Probabilistic Analysis Techniques Applied to Lifetime Reliability Estimation of Ceramics Stefan Reh...
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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 LifeDecember 1-5, 2003,
Colorado Springs, Colorado
Glenn Research Centerat Lewis Field
Outline
Objective Background
- Why probabilistics…
- CARES/Life
- ANSYS Probabilistic Design System (PDS)
- ANSYS/CARES/PDS
Example
- Silicon nitride turbine stator blade
Conclusions
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.
“Dual-Use” Ceramics Design Examples“Dual-Use” Ceramics Design Examples
Turbocharger Rotor
Turbine BladeHip JointThree-Unit Bridge
MEMS MicroturbineTV Picture Tube
Radome SOFH Fuel Cell
Oxygen Transport Membrane Thermal Protection System
Ceramic Gun Barrel Micro-Rocket
Brittle material strength is highly stochastic(Pressure membrane fracture strength vs: probability of failure)
3C-SiC – Recipe 1a &1b (Effect of changing suseptor)
3C-SiC - Recipe 2 (Double growth rate)
Amorphous Si3N4
Polycrystaline SiC
Unfailed specimens (200 psi)
Why Probabilistics…
Strength as a function of time is highly stochasticG. 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
Why Probabilistics…
MaterialRecipe
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
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)
Measured variation infilm thickness can be significant for MEMS devices
Measured variation infilm thickness can be significant for MEMS devices
Std. Dev.
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
Specimen rupture tests• Characterize material stochastic response
Complex componentlife prediction
Weibull-BatdorfStress-Volume Integration
Weibull-BatdorfStress-Volume Integration
• Volume flaw & surface analysis• PIA & Batdorf multiaxial models• Fast fracture reliability analysis
•Time-/Cycle-dependent analysis• Multiaxial proof testing• Works with transient FE analysis
CARES/Life Schematic & Capabilities
Reliability Evaluation• Component probability of survival• Component “hot spots” - high risk of failure
Parameter EstimationWeibull and fatigue parameter
estimates generated fromspecimen rupture data
Finite Element InterfaceOutput from FEA codes
(stresses, temperatures, volumes)read and printed toNeutral Data Base
Time-Dependent Life Prediction Theory -Slow Crack Growth and Cyclic Fatigue Crack Growth Laws
Power Law: - Slow Crack Growth (SCG)
NIeqAK=
dt
da
Combined Power Law & Walker Law: SCG and Cyclic Fatigue
K )R1(fA
K gA = dt
da
NIeq
Qc2
NIeq1
Life Prediction TheoryFor 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
Transient Life Prediction Theory -Power Law
}]d]B
tZ....
]B
tZ]
B
tZ
)([[...[[4V{-exp)Zt(P
i11
2N1B0
1N
1,Ieq
)1k(
)1k(2N)1k(B0
)1k(N
)1k(,Ieqk
k2N
Bk0
kN
k,Ieq
2N
Bk0
maxT,k,Ieqin
1=ikS
21N1m
1
1
)2)1k(N()2k(m
)2)2k(N()1k(m
)1k(
)1k()2kN()1k(m
)2)1k(N(km
k
k
k
General reliability formula for discrete time steps:
k = number of time steps
over the cycle
n = number of elements Z = number of cycles
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.
Random inputvariables
Random inputvariables
Finite-ElementModel
Finite-ElementModel
Material• Strength• Material
Properties
Loads• Thermal• Structural
Geometry/Tolerances
Boundary Conditions
• Gaps• Fixity
ANSYS/PDS (Probabilistic Design System)
Bringing Probabilistic Design into Finite Element Analysis
PDS SimulationsPDS Simulations• Deformations• Stresses• Lifetime
(LCF,...)
Statistical analysis of output parameters
Statistical analysis of output parameters
• 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
ANSYS/PDS (Probabilistic Design System)
Capabilities
Fracture Test Data
Parameter EstimationCARES/Pest
Component Geometry &Boundary Conditions:FEA Model Generation
Heat Transfer Analysis
Ceramics Analysis and Reliability Evaluation of Structures
CARES/Life
ANSYS Stress Analysis
CARES/PDS Integrated Design ProgramCARES/PDS Integrated Design Program
ANSCARES Interface
Report and Statistical Analysis
ANSYSLoop
Fracture Test Data
Parameter EstimationCARES/Pest
Component Geometry &Boundary Conditions:FEA Model Generation
Heat Transfer Analysis
Ceramics Analysis and Reliability Evaluation of Structures
CARES/Life
ANSYS Stress Analysis
CARES/PDS Integrated Design ProgramCARES/PDS Integrated Design Program
ANSCARES Interface
Report and Statistical Analysis
ANSYSLoop
ANSYS/CARES/PDS – Probabilistic Component Life Prediction
ANSYS macros were developed toallows CARES/Life to run within PDS
ANSYS macros were developed toallows CARES/Life to run within PDS
CARES/Life uses results fromDeterministic FEA. Enabling CARES/Life to work with PDSAllows the effects of componentStochastic 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
EXAMPLE: Simplified Turbine Stator Vane in Startup and Shutdown
DATA: Material: A generic silicon nitride
MODEL: • ANSYS FEA analysis using 24151 solid tetrahedral elements
• CARES/Life analysis - volume flaw failure mode & 17 time steps
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
Height 110 mm
Chord Length
70 mm
Width 40 mm
Red = clamped areas
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
Temperature Dependent Material Properties
Density: 3300 [kg/m3 ] Poisson’s ratio: 0.28
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Time [100 sec.]
