘ config test it p01)’111 in a U the tu sure t< with 2 tubulz The gitudi check Both mens cess.‘ groun using 15 or radiu: axis c Th 0f tul figun speci nngs speci 0.005 the f: O-rir outer as-re (~1C were for t long ules teste feret mac} ends of 0 load tion the spec The 6D ‘* Penn 7Ir 8Ir satanic): wanamm its: if t I l SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II where the small grain sizes range from 2 to 5 pm and the large grain sizes range from 25 to 75 pm. This material contains 19 , vol% of free silicon which completely infiltrates the silicon car- } bide compact. An optical micrograph of the microstructure is, t exhibited in Fig. 1, where the light phase is the silicon and the l. l I, a I s and the failure ' :d in this study. gs, O-rings, and a: simple geom_ ed flaw popu1a_ from which the ‘1‘ were machined ore testing The widespread use :ngth prediction r characterizin eibull analysis, However, other stresses and/or 12—18]. Due to characterizing ibull statistical lustrial tubular rom laboratory is investigators r of one simple Ill statistics, no :ure discussing iponents from ' for a dis- derivation of 5 used in this ita to demon- ts can be pre- iper will apply :rength distri— ions (e.g., G h distribution is paper will suits, and the )redicting the ‘5 was used in : distribution pany, Golden, and Materials 1 '5 l u l l l I i a; 4 i 1 ill. dark phase is the bimodally sized silicon carbide grains, Four simple specimen configurations and two tubular config- “rations were selected for this study. The simple specimen con- figurations were: (1) four-point bend bars (V4 point loading), (2) C—rings tested in compression, (3) C-rings tested in tension, and (4) diametrally compressed O-rings (Fig. 1, [7]). The tubular configuratiOns were used for: (1) a short tube internal pressure test in which a ~101 mm long tube was filled with an RTV polyurethane rubber6 that was then axially compressed, resulting in a uniform radial pressure being applied to the inner walls of the tube (see [7] for analysis); and (2) a long tube internal pres— sure test in which a ~660 mm long tube was internally pressurized with argon gas to failure. The dimensions of the simple and tubular specimens are listed in Table 1. The flexure bars were cut from plate stock in directions lon- gitudinal and transverse to the axis of the plate, respectively, to check for strength anisotrophy due to the fabrication process. Both the plate stock and the tubes, from which the other speci- mens were machined, were manufactured via a slip casting pro— cess. The tensile surfaces of the flexure bars were subsequently g H ground to within 0.076 to 0.127 mm (0.003 to 0.005 in.) oversize using a 220 grit wheel and lapped to final size (Table 1) using a 15 um lapping medium. The edges were rounded to 0.15 mm radius. The grinding direction was always parallel to the long axis of the flexure specimens. The C-ring and O—ring specimens were cut from the same lot of tubes to be tested in the form of short and long tubular con- figurations to ensure that the study was not biased by testing specimens from different lots. Both sides (surfaces where the rings were sectioned from the tubes) of the C-rings and O-ring specimens were ground to within 0.0762 to 0.127 mm (0.003 to , 0.005 in.) oversize and lapped using 15 um lapping medium to the final dimensions (Table 1). While the sides of the C~ring and O-ring specimens were machined and polished, the inner and outer surfaces of these specimens (test surfaces) remained in the as—received condition,‘ of the original tubes. Both the short tube (~101 mm length) and long tube (~660 mm length) specimens were machined from 1.83 m (6 ft) long tubes. Unlike the case for the short tubes, additional machining was required for the long tubes on the inside surface to remove residual silicon nod- ules and to accommodate the end seals of the internal pressure tester [19]. The radial edges on the ends of the tubes were cham— fered, and the inside diameters on both ends of the tubes were machined to roundness to a distance of 50.8 mm (2 in.) from the ends. ' A universal testing machine7 operating at a cross-head speed 0f 0.508 mm/min (0.02 in./min) was used to test all uniaxially loaded specimens. To redistribute the load and reduce the fric- tion, a 1.5 mm thick piece of alumina felt was placed between the specimen and the alumina loading rams. The short tube Specimens were tested on another universal testing machine.8 The long tube specimens were tested in a tube burst tester where P “Devcon Flexane 80 Liquid, Norcen Industries, Inc., Jersey Shore, enn. 7Instron Model 4202, Instron Corporation, Canton, Mass. 81lnstron Model 4606, Instron Corporation, Canton, Mass. 198 FIG. 1—0ptical micrograph showing the microstructure of the sili— conized silicon carbide material (SCRBZJO). the tube specimen was mounted vertically using internal pressure seals. The pressure seals consisted of a compression viton O—ring seal; this design prevented the tube from experiencing axial con- straint. Thus the tube was essentially free floating on the O—ring seals. The tube specimen was contained in a steel test chamber and was internally pressurized with argon gas until failure oc« curred. The tube fragments were contained by insulation which served as a witness shield; thus the fragments could be collected for fractographic examination. Results and Discussion Results are presented in the form of unadjusted and adjusted Weibull strength distributions, using volume and area analyses. The unadjusted distributions show the distributions obtained ex— perimentally for each simple and tubular specimen configuration investigated (see Appendix for sample calculation for O-ring specimen configuration). The adjusted distributions were ob- tained by using the following equations (see [7] for derivation): l/In (Tim _ K2V2 7 0” llm at“ _ K2142 ' (Kai) (1b) where K V and KA are the effective volume and effective area, respectively. As discussed in Part I [7], the KV and KA terms are also a function of the Weibull modulus. The shift factor (the ratio of the effective volumes (areas) raised to 1/m) was employed to adjust the experimentally obtained distributions for the simple and short tube test configurations to the same expected volume (area) as that of the long tube, thereby accounting for differences in stress distribution and volume (area) of material stressed in tension. The long tube data were used as a baseline, since this configuration most closely resembles tubular components used in industrial applications (e.g., radiant tubes and heat exchan- gers). A 95% confidence interval [20, 21] was constructed around the strength distribution of the long tube specimens to justify 194 JOURNAL OF TESTING AND EVALUATION TABLE 1—Dimensions of simple and tubular specimen configurations. Specimen b h Flexure Bars 4.0 3.0 C-ring (Tension) 9.5 C—ring (Compression) 9.5 O~ring 9.5 Short Tube Long Tube Dimensions* (mm) L ,. rol' ri‘l’ 5.8 . . . . . . 21.9 i 0.4 17.1 i 0.4 21.9 i 0.4 17.1 i 0.4 ... _ 21.9 i 0.4 17.1 i 0.4 101.6 21.9 i 0.4 17.1 i 0.4 660.4 21.9 t 0.4 17.1 i 0.4 *b = width, h = thickness, L = length, r0 = outer radius, ri = inner radius. T = as—received. STRENGTH (MP3) 20.1 54.6 148.4 403.4 1096.6 .999 .934 .632 + BEND LONG. 9" LL V t o BENDTRANS. :1 Q -308 E x C-RING COMP. LL u “1' v- m U ;: I:0: 0 came TENS. V In. El .127 Lu c a: D O-RING ..J D d .x A SHORTTUBE U. .049 0 LONG TUBE .018 .007 3 4 5 6 7 LN STRENGTH (MPa) FIG. Z—Weibull plot of unadjusted strength distributions for all spec- imen configurations tested at room temperature. the success of the analysis used to scale the strength distributions of the simple specimen configurations. Figure 2 shows the unadjusted Weibull strength distributions for the SCRB210 material. As expected, different strength dis- tributions were obtained for each specimen configuration due to differences in stress distributions and volume of material stressed in tension. The bend bars (lowest volume) exhibit the highest strengths and show no evidence of strength anisotrophy due to processing. The C-ring and O-ring configurations (intermediate volumes) exhibit intermediate strengths and give somewhat sim- . . . . . ,4 wmexxmmfismga ..tfiz‘zktfitfl ilar results at higher strengths. The tubular specimens (largest volumes) display the lowest strengths. Scaling the strength distribution of a smaller to larger specimen configuration is accomplished using Eq 1, where the ratio of the effective volume (area) terms are raised to 1/m. In addition, both effective volume (area) terms are evaluated as a function of the same m. However, what is the proper value of m to use in Eq‘ 1? Two methods for evaluating the shift factor will be dis cussed below. The first assumes that the individual in value for each configuration is correct and the same as that of the long tube. The second method uses an equivalent modulus obtained by combining Weibull data. The latter method enables one to sample more specimens and combines information from speci- mens of different sizes