Modeling Reliability of Ceramics Under Transient Loads and

Modeling Reliability of Ceramics Under Transient Loads and Temperatures Noel N. Nemeth Osama M. Jadaan Eric H. Baker The 26 th Annual International Conference on Advanced Ceramics & Composites January 13 -18, 2002, Cocoa Beach Florida E-mail: Noel. N. Nemeth@grc. nasa. gov Glenn Research Center Life Prediction Branch at Lewis Field

Outline q Objective q Background - CARES/Life - References to previous work q Theory - Power law & Walker law - Computationally efficient method for cyclic loading q Examples - Diesel exhaust valve - Alumina static fatigue q Conclusions

Objective q Develop a methodology to predict the time-dependent reliability (probability of survival) of brittle material components subjected to transient thermomechanical loading, taking into account the change in material response with time. q Transient reliability analysis

Fully Transient Component Life Prediction MOTIVATION: To be able predict brittle material component integrity over a simulated engine operating cycle REQUIRES: • Life prediction models that account for: - transient mechanical & temperature loads - transient Weibull and fatigue parameters (temperature/time) • Interface codes that transfer transient analysis finite element results into life prediction codes (CARES/Life)

CARES/Life (Ceramics Analysis and Reliability Evaluation of Structures) Software For Designing With Brittle Material Structures Ø CARES/Life – Predicts the instantaneous and time-dependent probability of failure of advanced ceramic components under thermomechanical loading Ø Couples to ANSYS, ABAQUS, MARC, NASTRAN

CARES/Life Structure Parameter Estimation Weibull and fatigue parameter estimates generated from failure data Finite Element Interface Output from FEA codes (stresses, temperatures, volumes) read and printed to Neutral Data Base Reliability Evaluation Component reliability analysis determines “hot spots” and the risk of rupture intensity for each element

Some References Regarding Transient Reliability Analysis q Paluszny and Nicholls (1978) -- Discrete time steps, SCG, Weibull and fatigue parameters were constant: Paluszny, A. , and Nicholls, P. F. , “Predicting Time-Dependent Reliability of Ceramic Rotors, ” Ceramics for High Performance Applications-II, edited by Burke, J. , Lenoe, E. , and Katz, N. , Brook Hill, Chesnut Hill, Massachuestts, 1978. q Jakus and Ritter (1981) -- Probabilistic parameters for both applied stress (truncated Gaussian distribution) and component strength (Weibull distribution): Jakus, K. , and Ritter, J, “Lifetime Prediction for Ceramics Under Random Loads, ” Res Mechanica, vol. 2, pp. 39 -52, 1981. q Stanley and Chau (1983) – Failure probability for non-monotonically increasing stresses (maximization procedure): Stanley, P. , and Chau, F. S. ; “A Probabilistic Treatment of Brittle Fracture Under Non-monotonically Increasing Stresses, ” Int. J. of Frac. , vol. 22, 1983, pp. 187 -202. q Bruckner-Foit, A. , and Ziegler (1999) – 3 Reliability formulations; no SCG, SCG governed by a power law, and SCG governed by a power law with a threshold: (1) Bruckner-Foit, A. , and Ziegler, C. , “Design Reliability and Lifetime Prediction of Ceramics, ” Ceramics: Getting into the 2000’s, edited by Vincenzini, P. , 1999. (2) Bruckner-Foit, A. , and Ziegler, C. , “Time-Dependent Reliability of Ceramic Components Subjected to High-Temperature Loading in a Corrosive Environment, ” ASME paper number 99 -GT-233, International Gas Turbine and Aeroengine Congress and Exhibition, Indianapolis, Indiana, 1999. q Ziegler (1998) -- SCG parameters vary with temperature/time: Ziegler, C. , Bewertung der Zuverlassigkeit Keramischer Komponenten bei zeitlich veranderlichen Spannungen und bei Hochtemperaturbelastung, Ph. D. Thesis, Karlsruhe University, 1998. q Jadaan and Nemeth (2001) – Cyclic loading + Weibull and SCG parameters vary with temperature/time: (1) Jadaan, O, and Nemeth, N. N. ; ”Transient Reliability of Ceramic Structures. ” Fatigue & Frac. Of Eng. Mater. Struct. , vol. 24, pp. 475 -487. (2) Nemeth, N. N. , and Jadaan, O. ; “Transient Reliability of Ceramic Structures For Heat Engine Applications, ” Proceedings of the 5 th Annual FAA/Air Force/NASA/Navy Workshop on the Application of Probabilistic Methods to Gas Turbine Engines, June 11 -14, 2001, Westlake Ohio.

