Improved Probabilistic Modeling of Multi-Site Fatigue Cracking
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Published
Nov 3, 2020
Abdallah Al Tamimi
Mohammad Modarres
Abstract
The purpose of this paper is to investigate the effect of fatigue, in the presence of neighboring cracks, and to integrate that into a model that could be used to predict crack growth. A total of 20 fatigue experiments were performed at different loading conditions using dog-bone samples of API-5L grade B carbon steel containing neighboring cracks. The impact of the neighboring cracks dimensions and the loading conditions on the interaction, coalescence and growth of cracks were investigated. A design of experiment approach to study neighboring cracks interactions and growth in carbon steel materials is also presented. Simulation efforts were performed to assess the Stress Intensity Factor (SIF) around neighboring cracks. Models discussing how the SIF of single semi-elliptical crack could be corrected to account for the neighboring cracks interaction were discussed in order to better understand the fatigue behavior. A combination of these models was integrated to find the SIF values necessary for the probabilistic life prediction modeling purposes. Finally, a multi-site fatigue crack growth rate model was developed and its parameters including their uncertainties were estimated. A Bayesian approach was adopted to perform uncertainty characterization and model validation.
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References
Ahmad, Z. (2007). Corrosion Fatigue . In Principles of Corrosion Engineering and Corrosion Control (pp. 221-241). Elsevier.
Al Tamimi, A. (2014). Improved probabilistic remaining useful life estimation in engineering structures: modeling multi-site fatige cracking, PhD Thesis. College Park: University of Maryalnd.
Al Tamimi, A., & Modarres, M. (2014). Coalescence and Growth of Two Semi-elliptical Coplanar Cracks in API-5L Grade B steel . SEM 2014 Annual Conference & Exposition on Experimental and Applied Mechanics. Greenville, SC, USA.
Alabama Specialty Products. (2014). (Metal Samples ) Retrieved from http://www.alspi.com/MS-machining.htm
Alseyabi, M. C. (2009). Structuring a Probabilistic Model for Reliability Evaluation of Piping Subject to Corrosion-fatigue Degredation, PhD thesis. College Park: University of Maryland.
ASME. (1992). Rules for inservice inspection of nuclear power plant components. In Boiler and Pressure Vessel Code (pp. Appendix A, Section XI). ASME.
Azarhkail, M., & Modarres, M. (2007). Markov chain Monte Carlo simulation for estimating accelerated life model parameters. Annual Reliability and Mainitainability Symposium (RAMS) Proceedings. Orlando.
Azarkhail, M., & Modarres, M. (2007). A Novel Bayesian Framework for Uncertainity Management in Physics-Based Reliability Models. ASME International Mechanical Engineering Congress and Expositon . Seattle, Washignton, USA.
Azarkhail, M., & Modarres, M. (2012). The Evolution and History of Reliability Engineering: Rise of Mechanistic Reliablity Modeling. International Journal of Performability Engineering , 8, 35-47.
Azarkhail, M., Ontiveros, V., & Modarres, M. (2009). A Bayesian Framework for Model Uncertainity Consideration in Fire Simulation Codes. 17th International Conference on Nuclear Engineering . Brussels.
Bayley, C. (1997). Parametric Investigation on the Coalescence of Coplanar Fatigue Cracks, Masters' thesis. Ottawa: Carelton University .
Beretta, S., Carboni, M., Cantini, S., & Ghidini, A. (2004). Application of fatigue crack growth algorithms to railway axles and comparison of two steel grades. Journal of Rail and Rapid Transit , 218 (4), 317-326.
BSI. (1991). Guidance on methods for assessing the acceptability of flaws in fusion welded joints. British Standard Institute.
Carlton, R., Lyman, c., & Roberts, J. (2004). Accuracy and Precision of Quantitative Energy Dispersive X-ray Spectrometry in the Environmental Scanning Electron Microscope. Scanning 26 , 167-174.
