Virtual Testing and Predictive Modeling
Bahram Farahmand Editor
Virtual Testing and Predictive Modeling For Fatigue and Fracture Mechanics Allowables
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Editor Bahram Farahmand TASS – Americas, a subsidiary of TASS Inc. 12016 115th Ave NE Suite 100 Kirkland, WA 98034 USA
[email protected] ISBN 978-0-387-95923-8 e-ISBN 978-0-387-95924-5 DOI 10.1007/978-0-387-95924-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009921172 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Acknowledgments
The author is grateful to all co-authors who contributed to this book. Their dedication and effort for submitting their chapters on time are greatly appreciated. This book will be dedicated to my dear mother Gohartaj and my lovely wife Vida. My great appreciation goes to my son, Houman, and my daughter, Roxana, for being extremely helpful with their support during putting sections of this book together.
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The materials used in manufacturing the aerospace, aircraft, automobile, and nuclear parts have inherent flaws that may grow under fluctuating load environments during the operational phase of the structural hardware. The design philosophy, material selection, analysis approach, testing, quality control, inspection, and manufacturing are key elements that can contribute to failure prevention and assure a trouble-free structure. To have a robust structure, it must be designed to withstand the environmental load throughout its service life, even when the structure has pre-existing flaws or when a part of the structure has already failed. If the design philosophy of the structure is based on the fail-safe requirements, or multiple load path design, partial failure of a structural component due to crack propagation is localized and safely contained or arrested. For that reason, proper inspection technique must be scheduled for reusable parts to detect the amount and rate of crack growth, and the possible need for repairing or replacement of the part. An example of a fail-safedesigned structure with crack-arrest feature, common to all aircraft structural parts, is the skin-stiffened design configuration. However, in other cases, the design philosophy has safe-life or single load path feature, where analysts must demonstrate that parts have adequate life during their service operation and the possibility of catastrophic failure is remote. For example, all pressurized vessels that have single load path feature are classified as high-risk parts. During their service operation, these tanks may develop cracks, which will grow gradually in a stable manner. To avoid catastrophic failure, a thorough nondestructive inspection, a proof test prior to service usage, and a comprehensive fracture mechanics analysis (i.e., safe-life analysis) are requested by the customer. To demonstrate that structural failure of single load path component does not occur and that the part has adequate life during its entire operation, a comprehensive fatigue and fracture mechanics analysis using linear elastic fracture mechanics must be performed. In conducting safe-life analysis, full fracture mechanics data for the material must be available. These data are generated based on the ASTM testing standards. Because fracture toughness is thickness dependent and structures have components that have different sizes and thicknesses, numerous fracture toughness tests must be conducted to include the plane strain, plane stress, and the mixed-mode conditions. In addition to fracture toughness values, the fatigue crack growth rate data must also be available to analysts in order to conduct a meaningful safe-life vii
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analysis. These tests are costly and time consuming, and the cost and time of testing will increase substantially when scatter in fracture allowable, as the result of material variation, need to be considered. Therefore, any method that can reduce the number of tests will be useful to the industry to avoid unnecessary costs when fracture allowables are needed as an input to the life-assessment analysis. The safe-life analysis of high-risk components will demonstrate the ability or tolerance of parts at the presence of existing crack under the load-varying environment. For this reason the fracture mechanics analysis in many cases is called “damage tolerance” analysis. In reality, the total life of structural components is the sum of crack initiation cycles and crack propagation. In aircraft industry, the number of cycles to crack initiation must be accounted for when assessing the total life of the part. Figure 1 shows the process of creating a crack growth/residual strength analysis with emphasis on the comparison (feedback) of analysis with practice. That is, crack growth analysis should be checked against results obtained from the field experience through the interval inspection. Airlines already implemented this approach through their Reliability Centered Maintenance program to adjust their interval inspection period of the entire fleet. As indicated in the figure, the total life analysis is incomplete without having fatigue and fracture allowables through testing.
Fig. 1 Comparison (feedback) of analysis with practice. Analysis is incomplete without fatigue and fracture data
Because the induced stresses in aircraft components must be kept in the elastic range, the high cycle fatigue data generated through the stress to life (S–N) can be useful in the fatigue assessment analysis (also called durability analysis). These data are stress ratio (R) dependent and require considerable time and cost to generate the full range of the S–N curve. Any analytical technique that can be used to generate the S–N data without conducting the traditional laboratory coupon tests will be helpful to the aircraft and aerospace industry to avoid tests.
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The sole purpose of this book is to provide the structural engineers with the present and near-future approaches to the virtual testing techniques, where fatigue and fracture mechanics data can be generated rapidly with minimum amount of tests. As mentioned before, fatigue and fracture allowables are needed as an input to the damage tolerance and durability assessment of fracture critical parts. In many instances, analysts do not have fatigue fracture mechanics values and the budget is not adequate to generate data through testing approach. In other instances, the time does not allow to conduct tests, because of deadlines that designers must meet and laid down by the customers. Therefore, the virtual testing is the right tool to have when both the budget and time do not allow engineers to conduct tests for durability and damage tolerance analysis. The virtual testing technique for generating fatigue and fracture allowables will be presented in this book through two unique techniques. The first approach will use the conventional continuum mechanics approach, which will allow engineers to generate the S–N; fracture toughness, Kc; and fatigue crack growth rate data (da /dN versus ΔK) through analytical approach. The second method of generating these data is based on the fundamental laws of physics (i.e., the ab initio), where material will be assessed from the bottom-up approach. Both approaches to the virtual testing will be presented in this book. The latter utilizes the multiscale modeling and simulation technique to predict the material properties. For this reason the author chooses to use “Virtual Testing and Predictive Modeling” as the title of this book. Chapters 1, 2, 3, 4, 5, and 6 of this book will be dedicated to virtual testing using the continuum mechanics approach. Both metallic and composite materials will be addressed with numerous examples related to the aerospace and aircraft parts. Chapters 7, 8, 9, 10, and 11 will discuss the multiscale modeling and simulation technique. Both quantum mechanics and molecular dynamics approaches will be used to conduct the predictive modeling analysis. Because of outstanding mechanical properties of nanoparticles, there is a strong future demand for their application in aerospace and aircraft structural parts. These particles when combined with polymer matrix will enhance the mechanical properties of polymer, which is an important factor in reducing the weight of the structural parts in modern airplanes. The implementation of multiscale modeling and simulation at the interface region between nanoparticles and matrix is challenging and proper chemistry between nanoparticles and polymer is needed to provide a good bond at the interface region. To make nanoparticles more easily dispersible in polymer, it is necessary to physically or chemically attach certain molecules, or functional groups, to their smooth sidewalls without significantly changing the nanoparticle’s desirable properties. This process is called functionalization. The production of robust composite materials that can allow the transfer of load without causing localized damage requires strong covalent chemical bonding between the filler particles and the polymer matrix that can be achieved through the functionalization process. Chapter 12 will address the most recent approach to the functionalization technique that can be useful to bond dissimilar material with achieving adequate interface strength.
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Finally, Chapter 13 will be allocated to the verification technique using the stateof-the art approach to verify the interface region between nanoparticles and the matrix by applying the transmitted electron microscope (TEM) and atomic force microscope (AFM). The experimental flexibility of these techniques will provide insights into the fundamental structure and deformation processes of nanoscales materials. The in situ measurement of interface region while material under stress will be discussed. Kirkland, Washington
Bahram Farahmand
Contents
1 Virtual Testing and Its Application in Aerospace Structural Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bahram Farahmand 1.1 Introduction to the Virtual Testing . . . . . . . . . . . . . . 1.2 Virtual Testing Theory and Fracture Toughness . . . . . . . 1.3 The Extended Griffith Theory and Fracture Toughness . . . 1.4 Extension of Farahmand’s Theory to Fatigue Crack Growth Rate Data . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Accelerated Region and Fracture Toughness . . 1.4.2 The Paris Constants, C and n . . . . . . . . . . . . 1.4.3 The Threshold Value (Region I) . . . . . . . . . . 1.4.4 The da/dN Versus ΔK from Virtual Testing Against Test Data . . . . . . . . . . . . . . . . . . 1.5 Application of Virtual Testing in Aerospace Industry: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Background . . . . . . . . . . . . . . . . . . . . . 1.5.2 Manufacturing Process and Plastic Deformation of COPV Liner . . . . . . . . . . . . . . . . . . . 1.5.3 Generating Fracture Allowables of Inconel 718 of COPV Liner Through Virtual Testing Technique 1.5.4 Generating Fracture Allowables of 6061-T6 Aluminum Tank Through Virtual Testing Technique 1.6 Summary and Future Work . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Tools for Assessing the Damage Tolerance of Primary Structural Components . . . . . . . . . . . . . . . . . . . . . . . . R. Jones and D. Peng 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 An Equivalent Block Method for Predicting Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fatigue Crack Growth under Variable Amplitude Loading . . . 2.3.1 Fatigue Crack Growth in an F/A-18 Aircraft Bulkhead
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Crack Growth in Mil Annealed Ti–6AL–4V under a Fighter Spectrum . . . . . . . . . . . . 2.4 A Virtual Engineering Approach for Predicting the S–N Curves for 7050-T7451 . . . . . . . . . . . . . . . . . . 2.4.1 Computing the Endurance Limit . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Formulae for Computing the Crack Opening Stress . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Cohesive Technology Applied to the Modeling and Simulation of Fatigue Failure . . . . . . . . . . . . . . . . . . . Spandan Maiti 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Models for the Prediction of Threshold Fatigue Crack Behavior . . . . . . . . . . . . . . . . . . . . . 3.2.2 Models for the Prediction of Fatigue Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cohesive Modeling Technique . . . . . . . . . . . . . . . . . . 3.3.1 Reversible Cohesive Model . . . . . . . . . . . . . . . 3.3.2 A Bilinear Cohesive Law . . . . . . . . . . . . . . . . 3.3.3 A Cohesive Model Suitable for Fatigue Failure . . . . 3.3.4 Incorporation of Threshold Behavior . . . . . . . . . . 3.3.5 Finite Element Implementation . . . . . . . . . . . . . 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Paris Curve Simulation . . . . . . . . . . . . . . . . . 3.4.2 Prediction of Threshold Limit of Fatigue Crack Growth 3.4.3 Effect of on the Threshold Limit . . . . . . . . . . . . 3.4.4 Effect of Load Ratio R on Fatigue Crack Threshold . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fatigue Damage Map as a Virtual Tool for Fatigue Damage Tolerance . . . . . . . . . . . . . . . . . . . . . . Chris A. Rodopoulos 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Basic Understanding of Fatigue Damage . . . . . 4.2.1 Development of Fatigue Cracks and Fatigue Damage Stages . . . . . . . . . . . . . . . . 4.2.2 Stage II Fatigue Cracking . . . . . . . . . . . 4.2.3 Stage I Fatigue Cracking . . . . . . . . . . . 4.2.4 Stage III Fatigue Cracks . . . . . . . . . . . . 4.3 Fatigue Damage Map the Basic Rationale – The Navarro–de los Rios Model . . . . . . . . . . . . . . 4.3.1 Fatigue Damage Map – Defining the Stages of Fatigue Damage . . . . . . . . . . . . . . . .
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Fatigue Damage Map – Defining the Propagation Rate of Fatigue Stages . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Predicting Creep and Creep/Fatigue Crack Initiation and Growth for Virtual Testing and Life Assessment of Components K.M. Nikbin 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Background to Life Assessment Codes . . . . . . . 5.1.2 Creep Analysis of Uncracked Bodies . . . . . . . . 5.1.3 Physical Models Describing Creep . . . . . . . . . 5.1.4 Complex Stress Creep . . . . . . . . . . . . . . . . 5.1.5 Influence of Fatigue in Uncracked Bodies . . . . . 5.2 Fracture Mechanics Parameters in Creep and Fatigue . . . . 5.2.1 Creep Parameter C ∗ Integral . . . . . . . . . . . . 5.3 Predictive Models in High-Temperature Fracture Mechanics 5.3.1 Derivation of K and C ∗ . . . . . . . . . . . . . . . 5.3.2 Example of CCG Correlation with K and C∗ . . . . 5.3.3 Modelling Steady-State Creep Crack Growth Rate . 5.3.4 Transient Creep Crack Growth Modelling . . . . . 5.3.5 Predictions of Initiation Times ti Prior Onset of Steady Creep Crack Growth . . . . . . . . . . . . 5.3.6 Consideration of Crack Tip Angle in the NSW Model . . . . . . . . . . . . . . . . . . . . . 5.3.7 The New NSW-MOD Model . . . . . . . . . . . . 5.3.8 Finite Element Framework . . . . . . . . . . . . . 5.3.9 Damage Accumulation at the Crack Tip . . . . . . 5.3.10 Elevated Temperature Cyclic Crack Growth . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Nomenclatures and Abbreviations . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Computational Approach Toward Advanced Composite Material Qualification and Structural Certification . . . Frank Abdi, J. Surdenas, Nasir Munir, Jerry Housner, and Raju Keshavanarayana 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 6.2 Background . . . . . . . . . . . . . . . . . . . . . . 6.2.1 FAA Durability and Damage Tolerance Certification Strategy . . . . . . . . . . . . 6.2.2 Damage Categories and Comparison of Analysis Methods and Test Results . . . . . 6.2.3 FAA Building-Block Approach . . . . . . . 6.2.4 Test Reduction Process . . . . . . . . . . .
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Computational Process for Implementing Building-Block Verification . . . . . . . . . . . . . . . . . 6.3.1 Multiple Failure Criteria . . . . . . . . . . . . . . 6.3.2 Micro- and Macro-Composite Mechanics Analysis 6.3.3 Progressive Failure Micro-Mechanical Analysis . . 6.3.4 Calibration of Composite Constitutive Properties . 6.3.5 Composite Material Validation . . . . . . . . . . . 6.3.6 Material Uncertainty Analyzer (MUA) . . . . . . . 6.4 Establish A- and B-Basis Allowables . . . . . . . . . . . . 6.4.1 Combining Limited Test Data with Progressive Failure and Probabilistic Analysis . . . . . . . . . 6.4.2 Examples of Allowable Generation for Unnotched and Notched Composite Specimens . . 6.5 Certification by Analysis Example . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 Multiscale Modeling of Nanocomposite Materials . . . . . . . . . . Gregory M. Odegard 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Modeling of Multiscale Fatigue Crack Growth: Nano/Micro and Micro/Macro Transitions . . . . . . . . . . . . . . . . . . . G.C. Sih 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Scale Implications Associated with Size Effects . . . . . . . 7.2.1 Physical Laws Change with Size and Time . . . . . 7.2.2 Surface-to-Volume Ratio as a Controlling Parameter 7.2.3 Strength and Toughness: Nano, Micro and Macro . 7.3 Form Invariant of Two-Parameter Crack Growth Relation . . 7.4 Dual-Scale Fatigue Crack Growth Rate Models . . . . . . . 7.4.1 Micro/Macro Formulation . . . . . . . . . . . . . 7.4.2 Nano/Micro Formulation . . . . . . . . . . . . . . 7.5 Micro/Macro Time-Dependent Physical Parameters . . . . . 7.5.1 Macroscopic Material Properties . . . . . . . . . . 7.5.2 Microscopic Material Properties . . . . . . . . . . 7.6 Nano/Micro Time-Dependent Physical Parameters . . . . . 7.6.1 Nanoscopic Material Properties . . . . . . . . . . . 7.6.2 Nanoscopic Fatigue Crack Growth Coefficient . . . 7.7 Fatigue Crack Growth and Velocity Data . . . . . . . . . . 7.7.1 Predicted Micro/Macro Results . . . . . . . . . . . 7.7.2 Predicted Nano/Micro Results . . . . . . . . . . . 7.8 Validation of Nano/Micro/Macro Fatigue Crack Growth Behavior . . . . . . . . . . . . . . . . . . . . . . . 7.9 Implication of Multiscaling and Future Considerations . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Computational Modeling Tools . . . . . . . . . . . . . . Equivalent-Continuum Models . . . . . . . . . . . . . . . 8.3.1 Representative Volume Element . . . . . . . . . 8.3.2 Equivalent Continuum . . . . . . . . . . . . . . 8.3.3 Equivalence of Averaged Scalar Fields . . . . . . 8.3.4 Kinematic Equivalence . . . . . . . . . . . . . . 8.4 Equivalent-Continuum Modeling Strategies . . . . . . . . 8.4.1 Crystalline and Highly Ordered Material Systems 8.4.2 Fluctuation Methods . . . . . . . . . . . . . . . 8.4.3 Static Deformation Methods . . . . . . . . . . . 8.4.4 Dynamic Deformation Methods . . . . . . . . . 8.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Silica Nanoparticle/Polymer Composites . . . . . 8.5.2 Nanotube/Polymer Composites . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Predictive Modeling . . . . . . . . . . . . . . . . . . . . . . Michael Doyle 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Nanocomposites . . . . . . . . . . . . . . . . . . . . . 9.2.1 Nanotechnology and Modeling . . . . . . . . . 9.2.2 Composites . . . . . . . . . . . . . . . . . . . 9.2.3 The Interface Region . . . . . . . . . . . . . . 9.2.4 Functionalization of Interface Region . . . . . 9.2.5 Modeling Approaches . . . . . . . . . . . . . . 9.2.6 Method Developments . . . . . . . . . . . . . 9.3 Multiscale Modeling . . . . . . . . . . . . . . . . . . . 9.4 Continuum Methods . . . . . . . . . . . . . . . . . . . 9.4.1 Predicting Material Properties from the Top-Down Approach . . . . . . . . . . . . . . 9.4.2 Analytical Continuum Modeling . . . . . . . . 9.4.3 Computational Continuum Modeling . . . . . . 9.5 Materials Engineering Simulation Across Multi-Length and Time Scales . . . . . . . . . . . . . . . . . . . . . 9.5.1 Predicting Material Properties from the Bottom-Up Approach . . . . . . . . . . . . . . 9.5.2 Quantum Scale . . . . . . . . . . . . . . . . . 9.5.3 Molecular Scale . . . . . . . . . . . . . . . . . 9.5.4 Molecular Dynamics . . . . . . . . . . . . . . 9.6 Extension of Atomistic Ensemble Methods . . . . . . . 9.6.1 Combining the Top-Down and Bottom-Up Approaches . . . . . . . . . . . . . . . . . . . 9.7 Future Improvement . . . . . . . . . . . . . . . . . . . 9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Multiscale Approach to Predicting the Mechanical Behavior of Polymeric Melts . . . . . . . . . . . . . . . . . . R.C. Picu 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Single and Multiscale Modeling Methods: Limitations and Tradeoffs . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Atomistic and Atomistic-Like Models . . . . . 10.2.2 Molecular Models . . . . . . . . . . . . . . . . 10.2.3 Continuum Models . . . . . . . . . . . . . . . 10.3 Two Information-Passing Examples . . . . . . . . . . . 10.3.1 General Strategy . . . . . . . . . . . . . . . . . 10.3.2 Calibration of Rheological Constitutive Models 10.3.3 Developing Coarse-Grained Models of Polymeric Melts . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Prediction of Damage Propagation and Failure of Composite Structures (Without Testing) . . . . . . . . . . . . . G. Labeas 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basics of Progressive Damage Modelling methodology . . . 11.2.1 PDM – An Overview . . . . . . . . . . . . . . . . 11.2.2 Multiscale Computational Model . . . . . . . . . . 11.2.3 Prediction of Local Failure at Different Scale Levels 11.2.4 Behaviour of Damaged Material . . . . . . . . . . 11.3 Buckling and Damage Interaction of Open-Hole Composite Plates by PDM . . . . . . . . . . . . . . . . . . 11.3.1 Composite Panel with Circular Cut-Out . . . . . . 11.3.2 Computational Model for the Open-Hole Panel Problem . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Interaction Effects Between Damage Failure and Plate Buckling . . . . . . . . . . . . . . . . . 11.4 Implementation of PDM in Composite Bolted Joints . . . . 11.4.1 Description of Composite Bolted Joint Problem . . 11.4.2 Damage Initiation and Progression Within the Bolted Joint . . . . . . . . . . . . . . . . . . . . . 11.5 Implementation of PDM in Composite Bonded Repairs . . . 11.5.1 Description of the Composite Repair Patch Problem 11.5.2 Details of PDM Model for Composite Repair Patch Analysis . . . . . . . . . . . . . . . . . . . . 11.5.3 Effects of Composite Patch Geometry and Material on the SIF . . . . . . . . . . . . . . . . . 11.6 Multi-Scale Modeling of Tensile Behavior of Carbon Nanotube-Reinforced Composites . . . . . . . . . . . . . . 11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Functional Nanostructured Polymer–Metal Interfaces Niranjan A. Malvadkar, Michael A. Ulizio, Jill Lowman, and Melik C. Demirel 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . 12.2 Oblique-Angle Polymerization . . . . . . . . . . . 12.2.1 Nanostructured Polymer growth . . . . . 12.2.2 Control of Morphology and Topography . 12.3 Metallization of Nanostructured Polymers . . . . . 12.3.1 Electroless Metal Deposition . . . . . . . 12.3.2 Vapor Phase Metal Deposition . . . . . . 12.3.3 Nanoparticle Assembly . . . . . . . . . . 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
. . . . . . .
357
. . . . . . . . . .
357 358 358 360 361 362 363 364 366 368
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
Advanced Experimental Techniques for Multiscale Modeling of Materials . . . . . . . . . . . . . . . . . Reza S. Yassar and Hessam M.S. Ghassemi 13.1 Atomic Force Microscopy (AFM) . . . . . . . . . . . . . . . . 13.1.1 Principles of AFM . . . . . . . . . . . . . . . . . . . 13.1.2 AFM Operation . . . . . . . . . . . . . . . . . . . . . 13.1.3 Application of AFM . . . . . . . . . . . . . . . . . . . 13.1.4 Modeling and Simulation . . . . . . . . . . . . . . . . 13.2 X-Ray Ultra-Microscopy . . . . . . . . . . . . . . . . . . . . . 13.2.1 Principles of XuM . . . . . . . . . . . . . . . . . . . . 13.2.2 Phase Contrast and Absorption Contrast . . . . . . . . 13.2.3 3D Imaging and Multiscale Modeling Applications . . 13.3 In Situ Micro-Electro-Mechanical-Systems (MEMS) Introduction 13.3.1 Principle and Design of MEMS Devices . . . . . . . . 13.3.2 Application of MEMS Devices for Materials Modeling 13.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371 372 372 374 375 379 382 382 384 385 388 389 392 395 396 399
Contributors
Frank Abdi Alpha STAR Corporation, Long Beach, CA, USA,
[email protected] Melik C. Demirel The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA,
[email protected] Michael Doyle Principal Solution Scientist, Materials Science @ Accelrys, USA,
[email protected] Bahram Farahmand Taylor Aerospace (TASS – Americas) Inc., Kirkland, WA, USA,
[email protected] Hessam M.S. Ghassemi Mechanical Engineering-Engineering Mechanics Department, Michigan Technological University, 1400 Townsend Dr., Houghton, MI 49931, USA Jerry Housner Analytical Enterprises, Arlington, VA, USA,
[email protected] R. Jones DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical and Aerospace Engineering, P.O. Box 31, Monash University, Victoria, 3800, Australia,
[email protected] Raju Keshavanarayana Department of Aerospace Engineering/National Institute for Aviation Research (NIAR), Wichita State University, Wichita, KS, USA,
[email protected];
[email protected] G. Labeas Laboratory of Technology and Strength of Materials, University of Patras, Panepistimioupolis Rion, 26500 Patras, Greece,
[email protected] Jill Lowman The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA Spandan Maiti Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton MI 49931, USA,
[email protected] Niranjan Malvadkar The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA,
[email protected] xix
xx
Contributors
Nasir Munir North Grumman Corporation, El Segundo, CA, USA,
[email protected] K.M. Nikbin Mechanical Engineering Department, Imperial College, London SW7 2AZ, UK,
[email protected] Gregory M. Odegard Department of Mechanical Engineering – Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA,
[email protected] D. Peng DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical and Aerospace Engineering, P.O. Box 31, Monash University, Victoria, 3800, Australia R.C. Picu Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA,
[email protected] Chris A Rodopoulos Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Greece,
[email protected] G.C. Sih School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China; Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA,
[email protected],
[email protected] J. Surdenas Alpha STAR Corporation, Long Beach, CA, USA,
[email protected] Michael Ulizio The Pennsylvania State University, 212 Earth Engineering Sciences Bldg, University Park, PA 16802, USA Reza S. Yassar Mechanical Engineering-Engineering Mechanics Department, Michigan Technological University, 1400 Townsend Dr., Houghton, MI 49931, USA,
[email protected] Introduction
Recent failures of several major structures, such as pressure vessels, storage tanks, ships, aircrafts, gas pipe lines, bridges, dams, and many welded parts, have raised concern on issues like loss of life, environmental safety, and high costs associated with repair and replacement of components that have been estimated to be in millions of dollars. Almost in all cases these failures occurred during structural usage where cracks have initiated and advanced in a stable manner to failure under the fatigue load environment well below the material yield allowable. In-depth scientific investigation into the nature of these failures indicated that poor structural design practices, such as the presence of stress concentrations, insufficient material ductility, residual stresses during the fabrication and manufacturing phase of hardware, lack of adequate NDI inspections, corrosive environment, and material degradation in low-temperature environment, each can contribute to an accelerated crack growth, resulting in catastrophic failure and in some cases loss of life. In designing components of commercial aircraft or space vehicles, the induced stresses in the components, as the result of environmental load, must fall below the material yield value. That is, the bulk of the structure must stay elastic and any plastic deformation has to be localized. By enhancing the mechanical properties of material (i.e., design allowables), thickness of structural parts can be reduced and considerable weight saving is possible. The weight saving can be achieved by selecting suitable materials that possess high strength, adequate durability, as well as good ductility. It has been always the goal of the aerospace industry to minimize the weight of components without compromising the structural integrity of the system they include. Over the past few years, more attention has been given to composite materials as a possible solution to the design of aerospace and aircraft parts because of their weight and superior static and fatigue properties over the traditional aluminum alloys. In typical composite materials, high-strength fibers provide reinforcement to the polymer matrix so that they become capable of carrying the high environmental and mechanical loads desired. Recently, reinforcing nano-particles (nano-fillers) in the aerospace polymer resin have shown promise to provide higher mechanical and fatigue properties to composite materials. However, execution of right functionalization at the interface and uniform dispersion of these particles in the polymer matrix are challenging, but both must be implemented correctly in order to have enhancement in mechanical properties. To verify the improvement in xxi
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Introduction
mechanical properties, limited amount of mechanical coupon testing is required. Moreover, to establish material allowables for design purposes, extensive static, fatigue, and fracture mechanics tests must be performed in order to achieve some level of confidence due to material variability. In the case of composite material, it is common to use the “B-basis” allowables for static analysis and average values for fatigue and fracture mechanics allowables. To establish material allowables for both metals and composite parts, it requires extensive tests that carries with it significant cost and time for specimens preparation and data gathering. Therefore, any method that can reduce the number of tests will be extremely useful to the industry to avoid unnecessary costs when fatigue and fracture allowables are required for designing components of structures. The importance of virtual testing and its application in aerospace and aircraft industry was realized by the author since early 1990s while working on the International Space Station (ISS) program. As part of the requirements outlined by customers and written in the ISS fracture control plan, all high-risk parts (fracture critical parts) must shown by analysis to have adequate life during their entire service usage. The safe-life analysis of fracture critical parts requires having fracture allowables (i.e., fracture toughness and fatigue crack growth rate data). On several occasions, the author experienced that these data were not available to analysts for the material under study. Efforts were always made to obtain allowables first through the literature search method and if that failed, the alternative approach was to conduct numerous tests. These tests are labor intense and time consuming and designers had been frustrated because of budgetary constraint to conduct tests. In other instances, the time needed to conduct tests and to establish allowables would exceed the deadline laid down by customers. This dilemma was experienced by the author in several occasions during his involvement with the aerospace industry. The virtual testing methodology discussed in this book is based on the continuum mechanics approach and has been discussed previously in [1, 2, 3]. Data estimated by the virtual-testing technique were in excellent agreement with data generated by the ASTM testing standards. It should be noted that in generating fracture allowables, K c and the da/dN, test coupons were first prepared with a sizable notch machined in the specimen. Subsequently, a sharp natural crack at the tip of the notch is introduced in the specimen through cyclic loading technique. Therefore, all fracture mechanics allowables used in safe-life analysis were based on large crack behavior. In real situation, cracks embedded in material prior to parts service usage are much smaller in length, and linear elastic fracture mechanics concepts and simulative law may not apply to their growth behavior under cyclic loading. With the current approach to the safe-life analysis, the initial flaw size assumption in material is estimated by assuming parts have defects that can be detected through the traditional Non-Destructive Inspections (NDI) techniques. The largest flaw size that may escape from detection can be estimated and used as the initial flaw size in assessing the remaining life of components. The life assessment analysis results based on this approach are too conservative and in some cases lead to redesigning the part and adding unnecessary weight to the structure. Experimental and analytical work showed that the typical crack-like defects found on the part, after machining
Introduction
xxiii
operation, are marks that are much smaller in size than the standard NDI capability. These machining marks on the surface of parts can be as small as a few microns in depth. In contrast to the aerospace industry, the life assessment of aircraft components is divided into two parts: (1) the number of cycles to initiate a crack, where crack reaches to a visible measurable size ∼0.1 in. in length, and (2) the number of cycles from initiation to final failure. The first part of life analysis is called ‘durability’ and the latter part (i.e., the remaining life) is referred to as ‘damage tolerance’ analysis. The durability analysis portion of life assessment uses the traditional high cycle fatigue data (the S–N data) that must be generated through the ASTM testing standards, which require conducting numerous tests to cover all regions of the S–N curve. Just like fracture mechanics allowables, the high cycle fatigue tests are also costly and time consuming to the aerospace and aircraft industry. Therefore, the application of virtual testing technique will be extremely useful to eliminate unnecessary tests and to reduce cost and time associated with generating allowables for structural life analysis.