Lo
ad
Fa
cto
r [-
]
1.Start-up Time 2.Hold Time 3.Shut-down Time 4.Hold Time
0
200
400
600
800
1000
1200
1400
1600
0 500 1000 1500 2000
Time [sec.]
Te
mp
era
ture
[C
]
0
50
100
150
200
250
300
350
400
450
500
0 500 1000 1500 2000
Time [sec.]
Ma
x. P
rin
cip
le S
tre
ss
[M
Pa
]
Maximum vane temperature & principle stress as a function of timeMaximum vane temperature & principle stress as a function of time
Transient FEanalysis performedusing 17 time steps
Time profile of the transient thermal loadsTime profile of the transient thermal loads
Steady state temperatures [°C]at time 75 seconds
Steady state temperatures [°C]at time 75 seconds
Location of maximumprincipal stress [Pa] at time 75 seconds
Location of maximumprincipal stress [Pa] at time 75 seconds
Temperature[C]
Weibull modulus,
mV
Weibull Scale Parameter, oV,
Fatigue exponent,
NV
Fatigue constant, BV [MPa2 Sec]
20 21 864 100 1.28115E+08
1315 24 620 12 1.29023E+08
1371 34 573 11 6.55017E+07
]mmMPa[ Vm/3
CARES/Life Predictions From Deterministic Finite Element Analysis
1.E-151.E-141.E-131.E-121.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00
1 10 100 1000 10000 100000Number of Cycles
Co
nd
itio
na
l Pro
ba
bili
ty o
f F
ailu
re
Conditional probability of failureas a function of number ofload cycles from CARES/Life anddeterministic finite element analysis
Conditional probability of failureas a function of number ofload cycles from CARES/Life anddeterministic finite element analysis
Weibull and fatigue parameters of the silicon nitride ceramic material
Weibull and fatigue parameters of the silicon nitride ceramic material
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 - Gaussian 1.0 0.05
Factor on the thermal conductivity curve
- Gaussian 1.0 0.05
Shift of the hot gas bulk temperature C Gaussian 0.0 30.0
Factor on the heat transfer coefficient on hot gas side
- Lognormal 1.0 0.2
Factor on the hot gas mass flow - Lognormal 1.0 0.03
Start-up time of transient load cycle sec. Lognormal 50.0 5.0
Factor on the Weibull exponent - Gaussian 1.0 0.04
Factor on the Weibull scale parameter - Gaussian 1.0 0.04
Factor on the fatigue exponent - Gaussian 1.0 0.04
Factor on the fatigue constant - Gaussian 1.0 0.04
Random input variables for the PDS analysis
CARES/Life With PDS Analysis
0.001-4
0.1
1.0
10
3050709099
99.999.999
-8 -6-10-12-14-16
Cu
mu
lati
ve
Pro
ba
bili
ty [
%]
Cu
mu
lati
ve
Pro
ba
bili
ty [
%]
log. of Conditional Failure Probability
Conditional Failure Probability for 1000 cycles
0.0010
0.1
1.0
10
3050709099
99.999.999
log. of Conditional Failure Probability
-4-6-8-10-14
Cu
mu
lati
ve
Pro
ba
bili
ty [
%]
Cu
mu
lati
ve
Pro
ba
bili
ty [
%]
-2-12
Conditional Failure Probability for 30000 cycles
Cumulative Distribution of the Conditional Failure Probability for 1,000 and 30,000 Cycles
Monte Carlo simulation method (400 simulations)
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
100 1000 10000 100000Number of Cycles
Pro
ba
bili
ty o
f F
ailu
re
Pf,Conditional
Pf,Total with MCS
Pf,Total with RSM
Failure Probability From Deterministic FE AnalysisVersus Total Probability From PDS Analysis
for 400 Simulations
MCS =Monte Carlo
RSM =Response SurfaceMethod
0.0E+00
2.0E-07
4.0E-07
6.0E-07
8.0E-07
1.0E-06
1.2E-06
1.4E-06
0 100 200 300 400Number of simulations
To
tal F
ailu
re P
rob
ab
ility
0.0E+00
2.0E-02
4.0E-02
6.0E-02
8.0E-02
1.0E-01
1.2E-01
0 100 200 300 400Number of simulations
To
tal F
ailu
re P
rob
ab
ility
Convergence Behavior of the Monte Carlo Simulation Results
1000 cycles 30,000 cycles
• Convergence behavior is significantly better at 30,000 cycles
Factor on the heat transfer coefficient on hot gas side
Factor on the Young’s Modulus
curve
Factor on the fatigue
exponent
Factor on the thermal
expansion
Shift of the hot gas bulk
temperature
Factor on the Weibull
exponent
Factor on the fatigue
constant
Factor on the hot gas mass
flow
Factor on the thermal
conductivity
Factor on the Weibull scale
parameter
Start-up time of transient load cycle
Sensitivity of Conditional Failure Probability
1,000 load cycles with Monte Carlo simulation
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.