Cavanaugh, J. (2012, September 25). Lecture V: The Bayesian Information Criterion. (University of Iowa) Retrieved August 29, 2014, from http://myweb.uiowa.edu/cavaaugh/ms_lec_5_ho.pdf
Chang, R. (1982). On Crack-Crack Interaction and Coalescence in Fatigue . Engineering Fracture Mechanics , 16, 683-693.
Chaussumier, M., Shahzad, M., Mabru, C., Chieragatti, R., & Rezai-Aria, F. (2010). A Fatigue Multi-Site Cracks Model Using Coalescence, Short and Long Crack Growth Laws, for Anodized Aluminum Alloys.
Cortie, M., & Garrett, G. (1988). On the correlation between the C and m in the Paris equation for fatigue crack propagation. Engineering Fracture Mechanics , 30 (1), 49-58.
DeBartolo, E., & Hillberry, B. (1998). Effects of Constituent Particle Clusters on Fatigue Behavior of 2024-T3 Aluminum Alloy . Int. J. Fatigue , 20, 727-735.
Dowling, N., Calhoun, C., & Arcari, A. (2008). Mean stress effects in stress-life fatigue and the Walker equation . Fatigue and Fracture of Engineering Materials and Structures , 32, 163-179.
Dwivedi, P., & Green, D. (1995). Indentation Crack Shape Evolution During Subcritical Crack Growth. Journal of the American Ceramic Society , 78 (5), 1240-1246.
Erdogan, F. (1962). On the stress distribuion in plates with collinear cuts under arbitary loads. 4th US Nat. Cong. Appl. Mech., (pp. 547-53). PP. 547-553.
Ewalds, H., & Wanhill, R. (1984). The three modes of loading. Retrieved November 1, 2013, from http://thediagram.com/12_3/thethreemodes.html
Fatemi, A. (n.d.). Fundamentals of LEFM and Applications to Fatigue Crack Growth, Chapter 6. Toledo: University of Toledo.
Fernandes, J. (2002). Uma metodologia para a analise e modelagem, PhD thesis. Rio de Jeneiro, Brasil: PUC.
Forsyth, P. (1983). A unified Description of Micro and Macro-scopic Fatigue Crack Behaviour . International Journal of Fatigue , 5, 3-14.
Georgsson, P. (2000). The determinatino of uncertainties in fatigue crack growth measurements. Trollhattan, Sweden : Volvo Aero Corporation .
Hamam, R., Pommier, S., & Bumbieler, F. (2007). Variable amplitude fatigue crack growth, experimental results and modeling . 29 (1634-1646).
Harrington, D. (1995). Fatigue Crack Coalescence and Shape Development and Experimental Invistigation, Masters Thesis. Carleton University, Department of Mechanical and Aerospace Engineering .
Ho, K., & Newman, S. (2003). State of the Art Electrical Discharge Machining (EDM). International Journal of Mchine Tools and Manufacture , 43 (13), 1287-1300.
Iida, K. (1983). Shapes and Coalescence of Suface Fatigue Cracks. I.C.F. International Symposium on Fracture Mechanics. Beijing.
Irwin, G. (1962). Crack Extension Force for a Part Through Crack in a Plate . Journal of Applied Mechanics , 29, 651-654.
ISO TAG4. (1993). In Guide to the expression of uncertainty in measurement (pp. ISBN 92-67-10188-9). Geneva, Switzerland: International Organization for Standardisation.
Keshtgar, A. (2013). Acoustic emission-based structural health management and prognostics subject to small fatigue cracks, PhD thesis . College Park: University of Maryland.
Keshtgar, A., & Modarres, M. (2013). Detection of crack intitation based on acoustic emission. Chemical engineering 33 , DOI: 10.3303/CET1333092.
Keshtgar, A., & Modarres, M. (2013). Probabilistic model developement for fatigue crack detection using acoustic emission technology. International Topical Meeting on Probabilistic Safety Assesement and Analysis. Columbia, SC: ANS PSA.
Kishimoto, K., Soboyejo, W., Smith, R., & Knott, J. (1989). A Numerical Invistigation of the Interaction and Coalescence of Twin Coplanar Semi Elliptical Fatigue Cracks. International Journal of Fatigue , 11 (2), 91-96.
