Weibar Tie Dependent Theory Guide Dynaic Fatigue araeter Estiation COECTICUT RESERE TECHOLOGIES, IC. Copyright 998-5
Weibar Theory Tie Dependent Introduction Ceraic aterials can easily exhibit coplex theroechanical behavior that is both inherently tie-dependent and hereditary in the sense that current behavior depends not only on current conditions, but also on theroechanical history. The ability o a brittle aterial coponent to sustain load degrades over tie due to a variety o eects such as oxidation, creep, stress corrosion, and atigue. The design ethodology presented in this report cobines the statistical nature o strength-controlling laws with the echanics o crack growth. Using the ast racture and atigue paraeters, as well as the results obtained ro a inite eleent analysis, coponent reliability as a unction o tie can be coputed. The S CRES/Lie algorith ebodies these concepts and this sotware was utilized in analyzing the odel paraeters and in deterining coponent lie. Lie can be assessed in ters o tie to ailure, or in ters o cycles to ailure. With this type o reliability algorith, the gun barrel design engineer can ake appropriate design changes until an acceptable coponent lie has been coputed. In addition to the ast racture paraeters, i.e, the Weibull shape and scale paraeters, the analysis o tiedependent reliability necessitates evaluation o distinct paraeters relating to atigue crack growth and atigue lie. The theoretical developent that supports the estiation o atigue paraeters is outlined in this report. Under this task estiation ethods or the geoetry independent paraeters needed or tie dependent (sub-critical crack growth) coponent reliability analyses were established or data obtained using either the C-ring and the sectored lex bar. These geoetry independent paraeters ( and ) are siilar to the aterial speciic characteristic strengths (s and s ) paraeters needed to conduct a ast racture coponent analysis. The S CRES/Lie algorith has estiation ethods established or and that are associated with a liited nuber o specien conigurations, but these specien conigurations do not include the C-ring or the sectored lex bar. It was proposed that estiates or the power law exponent, and the geoetry independent paraeter, be extracted ro the tie dependent ceraic data generated by RL. However, at the tie this report was prepared the requisite tie dependent ailure data was not available. Establishing conidence bounds on the tie dependent aterial paraeters using bootstrap techniques is briely discussed along with the easibility o incorporating the requisite coputational ethods into a paraeter estiation coputer algorith. Finall since a liited aount o tie dependent ceraic ailure data will be ade available early in the project, an eort to explore and establish pooling procedures or tie dependent paraeter estiation will be conducted. Subcritical Crack Growth Ceraics exhibit the phenoenon o delayed racture or atigue. With load histories at stress levels that do not induce ast racture, there is a regie where subcritical crack growth (SCG) occurs. Subcritical crack growth involves a cobination o siultaneous, deleterious and synergistic ailure echaniss. These can be grouped into two categories: () crack growth due to stress corrosion, and () crack growth due to echanical eects arising ro cyclic loading. Stress corrosion relects a stress-dependent cheical interaction between the aterial and its July 5 age
Weibar Theory Tie Dependent environent. Water, or exaple, has a pronounced deleterious eect on the strength o ceraics. Higher teperatures also tend to accelerate this process. Models or SCG that have been developed tend to be sei-epirical and approxiate the behavior o crack growth phenoenologically. Experiental data indicates that crack growth rate is a unction o the applied stress intensity actor (or the range in the stress intensity actor). Curves o experiental data show three distinct regies or regions o growth when the data is graphically depicted as the logarith o the rate o crack growth versus the logarith o the ode I stress intensity actor (reerence Figure ). The irst region indicates threshold behavior o the crack, where below a certain value o stress intensity the crack growth is zero. The second region shows an approxiately linear relationship o stable crack growth. The third region indicates unstable crack growth as the aterials critical stress intensity actor is approached. III log Crack elocit da/dt I II K th K Ic log Stress Intensity Figure Crack elocity as a Function o Stress Intensity (K) in Monolithic Ceraics. Crack Initation Does ot Occur when K<K th. SCG can occur when K th <K<K Ic. Fracture occurs when K> K Ic. (Regions I, II, and III) July 5 age 3