References 1. B. Farahmand, “Application of Virtual Testing For Obtaining Fracture Allowables of Aerospace And Aircraft Materials,” Book Chapter to “Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials,” by G. Sih, Springer, 2008 2. B. Farahmand, “Predicting Fracture and Fatigue Crack Growth Properties Using Tensile Properties,” Eng. J. Fract. 2007 3. B. Farahmand, “Analytical Development of Fatigue Crack Growth Curve without Conducting ASTM E647 Tests,” AIAA Conference, Houston Texas, May 2006
Chapter 1
Virtual Testing and Its Application in Aerospace Structural Parts Bahram Farahmand
Abstract In many occasions, metallic parts will undergo plastic deformation either during their service usage or prior to their actual operation in the manufacturing and assembly phases. Under these circumstances, material properties may change considerably and must be accounted for when estimating the residual strength capability of parts. The change in material properties will occur when load has been removed and the work-hardening phenomenon has caused increase in material yield value, reduction in percent elongation, and degradation in fracture allowables. For these reasons, new static and fracture data must be generated if safe-life assessment must be performed on fracture critical parts. Fracture data are currently obtained through the ASTM testing standards, which are costly and labor intense. Difficulties associated with preparing the specimen, precracking the notch, recording and monitoring the data, obtaining the variation of fracture toughness versus part thickness, capturing data in the threshold region, and repeating the test in many cases due to invalid results make the virtual testing technique extremely helpful to overcome the time and cost related to the above-mentioned testing difficulties. Farahmand (Fatigue & Fracture Mechanics of High Risk Parts, Chapman & Hall, 1997, Chapter 5; Fracture Mechanics of Metals, Composites, Welds, and Bolted Joints, Kluwer Academic Publisher 2000, Chapter 6) utilized the extended Griffith theory to obtain a relationship between the applied stress and half a crack length by using the full stress– strain curve for the material under consideration. The extension of this work led to estimation of material fracture toughness and construction of full fatigue crack growth rate curve (Farahmand, J. Fract., 2007, Vol. 75, pp 2144–2155). This technique will be very useful to implement when materials undergo plastic deformation (the work-hardening phenomenon) and fracture allowables are needed to conduct a meaningful life analysis assessment of parts. First, Farahmand’s virtual testing theory is described briefly and, subsequently, the application of this methodology to two cases of aerospace pressurized tanks, exposed to plastic deformation, presented in this chapter.
B. Farahmand (B) Taylor Aerospace (TASS – Americas) Inc., Kirkland, WA, USA e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 1,
1
2
B. Farahmand
1.1 Introduction to the Virtual Testing It is estimated that the cost of fully characterizing mechanical properties of a new material, which must be used as part of aerospace or aircraft structures, can be substantial. The static, fatigue, and fracture mechanics data have to be available in order to conduct a meaningful stress, durability, and damage tolerance analysis of highrisk components. The successful prediction of mechanical properties through the virtual testing technique is the goal of aerospace and aircraft industry. The proposed technique utilizes the basic data from a simple static tensile test to estimate the fracture mechanics allowables through the extended Griffith theory [1, 2]. The fracture toughness values can be obtained by taking the energy per unit volume under the uniform and nonuniform portions of the stress–strain curve to calculate the energy consumed at the crack tip up to the failure. Section 1.2 discusses the virtual testing method of obtaining fracture toughness. The fracture toughness value obtained through this approach can be used for the upper portion of the fatigue crack growth rate curve. The upper portion of the da/dN curve is related to the critical value of the stress intensity factor; therefore, the fracture toughness value, estimated through the virtual testing technique, can be used to establish the region III of the da/dN versus ΔK data. Other regions of fatigue crack growth rate can be obtained separately and are presented in Section 1.3 [3–5].
1.2 Virtual Testing Theory and Fracture Toughness Metallic material used for the design of aircraft and aerospace parts exhibits three distinct regions when subjected to static loading. The area under the load versus displacement curve (up to the final failure) will represent the degree of ductility that the material possesses. Under monotonic loading, purely elastic materials (Fig. 1.1a) show no ductility at the onset of failure and are strictly prohibited to be used in the design of aerospace and aircraft parts, where safety of vehicle is of great concern. As part of the aerospace design requirement, material selection must be limited to those alloys that exhibit elongation larger than 4%. Aluminum and titanium alloys are excellent candidates for aircraft and aerospace parts because of their attractive strength-to-weight ratio, as well as their great elongation that almost in all cases exceeds the design requirement of 4%. Since ductility and toughness are directly related to each other, the extended Griffith theory is used to relate the area under the stress–strain curve to the material fracture toughness via Farahmand’s theory. Figure 1.1 shows different stress–strain curves for metallic materials where the area under the curve, WF and WU , can be shown to be related to plastic deformation at the crack tip. The critical value of stress intensity factor, K (i.e., Kc), can be calculated by using the energy under the stress–strain curve and will be used as an input in the safe-life analysis of parts. The results of analysis using Farahmand’s approach indicated that materials with uniform plastic deformation (case b in Fig. 1.1, where WU = 0 and WF = 0) possess lower fracture toughness value when compared with
WF = 0 WU = 0
Stress
Fig. 1.1 Typical stress–strain curves for different materials. The quantity W is the energy per unit volume represented by the area under the curve
3
Stress
1 Virtual Testing and Its Application
(a)
(b) Strain
WF ≠ 0 WU ≈ 0
(c)
Stress
Strain
Stress
WF = 0 WU ≠ 0
WU ≠ 0
WF ≠ 0
(d) Strain
Strain
case (d) where materials display both uniform and nonuniform plastic deformations (WF = 0 and WU = 0). The presence of necking shows materials exhibit considerable plastic deformation and thus have higher resistance to fracture before their final failure. Section 1.3 discusses the methodology and derivation of fracture toughness through the information available via the full stress–strain curve and the extended Griffith theory.
1.3 The Extended Griffith Theory and Fracture Toughness The materials selected by engineers to manufacture fracture-critical hardware must exhibit some amount of plastic deformation and stable slow crack growth at the crack tip region prior to their final failure. The amount of energy consumed at the crack tip for plastically deforming material is largely due to material resistance in that region prior to final failure, which is not properly accounted for in the linear elastic fracture mechanics analysis. Irwin and Orowan [6, 7] independently observed that, for tough material, the amount of energy dissipated at the crack tip for plastic deformation, UP , is much larger than the energy consumed for the formation of two crack surfaces, US . Farahmand showed that the energy consumed at the crack tip for straining material is of two kinds: local nonuniform strainability, UF , and uniform strainability, UU [1]. Thus, fracture behavior can be characterized by two energyreleased terms representing plastic deformation at and near the crack tip. Both terms can be derived by using the areas associated with uniform and nonuniform plastic deformation under the full uniaxial stress–strain curve for the alloy under consideration, Fig. 1.1. The extended Griffith theory in terms of UU and UF , including the two new crack surfaces term, 2T, can be written as
4
B. Farahmand
πσ 2c E
= 2T +
Energy rate associated with final fracture
∂UU ∂c
+
∂UF
(1.1)
∂c
Energy rate associated with uniform straining
Energy rate associated with non-uniform straining
For ductile metals, the two terms associated with uniform and nonuniform plastic deformation are much greater than the two new crack surfaces term, 2T [7]. If quantities to the right of Equation (1.1) are determined, then a relationship between applied stress and half a crack length at the onset of failure can be established. The U term can be derived by assuming that all the crack energy rate associated with ∂U ∂c tip energy due to uniform straining up to the ultimate failure point has been consumed by the amount WU to create permanent slip of height hU : ∂UU = WU h U ∂c
(1.2)
The unrecoverable energy per unit volume can be calculated by integrating the area under the curve from the limit stress up to the ultimate: σT U WU =
σT dεT P
(1.3)
σT L
Equation (1.3) is written in terms of true stress and strain, where for metals the true stress versus true plastic-strain curve can be approximated by the Ramberg– Osgood equation, when fitted at true stress at the limit, σ TL , and ultimate, σ TU . The true stress, σ T , in terms of true plastic strain, εTP , can be written as σT = σT U
εT P εT U
1/n (1.4)
where n is the Ramberg–Osgood exponent. The true plastic strain, εTP εT P = εT U dεT P = nεT U
σT σT U
After integration Equation (1.3) becomes
σT n σT U n−1
(1.5) dσT σT U
(1.6)
1 Virtual Testing and Its Application
5
n σT L n+1 εT U σT U 1 − WU = n+1 σT U
(1.7)
The quantity hU defines the height of the plastic deformation in the uniform strain region near the crack tip. It can be formulated through linear elastic fracture mechanics [1] and in its final format it is n−1 εT U n εT F εT L ∗ n+1 h −1 β hU = n−1 εT U εT εT L
(1.8)
where β for the plane stress and plane strain conditions is 1.3 and 0.127, respecF can be derived by assuming that all the tively. The energy term associated with ∂U ∂c crack tip energy due to uniform straining up to the failure has been consumed by the amount WF to create permanent coarse slip of height hF : ∂U F = WF h F ∂c
(1.9)
where the quantity WF in Equation (1.9) is the area under the stress–strain curve from necking up to fracture (see Fig. 1.1) and is approximately equal to [1]: W F = σ¯U F ε P N
(1.10)
Moreover, the amount of deformation due to slip mechanism, hF , is directly related to the material ability to absorb energy, WF , and can be written as h F = γ (E 2 /3 σU )W F
(1.11)
where E is the modulus of elasticity and σ U is the material ultimate value. The correction factor γ = (8/π )αhmin is a material constant. The minimum value of hF is designated by hmin = 0.000557 in. and α is the material atomic spacing. The constant γ for most metallic material can be approximated as ∼3.5 × 10–8 . Combining Equations (1.2) and (1.9), the energy balance equation for the extended Griffith theory can be written as [1] n σT n+1 E σT U εT U 1 − h min c= 2T + σ¯U F ε P N h F k + π σ 2μ n−1 σT U ⎡ ⎤ ⎫ ⎪ ∗ n − 1 ⎬ εT F εT L ⎢ εT U n − 1⎥ β ⎣ ⎦ ⎪ εT U εT εT L ⎭ (1.12) Equation (1.12) relates half a crack length, c, to the applies stress, σ, in an infinite plate at failure. Other quantities in Equation (1.12) are defined as
6
B. Farahmand
σ¯U F = Average stress between ultimate and final fracture σT U = True stress at ultimate σT F = True stress at failure σT L = True stress at limit εT U = True strain at ultimate εT F = True strain at failure εT L = True strain at limit ε P N = Plastic strain at necking Thickness parameters describing the plane stress and strain conditions for each term of Equation (1.12) are designated by μ, k, and β [1]. As mentioned earlier, by calculating half a crack length, c, for a given applied stress, σ (Equation 1.12), the critical value of the stress intensity factor, Kc , can easily be calculated and will be used for the region III of the da/dN versus ΔK. Detailed discussion related to this topic is presented in Section 1.4.
1.4 Extension of Farahmand’s Theory to Fatigue Crack Growth Rate Data To obtain the da/dN versus ΔK data for all regions of fatigue crack growth rate curve, the virtual testing technique proposed by Farahmand [3] can be extremely useful to apply when there are budget or time limitations to conduct coupon tests. Farahmand’s unique approach is based on establishing each region of the da/dN curve separately and relating them through the Forman–Newman fatigue crack growth equation [8]: th p c(1 − f )n ΔK n (1 − ΔK ) da ΔK = ΔK dN (1 − R)n (1 − (1−R)K c )q
(1.13)
Traditionally, all fatigue crack growth constants (c, n,Kc , and ΔKth ), shown in Equation (1.13), will be provided through the ASTM tests, which are costly and labor intense. However, by using the virtual testing concepts, all constants can be estimated and the da/dN versus ΔK, from Equation (1.13), can be plotted.
1.4.1 The Accelerated Region and Fracture Toughness The region I of the da/dN versus ΔK curve is related to the fracture toughness, Kc , and it is thickness dependent. As mentioned previously, the quantity Kc can be estimated via Equation (1.12) for all ranges of material thickness. Figure 1.2 is the plot of stress–strain curves for 2219-T6 and 6061-T6 aluminums that can be used to estimate fracture toughness [9]. Figure 1.3 is a plot of fracture toughness versus material thickness for the above-mentioned aluminums (2219-T6 and 6061-T6). In all cases the fracture toughness variation with respect to thickness calculated by the virtual testing technique is in good agreement with the NASGRO database generated
1 Virtual Testing and Its Application
7 60
50
50
40
40
30
Stress, ksi
Stress, ksi
60 2219-T6
Longitudinal And Long Transverse
6061-T6
30 20
20
10
10 Typical Thickness: 0.125 –2.00 in.
0.0 0.00
0.02
0.04
0.06
0.08
Typical
0.10
0.12
0.0 0.00
0.02
0.04
Strain, in./in.
0.06
0.08
0.10
0.12
Strain, in./in.
Fig. 1.2 Typical full stress–strain curves for 2219-T6 and 6061-T6 aluminums
Virtual Testing
70 60 50 40 30 20 10 0
NASGRO
KIc
0
0.5
1 1.5 2 2.5 Thickness (Inch)
3
6061-T6 Aluminum Alloy - RT
Fracture Toughness ksi (in.)^0.5
Fracture Toughness ksi (in.)^0.5
2219-T6 Aluminum Alloy - RT
3.5
Virtual Testing
60
NASGRO
50 40
KIc
30 20 10 0 0
0.5
1 1.5 2 2.5 3 Thickness (Inch)
3.5
Fig. 1.3 Virtual testing versus test data for 2219-T6 and 6061-T6 aluminums
by test data [8]. Therefore, by using this technique, the region I of the da/dN versus ΔK curve can be obtained without conducting tests.
1.4.2 The Paris Constants, C and n A significant number of fatigue crack growth rate data for numerous aluminum alloys were extracted from the NASGRO database [8]. The da/dN versus ΔK for all these alloys were plotted, and, as the result, two important observations were obtained that were helpful to establish the Paris region of the da/dN curve. The lower point in the Paris region of the fatigue curves (Fig. 1.4) has a material-independent property so that the ratio of the stress intensity factor at the lower-bound point and the threshold value (ΔK/Kth for R = 0) is ∼1.125 for the crack growth rate per cycle, da/dN∼1.0E-7 in./cycle (∼2.54E-6 mm/cycle). In the upper region of the da/dN
8
B. Farahmand • For da/dN~1.0E–7 inch/cycle, ΔK/Kth~1.125 • For da/dN~0.005 inch/cycle, ΔK/Kc~0.9 2014-T6-L-T
1.00E+00
2014-T651-L-T 2020-T651-L-T 2024-T3-L-T
1.00E–01
2024-T351-L-T 2024-T6-L-T
1.00E–02
2024-T81-L-T 2219-T6
da/dN, in./cycle
2219-T851
1.00E–03
da/dN
2219-T87 6061-T6
Paris Region
7005-T6
1.00E–04
7010-T7365
Accelerated Region
7050-T74
1.00E–05
7050-T745
Threshold Region
7050-T765 7075-T6
1.00E–06
7075-T65
ΔK
7075-T73
1.00E–07
7075-T765 7079-T651 7178-T6
1.00E–08
7178-T765 7475-T61 7475-T651
1.00E–09 1
10
100
7475-T7351
ΔK, ksi-(in.)^0.5
Fig. 1.4 The upper and lower regions of the Paris crack growth have special characteristics common to many alloys
curve (at the end of the Paris region, Fig. 1.4), the ratio of the upper-bound stress intensity factor and its critical value, Kc, (ΔK/Kc for R = 0) is found to be ∼0.9 for da/dN ∼0.005 in./cycle (∼0.127 mm/cycle). These two points are useful to plot the entire region II. The fatigue crack growth curve can then be plotted using Equation (1.13), where the fracture parameters and constants are taken from the estimated Kc , Kth , and the Paris constants C and n values for the case of R∼0. Other ranges of R-ratios can be plotted by using the Newman closure equation, f. Section 1.4.3 discusses the virtual testing approach to obtain the threshold value of region III.
1.4.3 The Threshold Value (Region I) The threshold region of the da/dN versus ΔK curve is difficult to obtain. The complexity is due to plasticity and surface toughness closure phenomenon that makes the actual measurement of the threshold value difficult. The threshold stress intensity factor measurement is time consuming and sometimes may take several days
1 Virtual Testing and Its Application
9
to make a few measurements of the ΔKth value. For these reasons, the approximate value of ΔKth is of interest to engineers when conducting life assessment. Farahmand [10] was able to establish a reasonable relationship between the threshold stress intensity factor ΔKth (for the case of stress ratio, R∼0) and the plane strain fracture toughness, KIc . The result of his observation on the threshold values of more than 100 metallic alloys was such that the quantity ΔKth falls between KIc /4π and KIc /3π (for R = 0), Fig. 1.5. Farahmand argued that materials with high KIc value also possess higher Kth value and, conversely, the lower Kth value belongs to low KIc . Table 1.1 shows the calculated threshold values and the corresponding test data extracted from the NASGRO database. Table 1.1 shows that in most cases the test values fall between the two above-mentioned KIc values. In all cases the estimated Kth values based on KIc /4π are lower than the test values and can be considered as the lower-bound values of Kth (only a few threshold values were equal to KIc /4π , see also Table 1.1). When test data are not available, the lower-bound value of Kth = KIc /4π can be used in life estimation of structural parts. Currently, work is in progress to obtain better fit between the calculated versus test values by incorporating the KIc /Fty ratio to account for material yield value, Fty .
Kth Value ksi-(in.)^0.5
Kth Values For 2000 Series Aluminum (Test Data Versus KIc/3π & KIc/4π) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
Test KIc/4π KIc/3π 1
4
7
10
13
16
19
22
25
28
31
34
37
40
Number of Data
Fig. 1.5 Bounding threshold values between two values, KIc /4π and KIc /3π
1.4.4 The da/dN Versus ΔK from Virtual Testing Against Test Data As mentioned previously, all regions of da/dN versus ΔK must be estimated separately before being able to plot the fatigue crack growth rate curve. Equation (1.13) (the da/dN versus ΔK) can be plotted by calculating its constants through regions I, II, and III defined in Sections 1.4.1, 1.4.2, and 1.4.3. Figures 1.6, 1.7, and 1.8 are only a few selected cases for aluminum (2014-T3 and 7075-T73), titanium (Ti6Al-4 V, ST(1750) + A(1000F-4 hr) and Ti-6Al-4 V, MA(1350-2 hr) Extruded), and ferrous alloys (Custom 455 H1000 and A286 (AISI 660) 160 UTS), where fatigue crack growth rate data were plotted for stress ratio, R = 0, and checked against
KIc
27 22.5 33 29 33 29 36 30 28 23 35 30 30 26 30 33 30
Material
2014-T6 (L-T) 2020-T651 (L-T) 2024-T3 (L-T) 2024-T3 (T-L) 2024-T3 (L-T) 2024-T351 (T-L) 2024.T62 (L-T) 2024-T62 (T-L) 2024-T852 2024-T861 (L-T) 2048-T851 (L-T) 2048-T851 (T-L) 2124-T851 (L-T) 2124-T851 (T-L) 2219-T62 (-320F) 2219-T851 (L-T) 2219-T87 (L-T)
2.1 1.8 2.6 2.3 2.6 2.3 2.9 2.4 2.2 1.8 2.8 2.4 2.4 2.1 2.4 2.6 2.4
Kth (KIc/4π 2.7 2.2 2.9 2.9 2.9 2.6 2.9 2.9 2.9 2.2 2.7 2.7 2.7 3 2.9 3 2.9
Kth (Test) 3.0 2.4 3.5 3.1 3.5 3.1 3.8 3.2 3.0 2.4 3.7 3.2 3.2 3.2 3.2 3.5 3.2
Kth (KIc/3π ) 2219-T87 (T-L) 7005-T6 & T63 (T-L) 7010-T73651 7050-T736 & T74 (T-L) 7050-T76511 (T-L) 7075-T651 (L-T) 7075-T6510 (L-T) 7075-T6511 (L-T) 7075-T73 (L-T) 7075-T7351 (L-T) 7075-T73510 (L-T) 7075-T73511 (L-T) 7075-T7352 (L-T) 7075-T7651 (T-L) 7079-T651 (L-T) 7149-T73511 (L-T) 7178-T7651 (L-T)
Material 27 40 31 24 24 28 28 28 28 29 31 33 33 23 26 31 28
KIc
Table 1.1 Threshold values for several aluminums (test versus analysis)
2.1 3.2 2.5 1.9 1.9 2.2 2.2 2.2 2.2 2.3 2.5 2.6 2.6 1.8 2.1 2.5 2.23
Kth (KIc/4π 2.9 3.4 2.5 2.3 2 3 3 3 3 3 3 3 3 2.4 2 3 3
Kth (Test)
2.9 4.2 3.3 2.5 2.5 3.0 3.0 3.0 3.0 3.1 3.3 3.5 3.5 2.4 2.8 3.3 2.97
Kth (KIc/3π )
10 B. Farahmand
11
2024-T3 L-T, R = 0
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
7075-T73 Aluminum, R = 0
NASGRO
da/dN –In./Cycle
da/dN – In./Cycle
1 Virtual Testing and Its Application
Virtual Testing
1
10
1.0E+00 1.0E–01 1.0E–02 1.0E–03 1.0E–04 1.0E–05 1.0E–06 1.0E–07 1.0E–08 1.0E–09 1.0E–10
NASGRO Virtual Testing
1
100
10
ΔK–ksi (in.)^0.5
Fig. 1.6
Virtual testing results versus test data (aluminum alloys)
Ti-6Al-4V, MA (1350-2hr) Extruded, R = 0
Ti-6Al-4V, ST (1750) + A (1000F/4hr), R = 0 1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
NASGRO
da/dN –In./Cycle
da/dN- in/cycle
100
ΔK–ksi (in.)^0.5
Virtual Testing
1
10
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
100
NASGRO Virtual Testing
1
10
ΔK-ksi (in)^0.5
100
ΔK–ksi (in.)^0.5
Fig. 1.7 Virtual testing results versus test data (titanium alloys) A-286 (AISI 660) 160 UTS, R = 0
Custom 455 H1000, R = 0
Virtual Testing
da/dN –In./Cycle
da/dN-in./cycle
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1
10
100
1000
NASGRO Virtual Testing
1.00E–00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1
ΔK-ksi (in.)^0.5
10
100
1000
ΔK–ksi (in.)^0.5
Fig. 1.8 Virtual testing results versus test data (ferrous alloys)
available test data [8]. Excellent agreement between test data and virtual testing can be seen. The appendix has collections of more fatigue crack growth curves that have been generated and compared with test data obtained from NASGRO database. In all cases, good agreement between the virtual and test data was obtained. For other stress ratios, R, the fatigue crack growth rate curve can be established by using Equation (1.13), where the quantities R and f are two parameters that will be able to shift the fatigue crack growth curve of R = 0 to the right for R0. Figure 1.9 shows the fatigue crack growth curve for 2024-T3 and Ti-6Al-4 V, ELI-BA (1900F, 0.5 hr+1325F) alloys with R = 0.5 (R>0). Good
12
B. Farahmand
fit between the virtual testing and test data from NASGRO database can be seen. In addition, the case of R = –1 is shown in Fig. 1.10 for 7075-T73 and 7050-T74511 aluminum alloys. Good correlation between the virtual testing results and test data can be seen.
1.00E+ 00 1.00E– 01 1.00E– 02 1.00E– 03 1.00E– 04 1.00E– 05 1.00E– 06 1.00E– 07 1.00E– 08 1.00E– 09 1.00E– 10
Ti-6Al-4V, ELI-BA (1900F, 0.5hr+1325F) –R = 0.5
NASGRO Virtual Testing
da/dN-In./Cycle
da/dN-In./Cycle
2024-T3 L-T, R = 0.5
1
10
NASGRO
1.00E+ 00 1.00E– 01 1.00E– 02 1.00E– 03 1.00E– 04 1.00E– 05 1.00E– 06 1.00E– 07 1.00E– 08 1.00E– 09 1.00E– 10
100
Virtual Testing
1
10
ΔK–ksi(in.)^0.5
100
ΔK–ksi(in.)^0.5
Fig. 1.9 Virtual testing data versus tests for stress ratio other than R = 0 (R = 0.5)
7050-T74511 Aluminum, R = –1 NASGRO
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11
Virtual Testing
1
10
da/dN-In./Cycle
da/dN-In./Cycle
7075-T73 Aluminum, R = –1
100
ΔK–ksi(in.)^0.5
1000
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
1000
ΔK–ksi(in.)^0.5
Fig. 1.10 Virtual testing data versus tests for stress ratio other than R = 0 (R = −1)
1.5 Application of Virtual Testing in Aerospace Industry: Introduction The mechanical properties of the materials selected for manufacturing aerospace and aircraft parts must be fully characterized before their consideration for space application. These properties are needed when assessing the static and fracture mechanics analysis of critical parts. In typical static analysis, stresses throughout the body are estimated by the finite element method and they will be incorporated into the fatigue and fracture mechanics assessment of high-risk parts. These stresses will be used as the far-field stress when conducting the fracture mechanics analysis. To establish the static allowables, numerous dog bone tensile specimens will be prepared, pulled to failure, and, based on statistical analysis, the A basis value for the modulus, yield, and ultimate (E, Fty, Ftu) as well as final elongation will be recorded for the material under consideration. The same is true for fracture allowables. However, because of the cost and time associated with generating these allowables, the
1 Virtual Testing and Its Application
13
number of tests will be limited typically to five specimens per case and the average values from test data are required to be used for safe-life analysis. The average or typical values are acceptable to be used for analysis in order to avoid impacts on structure weight. The lower-bound value of fracture allowables can increase the structural weight and cause economical impact on the program. From a safety perspective, the typical value can be justified because of other factors of safety already embedded in the static and life analysis of parts that are documented in the fracture control plan provided by the customers [11]. Nonetheless, the overall cost and time consumed for establishing these allowables through testing must be avoidable to minimize its effect on the program. The following addresses the application of the virtual testing on one of the space programs where the structure must be launched on time in order to meet the deadline set by the customer. The analyst must demonstrate, through fracture mechanics analysis, that the number of cycles to failure is adequate for the part to survive during its entire service usage. However, the fracture allowable for the material must be first established through testing. The cost of testing in the example shown below may not be an issue, but the time required for completing the test is not tolerable by the customer. For this reason, the application of the virtual testing is of great help to the program, which can be implemented in order to meet the deadline specified by the customer.
1.5.1 Background In many instances, fracture critical parts of aircraft or space structures go through high-stress magnitude cycles during or prior to their service usage. The plastic deformation can occur in localized areas that already have high stresses or the net section area (net section yielding), and in some cases the whole structure can be plastically strained if the load magnitude is above the material yield value. For example, a pressurized tank of a reusable (or expendable) space vehicle can locally undergo severe plastic deformation during the proof test process in the boss area of the doom region. When proof test is completed and pressure is removed, the work hardening causes degradation in material fracture toughness, which shortens the remaining life of the tank during its subsequent operation. In other instances, structural parts are exposed to manufacturing loads that can strain the material above the yield value or, during the operational environment, when the structural components see a few repeated cycles with the stress magnitude exceeding the allowable material yield. In all the above-mentioned cases, both the fracture toughness and the fatigue crack growth rate properties have lower values and must be reevaluated through testing. This is where virtual testing proves to be a valuable tool to implement in order to avoid costly and time-consuming tests. As mentioned previously, the fatigue crack growth rate, as well as the fracture toughness tests, requires specimen preparation and pre-cracking that are time consuming. It can delay life assessment analysis for weeks. The following is an example of a real case in which the virtual testing technique was helpful in calculating the fracture allowables for a safe-life analysis
14
B. Farahmand
of composite over-wrapped pressure vessel. Not using this technique would have delayed the launching of the satellite for weeks.
1.5.2 Manufacturing Process and Plastic Deformation of COPV Liner Composite over-wrapped pressure vessels (COPVs) are used extensively in aerospace industry for their light-weight feature [12]. These tanks have metallic liners that undergo welding process and are wrapped in composite material. In general, metallic pressurized vessels are assembled through the welding process, where the shell (cylindrical) and dome components are joined together at both ends through the circumferential welding technique. In addition, the two halves of cylindrical portion of the tank are also joined together by the longitudinal welding technique (Fig. 1.11). The welding technique can be performed by the traditional fusion welding (the VPPA or GTA techniques). The presence of residual stresses in the heataffected zone (HAZ) and the formation of porosity in the nugget area by the fusion welding are not desirable and therefore must be minimized at all costs to maintain the safety of the structure. The new state-of-the-art welding technique called the friction stir welding (FSW) is recently used throughout the aerospace industry in order to avoid premature failure due to inherent residual stresses embedded in the fusion welding process if post-heat treatment is not feasible. For the COPV metallic liner, the post-heat treatment process is required by the program in order to eliminate the residual stresses. However, the presence of unshaved welds is another main source Outlet tube Dome Vertical weld
Circumferential weld cylinder Circumferential weld Dome
The composite Over-wrapped Pressure vessel
Fig. 1.11 Longitudinal and circumferential welding of a pressurized cylinder [12]
1 Virtual Testing and Its Application Fig. 1.12 Crack initiation and damage on the liner as the result of after the auto-frettage process
15
FM73 adhesive
Bulge inward as the result of compressive residual stresses
Damage on the liner as the result of dis-bond and compressive residual stresses after the auto-frettage process
of failure because of the localized buckling due to the presence of large compressive residual stresses that can occur after depressurization cycles. Under the compressive residual stresses, the occurrence of buckling is assumed to be as a result of (1) the geometrical imperfection due to the presence of a crown at the weld (unshaved weld can act as the source of geometrical imperfection when compared with the parent material) and (2) inadequacy of bond between the liner and the FM73 adhesive material (Fig. 1.12). It should be noted that the problem with shaving the weld is the undercutting issue that may further damage the liner, where the skin thickness can be less than the 0.05 in. It is essential for the COPV tanks to be exposed to several pressurization and depressurization cycles prior to their service usage. These cycles are the autofrettage process cycle, one proof test cycle, and several maximum operating pressure cycles, depending on the program requirement. The auto-frettage process induces global plastic deformation on the liner that can be as high as 5% straining (beyond the yielding) and, upon depressurization part of the cycle, the fracture properties of the liner degrade due to the work-hardening phenomenon. The material degradation, localized buckling along the length of the weld and localized de-bond between the liner and over-wrapped material can reduce the life of the liner considerably. In this work, the remaining safe-life capability of COPV metallic liner is assessed through the fracture mechanics approach. Several assumptions are made regarding the fracture allowables after the auto-frettage process, where the metallic liner material undergoes a severe global plastic deformation. It is assumed that after the auto-frettage process, crack initiation can occur and upon the depressurization phase, when stresses are compressive in the liner, localize buckling will take place (Fig. 1.12). However, for the remaining part of this chapter, discussions are allocated to generating fracture allowables of the liner (made of Inconel 718) after the global plastic deformation (subsequent to the auto-frettage process) by applying the virtual testing technique. As indicated earlier, due to the auto-frettage process, material properties change and there is a need to obtain new allowables for the safelife analysis of COPV liner. Due to intolerable amount of time that is needed to obtain fracture allowables through traditional testing, the virtual testing is a logical approach to implement in order to meet the deadline set forth by the program.