and stress states. This combined method is acceptable, provided the fracture initiating flaws for the speci- mens are from the same population. V The results of the adjusted strength distributions for the simple and short tube specimen configurations are illustrated in Fig. 3. The adjusted strength distributions were obtained via Eq 1a (vol- ume analysis) by using a Weibull modulus obtained for each configuration. The adjusted strength distributions for the short tube and O-ring specimens accurately predict the strength dis- tribution of the long tube. The adjusted strength distribution for the C—rings tested in compression predicted strengths higher than the tubular distributions and did not compare well with the tensile C—ring distribution. This lack of fit was expected because the failures for C—rings tested in compression originated at the outer surface unlike the failures for the C-rings tested in tension and the short and long tubes which originated from flaws at the inner surface, thus yielding a different Weibull modulus and charac- teristic strength. The bend bars showed poor adjusted strength distributions because the surface grinding changed the flaw pOP- a: ulation and resulted in different Weibull moduli and character- istic strengths when compared with the as—received simple and tubular configurations. To explain why the data for the C—rings tested in tension did not overlap the O—ring and tubular data, the fracture origins for the C—ring and O-ring specimens were located and identified. Figure 4 shows the unadjusted strength distributions for these two specimen configurations, in which the fracture origins are identified by the number 1 or 2, where 1 represents a failure associated with residual silicon nodules on the inside surface and 2 represents a failure from a “natural” flaw (e.g., processing, handling, or machining flaws). In addition, specimens in which failure occurred at the corner are indicated by the letter c. It 15 noted that these two distributions basically overlap except for 33!? V. 151332515 \ l I i v LN LN(1t(1-F)‘j FIG. unadju 95% cc the lo eters test 21' C—rinj . from deper Eq 1) result noted limite specii conve numt m, fifz‘tatatéanstmm )ecimens (largegf‘ 3 larger specimen , ‘e the ratio of the i In addition, borh a function of the of m to use in ictor will be dis; 7 dual m value for i that of the long iodulus obtained 1 enables one to tion from speci- ‘ )mbined method ,ws for the speci- l. 1 ns for the simple l strated in Fig. 3. (1 via Eq 1a (vol- )tained for each ms for the short :he strength dis- 1 distribution for ‘ gths higherthan l l with the tensile ted because the e tted at the outer i in tension and aws at the inner ilus and charac- ijusted strength ‘ 3d the flaw pop~ _. i and character- '4 ved simple and l in tension did ’ ture origins for . and identified. tions for these a are origins are a! :sents a failure ide surface and , g., processing, l mens in which ° .e letter c. It iS "lap except for I SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II 195 STRENGTH (MP2) 7_4 20.1 54.6 148.4 403.4 .999 .934 + BEND (Long) .532 A u. ‘-'° BEND (Tran) h g u C-FlING (Comp.) 6: .308 m I "I 5 m o C—HING (rem) : 0 v a E _127 Lu D O-RING I C: J D A SHORTTUBE d 4; ~049 “- 0 LONG TUBE .018 .007 LN STRENGTH (MPa) FIG. 3—Weibull plot of adjusted strength distributions compared with unadjusted long tube strength distribution. The curved lines represent the 95% confidence bands constructed around long tube strength distribution. the lower strength specimens. Table 2 lists the Weibull param- eters for all specimen configurations. Based on a two-tailed t- test at a 95% level of significance, the Weibull modulus for the C-rings tested in tension (4.3 i 0.7) is not significantly different from that of the O-rings (5.5 i 08). Because of the strong dependence of the shift factor on the Weibull modulus (see Eq 1), the lower Weibull modulus for the C-rings tested in tension resulted in a more conservative strength prediction. It should be noted also that the number of specimens tested is somewhat limited and that convergence to a single Weibull modulus for all Specimens failing at the inner surface was not observed. Perhaps Convergence to a single Weibull modulus would result if a large number of specimens for each configuration were tested. STRENGTH (M Pa) 99.5 148.4 221.4 330.3 .999 El O-RING 0 C-RlNG TENS. .934 .632 .308 .127 Ln Ln (1t(1—F)) .049 1- Associated with St on tensile sudace 2- Natural Flaw .015 c- corner failure FAILURE PROBABILITY (F) .007 4.5 4.8 5.0 5.2 5.4 5.6 5.8 6.0 LN STRENGTH (MPa) FIG. 4—Weibull plot of unadjusted strength distributions for O-rings and C—rings tested in tension with identification of failure sources for all specimens. Recently, Johnson and Tucker [22] discussed a method of estimating Weibull parameters (i.e., m and 0'0) by combining data from specimens of different loading factors and multiple specimen sizes. This approach represents all the data from the different Specimens on a plot of fracture stress versus the effective volume. Figure 5 demonstrates the method used to determine an equivalent or combined Weibull modulus. The open symbols represent the original strength data as a function of that speci— men’s effective volume (KV). The associated solid symbols were obtained by transforming each original effective volume (KV) to that effective volume (KV') that would be expected to yield the observed strength at a particular probability level (e.g., F’ = 0.5). This transformation is accomplished via the following expression [22]: 7 _ ln (1 — F’) KV — KV————-—In (1 _ F) (2) From this procedure a linear regression fit to the data can be used to estimate a combined or equivalent Weibull modulus, since the slope of the line is equal to the negative of the inverse of the Weibull modulus (i.e., —1/m). In addition, the position TABLE 2—Weibull parameters for simple and tubular specimens. - N0. of Weibull Modulus Characteristic Strength Specimen Samples (m) (MPa) (0'0) Flexure (Longitudinal) 17 26.9 t 5.1 280.4 : 13.1 Flexure (Transverse) 26 20.6 i 3.2 266.6 1— 14.5 C-ring (Compression) 22 7.2 17 1.2 118.5 : 17.9 C-ring (Tension) 26 4.3 t 0.7 75.8 i 17.9 O~ring ‘ 26 5.5 i— 0.8 86.1 i 158 Short Tube 10 6.0 1- 1.5 85.4 i 14.5 Long Tube 9 4.5 t 1.2 ' 106.1 x 13.8 “WmammeMKMWFAM‘ NM mammwe "smears JWWAMWWWYMwWWWWk ' -- -'. 196 JOURNAL OF TESTlNG AND EVALUATION Weibull Modulus=6.1 LO—RING O-RING C~RING (Tens) C—RING trans.) SHORT TUBE SHORT TUBE LN STRENGTH (MPa) A LONG TUBE LONG TUBE LN EFFECTIVE VOLUME (KV) FIG. 5—Strength as a function of effective volume to determine an equivalent Weibull modulus for SCRBZIO specimens tested at room tem- perature (based on analysis in Ref 22). of the line can also be used to estimate a combined or equivalent characteristic strength (00). Batdorf and Sines [23] proposed an— other method for combining data for estimating Weibull param- eters based on a least squares approach for obtaining a combined m value. The Johnson and Tucker method has an advantage over that proposed by Batdorf and Sines in that a single equivalent on can be estimated. Although the latter method will produce equivalent Weibull moduli, separate values of 00 are obtained for each specimen distribution. The method proposed by Johnson and Tucker was used to determine an equivalent Weibull modulus for all specimen con- figurations where fracture initiates at the inner surface (C-rings tested in tension, O-rings, and both tubular configurations). Fig- ure 6 shows the adjusted volume strength distributions as a result of using the equivalent Weibull modulus of 6.1 in Eq 1. Unlike the data in Fig. 3, the data for the C—rings tested in tension tend to overlap the other strength distributions and are within the 95% confidence bounds. Thus the effective volume expression for these specimens can be used to predict the strength distri- bution of industrial size components (e. g, open—ended tube). The discrepancy between the results for the C-rings tested in tension and C—rings tested in compression can be explained by the presence of different flaw distributions on the inner and outer tube walls. As a result of the processing of these tubes, both the inner and outer surfaces contain residual silicon nodules [24]. STRENGTH (MPa) 7.4 20.1 54.6 148.4 403.4 .999 SCRBZ10 .934 .632 .308 .127 LN LN(1t(1-F)) .049 FAILURE PROBABILITY (F) 00 .018 Volume Weibull Analysls Room Temperature .007 2 a 4 5 5 LN STRENGTH, (MP3) FIG. 6— Weibull plot of adjusted volume strength distributions com- pared with unadjusted long tube strength distribution where the combined Weibull modulus (m = 6.1) was used in Eq 14 to scale the strength data. The curved lines represent the 95% confidence bands constructed around long tube strength distribution. The silicon nodules on the outer surface were removed via a sandblasting procedure by Coors before shippings. However, the inner tube wall, which did not receive the sandblasting treatment, is often highly decorated with silicon nodules remaining from the tube