Weibar Theory Tie Dependent The second region typically doinates the lie o the aterial. For the stress corrosion ailure echanis, these curves are aterial and environent sensitive. The ost oten cited odels in the literature regarding SCG are based on power law orulations or the rate equations. ower law orulations are used to odel the second region o the experiental data cited above or both the stress corrosion and the atigue. This power law orulation is expressed as da( x, z, t) dt K Ieq Ieq ( x, z,t) ( x, z, t) Y a( x, z, t) () where and (crack growth exponent) are aterial/environental constants and Ieq (x,z,t) is the ode I stress. In order to utilize this relationship in a reliability setting, the ode I stress Ieq (x,z,t ) at the tie o ailure (t t ) is transored to its critical eective stress distribution at tie t. The transoration is obtained by solving the ollowing classical racture echanics expression or the crack length a K Ieq ( x, z,t) ( x, z, t) Y a( x, z,t) () where Y is a unction o crack geoetr and a(x,z,t) represents the crack length at tie t. Dierentiating the resulting expression or a with respect to tie, and setting this dierential equation equal to the right hand side o equation yields the ollowing expression where Ieq t (-) (,,,t) dt Ieq x y z - Ieq, ( x, z,t) Ieq ( x, z,t) (3) Y K IC ( ) (4) is a aterial/environental atigue paraeter, K Ic is the critical stress intensity actor, and Ieq (x,z,t ) is the equivalent stress distribution in the coponent at tie t t. The paraeter has units o stress tie. ote that Y is assued constant or subcritical crack growth. The paraeter, Y and K Ic are never coputed directl but are aniested through the paraeter via the expression above when a value or the paraeter is estiated ro experiental data. The transoration given by equation 3 perits the use o the Weibull ast racture expressions in evaluating tie dependent coponent reliability. The use o equation 4 strongly iplies that the paraeter is inherently tied to a speciic deect population through the use o Y and K Ic. The subcritical crack growth odel is cobined with a two-paraeter Weibull cuulative distribution unction to characterize the coponent ailure probability as a unction o service lietie. This is accoplished by equating e with Ieq,. Multiaxial stress ields are included by July 5 age 4
Weibar Theory Tie Dependent using the principle o independent action (I) odel, or the atdor theory. These ultiaxial reliability expressions were outlined in the previous discussion on tie-independent reliability analysis odels. SCG Model or Creep (Static Fatigue) Tensile Specien Creep loading, also known as static atigue to the ceraist, is deined by the engineer as the application o a constant load over tie. To the ceraist creep deines a speciic type o deoration echanis. This report adopts the engineer s viewpoint as well as the engineering noenclature or describing tie dependent load histories in general. For this load case the ode I equivalent stress, Ieq (x,z,t), is independent o tie and is thus denoted by Ieq (x,z). Integrating the right hand side o equation 3 with respect to tie yields ( x, z) t Ieq Ieq, ( x, z, t ) Ieq ( x, z) (5) SCG Model or Monotonic Load (Dynaic Fatigue) Tensile Specien Monotonically increasing uniaxial loading, also know as dynaic atigue to the ceraist, is deined as the application o a constant stress rate & (x, z) over a period o tie, t. ssuing the applied stress is zero at tie t, then d ( x, z ) Ieq( x, z,t) t (6) dt Substituting equation 6 into equation 3 results in the ollowing expression or eective stress ( x,z,t t ) Ieq t Ieq, ( x,z) Ieq ( x,z,t t ) (7) ( ) araeter Estiation Tensile Dynaic Fatigue Data The approach or coputing the tie dependent strength paraeters and ro ailure data is presented in this section or a speciic specien geoetr i.e., a uniaxial test specien. Hence the stress state is unior throughout the specien, and is not spatially dependent. This is not the case or C-ring speciens, or sectored lex bars (see next section) that will be used to characterize aterial properties in this progra. egin by expressing the uniaxial orulation o the probability o ailure as July 5 age 5
