Contents

Cover

Title page

Copyright page

Preface

Chapter 1: Overview of Probabilistic Physics-of-Failure Approach to Reliability

1.1. Introduction
1.2. Overview of Physics-of-Failure Modeling
1.3. Important Forms of PoF Models
1.4. PPoF Approach to Life Assessment
1.5. Accelerated Testing in PPoF Model Development
1.6. Organization of the Book
References

Chapter 2: Summary of Mechanisms of Failure and Associated PoF Models

2.1. Introduction
2.2. Fatigue
2.3. Wear
2.4. Creep
2.5. Corrosion
References

Chapter 3: Types of Accelerated Testing and Modeling Concepts

3.1. Introduction
3.2. Types of Accelerated Testing – Qualitative and Quantitative
3.3. Qualitative Accelerated Tests
3.4. Quantitative Accelerated Tests
References

Chapter 4: Analysis of Accelerated Life Testing Data and Physics-Based Reliability Model Development

4.1. Introduction
4.2. Accelerated Life Data Analysis Methods
4.3. Basics of ALT Data Analysis
4.4. Types of Collected Accelerated Life Test Data
4.5. Life-Stress Models
4.6. Probability Plotting Method for ALT Model Estimation
4.7. Maximum Likelihood Estimation Approach to ALT Data Analysis
4.8. Confidence Intervals for MLE
4.9. MLE Approach to Estimating Parameters of Common Distributions
4.10. MLE-Based Parameter Estimation for Different Life-Stress Models
4.11. Proportional Hazards (PH) Model
4.12. Bayesian Estimation Approach to ALT Model Parameter Estimation
4.13. Determining Stress Dependencies
4.14. Summary of the ALT Steps and Common Problems in Practice
4.15. Time Varying Stress Tests
4.16. Step-Stress Analysis and Model Development
References

Chapter 5: Analysis of Accelerated Degradation Data and Reliability Model Development

5.1. Introduction
5.2. Degradation Models
References

Chapter 6: Accelerated Test Planning

6.1. Introduction
6.2. Issues to Consider Prior to Accelerated Testing
6.3. Planning for Accelerated Life Tests
6.4. Planning for Accelerated Degradation Tests
References

Chapter 7: Accounting for Uncertainties and Model Validation

7.1. Introduction
7.2. Uncertainties in Evidence
7.3. PPoF Model Uncertainties, Errors, and Validation
7.4. Applications of Model Validation in ADT
References

Index

List of Illustrations

Chapter 1

Figure 1.1: System hierarchy used in PoF analysis
Figure 1.2: Stress-strength modeling
Figure 1.3: Damage-endurance model
Figure 1.4: Performance-requirement model
Figure 1.5: System hierarchy in probabilistic-mechanistic reliability life model
Figure 1.6: Conceptual acceleration of stress agents at two overstress conditions and corresponding data points generated from ALT
Figure 1.7: Conceptual PPoF model development and extrapolation in ALT

