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

(Noel.N.Nemeth@nasa.gov)

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.