1. Probabilistic Analysis Techniques Applied to Lifetime Reliability Estimation of Ceramics
Stefan Reh
Tamas Palfi
Noel Nemeth*
JANNAF Interagency Propulsion Committee –
NGLT Advanced Materials and Safe Life
December 1-5, 2003,
Colorado Springs, Colorado
Glenn Research Center
at Lewis Field

2. Outline Objective Background - Why probabilistics… Example
- CARES/Life
- ANSYS Probabilistic Design System (PDS)
- ANSYS/CARES/PDS
Example
- Silicon nitride turbine stator blade
Conclusions

3. Objective
To predict the lifetime reliability (probability of survival) of brittle material components subjected to transient thermomechanical loading, taking into account stochastic variables such as loading, component geometry, and material properties.

4. Radome
SOFH Fuel Cell
Oxygen Transport Membrane
Thermal Protection System
Ceramic Gun Barrel
Micro-Rocket

5. Why Probabilistics… Brittle material strength is highly stochastic
(Pressure membrane fracture strength vs: probability of failure)
3C-SiC – Recipe 1a &1b (Effect of changing suseptor)
Amorphous Si3N4
Unfailed specimens (200 psi)
3C-SiC - Recipe 2 (Double growth rate)
Polycrystaline SiC

6. Strength as a function of time is highly stochastic
Why Probabilistics…
Strength as a function of time is highly stochastic
G. D. Quinn, “Delayed Failure of a Commercial Vitreous Bonded Alumina”;
J. of Mat. Sci., 22, 1987, pp
Static Fatigue Testing of Alumina (4-Point Flexure)
10000 C

7. MEMS devices - tolerance control of dimensions
Why Probabilistics…
For many applications the variability of other quantities or properties on component lifetime can be significant
MEMS devices - tolerance control of dimensions
Batch-to-batch variations in material properties
Probabilistic loading.
- magnitude of loads & loading directions (dental prosthetics)
- random vibrations (engine parts)
Material
Recipe
film width (mm)
thickness (m) 1a
1.097 0.041
1.60 0.09 1b
1.040 0.033
1.64 0.09 2
1.049 0.035
2.69 0.17
Poly SiC
1.045 0.038
2.86 0.34
Si3N4
1.060 0.030
0.20 0.00
Measured variation in
film thickness can be
significant for MEMS
devices
 Std. Dev.

8. 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 commercial finite element software  ANSYS
Weibull-Batdorf
Stress-Volume Integration
Specimen rupture tests
Characterize material stochastic response
Complex component
life prediction

9. CARES/Life Schematic & Capabilities Finite Element Interface
Parameter Estimation
Weibull and fatigue parameter estimates generated from
specimen rupture data
Finite Element Interface
Output from FEA codes
(stresses, temperatures, volumes)
read and printed to
Neutral Data Base
Reliability Evaluation
Component probability
of survival
Component “hot spots”
- high risk of failure
Volume flaw & surface analysis
PIA & Batdorf multiaxial models
Fast fracture reliability analysis
Time-/Cycle-dependent analysis
Multiaxial proof testing
Works with transient FE analysis

10. Power Law: - Slow Crack Growth (SCG)
Time-Dependent 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

11. Life Prediction Theory For Transient Mechanical & Thermal Loads
Methodology:
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 between 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

12. Transient Life Prediction Theory - Power Law
General reliability formula for discrete time steps:
k = number of time steps
over the cycle
n = number of elements
Z = number of cycles

13. CARES/Life Uses Results From Deterministic Finite Element Analysis
CARES/Life predicts component lifetime probability of survival based on stochastic strength. It does not assess the effect on probability of survival from other stochastic variables related to the component - such as loads, geometry, and material properties.

14. Statistical analysis of output parameters
ANSYS/PDS
(Probabilistic Design System)
Bringing Probabilistic Design into Finite Element Analysis
Random input
variables
Finite-Element
Model
Statistical analysis of output parameters
Material
Strength
Material Properties
PDS Simulations
Loads
Thermal
Structural
Deformations
Stresses
Lifetime
(LCF,...)
Geometry/
Tolerances
Boundary Conditions
Gaps
Fixity

15. (Probabilistic Design System)
ANSYS/PDS
(Probabilistic Design System)
Capabilities
General
- Free for ANSYS users
- works with any kind of ANSYS finite element model – including transient,
static, dynamic, linear, non-linear, thermal, structural, electro-magnetic, CFD ..
Probabilistic preprocessing
- Allows large number random input and output parameters
- modeling uncertainty in input parameters – Gaussian, log-normal, Weibull…
- random input parameters can be defined as correlated data
Probabilistic methods
- Monte Carlo  Direct & Latin Hypercube Sampling
- Response Surface Method  Central Composite & Box-Behnken Designs
Probabilistic postprocessing
- Histograms
- Cumulative distribution functions
- Sensitivity plots
Parallel, distributed computing