16
B. Farahmand
1.5.3 Generating Fracture Allowables of Inconel 718 of COPV Liner Through Virtual Testing Technique To generate fracture toughness and the fatigue crack growth curve for the Inconel 718 liner, the full stress–strain curve for the material must be available. First, the fracture toughness will be estimated and it will be utilized for generating the da/dN versus ΔK curve. The stress–strain curve prior to plastic deformation for the Inconel 718 is shown in Fig. 1.13, where it is taken from the Metallic Materials handbook [9]. The 5% permanent deformation after the auto-frettage process is also shown in the same figure. Note that the 5% straining is above the yielding point of material (i.e., 2%). After the 5% plastic deformation, when the load has been removed, the Inconel 718 will undergo work-hardening phenomenon. The new stress–strain curve after load removal is plotted as part of Fig. 1.13. Therefore, the new stress–strain curve after load removal will be different and is shown in Fig. 1.14, where the total strain is 0.07 in./in. versus 0.12 prior to the plastic deformation.
180
Stress-strain curve for Inconel 718
Stress, ksi
150
Stress-strain curve after 5% permanent deformation & unloading after autofrettage process
5%
0.12
Strain, in./in. Fig. 1.13 Stress-strain curve before and after the 5% plastic deformation
The area under the stress–strain curve is smaller in Fig. 1.14, which is an indication that the fracture toughness value after the auto-frettage process is lower. For this reason the analyst must use the correct value of Kc in the safe-life analysis assessment. Not considering this fact, the analysis results can be overestimated, which can result in catastrophic failure of liner and loss of the space vehicle during its service usage. Under this situation, the new fracture allowables can best be obtained through the virtual testing technique. Using this approach will help analysts to perform safelife analysis in a short time without compromising the safety of the structure. Based on the extended Griffith theory, values of both the plane strain and plane stress fracture toughness can be calculated from Equation (1.12). This equation can be easily programmed in an Excel sheet where the area under the stress–strain curve must be available as an input to the program.
1 Virtual Testing and Its Application 180 New yield value 170
Stress, ksi
Fig. 1.14 The area under the curve is reduced after the auto-frettage process (5% plastic deformation)
17
Estimated stress-strain curve after 5% plastic deformation
0.035
0.07
Strain, in./in.
Fig. 1.15 The variation of fracture toughness for the Inconel 718 before and after 5% plastic deformation
Fracture Toughness For Inconel 718 (Before & After 5% Plastid Deformation)
Fracture Toughness ksi (in.)^0.5
No Deformation
120
Analysis (5%)
100
KIc = 81 KIc = 70
80 60 40 20 0 0
1
2
3
4
Thickness, in.
Both the fracture toughness and the da/dN versus ΔK data can be generated from Fig. 1.14. Figure 1.15 is the variation of fracture toughness for the Inconel 718 before and after 5% plastic deformation. The plane strain fracture toughnesses before and after 5% straining were calculated by the virtual testing technique and are KIc = 81 and 70 ksi (in.)ˆ0.5, respectively. The “ksi (in.)ˆ0.5” refers to the unit of fracture toughness, KIc. Note that the yield value for the material after the workhardening process is higher than prior to the 5% straining due to the auto-frettage process (170 ksi vs. 150 ksi) and moreover, the total elongation is lower (12% vs. 7%), respectively. Both the increase in the yield value and reduction in percent elongation are main contributors to the drop in the fracture toughness value. The fatigue crack growth rate data must also be available for the case of 5% plastic deformation. In the upper portion of the da/dN versus ΔK data, where crack accelerates, the quantity ΔK approaches the material fracture toughness, Kc. Of course, the value of this quantity depends on the thickness of test specimen. The
18
Inconel 718 (Before & After Plastic Deformation) 0% Deformation
da/dN - In./Cycle
Fig. 1.16 Fatigue crack growth curve for the Inconel 718 before and after 5% Plastic deformation (0% plastic deformation data is provided from NASGRO database)
B. Farahmand
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E+00
5% Deformation
1.00E+01
1.00E+02
1.00E+03
ΔK-ksi (in.)^0.5
variation of Kc versus thickness is already shown in Fig. 1.15. The lower portion of the da/dN versus ΔK curve is related to the threshold value. As indicated previously (see Table 1.1), this value falls between Kth = KIc /4π and Kth = KIc /3π values. Conservatively, the value of Kth = KIc /4π is selected for this analysis. This is the lower-bound value of Kth for the stress ratio of R = 0. Based on the above statement, the threshold value before and after 5% plastic deformation should be 6.4 ksi-(in.)ˆ0.5 and 5.5, respectively. The following fracture mechanics analyses are conducted to check whether the liner is suitable for flight with the assumption that fracture allowables are lower after the 5% plastic deformation. Since the access to the liner is difficult, it is assumed that after the auto-frettage process with 5% plastic deformation, there is a possibility that cracks have been introduced in the material as deep as 95% through the thickness. Figure 1.17 shows the part through crack that is almost becoming a through crack as the result of plastic deformation. In the analysis, it is assumed the crack has initiated in the weld region. The weld will locally bulge when the applied load (causing the 5% plastic deformation) is removed, resulting in the induced residual compressive stresses through the liner. During the operation, when tank is pressurized for the operation, the bulge area may be eliminated and its original geometry Crack is 95% through the thickness
After unloading condition
2c
Fig. 1.17 The part through crack assumption (95% through the thickness) after the plastic deformation
Loading condition
a
0.025”
1 Virtual Testing and Its Application
19 Table 1.2
may be maintained. The safe-life analysis will assume (1) the initial flaw size is 95% through the thickness, (2) fracture allowables are reduced, and (3) the induced hoop stress is 150 ksi. The yield value is higher due to the work-hardening process, and its new value is 170 ksi as shown in Fig. 1.14. Tables 1.2 and 1.3 show results of life analysis conducted by the NASGRO computer code [8], where the abovementioned assumptions were used as the required input in the code. Two extreme cases of crack geometry were considered: (1) a circular crack with the aspect ratio of a/c = 1 and (2) the shallow crack with a/c = 0.2 (Fig. 1.18). The latter crack geometry has shorter life because the depth will advance faster in order to catch up with the crack length (Table 1.3). As shown in Table 1.3, the number of cycles to failure is zero and crack will propagate catastrophically upon the first load cycle with maximum stress of 150 ksi. However, if the crack geometry is a circular crack (a/c = 1) with the same depth, the analysis results show the tank will be able to handle additional 99 cycles prior to becoming a through crack. The following will be an additional example that shows the usefulness of the virtual testing technique when fracture allowable cannot be obtained through the traditional testing because the program is required to meet the customer’s deadline.
20
B. Farahmand Table 1.3
1.5.4 Generating Fracture Allowables of 6061-T6 Aluminum Tank Through Virtual Testing Technique A pressurized container is made of 6061-T6 aluminum alloy and is used as an oxygen tank for a space structure. As part of the requirement, the tank must undergo a proof test and several subsequent pressurization cycles equal to the maximum expected pressure cycle at the start of operation. The proof test is defined as one pressurization and depressurization cycle in excess of maximum expected operating pressure (MEOP). This is done in order to verify the structural integrity of tank [13]. During the proof test operation, the amount of pressure exceeded the required value and caused plastic deformation in most part of the tank. A decision was made by the program manager to use the tank if the analyst could show that it has adequate life during its service operation. Because of induced plastic deformation during the proof test, both the static and fracture properties of 6061-T6 tank are different and must be taken into the consideration when conducting static and fracture analyses. The amount of plastic deformation was estimated to be 5% (above the material yield value, 2%). Figure 1.19 shows the stress–strain curve before and after plastic deformation. Based on our previous discussions related to the COPV liner in
1 Virtual Testing and Its Application
21 The same depth but different length
Fig. 1.18 Two extreme cases of crack geometry were considered (a/c=1 and a/c=0.2) for life assessment analysis
Long Crack, a/c = 0.2
Circular Crack, a/c = 1
t = 0.025 in.
t = 0.025 in.
6061-T6 Stress-Strain Curve -RT (0% & 5% Work Hardening)
Stress, ksi
Fig. 1.19 The stress-strain curve before and after 5% plastic deformation
50 40 30 20 10 0
Longitudinal 5% Plastic Deformation
0
0.05 0.1 Strain, in./in.
0.15
Section 1.5.3, the fracture allowables for the 6061-T6 tank have been degraded due to the work-hardening phenomenon. In Section 1.5.3, the yield value of Inconel 718 has been elevated, but the area under the stress–strain curve was reduced. The same can be said about the above-mentioned tank. The fracture toughness value of 6061-T6 after the 5% plastic deformation can be estimated through the virtual testing technique and, subsequently, the da/dN versus ΔK data can be generated. Figure 1.20 shows the variation of fracture toughness versus material thickness. In part A, the fracture toughness versus thickness for the 0% work-hardening case (no plastic deformation) is compared with the experimental data extracted from the NASGRO database. Excellent agreement between test data and virtual testing can be seen. In part B, the fracture toughness versus thickness for two cases of 5% plastic deformation and prior to 5% plastic deformation (0% work hardening) are also plotted. It can be seen that for the case of 5% plastic deformation, the corresponding KIc value is lower (KIc = 27 versus 22 ksi (in.)ˆ0.5). The fatigue crack growth rate data for the 6061-T6 before and after 5% plastic deformation is shown in Fig. 1.21. The purpose of this plot is to show the capability and accuracy of virtual testing when compared with test data. This is shown in part A where the da/dN versus ΔK data for the material prior to the plastic deformation (0% work hardening) is compared with the NASGRO database. As can be seen from Fig. 1.21, the two data (virtual testing and test data) are almost on top of each other. In addition, the fatigue crack growth rate curves for the 5% plastic deformation and before plastic deformation are plotted in part B. It is clear that the da/dN curve for 5% plastic deformation is shifted to the left, which is an indication that fracture allowable has been reduced. Therefore, in safe-life analysis of pressurized tank after
22
B. Farahmand 6061-T6 Aluminum Alloy -RT
60
0% Work Hardening
0% Work Hardening
50
Fracture Toughness ksi (in.)^0.5
Fracture Toughness ksi (in.)^0.5
6061-T6 Aluminum Alloy -RT
A
40
KIc = 27
30 20
Virtual Testing
10
NASGRO Data
0 0
0.5
1
1.5
2
2.5
3
3.5
Thickness (in.)
60
5% Work Hardening
50 40
B
30 20 KIc = 22
10 0
0 0.5 1 1.5 2 2.5 3 3.5 Thickness (in.)
Fig. 1.20 The variation of fracture toughness versus material thickness before and after 5% plastic deformation
Fig. 1.21 Fatigue crack growth curves before and after 5% plastic deformation
the plastic deformation, the reduced fracture properties must be incorporated in the life analysis computer code.
1.6 Summary and Future Work The virtual testing technique and its application to the aerospace industry are discussed in detail. The method can be used to establish fracture allowables (fracture toughness and da/dN versus ΔK data) when both time and budget do not allow performing tests. These allowables must be available in order to conduct safe-life analysis of fracture critical parts. Two examples were provided by the author that demonstrates the need for this technology when the program must deliver the space hardware reliably to the customer on time. Numerous alloys were used in order to establish the validity of this technique. In all cases, excellent correlation between test data and the virtual testing method were found. Appendix shows several cases where virtual testing technique and test data were compared for randomly selected alloys extracted from NASGRO database. Only in the case of welds a few discrepancies were noticed between the two approaches. This was expected because of errors
1 Virtual Testing and Its Application
23
that can be encountered in testing due to (1) residual stresses and (2) porosity in the weld. Lack of data in the threshold region makes this technique desirable for engineers when designing parts for infinite life.The lack of test data in the threshold region is mainly because of time and cost associated with running the test in that region of the da/dN curve. In most cases, the threshold crack growth reading becomes complicated due to plasticity closure, surface roughness, and environmental effects (oxide particles) at the crack tip. The virtual testing technique utilizes the extended Griffith theory by estimating the amount of energy that has been consumed for straining material at the crack tip. A relationship between the applied stress and critical crack length was formulated, which can be used to calculate the critical value of stress intensity factor, Kc. The proposed technique uses the area under the stress–strain curve and relates the energy per unit volume of a uniaxial tensile test to the energy consumed at the crack tip. Therefore, this technique is required to have only the full stress–strain curve as an input to the theory for the material under study. As already emphasized throughout this chapter, the application of this technique in aerospace industry can be extremely useful. It can significantly reduce the cost and time of testing. It clearly indicates how the virtual testing technique can accurately estimate fracture allowables when they are needed to program for safe-life assessment of structural parts in order to meet the deadline set forth by the customer. Because of the dependency of this method on the stress–strain curve, the author believes there is a need to generate the full stress–strain curve through the multiscale modeling and simulation approach, therefore making the proposed method to become free from ASTM testing. This can be part of the future work in the field of virtual testing. One approach to the multi-scale modeling is the electron density functional theory that imposes deformation constraints to model the energetics of the stress–strain curve in metal and metal alloy systems [14]. In general, multiscale modeling efforts rely on atomistic- or molecular-level information to predict the behavior of the material in response to applied loadings and environmental conditions. The effects of grain boundaries and dislocation are directly incorporated into the prediction of large-scale material behavior using an appropriately chosen atomic potential [15]. Because the prediction of stress–strain behavior of aerospace alloys from molecular-based multi-scale approaches has not been rigorously pursued, future efforts must be focused on developing models for these materials.
Appendix To appreciate the usefulness of the virtual testing technique, numerous fatigue crack growth rate data (da/dN versus ΔK curve) for several aerospace alloys were plotted using the virtual testing approach. In all cases, data generated by this technique were compared with test data extracted from the NASGRO database. Excellent
24
B. Farahmand 2014-T6 ALUMINUM - R=0
2219-T87 Aluminum Alloy, R=0 Virtual Testing
1
da/dN-In./Cycle
da/dN (in./Cycle)
NASGRO Data
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09
10
100
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11
Virtual Testing
1
Δ K-ksi(in.)^0.5
10
100
Δ K -ksi (in.)^0.5
Fig. 1.22 Fatigue crack growth by virtual testing versus NASGRO database (2000 series aluminums) 6082-T651 Aluminum, R = 0
6061-T651 Aluminum, R = 0 NASGRO
Virtual Testing
1.00E–01 1.00E–03 1.00E–05 1.00E–07 1.00E–09 1.00E–11
da/dN -in./cycle
da/dN -in./cycle
NASGRO
1
10 Δ K ksi (in.)^0.5
100
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
ΔK -ksi (in.)^0.5
Fig. 1.23 Fatigue crack growth by virtual testing versus NASGRO database (6000 series aluminums)
agreement between the test data and virtual testing technique can be seen. The following is a list of a few alloys that have been used in manufacturing aerospace parts. Figures 1.22, 1.23 and 1.24 are 2000, 6000, and 7000 series aluminum alloys. Figures 1.25, 1.26 and 1.27 are unalloyed and binary, ternary, and quaternary titanium alloys. Figures 1.28, 1.29 and 1.30 are magnesium, copper–bronze, and Russian aluminum alloys. Figures 1.31, 1.32, and 1.33 are miscellaneous super-alloys, miscellaneous corrosion- and heat-resistance steel, and Ni alloys. In all cases, the fracture toughness data from the test data were used for the accelerated region to generate the da/dN versus ΔK curve. If the fracture toughness value is not available for the material, the extended Griffith theory can be used to provide the needed Kc value. However, as mentioned previously, the full stress–strain curve is required for the virtual testing analysis.
1 Virtual Testing and Its Application
25 7175-T7452Aluminum, R = 0 NASGRO
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11
Virtual Testing
da/dN -in./cycle
da/dN-In./Cycle
7010-T7451 T-L Aluminum, R = 0
1
10
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
100
Δ K -ksi (In.)^0.5
10
100
Δ K -ksi (in.)^0.5
Fig. 1.24 Fatigue crack growth by virtual testing versus NASGRO database (2000 series aluminums)
Unalloy Titanium (σ σ ys = 70ksi)
NASGRO
1.00E+00 1.00E–02 1.00E–04 1.00E–06 1.00E–08 1.00E–10
Virtual Testing
1
10 Δ K-ksi (in.)^0.5
da/dN-in/cycle
da/dN-in./cycle
Ti-2.5 Cu Titanium
100
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
Δ K-ksi (in)^0.5
Fig. 1.25 Fatigue crack growth by virtual testing versus NASGRO database (binary and unalloyed titanium alloys)
Ti-5Al-2.5Sn Titanium Annealed, R=0
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09
Virtual Testing
1
1
10
Δ K-ksi (In)^0.5
Fig. 1.26 alloys)
NASGRO
Virtual Testing
da/dN-in/cycle
da/dN, In./Cycle
Ti-3Al-2.5V, Titanium, CW +SR (750F)
100
10
100
1000
ΔK-ksi (in)^0.5
Fatigue crack growth by virtual testing versus NASGRO database (ternary titanium
26
B. Farahmand Ti-6Al-6V-2Sn, RA Tiatanium, R = 0
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
da/dN -in./cycle
da/dN-in/cycle
Ti-4.5Al-5Mo-1.5Cr Titanium -STA, R = 0
Δ K-ksi (in)^0.5
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09
Virtual Testing
1
10 ΔK -ksi (in)^0.5
100
Fig. 1.27 Fatigue crack growth by virtual testing versus NASGRO database (quaternary titanium alloys)
AZ-31B-H24-Magnesium, R = 0
AM 503 Magnesium as wrought, R = 0
NASGRO
Virtual Testing
da/dN-In./Cycle
da/dN-In/Cycle
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1
10
100
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1.00E–11
Virtual Testing
1
10
Δ K -ksi(In.)^0.5
100
Δ K -ksi(In.)^0.5
Fig. 1.28 Fatigue crack growth by virtual testing versus NASGRO database (magnesium alloys)
AMg6M Russian Alloy-Annealed, R=0
ZW1 Magnesum as wrought, R=0
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
Δ K -ksi(In.)^0.5
da/dN-In./Cycle
da/dN-In./Cycle
NASGRO
100
NASGRO Virtual Testing
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1
10
100
ΔK -ksi(In.)^0.5
Fig. 1.29 Fatigue crack growth by virtual testing versus NASGRO database (Russian and magnesium alloys)
1 Virtual Testing and Its Application
27 Russian Alloy AMg6H Ann, R = 0
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
NASGRO
Virtual Testing
da/dN -In/Cycle
da/dN-In./Cycle
Al-Bronze CDA 630-R = 0
1.00E+00
Virtual Testing
1.00E–02 1.00E–04 1.00E–06 1.00E–08 1.00E–10 1
1
10
10
100
100
ΔK -ksi (in)^0.5
Δ K -ksi(In.)^0.5
Fig. 1.30 Fatigue crack growth by virtual testing versus NASGRO database (Al-Bronze and Russian zinc alloys)
Nitronic 50 Alloy -Anneal, R = 0 NASGRO
NASGRO Virtual Testing
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10 1
10
100
Δ K-ksi (in)^0.5
da/dN -in./cycle
da/dN-in/cycle
MP35N STA Super Alloy, R = 0
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09
Virtual Testing
1
10
100
1000
ΔK-ksi (in.)^0.5
Fig. 1.31 Fatigue crack growth by virtual testing versus NASGRO database (super alloy and corrosion resistance Nitronic alloys)
PH13-8Mo H1000 Alloys, R = 0
15-5PH H900 Alloy, R = 0
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09
Virtual Testing
da/dN-in/cycle
da/dN-in/cycle
NASGRO
1
10
100
Δ K-ksi (in)^0.5
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
ΔK-ksi (in)^0.5
Fig. 1.32 Fatigue crack growth by virtual testing versus NASGRO database (corrosion resistance alloys)
28
B. Farahmand Rene 41 ST(1950F); A(1400F 16h) Alloy, R = 0 NASGRO
NASGRO
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
Δ K-ksi (in)^0.5
da/dN-in/cycle
da/dN-in/cycle
17-4PH H1025 Alloy, R = 0
100
1.00E+00 1.00E–01 1.00E–02 1.00E–03 1.00E–04 1.00E–05 1.00E–06 1.00E–07 1.00E–08 1.00E–09 1.00E–10
Virtual Testing
1
10
100
1000
ΔK-ksi (in)^0.5
Fig. 1.33 Fatigue crack growth by virtual testing versus NASGRO database (corrosion resistance and super alloys)
References 1. B. Farahmand, Fatigue & Fracture Mechanics of High Risk Parts, Chapman & Hall, 1997, Chapter 5 2. B. Farahmand, Fracture Mechanics of Metals, Composites, Welds, and Bolted Joints, Kluwer Academic Publisher, November 2000, Chapter 6 3. B. Farahmand, “Predicting Fracture and Fatigue Crack Growth Properties Using Tensile Properties, Engineering,” J. Fract., Vol.75, 2007, pp. 2144–2155 4. B. Farahmand, De Xie, F. Abdi, “Estimation of Fatigue and Fracture Allowables for Metallic Materials Under Cyclic Loading,” AIAA-2007 5. B. Farahmand, “Multiscaling of Fatigue” Application of virtual testing for obtaining fracture allowables of aerospace and aircraft materials. Springer, 2008 (A Book Chapter by G. Sih) 6. G. Irwin, “Fracture Dynamics,” Fracture of Metals, ASM, 1948, p. 147 7. E. Orowan, “Fracture and Strength of Solids,” Rep. Prog. Phys., Vol. 12, 1949, pp. 185–232 8. Fatigue Crack Growth Computer Program “NASGRO 4.0”, JSC, SRI, ESA, and FAA, 2002 9. MIL-HDBK-5H “Military Handbook Metallic Materials and Elements for Aerospace Vehicle Structure” 10. B. Farahmand, “Virtual Testing for Estimating Material Fracture Properties (Reducing Time & Cost of Testing),” 11th International Conference on Fracture (ICF11), Turin Italy, March 2005 11. NHB 8071 “Fracture Control Requirements for Payloads Using the National Space Transportation System (NSTS),” NASA, Washington, DC, 1985 12. Metal lined Composite Overwrapped Pressure Vessels (COPVs), Arde Inc., 875 Washington Avenue, Carlstadt, New Jersey 07072 13. Military Standard 1522A, “Standard General Requirements for Safe Design Operation of Pressurized Missile and Space System,” USA, May 1984 14. B. Farahmand and M. Doyle, “Obtaining Fracture Properties by Virtual Testing and Multiscale Modeling,” The Aging Aircraft Conference, 2009 15. B. Farahmand and G. Odegard, “Obtaining Fracture Properties by Virtual Testing and Molecular Dynamics Techniques,” The NSTI Conference, 2009
Chapter 2
Tools for Assessing the Damage Tolerance of Primary Structural Components R. Jones and D. Peng
Abstract Fatigue considerations play a major role in the design of optimised flight vehicles, and the ability to accurately design against the possibility of fatigue failure is paramount. However, recent studies have shown that, in the Paris Region, cracking in high-strength aerospace quality steels and Mil Annealed Ti–6AL–4V titanium is essentially R ratio independent. As a result, the crack closure and Willenborg algorithm’s available within commercial crack growth codes are inappropriate for predicting/assessing cracking under operational loading in these materials. To help overcome this shortcoming, this chapter presents an alternative engineering approach that can be used to predict the growth of small near-micron-size defects under representative operational load spectra and reveal how it is linked to a prior law developed by the Boeing Commercial Aircraft Company. A simple method for estimating the S–N response of 7050-T7451 aluminium is then presented. Keywords Fatigue crack growth · Fatigue modelling · Life prediction · Similitude
2.1 Introduction To achieve their design requirements, modern military make extensive use of aluminium, high-strength steels, that is, 4340 and D6ac, and titanium. The Joint Strike Fighter (F-35), the Super Hornet and the F/A-18 make extensive use of 7050-T7451 aluminium. However, there has been an increasing use of titanium in primary structural members due to its high strength, light weight, and good fatigue and fracture toughness properties. As a result, bulkheads in the F-22, the Super Hornet, the Swiss F/A-18, and the Joint Strike Fighter are made of titanium. In the F-22, titanium accounts for ∼36%, by weight, of all structural materials used in the aircraft. R. Jones (B) Department of Mechanical and Aerospace Engineering, DSTO Centre of Expertise for Structural Mechanics, Monash University, Victoria 3800, VIC, Australia e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 2,
29
30
R. Jones and D. Peng
Until recently, it had been thought that fatigue crack growth in 7050-T7451, highstrength aerospace steels and titanium was well understood. However, in his review of fatigue crack growth under variable amplitude loading, Skorupa [1] concluded that, viz: Experimental results also suggest that the underlying causes of load interaction phenomena are not necessarily similar for different groups of metals, e.g. steels of and Al and Ti alloys.
Furthermore, as a result of the Australian Defence Science and Technology Organisation’s Flaw IdentificatioN through the Application of Loads (FINAL) testing program [2] it is now known [3, 4] that similitude-based concepts on which the crack growth programs AFGROW, NASGRO, and FASTRAN are based cannot be used to accurately predict the growth of near-micron-size flaws in 7050-T7451 aluminium alloy under representative in-flight loading. In this context it should also be noted that Forth, James, Johnston, and Newman [5] have reported that crack growth data obtained for D6ac and 4340 steels using compact tension (CT) specimens tested in accordance with the ASTM standards exhibited no R ratio dependency and hence no closure in the Paris region (Region II), see Fig. 2.1. 1.00E–05
Fig. 2.1 Fatigue crack growth data from D6AC steel. Plot reproduced from [5]
R = 0.7 1.00E–06
R = 0.7
da/dN (m/cycle)
R = 0.7 1.00E–07 R = 0.1 1.00E–08
R = 0.1 R = 0.3
1.00E–09 R = 0.3 1.00E–10
1.00E–11 1
10
100
ΔK MPa √ m
This behaviour, that is, the da/dN versus ΔK relationship appearing to be R ratio dependent in Region I but showing no R ratio dependence and hence no closure in the Paris region, is also evident in the work of James and Knott [6] who studied cracking in QIN (HY80) steel, see Fig. 2.2. As such, the various closure-based models and the Willenborg crack growth law, which models load interaction and sequence effects by modifying the effec-
2 Tools for Assessing the Damage Tolerance
31
1.00E–04
Fig. 2.2 Fatigue crack growth data from QIN (HY80) steel. Plot reproduced from [6]
R = 0.7 R = 0.5
da /dN (m/cycle)
1.00E–05
R = 0.35 R = 0.2 1.00E–06
1.00E–07
1.00E–08 1
10
100
Δ K MPa √m
tive R ratio, available within these codes cannot be used to accurately predict crack growth in high-strength steels. Jones, Farahmad, and Rodopoulos [7] have revealed that Mil Annealed Ti–6AL–4V titanium has a similar (near) R ratio independence. As such the various closure-based models and the Willenborg crack growth law cannot be used to accurately predict crack growth in Ti–6AL–4V. (This R ratio independence has also been seen in crack growth in rail steels [8] which have also been found to conform to the generalised Frost–Dugdale crack growth law [8, 9].) When addressing the question of crack growth under representative in-service loading it should also be noted that in the review paper on crack growth and similitude Davidson [10] concluded that similitude was lost during fatigue crack growth under variable amplitude loading and stated that: “Detailed measurements of fatigue cracks undergoing simple load spectra confirm that when ΔKeff is based on Kopen , good correlations are achieved with large crack growth data.This understanding, although useful, does not easily translate to an engineering method for computing crack growth rate under complex variable amplitude loading.” The question thus arises: How can a valid virtual assessment of the performance of an aircraft/rail component under representative operational loading be performed if the fundamental concepts inherent in the existing crack growth codes, viz: AFGROW, FASTRAN, and NASGROW, do not apply to the materials from which the component fabricated, that is, for components made out of 4340 and D6ac steel, QIN (HY80) steel, rail steels, Mil Annealed Ti–6AL–4V, STOA Ti– 6AL–4V, etc.? This chapter presents one possible approach which is based on the equivalent block formulation presented in [8, 11, 18] and reveals how it is linked to spectra where the constant amplitude Region II growth mechanism tends to be suppressed and a single value of C∗ can be used to predict the crack length versus cycles history.