fabrication process. Figure 7 shows the fracture origin of a C—ring tested in tension, in which the failure origin is associated with a residual silicon nodule on the inside tensile surface. Local residual stresses are most likely associated with these silicon nodules due to thermal expansion mismatches during the pro- cessing of this material. Thus these silicon nodules can act as local stress concentrators, thereby reducing the stress level nec— essary to cause fracture. Failures associated with these silicon nodules were also observed for some C-ring and tubular (both short and long) specimens. In fact, four out of the nine long tube specimens had failures associated with these residual silicon nod- ules.‘ Figure 8 shows the‘reconstruction of a long tube specimen which failed at a fracture stress of ~63 MPa. It is noted that this tube‘ is decorated to a considerable extent with these residual silicon nodules. Upon reassembling this tube, the crack paths were used to located the fracture origin shown in Fig. 9. The critical flaw was measured to be ~2175 um in size. Upon inspection of Figs. 3 and 6, it is apparent that the tWO lowest data points of the long tube specimens have deviated from the rest of the strength distribution. Fractographic examination after failure did not reveal the location of the fracture origin within the unmachined section of the, tube. These two tubeS apparently failed at the ends where the internal surfaces of the tubes were machined before testing (to remove the silicon nod— ules decorating the internal surfaces of the tubes) in order to fit .,.,_.~v_~ -. we <~ ——. 1..— FI( C -rin; Fuilu: surfai ~ Arrot the t devie size. Th the d son [. for t more crem éuwlmmuieniez 'l'l ' '3 0~RING O GRING (TENS) 0...... u...“ 0 LONG TUBE ttributionx Com. re the combined 18 strength data. structed around imoved via 3 However, the ng treatment, ning from the re origin of a , is associated urface. Local these silicon ring the pro- es can act as ess level nec~ these silicon ubular (both ine long tube .1 silicon nod— lbe specimen )t'ed' that this iese residual crack paths Fig. 9. The that the two eviated from examination lCtLll‘C origin 6 two tubes rfaces of the silicon nod- 1 order to fit A SHORT TUBE x I :- =(N+1)—-n,- I, l, l 1) \ FIG. 7~Scanning electron micrograph: of the fracture surface for a C-ting specimen tested in tension tliatfai/cd at a stress of 180 MPa. Note: y Failure origin is associated with a residual silicon nodule on the tensile surface: (a) secondary electron image, (b) backseattered electron image. ~ Arrows indicate size of critical flaw (226 um). g the test fixture. This finding is a plausible explanation for the ' fiffiViation of these two data points from the rest of the sample -. Size. The two outliers (outlying data points) were eliminated from ’ the distribution via the censored data method proposed by John— 5011 [25]. Essentially, this method determines the ranking number , f0r the strengths by calculating a new increment when one or w more censored strengths are encountered in the data. This in- Crement, A, is expressed as 4 A (3) 1+n,, SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART ll 197 where N is the total number of specimens, n,- is the previous ranking, and n, is the number of specimens beyond the present censored set. The new ranking is obtained by adding the new increment, A, to the previous ranking, n,. This new ranking is then used to determine the failure probability using the same Weibull statistical method (see Eq 2 in Since failures predominantly occurred at the surface, an area Weibull analysis was performed. The adjusted strength distri— butions are shown in Figs. 10 and 11 based on the individual modulus and combined Weibull modulus methods, respectively. Based on the area analysis using the individual modulus for each configuration, the strength distributions for the C—rings tested in tension, the O—rings, and the short tube fall Within the 95% confidence bounds. As was the case for the volume Weibull analysis (Fig. 2), the C~rings tested in compression failed to predict the long tube. A combined or equivalent Weibull modulus of 4.7 was ob- tained by the foregoing procedure, except that strength was plot— ted as a function of area instead of volume (Fig. 12). The results for the adjusted area strength distributions using the combined modulus (Fig. 11) were analogous to those obtained using the volume analysis. Namely, all specimen configurations that sampled the flaw population on the inner surface of the