16. ANSYS/CARES/PDS – Probabilistic Component Life Prediction
CARES/Life uses results from
Deterministic FEA. Enabling
CARES/Life to work with PDS
Allows the effects of component
Stochastic variables to be
Considered in the life prediction
Stochastic loads, geometry &
material properties
Stochastic Weibull and fatigue
parameters  Simulates batch-to-batch
material variations or uncertainty in
measured parameters from specimen
rupture data
ANSYS macros were developed to
allows CARES/Life to run within PDS

17. Simplified Turbine Stator Vane in Startup and Shutdown
EXAMPLE:
Simplified Turbine Stator Vane in Startup and Shutdown
OBJECTIVE:
Explore the failure probability response of a turbine stator vane model from repeated startup/shutdown thermal loading - assuming stochastic thermal loads, material parameters, and Weibull & fatigue parameters
DATA:
Material: A generic silicon nitride
MODEL:
ANSYS FEA analysis using solid tetrahedral elements
CARES/Life analysis - volume flaw failure mode & 17 time steps
Height
110 mm
Chord Length
70 mm
Width
40 mm
Red = clamped areas

18. Thermal expansion [1e-6 1/K]
Temperature Dependent
Material Properties
Temperature [C]
Specific heat [J/kgK]
Thermal cond. [W/mK]
Thermal expansion [1e-6 1/K]
Young's modulus [Pa] 20
3.10 E+11 23
680
66.3
100
773
55.4
117
1.58
200
891
48.3
217
1.99
300
959
42.4
317
2.28
400
1023
39.3
417
2.5
500
1099
37.0
517
2.67
600
1120
33.1
617
2.82
700
1155
30.3
717
2.95
800
1180
28.0
817
3.08
900
1203
26.0
917
3.18
982
2.97 E+11
1000
1223
24.0
1017
3.35
1117
3.54
1200
1225
20.2
1204
2.93 E+11
1217
3.90
1400
1280
18.1
1417
4.89
Density: [kg/m3 ]
Poisson’s ratio: 0.28

19. Time profile of the transient thermal loads
Transient FE
analysis performed
using 17 time steps
Maximum vane temperature & principle stress as a function of time

20. Steady state temperatures [°C]
at time 75 seconds
Location of maximum
principal stress [Pa]
at time 75 seconds

21. CARES/Life Predictions From Deterministic Finite Element Analysis
Temperature [C]
Weibull modulus, mV
Weibull Scale Parameter, oV,
Fatigue exponent, NV
Fatigue constant,
BV [MPa2  Sec] 20 21
864
100
E+08
1315 24
620 12
E+08
1371 34
573 11
E+07
Weibull and fatigue parameters of the silicon nitride ceramic material
Conditional probability of failure
as a function of number of
load cycles from CARES/Life and
deterministic finite element analysis

22. CARES/Life With PDS Analysis
Random input variables for the PDS analysis
Random Input Parameter
Unit
Distribution Type
Mean Value
Standard Deviation
Factor on the Young’s Modulus curve -
Gaussian
1.0
0.04
Factor on the thermal expansion curve
0.05
Factor on the thermal conductivity curve
Shift of the hot gas bulk temperature C
0.0
30.0
Factor on the heat transfer coefficient on hot gas side
Lognormal
0.2
Factor on the hot gas mass flow
0.03
Start-up time of transient load cycle
sec.
50.0
5.0
Factor on the Weibull exponent
Factor on the Weibull scale parameter
Factor on the fatigue exponent
Factor on the fatigue constant

23. Cumulative Distribution of the Conditional Failure Probability
for 1,000 and 30,000 Cycles
Monte Carlo simulation method (400 simulations)

24. Failure Probability From Deterministic FE Analysis
Versus Total Probability From PDS Analysis
for 400 Simulations
MCS =
Monte Carlo
RSM =
Response Surface
Method

25. Convergence Behavior of the Monte Carlo Simulation Results
1000 cycles
30,000 cycles
Convergence behavior is significantly better at 30,000 cycles

26. Sensitivity of Conditional Failure Probability
1,000 load cycles with Monte Carlo simulation

27. Conclusions
A coupling of the NASA CARES/Life and the ANSYS Probabilistic Design System has been demonstrated for brittle material component life prediction.
This methodology accounts for stochastic variables such as loading, component geometry, material properties, and lifing parameters on component probability of survival over time.
The turbine vane example demonstrated that ignoring stochastic effects can lead to un-conservative design
Acknowledgments:
The authors would like to acknowledge NASA Next Generation Launch Technology (NGLT) program Propulsion Research & Technology (PR&T) project program manager, Mark D. Klem and Safe Life Design Technologies subproject manager, Rod Ellis. We also would like to acknowledge the generous cooperation and support of ANSYS Incorporated.