32
R. Jones and D. Peng
2.2 An Equivalent Block Method for Predicting Fatigue Crack Growth It is now known that the mechanisms underpinning crack growth under variable amplitude load differ from those seen under constant amplitude loading [12]. It is also known that many materials either follow a non-similitude-based crack growth law [3, 4, 8, 9], lose similitude as the crack grows [10], or exhibit a near R ratio independence in the Paris Region [5–8]. In these cases, crack growth under representative operational loading cannot be predicted using the concepts inherent in the existing crack growth codes, viz: AFGROW, FASTRAN, and NASGROW, since they do not apply to the materials from which the component is fabricated, and since the data used in these calculations are obtained from constant amplitude tests that may not reflect the mechanisms driving growth under the spectrum of interest [12]. However, many practical engineering problems, that is, cracking in rail and aircraft structures, involve complex load spectra that can be approximated by a number of repeating load blocks. Schijve [13], Gallagher, and Stalnaker [14], Miedlar, Berens, Gunderson, and Gallagher [15], Barsom and Rolfe [16] and Miller, Luthra, and Goranson [17] revealed that these repeated blocks of loads can, in certain circumstances, be treated as equivalent to load cycles. We now show how this concept, that is, an equivalent block approach, can be used to describe crack growth in Mil Annealed Ti–6AL–4V and D6ac steel under complex variable amplitude loading. To this end let us consider the case of block loading, where each block consists of a spectrum with n cycles that have peak stresses of σ i , i = 1. . . n, with the associated cyclic ranges being Δσ i , i = 1. . . n. Let us also assume that: (i) The slope of the a versus block curve has a minimal number of discontinuities. (ii) There are a large number of blocks before failure. With these assumptions, Jones and Pitt [18] derived an “equivalent block” variant of the generalised Frost–Dugdale crack growth law [4, 8] to account for the crack growth per block, da/dB, viz da/dB = C˜ K max γ a 1−γ/2
(2.1)
where C˜ is a spectra-dependent constant and Kmax is the maximum value of the stress intensity factor in the block. (The precise relationship between C˜ and the constant of proportionality in the Paris crack growth law is yet to be determined.) Jones, Molent, and Krishnapillai [11] subsequently extended this “equivalent block” law to have a form consistent with regions I, II, and III, viz ˜ 1−γ/2 K max γ − da/dBo )/(1.0 − K max /K c ) da/dB = (Ca
(2.2)
where a is now the average crack length in the block, and Kc is the apparent cyclic fracture toughness. Here, as described in [11], the term da/dBo reflects the both nature of the discontinuity from which the crack initiates and the apparent fatigue
2 Tools for Assessing the Damage Tolerance
33
threshold for this particular block loading spectra. However, it should (again) be stressed that this variant of the generalised Frost–Dugdale law is only applicable to crack growth data where the slope of the a versus block curve has minimal discontinuities and there are a large number of blocks to failure, see [8, 11, 18]. This formulation is (extensively) validated in [8, 11]. At this stage it is important to note that Miller, Luthra, and Goranson [17], at the Boeing Commercial Aircraft Company, have also developed a related (nonsimilitude) approach whereby instead of Equation (2.2) da/dB was expressed as
da/dB = C(K /g(a/t))m
(2.3)
where the function g(a/t), which is a function of ratio of the crack length (a) to the thickness (t) of the specimen, was experimentally determined and its functional form is presented in [17]. This formulation was necessary to enable the predictions to match the measured crack length histories. However, Jones, Pitt, and Peng have shown [8] that the experimental test data used in [17] to determine the function g(a/t) followed the generalised Frost-Dugdale crack growth law so that the two methodologies essentially coincide. In the next section we present three examples that illustrate how the present nonsimilitude approach, that is, Equation (2.2), can be used to accurately predict crack growth in 7050-T7451, D6ac steel, and Mil Annealed Ti–6AL–4V aluminium specimens subjected to complex variable amplitude load spectra.
2.3 Fatigue Crack Growth under Variable Amplitude Loading The first problem considered is that of crack growth in the 1969 General Dynamics, now Lockheed Martin Tactical Aircraft Systems (LMTAS), F-111 wing fatigue tested under a representative F-111 usage spectra. (An early F-111 in-flight failure was largely responsible for the USAF adopting a damage tolerance approach.) In this test, cracking was measured at a cut-out location designated as fuel flow hole 58 [19] on the lower (tension) surface of the D6ac steel wing pivot fitting, see Figs. 2.3 and 2.4. Before attempting to predict crack growth in the pivot fitting we first confirmed that growth in D6ac steel conformed to the generalised Frost–Dugdale law. This was done via a collaborative project with Dr. Scott Forth at NASA [20]. As part of this project we examined the results of a detailed NASA study into crack growth in D6ac steel CT specimens. The test matrix evaluated is given in Table 2.1. In this study it was found, √ see Fig. 2.5, that √ if we restrict ourselves to regions where Kmax < 115.0 Ksi in ( = 125 MPa m) then the data conforms to the generalised Frost–Dugdale crack growth law, viz da/dN = 8.12 × 10−9 a (1−γ/2) (Δκ)γ − 2.79 × 10−7
(2.4)
34
R. Jones and D. Peng
Fig. 2.3 Full 3D F-111 model, from DSTO 734247 694766 655285 615804 576323
Mousehole 58
536842 497361 457880 418399 378917 339436 299955 260474 220993 181512 142031 102550 63069
Fig. 2.4 Interior of the DSTO 3D F111 model
where the value of γ = 2.6 was taken from Murtagh and Walker [19] and where as per Walker [21] we have defined the crack driving force as Δκ = K max (1− p) ΔK p
(2.5)
where a value of p = 0.95 was found to best collapse the data. This low value of p confirmed the finding reported in [5] that the crack increment per cycle (da/dN) essentially has no R ratio dependency. Having established that crack growth in D6ac steel conforms to the Generalised Frost–Dugdale law we assumed that in the 1969 wing tests there was, as reported in
2 Tools for Assessing the Damage Tolerance
35
Table 2.1 Test matrix Test frequency Hz Ct3-5-tl Ct3-10b-lt Ct3-12-lt Ct3-25-lt Ct3-27-lt Ct3-29-lt Ct3-46-lt Ct3-47-lt
Constant Kmax = 15 Constant R = 0.3 LI Constant R = 0.9 LI Constant R = 0.7 LI Constant R = 0.9 LI Constant R = 0.3 LI R = 0.1 LI R = 0.8 LI
18 20 20 20 22 10 20 10
LI = Load increasing test, Kmax = constant Kmax test. 1.0E–02
Fig. 2.5 Crack growth in D6ac steel, from [20]
C3-5-lt Ct3-12-lt
1.0E–03
da /dN (mm/cycle)
Ct3-27-lt
1.0E–04
Ct3-25-lt Ct3-47-lt
1.0E–05 Ct3-46-lt Equation (4)
1.0E–06
y = 8.12E–09x – 2.79E–07 R2 = 1.00
Ct3-29-lt
1.0E–07 Ct3-10b-lt
1.0E–08 1
10
(ΔK
100 0.95 K
1000
10000 100000 1E+06
0.05)2.6/a 0.3 MPa2.6 m
max
[19], an initial 0.19-mm semi-circular flaw. At each increment of crack growth, the stress intensity factors were computed using a weight function technique together with the stress field determined from the finite element model shown in Figs. 2.3 and 2.4. Crack √ growth was then predicted using Equation (2.2) with γ = 2.6 and KC = 87 MPa m, as given in [19], and C˜ = 3.0 × 10−6 . The load spectra used in the 1969 test, and in this study, was provided by the Australian Defence Science and Technology Organisation (DSTO) and corresponds to that used in [19]. The resultant predicted crack depth histories are presented in Fig. 2.6 where we see good agreement between the predicted and the measured crack depth histories. In this example, when using Equation (2.2) to compute crack growth at the deepest point of the semi-elliptical surface flaw the quantity ‘a’ on the left- and the righthand sides of Equation (2.2) is the crack depth. Similarly, when using Equation (2.2) to compute crack growth at the surface points, the quantity ‘a’ on the left- and the
36
R. Jones and D. Peng 10 LMTAS Experimental
Crack depth (mm)
Equivalent Block
1
0.1 0
1000 2000 3000 4000 5000 6000 7000 8000 Simulated Flight Hours
Fig. 2.6 Measured and predicted crack growth in the 1969 F-111 wing test
right-hand side of Equation (2.2) is the half crack surface length. In this fashion, we allow for the variation of the crack aspect ratio during crack growth.
2.3.1 Fatigue Crack Growth in an F/A-18 Aircraft Bulkhead The next problem considered involved cracking in an F/A-18 Y488 bulkhead tested as part of the DSTO Flaw IdeNtification through the Application of Loads (FINAL) test program, see Dixon et al. [2]. This test program utilised ex-service Canadian Forces (CFs) and US Navy (USN) wing attachment centre barrel (CB) sections loaded using an industry-standard-modified mini-FALSTAFF spectrum, see [2], which is representative of flight loads seen by fighter aircraft. Since cracking in the bulkhead was three-dimensional, a three-dimensional FE model was required, see Figs. 2.7 and 2.8. The location of the crack is shown in Fig. 2.8, where node 4390 represents the centre of the initial semi-elliptical surface flaw. This problem had previously been studied using a cycle-by-cycle approach [4] and it was known that cracking in 7050-T7451 conformed to the generalised Frost–Dugdale law, [4, 22]. As in [4] we again used a weight function technique together with the stress field as determined from the FE model of the bulkhead to compute the associated stress intensity factors. The crack growth history from initial equivalent pre-crack sizes (EPS) of√0.003 mm was predicted using Equation (2.2) with γ = 3.36 and Kc of 35.4 MPa m as given in [4] and C˜ = 2.25 × 10−10 .
2 Tools for Assessing the Damage Tolerance
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Fig. 2. 7 The bulkhead structure
Node 4390
Fig. 2.8 The local mesh
The predicted crack depth history, allowing for changes in the aspect ratio of the flaw as the crack grows, is shown in Fig. 2.9 together with the associated experimental test result, where we see that there is very good agreement. Figure 2.9 also contains a comparison with predictions, presented in [4], made using FASTRAN II. Here we see that FASTRAN II predicted a very long fatigue life. Furthermore,
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Fig. 2.9 Experimental and predicted crack growth histories
10
Crack Depth (mm)
Y488 Experimental Results 1
FASTRAN Average Block Solution
0.1
0.01
0.001 0
20000
40000
60000
80000
Flight Hours
the shape of the crack depth versus cycles curve predicted by FASTRAN II differed markedly from the test data. In Fig. 2.9 we see that the experimental and predicted (from Equation 2.2) crack depth histories show a behaviour that is consistent across three decades of crack lengths, that is, from 0.003 mm to more than 5 mm.
2.3.2 Crack Growth in Mil Annealed Ti–6AL–4V under a Fighter Spectrum Jones, Farahmad, and Rodopoulos [7], who analysed the data presented in [23, 24], found that crack growth in Mil Annealed Ti–6AL–4V titanium was essentially R ratio independent, see Fig. 2.10. Figure 2.10 shows that cracking in Mil Annealed Ti-6AL-4V also appears to conform to the generalised Frost–Dugdale law, viz da/dN = C ∗ a (1−γ/2) (Δκ)γ − da/dN0
(2.6)
with C∗ ∼ 2.5 10–11 , γ = 2.5, p = 0.08, Kc = 100 MPa √ Δκ as given in Equation (2.5) –9 m and da/dN0 = 4.45 × 10 . As explained in [4, 8, 9] the term da/dN0 reflects both the nature of the discontinuity from which the crack initiates and the apparent fatigue threshold. The small value of p reveals that crack growth in Mil Annealed Ti–6AL–4V titanium has a very weak R ratio dependency. We also see that this relationship, that is, Equation (2.6), holds over 3 orders of magnitude, that is, 2 × 10–9 < da/dN < 2 × 10–6 . It should also be noted that this value of γ compares well with that of γ = 2.6 obtained by Zhuang et al. [25] for Mil Annealed Ti–6AL–4V tested under spectrum loading. With this in mind let us now examine the crack growth data presented by Northrop-Grumman [26] who studied crack growth in 6-inch-wide and 0.289-inchthick centre cracked Mil Annealed Ti-6AL–4V panels subjected to a fighter load
2 Tools for Assessing the Damage Tolerance
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1.00E–05
Fig. 2.10 Crack growth in Mil-Annealed Ti–6AL–4V, from [7]
y = 2.56E–11 x – 4.45E–09 R2 = 0.984
da/dN (m/cycle)
1.00E–06
1.00E–07 R = 0.85 R= 0.66 R = 0.43 R = 0.25
1.00E–08
1.00E–09 100
1000
10000
100000
1000000
(Δ K /(1–R)0.08)2.5/a 0.25 MPa2.5 m
spectrum with a peak remote stress of 103 ksi (710 MPa). The resultant predictions are shown in Fig. 2.11 where we again see an excellent agreement between the measured and the computed crack length histories. In this case, the left hand √ side ˜ = 2.83 × 10−10 , γ = 2.5, and Kc = 150 ksi in of Equation (2.2) is da/dBlock, C √ (163 MPa m). 100
Measured
a (mm)
Fig. 2.11 Grumman centre cracked panel crack growth under a fighter spectra
Predicted
10 240
260
280 Blocks
300
320
The above examples illustrate how the equivalent block method may be used to simulate crack growth under variable amplitude loading both for aluminium alloys and for the materials that exhibit minimal R ratio dependency. However, it must be stressed that this approach has a number of fundamental requirements, viz:
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R. Jones and D. Peng
(i) There are a large number of blocks before failure. (ii) The slope of the a versus block curve has a minimal number of discontinuities. Applications of this methodology to a range of aluminium alloys as well as to cracking under a Helicopter load spectra and spectra corresponding to several control points in the Joint Strike Fighter are given in [8, 11, 27, 28]. White, Barter, and Molent [12] studied block loading which consisted of a large number of variable amplitude loads interspersed with a single block of constant amplitude loading. They found that at the onset of the constant amplitude loading, the crack changed planes and subsequently reverted back to its original plane after the constant amplitude loading ceased. This indicated that the mechanism’s driving constant amplitude and variable amplitude loading differed and that, in the Paris region, the constant amplitude mechanism was suppressed during variable amplitude loading. This observation explains why, in the examples presented above, only one value of C∗ is needed to represent crack growth. At this stage it should be noted that Liu [29] has shown that the Frost–Dugdale law has different slopes in regions I and II. Tiong and Jones [30] revealed that for aluminium alloys the value of C∗ in Region II is approximately 5 times its Region I value. However, when the Region II growth mechanism is suppressed crack growth can be predicted using the C∗ value associated with Region I.
2.4 A Virtual Engineering Approach for Predicting the S–N Curves for 7050-T7451 Section 2.1 when taken together with the cycle-by-cycle study presented by Jones, Molent, and Pitt [4] illustrates the ability of the Generalised Frost–Dugdale law to simulate the growth of near-micron-size flaws in 7050-T7451 aluminium alloy. As a result, it is possible to use this formulation to derive the S–N curve for 7050T7451. To illustrate this approach let us assume that the material contains a small semi-circular surface initial defect and that it retains this semi-circular shape during growth. Then √ ΔK = FΔσ (aπ)
(2.7)
where Δσ is the remote stress, F is a geometry factor, also termed β, which is also commonly called a boundary correction factor. For a small three-dimensional semielliptical surface flaw we can approximate F as F = 2 × 1.12/π
(2.8)
√ ΔK = (2 × 1.12/π)Δσ (a/π)
(2.9)
so that
2 Tools for Assessing the Damage Tolerance
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To account for R ratio effects in aluminium alloys under constant amplitude loading, we will adopt Newman’s [31] proposal that ΔK be replaced by ΔKeff , which for the present problem we can express as √ ΔK eff = (1 − σo /σmax )(2 × 1.12 σmax (a/π))
(2.10)
where σ o is the so-called “crack opening stress”, see Appendix. However, when performing crack growth calculations for surface flaws, FASTRAN-II [31] only uses 0.9 of this value, that is, ΔK eff (in calcs.) = 0.9(1 − σo /σmax )(2 × 1.12 σmax
√
(a/π))
(2.11)
so that √ da/dN = C ∗ (0.9(1 − σo /σmax )(2 × 1.1.2 σmax / π)))γ a
(2.12)
Integrating Equation (2.12) gives √ N = ln(af /ai )/C ∗ (2 × 0.91.12 × (1 − σo /σmax )σmax / (π))γ
(2.13)
where ai is the initial defect size, which as shown by Molent et al. [32], for 7050T7451 aluminium has a mean value of ∼10 microns and af is the crack size at failure.
2.4.1 Computing the Endurance Limit If we say that there will be no growth if the computed value of da/dN (at the initial flaw size ai ) is less than a critical value then this will give an endurance stress. In this work we will take this value to be between 1–2 × 10–10 m/cycle. This produces a different endurance limit for each stress. For 7050-T7451 C∗ = 1.21 × 10–12 and γ = 3.36. The resultant predicted S–N curve is plotted in Fig. 2.12 along with the associated Mil Handbook 5 S–N curve. Note that the yield stress for this material in the thick plate condition is in the range 455–496 MPa (66–72 ksi).
2.5 Conclusion The Australian Defence Science and Technology Organisation’s Flaw IdentificatioN through the Application of Loads (FINAL) testing program revealed that the crack growth programs AFGROW, NASGRO, and FASTRAN cannot be used to accurately predict the growth of near-micron-size flaws in 7050-T7451 aluminium alloy under representative in-flight loading. This paper has shown that the Region II crack growth data reveals that cracking in high-strength aerospace quality steels and
42
R. Jones and D. Peng Predicted R = 0 Predicted R = 0.5 Mil Hndbk R = 0 Mil Hndbk R = 0 Fit Mil Hndbk R = 0.5 data Mil Hndbk R = 0.5 Fit Predicted R = –1 Mil Hndbk R = –1 Mil Hndbk R = –1 Fit
S max (ksi)
100
Predicted endurance limit(s) if we impose the requirement that da/dN must be greater than 1- 210– 10
m/cycle
10 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 N, Cycles Fig. 2.12 Measured and predicted S–N curves for 7050-T7451 aluminium alloy
Mil Annealed Ti–6AL–4V titanium is essentially R ratio independent. As a result, the crack closure and Willenborg algorithm’s available within commercial crack growth codes are also inappropriate for predicting/assessing cracking under operational loading in these materials. To help overcome this shortcoming this chapter has presented an alternative engineering approach that is linked to the formulation developed by the Boeing Commercial Aircraft Company, which can be used to predict the growth of small near-micron-size defects under representative operational load spectra. This approach: i. is generally consistent with experimental results, ii. can be used to predict crack growth from near-micron-size initial flaws, and iii. has the potential to accurately predict crack growth in real aircraft structures under complex load spectra. However, it should be stressed that this variant of the Generalised Frost–Dugdale law is only applicable to crack growth data where the slope of a versus block curve has minimal discontinuities and there are a large number of blocks to failure. In such cases the constant amplitude Region II growth mechanism tends to be
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suppressed and a single value of C∗ can be used to predict the crack length versus cycles history. Acknowledgments This work was performed under the auspices of the DSTO Centre of Expertise in Structural Mechanics which is supported by the RAAF Directorate General Technical Airworthiness Air Structural Integrity Section. We specifically acknowledge the support given by Lorrie Molent, Functional Head Combat Aircraft (Structural Integrity), Dr Weiping Hu, Science Team Leader: Structural Lifing Methods and Tools, Dr. Scott Forth at the NASA Johnson Space Center, Prof. Chris Rodopoulos at the University of Patras, Greece, and the Materials and Engineering Research Centre, Sheffield Hallam University, England, and Dr. Bob Farahmand at TASS (Los Angeles).
Appendix: Formulae for Computing the Crack Opening Stress Newman [31] defined an opening load, which he denoted as S0 , as: S0 /S max = A0 + A1 R + A2 R 2 + A3 R 3 for R ≥ 0
(2.14)
S0 /S max = A0 + A1 = R for R < 0
(2.15)
and
for Smax < 0.8σ0 , Smin > −σ0 , where Smax and Smin are the maximum minimum stress in the cycle and σ 0 is the yield stress. If S0 /Smax is less than R then S0 = Smin , whilst if S0 /Smax is negative then S0 /Smax = 0.0. The A j coefficients in Equations (2.14) and (2.15) are functions of α, the constraint factor, and Smax /σ0 and are given in [31] as: Ao = (0.825 − 0.34α + 0.05α 2 )[COS(π Smax F/2σ0 ]1/α
A1 = (0.415 − 0.071α)Smax F/σ0
A2 = 1−A0 − A1 −A3
A3 = 2A0 + A1 − 1
(2.16)
for α = 1 to 3. The boundary correction factor, F, accounts for the influence of finite width on the stresses required to propagate the crack. For 3D small 3D surface cracks we can approximate F as F ∼ 2 × 1.12/π.
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References 1. M. Skorupa, “Load Interaction Effects During Fatigue Crack Growth Under Variable Amplitude Loading—A Literature Review. Part II: Qualitative Interpretation,” Fatigue Fract. Eng. Mater. Struct., Vol. 22, 1999, pp. 905–926. 2. B. Dixon, L. Molent, and S.A. Barter, “The FINAL program of enhanced teardown for agile aircraft structures,” Proceedings of 8th NASA/FAA/DOD Conference on Aging Aircraft, Palm Springs, 31 Jan–3 Feb, 2005. 3. L. Molent, R. Singh, and J. Woolsey, “A method for evaluation of in-service fatigue cracks,” Eng. Fail. Anal., Vol. 12, 2005, pp. 13–24. 4. R. Jones, L. Molent, and S. Pitt, “Crack growth from small flaws,” Int. J. Fatigue, Vol. 29, 2007, pp. 658–1667. 5. S.C. Forth, M.A. James, W.M. Johnston, and J.C. Newman, Jr., “Anomolous Fatigue Crack Growth Phenomena in High-strength Steel,” Proceedings Int. Congress on Fracture, Italy, 2007. 6. M.N. James and J.F. Knott, “An Assessment of Crack Closure and the Extent of the Short Crack Regime in QlN (HY80) Steel,” Fatigue Frac. Eng. Mater. Struc., Vol. 8, No. 2, 1985, pp. 177–191. 7. R. Jones, B. Farahmand, and C. Rodopoulos, “Fatigue crack growth discrepencies with stress ratio,” Theor. Appl. Frac. Mech., doi: 10.1016/tafmec.2009.01.004. 8. R. Jones, S. Pitt, and D. Peng, “The Generalised Frost–Dugdale Approach to Modeling Fatigue Crack Growth,” Eng Fail Anal, 15, 2008, pp. 1130–1149. 9. R. Jones, B. Chen, and S. Pitt, “Similitude: Cracking in Steels,” Theor. Appl. Frac. Mech., Vol. 48, No. 2, pp. 161–168. 10. D.L. Davidson, “How Fatigue Cracks Grow, Interact with Microstructure, and Lose Similitude,” Fatigue and Fracture Mechanics: 27th Volume, ASTM STP 1296, R.S. Piascik, J.C. Newman, and N.E. Dowling, Eds., American Society for Testing and Materials, 1997, pp. 287–300. 11. R. Jones, L. Molent, and K. Krishnapillai, “An Equivalent Block Method for Computing Fatigue Crack Growth,” Int. J. Fatigue, Vol. 30, 2008, pp. 1529–1542. 12. P. White, S.A. Barter, and L. Molent, “Observations of Crack Path Changes Under Simple Variable Amplitude Loading in AA7050-T7451,” Int. J. Fatigue, Vol. 30, 2008, pp. 1267–1278. 13. J. Schijve, “Fatigue Crack Growth Under Variable-Amplitude Loading,” Eng. Frac. Mech., Vol. 11, 1979, pp. 207–221. 14. J.P. Gallagher and H.D. Stalnaker, “Developing Normalised Crack Growth Curves for Tracking Damage in Aircraft, American Institute of Aeronautics and Astronautics,” J. Aircraft, Vol. 15, No. 2, pp. 114–120. 15. P.C. Miedlar, A.P. Berens, A. Gunderson, and J.P. Gallagher, “Analysis and Support Initiative for Structural Technology (ASIST),” AFRL-VA-WP-TR-2003-3002, 2003. 16. J.M. Barsom and S.T. Rolfe, “Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics,” Butterworth-Heinemann Press, 1999. 17. M. Miller, V.K. Luthra, and U.G. Goranson, “Fatigue Crack Growth Characterization of Jet Transport Structures,” Proc. of 14th Symposium of the International Conference on Aeronautical Fatigue (ICAF), Ottawa, Canada, 1987. 18. R. Jones, and S. Pitt, “Crack Patching: Revisited,” Comp. Struct., Vol. 32, 2006, pp. 218–223. 19. B.J. Murtagh and K.F. Walker, “Comparison of Analytical Crack Growth Modelling and the A-4 Wing Test Experimental Results for a Fatigue Crack in an F-111 Wing Pivot Fitting Fuel Flow Hole Number 58”, DSTO-TN-0108, 1997. 20. R. Jones and S.C. Forth, “Cracking In D6ac Steel,” Submitted J. Theor. Appl. Fract. Mech., 2008 (in press).
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21. E.K. Walker, “The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7076-T6 Aluminium.” In: Effect of Environment and Complex Load History on Fatigue Life, ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp. 1–14. 22. R. Jones, C. Wallbrink, S. Pitt, and L. Molent, “A Multi-Scale Approach to Crack Growth,” Proceedings Mesomechanics 2006: Multiscale Behavior of Materials and Structures: Analytical, Numerical and Experimental Simulation, Porto, Portugal, 2006. 23. C.M. Hudson, “Fatigue-Crack Propagation in Several Titanium and One Superalloy StainlessSteel Alloys, NASA TN D-2331, 1964. 24. T.R. Porter, “Method of Analysis and Prediction for Variable Amplitude Fatigue Crack Growth,” Eng. Fract. Mech., Vol. 4, 1972, pp. 717–736. 25. W. Zhuang, S. Barter, L. Molent, “Flight-By-Flight Fatigue Crack Growth Life Assessment,” Int J Fatigue, Vol. 29, 2007, pp. 1647–165. 26. P.D. Bell and M. Creager, “Crack Growth Analysis For Arbitrary Spectrum Loading,” Volume I – Results and Discussion, Final Report: June 1972 – October 1974, Technical Report AFFDL-TR-74-129, 1974. 27. R. Jones, S. Pitt, and D. Peng, “An Equivalent Block Approach to Crack Growth,” In: Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials: Structural Integrity and Microstructural Worthiness, G.C. Sih, Ed., ISBN 978-1-4020-8519, Springer Press, 2008. 28. L. Molent, S. Barter, and R. Jones, “Some Practical Implications of Exponential Crack Growth,” In: Multiscale Fatigue Crack Initiation and Propagation of Engineering Materials: Structural Integrity and Microstructural Worthiness, G.C. Sih, Ed., ISBN 978-1-4020-8519, Springer Press, 2008. 29. H.W. Liu, Crack Propagation in Thin Metal Sheet Under Repeated Loading, Wright Air Development Center, WADC TN, 1959, pp. 59–383. 30. U.H. Tiong and R. Jones, “Damage Tolerance Analysis of a Helicopter Component,” Int. J. Fatigue, 2008 doi:10.1016/j.ijfatigue.2008.05.012 31. J.C. Newman, Jr., FASTRAN-II- A fatigue Crack Growth Structural Analysis Program, NASA Technical Memorandum 104159, 1992. 32. L. Molent, Q. Sun and A.J. Green, “Characterisation of equivalent initial flaw sizes in 7050 aluminium alloy,” Fatigue Fract. Engng. Mater Struct., Vol. 29, 2006, pp. 916–937.
Chapter 3
Cohesive Technology Applied to the Modeling and Simulation of Fatigue Failure Spandan Maiti
Abstract Estimation of fatigue and fracture properties of materials is essential for the safe life estimation of aging structural components. Standard ASTM testing procedures being time consuming and costly, computational methods that can reliably predict fatigue properties of the material will be very useful. Toward this end, we present a computational model that can capture the entire Paris curve for a material. The model is based on a damage-dependent irreversible cohesive failure formulation. The model relies on a combination of a bilinear cohesive failure law and an evolution law relating the cohesive stiffness, the rate of crack opening displacement, and number of cycles since the onset of failure. Threshold behavior of the fatigue crack propagation is determined by the initial value of the damage parameter of the cohesive failure law, while the accelerated region is the natural outcome of the cohesive formulation. The Paris region can be readily calibrated with the two parameters of the proposed cohesive model. We compare the simulation results with the NASGRO material database, and show that the threshold region is adequately captured by the proposed model. We summarize a semi-implicit implementation of the proposed model into a cohesive-volumetric finite element framework, allowing for the simulation of a wide range of fatigue problems.
3.1 Introduction Fatigue failure of materials has been a topic of investigation for more than 150 years till date, but still complete physical understanding of this failure process is not understood. The critical aspect of fatigue failure is that it occurs at driving force levels less than that required for monotonic failure of the same material. Fatigue crack propagation behavior for materials shows four different fatigue S. Maiti (B) Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 3,
47
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allowables needed to quantify fatigue crack behavior for any material, that is, threshold limit (ΔKth ), slope (m), intercept (C), and accelerated crack propagation limit (KIC ). Under the defect tolerant design for fatigue failure, all components are assumed to be initially flawed creating stress concentration under loading that might lead to failure. Thus, for designing components for infinite life, stress concentration produced by largest flaw in the component should be kept lower than the threshold limit for fatigue crack propagation. Critical parts in airplanes like fuselage are periodically tested for the largest crack length present in them. Knowledge of the fatigue allowable is necessary for the calculation of critical crack length below which crack growth rate will not be dangerously fast to call the part unusable as well as estimation of remaining life of that part. Currently in the industry, experimental analysis are carried out to determine this fatigue allowable; but experimental procedures are generally expensive and time consuming and are not feasible while doing design revisions or using mathematical techniques like optimization. Computational models for fatigue failure with predictive capabilities would appreciably help in such scenarios. Particularly in case of threshold behavior, which is critical for the life estimation of a component, a quick and robust model will be of immense help. However, numerical methods currently available treat crack initiation and crack propagation regimes under separate mathematical models even though they are the part of the same phenomenon [1]. The main focus of this chapter is to present a unified model for the prediction of stage I crack initiation and stage II and stage III crack propagation under the same numerical scheme. To do so, a fatigue crack growth simulation scheme using cohesive elements for stage II and stage III crack propagation has been modified to be used for the study of fatigue crack propagation in metals, and crack initiation (stage I) prediction capability was added to it [2]. In the current study, we are particularly interested in the predictive modeling of fatigue crack behavior in various materials important to aerospace industry. In organization of this chapter, the second section discusses various modeling efforts undertaken in the literature. Section 3.3 presents our modeling framework with implementation details. Fatigue crack growth simulation results along with a parametric study are presented in Section 3.4. We conclude our discussions in Section 3.5. With help of the developed unified model for fatigue crack propagation, we show that the presented model can predict the complete fatigue crack propagation curve including initiation and propagation regimes for various alloys successfully. We relate the threshold for the fatigue crack propagation with fracture toughness of the material with a new parameterSinit . Comparing the predicted value of Sinit with the experimentally evaluated values of lower threshold and fracture toughness of a number of aerospace alloys, we find that it varies between 0.98 and 1.0 for most of the materials studied.