Transient Life Prediction Theory For Slow Crack Growth Assumptions: • Component load and temperature history discretized into short time steps • Material properties, loads, and temperature assumed constant over each time step • Weibull and fatigue parameters allowed to vary over each time step – including Weibull modulus • Failure probability at the end of a time step and the beginning of the next time step are equal

Transient Life Prediction Theory Slow Crack Growth and Cyclic Fatigue Crack Growth Laws Power Law: - Slow Crack Growth (SCG) Combined Power Law & Walker Law: SCG and Cyclic Fatigue - Denotes location and orientation

Transient Life Prediction Theory Power Law General reliability formula for discrete time steps:

Binomial Series Approximation Used to Derive Computationally Efficient Solution For Cyclic Loading Binomial Series Expansion: When x>>y the series can be approximated as a two term expression (x + y)n xn + nxn-1 y , when x >> y

Computationally efficient transient reliability formula for cyclic loading - full solution

Transient Life Prediction Theory - Slow Crack Growth Modeled With Power Law load time T T 2 T ZT Computationally efficient transient reliability formula for cyclic loading - simplified version

Combined Walker Law & Power Law for cyclic fatigue - Computationally efficient version with Z factor multiple

Example – Tradeoff Between Accuracy and Computational Efficiency For a Cyclic Load 10 step transient uniaxial loading for a single load block – single element problem Time step # Time Ieq Temp 1 25 100 2 50 90 200 Temp m o N B 3 75 80 300 100 5 230 40 0. 0021 4 100 70 400 500 9 226 36 0. 021 5 125 60 500 1000 14 221 31 0. 21 6 150 70 600 7 175 80 700 8 200 90 800 9 225 95 900 10 250 1000 Temperature vs: material properties

Exact solution versus the Z approximation method for one solution increment (n = 1) • The results for one solution increment represent the least accurate but most computationally efficient answer. Note: Load factor is 0. 5 Number of cycles Pf , Exact solution Pf , Z method % Error 1 4. 6620 E-3 0. 0 10 6. 3066 E-3 -2. 0 E-4 100 8. 5288 E-3 -4. 0 e-4 1000 0. 011530 -6. 0 E-4 10, 000 0. 015578 -0. 001 100, 000 0. 021032 -0. 002 1, 000 0. 028369 0. 028368 -0. 005 Pf = Failure Probability

Exact solution versus the Z approximation method for one solution increment (n = 1) • The results for one solution increment represent the least accurate but most computationally efficient answer. Note: Load factor is 1. 0 Number of cycles Pf , Exact solution Pf , Z method % Error 1 0. 16428 0% 10 0. 21701 0. 21571 -0. 6 100 0. 28955 0. 28037 -3. 2 1000 0. 41831 0. 36997 -11. 6 10, 000 0. 70425 0. 68330 -3. 0 100, 000 0. 96954 0. 96850 -0. 1 1, 000 0. 99997 9. 99997 -1. 0 E-4 Pf = Failure Probability

Example of Z approximation method for various values of n. The solution increments are equally spaced (Zi = Zj= Zn). Percent error from exact solution versus number of load blocks for a failure probability prediction of 1000 cycles Increasing Computational Effort n = Number of discrete load blocks Cycles Pf Pf exact solution n=1 n=2 n=5 n = 100 n = 500 n = 1000 1, 000 0. 41831 0. 36997 0. 39447 0. 40958 0. 41420 0. 41796 0. 41827 0. 41831 100, 000 0. 96954 0. 96850 0. 96864 0. 96877 0. 96884 0. 96924 0. 96948 0. 96951 Pf = Failure Probability

EXAMPLE: Diesel Engine Si 3 N 4 Exhaust Valve (ORNL/Detroit Diesel) OBJECTIVE: DATA: Contrast failure probability predictions for static loading Versus transient loading of a Diesel engine exhaust valve for the power law and a combined power & Walker law Material: Silicon Nitride NT 551 Information Source: Andrews, M. A. , Wereszczak, A. A. , Kirkland, T. P. , and Breder, K. ; “Strength and Fatigue of NT 551 Silicon Nitride and NT 551 Diesel Exhaust Valves, ” ORNL/TM 1999/332. Available from the Oak Ridge National Laboratory 1999 Corum, J. M, Battiste, R. L. , Gwaltney, R. C. , and Luttrell, C. R. ; “Design Analysis and Testing of Ceramic Exhaust Valve for Heavy Duty Diesel Engine, ” ORNL/TM 13253. Available from the Oak Ridge National Laboratory, 1996 MODEL: • ANSYS FEA analysis using axisymmetric elements • Combustion cycle (0. 0315 sec. ) discretized into 29 load steps • A 445 N (100 lb) spring pre-load applied to valve stem in open position. 1335 N (300 lb) on valve stem on closure. Thermal stresses superposed with mechanical stresses • Volume flaw failure assumed