Krishnaprasad, K., & Prakash, V. (2009). Fatigue crack growth behavior in dissimilar metal weldment of staniless steel and carbon steel. World Academy of Science, Engineering and Technology , 3 , 8-22.
Leek, T. (1990). The Interaction and Growth of Two Surface Cracks Under Fatigue Loading, PhD Thesis. UK: University of Sheffield.
Leek, T., & Howard, I. (1996). An examination of methods of assessing interacting surface crack by comparison with experimental data. International Journal of Pressure Vessels and Piping , 181-201.
Leek, T., & Howard, I. (1996). An Examination of Methods of Assessing Interacting Surface Cracks by Comparison with Experimental Data. International Journal of Pressure Vessels and Piping , 68, 181-201.
Leek, T., & Howard, I. (1994). Estimating the Elastic Interaction Factors of Two Coplanar Surface Cracks Under Mode I load. International Journal of Pressure Vessels and Pipeing , 60, 307-321.
Leek, T., & Howard, I. (1994). Rules for the Assessment of Interacting Suface Cracks Under Mode I Load. Internationl Journal of Pressure Vessels and Piping , 60, 323-339.
Li, C. (1996). Probabilistic Modeling for Corrosion-Fatigue Crack Growth, PhD Dissertation. Lehigh University .
Lin, X., & Smith, R. (1995). Numerical Prediction of Fatigue Crack Growth of a Surface Defect. Fatigue and Fracture of Engineering Materials and Structures , 18 (2), 247-256.
Materials Evaluation and Engineering Inc. (2009). Energy Dispersive X-ray Spectroscopy. (Materials Evaluation and Engineering, Inc.) Retrieved December 02, 2013, from http://mee-inc.com/eds.html
Melin, S. (1983). Why do Cracks Avoid Each Other. International Journal of Fracture , 23, 37-45.
Modarres, M. (2012). Accelerated Life Testing, ENRE64, Class Presentation Slides . College Park: University of Maryland.
Modarres, M. (2008). Accelerated Testing, ENRE 641. College Park : Center of Risk and Reliability, UMD, College Park, USA.
Modarres, M., Kaminskiy, M., & Krivtsov, V. (2010). Reliability engineering and risk analysis. CRC Press.
Murakami, Y., & Nemat-Nasser, S. (1982). Interacting Dissimilar Semi-elliptical Surface Flaws Under Tension and Bending . Engineering Fracture Mechanics , 16, 373-386.
Murakami, Y., & Nisitani, H. (1981). Stress Intensity Factors for Interacting Two Equal Semi-elliptical Surface Cracks in Tension. Trans. Japan Soc. Mech. Engrs SerA , 47, 295-303.
National Research Council. (2013). Effects of Diluted Bitumen on Crude Oil Transmission Pipeline, Special Report 311. National Research Council.
Neves Beltrao, M., Castrodeza, E., & Bastian, F. (2010). Fatigue crack propagation in API 5L X-70 pipeline steel longitudinal welded joints under constant and variable amplitude. Fatigue and Fracture of Engineering Materials and Structures , 34, 321-328.
Newman, J. (1979). A Review and Assessment of the Stress Intensity Factors for Surface Cracks. ASTM STP 687.
Newman, J., & Raju, I. (1981). An Emperical Stress Intensity Factor Equation for the Surface Crack. Engineering Fracture Mechanics , 15.
Newman, J., & Raju, I. (1984). Stress Intensity Factor Equations for Cracks in Three Dimensional Finite Bodies Subjected to Tension and Bending Loads. Hampton, Virginia: NASA.
Newman, J., & Raju, I. (1979). Stress intensity Factors for a Wide Range of Semi-Elliptical Suface Cracks in Finite Width Plates. Engineering Fracture Mechanics , 11.