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3.2 Background The steps to fatigue failure can be typically described as (i) accumulation of permanent damage at the microstructure level, (ii) creation of microscopic cracks at the damages sections, (iii) growth and coalescence of these microcracks to form dominant crack, (iv) stable propagation of the dominant crack, and (v) final complete failure [3]. Any numerical model that aims to predict fatigue crack growth behavior should incorporate these steps into the model. One of the oldest and most used numerical techniques for fatigue crack propagation is Paris law [4]. This method is phenomenological in nature and can successfully predict fatigue crack growth in stage II, known also as Paris region. However, the original Paris law is not useful for the prediction of fatigue crack growth in stage I or stage III. Even with subsequent modifications with a number of researchers [5], the inherent problem with this methodology remains in its phenomenological nature and consequent lack of true predictive capability.
3.2.1 Models for the Prediction of Threshold Fatigue Crack Behavior Observation of crack closure at the low-stress-intensity levels has given rise to a number of micromechanical models for the prediction of threshold behavior of metals [6–9]. Basic assumption in these models is that materials do not possess any intrinsic lower threshold: presence of the lower threshold is solely due to crack closure. But with presence of fatigue crack threshold even in vacuum, it has been realized that crack closure may not be the only reason for the presence of threshold. Weertman’s [10] model for prediction of lower threshold was based on the rupture energy of atomic bonds in metal crystals. He argued that the slowest possible crack growth will be of the order of the interatomic distance per cycle and found the following relationship for the threshold: ΔK th ≈ βGb1/2
(3.1)
with ΔK th as the threshold stress intensity range, G as the shear modulus, and b as the interatomic distance. The parameter β in the original model was taken to be the order of unity. To compare this model with experiments, β has been computed for five different aluminum alloys (Table 3.1). Aluminum has a cubic closed-packed structure and the interatomic distance for an aluminum crystal is 404.95 pm. The value of β is calculated as a ratio of experimental Kth and the value predicted from the above model [11]. It can be seen that β is about 5 for all the cases chosen. The application of discrete dislocation dynamics to the problem of fatigue threshold calculation was presented by Deshpande and Needleman [12–15]. Along with discrete dislocation dynamics, reversible and irreversible cohesive laws were used by the authors to simulate fatigue threshold value for metals. They were also
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S. Maiti
Table 3.1 Evaluation of Weertman’s model for threshold prediction. Experimental values are taken from the NASGRO database [11] Material (Aluminum alloys)
Shear modulus (Pa)
6061-T62 2014T6 6061-T6 2024-T3
2.59E+10 2.72E+10 2.59E+10 2.75E+10
KIc (Pa-m1/2 )
Kth (from NASGRO) (Pa-m1/2 )
Predicted Kth = Gb∧ 0.5 (Pa-m1/2 )
β = Experimental/ predicted threshold
2.55E+007 3.00E+007 2.70E+007 2.90E+007
2.70E+006 2.60E+006 3.00E+006 2.90E+006
5.21E+05 5.48E+05 5.21E+05 5.53E+05
5.18 4.75 5.76 5.24
successful in demonstrating the effect of various parameters: loading parameters such as load ratio, tensile overloads, and microstructural parameters such as obstacle density and slip geometry. Recently, Farkas et al. [16], for the first time, simulated fatigue crack growth using atomistic simulations for nanocrystalline materials. They were successful in simulating the Paris curve and predicting lower threshold value to a good degree of accuracy.
3.2.2 Models for the Prediction of Fatigue Crack Propagation Finite elements codes incorporating fracture mechanics concepts have been a natural choice for the simulation of fatigue crack growth. A number of finite element studies have been performed to simulate the fatigue crack propagation in stages II and III [17]. However, most of these models accomplish the fatigue crack growth in an element-by-element manner by releasing nodes once in each cycle [18–20]. These models do not incorporate the damage event occurring at the crack tip, as described earlier, and thus may not represent the physics involved in a sound manner. Recently, cohesive modeling techniques suitable for fatigue crack propagation have appeared in the literature. These techniques are appealing compared to the earlier techniques due to their capability to incorporate progressive damage of the material naturally into the modeling framework. These models are also capable of predicting arbitrary crack propagation, branching, and coalescence in a solid domain, thus representing the physical phenomenon closely. Cohesive modeling technique has been applied for fatigue crack growth in metals [12–15, 21, 22], along interfaces [23] and in quasi-brittle materials [24]. In one of the early models proposed in [21], no distinction was made between the loading and unloading paths, but a damage parameter was assumed. The evolution of this parameter with the number of load cycles was prescribed explicitly in the model. The presence of plasticity in the bulk material around the crack tip influenced the crack closure and hence the failure of the material. But it was found that the crack ceases to grow after a few cycles due to plastic shakedown [22]. It has since been identified that a distinction needs to be made between the loading and unloading paths allowing for hysteresis so that subcritical crack growth becomes possible. Nguyen
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and co-workers have worked out a one-parameter cohesive model for metals [22], which is able to capture experimental Paris curves quite well and, in particular, the slope m of the curves (equal to approximately 3 for most metals). In the irreversible cohesive model mentioned in [12–15], the unloading path is assumed to be parallel to the previous loading path, thus leaving certain amount of residual separation in the cohesive zone after each cycle. It is argued that the environment-assisted oxidation of the crack faces can give rise to this kind of behavior. But this type of crack closure effect is not very common in polymers. The fatigue model for interface cracks presented by Roe and Siegmund [23] is based on damage mechanics, where a history-dependent damage parameter gives rise to irreversibility. In this model also, the unloading path is not toward the origin. Values of the slope m of the Paris curves up to 3.1 have been reported in that study. Fatigue crack growth in quasibrittle materials has also been studied by cohesive techniques in [24], where the irreversibility of the loading and unloading paths is taken into account. A polynomial expression for the cyclic behavior is postulated in that study. A special case of Paris law, where the multiplicative constant is functionally dependent on the maximum loading, has been reported by these authors. Maiti and Geubelle have proposed a two-parameter cohesive model for the simulation of fatigue crack growth in stages II and III for polymers [25]. This particular model has been successfully used to study the crack closure behavior for polymeric materials [26]. A multiscale modeling technique for self-healing polymers has also been suggested by these researchers [27]. It becomes clear from the previous discussion that most of these models, barring a few, treat the threshold regime and fatigue crack propagation regime in separate computational frameworks. Models capable of simulating stage I and stage II in a single framework [12–16] are typically computationally costly, and may not be able to perform simulations up to stage III of fatigue crack propagation. We outline a cohesive methodology-based unified model capable of predicting all the stages of fatigue crack propagation in subsequent sections.
3.3 Cohesive Modeling Technique The first model for the computation of crack extension in solids was postulated by Griffith [28]. He postulated that when a crack passes through a body, due to the effect of applied stresses, it causes decrease in the potential energy of the system. This decrease in the potential energy is influenced by the displacement of the outer boundaries and changes to the stored elastic energy. The decrease in the potential energy should be balanced by the increase in the surface energy released due to crack extension. Later, Dugdale [29] envisioned a narrow strip of plastic region ahead of the crack tip which is of near-zero thickness and extends to a distance rp ahead of the crack tip. The damage resulting into new fracture surface is supposed to be happening exactly in front of the crack tip in this strip-like zone. This particular assumption eliminated the presence of an infinite stress at the crack tip, as predicted
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S. Maiti
by linear elastic fracture mechanics (LEFM), and provided a structure to the crack process zone. As per his model, the process zone rp ahead of the crack tip can be calculated as rp =
π 8
kI 2 , σy
(3.2)
where KI is the stress intensity factor in the opening mode and σ y is the yield stress of the material. The model also assumes the presence of an opening displacement δ = 2v at the crack tip. Figure 3.1 conceptually describes Dugdale’s strip yield model.
σ farfield
Fig. 3.1 Dugdale’s strip failure model
Crack Tip σy
2v Cohesive Zone Tip
Cohesive Zone
σ farfield
The strip yield model, developed by Barenblatt [30], is analogous to the Dugdale model. Barenblatt added that the crack proceeds when the crack face traction σ y reaches a critical value σ th where σ th is the theoretical bond rupture strength, and the cohesive zone size reaches a critical value rco. The energetic relationship for fracture can be expressed in terms of critical cohesive zone size rco or critical crack opening displacement δc = 2νc :
νc
Gc = 2 0
σ y dv =
8σth 2rc0 = 2γs πE
(3.3)
Idealistic scenario of Barenblatt’s strip yield model can be seen in Fig. 3.2, where the dark strip of material (as shown in the figure) is the strip of material with nearzero thickness undergoing failure process.
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53
Fig. 3.2 Barenblatt’s strip yield model
3.3.1 Reversible Cohesive Model From the models of Dugdale and Barenblatt, a computational scheme for failure of materials has been formulated [31, 32] which is generically known as the cohesive failure methodology. The failure of any material under this theory is governed by two parameters, cohesive traction σ max and the critical crack opening displacement Δnc . Here σ max is analogous to the σ th , that is, the bond rupture strength, and Δnc is analogous to the critical crack opening displacement δc from the Barenblatt model. For explanation purpose consider a cracked body put under loading perpendicular to the crack length as seen in Fig. 3.3. Let the far field stress be σ . Because of the presence of geometric discontinuity in the domain in the form of the crack, there will be a stress concentration K in front of the crack tip. Hence the yield strip (cohesive zone) ahead of crack tip will be experiencing a traction Tn that is much higher that
σ
Crack Tip Cohesive Zone Tip
Δn Δnc
Tn
a Cohesive Zone Length σ
Fig. 3.3 Cohesive failure modeling scheme
54
S. Maiti
far field stress σ . Now as per Dugdale and Barenblatt strip yield model, this traction will create a cohesive crack opening displacement Δn at the crack tip. The cohesive opening Δn is a function of Tn and the material properties for that material. Once the crack opening displacement reaches a critical value Δnc, the traction on the surface vanishes and new cracked surface is created with complete failure of the material in question. It is worthy to note that if crack is unloaded without crack opening displacement reaching Δnc, failure will not occur and the crack will not advance. The crux of the cohesive methodology of crack propagation is the relationship between the traction acting on the cohesive surface and the displacement jump between two opening sides. The traction–separation law, also known as the cohesive law, can be derived from a potential function. A number of cohesive laws have been postulated that differ in the description of the envelope of the traction–separation relationship. For all these cohesive laws, energy U spent for the creation of new fracture surface is given by U=
Δnc
Tn dΔn
(3.4)
0
Where Tn is the traction on the cohesive surface in the cohesive zone and Δn is the crack opening displacement over the cohesive zone and are integrated over cohesive process zone length rp . For implementation into finite element scheme, cohesive elements are introduced at the boundary of volumetric elements where failure is expected. The cohesive elements allow spontaneous initiation and propagation of crack through them when enough amount of energy is supplied to them. The cohesive elements act like nonlinear springs during the failure process, where the springs resist opening under applied load. The resistance of the springs is instantaneous stiffness of that cohesive element. This stiffness is governed by the specific traction–separation relationship. Eventually the opening displacement reaches the critical value and the traction provided by the cohesive element reduces to zero, thus creating two separate bodies on each side of the cohesive element.
3.3.2 A Bilinear Cohesive Law The bilinear rate independent but damage-dependent cohesive model discussed by Geubelle and Baylor [33] is used as the base cohesive model for this study and hence will be explained in details here after. It couples the normal and tangential components of traction and also allows relatively easy implementation. The damage achieved during any phase of simulation is remembered through a special ‘damage parameter’ S defined by ˜ 2 S = 1 − |Δ|
(3.5)
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˜ is defined as following Where | |2 denotes Euclidean norm and Δ ˜ = Δ
˜n Δ ˜t Δ
=
Δn /Δnc Δt /Δtc
with subscripts n and t denoting normal and tangential components, respectively. The strength parameter S is initially assigned to an initial value Sinit . As the tractions are applied on the cohesive elements, S monotonically decreases from the initial value to zero. The permanence of damage is achieved by storing the minimum value of S achieved so far in the simulation. The expression for S is as follows: ˜ 2) . S = min S pr ev , max(0, 1 − |Δ|
(3.6)
The cohesive traction–separation law in mode I for this particular model takes the following form: Tn =
S Δn σmax 1 − S Δnc Sinit
(3.7)
where σmax is the failure strength of the cohesive element and Δnc is the critical normal displacement jump. Similar expression can be derived for the tangential traction and separation as well. The traction separation law is depicted in Fig. 3.4.
Fig. 3.4 Bilinear traction–separation law for cohesive failure
The energy expended for failure is area under the traction–separation law envelope curve which can be equated to fracture energy for that material so that GIC =
1 σmax Δnc . 2
(3.8)
Since the damage is stored in parameter S, the path traced on reloading takes into consideration the energy already spent for partial damage achieved during first
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S. Maiti
loading phase. Hence, even though the failure does not occur in one loading phase, the energy used for failure will always be equal to G I C . Generally, KIC , critical stress intensity factor is used for characterization of fracture properties for any material. KIC is related to GIC through following relationship:
GIC =
K I2C . E
(3.9)
Here E is the Young’s modulus of the material. Typically for metals, σmax is taken as the yield strength of the material and Δnc is calculated as follows: Δnc =
2G I c . σmax
(3.10)
3.3.3 A Cohesive Model Suitable for Fatigue Failure Although it prevents healing of the fracture surfaces through the enforcement of monotonic decay of the damage parameter, the cohesive model discussed in the previous section leads to similar unloading and reloading paths in the traction– separation curve. This behavior is illustrated in Fig. 3.4. This characteristic prevents crack growth under subcritical cyclic loading due to the progressive degradation of the cohesive properties in the cohesive failure zone. This limitation suggests the need for an evolution law to describe the changes incurred by the cohesive strength under fatigue. The cohesive model for mode I fatigue, as developed by Maiti and Geubelle [25], is developed with some modification to the monotonic cohesive crack propagation model. The main difference between the monotonic crack propagation and fatigue crack propagation is the method for application of load and total energy supplied to the system. For fatigue crack propagation the load is applied in sequence of loading and unloading patterns. The load applied is not large enough to create stress intensity factor equal to or more than the stress intensity factor KIc required for monotonic crack propagation. In addition, the energy required for fatigue crack propagation is typically less than its quasi-static counterpart. The mechanism for fatigue crack propagation, as explained by Ritchie [34], reveals that the crack propagation below the fracture toughness value for any ductile solid on a simplistic level involves cyclic damage accumulation due to cyclic plastic deformation in the plastic zone ahead of the crack tip. In implementation of cohesive law for fatigue crack propagation, generally few elements ahead of the crack tip as undergoing cyclic fatigue failure which can be related to the plastic zone ahead of the crack. The evolution law for the instantaneous cohesive stiffness kc , that is, the ratio of the cohesive traction Tn to the displacement jump Δn during reloading can be expressed in the general form as [25]
3 Cohesive Technology Applied to the Modeling and Simulation
Kc =
57
dTn = F(N f , Tn ). dΔn
where Nf denotes the number of loading cycles experienced by the material point since the onset of failure, that is, at the time cohesive traction Tn first exceeded the failure strength σ max . Maiti and Geubelle adopted a separable form [25]: dTn = −γ (N f )Tn dΔn
(3.11)
leading to an exponential decay of the cohesive strength, with the rate of decay controlled by the parameter γ . In the present study, we use the following two-parameter power–law relationship [25]: γ =
1 Nf a
−β
(3.12)
where α and β are material parameters describing the degradation of the cohesive failure properties. Here α has the dimension of length and β denotes the history dependence of the failure process. The proposed evolution law for the cohesive model can also be expressed as 1 ˙n >0 ˙ n if Δ K˙ c = − (N f )−β K c Δ a ˙n ≤0 = 0 if Δ ˙ n is rate of change of normal separation. where Δ With these modifications, the traction separation law during cyclic loading can be described with 3.5. The cohesive law during the first loading cycle behaves like monotonic loading. Once the traction goes beyond σmax , the process of damage accumulation begins at that location. In the next loading cycle the evolution law for the cohesive stiffness will take effect and the stiffness at the integration point will reduce as shown in Fig. 3.5. The amount of degradation in the cohesive stiffness is dictated by (3.13), and can be calibrated with the actual physical damage occurring ahead of the crack tip. Observe from this figure that no stiffness degradation occurs during the unloading process. So, the developed model incorporates the physics of the fatigue crack growth into its formulation, and is expected to exhibit good predictive capability. Another important observation is that the cohesive model does not explicitly include any particular set of material properties, thus enabling it to simulate fatigue crack growth in a wide array of materials. Material properties are embedded only in the volumetric finite elements so that the specification of relevant constitutive law is required only for these elements. Cohesive elements include only the failure properties of the material that can be evaluated separately. Though the original model was developed for polymeric materials, we will show that it is equally successful in capturing fatigue crack growth behavior in metals and alloys through a proper calibration of the model parameters. Finally, note that the cohesive
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S. Maiti
Fig. 3.5 Behavior of the traction–separation law for a typical fatigue crack growth simulation scenario
stiffness gradually reduces to zero, at which point the fatigue crack is said to have propagated through the integration point under observation. This event is not dictated by any pre-mediated cycle-by-cycle node release procedure; it rather depends on the loading and geometric conditions as well as the material properties of the physical system under study. Typically, the displacement jump at this event does not reach the critical displacement jump Δnc as to be expected for the crack propagation under quasi-static loading. Also observe that the energy expended in the process, the area under the traction–separation law, is much lesser than that for the monotonic case. This observation is critical for the success of irreversible cohesive models, as this particular property of this class of models enables them to simulate subcritical crack growth.
3.3.4 Incorporation of Threshold Behavior In contradiction to the closure-based theory for lower-threshold prediction, in the current formulation, it is assumed that all materials show an intrinsic lower threshold. This is particularly true as lower threshold for various metals has been even seen in vacuum [35] which cannot be explained with closure-based theory. Stress intensity factor applied below this intrinsic value will not produce crack initiation. As discussed earlier, the cyclic stiffness degradation of cohesive element does not start till the traction on it reaches σmax . Hence, as a consequence, if the loading on the geometry is such that the traction on any cohesive element is less than σmax at any point of time in loading cycle, there will not be any degradation of the cohesive
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59
stiffness. So the loading and unloading path for this type of loading will be following the initial loading path for the cohesive element for infinite number of cycles, that is, the fatigue crack will not initiate under the current scheme of simulation. The maximum value of loading on a given geometry at which the fatigue crack will not grow (traction is less than σmax ) will give us the lower threshold for fatigue crack propagation. For detection of threshold value, crack opening displacement corresponding to the cohesive strength σmax is monitored continuously. Loading at which the crack opening displacement reaches Δnth is the threshold load value for a particular geometry. Threshold stress intensity factor can be calculated from the knowledge of threshold loading and geometry of the specimen. The equation for traction under cohesive scheme is given by Tn =
S Δn σmax 1 − S Δnc Sinit
Now to find out expression forΔnth , we put Tn equal to σmax . Also note that the value S is still Sinit as the traction is less than σmax . Hence, putting these values in above equation we get Δnth = (1 − Sinit )Δnc
(3.14)
Hence to detect the achievement of threshold value, crack opening displacement at all integration points is checked in the simulation code. The simulation starts with a low value of load applied on the DCB specimen, this load is incrementally increased till the threshold is reached. A close examination of (3.14) reveals that the threshold value depends on two parameters: the critical displacement jumpΔnc , which in turn depends on the fracture toughness of the materials, and the parameter Sinit . The last parameter controls the location of the peak of the traction–separation curve. So, it can be said that the peak of the cohesive law envelope is critical for the determination of the threshold limit of fatigue crack propagation. We will show later that the value of Sinit is quite close for a number of materials.
3.3.5 Finite Element Implementation The cohesive model described above is readily implemented in a finite element framework, normally called the cohesive volumetric finite element (CVFE) scheme, using the principle of virtual work:
Ω
S : δEdΩ −
Γex
Tex δudΓex −
Γc
Tn δΔn dΓc = 0
(3.15)
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where u is the displacement vector, S and E denote the internal stress and strain tensors, respectively, Tex is the externally applied traction, and Ω, Γe , and Γex , respectively, denote volume, cohesive boundary, and exterior boundary of the deformable body. The last term corresponds to the virtual work done by cohesive traction Tn for a virtual separation of δΔn . The expression for internal component of virtual work depends upon the type of volumetric element used in the analysis, which in turn affects the expression used for calculation of internal stress S and virtual strain δE. For general implementation of cohesive elements into the finite element scheme, these elements are modeled as if they are sandwiched between two volumetric elements between which the failure is supposed to occur. In addition, these elements do not have any thickness associated with them to start with. For the cohesive formulation used here, the cohesive elements are inserted before the simulation, pre-specifying the flow of the crack through the material. As discussed earlier, the traction at an integration point is given by (3.7). Value of S does not degrade till the traction value reaches σmax , which is the threshold of fatigue crack propagation as explained in the last section. Crack opening displacement at a particular cohesive integration point is calculated throughout the simulation and is checked against Δnth . If the crack opening displacement is more than Δnth , damage parameter S is evolved using (3.6). Minimum value of S achieved at each integration point is stored and new calculated value at each step in cycle is checked against previously achieved value. The expression for cyclic cohesive stiffness degradation, as given by (3.13), has been discretized for the finite element implementation using two different methods, namely, forward difference method and Tustin transformation. Forward difference method was used in earlier studies by Maiti and Geubelle [25–27]. But this particular method is stable only at very small time-step values. With the required reduction in the time steps, total time required for solution becomes very high. Since the simulations done here are cycleby-cycle simulations for the entire fatigue crack life, increase in the number of load steps per cycle increases the total time considerably. Tustin transformation is known for its stability so that larger time steps can be used. However, too large time steps can compromise the accuracy of the solution considerably. A convergence study was undertaken by us to find the stable yet accurate time step for both the methods. Our study shows a significant increase in the acceptable time step for the Tustin transform. A quasi-implicit load stepping scheme has been used to form equations of equilibrium in this investigation. The time increment at each step is calculated from the frequency of loading and number of steps per cycle. The discretization of the cohesive degradation expression with the forward difference method is as follows: K cj+1
1 j β j+1 1 σmax sj j 1− Nf Δn − Δn . = 1 − s j Δnc sinit α
Δnj+1 ≥ Δnj .
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With Tustin transform the expansion is as following K cj+1
=
2α − N j
−β
2α + N f
−β
(Δn+1 − Δn ) (Δn+1 − Δn )
K cj .
Crack location is detected by finding out the coordinate of the newly completely failed cohesive element (S = 0).
3.4 Simulation Results Finally, we are in a position to simulate the fatigue crack growth behavior in different materials by the developed model. We restrict our simulations to a simple double cantilever beam (DCB) arrangement. Due to symmetry of the problem on hand, only the half portion of the DCB specimen is modeled. The calculation of crack opening displacement for cohesive elements is taken care of accordingly. The length of the DCB specimen is taken to be 150 mm with an initial crack of 50 mm. The length of the cohesive elements depends on the materials properties [25], and has been calculated accordingly.
3.4.1 Paris Curve Simulation First we show the capability of the presented cohesive model in simulating all three stages of the Paris curve. For this purpose, the simulations are run for two Aluminum alloys 6061-T62 and 2024 T3 and one Titanium alloy Ti-2.5Cu. The material properties for these alloys are as given in Table 3.2, and are taken from NASGRO database [11]. Table 3.2 Material properties for simulated alloys Material
Young’s Modulus Yield stress (GPa) KIc (MPa-m1/2 ) (MPa)
GIc =
6061 –T62 Al 2024 T3 Al T1-2.5Cu STA
68.9 73.1 105
9437.59071 J 68.4 μm 11504.788 J 69.6 μm 19716.6667 J 59.0 μm
25.5 29.0 45.5
276 331 668
K I cz Z
Δnc =
2G I c σmax
The experimental values for fatigue properties for these three materials taken from NASGRO database [11] are provided in Table 3.3. All the simulations were performed for a frequency of loading of 1 Hz while the load ratio was kept constant at R = 0. Number of steps per cycle was set to be 50 for the load stepping scheme. The output for crack extension versus number of cycles for one of the materials, Al 6061-T62, is shown in Fig. 3.6.
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S. Maiti Table 3.3 Experimental values of fatigue properties of the alloys described in Table 3.2 Material
ΔKth (from NASGRO) (MPa)
Intercept C (m)
Slope n
6061 -T62 Al 2024 T3 Al T1-2.5Cu STA
2.70 2.90 4.60
5.50E-010 8.4E-010 2.5E-10
2.8 2.75 2.6
Fig. 3.6 Crack extension versus number of cycles for Al 6061-T62
Notice that the cohesive model does not possess any in-built assumption about the Paris curve of the material. The crack extension data was post-processed to derive the Paris curve and is presented in Fig. 3.7. The slope of the Paris curve from the simulation is 2.82 and the intercept is 4.4e-10 m, which is very close to the experimental values. In addition, we can note that the threshold limit as well as the accelerated region is also very well captured by the simulation. Two thick vertical lines at the bottom of the plot denote these two limits for this particular alloy. The values of α and β used for this simulation are 8 μm and 0.1, respectively, while Sinit was kept constant at 0.988. Figure 3.8 shows the simulated Paris curve for 2024 T3 aluminum alloy. The Paris curve for the Titanium alloy Ti–2.5Cu is depicted in Fig. 3.9. It can be observed from these plots that the limits are very well captured as was the case with the first material. The simulation values of slope and intercept for various alloys are tabulated in Table 3.4. On comparison with the experimental values from NASGRO database [11] in Table 3.5, we can see that the results of the simulation are fairly accurate even for the Ti–2.5Cu, which has far higher values of fracture toughness, yield strength, and Young’s modulus compared to two other alloys under study. This exercise shows that the model can be used for various types of alloys regardless of their material properties. Our model can predict the fatigue crack growth in all
3 Cohesive Technology Applied to the Modeling and Simulation Fig. 3.7 Paris curve for Al 6061
63
da/dN (mm/cycle)
10–2
10–3
10–4
10–5 0 10
101
102
ΔK (MPa-(m)1/2) Fig. 3.8 Simulated Paris curve for 2024 T3 Al alloys
da/dN (mm/cycle)
10–2
10–3
10–4
10–5
101
102
ΔK (MPa-m1/2)
three stages for a number of materials in a single computational framework. The model calibration parameters α and β are shown in Table 3.6. These parameters can also be evaluated from fatigue crack growth experiments for stage II. A look at the numerical values of these parameters shows that while the parameter α can change appreciably for different material, β keeps an almost constant value. As
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Fig. 3.9 Simulated Paris curve for Ti–2.5Cu STA alloy
da/dN (mm/cycle)
10–2
10–3
10–4
10–5 0 10
Table 3.4 Simulation values of slope and intercept for the three materials described in Table 3.2
Table 3.5 Experimental values of slope and intercept for the three materials described in Table 3.2. [11]
101 Δ K (MPa-m1/2)
102
Material
ΔKth (MPa)
Intercept C (m)
Slope m
6061 -T62 Al 2024 T3 Al T1-2.5Cu STA
2.70 2.90 4.60
5.0E-010 8.3E-010 2.35E-10
2.8 2.75 2.65
Material
ΔKth (MPa)
Intercept C (m)
Slope m
6061 -T62 Al 2024 T3 Al T1-2.5Cu STA
2.70 2.90 4.60
5.0E-010 8.3E-010 2.35E-10
2.8 2.75 2.65
shown by Maiti and Geubelle [25], the first parameter is responsible for the intercept of the Paris curve while the second parameter changes its slope. As the slopes of the experimental Paris curves for all the alloys are very close, the parameter β is also similar in magnitude for all the cases. The wide variation in the intercept of the experimental Paris curves is reflected by the variation in the parameter α in Table 3.6.
3 Cohesive Technology Applied to the Modeling and Simulation Table 3.6 α and β values used for the simulation of three materials listed in Table 3.2
65
Material
α
β
6061 -T62 Al 2024 T3 Al T1–2.5Cu STA
8 μm 13 μm 17 μm
0.1 0.105 0.18
3.4.2 Prediction of Threshold Limit of Fatigue Crack Growth In this section, we turn our attention exclusively to the simulation of threshold limit for different aerospace alloys. As per the procedure explained previously, threshold values for few aluminum alloys are evaluated. The parameter Sinit is chosen to be 0.9828 for all the simulation cases. As we can see from Fig. 3.10, the values predicted by the simulation as very close to the actual experimentally evaluated values except for 6061-T6 GTA weld. For rest of the materials, the simulated lower threshold value is within 10% of the experimental value. Hence with the input values of material stiffness, fracture toughness and yield stress, and one model parameter, Sinit , lower threshold stress intensity factor for most of the materials can be simulated using this model. A proper calibration of this parameter produces simulated threshold limits still closer to the experimental values, as listed in Table 3.7. For the outlying value of 6061-T6 plt:GTA weld, the
Fig. 3.10 Simulated threshold stress intensity range for different aluminum alloys
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S. Maiti Table 3.7 Simulated and experimental lower threshold values with other material properties
Material 6061 -T62 Al 2014T6 6061-T6 plt; T-L 6061-T6 plt;GTA weld 2024-T3clad plt sht t-l DW
Young’s modulus (GPa)
Kic (MPa-m1/2 )
Yield Stress ΔKth (NASGRO) (MPa-m1/2 ) (MPa)
Simulated ΔKth (MPa-m1/2 )
Sinit
68.9 72.4 68.9
25.5 30.0 27.0
276 414 282
2.7 2.6 3
2.46 2.72 2.76
0.989 0.992 0.988
68.9
27.0
158
4.5
3.20
0.972
73.1
29.0
331
2.9
2.79
0.990
difference in the simulated and experimental value could be due to effect of parameters like material microstructure which are explicitly not taken into consideration in this model.