Weibar Theory Tie Dependent exp ~ exp ( t ) [ ] Ieq, x, z, t d d (8) or law distributions distributed through the volue o all test speciens (a siilar expression exists or surace law distributions). Here ~ is reerred to the inert strength in the ceraics literature (e.g., ppendix in STM C 368). Under the assuption that ~ << (9) where is the stress at ailure. For onotonically increasing stress tests (dynaic atigue) equation 7 sipliies to ~ ( t ) ( ) t ( ) ( ) ( ) & () under the assuption indicated in equation 9. Thus the stress at ailure can be expressed as ( ) ( ~ ) & () ow let then ( ~ ) D () ( ) & Taking the natural log o both sides o equation 3 yields D (3) July 5 age 6
Weibar Theory Tie Dependent July 5 age 7 & ln ln ln D (4) Thus plotting the log o the stress at ailure against the log o the applied stress rate & should yield a straight line with a slope o [/()]. Typically linear regression techniques are used to deterine the paraeters and D. Once these paraeters are deterined ro the tie dependent ailure data this inoration would be cobined with the Weibull distribution paraeter estiates and the paraeter would be coputed ro the expressions developed below. Substitution o equation into equation 8 yields gage d exp exp & & (5) Solving this expression or yields & ln gage (6) Coparing equation and 6 leads to D gage ln (7) and solving or ro this last expression yields ln D gage (8) In order to copute the paraeter ro equation 8 a probability o ailure ust be utilized. The siplest approach would be to take
Weibar Theory Tie Dependent.5 (9) since linear regression ethods are used. lso note that the paraeter will be independent o the specien geoetry used to generate tie to ailure data, whereas the paraeter D is strongly dependent on the specien geoetry. This is siilar to the ast racture Weibull paraeters, i.e., is not dependent on the specien geoetry where θ is. oting that D is specien dependent, keep in ind that the developent above was presented or creep in a uniaxial test specien. The next section develops an expression or or generic test speciens. araeter Estiation Dynaic Fatigue Data, rbitrary Specien Geoetry The approach or coputing the tie dependent strength paraeters and ro ailure data or an arbitrary specien geoetry is presented in this section. Here the stress state and stressing rate are assued to vary spatially throughout the specien. egin by expressing the probability o ailure as exp exp [ ] Ieq, x, z, t [ ~ ( t )] d d () This uniaxial stress orulation will be expanded to include ultiaxial stress states oentarily. The uniaxial orulation here is strictly or convenience. oting that and the stressing rate are spatially dependent, but ( x, z) () t () & is constant throughout the specien geoetr i.e., t is not spatially dependent. Inserting the results ro equations and into equation yields July 5 age 8
Weibar Theory Tie Dependent July 5 age 9 d t d exp exp & (3) I we identiy (4) and (5) then d t exp (6) ext, identiy ax as the axiu stress in the arbitrary coponent associated with the ailure stress distribution z y x,,, then d t ax ax exp (7) In the Task # Report entitled "Evaluation o Eective olue & Eective rea or C Ring Test Specien" prepared under the previous contract the integral appearing in equation 7 was identiied as an eective volue, i.e., T d k ax (8) Here the subscript T ephasizes that this eective volue is a "teporal" eective volue. The odulus is not the ast racture Weibull odulus as it was in the previous report. This odulus is a unction o the ast racture odulus and the power law exponent, as indicated in equation 5.