tube (C—ring tested in tension, O-ring, and short tube) gave accurate (within 95% confidence) predictions of the long tube strength distribution. Conclusions For the SCRB210 material, the O-ring and short tube speci— mens best predicted the strength distribution of the long tube components using either the volume or area Weibull analysis. The C-rings tested in tension yielded accurate predictions based on the area analysis, but gave conservative predictions based on a volume analysis. With a combined Weibull modulus to scale the strength distributions, analogous accurate strength predic- tions for the long tube strength distribution were obtained using volume and area analyses. Thus the effective volume expressions for the C-rings tested in tension, O-rings, and tubular specimen configurations produce acceptable predictions (within a 95% confidence interval) of the strength ofindustrial size components. The C-rings tested in compression overestimated the long tube fracture strength because failures initiated at the outer surface, where the flaw population was altered due to sandblasting. The SCRBZIO tubes possessed a more severe flaw population at the inner surface due the existence of silicon nodules, which cause local stress concentrations due to residual stresses from thermal expansion mismatch during processing. The bend bar specimens resulted in poor predictions because they were ground before testing, thus altering their inherent flaw populationg Previously, a simplified stress solution for a diametrally com- pressed O-ring was investigated to measure the strength of tubes as a function of the length-to-radius ratio [26]. The simplified analysis used in Ref 26 applies only to thin-walled tubes and a plane stress condition. As the length-to-radius ratio is increased, the stress condition changes from plane stress to plane strain. These conditions were not incorporated into the stress analysis used in Ref 26, and as such it was concluded that the analysis used to evaluate the strength of ceramics subject to diametral compression was inadequate. Based on the results of our study, 198 JOURNAL OF TESTING AND EVALUATION FIG. STReconstt-uction of long tube specimen that failed at a stress of 63 MPa. Arrows indicate location of failure origin, which is associated with . residual silicon nodules on the tnszde surfaCe of the tube. Cracks propagated in an axial direction, indicating that the hoop stress was the maximum " stress leading to failure of the tube. FIG. 9—Qplical photograph of the failure origin shown in Fig. 8; arrows indicate criticalflaw Size (2175 um) on fracture surface The silicon nmlll/L’ assoctated With the failure origin and the tensile surface of the tube fragment are indicated by labels and arrows. ‘ the analysis presented in Ref 7 can be used to predict the strength of large tubular components. As discussed in Part I, as the wall thickness of the ring decreases (ri/r0 increases), the exact stress distributiOn (Eqs 5a to 8a, [7]) approaches the approximate so lution obtained through strain energy-straight beam theory con- siderations (Eq 9a [7]). ~ Based on this analysis and the results of this study, both the C—ring Specimens tested in tension and the O-ring specimens lead to accurate strength predictions of full-scale tubular components in which failures initiate on the inside surface. However, the O-ring specimen has two major advantages over the C—ring speci- men tested in tension; namely, less machining is required and a ‘J LN LN(1I(1—F)) M J l _, i t more stable loading configuration is obtained. Loading a SpCCi‘ men in compression is generally much easier than loading in tension, which often requires complicated instrumentation 10 insure proper alignment of loading pins and/orgrips. Thus We » recommend the O-ring loaded in diametral compression as the best configuration to evaluate the strength of large tubular com- ponents in which failure initiates at the inner tube surface. , For predicting the strengths of full—scale tubular componentS l in which failure initiates on the outer surface, we would ref? ’ ommend the compressive C-ring specimen configuration for 1‘5 easy machining and loading. Although fracture initiated on the inside surfaces of the long tubes in this study, actual in—servlce “AV-luau.” ,~—- g— . - .s 1 r0 - U FIG. individu pared w. The mm long tub 7. 2 1 0 -1 r el l l LN LN(1!