Chapter 2

Figure 2.1: Classification of material degradation
Figure 2.2: Nomenclature for constant amplitude cyclic loading
Figure 2.3: Idealized S-N diagram
Figure 2.4: Effect of mean stress on fatigue life
Figure 2.5: S-N diagram for example problem
Figure 2.6: Biaxial stress state in thin-walled pressure vessel
Figure 2.7: State of stress at a point represented by stress element
Figure 2.8: Cylindrical specimen under monotonic load
Figure 2.9: Typical monotonic and cyclic load stress-strain curves
Figure 2.10: Cyclic stress-strain behavior
Figure 2.11: Massing’s hypothesis showing how stabilized hysteresis loop is obtained
Figure 2.12: Bauschinger effect
Figure 2.13: Typical strain-life curve
Figure 2.14: Morrow’s mean stress correction to the strain-life curve for a tensile mean stress (Bannantine, et al., 1997)
Figure 2.15: Strain history
Figure 2.16: Stress-strain response for given example
Figure 2.17: Strain reading: a) zero mean stress, b) tensile mean stress
Figure 2.18: Variable amplitude loading consisting of constant amplitude “blocks”
Figure 2.19: S-N plot showing spectrum of loading where n, cycles are accrued at each of the different corresponding stress levels S_i, and the N_i are cycles to failure at each S_i
Figure 2.20: Juvinall method for modification for the S-N curve for notched components
Figure 2.21: Notched bar configuration for Example 2.7
Figure 2.22: S-N design curve for Example 2.7
Figure 2.23: Nominal and local stress-strain regions
Figure 2.24: Crack initiation and propagation portions of fatigue life
Figure 2.25: Fatigue crack length vs. applied cycles (x marks the fracture)
Figure 2.26: Typical curve for crack growth rate vs range of stress intensity factor
Figure 2.27: Effect of initial crack size on fatigue life
Figure 2.28: Diagram for Example 2.10
Figure 2.29: A shallow notch
Figure 2.30: A deep notch
Figure 2.31: Effect of frequency, f on low-cycle fatigue life, a) crack growth b) fatigue life
Figure 2.32: Fatigue life reduction due to harsh environment and/or elevated temperature
Figure 2.33: Different modes of wear: (a) adhesive wear by adhesive shear and transfer (b) abrasive wear by microcutting of ductile bulk surface (c) flow wear by accumulated plastic shear flow (d) fatigue wear by crack initiation and propagation (e) corrosive wear by shear fracture of ductile tribofilm (f) corrosive wear by shaving of soft tribofilm; (g) corrosive wear by accumulated plastic shear flow of soft tribofilm (h) corrosive wear by delamination of brittle tribofilm (i) melt wear by local melting and transfer or scattering (Kato, 2000; with permission)
Figure 2.34: Transient and steady-state wear regime for adhesive wear mechanism
Figure 2.35: Change of wear coefficient with PV (Bayer, 2004)
Figure 2.36: Pin-on-disk tribosystem
Figure 2.37: Schematic of a hard cone plows a groove in a soft material (Adopted form Engel, 1993)
Figure 2.38: Impact wear regimes (adapted from Wear Control Handbook, 1980)
Figure 2.39: Two parallel cylinders in rolling-sliding contact
Figure 2.40: Effect of slide-to-roll ratio on life curves
Figure 2.41: Schematic of a ball bearing with a contact angle between ball and its race
Figure 2.42: Bearing life multiplying factors, a) load factor b) lubricant factor c) water contamination factor
Figure 2.43: Static and dynamic seals (source: Handbook of Reliability Prediction Procedures for Mechanical Equipment, 1994)
Figure 2.44: Wear rate vs. PV factor in seal tests (Adopted form Wear Control Handbook, 1980)
Figure 2.45: Schematic for the definition of viscosity
Figure 2.46: Stribeck curve
Figure 2.47: Expected useful life of hydraulic and turbine oils as a function of temperature
Figure 2.48: Schematic of creep curves for different stress levels or temperatures
Figure 2.49: Thermal acceleration method for creep testing, Collins (1993)
Figure 2.50: Creep rupture curves for Cr-Mo-V steel
Figure 2.51: Creep curves for three materials plotted on log-log coordinates (From Collins, 1993, with permission-original from Sturm, et. al, 1936)
Figure 2.52: Various forms of corrosion (Jones, 1996; with permission)

Chapter 3

Figure 3.1: Types of accelerated testing
Figure 3.2: Areas of life where HALT and HASS are useful
Figure 3.3: Stress vs. life relationship
Figure 3.4: Impact of aging over time as described by damage-endurance modeling
Figure 3.5: Cumulative damage as a truncated normal pdf overlapping the endurance limit described as a Gaussian probability density function
Figure 3.6: Histogram of the time-to-failure distribution

