1. 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 Glenn Research Center at Lewis Field E-mail: Noel.N.Nemeth@grc.nasa.gov Life Prediction Branch

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

3. Objective  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.  Transient reliability analysis

4. 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)

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

6. CARES/Life Structure Reliability Evaluation Component reliability analysis determines “hot spots” and the risk of rupture intensity for each element 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

7.  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.  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.  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.  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.  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.  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. Some References Regarding Transient Reliability Analysis

8. 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

9. 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

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

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

12. Computationally efficient transient reliability formula for cyclic loading - full solution Computationally efficient transient reliability formula for cyclic loading - full solution

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

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

15. 10 step transient uniaxial loading for a single load block – single element problem Time step #Time  Ieq Temp 125100 25090200 37580300 410070400 512560500 615070600 717580700 820090800 922595900 102501001000 Tempm oo NB 1005230400.0021 5009226360.021 100014221310.21 Temperature vs: material properties Example – Tradeoff Between Accuracy and Computational Efficiency For a Cyclic Load

16. 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. Number of cycles P f, Exact solution P f, Z method % Error 14.6620E-3 0.0 106.3066E-3 -2.0E-4 1008.5288E-3 -4.0e-4 10000.011530 -6.0E-4 10,0000.015578 -0.001 100,0000.021032 -0.002 1,000,0000.0283690.028368-0.005 Note: Load factor is 0.5 P f = Failure Probability

17. 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. Number of cycles P f, Exact solution P f, Z method % Error 10.16428 0% 100.217010.21571-0.6 1000.289550.28037-3.2 10000.418310.36997-11.6 10,0000.704250.68330-3.0 100,0000.969540.96850-0.1 1,000,0000.999979.99997-1.0E-4 Note: Load factor is 1.0 P f = Failure Probability

18. Cycles P f exact solution P f n = 1 P f n = 2 P f n = 5 P f n = 10 P f n = 100 P f n = 500 P f n = 1000 1,0000.418310.369970.394470.409580.414200.417960.418270.41831 100,0000.969540.968500.968640.968770.968840.969240.969480.96951 Example of Z approximation method for various values of n. The solution increments are equally spaced (Z i = Z j = Z n ). Percent error from exact solution versus number of load blocks for a failure probability prediction of 1000 cycles Percent error from exact solution versus number of load blocks for a failure probability prediction of 1000 cycles n = Number of discrete load blocks Increasing Computational Effort P f = Failure Probability

19. EXAMPLE:Diesel Engine Si 3 N 4 Exhaust Valve (ORNL/Detroit Diesel) DATA: Material: Silicon Nitride NT551 Information Source: Andrews, M. A., Wereszczak, A. A., Kirkland, T. P., and Breder, K.; “Strength and Fatigue of NT551 Silicon Nitride and NT551 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 OBJECTIVE: 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

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

21. Silicon Nitride NT551 Fast Fracture and SCG Material Properties T (  C)m  0V (MPa.mm 3/m ) Average strength (MPa) NB (MPa 2.sec) QA2A1 209.4105480631.65.44e5 3.20.65 7009.6773593871.12e43.20.65 8508.4790577191.13e63.20.65 Power Law Parameters (NT551): N and B Cyclic Fatigue Parameters: Q and A2A1 Note: Cyclic fatigue parameters are assumed values for demonstration purposes only

22. 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 = 1.14E+8 cycles)

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

24. 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

25. EXAMPLE:Predict material reliability response of an alumina assuming time varying Weibull & Fatigue Parameters DATA: Material: Alumina Specimen: 4-pt flexure (2.2mm x 2.8mm x 50mm -- 38mm and 19mm bearing spans) Test Type: Static Fatigue Temperature: 1000 0 C Source: G. D. Quinn – J. Mat. Sci. – 1987 MODEL: Single element model of specimen inner load span (2.8mm x 19mm) 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.

26. 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) 1000 0 C

27. 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 = 1.0E+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

28. 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 Parameters interpolated with log of time - No extrapolation outside of range

29. Conclusions  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).  This methodology accounts for varying stresses as well as varying Weibull and fatigue parameters with time/temperature.  FORTRAN routines have been coded for the CARES/Life (version 6.0), and examples demonstrating the program viability & capability were presented.

30. Future Plans  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)  CARES/MEMS - Single crystal reliability - Edge recognition macro within ANSYS - Edge flaw reliability model