3.4.3 Effect of on the Threshold Limit Fracture energy for the material is embedded into the cohesive law such that GIC =
1 σmax .Δnc 2
We assume that the energy required to start a fatigue crack has a threshold value below which no crack propagation is possible. We further assume that this “minimum energy” G th is an intrinsic property of the material. An expression for this energy can be found from the observation of the cohesive traction–separation curve, and can be expressed as G th =
1 σmax .Δnth 2
But as seen previously from (3.14), Δnth = (1 − Sinit )Δnc Combining these equations, G th = (1 − Sinit ). GIC
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67
But G th = ΔK th 2 /E (at R = 0) and G I C = K I C 2 /E Hence we get ΔK th 2 = (1 − Sinit ). KIC2
(3.16)
Now let us look at the data from NASGRO database [11] for threshold values of different aerospace materials. We can calculate the Sinit value for various materials from the above derived equation. Figure 3.11 shows the plot for ΔKth /KIC for close to 200 aerospace materials taken from the NASGRO database.
Fig. 3.11 ΔKth /KIC for various materials calculated from NASGRO database
It is interesting to observe from Fig. 3.11 that except for just a few materials, most of the materials have Sinit values ranging between 0.98 and 1.0. We observe a very tight bound on the parameter, Sinit . So, this particular parameter can be taken as a material constant, valid at least for aerospace alloys. In the absence of detailed experimental analysis, a choice of Sinit close to 1.0 will yield satisfactory value of the fatigue crack propagation threshold for these materials. This observation points to the ability of our model to predict the fatigue lower threshold stress intensity range for most of the materials relevant for aerospace applications. Equations (3.14) and (3.16) are the key results presented in this discussion; (3.16) relates the experimental fatigue threshold stress intensity range with the model
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parameter Sinit with a very tight bound for this parameter over a range of materials, whereas (3.14) shows that this particular parameter can also be related to another set of failure properties, namely crack opening displacement (COD). There exist a number of analytical approaches to relate the crack tip opening displacement to the geometry of the specimen, and the external loading condition. So, enumeration of COD, and consequently Sinit , can be performed in a computational framework without taking recourse to further experiments.
3.4.4 Effect of Load Ratio R on Fatigue Crack Threshold Experimental results on the effect of load ratio R on the threshold value normally exhibit a large scatter. But the general trend is that the threshold limit decreases linearly with R for most of the materials. Blacktop and Brook [36] noted that threshold value should touch zero when the load ratio assumes a value of 1. On the other hand, other researchers have observed a leveling off of the threshold value at higher load ratio [37]. Mechanistic explanation for this effect is [38] a. Presence of crack closure at low values of stress intensity Kmin, b. Presence of static fracture modes as the maximum stress intensity Kmax approaches fracture toughness for the material. When only the effect of crack closure is taken into consideration, Schmidt and Paris [39] hypothesized that the threshold value will reduce with load ratio till the minimum stress intensity Kmin is less than the crack closure stress intensity Kcl . After some critical value of R (beyond which Kmin > Kcl ) effect of load ratio on the fatigue threshold limit will not be observed any more. Oxide-induced crack closure was cited as one of the mechanisms resulting in the effect of load ratio [40]. Boyce and Ritchie [38] also confirmed these results for Ti–6Al–4 V alloy, where they observed that there is heavy reduction in ΔKth till R = 0.5 but after that point the slope of ΔKth vs. R curve does not become zero. They tried to explain this phenomenon by predicting the presence of sustained load cracking mechanism. They also cited observations of Davidson [41] who reported the presence of closure in the local region of crack tip (within ∼10 μm). Our computational model does not incorporate the effect of crack closure, and hence is not expected to exhibit the leveling off of the ΔKth vs. R curve, as discussed above. We rather choose to compare our simulation results with the experimental trend obtained for a variety of metal alloys [42] that shows a linear reduction for this curve. The results from the simulations for Al 6061–T62 alloy with varying load ratio R when plotted on above scale exactly matches the experimental data (experimental values are shown by filled circles and simulated results are shown by the black line) as seen in Fig. 3.12. So, in the absence of crack closure, our model is capable of reproducing the experimental variation of the fatigue threshold limit with R quite accurately.
Ratio of Threshold Stress Intensity Range at R to that at R = 0.5
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2.50 2.00 1.50 1.00 0.50 0.00
0
0.5 R-Ratio
1
Fig. 3.12 Effect of load ratio R on the threshold value of fatigue crack propagation. Experimental data (filled circles) have been taken from [42]
3.5 Conclusions We have presented a cohesive law-based fatigue crack propagation model that can simulate and predict all the stages of fatigue crack propagation in aerospace alloys. The model was used to simulate fatigue crack growth for five different materials. Except for the threshold limit of 6061-T6 GTA weld, it could successfully predict the fatigue allowables for rest of the materials. We hypothesize that the variation in simulated results for this particular material is due to the microstructural changes caused by the welding process as the model could predict accurate threshold value for 6061-T6 alloy without weld. The scope of future improvement for the presented model lies in the incorporation of microstructural effects. We have also observed that the position of peak of the cohesive law envelope plays an important role in determining the threshold limit. Model parameter Sinit determines the position of this peak, and generally lies between 0.98 and 1.0 for most of the materials studied. The tight bound of this parameter leads us to conclude that Sinit may be an intrinsic material parameter. Finally, we have demonstrated that the model can also predict the effect of load ratio on the fatigue crack threshold behavior quite accurately.
References 1. R. Sunder, “A unified model of fatigue kinetics based on crack driving force and material resistance,” Int. J. Fatigue, Vol. 29, No. 9–11, 2007, pp. 1681–1696. 2. B. Farahmand and S. maiti, “Estimation of threshold of fatigue crack growth curve for aerospace alloys,” Aging Aircraft 2008, Phoenix, 2008. 3. S. Suresh, Fatigue of Materials, 2nd edition, Cambridge University Press, 1991. 4. P. Paris and F. Erdogan, “A Critical Analysis of Crack Propagation,” J. Basic Eng., Vol. 85, 1963, pp. 528–34.
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5. A. Vasudevan, K. Sadananda, and N. Louat, “A Review of Crack Closure, Fatigue Crack Threshold and Related Phenomena,” Int. J. Fatigue, Vol. 18, No. 1, 1996, p. 62. 6. W. Elber, “Fatigue Crack Closure Under Cyclic Tension,” Eng. Frac. Mech., Vol. 2, 1970, pp. 37–45. 7. J.C. Newman, “A Finite Element Analysis of Fatigue Crack Closure,” ASTM STP 590, 1976, pp. 281–301. 8. B. Budiansky and J.W. Hutchinson, “Analysis of Closure in Fatigue Crack Growth,” J. Appl. Mech., Vol. 45, 1978, pp. 267–276. 9. J.T. Gray, J.C. William, and A.W. Thompson, “Roughness Induced Crack Closure; Explaination of Microstructurally Sensitive Fatigue Crack Behavior,” Metallurgical Trans., Vol. 14A, 1983, pp. 421–433. 10. J. Weertman, “The Paris Exponent and Dislocation Crack Tip Shielding,” In: High Cycle Fatigue of Strutcutral Materials, TMS Publication, 1997, pp. 41–48. 11. B. Farahmand, “Fatigue and Fracture Mechanics of High Risk Parts,” Chapman and Hall, 1997. 12. V.S. Deshpande, A. Needleman, and E. Van Der Giessen, “A Discrete Dislocation Analysis of Near Threshold Fatigue Crack Growth,” Acta Materialia, Vol. 49, No. 16, 2001, pp. 3189–3203. 13. V. Deshpande, A. Needleman, and E. Van der Giessen, “Discrete Dislocation Modeling of Fatigue Crack Propagation,” Acta Materialia, Vol. 50, No.4, 2002, pp. 831–846. 14. V. Deshpande, A. Needleman, and E. Van der Giessen, “Discrete Dislocation Plasticity Modeling of Short Cracks in Single Crystals,” Acta Materialia, Vol. 51, No. 1, 2003, pp. 1–15. 15. V. Deshpande, A. Needleman, and E. Van der Giessen, “Scaling of Discrete Dislocation Predictions for Near-Threshold Fatigue Crack Growth,” Acta Materialia, Vol. 51, No. 15, 2003, pp. 4637–4651. 16. D. Farkas, M. Willemann, and B. Hyde, “Atomistic Mechanisms of Fatigue in Nanocrystalline Metals,” 10.1103/PhysRevLett.94.165502, 2005. 17. S. Roychowdhury and R.H. Dodds, Jr., “A Numerical Investigation of 3-D Small-Scale Yielding Fatigue Crack Growth,” Eng. Frac. Mech., Vol. 70, 2003, pp. 2363–2383. 18. S. Roychowdhury and R. H. Dodds, Jr., “Effect of T-stress on Fatigue Crack Closure in 3-D Small-Scale Yielding,” Int. J. Solids Struct., Vol. 41, 2004, pp. 2581–2606. 19. J. Wu and F. Ellyin, “A Study of Fatigue Crack Closure by Elasto-Plastic Finite Element Analysis for Constant-Amplitude Loading,” Int. J. Fract., Vol. 82, 1996, pp. 43–65. 20. A.G. Carlyle and R.H. Dodds, Jr., “Three-Dimensional Effects on Fatigue Crack Closure under Fully-Reversed Loading,” Eng. Frac. Mech., Vol. 74, 2007, pp. 457–466. 21. A. de-Andres, J.L. Perez, ´ and M. Ortiz, “Elastoplastic Finite Element Analysis of ThreeDimensional Fatigue Crack Growth in Aluminium Shafts Subjected to Axial Loading,” Int. J. Solids Struct., Vol. 36, 1999, pp. 2231–2258. 22. O. Nguyen, E.A. Repetto, M. Ortiz, and R.A. Radovitzky, “A Cohesive Model of Fatigue Crack Growth,” Int. J. Frac., Vol. 110, 2001, pp. 351–369. 23. K.L. Roe and T. Siegmund, “An Irreversible Cohesive Zone Model for Fatigue Crack Initiation,” Eng. Frac. Mech., Vol. 70, 2002, pp. 209–232. 24. B. Yang, S. mall, and K. ravi-Chandar, “A Cohesive Zone Model for Fatigue Crack Growth in Quasibrittle Materials,” Int. J. Solids Struct., Vol. 38, 2001, pp. 3927–3944. 25. S. Maiti and P.H. Geubelle, “A Cohesive Model for Fatigue Failure of Polymers,” Eng. Frac. Mech., Vol. 72, 2004, pp. 691–708. 26. S. Maiti and P.H. Geubelle, “Cohesive Modeing of Fatigue Crack Retardationin Polymers: Crack Closure Effect,” Eng. Frac. Mech., Vol. 73, 2006, pp. 22–41. 27. S. Maiti, C. Shankar, P.H. Geubelle, and J. Kieffer, “Continuum and Molecular-Level Modeling of Fatigue Crack Retardation in Self-Healing Polymers,” J. Eng. Mat. Tech., Vol. 128, 2006, pp. 595–602. 28. A.A. Griffith, “The Phenomenon of Rupture and Flow in Solids,” Philos. Trans. R Soc. Lond., Vol. 221, 1920.
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29. D.S. Dugdale, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, Vol. 8, No. 2, 1960, pp. 100–104. 30. G.I. Barenblatt, “The Mathematical Theory of Equilibrium of Cracks in Brittle Fracture,” Adv. Appl. Mech., Vol. 7, 1962, pp. 55–129. 31. G.T. Camacho and M. Ortiz, “Computational Modelling of Impact Damage in Brittle Materials,” Int. J. Solids Struc., Vol. 33, 1996, pp. 2899–2938. 32. X. Xu and A. Needleman, “Numerical Simulations of Fast Crack Growth in Brittle Solids,” J. Mech. Phys. Solids., Vol. 42, No. 9, 1994, pp. 1397–1407. 33. P. Geubelle and J. Baylor, “Impact-Induced Delamination of Composites: a 2 D Simulation” Composites Part B, Vol. 29B, 1998, pp. 589–602. 34. R.O. Ritchie, “Mechanisms of Fatigue-Crack Propagation in Ductile,” Int. J. Frac., Vol. 100, 1999, pp. 55–83. 35. R.O. Ritchie, V. Schroeder, and C.J. Gilbert, “Fracture, Fatigue and Environmentally-Assisted Failure of a Zr-based Bulk Amorphous Metal,” Intermetallics, Vol. 8, No. 5–6, 2000, pp. 469–475. 36. J. Blacktop and R. Brook, “Compendium,” Eng. Fract. Mech., Vol. 12, 1979, pp. 619–20. 37. T.M. Ahmed and D. Tromans, “Fatigue Threshold Behavior of Alpha Phase Alloys in Desiccated Air: Modulus Effect,” Int. J. Fatigue, Vol. 21, 2004, pp. 641–649. 38. B.L. Boyce and R.O. Ritchie, “Effect of Load Ratio and Maximum Stress Intensity on the Fatigue Threshold in Ti-6Al-4 V,” Eng. Frac. Mech., Vol. 68, 2000, pp. 129–147. 39. R. Schmidt and P. Paris, “Threshold for Fatigue Crack Propagation and the Effects of Load Ratio and Frequency,” Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, ASTM, Philadelphia, 1973, pp. 79–94. 40. S. Suresh, G.F. Zamiski, and R.O. Ritchie, “Oxide-Induced Crack Closure: An Explanation for Near-Threshold Corrosion Fatigue Crack Growth Behavior,” Metallurgical Mater. Trans. A, Vol. 12, No. 8, 1981, pp. 1435–1443. 41. D. Davidson, Damage Mechanisms in High Cycle Fatigue. AFOSR Final Report, Project 068243. Southwest Research Institute, 1998. 42. L. Lawson, E.Y. Chen and M. Meshii, “Near-Threshold Fatigue: A Review,” Int. J. Fatigue, Vol. 21, 1999, pp. S15–S34.
Chapter 4
Fatigue Damage Map as a Virtual Tool for Fatigue Damage Tolerance Chris A. Rodopoulos
Abstract Using only readily available material properties and the concept of dislocation density evolution ahead of the crack tip, the fatigue damage map attends to develop a virtual tool able to predict the limits and the corresponding crack tip propagation rates characterising each of the fatigue stages, namely crack arrest, microstructurally and physically short crack (Stage I), long crack growth (Stage II), and Stage III growth.
4.1 Introduction In principle, the method of damage tolerance fatigue design is trying to fulfil two needs. The first is the description of “What is happening when I have a crack?” Herein, elastic solutions representing fatigue damage are transformed into the wellknown stress intensity factor, K [1]. In other words, fatigue damage is limited to the single geometric parameter of length and the tensor of stress. The second need is “When this crack will start causing problems?” The latter has been approached with a number of mathematical models [2–5], all of which have been based on the similitude concept. The concept assumes that each material inherently delivers a specific crack growth rate, da/dN. Herein, a is the crack length and N is the number of loading cycles. With the above in mind and considering the particulars emanating from having an elastic solution, it was assumed that once the value of K is known only the material can affect the growth rate. In other words, same material and same K value will always deliver the same crack growth rate. Despite its success, the similitude concept has created a number of additional problems. Large crack growth databases were needed in order to provide the designer with a relationship between K and da/dN. Towards such experimental requirement, especially the aerospace companies have heavily invested in creating C.A. Rodopoulos (B) Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Greece e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 4,
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testing procedures. The issue was further reinforced by ASTM standards to further normalise crack growth testing [6]. Testing revealed a number of unprecedented problems: (a) only a small portion of the testing results was likely to follow the similitude concept and (b) the material seemed to play a more vital role in controlling this portion (see Fig. 4.1).
Fig. 4.1 Typical representation of crack growth stages
All the above forced the industry to seek a solution able to fulfil a number of prerequisites: (a) the effect of the material properties should be acknowledged; (b) testing effort should be kept at minimum; (c) the solution should acknowledge complex conditions, that is, environmental effects, variable amplitude loading, notches, etc.; (d) the scatter emanating from each material should be addressed; and (e) finally the solution should have the capability to operate in a predictive mode. The quest to such approach lasted for over 20 years. This chapter examines the basic assumptions, the steps followed, the problems ahead, the solution, and through that an overall demonstration of the applicability of the fatigue damage map method. In order for the reader to understand the classification of fatigue stages, in order to better acknowledge the necessity of being able to distinguish and classify them, the first section of the chapter provides a brief review.
4.2 The Basic Understanding of Fatigue Damage 4.2.1 Development of Fatigue Cracks and Fatigue Damage Stages Research into the creation of fatigue cracks on free surfaces has confirmed that the damage process is related to the forward and backward motion of dislocations
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along the slip planes of metallic crystals, that is, the reversed plastic flow. A consequence of these repeated dislocation movements are the creation of small-localised deformations called extrusions and intrusions at sites where the persistent slip bands emerge at the material surface [7]. In this respect, a large number of models have been formulated to explain nucleation of microcracks [8], thereby making a distinction between intrusions and microcracks. Furthermore, as a consequence of microscopic material defects, such as second-phase particles, inclusions or precipitates, surface notches, and machining marks; microstructural features, such as grain boundaries, triple points, and twin boundaries [9, 10] and also environmental effects like pitting corrosion [11] a local concentration of stress which may exceed the yield strength of the material is developed. Consequently, cyclic plastic deformation due to the higher stresses of these stress concentration sites and also due to the lower degree of constraint of the near-surface volumes from a cyclically loaded material is developed. It is well known that fatigue damage occurs only when cyclic plastic strains are generated [12]. Therefore, given the intrinsic heterogeneity of polycrystalline metals, the above conditions, separately or in a variety of combinations, can lead to the nucleation of microcracks. The preferred mechanism of initiation will, therefore, depend on the microstructure and manufacturing process of the material, the type of loading, and shape of the component. Once nucleated, a microcrack may be arrested by a microstructural barrier or may propagate until reaching a critical size, causing the final failure [13]. Cyclic crack growth is found to be generally divided into three stages. Stage I fatigue crack growth occurs by a shear mechanism in the direction of the primary slip system over a few grains. The crack propagates on planes oriented at approximately 45◦ to the stress axis1 , that is, the crack follows the best-orientated grain path, for example, a high-angle grain boundary where, due to the high local stress concentration, it will form a new slip band in the next grain giving raise to crack extension [14]. The effect of this crack growth mode is the characteristic zig-zag crack path as defined by Forsyth [15]. At this stage, the crack length may reach a length of a few grains. However, the zone of near-tip plasticity is smaller than the grain dimensions [7]. As such, if the grain is considered as the local self-equilibrating medium, then conditions of large plasticity prevail. Such case negates the application of the stress intensity factor and hence the similitude concept. Favourable conditions and higher stress intensity range values allow the crack to grow longer, as the plastic zone size increases, and also to be able to overcome the resistance offered by successive barriers. After a certain growth, there is a transition from stage I to stage II crack propagation, where the crack overcomes microstructural barriers with ease and simultaneous shear planes develops [15]. At stage II, also known as the steady-state or Paris regime, crack propagation is driven by the stress normal to the crack face and the mode will change to mode I. It is proposed to the Schmid’s law (τ c = σ cos θ cos ϕ, where for θ and ϕ equal to 45◦ , τ c is a maximum), plastic deformation occurs when, the applied tensile stress σ , resolved as τ c on a particular slip plane, exceeds a determined shear stress τ y .τ y = σ y /2 if the material observes the Tresca’s principles, where σ y is the corresponding yield stress.
1 According
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that at the transition from stage I to stage II, the plastic zone will be of the order of the grain size [16]. Fatigue stages are shown in Fig. 4.2. When the crack tip stress intensity reaches values close to those required for unstable crack propagation, stage III begins [17]. tm
ax
t
s
Grains with different crystallographic orientation Plastic Zone
Stage I (shear crack)
Stage II (tensile crack)
Stage III (meandering crack)
s Fig. 4.2 Schematic representation of the fatigue crack growth stages. As indicated (on the left) is a general dislocation model of crack nucleation from the free surface, at the largest and bestorientated grain, as described by Mutoh [18]. The circles in stage III represent voids
The transition from stage II to stage III crack growth has not been widely studied, but it has been observed regularly by E. Hay and M.W. Brown [19]. At this stage, a tearing mechanism is dominant, and the crack branches from the main crack path (crack meandering) [20] as illustrated in Fig. 4.3. In high-cycle fatigue, Stage III represents an insignificant proportion of life, that is, failure follows very rapidly.
Fig. 4.3 Crack meandering from the main crack path in stage III of a shot peened specimen (Al 2024–T351) fatigued under four-point bending constant amplitude loading. The general direction of crack extension is indicated by the arrow
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Despite the fact that short or stage I fatigue cracks will precede to long or stage II cracks, it is imperative to start with the latter in order to highlight differences both in their intrinsic mechanism as well as in the way they can be modelled.
4.2.2 Stage II Fatigue Cracking The theory of fracture mechanics provides the invaluable concept of the stress intensity factor (SIF), denoted by K. As aforementioned, SIF is a measure of the intensity of the near-tip stress fields. However, when such intensity is under linear elastic conditions, the approach used is the so-called linear elastic fracture mechanics (LEFM). This approach can readily be used to analyse and/or to predict fracture, provided the plastic zone surrounding the crack tip is sufficiently small (small-scale yielding) in a way that the K-elastic stress field or K-dominance is not significantly altered. In this respect, and according to Irwin, the singular solutions of the near-tip stress fields, σ ij , are correlated with K as follows: σij = √
1 2πr
K I f ijI (θ ) + K II f ijII (θ ) + K III f ijIII (θ )+ . . . ,
(4.1)
where r is the distance from the crack tip, θ the polar angle measured from the crack plane, fij a dimensionless function of θ at different modes of fracture. KI, II, III are the SIFs for each loading mode, which is generally expressed for mode I as: √ K = Yσ πa,
(4.2)
where Y is a non-dimensional function of the loading and crack geometry, σ is the uniform applied stress remote from the crack and a is the real crack length. Long crack growth rate has been successfully characterised in terms of SIF, K by P.C. Paris & Erdogan [21, 22]. For a cyclic variation of the imposed stress field under LEFM (quasi-elastic) conditions, the empirical relationship between crack growth increment da/dN and K is given by the power law: da = CΔK m dN
(4.3)
where C and m are experimental constants depending on material microstructure, min min = KKmax ; and ΔK is frequency, environment, temperature, and stress ratio R = σσmax the stress intensity factor range, defined as ΔK= Kmax – Kmin . It is important to indicate at this point that in LEFM, the initiation of crack propagation under monotonic, quasi-static loading conditions is characterised by the critical value of the SIF, KC . When the critical value is obtained for mode I stress intensity factor, it is known as fracture toughness and is universally denoted as KIC . In the shearing and tearing modes, fracture toughness is referred to as KIIC and KIIIC , respectively.
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It is well recognised that in Irwin’s analysis the plasticity ahead of the crack tip is assumed negligible. However, when the extent of plasticity is not very small compared to the crack length, the application of LEFM can lead to imprecise predictions. In this respect, views concerning the extent to which the elastic-based theory is applicable have long been documented, among these are: i. The maximum plastic zone at the crack tip taken to be one-fiftieth of the crack length provides a small-scale yielding [23]. ii. The applied stress should be up to two-thirds of the cyclic yield stress [17]. In cases where the plastic zone size is comparable to the crack length, that is, small-scale yielding conditions are not met, the application of elastic–plastic fracture mechanics is the appropriate approach to employ. Discrepancies or offset predictions delivered by Equation 4.3 forced a number of researchers to further analyse the mechanism of crack growth. Perhaps the most known and yet disputable scenario emanates from experimental works indicating that the crack faces do not open immediately with the application of stress. The crack closure effect, first discussed by Elber [24], has increasingly concerned researchers of fatigue crack growth behaviour, particularly in the near-threshold stress intensity levels of long cracks [25, 26]. Herein, it is important to note that the near-threshold behaviour should be considered as being governed by similar to Stage I mechanisms. The differentiation is due to the fact that near-threshold conditions of growth can only be found when the crack emanates from a through thickness notch/slit. This is because the fractured surfaces in the wake of an advancing crack tip close when the far-field load is still tensile before the attainment of the minimum load. Premature contact of the fracture flanks occurs and, as a result, the crack tip does not experience the full range of ΔK, that is, the real driving force or effective stress intensity range, ΔKeff (ΔKeff = Kmax – Kop ) is lower than the nominal ΔK and, therefore, a lower da/dN is expected. Here, Kop is the stress intensity when the crack is fully opened (≥ Kmin ). The most accepted mechanism for such an effect is that of the constraint effect on the residual plastically stretched material which is left on the wake of the crack front by the elastic material which surrounds it, when the crack tip continues advancing through the plastic zone. However, several other mechanisms are envisaged as possible, e.g., due to the presence of corrosion debris within the crack (oxide-induced closure) [27] and due to the contact between rough fracture surfaces (roughnessinduced closure) [28]. Figure 4.4, shows a typical fracture surface corresponding to stage II crack growth.
4.2.3 Stage I Fatigue Cracking It is well documented that short fatigue cracks (SFCs) behave markedly different to long cracks. Their growth could occupy a significant portion of the total fatigue
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Fig. 4.4 Striations found on the fracture surface of 2024-T351 represent clear and welldocumented evidence of steady state crack growth. Note that striations are not always visible or uniform along the propagation plane indicating conditions or strong anisotropy
log da/dN
life of several structures/components. The SFC problem was originally proposed by Pearson [29] with his experimental work performed on aluminium alloys. In this pioneer work it was realised that cracks of the order of the grain size tend to propagate at rates far higher than LEFM predictions suggest. In contrast, experimental evidence on 7075-T6 aluminium alloys [30] indicates that SFCs propagate at rates slower than that of long cracks subjected to the same nominal ΔK. Moreover, some SFCs grow at stress intensities well below ΔK threshold. The crack growth rate decelerates or even arrests in some other SFCs. Others cracks are known to reverse such trends and accelerate as much as two orders of magnitude higher than those of corresponding large cracks. The general ‘anomalous’ behaviour of SFCs is depicted in Fig. 4.5.
Short Crack Growth LEFM Long Crack Growth
Above Yield Stress At Yield Stress
Fig. 4.5 Schematic representation of short crack growth behaviour at different stress levels in 7075 aluminium alloy [30, 31]
At Fatigue Limit Below the Fatigue Limit
log ΔK
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Acceleration, deceleration and crack arrest is commonly attributed as the ‘anomalous’ behaviour of SFCs [32]. Lankford [30] suggested that such behaviour is caused by the difficulty of cracks to nucleate microplasticity in certain crystallographic orientations and/or smaller grains. A comprehensive review on short fatigue crack behaviour has been published by Miller [33, 34]. Other researchers indicated that SFCs may be divided into two primary zones of interest [35]: i. The microstructurally short crack zone/regime (MSC) in which the crack is small in relation to the surrounding microstructural features (e.g. cracks which are comparable to the grain size). Crack growth is strongly influenced by microstructure. The micromechanical description of its propagation is expressed by means of the microstructural fracture mechanics (MFM). Figure 4.6a. ii. The physically small cracks (PSC), which are significantly larger than the microstructural dimension and the scale of local plasticity, but are physically small with length typically smaller than a millimetre or two. Here, the microstructure is not the main parameter affecting their propagation but rather, PSC are strongly dependent on the stress level. PSC are conveniently described in terms of Elasto Plastic Fracture Mechanics (EPFM). Figure 4.6b.
A) MICROSTRUCTURALLY SHORT CRACK
B) PHYSICALLY SMALL CRACK a = 0.1 mm
a
C) LONG CRACK a = 10 mm
a > 100 grains a < 1 grain
FEATURES HIGH STRESS MFM-MODE II AND III STAGE I (SHEAR) CRACK a/rp >1
Fig. 4.6 Classification of SCs: (a) MSC, (b) PSC and for comparison (c) long crack. A briefly description of their main features according to Miller [14] is given (where, a is the crack length and rp is the plastic zone)
Suresh and Ritchie [36], in turn, suggested a comprehensive classification of SCs which include: microstructurally, mechanically, physically and chemically small cracks. A significant contribution to the understanding of SFCs was put forward by Kitagawa and Takahashi [37] who developed the well known Kitagawa–Takahashi diagram (K-T), shown in Fig. 4.7. This diagram shows, for a number of metals, that there is a dividing line, considered to be the bounding condition between propagation leading to failure and non-propagating cracks or crack arrest.