Chapter 4

Figure 4.1: Simple linear extrapolation for eliciting use-level information from ALT data
Figure 4.2: Complete data
Figure 4.3: Censored data (multiply censored)
Figure 4.4: Singly censored data
Figure 4.5: Interval or grouped censored data
Figure 4.6: Wear (damage) life-stress modeling concept
Figure 4.7: Lognormal data plotting at each stress level
Figure 4.8: Weibull data plotting at each stress level
Figure 4.9: Least squares regression for Arrhenius model
Figure 4.10: Example of rudimentary approach to estimating confidence interval using graphical method for data analysis
Figure 4.11: Weibull multi plot for Arrhenius-Weibull life-stress model
Figure 4.12: Multi plot for exponential life-stress with Weibull life
Figure 4.13: Extrapolation for exponential life-stress with Weibull time to failure
Figure 4.15: Schematic of Bayesian approach to estimation of ALT model parameters
Figure 4.16: Marginal posterior distributions of the parameters
Figure 4.17: Normal Use Weibull Plots from Example 4.12 & Example 4.14, showing the effect of the interaction term.
Figure 4.18: Step-stress profile and corresponding life distributions
Figure 4.19: Example stress profile for a step-stress test
Figure 4.20: Weibull multi plot for step 2 of example
Figure 4.21: Final Weibull plots for the corrected data of step-stress test example
Figure 4.22: Weibull multi plot for step 2 of example
Figure 4.23: Step stress profile and corresponding life distributions
Figure 4.24: A conceptual step-stress test

Chapter 5

Figure 5.1: Example of a degradation path
Figure 5.2: Example of degradation paths from an accelerated degradation test for three stress levels
Figure 5.3: Possible shapes for univariate cumulative degradation vs. time
Figure 5.4: A linear degradation model
Figure 5.5: General degradation path model for unit i
Figure 5.6: Weibull probability plot for calculated failure times
Figure 5.7: Lognormal probability plot for calculated failure times
Figure 5.8: Plot of weight loss in micrograms vs. hundreds of cycles
Figure 5.9: Reconstruction of trendline data and best fit line using MATLAB and MLE
Figure 5.10: The original data along with the best fit lines without and with assumption of 10% measurement error correction
Figure 5.11: Square root to linear plot
Figure 5.12: Reconstruction of Figure 5.11 with trend lines for each degradation path (using Matlab)
Figure 5.13: Weibull distribution fit for the pseudotimes given in Table 5.5
Figure 5.14: Life-stress plot (top), and estimated life distribution at the use temperature of 50 °C (323 °K) (bottom)
Figure 5.15: Performance degradation data from Table 5.7.
Figure 5.16: Trendlines of performance degradation obtained through MATLAB and Excel.
Figure 5.17: Weibull distribution fit with characteristics at the given conditions and the use conditions of 30 °C and 400 mA
Figure 5.18: Mean and 80% confidence bounds of the performance degradation at the use conditions of 30 °C and 400 mA
Figure 5.19: Relative luminosity of initial luminosity data
Figure 5.21: Temperature vs. time to failure relationship plot
Figure 5.20: Trendlines of relative luminosity based on parameter estimates obtained from the Matlab Bayesian estimation routine
Figure 5.22: Weibull distribution fit with characteristics at the given conditions and the use conditions of 20°C

Chapter 6

Figure 6.1: Possible results for a typical temperature-accelerated failure mode on an IC device (Meeker, 1998)
Figure 6.2: Failure mode 2 with lower activation energy, masked at high temperature and unmasked at lower temperature (Meeker, 1998)
Figure 6.3: Well-behaved comparison of two items (Meeker, 1998)
Figure 6.4: Log-variance vs. stress changing time

Chapter 7

Figure 7.1: Degradation model prediction D_m vs. degradation measurement D_e
Figure 7.2: Comparison of AE model prediction and experimental results
Figure 7.3: Model prediction with multiplicative error