Nuhi, M., Abu Seer, T., Al Tamimi, A., Modarres, M., & Seibi, A. (2011). Reliability Analysis for Degredation Effects of Pitting Corrosion in Carbon Steel Pipes. ICM11. Como .
O'Donoghure, T., Nishioka, P., & Atluris, N. (1984). Multiple Surface Cracks in Pressure Vessels. Engineering Fracture Mechanics , 20 (3), 545-560.
Ontiveros, V., Cartillier, A., & Modarres, M. (2010). An Integrated Methodology for Assessing Fire Simulation Code Uncertainity. Nuclear Science Engineering , 166, 179-201.
Paris, P., & Erdogan, F. (1963). A critical analysis of crack propagation laws . Journal of Basic Engineering , 528-534.
Pitt, S., & Jones, R. (1997). Multiple-Site and Widespread Fatigue Damage in Aging Aircrafts. Engineering Failure Analysis , 4, 237-257.
Roberge, P. (2012 ). Corrosion Pit Shapes. (Corrosion Doctors ) Retrieved June Tuseday, the 26th , 2012, from http://corrosion-doctors.org/Forms-pitting/shapes.htm
Savin, G. N. (1981). Stress concentration around holes. Oxford: Pergamon Press.
Shi, Y., Chen, B., & Zhang, J. (1999). Effects of welding residual stresses on fatigue crack growth behaviour in butt welds of a pipline steel. Engineering Fracture Mechanics , 36, 893-902.
Soboyejo, W., & Knott, J. (1990). Fatigue Crack Propagation of Copalanr Semi-elliptical Cracks in Pure Bending. Engineering Fracture Mechanics , 37 (2), 323-340.
Spiegelhalter, D., Thomas, A., Best, N., & Lunn, D. (2003). WinBUGS 1.4 manual .
Stephens, R., Fatemi, A., Stephens, R., & Fuchs, H. (2000). Metal fatigue in engineering. John Wiley & Sons .
Terrell, J. (1988). Effect of Cyclic Frequency on the Fatigue Life of ASME SA-106-B Piping Steel in BWR Environments. Journal of Material Engineering , 10, 193-203.
Twaddle, J., & Hancock, B. (1988). The Development of Cracks by Defect Coalescence. Engineering Materials Advisory Service , 185-198.
Willard, S. A. (1997). Use of of Marker Bands for Determination of Fatigue Crack Growth Rates and Crack Front Shapes in Pre-Corroded Coupons. Hampton, Virginia : NASA.
Yokobori, T., Ohashi, M., & Ichikawa, M. (1965). The interaction of two collinear asymmetrical elastic cracks. Vol. 1, PP. 33-39: Tohoku University .
Zhang, H. (1993). Fatigue Crack Growth and Coalescence Study. Ontario: Carleton University.
Al Tamimi, A. (2014). Improved probabilistic remaining useful life estimation in engineering structures: modeling multi-site fatige cracking, PhD Thesis. College Park: University of Maryalnd.
Al Tamimi, A., & Modarres, M. (2014). Coalescence and Growth of Two Semi-elliptical Coplanar Cracks in API-5L Grade B steel . SEM 2014 Annual Conference & Exposition on Experimental and Applied Mechanics. Greenville, SC, USA.
Alabama Specialty Products. (2014). (Metal Samples ) Retrieved from http://www.alspi.com/MS-machining.htm
Alseyabi, M. C. (2009). Structuring a Probabilistic Model for Reliability Evaluation of Piping Subject to Corrosion-fatigue Degredation, PhD thesis. College Park: University of Maryland.
ASME. (1992). Rules for inservice inspection of nuclear power plant components. In Boiler and Pressure Vessel Code (pp. Appendix A, Section XI). ASME.
Azarhkail, M., & Modarres, M. (2007). Markov chain Monte Carlo simulation for estimating accelerated life model parameters. Annual Reliability and Mainitainability Symposium (RAMS) Proceedings. Orlando.
Azarkhail, M., & Modarres, M. (2007). A Novel Bayesian Framework for Uncertainity Management in Physics-Based Reliability Models. ASME International Mechanical Engineering Congress and Expositon . Seattle, Washignton, USA.