(1-F)) -3 ,4 .sl 2 FIG. O-ring, bined Vl distribu , 95% c0 1 ts associated with was the maximum 2. The Silicon nodule Loading a Speci— ' than loading in strumentation to ’r grips. Thus we impression as the irge tubular com- ZUbe surface. rular components 3, We would rec- lfiguration for its 3 initiated on the actual in-service “fl—r SHELLEMAN ET AL. ON CERAMIC TUBULAR COMPONENTS: PART II STRENGTH (MPG) 7.4 20.1 54.8 148.4 403.4 .999 SCRB210 Room Temperature °+ 1; .934 §: 3* rs+ BEND (Lang) g .632 E; 4» BEND . gt E (Tran) .4 E w 308 a n GRING(Oomp.) I ° ' - ¢+ 4: 3 9+ m o C-RJNG crane.) .- o O u + Cl: 1 ° .127 D-EJ O—RING —' Lu 5 g A SHORTTUBE .1 -°49 E o LONGTUBE(C) .018 .007 LN STRENGTH (MP3) . F-IQ. lO—Weibull plot of the adjusted area strength distributions using tndtvzdual Weibull modulus for the simple and short tube specimens com— pared wtth unadjusted long tube strength distribution at room temperature. The curved lines represent the 95% confidence bands constructed around long tube strength distribution. STRENGTH (MPa) 20.1 54.6 148.4 403.4 .999 .934 r. .532 5;, t D O»RING 9‘ a .u': .306 g 0 Game (TENS.) V m E» to: A SHORTTUBE z 127 °- Tl ‘ Lu 0 LONG TUBE 2 cc ‘J D .1 .049 '3; LL .018 Area Weibull Analysls Room Temperature .007 2 3 4 5 B LN STRENGTH (MPa) FlG. ll—Weibull plot of the adjusted area strength distributions for offing, C-ring tested in tension, and short tube specimens where the com- b‘Pfl'i Weibull modulus (m = 4% was used in Eq 1b to scale the strength ggttrtbutto'ns obtained at room temperature. The curved lines represent the % confidence bands constructed around long tube strength distribution. ‘ W WWWWWLHMWaMmi-«m 1w ’ ~ w .W- ‘WW. ~ 199 Weibull Modulus - 4.7 CI O-RING I O—RING o C-RING (TENS) H «s o. a '. ‘CIRtN'G mNs.) I E . ,A T e z SHOR TU E E 1.. A SHORT ruse U1 2 .s 0 LONG TUBE 0 LONG ruse LN EFFECTIVE AREA (KA) FIG. 12—Strength as a function of effective area toldetermine an equiv- alent Weibull modulus (m = 4.7) for SCRBZJO specimens tested at room temperature (based on analysis in Ref 22). components such as radiant tubes and heat exchangers can ex— perience complex stress states due to internal gas pressures as well as thermal stresses which develop on transient heatup of such components. In addition to thermal stresses, a constrained tube can also develop large tensile stresses on the outer surface. Therefore it is important to sample both the inner and outer flaw distributions to evaluate thoroughly the strength behavior of tu— bular components subjected to both thermal and mechanical' (internal gas pressure) stresses. Acknowledgments This research was supported by the Gas Research Institute. The authors would like to than\k Darryl Butt, Dr. R. E. Tressler, and Dr. J. R. Hellmann for their advice and assistance in de- signing the tube burst test apparatus used in this study. Appre- ciation is also extended to Dr. C. A. Johnson for helpful dis- cussions throughout this study. APPENDIX Table 1 [7] lists the appropriate stress equations for calculating the maximum tensile stresses for all specimen configurations used in this study. A sample calculation for the O—ring specimen con- figuration is given below. ' The maximum tensile stress (tangential stress, 0'0) is a function of failure load (P), the width of the specimen (b), the outer radius of the specimen (r0), and the tangential stress magnifi— ' ' W33}; 200 JOURNAL OF TESTING AND EVALUATION cation factor {Q(r, 111)} and is expressed as P bnro max _ 0'3 .‘ Qtno) _ <4) The tangential stress magnification factor is given by Eq 22 in Ref 7. In order to determine the maximum stress, L]! is set equal to 0. This equation was programmed into a Vax 11/80 computer. For P = 296.0 lb, b = 0.375 in., r0 = 0.866 in., and Q(r,0) = 143.0, the maximum tangential stress is calculated to be 37.8 ksi or 260.5 MPa. References [1] Gyekenyesi, J. P., “SCARE: A Postprocessor Program to MSC/ NASTRAN for Reliability Analysis of Structural Ceramic Com- ponents,” Journal of Engineering for Gas Turbines and Power, Transactions ofASME, Vol. 108, 1986, pp. 540—546. [2] Batdorf, S. B., “Fundamentals of the Statistical Theory of Failure," in Fracture Mechanics of Ceramics, Vol. 3, R. C. Bradt, D. P. H. Hasselman, A. G. Evans, and F. F. Lange, Eds, Plenum Press, New York, 1978, pp. 1—29. [3] Deslavo, G. J., “Theory and Structural Design Applications of Weibull Statistics,” WANL-TME-2688, Westinghouse Electric Corp, 1970. [4] Shih, T., “An Evaluation of the Probabilistic Approach to Brittle ' Design,” Engineering Fracture Mechanics, Vol. 13, 1980, pp. 257—271. [5] Johnson, C. A., “Fracture Statistics in Design and Applications,” General Electric Report No. 79CRD212, 