List of Tables

Chapter 2

Table 2.1: Summary of common failure modes (Collins (1993), Arunajadai, Stone and Tumer (2002) and Khonsari and Amiri (2012))
Table 2.2: Some values of wear coefficient (from Engel (1993))
Table 2.3: Parameters used in development of wear models
Table 2.4: Typical bearing failure mechanisms (adopted from the Handbook of Reliability Prediction Procedures for Mechanical Equipment, 1994)
Table 2.5: Values of f_c for radial and angular ball bearings (adopted from Handbook of Reliability Prediction Procedures for Mechanical Equipment, 1994)
Table 2.6: Typical bearing failure mechanisms (adopted from Wear Control Handbook, 1980)
Table 2.7: Coefficient of friction for various seal materials (adopted from Wear Control Handbook, 1980)
Table 2.8: Fluid viscosity multiplying factor (adopted from Handbook of Reliability Prediction Procedures for Mechanical Equipment, 1994)
Table 2.9: Contaminant multiplying factor (adopted from Handbook of Reliability Prediction Procedures for Mechanical Equipment, 1994)
Table 2.10: Typical values for k₁ in oxidation life prediction of industrial mineral oils (adopted from Booser, 1997)
Table 2.11: Equivalent conditions based on Larson-Miller parameter
Table 2.12: Various constants for the Manson-Haferd equation
Table 2.13: Creep rupture data for Cr-Mo-V steel
Table 2.14: Constants for log-log stress-time creep law

Chapter 4

Table 4.1: Times to failure observed in an ALT
Table 4.2: Plotting points using rank adjustment method
Table 4.3: Time to failures observed in an ALT
Table 4.4: CDF by rank adjustment method
Table 4.5: Stress levels and failure times
Table 4.6: Data from fatigue test of material (prior data) and component ALT data
Table 4.7: Time to failure
Table 4.8: Details of the step-stress example and failures observed
Table 4.9: Estimate of the parameters of Weibull distribution
Table 4.10: Acceleration factor between all steps
Table 4.11: Computation of equivalent time that produces the same amount of damage
Table 4.12: Adjusted acceleration factor
Table 4.13: Final results of equivalent time computation after a few iterations
Table 4.14: Final acceleration factors and estimation of the parameters of life-stress model
Table 4.15: Step stress equivalent times for Weibull plot
Table 4.16: Comparisons of original estimates and final estimates after correcting for cumulative damage
Table 4.17: Tentative step-stress test plan
Table 4.18: Hypothetical results summary
Table 4.19: Detail of testing result 2
Table 4.20: Estimate of the parameters of Weibull distribution
Table 4.21: Acceleration factor between all steps
Table 4.22: Computation of equivalent time that produces the same amount of damage
Table 4.23: Adjusted acceleration factor
Table 4.24: Step-stress test data on cable 1 (* Before step 1, each specimen was held 10 minutes each at 5, 10, 15, and 20 kV; + denotes a right censored)
Table 4.25: Prior distribution results of cable 2
Table 4.26: Posterior distribution results from data of cable 1 and priors of cable 2

Chapter 5

Table 5.1: Times-to-failure corresponding to the recorded weight losses
Table 5.2: Accelerated degradation test data
Table 5.3: Pseudo failure times corresponding to the recorded weight
Table 5.4: Resistance over time of carbon-film resistors tested at three different levels of temperature
Table 5.5: Pseudo failure times corresponding to the recorded weight
Table 5.6: Three condition values as well as their corresponding temperature, forward current, and number of LEDs
Table 5.7: Temperature and current at four given time periods for 36 units
Table 5.8: Failure times corresponding to the recorded temperature and current and the failure conditions of 70% degradation
Table 5.9: Confidence intervals for degradation parameters
Table 5.10: LED luminosity data of Unit 1 at 25 °C, 65 °C, and 105 °C
Table 5.11: Posterior degradation parameters
Table 5.12: Failure times corresponding to the recorded temperature and the failure conditions of 50% degradation of luminosity

Chapter 6

Table 6.1: Expected variance of each plan

Chapter 7

Table 7.1: Jeffreys’ Guideline for Bayes factor of Model M₁ vs. Model M₂

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Performability Engineering Series
Series Editors: Krishna B. Misra (kbmisra@gmail.com)
and John Andrews (John.Andrews@nottingham.ac.uk)

Scope: A true performance of a product, or system, or service must be judged over the entire life cycle activities connected with design, manufacture, use and disposal in relation to the economics of maximization of dependability, and minimizing its impact on the environment. The concept of performability allows us to take a holistic assessment of performance and provides an aggregate attribute that reflects an entire engineering effort of a product, system, or service designer in achieving dependability and sustainability. Performance should not just be indicative of achieving quality, reliability, maintainability and safety for a product, system, or service, but achieving sustainability as well. The conventional perspective of dependability ignores the environmental impact considerations that accompany the development of products, systems, and services. However, any industrial activity in creating a product, system, or service is always associated with certain environmentalimpacts that follow at each phase of development. These considerations have become all the more necessary in the 21st century as the world resources continue to become scarce and the cost of materials and energy keep rising. It is not difficult to visualize that by employing the strategy of dematerialization, minimum energy and minimum waste, while maximizing the yield and developing economically viable and safe processes (clean production and clean technologies), we will create minimal adverse effect on the environment during production and disposal at the end of the life. This is basically the goal of performability engineering.