Azarkhail, M., & Modarres, M. (2012). The Evolution and History of Reliability Engineering: Rise of Mechanistic Reliablity Modeling. International Journal of Performability Engineering , 8, 35-47.
Azarkhail, M., Ontiveros, V., & Modarres, M. (2009). A Bayesian Framework for Model Uncertainity Consideration in Fire Simulation Codes. 17th International Conference on Nuclear Engineering . Brussels.
Bayley, C. (1997). Parametric Investigation on the Coalescence of Coplanar Fatigue Cracks, Masters' thesis. Ottawa: Carelton University .
Beretta, S., Carboni, M., Cantini, S., & Ghidini, A. (2004). Application of fatigue crack growth algorithms to railway axles and comparison of two steel grades. Journal of Rail and Rapid Transit , 218 (4), 317-326.
BSI. (1991). Guidance on methods for assessing the acceptability of flaws in fusion welded joints. British Standard Institute.
Carlton, R., Lyman, c., & Roberts, J. (2004). Accuracy and Precision of Quantitative Energy Dispersive X-ray Spectrometry in the Environmental Scanning Electron Microscope. Scanning 26 , 167-174.
Cavanaugh, J. (2012, September 25). Lecture V: The Bayesian Information Criterion. (University of Iowa) Retrieved August 29, 2014, from http://myweb.uiowa.edu/cavaaugh/ms_lec_5_ho.pdf
Chang, R. (1982). On Crack-Crack Interaction and Coalescence in Fatigue . Engineering Fracture Mechanics , 16, 683-693.
Chaussumier, M., Shahzad, M., Mabru, C., Chieragatti, R., & Rezai-Aria, F. (2010). A Fatigue Multi-Site Cracks Model Using Coalescence, Short and Long Crack Growth Laws, for Anodized Aluminum Alloys.
Cortie, M., & Garrett, G. (1988). On the correlation between the C and m in the Paris equation for fatigue crack propagation. Engineering Fracture Mechanics , 30 (1), 49-58.
DeBartolo, E., & Hillberry, B. (1998). Effects of Constituent Particle Clusters on Fatigue Behavior of 2024-T3 Aluminum Alloy . Int. J. Fatigue , 20, 727-735.
Dowling, N., Calhoun, C., & Arcari, A. (2008). Mean stress effects in stress-life fatigue and the Walker equation . Fatigue and Fracture of Engineering Materials and Structures , 32, 163-179.
Dwivedi, P., & Green, D. (1995). Indentation Crack Shape Evolution During Subcritical Crack Growth. Journal of the American Ceramic Society , 78 (5), 1240-1246.
Erdogan, F. (1962). On the stress distribuion in plates with collinear cuts under arbitary loads. 4th US Nat. Cong. Appl. Mech., (pp. 547-53). PP. 547-553.
Ewalds, H., & Wanhill, R. (1984). The three modes of loading. Retrieved November 1, 2013, from http://thediagram.com/12_3/thethreemodes.html
Fatemi, A. (n.d.). Fundamentals of LEFM and Applications to Fatigue Crack Growth, Chapter 6. Toledo: University of Toledo.
Fernandes, J. (2002). Uma metodologia para a analise e modelagem, PhD thesis. Rio de Jeneiro, Brasil: PUC.
Forsyth, P. (1983). A unified Description of Micro and Macro-scopic Fatigue Crack Behaviour . International Journal of Fatigue , 5, 3-14.
Georgsson, P. (2000). The determinatino of uncertainties in fatigue crack growth measurements. Trollhattan, Sweden : Volvo Aero Corporation .
Hamam, R., Pommier, S., & Bumbieler, F. (2007). Variable amplitude fatigue crack growth, experimental results and modeling . 29 (1634-1646).
Harrington, D. (1995). Fatigue Crack Coalescence and Shape Development and Experimental Invistigation, Masters Thesis. Carleton University, Department of Mechanical and Aerospace Engineering .