Loading and Stress Solution of Diesel Engine Exhaust Valve Thermal distribution Pressure load applied to face of a ceramic valve over the combustion cycle First principal stress at maximum applied pressure (MPa)

Silicon Nitride NT 551 Fast Fracture and SCG Material Properties Power Law Parameters (NT 551): N and B Cyclic Fatigue Parameters: Q and A 2 A 1 T ( C) m s 0 V (MPa. mm 3/m) Average strength N B Q A 2 A 1 (MPa 2. sec) (MPa) 20 9. 4 1054 806 31. 6 5. 44 e 5 3. 2 0. 65 700 9. 6 773 593 87 1. 12 e 4 3. 2 0. 65 850 8. 4 790 577 19 1. 13 e 6 3. 2 0. 65 Note: Cyclic fatigue parameters are assumed values for demonstration purposes only

Diesel Engine Si 3 N 4 Exhaust Valve Batdorf, SERR criterion with Griffith crack Transient and static probability of failure versus combustion cycles (1000 hrs = 1. 14 E+8 cycles)

Diesel Engine Si 3 N 4 Exhaust Valve Transient reliability analysis with proof testing capability for combined Walker & power law

Diesel Engine Si 3 N 4 Exhaust Valve Transient reliability analysis with proof testing capability Proof test: 10, 000 cycles at 1. 1 load level

EXAMPLE: Predict material reliability response of an alumina assuming time varying Weibull & Fatigue Parameters DATA: Material: Specimen: Test Type: Temperature: Source: Alumina 4 -pt flexure (2. 2 mm x 2. 8 mm x 50 mm -- 38 mm and 19 mm bearing spans) Static Fatigue 10000 C G. D. Quinn – J. Mat. Sci. – 1987 MODEL: • Single element model of specimen inner load span (2. 8 mm x 19 mm) with uniform uniaxial stress state (surface flaw analysis) • Loading is static (non-varying) over time • Weibull and fatigue parameters vary with the log of the time PROCEDURE: A single element CARES neutral file is constructed with discrete time steps (10 steps per decade on a log scale) spanning 8 orders of magnitude. Applied load is constant but Weibull and fatigue parameters allowed to vary with each time step.

EXAMPLE: Time Dependent Weibull & Fatigue Parameters G. D. Quinn, “Delayed Failure of a Commercial Vitreous Bonded Alumina”; J. of Mat. Sci. , 22, 1987, pp 2309 -2318. Static Fatigue Testing of Alumina (4 -Point Flexure) 10000 C

Parameters interpolated with log of time No extrapolation outside of range t= 1. 6 sec. , m = 29. 4, 0 = 165. 8, N = 6. 7, B = 2711. 1 t = 31. 6 sec. , m = 15. 8, 0 = 152. 7, N = 13. 2, B = 9707. 7 t = 1. 0 E+5 sec. , m = 13. 1, 0 = 127. 3, N = 36. 4, B = 2276. 2

Parameters interpolated with log of time No extrapolation outside of range t = 1. 6 sec. , m = 29. 4, 0 = 165. 8, N = 6. 7, B = 2711. 1 t = 31. 6 sec. , m = 7. 4, 0 = 263. 3, N = 8. 0, B = 2395. 9 t = 316. 2 sec. , m = 4. 5, 0 = 870. 1, N = 9. 0, B = 10, 389. 0

Conclusions q A computationally efficient methodology for computing the transient reliability in ceramic components subjected to cyclic thermomechanical loading was developed for power law (SCG), and combined power & Walker law (SCG & cyclic fatigue). q This methodology accounts for varying stresses as well as varying Weibull and fatigue parameters with time/temperature. q FORTRAN routines have been coded for the CARES/Life (version 6. 0), and examples demonstrating the program viability & capability were presented.

Future Plans q Goal to release CARES/Life 6. 0 to engine companies for evaluation/beta testing by 9/30/02 - Continuing benchmarking activities - Continue developing GUI - Complete ANSYS and ABAQUS interfaces - User guide with example problems ? ? (FY’ 02 - FY’ 03) q CARES/MEMS - Single crystal reliability - Edge recognition macro within ANSYS - Edge flaw reliability model