Weibar Theory Tie Dependent ow equation 7 can be written as t exp ax kt (9) and with ( x, z) ax ( x, z) & ax t & (3) then exp ( ) ax ( ) & ax kt (3) Fro equation 3, one can deduce that ( ) ax D (3) & ax which urther sipliies equation 3, i.e., exp ( D) kt (33) oting that this derivation has ocused on strength liiting laws distributed through the volue, at this point the notation, and D are adopted. This iners that these tie dependent paraeters are associated with a volue law population. Hence this last expression can be solved or the paraeter such that k ln ( D ) T ( ) ( ) (34) siilar derivation or strength liiting laws distributed along the surace o test speciens would lead to the ollowing expression July 5 age
Weibar Theory Tie Dependent k ln ( D ) T ( ) ( ) (35) where k T is the "teporal" eective area or the specien being analyzed. Data nalysis t the tie this report was written tie dependent data provided by RL was not available or any o the candidate ceraics under consideration as gun barrel aterial. In order to deonstrate how the approach outlined above can be ipleented, a silicon nitride data base (T55 - Saint Gobain/orton Industrial Ceraics) available in the open literature (ndrews et al., 999) was utilized. Data tables associated with the Saint Gobain aterial are included in this report to help clariy the procedure, and this inoration is reerred to as the "ndrews data." This ailure data includes several teperatures, stressing rates and specien geoetries. However, this report ocused on the roo teperature bend bar data (STM 6 specien geoetry) in order to assure the data under consideration was attributable to the sub-critical crack growth ailure echanis, and not a creep ailure echanis which can occur at elevated teperatures. Three stressing rates are available at roo teperature in the ndrews data. Fast racture Weibull paraeters were estiated ro the highest stressing rate data (3 Ma/sec see Table I). The ndrews data provided at the highest stressing rate were associated with surace deects. However, this data was censored due to the presence o exclusive laws. Eight ailure strengths ro the highest stressing rate data were identiied as belonging to an "exclusive law population." The ast racture paraeter estiation techniques available in STM C 39 are only applicable to "concurrent law populations." Thus these eight pieces o inoration were not used in the calculations o the Weibull paraeters. Fro this censored inoration an unbiased Weibull odulus o 8. 98 and a characteristic strength o ( θ ) 85. 86 Ma were established. ased on the -specien bend bar geoetry tested under speciications set orth in STM 6, the aterial speciic strength was established at ( ).,337. 4 Ma. s indicated in the discussions in the previous section, the ast racture Weibull paraeters ust be coputed prior to establishing values or the tie dependent paraeters. ext, the tie dependent paraeters and D were coputed eploying linear regression techniques, and utilizing the data listed in Tables I through III. ote careully that the data in all three tables have been censored. Two bend bar speciens ro the interediate stressing rate data (.3 Ma/second) and ive bend bar speciens ro the low stressing rate data (.3 Ma/second) ailed ro volue deects. sixth bend bar specien ro the low stressing rate data ailed outside the gage section. These data were censored in a very siple inded ashion, i.e., they were reoved ro the database prior to coputing and D. This July 5 age