LOG STRESS LEVEL RANGE, Δσ
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Δσ = Fatigue Limit Range
Microstructurally Short Cracking
LEFM Type Cracking
Physically Short Cracking
Δσ =
MFM
ΔΚ th Υ πa
EPFM
a1
ao
a2
LOG CRACK LENGTH, a
Fig. 4.7 Schematic representation of schematic Kitagawa–Takahashi diagram [37, 32, 38]
In Fig. 4.7, the crack lengths a1 and a2 define the deviation of the constant stress and constant stress intensity behaviour, respectively. Crack lengths between a1 and a2 can be expected to propagate faster and to have lower ΔKth , than cracks larger than a2 . This latter crack length represents the point below which the use of LEFM and the Paris law predictions are non-conservative. On the other hand, the point a1 represents the crack length below which there is no crack length effect on fatigue strength [39]. The K-T may be approximated by two straight asymptotic lines. The line given by ΔKth represents the low-stress threshold condition above which a crack should propagate according to LEFM given by [40, 41]: ΔK th Δσ = √ Y πa
(4.4)
It follows that, log Δσ = log
ΔKth √ Y π
−
1 log (a) 2
(4.5)
where Y is the geometric factor of both the geometry of loading and the geometry of the cracked specimen and a is the crack length. The horizontal line is the fatigue limit itself, i.e., the limiting conditions for the propagation of a crack in a plane specimen. In many works it was suggested that the intersection of the fatigue endurance and the LEFM threshold lines takes place at a critical crack length of ≈ 10 grains (PSC) [38, 39]. This assumption supports the fact that a crack must be very much larger than the microstructural features for a
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LEFM concept to be valid. More accurately, the intersection occurs when the crack length is a2 =
1 π
ΔKth YΔσ o
2 (4.6)
for short cracks, Δσ o = Δσ f l . This critical crack length ‘a2 ’ (transition crack size) has been employed as an empirical parameter to account for the differences in propagation rates between long and short cracks even when they are under the same driving forces from a LEFM standpoint [7]. In this sense, K-T may well illustrate the fracture behaviour under different combinations of stress range levels and crack lengths. The relevant K-T has been further extended by Brown [17]. The interfaces of several crack propagation mechanisms occurring at different stress levels have been eloquently identified in a fatigue damage map as depicted in Fig. 4.8. From the socalled Brown map, it is established that cracks may initiate and propagate at stress levels below the fatigue limit and LEFM threshold. These cracks eventually decelerate until they arrest just below the fatigue limit as a consequence of the existing microstructure. Consequently, the Brown map is useful for applying more accurately the several MFM, EPFM and LEFM models since the dominant mode of growth is correctly established. Δσ (MPa)
0.40 STEEL d = 100μm da/dn = 1000 nm
MCG CRACKS MODE II STAGE I
103
/cycle
10
102
MODE II CRYSTALLOGRAPHIC CRACKING
PSB FORMATION
10 10
MICRO STRUCTURAL BARRIER
FATIGUE LIMIT
102
2 σu EPFM MODE III
EPFM CRACKS MODEI STAGE II
2 σy
0.1
2σ y /3 LEFM MODE I
TH
RE
NON PROPAGATING CRACKS
103
SH
OL
D
104
105
α (μm)
Fig. 4.8 The Brown map showing the boundary conditions between short, long and nonpropagating cracks. The fatigue fracture-mode map encompasses six zones, namely (i) LEFM mode I, (ii) EPFM mode I, (iii) EPFM mode III, (iv) mode II stage I, (v) mode II crystallographic cracks, and (vi) non-propagating modes I/II [17]
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The large number of papers dealing with the quantification of short crack propagation rate behaviour reported in the literature indicates the importance of this type of fatigue cracks in metallic materials. In this sense, modelling of SFC propagation rate certainly has contributed to a better understanding of MSC and PSC and subsequently to better fatigue-based designs. Furthermore, attributed to developments of SFCs microstructural analysis, it is now possible to incorporate materials effects in an explicit form within the crack system. The behaviour of SFCs has been formulated as an answer to the belief that LEFM principles are violated due to the relatively large cyclic plasticity at the crack tip, which modifies substantially the strength of the stress field ahead of the crack. In addition to the above it was determined that the problem can lead to significant overestimations with overall effect the premature failure of components. Figure 4.9 shows such a case.
Fig. 4.9 The significance of short crack problem is illustrated for an Astroloy. The plot shows the number of fatigue cycles to failure, estimated using LEFM and small crack growth kinetics, as a function of the initial flaw size. The dashed line comes from experimental results
A typical fracture surface paying significant tribute to the different mechanisms governing the propagation of short cracks is shown in Fig. 4.10. The problem of short cracking was originally approached through several LEFM modifications. For example, a strain intensity factor that took into account cracks propagating in cyclic plastic strain fields was proposed by Boettner et al. [42], which is believed to be the first attempt to model SFC propagation. A model dealing with the blocking action of grain boundaries was analysed by Chang [43], in which critical strain energy must be exceeded at the tip of a crack in order for the crack to propagate. As a result, it is argued that an incubation period arises when cracks encounter grain boundaries. A modified LEFM equation with disorientation between grains analysis was put forward by Chan et al. [43]. Tanaka [44], on the
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Fig. 4.10 Short fatigue cracking in 2024-T351 aluminium alloy. The almost smooth fracture surface (facets) indicates crystallographic crack propagation. Explanation for such behaviour is provided in Section 4.2
other hand, considered the effect of grain boundaries on the development of slip bands and formulated the relationship of SFC propagation behaviour as a function of crack tip displacement and closure. In the same work, SFCs are related to the LEFM threshold stress intensity for long cracks. The application of LEFM principles in the modelling of short cracking was the cause for several concerns regarding the ability of the models to accurately predict such complex physics. Hobson, Brown and de los Rios [45, 46] proposed empirical models to quantify both short and long crack propagation rates, which incorporated the effect of the microstructure in Aluminium 7075-T6 and steel. These empirical relationships were later extended by Angelova and Akid [47] in an attempt to describe more precisely short fatigue crack behaviour not only in air but in an aggressive environment.
4.2.4 Stage III Fatigue Cracks The propagation of stage III or unsteady cracks represents a small portion of the overall fatigue life of components. In general, the stress field ahead of the crack tip is considered as strong (large) enough as to negate the application of LEFM models [7]. In general, it is believed that the high stress field will cause the crack to propagate in an unsteady mode. That is the crack tip plasticity rate exceeds that of the crack. For many [48], the mechanism leading to such behaviour replicates necking in a typical static test. Herein, excess plastic energy, created by the inability of the material to transform all the energy into crack growth, is transported into
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the crack tip plastic zone causing the development of voids and subsequent void growth. It is generally believed that when the strain ahead of the crack tip reaches a critical value voids will initiate especially around second-phase particles [48, 49]. The voids being under a tri-axial stress state will expand and coalesce by a simple plastic deformation mechanism (internal necking) [50]. Figure 4.11, shows a typical stage III fracture surface.
Transition from Stage II to Stage III
Vo ids Vo th ow Gr id
Vo ids
Fig. 4.11 Fracture surface exhibiting stage III crack growth in 2024-T351. The arrows indicate voids and void growth
4.3 Fatigue Damage Map the Basic Rationale – The Navarro–de los Rios Model Based on dislocation theory [51], Navarro and de los Rios [52–54] developed a model which characterises short crack behaviour. The Navarro–de los Rios (N-R) model considers three different zones (see Fig. 4.12): (a) the crack itself; (b) the fatigue damage (crack length and crack tip plastic zone, length 2c = i∗ D) and (c) the grain boundary zone (the barrier to further spread of plastic deformation) which represents either the locked source or the boundary itself. A microstructural short crack is assumed to grow along a persistent slip band (PSB) in the most favourable grain in terms of size and crystallographic orientation ( in fcc) [55, 56]. The plastic zone (slip band ahead of the crack tip)
86
C.A. Rodopoulos σ
σyc
σ2
σ1
crack zone
Plastic zone
0
a
0
n1
Boundary zone
iD/2 iD/2+ro
n2
1
σ
Fig. 4.12 Three zones of the N-R model. D is the grain diameter, ro is the width of the grain boundary and i is the number of half grains, σ 1 is the crack closure stress, σ y c is the cyclic yield stress and σ 2 is the slip band stress concentration
is blocked by the first barrier (grain boundary) and remains blocked until the crack attains the critical length required to activate slip in the next grain by unpinning dislocations. Typical blocking of crack tip plasticity is shown in Fig. 4.13. Unpinning of the dislocations will cause the plastic zone to extend to the next grain boundary and the crack growth rate increases to a new maximum. The process of crack growth, deceleration, slip spreading to the next grain and acceleration repeats itself for several grains until, at the transition point between the short and long crack, the oscillation of the propagation rate ends and a period of monotonic increase in crack growth rate begins. According to the model, the crack and the associated plastic zone are presented by means of continuous distribution of infinitesimal dislocations. The initiation of a new slip band will occur whenever the stress concentration is sufficiently high to activate an appropriate dislocation source in a neighbouring grain a distance r0 away. The stress acting upon such a source is given approximately as 2 τ0 1 1 S(ξ0 ) τ0 1− =√ √ cos−1 n + τ π τ τ (ξ − 1) 2 0
(4.7)
where ζ = (r0 + c)/c > 1 : c is the extent of the damage zone (crack and plastic zone) whose end coincides at any instant with a grain boundary, i.e. c = iD/2 (D is the grain diameter and i is the number of half grains). If a is the crack length, the parameter
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Fig. 4.13 Early crack growth from a surface notch in aluminium alloy 2024-T351: Initiation of slip band in a next grain
n = a/c defines the position of the crack tip in relation to the grain boundary, τ is the applied stress and τ0 the friction stress which opposes the movement of dislocations (lattice friction). This friction stress was taken to be constant and lower than the applied stress [57]. This is reasonable in the early stages of crack growth. However, as deformation proceeds, the associated rise in dislocation density brought about by the general increase in the number of dislocations, both within the slip band in which the crack grows and some other systems activated in the vicinity required by the compatible deformation between grains must result in an increase in the frictional stress. This agrees with experimental evidence of the Hall–Petch relationship [58]: τ = τ0 + K ε D − 2 1
(4.8)
where τ 0 is the friction stress, D is the grain size, Kε is the strain-dependent slope. The operation of a dislocation source requires the attainment of a critical stress σ c that would allow the operation of critical resolved shear stress, τ c . Hence, the critical condition is given as S(ξ0 ) =
1 ∗ m τc 2
(4.9)
where m∗ is the grain orientation factor representing the difficulty of generating dislocations in different angled grains. Once the crack tip has reached a position close to the grain boundary, i.e. once a critical value of n = a/c is reached: n 1 = n 1 c , n 1 c = cos
π 2
σ − σ Li σcomp
(4.10)
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where
σF L σ Li = √ i
(4.11)
σ comp is an appropriate comparison stress greater than the applied stress σ and σ Li is the stress equivalent to the threshold stress for a non-propagating crack spanning over an arbitrary i number of half grains. At the instance that the critical stress is achieved, a new slip plane (dislocation source) is activated and the plastic zone will extend to the next grain. At this instance, the parameter n decreases to a new value ns n s i+2 = n c i
i for i = 1,3,5, · · · i +2
(4.12)
which is obtained by relating crack length to the two successive values of c, that is, iD/2 before slip extension and (i + 2)D/2 after slip, where i gives the position of the tip of the crack in terms of the number of half grain diameter (D/2). As the crack grows the value of n1 will show an intermittent, oscillating pattern. For a growing crack, the initiation of a new slip band is triggered off by the proximity of the crack tip and so there is a sustained increment in the local dislocation density in the crack tip plastic zone. The distribution of dislocations is i i i i i i −1 1 − n 1 ξ −1 1 + n 1 ξ − cosh σ cosh − σ 2 1 ni − ξ i ni + ξ i π2 A 1 1 i i i i i 1 − n 1 + n ξ ξ i −1 −1 2 2 + σ3 − σ2 cosh n i − ξ i − cosh ni + ξ i 2 2 i i 2 ξ 1 i −1 i i i −1 i i + 2 σ τ − σ sin sin − σ n + σ − σ n + 2 1 1 3 2 2 3 π A 1 − ξ i2 1/2 2
f (ξ i ) =
1
(4.13) where τ is the applied stress, σ1i is the crack closure stress, σ2i is the flow resistance of the material to plastic deformation and σ3i is the stress intensity due to dislocation pile-up at the boundary. A = Gb/2π for screw dislocations, or A = Gb/2π(1 – υ) for edge dislocations. G is the shear modulus, b the Burgers vector and υ the Poisson’s ratio. A graphical representation of bounded dislocations is shown in Fig. 4.14. The process of overcoming the microstructural barrier can be thought of as either pushing dislocations through the barrier zone or of unpinning the dislocation source within that zone. In the first case, and for a given combination of applied stresses and crack lengths, r0 varies in relation to the number of dislocations being pushed through the zone. On further crack growth, a critical situation is reached at n i1 = n ic when σ3i attains the required level to overcome the barrier and the slip spreads right across the next grain. In the second case, r0 is considered constant, but an increasing number of dislocations are compressed into the boundary zone as the crack propagates towards the barrier. When the crack reaches the critical length defined by n i1 = n ic , σ3i attains the level of the unpinning stress, slip spreads into the next grain and the crack is able to propagate across the barrier. This friction stress σ3i
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Fig. 4.14 Graphical presentation of dislocation distribution along a crack plane
representing the strength of the barrier, which is required to maintain equilibrium, may be calculated in such a way that no singularity appears into the stress field. Setting the condition so that infinite stresses vanish in Equation 4.13 (by equating to zero the term multiplying the singularity) then i 1 τ − σ3i π = 0 σ2 − σ1i sin−1 n i1 + σ3i − σ2i sin−1 n i2 + 2
(4.14)
and the resulting expression is σ3i =
1 cos−1 n i2
σ2i sin−1 n i1 +
1 τ − σ2i π 2
(4.15)
Equation 4.15 can be simplified as σ3i
=
i −1 i 1 i i −1 i σ2 − σ1 sin n 1 − σ2 sin n 2 + σ π 2 cos−1 n i2 1
(4.16)
The parameter n1 and n2 represent in a dimensionless form the crack length and the fatigue damage size. Equations 4.15 and 4.16 describe the geometrical relationship between crack and plastic zone.
4.3.1 Fatigue Damage Map – Defining the Stages of Fatigue Damage 4.3.1.1 Crack Arrest Within the concepts of dislocation pile-up and slip band nucleation described above, conditions for crack arrest are assumed when the slip band concentration, σ 3 , is unable to attain the level required to initiate a slip band in the next grain before the crack tip reaches the grain boundary. Figure 4.15 shows in a schematic way the
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C.A. Rodopoulos 1000 R=0 R = 0.5 R = –0.5 R = 0.7 R = –0.7
Δσarrest (MPa)
Plain Fatigue Limit
100
10 10–5
10–4
10–3
10–2
Crack Length (m)
Fig. 4.15 Stress ratio effect on crack arrest stress range for 2024-T351 according to Equation 4.24. The mechanical and physical properties used for the calculations are: σ FL(R = 0) = 200 MPa, mi = 1 + 0.35 ln(2a/D), D = 52 μm, α = 0.5 and σ 1 = 0
conditions for crack arrest. The conditions for arrest can be obtained from Equation 4.16, considering that n1 = n2 = 1 and σ2i = 0 (no crack tip plastic zone): 2 i c σ 3 cos−1 n 2 + σ i 1 = σarr est π
(4.17)
where σ3i c is the stress required for the development of a new slip-band (critical constraint effect) and σ1i is the crack closure stress. By employing the approximation cos–1 n2 ≈(2(1 – n2 ))1/2 and the condition n2 = a/(a + ro ), Equation 4.9 is rewritten as √ ro 2 2 c σ i 3 + σ i 1 = σarr est (4.18) π ro + a where a is the crack length and ro is the width of the grain boundary. In [59], Equation 4.18 was further simplified to yield σarr est =
m i σ F L − σ1 + σ1 , √ m i=1 i
i = 2a/D
(4.19)
mi is the grain orientation factor denoting the difficulty of spreading plasm i=1 ticity along different oriented grains, where σ arrest is the crack arrest or threshold where
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stress, a is the crack length, σ FL is the fatigue limit of the material (N > 107 cycles), D is the average transverse grain size and σ 1 is the crack closure stress. It should be noted that mi increases monotonically with crack length from m1 = 1 until it reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). The hypothesis m1 =1 is rationalised by the fact that, in many cases, crack nucleation takes place initially in grains that are most favourably oriented so that the resolved shear stress can easily reach the maximum value. The threshold stress defined by Equation 4.19 identifies two controlling parameters: (a) the strength of the grain boundary that is part of the σ FL factor (fatigue limit) and (b) the effect of the grain orientation, mi . These two parameters reflect the microstructural influence on crack arrest; (1) by relating the strength of the boundary to the threshold stress for crack propagation and (2) by incorporating the effect of the increasing number of grains transverses to the crack front as the crack grows. The above reflects the increasing probability of a “hard” grain being included in the plastic zone. As hard grain we denote grains with principle slip orientation which exhibit strong angle difference with those already within the plastic zone. Kujawski [60] proposed that the stress ratio, R, on the threshold stress intensity factor range can be given as ΔK th = (1 − R)α for R ≥ 0 ΔK th,R=0
(4.20)
ΔK th = (1 − R) for R ≤ 0 ΔK th,R=0
(4.21)
and
where ΔKth,R = 0 is the threshold stress intensity factor range corresponding to R = 0 and α is a fitting parameter ranging between 0 and 1. A value of α = 0.5 was suggested for aluminium alloys and martensitic steels [60]. Navarro et al. [61] suggested that the smooth specimen fatigue limit is related to the threshold stress intensity factor through K th = σ F L π
D 2
(4.22)
Using a simple substitution between Equations 4.21 and 4.22, R-ratio effect on the fatigue limit is given by ΔσFL = (1 − R)α for R ≥ 0 σFL(R=0)
(4.23a)
Δσ F L = (1 − R) for R ≤ 0 σ F L(R=0)
(4.23b)
and
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Substitution of the above into Equation 4.19, R-ratio effect on the crack arrest stress can be written as Δσarrest =
m i (1 − R)α σFL(R=0) − σ1 + σ1 for R ≥ 0 √ m1 2a/D
(4.24a)
Δσarrest =
m i (1 − R) σFL(R=0) − σ1 + σ1 for R ≤ 0 √ m1 2a/D
(4.24b)
and
Figure 4.15 shows a typical outcome of Equation 4.24 for an aluminium alloy. It is worth noting that the approach is accurate for opening mode growing cracks. In order to avoid the use of the experimentally determined parameters a, Equation 4.24 can be written according to Goodman as Δσarr est
2σ Fmax L(R=−1) σU T S
1 m i 2σU T S (1 − R) + 2 (1 + R) σ F L(R=−1) ≤ √ Y m1 2a/D
− σ1 +σ1 , -∞ < R1 4
(4.31)
where x is the number of half grains constituting crack tip plasticity (c = α + xD/2), Δσ II→III is the transition from stage II to stage III stress range and f is the elongation to failure. The absence of the parameter Y reflects the independency of the boundary condition due to the type of failure (large-scale yielding). Application of Equations 4.25, 4.30 and 4.31 leads to the creation of the fatigue damage map. A typical example is shown in Fig. 4.17. Herein the reader can see how the stages of the growth can change with the stress ratio and also identify the effect of stress ratio on short crack growth (after a particular stress ratio short crack diminishes it effect). The application of the fatigue damage map also allows the user to see how different materials respond to specific growth stages (Figs. 4.18 and 4.19). A typical example is shown in Fig. 4.20. The reader can easily identify by comparing Figs. 4.17 and 4.20 that the tendency of the Ti-alloy towards short cracking diminishes while other stages of growth increase.
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Stress Range (MPa)
1000.0
100.0
Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III Crack growth Toughness or Testing Failure
10.0 10
100
1000 Crack Length (microns)
10000
100000
10000
100000
Fig. 4.17 Fatigue damage map for 2024-T351 at R = 0
Stress Range (MPa)
1000.0
100.0
Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure
10.0 10
100
1000 Crack Length (microns)
Fig. 4.18 Fatigue damage map for 2024-T351 at R = 0.3
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Stress Range (MPa)
1000.0
100.0
Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure
10.0 10
100
1000 Crack Length (microns)
10000
100000
10000
100000
Fig. 4.19 Fatigue damage map for 2024-T351 at R = 0.5
Stress Range (MPa)
1000.0
100.0
Crack Arrest Low Threshold High Threshold Stage II Crack Growth Fast - Stage III crack growth Toughness or Tearing Failure
10.0 10
100
1000 Crack Length (microns)
Fig. 4.20 Fatigue damage map for a fine grained Ti-Alloy at R = 0
4.3.2 Fatigue Damage Map – Defining the Propagation Rate of Fatigue Stages In order for the fatigue damage map to provide damage tolerance indications, it is necessary to be able to predict the crack growth rate without the need for expensive testing.
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In [70], de los Rios et al. argued that the elasto-plastic crack tip opening displacement can describe the growth rate of the crack tip through the relationship da = φδti p dN
(4.32)
where da/dN is the crack tip propagation rate ,a is the crack length, N is the number of loading cycles and δ tip is the crack tip opening displacement. The factor φ is a dimensionless parameter representing the fraction of dislocations in the slip band participating in the process of crack extension and takes values between 0 and 1. Herein, the size of the slip band is idealised to that of crack tip plasticity. This assumption has been previously discussed in detail in a number of works [71–73]. The parameter φ should depend on the number of pre-existed dislocations which interact with on-growing dislocations emitted by the crack tip, which in turn depend on the grain size and other strong microstructural barriers able to confine plasticity, like twin boundaries or large unshearable precipitates [70]. In [63], it was reported that φ follows a linear association with the far-field stress range Δσ . Such finding is partially correct since it excludes the crack tip and crack tip plasticity size effects. Of course, Equation 4.32 requires experimental data to incorporate material and crack type variations and thus can be easily subjected to experimental errors during the data input stage. Such problems can be overcome if a pure analytical crack tip prediction rate model is employed. Such model has been presented and compared to numerous stage II experimental results by Nicholls [74]: 1 da = ΔK 4 y dN 4Eσc K I c 2
(4.33)
where σ c y is the cyclic yield stress, KIc is the fracture toughness under plane strain conditions, E√is the modulus of elasticity and ΔK is the stress intensity factor range (ΔK = Δσ πa with Δσ being the far-field stress range). The use of Equation 4.33 does not represent a panacea to prediction error but a rather controllable output through macroscopic and not experimental/fitting parameters. If linear elastic conditions are assumed, Equation 4.32 can be modified following [75] λΔK 2 da =φ y dN Eσc
(4.34)
where λ contains not physical meaning and it is rather a mathematical moderator primarily used to fit experimental results. As opposed to the original estimate from Wells [75] of λ = 4/π, Nicholls proposed a value of unity to best match experimental results. By equating the above formulas (Equations 4.33 and 4.34) and assuming a stress ratio, R = 0, the participating fraction of dislocations can be expressed as φ=
σ 2 πa 4K c2
(4.35)
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The introduction of the effect of crack tip plasticity size on crack tip propagation rate should reveal the fraction of the participating dislocations. However, due to the complexity of the issue and the lack of experimental evidence, such association is beyond an analytical solution. Nevertheless, such relationship, with some degree of error, can be sought through the effect of crack tip plasticity size on δ tip . Such, relationship was expressed as φ2 k da = dN S
√
1 − n2 σa n
(4.36)
where n is the dimensionless value of a over the fatigue damage, c (c = a + crack tip plasticity size), k = 1 or 1 – ν (where ν is the Poisson’s ratio) for screw or edge dislocations, respectively, and S is the stiffness modulus with values between the shear modulus G the elastic modulus E. Considering that Equation 4.33 delivers such a wide range of crack tip propagation rates, most likely to include rates corresponding to stage I up to stage III crack growth, the effect of crack tip plasticity size on crack tip propagation rate can be given as da = dN
σ 2 πa
2k S
4K c 2
√
1 − n2 σa n
(4.37)
Equation 4.37 can be used in the equations providing the transition from the fatigue damage stages leading to,
da dN
=
Δσarr est 2 πa
4K I c 2
arr est
& ' ' (1 − 2k S
a a+
a a+
2 D 25
Δσarr est a
(4.38)
D 25
crack growth rate for crack arrest.
da dN
da dN
=
P
4K c 2
M
=
Δσ M 2 πa
Δσ P 2 πa 4K c
2
2k S
*
) 2k S
1−
2 a a+D
a a+D
a 1− a + 2D a a + 2D
Δσ M a
(4.38a)
2 Δσ p a
(4.38b)
crack growth rate for the transition from microstructurally to physically to stage II crack growth and
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da dN
I I →I I I
=
Δσ I I →I I I 2 πa 4K c
2
& ' ' (1 −
a a + x D2 ⎛ ⎞
2k S
⎜ ⎝
2 Δσ I I →I I I a
(4.39)
⎟ D⎠ a+x 2 a
the crack growth rate for the transition from stage II to stage III. Typical examples are shown in Figs. 4.21 and 4.22.
Crack Growth Rate (m/cycle)
1.0E–7
1.0E–8 Crack Arrest Low Threshold
1.0E–9
High Threshold Long Crack
1.0E–10
1.0E–11
1.0E–12 10
100
10000
1000 Crack Length (microns)
100000
Fig. 4.21 Fatigue damage map and corresponding crack growth rates for a fine-grained Ti-Alloy at R = 0
Crack Growth Rate (m/cycle)
1.0E–7
1.0E–8 Crack Arrest Low Threshold
1.0E–9
High Threshold Long Crack
1.0E–10
1.0E–11
1.0E–12 10
100
1000
10000
100000
Crack Length (microns)
Fig. 4.22 Fatigue damage map and corresponding crack growth rates for a fine-grained Ti-Alloy at R = 0.3
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4.4 Conclusions The chapter aims to familiarise the reader with the principles of the fatigue damage map and to draw attention towards the fact that the response of the material is inherently included in the basic material properties. With knowledge of the parameters affecting fatigue damage, the propensity of the material towards fatigue loading is possible using only readily available information. The fatigue damage map can be used as: (a) a virtual tool for examining the response of the material towards fatigue damage and hence provide a measure for material comparison; (b) a virtual tool for predicting the corresponding crack growth rates without expensive and complex testing procedures; and (c) a virtual tool which in comparison with finite element stress analysis can provide a strong damage tolerance indications.
References 1. A.A. Griffith, “The Phenomenon of Rupture and Flow in Solids,” Philos. Trans. R. Soc. Lond. A, Vol. 221, 1921, pp. 163–197. 2. P.C. Paris and F.A. Erdogan, “Critical Analysis of Crack Propagation Laws,” J. Basic Eng. TRANS ASME, Vol. 85(Series D), 1963, pp. 528–534. 3. N.E. Frost and D.S. Dugdale, “The Propagation of Fatigue Cracks in Test Specimens,” J. Mech. Phys. Solids, Vol. 6, 1958, 92–110. 4. J. Schijve, Fatigue of Structures and Materials, Kluwer Academic Publishers, The Netherlands, 2001. 5. Fatigue Design Handbook AE-10, Society of Automotive Engineers, USA, 1988. 6. S.R. Swanson, Handbook of Fatigue Testing, ASTM STOP 566, 1974. 7. S. Suresh, Fatigue of Materials, 2nd edition, Cambridge, University Press, Cambrige, 1998. 8. P. Luk´asˇ, “Fatigue Crack Nucleation and Microstructure,” ASM Handbook Volume 19: Fatigue and Fracture, ASM International, Materials Park, Ohio, 1996, pp. 96–109. 9. J.R. Yates, “Fatigue of Engineering Materials. MSc in Structural Integrity (MPE603), course notes,” Department of Mechanical Engineering, The University of Sheffiel, Sheffield, U.K., 1999. 10. K.J. Miller, “Fundamentals of Deformation and Fracture,” In: Proc. Eshelby Memorial Symposium, Cambridge, U.K., B.A. Bilby, Cambridge University Press, 1985, pp. 477–500. 11. D.W. Hoeppener, “Model for Prediction of Fatigue Lives Based upon a Pitting Corrosion Fatigue Process. Fatigue Mechanisms,” In: Proc. of an ASTM-NBS-NSF Symposium, Kansas, City, Mo., J.T. Fong, Ed., ASTM STP 675, 1978, pp. 841–870. 12. R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd edition New York, USA, John Wiley & Sons, 1989. 13. F. Guiu, R. Dulniak, and B.C. Edwards, “On the Nucleation of Fatigue Cracks in Pure Polycrystalline a-iron,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, 1982, pp. 311–321. 14. K.J. Miller, “Materials Science Perspective of Metals Fatigue Resistance,” Mater. Sci. Technol., Vol. 9, 1993, pp. 453–462. 15. P.J.E. Forsyth, “A two stage process of fatigue crack growth, in crack propagation,” In: Proceedings of Cranfield Symposium, London, Her Majesty’s Stationery Office, 1962, pp. 76–94. 16. V.M. Radhakrishnan and Y. Mutoh, “On Fatigue Crack Growth in Stage I,” In: The Behaviour of Short Fatigue Cracks, EGP Pub. 1. Great Britain, K.J. Miller and E.R. de.los.Rios, Eds., Mechanical Engineering Publications Limited, 1986, pp. 87–99.