Ho, K., & Newman, S. (2003). State of the Art Electrical Discharge Machining (EDM). International Journal of Mchine Tools and Manufacture , 43 (13), 1287-1300.
Iida, K. (1983). Shapes and Coalescence of Suface Fatigue Cracks. I.C.F. International Symposium on Fracture Mechanics. Beijing.
Irwin, G. (1962). Crack Extension Force for a Part Through Crack in a Plate . Journal of Applied Mechanics , 29, 651-654.
ISO TAG4. (1993). In Guide to the expression of uncertainty in measurement (pp. ISBN 92-67-10188-9). Geneva, Switzerland: International Organization for Standardisation.
Keshtgar, A. (2013). Acoustic emission-based structural health management and prognostics subject to small fatigue cracks, PhD thesis . College Park: University of Maryland.
Keshtgar, A., & Modarres, M. (2013). Detection of crack intitation based on acoustic emission. Chemical engineering 33 , DOI: 10.3303/CET1333092.
Keshtgar, A., & Modarres, M. (2013). Probabilistic model developement for fatigue crack detection using acoustic emission technology. International Topical Meeting on Probabilistic Safety Assesement and Analysis. Columbia, SC: ANS PSA.
Kishimoto, K., Soboyejo, W., Smith, R., & Knott, J. (1989). A Numerical Invistigation of the Interaction and Coalescence of Twin Coplanar Semi Elliptical Fatigue Cracks. International Journal of Fatigue , 11 (2), 91-96.
Krishnaprasad, K., & Prakash, V. (2009). Fatigue crack growth behavior in dissimilar metal weldment of staniless steel and carbon steel. World Academy of Science, Engineering and Technology , 3 , 8-22.
Leek, T. (1990). The Interaction and Growth of Two Surface Cracks Under Fatigue Loading, PhD Thesis. UK: University of Sheffield.
Leek, T., & Howard, I. (1996). An examination of methods of assessing interacting surface crack by comparison with experimental data. International Journal of Pressure Vessels and Piping , 181-201.
Leek, T., & Howard, I. (1996). An Examination of Methods of Assessing Interacting Surface Cracks by Comparison with Experimental Data. International Journal of Pressure Vessels and Piping , 68, 181-201.
Leek, T., & Howard, I. (1994). Estimating the Elastic Interaction Factors of Two Coplanar Surface Cracks Under Mode I load. International Journal of Pressure Vessels and Pipeing , 60, 307-321.
Leek, T., & Howard, I. (1994). Rules for the Assessment of Interacting Suface Cracks Under Mode I Load. Internationl Journal of Pressure Vessels and Piping , 60, 323-339.
Li, C. (1996). Probabilistic Modeling for Corrosion-Fatigue Crack Growth, PhD Dissertation. Lehigh University .
Lin, X., & Smith, R. (1995). Numerical Prediction of Fatigue Crack Growth of a Surface Defect. Fatigue and Fracture of Engineering Materials and Structures , 18 (2), 247-256.
Materials Evaluation and Engineering Inc. (2009). Energy Dispersive X-ray Spectroscopy. (Materials Evaluation and Engineering, Inc.) Retrieved December 02, 2013, from http://mee-inc.com/eds.html
Melin, S. (1983). Why do Cracks Avoid Each Other. International Journal of Fracture , 23, 37-45.
Modarres, M. (2012). Accelerated Life Testing, ENRE64, Class Presentation Slides . College Park: University of Maryland.
Modarres, M. (2008). Accelerated Testing, ENRE 641. College Park : Center of Risk and Reliability, UMD, College Park, USA.
Modarres, M., Kaminskiy, M., & Krivtsov, V. (2010). Reliability engineering and risk analysis. CRC Press.
Murakami, Y., & Nemat-Nasser, S. (1982). Interacting Dissimilar Semi-elliptical Surface Flaws Under Tension and Bending . Engineering Fracture Mechanics , 16, 373-386.