It may be observed that the above-mentioned performance attributes are interrelated and should not be considered in isolation for optimization of performance. Each book in the series should endeavor to include most, if not all, of the attributes of this web of interrelationship and have the objective to help create optimal and sustainable products, systems, and services.

Publishers at Scrivener
Martin Scrivener (martin@scrivenerpublishing.com)
Phillip Carmical (pcarmical@scrivenerpublishing.com)

Probabilistic Physics of Failure Approach to Reliability

Modeling, Accelerated Testing, Prognosis and Reliability Assessment

Mohammad Modarres

Center for Risk and Reliability, University of Maryland, College Park, U.S.A.

Mehdi Amiri

Department of Mechanical Engineering, George Mason University, Fairfax, U.S.A.

Christopher Jackson

Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, U.S.A.

This edition first published 2017 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2015 and 2017 Center for Risk and Reliability (CRR), University of Maryland, U.S.A.
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Edition History
First published 2015; reissued in 2017

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Library of Congress Cataloging-in-Publication Data
ISBN 978-1-119-38863-0

Preface

This book is result of the compilation of class notes from several years of teaching a graduate course on physics-of-failure and accelerated testing to the graduate students pursuing Master of Science, Master of Engineering and PhD degrees in Reliability Engineering at the University of Maryland. The book provides probabilistic and highly technical approaches to the physics-of-failure and mechanistic based reliability prediction and assessment. It relies on various methods and techniques published in the open literature regarding the development and practice of physics-of-failure analysis, accelerated life testing and accelerated degradation testing. The authors first discuss the overall concepts, objectives and framework for accelerated life assessment through the use of formal probabilistic physics-of-failure models. They review important failure mechanisms to demonstrate the process of examining and developing appropriate physics and mechanistic models that describe the degradation and failure phenomena in accompanying accelerated testing and accelerated degradation testing methods, including step-stress testing. The book presents data analysis methods to evaluate the probabilistic physics-of-failure models based on the observed data obtained from accelerated reliability tests. Further, it discusses the steps and methods of probabilistic life assessment and integrity of structures, components and systems based on the probabilistic physics-of-failure models. Since the book is intended for graduate-level students and for highly trained reliability engineers, it provides supplementary solved examples to clarify complex technical topics within each chapter. Some of these examples are benefitted directly or with some modifications from other sources, including Bannantine, et al. (1997), Collins (1993), Stephens, et al. (2003), Meeker & Escobar (1998), Nelson (2004), and Dowling (1998), which are referenced extensively in the book. Although qualitative accelerated tests such as the Highly Accelerated Life Test (HALT) and Environmental Stress Screening (ESS) have been briefly reviewed, the book is mainly about the quantitative methods in probabilistic physics-based and accelerated testing life assessment of structures, components and systems. A companion website under the auspices of the Center for Risk and Reliability at the University of Maryland (www.crr.umd.edu) provides downloadable support files for additional information and computational tools in form of MATLAB, R and OpenBUGS scripts to perform some of the more involved computational analyses discussed in the book. These files will be updated and conformed to the most recent versions of these tools. The companion website also includes a section on testing equipment and resources needed for accelerated testing. This book benefitted from contributions of many students who enrolled in the accelerated testing courses over many years at the University of Maryland. Particularly, inputs and solved example from Wendell Fox, Jonathan DeJesus, Reuel Smith, Reza Azarkhail, Andrew Bradshaw, and Taotao Zhou have been significant and are much appreciated.

Mohammad Modarres
Mehdi Amiri
Christopher Jackson