Weibar Theory Tie Dependent ay not be rigorous, but currently censoring techniques or tie dependent paraeters have not been developed. I subcritical crack growth is a doinant ailure ode, then issues such as censoring tie dependent data should be addressed later in the project. Given the data in Tables I through III, then 76 and D 73. 77, where the units or D are consistent with Ma or stress, and or units o length. Failure strength as a unction o stressing rate or the data listed in Tables I through III, as well as the regression line based on the estiated values o and D listed above are plotted in Figure. Table I T55 Fast Fracture Data ro ndrews et al. (999) Test Teperature ( C) Stress Rate Ma/s Machining Direction (Transverse or Longitudinal) Strength (Ma) Flaw Type 3 Trans 75.47 SURFCE 3 Trans 77.3 SURFCE 3 Trans 7.6 SURFCE 3 Trans 75.86 SURFCE 3 Trans 773.6 SURFCE 3 Trans 777.69 SURFCE 3 Trans 778.47 SURFCE 3 Trans 779.89 SURFCE 3 Trans 79.33 SURFCE 3 Trans 793.37 SURFCE 3 Trans 796.46 SURFCE 3 Trans 84.46 SURFCE 3 Trans 8.7 SURFCE 3 Trans 83.3 SURFCE 3 Trans 83.43 SURFCE 3 Trans 834.38 SURFCE 3 Trans 869.7 SURFCE 3 Trans 869.3 SURFCE 3 Trans 87.8 SURFCE 3 Trans 888.87 SURFCE 3 Trans 9.8 SURFCE 3 Trans 96. SURFCE July 5 age
Weibar Theory Tie Dependent Table II T55 Failure Data at the Interediate Stressing Rate, ro ndrews et al. (999) Test Teperature ( C) Stress Rate Ma/s Machining Direction (Transverse or Longitudinal) Strength (Ma) Flaw Type.3 Trans 435.3 SURFCE.3 Trans 53.39 SURFCE.3 Trans 58.5 SURFCE.3 Trans 6.5 SURFCE.3 Trans 634.66 SURFCE.3 Trans 639.45 SURFCE.3 Trans 639.7 SURFCE.3 Trans 64.3 SURFCE.3 Trans 643.34 SURFCE.3 Trans 653.88 SURFCE.3 Trans 663.45 SURFCE.3 Trans 667.56 SURFCE.3 Trans 67.9 SURFCE.3 Trans 678.7 SURFCE.3 Trans 684.66 SURFCE.3 Trans 686.4 SURFCE.3 Trans 687.46 SURFCE.3 Trans 69.67 SURFCE.3 Trans 694.79 SURFCE.3 Trans 7.64 SURFCE.3 Trans 7.66 SURFCE.3 Trans 73.9 SURFCE.3 Trans 77.68 SURFCE.3 Trans 76.8 SURFCE.3 Trans 735.74 SURFCE.3 Trans 739.7 SURFCE.3 Trans 74.7 SURFCE.3 Trans 744.7 SURFCE.3 Trans 788.99 SURFCE.3 Trans 85.48 SURFCE July 5 age 3
Weibar Theory Tie Dependent Table III T55 Failure Data at the Lowest Stressing Rate, ro ndrews et al. (999) Test Teperature ( C) Stress Rate Ma/s Machining Direction (Transverse or Longitudinal) Strength (Ma) Flaw Type.3 Trans 57.67 SUR-MD.3 Trans 57.43 SUR-MD.3 Trans 584.76 SUR-MD.3 Trans 596.4 SUR-MD.3 Trans 6.7 SUR-MD.3 Trans 6.9 SUR-MD.3 Trans 556.7 SUR-MD.3 Trans 557.77 SUR-MD.3 Trans 577.89 SUR-MD.3 Trans 66.3 SUR-MD.3 Trans 57.3 SUR-MD.3 Trans 64.5 SUR-MD.3 Trans 66.8 SUR-MD.3 Trans 588.9 SUR-MD.3 Trans 588.55 SUR-MD.3 Trans 69.6 SUR-MD.3 Trans 6.8 SUR-MD.3 Trans 645.94 SUR-MD.3 Trans 6.49 SUR-MD.3 Trans 67. SUR-MD.3 Trans 634.75 SUR-MD.3 Trans 64.4 SUR-MD.3 Trans 65.3 SUR-MD.3 Trans 676.77 SUR-MD July 5 age 4
Weibar Theory Tie Dependent St. Gobain/orton Industrial Ceraics T55 Silicon itride - C atura Log o Stress at Failure 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6. 6. 6.. Stressing Rate (Ma/sec) 3 Ma/s Stress Rate.3 Ma/s Stress Rate.3 Ma/s Stress Rate Regression Line Figure Failure data as a unction o stressing rate, and regression line ooling Tie Dependent Data I subcritical crack growth is a doinant tie dependent ailure echanis in the analysis o ceraic gun barrels, then it would ake sense to develop pooling techniques where data ro various specien geoetries could be cobined into a coon data pool in order to estiate iproved paraeter values. However, up to this point in the project specien geoetries, e.g., C-ring and sectored lex bars, have been developed to interrogate speciic law populations. Even i ractography could identiy coon law populations between two or ore data sets, pooling techniques or censored data are not available or ast racture data, let alone tie dependent data. So the