Murakami, Y., & Nisitani, H. (1981). Stress Intensity Factors for Interacting Two Equal Semi-elliptical Surface Cracks in Tension. Trans. Japan Soc. Mech. Engrs SerA , 47, 295-303.
National Research Council. (2013). Effects of Diluted Bitumen on Crude Oil Transmission Pipeline, Special Report 311. National Research Council.
Neves Beltrao, M., Castrodeza, E., & Bastian, F. (2010). Fatigue crack propagation in API 5L X-70 pipeline steel longitudinal welded joints under constant and variable amplitude. Fatigue and Fracture of Engineering Materials and Structures , 34, 321-328.
Newman, J. (1979). A Review and Assessment of the Stress Intensity Factors for Surface Cracks. ASTM STP 687.
Newman, J., & Raju, I. (1981). An Emperical Stress Intensity Factor Equation for the Surface Crack. Engineering Fracture Mechanics , 15.
Newman, J., & Raju, I. (1984). Stress Intensity Factor Equations for Cracks in Three Dimensional Finite Bodies Subjected to Tension and Bending Loads. Hampton, Virginia: NASA.
Newman, J., & Raju, I. (1979). Stress intensity Factors for a Wide Range of Semi-Elliptical Suface Cracks in Finite Width Plates. Engineering Fracture Mechanics , 11.
Nuhi, M., Abu Seer, T., Al Tamimi, A., Modarres, M., & Seibi, A. (2011). Reliability Analysis for Degredation Effects of Pitting Corrosion in Carbon Steel Pipes. ICM11. Como .
O'Donoghure, T., Nishioka, P., & Atluris, N. (1984). Multiple Surface Cracks in Pressure Vessels. Engineering Fracture Mechanics , 20 (3), 545-560.
Ontiveros, V., Cartillier, A., & Modarres, M. (2010). An Integrated Methodology for Assessing Fire Simulation Code Uncertainity. Nuclear Science Engineering , 166, 179-201.
Paris, P., & Erdogan, F. (1963). A critical analysis of crack propagation laws . Journal of Basic Engineering , 528-534.
Pitt, S., & Jones, R. (1997). Multiple-Site and Widespread Fatigue Damage in Aging Aircrafts. Engineering Failure Analysis , 4, 237-257.
Roberge, P. (2012 ). Corrosion Pit Shapes. (Corrosion Doctors ) Retrieved June Tuseday, the 26th , 2012, from http://corrosion-doctors.org/Forms-pitting/shapes.htm
Savin, G. N. (1981). Stress concentration around holes. Oxford: Pergamon Press.
Shi, Y., Chen, B., & Zhang, J. (1999). Effects of welding residual stresses on fatigue crack growth behaviour in butt welds of a pipline steel. Engineering Fracture Mechanics , 36, 893-902.
Soboyejo, W., & Knott, J. (1990). Fatigue Crack Propagation of Copalanr Semi-elliptical Cracks in Pure Bending. Engineering Fracture Mechanics , 37 (2), 323-340.
Spiegelhalter, D., Thomas, A., Best, N., & Lunn, D. (2003). WinBUGS 1.4 manual .
Stephens, R., Fatemi, A., Stephens, R., & Fuchs, H. (2000). Metal fatigue in engineering. John Wiley & Sons .
Terrell, J. (1988). Effect of Cyclic Frequency on the Fatigue Life of ASME SA-106-B Piping Steel in BWR Environments. Journal of Material Engineering , 10, 193-203.
Twaddle, J., & Hancock, B. (1988). The Development of Cracks by Defect Coalescence. Engineering Materials Advisory Service , 185-198.
Willard, S. A. (1997). Use of of Marker Bands for Determination of Fatigue Crack Growth Rates and Crack Front Shapes in Pre-Corroded Coupons. Hampton, Virginia : NASA.
Yokobori, T., Ohashi, M., & Ichikawa, M. (1965). The interaction of two collinear asymmetrical elastic cracks. Vol. 1, PP. 33-39: Tohoku University .
Zhang, H. (1993). Fatigue Crack Growth and Coalescence Study. Ontario: Carleton University.
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