1979. [6] Ferber, M. K., Tennery, V., Waters, 5., "and Ogle, J., “Fracture Strength Characterization of Tubular Ceramic Materials Using a Simple C-ring Geometry,” Journal of Materials Science, Vol. 8, 1986, pp. 2628—2632. [7] Jadaan, O. M., Shelleman, D. L., Conway, J. C., Jr., Mecholsky, J. J. , Jr., and Tressler, R. E., “Prediction of the Strength of Ceramic Tubular Components: Part I—Analysis,” Journal of Testing and Evaluation, Vol. 19, No. 3, May 1991, pp. 181—191. [8] Weibull, W., “A Statistical Distribution Function of Wide Appli- cability,” Journal of Applied Mechanics, Vol. 18, 1951, pp. 293— 297. [9] Shetty, D. K., Rosenfield, A. R., and Duckworth, W. H., “Sta— tistical Analysis of Size and Stress State Effects ‘on the Strength of an Alumina Disk," in MethodsforAssessing the Structural Reliability of Brittle Materials, ASTM STP 844, American Society for Testing and Materials, Philadelphia, 1984, pp. 57-80. - emunM-rax- waning was»: :3 21‘s." ' “3-” "‘fiJL’HEL ' atria? animation ""21;‘ilifiiififis‘i‘t‘fifilfiz‘fitfitEmits; ‘itéte-r’ifufltfilbfl m: 2. [10] Bansal, G. K. and W. H. Duckworth, “Effect of Specimen Size on Ceramic Strength,” in Fracture Mechanics of Ceramics, V01. 3, Plenum Press, New York, 1978, pp. 189—204. [11] Weaver, G., “Engineering with Ceramics: Part I—The Weibun Model,” Journal of Materials Education, Vol. 5, No. 5, 1983. [12] Evans, A. G., “AGeneral Approach for the Statistical Analysis of Multiaxial Fracture,” Journal ofthe American Ceramic Society, Vol. 61, No. 7—8, 1978, pp: 302—308. [13] Giovan, M. N. and Sines, G., “Biaxial and Uniaxial Data for Sta. tistical Comparisons of a Ceramic’s Strength,” Journal oft/1e Amer. ican Ceramic Society, Vol. 62, No. 9—10, 1979, pp. 510—515. [14] Giovan, M. N. and Sines, G., “Strength of a Ceramic at High Temperatures under Biaxial and Uniaxial Tension," Journal of the American Ceramic Society, Vol. 64, N0. 2, 1981, pp. 68—73. [15] Petrovic, J. J. and Stout, M. G., “Fracture of A1203 in Combined Tension/Torsion: II, Weibull Theory,” Journal of the American Ce. ramic Society, Vol. 64, No. 11, 1981, pp. 661—666. [16] Stout, M. G. and Petrovic, J. J., “Multiaxial Loading Fracture of A1203 Tubes: 1, Experiments," Journal of the American Ceramic Society, Vol. 67, No. 1, 1934, pp. 14—23. [17] Lamon, J. and Evans, A. G., “Statistical Analysis of Bending Strengths for Brittle Solids: A Multiaxial Fracture Problem,” Journal of the American Ceramic Society, Vol. 66, No. 3, 1983, pp. 177—182. [18} Lamon, J., “Statistical Approaches to Failure for Ceramic Relia- bility Assessment,” Journal of the American Ceramic Society, Vol. 71, No. 2, 1988, pp. 106—112. [19] Shelleman, D. L., Jadaan, O. M., Mecholsky, J. J., Jr., and Com way, J. C., Jr., “Tube Burst Test Apparatus for High Temperature Strength Evaluation of Ceramics,” to be submitted to Journal of Testing and Evaluation. 1 [20] Abernethy, R. B., Breneman, J. E., Hedlin, C. H., and Reinman, G. L., Weibull Analysis Handbook, Final Report, AFWAL-TRv 2079, Pratt and Whitney Aircraft, West Palm Beach, Fla, 1983. [21] Srinivisan, R. and Wharton. R. M., “Confidence Bands for the Weibull Distribution,” Technometrics, Vol. 17, N0. 3, 1975. [22] Johnson, C. A. and Tucker, W. T., “Advanced Statistical Concepts of Fracture in Brittle Materials,” Ceram. Tech. Newsletter, No. 21, 1989. [23] Batdorf, S. B. and Sines, G., “Combining Data for Improved Wei- bull Parameter Estimation,” Journal of the American Ceramic So- ciety, V01. 63, No. 1—2, 1980, pp. 214—218. [24] Roy, D. W., Green, K. B., and Dobos, G. J., Ceramic Tube Ma- terials and Processing Development, Development of Large Reaction Bonded Silicon Carbide Comporzentsfor Gas-Fired Furnances, Top» ical Report, Gas Research Institute, GRI-88/0139, 1985—87. [25] Johnson, L. G., The Statistical Treatment of Fatigue Experiments, Research Laboratories, General Motors Corp, 1959, pp. 44—50. [26] De With, G., “Note on the Use of the Diametral Compression Test , for the Strength Measurement of Ceramics," Journal of Materials Science Letters, Vol. 3, No. 11, 1984, pp. 1000—1002. \ '" .flkfiifl’fikifilififilfdiflfi El L‘ARH‘E‘AEJ l t _, RE ~ fee Dis i Vc AB net str: 5m l unt Ex ela ¢ KI sit: mi Nom of M I Mont , 'lPr neeri 1 rue ( 3Pl . Polyt ©1E l l t i ; a’l‘eiczktmz’di ...
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Shelleman_pt2_Cring_ - D L Shelleman,1 O M Jadacm,2 J C Conway Jr,3 and J Mecholsky Jr.4 Prediction of the Strength of Ceramic Tubular Components

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