discussion that ollows ocuses on data sets with a single law population. Conceptually pooling data results in ore inoration available to estiate paraeters. araeter estiates based on a cobined data set thereore would yield iproved values over the estiates ro a single data set. For tie dependent ailure data (as well as ast racture data) the key to pooling data this lies in the ability to transor ailure data to a coon specien geoetry beore extracting paraeter estiates. The ost convenient specien geoetry to convert to is a tensile specien with a unit gage geoetry (volue or area). July 5 age 5
Weibar Theory Tie Dependent Focusing the discussion on data sets with surace deects, then with k T. equations 3 and 35 becoe and ( ) & D (36) ln ( D ) ( ) ( ) (37) For an arbitrary test specien with a surace deect population and ( ) & D (38) k ln ( D ) T ( ) ( ) (39) ssuing the sae law population is interrogated using either the siple tensile specien with a unit gage surace or an arbitrary test specien, the true value o would be sae or both geoetries. Thus equations 37 and 39 are equivalent, which leads to the ollowing relationship ( k ) ( ) D (4) D T To convert ailure data or an arbitrary test specien geoetr take the relationship ro equation 4 and insert it into equation 38. This yields the ollowing expression or converted stress values at ailure ( D )( & ) ( D) ( k ) ( ) ( k ) ( ) ( D ) ( & ) T CO ( D ) ( & ) July 5 age 6 T ( &) (4)
Weibar Theory Tie Dependent This conversion is siilar to the conversion that takes place when pooling ast racture data. The conversion and subsequent paraeter estiation coputations are not without their diiculties. Consider that or surace deects k T k T ax ax (,...) d d (4) and in addition ( k ) ( ) T (,...) CO CO (43) Thus to ake the conversion we need to know the paraeter estiate or a priori. However, the data is being pooled in order to iprove estiates in (as well as ). Thus a nuerical algorith ust be developed to iniize the su o the squares o the residuals, residuals that will be deined in ters o CO. ote that the regression equations or D and available in STM C 368 will not be applicable to pooled data. ew regression estiators are required. Suary ethodology has been established to extract the tie dependent paraeters and ro ailure data obtained using any type o specien geoetry. These two paraeters along with ast racture Weibull paraeters are necessary in conducting tie dependent reliability analyses using CRES/Lie. The paraeter estiation technique was deonstrated using a data base available in the open literature since tie dependent ailure data ro the gun barrel progra was not available when this report was copiled. ote that bootstrapping ethods can be ipleented through the use o equation 33. ounds on paraeter estiates using bootstrap techniques would be based on known values o the ast racture Weibull paraeters. Care ust be taken to aithully reproduce the sae nuber o ailure data at each stressing rate that appears in the experiental data set. Finall a pooling approach was outlined in the event that tie dependent ailure data ro ultiple specien geoetries representing a single law population are ade available. July 5 age 7
Weibar Theory Tie Dependent Reerences ndrews, M.J., Wereszczak,.., Kirkland, T.., and reder, K., Strength and Fatugue o T55 Silicon itride and T55 Diesel Exhaust alves, ORLTM-999/33, 999. STM Standard C 6-94, Standard Test Method or Flexural Strength o dvanced Ceraics at bient Teperature. STM Standard C 368-, Standard Test Method or Deterination o Slow Crack Growth araeters o dvanced Ceraics by Constant Stress-Rate Flexural Testing at bient Teperature. July 5 age 8