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17. M.W. Brown, “Interfaces Between Short, Long and Non-Propagatig Cracks,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 423–439. 18. Y. Mutoh and V.M. Radhakrishnan, “An Analysis of Grain Size and Yield Stress Effects on Stress at Fatigue Limit and Threshold Stress Intensity Factor,” J. Eng. Mater. Technol., Vol. 103, 1986, pp. 229–233. 19. E. Hay and M.W. Brown, “Initiation and Early Growth of Fatigue Cracks from a Circunferential Notch Loaded in Torsion,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 309–321. 20. R.O. Ritchie, F.A. McClintock, H. Nayeb-hashemi, and M.A. Ritter, “Mode III Fatigue Crack Propagation in Low Alloy Steel,” Metall. Trans., Vol. 13A, 1982, pp. 101–110. 21. P.C. Paris and F.J. Erdogan, “A Critical Analysis of Crack Propagation Law,” J. Basic Eng. Trans. ASME, Series D, Vol. 85, No. 4, 1963, pp. 528–535. 22. P.C. Paris, “Fracture Mechanics and Fatigue: A Historical Perspective,” Fatigue Fract. Eng. Mater. Struct., Vol. 21, 1998, pp. 535–540. 23. J.F. Knott, Fundamentals of Fracture Mechanics, Butterworths, London, 1973. 24. W. Elber, “Fatigue Crack Closure Under Cyclic Tension,” Eng. Fract. Mech., Vol. 2, 1970, pp. 37–45. 25. S. Suresh and R.O. Ritchie, “The Propagation of Short Fatigue Cracks,” Int. Met. Rev., Vol. 29, No. 6, 1984, pp. 445–501. 26. N. Louat, K. Sadananda, M. Duesbery, and A.K. Vasudevan, “A Theoretical Evaluation of Crack Closure,” Metall. Trans., Vol. 24-A, 1993, pp. 2225–2232. 27. R.O. Ritchie, S. Suresh, and C.M. Moss, “Near Threshold Fatigue Crack Growth in 21/4 Cr 1Mo Pressure Vessel Steel in Air and Hydrogen,” J. Eng. Mater. Technol. (Trans. ASME), Vol. 102, 1980, pp. 293–299. 28. S. Suresh and R.O. Ritchie, “A Geometric Model for Fatigue Crack Closure Induced by Fracture Surface Roughness,” Metall. Trans. A, Vol. 13A, 1982, pp. 1627–1631. 29. S. Pearson, “Initiation of Fatigue Cracks in Commercial Aluminium Alloys and the Subsequent Propagation of Very Short Cracks,” Eng. Fract. Mech., Vol. 7, 1975, pp. 235–247. 30. J. Lankford, “The Growth of Small Fatigue Cracks in 7075-T6 Aluminium,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, No. 3, 1982, pp. 233–248. 31. D. Kujawski and F. Ellyin, “A Microstructurally Motivated Model for Short Crack Growth Rate,” Short Fatigue Cracks, ESIS 13, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1992. 32. D. Taylor and J.F. Knott, “Fatigue Crack Propagation Behaviour of Short Cracks: The Effect of Microstructure,” Fatigue Fract. Eng. Mater. Struct., Vol. 4, No. 2, 1981, pp. 147–155. 33. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part I – A Review of Two Recent Books,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91. 34. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part II – A General Summary,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91. 35. K. Tokaji and T. Ogawa, “The Growth Behaviour of Microstructurally Small Fatigue Cracks in Metals,” Short Fatigue Cracks, ESIS 13, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1992, pp. 85–99. 36. S. Suresh and R.O. Ritchie, “The Propagation of Short Fatigue Cracks,” Int. Met. Rev., Vol. 29, No. 6, 1984, p. 445. 37. H. Kitagawa and S. Takahashi, “Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage,” In: 2nd International Conference on Mechanical Behaviour of Materials (ICM2), Boston, USA. American Society of Metals, Metal park, Ohio, 1976, pp. 627–631. 38. K.J. Miller, “The Behaviour of Short Fatigue Cracks and their Initiation. Part I – A Review of Two Recent Books,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 1, 1987, pp. 75–91.
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39. D. Taylor, “Fatigue of Short Cracks: The Limitations of Fracture Mechanics,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 479–490. 40. J.M. Kendall, M.N. James, and J.F. Knott, “The Behaviour of Physically Short Fatigue Cracks in Steels,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 241–258. 41. E.R. de.los.Rios, P. Mercier, and B.M. El-Sehily, “Short Crack Growth Behaviour Under Variable Amplitude Loading of Shot Peened Surfaces,” Fatigue Fract. Eng. Mater. Struct., Vol. 19, No. 2/3, 1996, pp. 175–184. 42. R.C. Boettner, C. Laird, and A.J. McEvily, “Crack Nucleation and Growth in High Strain-Low Cycle Fatigue,” Trans. Metall. Soc. AIME, Vol. 233, 1965, pp. 379–385. 43. K.S. Chan and J. Lankford, “A Crack Tip Model for the Growth of Small Fatigue Cracks,” Scr. Metall., Vol. 17, 1983, pp. 529–538. 44. K. Tanaka, “Modelling of Propagation and Non-Propagation of Small Cracks,” In: Small Fatigue Cracks, R.O. Ritchie and J. Lankford, Eds., Metallurgycal Society Inc. 1986, pp. 343–362. 45. P.D. Hobson, “The Formulation of a Crack Growth Equation for Short Cracks,” Fatigue Fract. Eng. Mater. Struct., Vol. 5, No. 4, 1982, pp. 323–327. 46. P.D. Hobson, M.W. Brown, and E.R. de.los.Rios, “Two Phases of Short Crack Growth in a Medium Carbon Steel,” In: The Behaviour of Short Fatigue Cracks, EGF Pub. Sheffield, U.K., K.J. Miller and E.R. delosRios, Eds., Mechanical Engineering Publications, London, 1986, pp. 441–459. 47. D. Angelova and R. Akid, “A Note on Modelling Short Fatigue Crack Behaviour,” Fatigue Fract. Eng. Mater. Struct., Vol. 21, 1998, pp. 771–779. 48. J.R. Rice, The Mechanics of Fracture, ASME AMD, Vol. 19, 1976, pp. 23–53. 49. R.M. McMeeking, “Finite Deformation Analysis of Crack Tip Opening in Elastic-Plastic Materials and Implications for Fracture,” J. Mech. Phys. Solids, Vol. 25, 1977, pp. 357–381. 50. J.F. Knot, “Microscopic Aspects of Crack Extension,” In: Advances in Elasto-Plastic Fracture Mechanics, L.H. Larsson, Ed., Applied Science Publishers, Essex, England, 1980. 51. B.A. Bilby, A.H. Cottrell, and K.H. Swinden, “The Spread of Plastic Yielding from a Notch,” Proc. R. Soc. Lond. A, Vol. 272, 1963, pp. 304–314. 52. A. Navarro and E.R. de.los.Rios, “A Model for Short Fatigue Crack Propagation with an Interpretation of the Short-Long Crack Transition,” Fatigue Fract. Eng. Mater. Struct., Vol. 10, No. 2, 1987, pp. 169–186. 53. A. Navarro and E.R. de.los.Rios, “Compact Solution for a Multizone Bcs Crack Model with Bounded or Unbounded End Conditions,” Philos. Mag. A, Vol. 57, No. 1, 1988, pp. 43–50. 54. A. Navarro and E.R. de.los.Rios, “Fatigue Crack Growth Modelling by Successive Blocking of Dislocations,” Proc. R. Soc. Lond. A, Vol. 437, 1992, pp. 375–390. 55. A.H. Cottrell and D. Hull, “Extrusion and Intrusion by Cyclic Slip in Copper,” Proc. R. Soc. Lond. A, Vol. 242, 1957, pp. 211–213. 56. P.J.E. Forsyth, “Slip Band Damage and Extrusion,” Proc. R. Soc. Lond. A, Vol. 242, 1957, pp. 198–202. 57. E.R. de los Rios, M.W. Brown, K.J. Miller, and H.X. Pei, “Fatigue damage accumulation during cycles of non-proportional straining,” In Basic questions in fatigue, Vol. 1, ASTM STP 924: 1988, pp. 194–213, ASTM, Philadelphia. 58. N.J. Petch, “The Cleavage Strength of Polycrystals,” J. Iron Steel Inst., Vol. 174, 1953, pp. 25–28. 59. E.R. de los Rios, “Dislocation Modelling of Fatigue Crack Growth in Polycrystals,” Eng. Mech., Vol. 5, No. 6, 1998, pp. 363–368. 60. D. Kujawski, “A Fatigue Crack Driving Force Parameter with Load Ratio Effects,” Inter. J. Fatigue, Vol. 23, 2001, pp. S236–S246. 61. A. Navarro, C. Vallellano, E.R. de los Rios, and X.J. Xin, Notch Sensitivity and Size Effects Described by a Short Crack Propagation Model,” In: Engineering Against Fatigue, J.H.
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C.A. Rodopoulos Beynon, M.W. Brown, T.C. Lindley, R.A. Smith and B. Tomkins, Eds., Balkema, Netherlands, 1999. E.R. de los Rios and A. Navarro, “Considerations of Grain Orientation and Work Hardening on Short-Fatigue Crack Modelling,” Phil. Mag. A, Vol. 61, No. 3, 1990, pp. 435–449. C.A. Rodopoulos, “Predicting the Evolution of Fatigue Damage Using the Fatigue Damage Map Method,” Theor. Appl. Fract. Mech., Vol. 45, 2006, pp. 252–265. M.W. Brown, “Interfaces Between Short, Long and Non-Propagating Cracks,” In: The Behaviour of Short Fatigue Cracks, K.J. Miller and E.R. de los Rios, Eds., Mechanical Engineering Publications, London, 1986, pp. 423–439. A.T. Winter, “Cyclic Deformation: the Two Phase Model,” Proceedings of the Eshelby Memorial Symposium, International Union of Theoretical and Applied Mechanics, B.A. Bilby, K.J. Miller and J.R. Willis, Eds., Cambridge University Press, Cambridge, 1984, pp. 573–582. S. Taira, K. Tanaka, and Y. Nakai, “A Model of Crack Tip Slip Band Blocked by Grain Boundary,” Mech. Res. Comm., Vol. 5, 1978, pp. 375–381. C.A. Rodopoulos and Al.Th. Kermanidis, “Understanding the Effect of Block Overloading on the Fatigue Behaviour of 2024-T351 Aluminium Alloy Using the Fatigue Damage Map,” Inter. J. Fatigue, Vol. 29, No. 2, 2006, pp. 276–288. G.R. Yoder, L.A. Cooley, and T.W. Crooker, “On Microstructural Control of Near-Threshold Fatigue Crack Growth in 7000-Series Aluminium Alloys,” Scr. Metall., Vol. 16, 1982, pp. 1021–1025. C.A. Rodopoulos, E.R. de los Rios, J.R. Yates, and A. Levers, “A Fatigue Damage Map for the 2024-T3 Aluminium Alloy,” Fatigue Fract. Eng. Mater. Struc., Vol. 26, No. 7, 2003, pp. 569–576. E.R. de los Rios, Z. Tang, and K.J. Miller, “Short Crack Fatigue Behaviour in a Medium Carbon Steel,” Fatigue Fract. Eng. Mater. Struct., Vol. 7, No. 2, 1984, pp. 97–108. R.M. Pelloux, “Crack Extension by Alternating Shear,” Eng. Fract. Mech., Vol. 1, 1970, pp. 697–704. F.A. McClintock, “Considerations for Fatigue Crack Growth Relative to Crack Tip Displacement,” In: Engineering Against Fatigue, J.H. Beynon, M.W. Brown, T.C. Lindley, R.A. Smith and B. Tomkins, Eds., A.A. Balkema Publishers, Rotterdam, Netherlands, 1999. J.N. Eastbrook, “A Dislocation Model for the Rate of Initial Growth of Stage I Fatigue Cracks,” Inter. J. Fract., Vol. 24, 1984, pp. R43–R49. D.J. Nicholls, “The Relation Between Crack Blunting and Fatigue Crack Growth Rates,” Fatigue Fract. Eng. Mater. Struct., Vol. 17, No. 4, 1994, pp. 459–467. A.A. Wells, “Application of Fracture Mechanics at and Beyond General Yielding,” Br. Weld. J., Vol. 10, 1963, pp. 563–570.
Chapter 5
Predicting Creep and Creep/Fatigue Crack Initiation and Growth for Virtual Testing and Life Assessment of Components K.M. Nikbin
Abstract Predicting creep and creep/fatigue crack initiation and growth, under static and cyclic loading, in engineering materials at high temperatures is an important aspect for improved life assessment in components and development of virtual test methods. In this context, a short overview of the present standards and codes of practice as well as experimental methods and models to predict failure are presented. Following a brief description of engineering creep parameters and basic elastic–plastic fracture mechanics methods, high-temperature fracture mechanics parameters are derived by analogy with plasticity concepts. Techniques are shown for determining the creep and fatigue fracture mechanics parameters K and C∗ to predict experimental crack growth using uniaxial creep data. An analytical ‘failure strain/constraint’ based ductility exhaustion model called the NSW model is presented. The model uses a ductility exhaustion argument and constraint at the crack tip is able to predict, within a range of as much as a factor of ∼30 crack initiation and growth at high temperatures, over the plane stress to plane strain regimes. By taking into account angular damage distribution around the crack tip NSW model is further refined as the NSW-MOD model. This model is able predict a much improved upper/lower bound of a factor of between ∼0.5–7, depending on the creep index n, in steady-state crack initiation and growth over the plane stress/strain region in components containing defects. The presence of cyclic load is assumed to introduce a cycle-dependent fatigue component with a linearly summed cumulative damage effect with the creep response. The prediction for the fatigue component can be handled using either the Paris law or the method proposed by Farahmand in this book. Hence, this chapter does not get into the details of the fatigue response. In conclusion, the creep/fatigue modeling presented can be used as a tool in component design of metallic parts, as well as in life assessment of cracked components at elevated temperatures and in predicting virtual cracking behavior in fracture mechanics specimens.
K.M. Nikbin (B) Mechanical Engineering Department, Imperial College, London SW7 2AZ, UK e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 5,
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5.1 Introduction Engineering life assessment and component lifetime prediction against fracture at high temperatures utilise models based on physical principles but which always need to be validated under practical and operational circumstances [1–9]. The use of computational methods to develop predictive methods in creep crack initiation and growth is important as this will allow to optimise the number of tests that will be needed as well as help improve life prediction methods in components. In this chapter a short description of the background to life assessment methodologies for cracked components which have their roots in creep analysis of uncracked bodies is presented. Following this the engineering creep parameters ranging from uniaxial to multiaxial states of stress are considered. However, by the very nature of the subject’s diversity this chapter cannot go into detail of the derivation of the parameters and models and the reader should follow the appropriate references for further detail [10–21]. The mechanism of time-dependent deformation is shown to be analogous to deformation due to plasticity. Therefore, elastic– plastic fracture mechanics parameters such as K or J are reviewed and linked to high-temperature fracture mechanics parameter C∗ and techniques are shown for determining the creep fracture mechanics parameter C∗ using experimental crack growth data, collapse loads and reference stress methods [3–9]. Finally, models for predicting creep crack initiation and growth in terms of C∗ and the creep uniaxial ductility are developed [22–32]. Cumulative damage concepts are used for predicting crack growth under static and cyclic loading conditions [34–37].
5.1.1 Background to Life Assessment Codes Components in the power generation and petro-chemical industry operating at high temperatures are almost invariably submitted to static and/or combined cycle loading. They may fail by net section rupture, crack growth or a combination of both. The development of codes in different countries has moved in similar direction and in many cases the methodology has been borrowed from a previously available code in another country. The early approaches to high-temperature life assessment used methodologies that were based on defect-free assessment codes. For example, ASME Code Case N-47 [1] and the French RCC-MR [2], which have many similarities, are based on lifetime assessment of uncracked structures. The materials properties data that are used for these codes are usually uniaxial properties and S–N curves for fatigue. However, with the development of non-destructive examination (NDE) methods and improvements in the ability to find smaller and smaller cracks in aging parent and welded components it has become clear that fracture mechanics should be considered in component design and predicting remaining life. More recent methods, therefore, make life assessments based on the presence of defects in the component. The codes dealing with defects [3–9] vary in the extent of the range of failure
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behavior they cover. However, they define a new direction for life assessment as well as component design based on the presence of cracks which should realistically improve predictions. This is especially so where welds or creep-damaged components which substantially aged are concerned so that the presence of a predetermined crack size would need to be assumed even if one is not physically observed due to NDE limitations. Therefore, fracture mechanics solutions and models dealing with creep and creep/fatigue interaction in initiation and growth of defects are treated in the codes in a robust and pragmatic way [3–9] so that a user can relate his/her findings to the plant life cycle. In terms of creep crack growth, all the codes propose similar approaches but use different formulae for the analysis of the crack under linear and non-linear conditions. This difference is most likely to affect the predictive solutions. In such codes, material properties dealing with crack growth data that are needed are more complex compared to uniaxial data both in terms of testing methods and derivation. Consequently, valid methods for material characterization coupled with advanced fracture mechanics modeling is an important part for design and life prediction in components.
5.1.2 Creep Analysis of Uncracked Bodies The time-dependent deformation mechanism occurring at elevated temperature that is generally non-reversible is defined as creep. Creep is most likely to occur in components that are subjected to high loads at elevated temperatures for extended periods of time. Creep may ultimately cause fracture or assist in developing a crack in components subjected to stresses at high temperatures. In the last 30 years, rapid development has taken place in the subject and references at the end of the chapter give an indication of this work. The phenomenon of creep is based on a time-dependent process whereby the material deforms irreversibly. Creep in polycrystalline materials occurs as a result of the motion of dislocations within grains, grain boundary sliding and diffusion processes. A creep curve can simply be split up into three main sections as shown in Fig. 5.1. All the stages of creep are not necessarily exhibited by a particular material for given testing conditions. In Fig. 5.2 the representation of the rupture times versus the applied stress covers the response of the material throughout the three regimes. The primary region is a period of decreasing creep rate where work-hardening processes dominate and cause dislocation motion to be inhibited. The secondary or steady-state region of creep deformation is frequently the longest portion and corresponds with a period of constant creep rate where there is a balance between workhardening and thermally activated recovery (softening) processes. The final stage is termed the tertiary region. This is a period of accelerating creep rate which culminates in fracture. It can be caused by a number of factors which include increase in stress in a constant load test, formation of a neck (which also results in an increase in stress locally), voiding and/or cracking and overageing (metallurgical instability in alloys).
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Fig. 5.1 The creep curve, showing definitions of creep strain rates
× ε&A
Creep Strain εc
Failure strain
ε&s Tertiary
Primary
Secondary
Time, t
tr
Fig. 5.2 Stress/time to rupture relation
Log σ −1/ν
Log tr
5.1.3 Physical Models Describing Creep A number of processes dominate the creep processes [10–21] producing a creep curve of creep strain versus time as schematically shown in Fig. 5.1. When secondary creep dominates, it is often possible to express the minimum secondary creep strain rate ε˙ s or ε˙ sc , in the form ε˙ sc ασ n exp (−Q/RT )
(5.1)
The existence of several creep processes indicates that in general n and Q in Equation (5.1) will change as a mechanism boundary is crossed. In addition, it has been established that a simple power law stress dependence is not always satisfactory. At high stresses an exponential expression of the form ε˙ sc α exp (βσ ) exp (−Q/RT )
(5.2)
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where β is a material-dependent constant is often more adequate. It has shown [11] that Equations (5.1) and (5.2) can both be encompassed by the relation ε˙ sc α (sinh ασ )n exp (−Q/RT )
(5.3)
provided β = αn. When αn < 0.8, Equation (5.3) reduces to Equation (5.1) and when ασ > 1.2 it reverts to Equation (5.2). No satisfactory physical model has yet been developed which produces expressions of the form of Equation (5.2) or (5.3). There are creep laws which deal with the time dependence of creep. The stress and temperature dependence of the material parameters introduced will not be examined. However, for the most part, they can be described by expressions that are similar to those used for secondary creep in the previous section. Model-based laws where creep strain is predicted from motion of dislocations give an understanding of the creep behaviour but are rarely useful for engineering purposes. As a result, empirical laws have been produced [11–12] to give more accurate descriptions of the observed shapes of creep curves. A representative selection is listed below with an indication of their ranges of applicability. Usually for T/Tm < 0.3, work-hardening processes dominate and primary creep is observed which can often be described by a logarithmic expression of the form εc = α ln (1 + βt)
(5.4)
where α and β are parameters which in general are functions of stress and temperature. Within the temperature range 0.3 0.5, Equation (5.5) can still be employed [11], but an alternative expression that has been used is −t + ε˙ sc t εc = εt 1 − exp τ
(5.6)
Other empirical laws have been proposed that have wider applicability than those just presented and which can also accommodate tertiary creep [11–13]. Two representative equations are 1
εc = αt 3 + βt + γ 2
(5.7)
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and εc = θ2 {1 − exp (−θ2 t)} + θ2 {exp (−θ4 t) − 1}
(5.8)
In their most general formulations, each of these parameters consists of a summation of terms involving stress and temperature. The equations describe primary, secondary and tertiary creep. They can correlate a wide spread of behaviour because of the number of disposable parameters that are used. They do however need a large body of data to identify all the terms. They are of most use in extrapolating experimental data to longer times. Details of describing and modeling the creep curve have been presented. However, in most circumstances and especially in the case of the modeling carried out in this chapter, the simple secondary creep strain rate ε˙ sc or the average creep strain rate ε˙ A (shown schematically in Figs. 5.1 and 5.2) are sufficiently accurate to describe the creep response of metals and structures. These are represented by a power law ε˙ sc = As σ n s
(5.9)
and the average creep rate obtained directly from creep rupture data (e.g. in Figs. 5.1 and 5.2) has been proposed to account for all three stages of creep as ε˙ A =
εf = ε˙ 0 tr
σ σ0
n A = A Aσ nA
(5.10)
This is particularly useful in modelling the crack tip where the combinations of the three regimes interact to develop damage, crack initiation and growth within a process zone. The models developed in this chapter make use of Equations (5.9), (5.10) to predict the creep response of structures containing cracks.
5.1.4 Complex Stress Creep Because creep deformation is not linearly dependent on stress, the effects of stresses that are applied in different directions cannot be superimposed linearly. However, it is found experimentally that: (i) hydrostatic stress does not affect creep deformation; (ii) the axes of principal stress and creep strain rate coincide; (iii) and no volume change occurs during creep. These observations are the same as those that are made for plastic deformation [15]. This is not surprising when both processes are controlled by dislocation motion. The observations imply that the definitions of equivalent stress and strain increment used in classical plasticity theory can be applied to creep provided strain rates are written in place of the plastic strain increments. Therefore, for creep, the Levy–Mises flow rule becomes
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ε˙ 3c ε˙ 1c ε˙ 2c ε¯˙ c = = = (5.11) [σ1 − 0.5(σ2 + σ3 )] [σ2 − 0.5(σ1 + σ3 )] [σ3 − 0.5(σ1 + σ2 )] [σ¯ ] Equation (5.11) satisfies the experimental observations (i) to (iii) provided appropriate definitions are chosen for σ¯ and ε˙ c . From (i) and the observation that dislocations are mainly responsible for creep, it may be inferred that shear stresses govern creep deformation so that either the Von-Mises or Tresca criterion can be employed. With the Von-Mises definition being given as 1 σ¯ = √ = [(σ1 − σ2 )2 + (σ2 − σ3 )2 (σ3 − σ1 )2 ]1/2 2
(5.12)
and √ + 2 2 2 ,.1/2 2 c ε˙ 1 − ε˙ 2c + ε˙ 2c − ε˙ 3c + ε˙ 3c − ε˙ 1c ε˙ = 3 c
Assuming the Tresca definition σ¯ = (σ1 − σ2 )
(5.14)
ε¯˙ = 2/3(˙ε1c − ε˙ 3c )
(5.15)
and
The Von-Mises definition can be regarded as a root mean square maximum shear stress criterion and the Tresca definition as a maximum shear stress criterion. Most investigations of equivalent stress criteria were carried out mainly on thin-walled cylinders subjected to different combinations of tension, torsion and internal pressure [16]. The case of internal pressure alone will now be considered as an application of the complex stress creep analysis. It can be shown for any complex stress state that the equivalent stress calculated using the Tresca definition is always greater than, or equal to, that determined from the Von-Mises definition. Use of the Tresca criterion will, therefore, always produce the same, or a higher, creep rate than is obtained from the Von-Mises criterion. As most experimental results usually fall between the two predictions [17–18] assumption of the Tresca criterion is therefore likely to be conservative. The list of creep laws and models presented for describing the time dependence of creep and multiaxiality effects are by no means extensive. For further reading the reader should go to the references which cover the range of models in depth [17–20].
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5.1.5 Influence of Fatigue in Uncracked Bodies Under combined static and cyclic loading at elevated temperatures creep and fatigue processes can take place together [34–36]. It is possible for these processes to occur independently or in conjunction depending upon the controlling mechanisms in each case. In the former case failure will be dominated by the process which occurs most rapidly and in the latter by the slowest process. When creep mechanisms govern, it is likely that the fracture surface will be intergranular and when fatigue processes dominate transgranular. When both mechanisms contribute to failure a mixed mode of fracture may be expected. Cumulative damage laws are available for dealing with superimposed mean and cyclic loading and for assessing the influence of variable amplitude cycling. Most engineering components experience variable amplitude loading during operation. It is usually supposed that fatigue damage incurred under these conditions can be accommodated using Miner’s cumulative damage law. This states that failure occurs when the fractional damage accumulated at each condition sums to one, that is, - i ti n3 n1 + + + . . . = 1 or =1 N1 T1 N3 N
(5.16)
The relation should be used as a general guide only as experimental evidence indicates that the fractional damage suffered at failure can range between about 0.5 and 2.0. This can usually be attributed to sequence of loading effects which are not accommodated in a simple expression like the Miner’s Law. It has been proposed that when creep and fatigue processes occur independently, the fractions of damage incurred by each mechanism can be summed separately to predict failure when -1 - n =1 + tr N
(5.17)
Other approaches have been proposed for dealing with more complicated interaction effects. The two most common are due to Coffin [34] and Manson [35]. In the former, a frequency term is incorporated to allow for time-dependent effects. In the latter, a strain range partitioning (SRP) approach is adopted for separating the total strain range of each cycle into creep, fatigue and mixed creep and fatigue components. With this method, a total of four components can be identified depending upon the type of loading cycle and whether hold times are present. Further reading to gain detailed insight into the fatigue of uncracked bodies can be found in the literature [10–21].
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5.2 Fracture Mechanics Parameters in Creep and Fatigue For situations where linear elastic conditions prevail (short times and/or low loads) the linear elastic stress intensity factor, K, may be used to predict creep crack growth either in creep or fatigue [22–23]. For non-linear conditions the J integral can be used to define a particular contour integral that is equal to the energy release rate in a non-linear elastic material as ∂u i W dy − Ti ds (5.18) J= ∂x Γ
where W is the non-linear elastic strain energy as shown in Fig. 5.3 is given as εi j W =
σi j dεi j
(5.19)
Γ
and Ti is given as Ti = σ ij nj and nj is the normal unit vector outside from the path Γ as shown in Fig. 5.4. This contour integral is a conservation integral and provides the path-independent values when contours are taken around a crack tip [24]. As the elastic energy release rate G is related to K, the J integral, which is also energy release rate in non-linear materials, can also characterise the stress and strain
ni x Γ
Fig. 5.3 Integration path for J integral
y
σ
W
Fig. 5.4 Strain energy W definition
0
ε ij
ε
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fields around a crack tip. When non-linear behaviour of a material is expressed by a power law as
ε εo
=
σ σo
N (5.20)
The stress and strain at a distance r are predicted by the following equations using the J integral. The stress and strain fields are often called the HRR field [25–26]. These are described as 1/(N +1) J σi j = σo σ˜ i j (5.21) In σo εo r εi j = εo
J In σo εo r
N /(N +1)
ε˜ i j
(5.22)
The calculated value of In [25] varies between 2–6 for a range of n between plane stress to plane strain [25]. Because the stress and strain fields given by Equations (5.21) and (5.22) are based on small-deformation theory, the true stress state will be different, close to a crack tip where the assumption of small-deformation theory is not adequate. For this reason, the stress or strain field characterised using the J integral is not the actual state at a crack tip itself. However, the J integral can still be considered as a valid fracture parameter as long as the fracture process zone is related to the surrounding J integral stress and strain fields. This is similar to the concept of applying the stress intensity factor for describing fracture in a material which undergoes small-scale yielding.
5.2.1 Creep Parameter C∗ Integral The arguments for high-temperature fracture mechanics essentially follow those presented above. For creeping situations, where elasticity dominates, the stress intensity factor K may be sufficient to predict crack growth. However, as creep is a non-linear, time-dependent mechanism even in situation where small-scale creep exists, linear elasticity may not be the answer. The behaviour in plasticity can be compared to the creep response of a body by observing that Equations (5.9) and (5.20) are similar. Hence, by analogy, using the J definition estimation procedures the C∗ parameter can be developed. Thus, in the case of large-scale creep where stress and strain rate determine the crack-tip field case, the parameter C∗ analogous to J has been proposed for this purpose. Substantial body of work exists in this respect [22–24] and the C∗ integral has been widely accepted as the fracture mechanics parameter for this purpose both in standards [3–8] as well as in Codes of Practices [4, 7, 9].
5 Predicting Creep and Creep/Fatigue Crack Initiation and Growth
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The theoretical definition of the C∗ integral, therefore, is obtained by substituting strain rate and displacement rate for strain and displacement of the J integral defined by equation as . / C ∗ = ∫ Ws∗ dy − Ti (δ u˙ i /δx )ds Γ
(5.23)
and W∗ is given as ε˙i j c
Ws∗ =
σi j d ε˙ icj
(5.24)
o
By analogy between non-linear elasticity and non-linear viscosity, the stress and strain rate in materials where (u˙ i = du i /dt) is the displacement rate. The other notations are the same as in the J integral definition. As the J integral characterises the stress and strain state, the C∗ integral is also expected to characterise the stress and strain rate around a crack. C∗ may also be interpreted as an energy-release rate analogous to the energy definition of J [22], so that C∗ = −
1 dU ∗ B da
(5.25)
This energy estimate of the C∗ integral has been widely used as a parameter for correlating CCG rate under steady-state creep conditions [22–23]. For a non-linear elastic material, it was shown that the asymptotic stress and strain fields are expressed by Equations (5.21) and (5.22). For creep in order to relate the uniaxial properties to the stress and strain rate distributions ahead of the crack tip, the asymptotic stress and strain rate fields in creep are used and they are thus expressed by [25] σi j = σo ε˙ i j = ε˙ o
C∗ In σo ε˙ o r C∗ In σo ε˙ o r
1/(n+1) σ˜ i j (θ, n)
(5.26)
ε˜ i j (θ, n)
(5.27)
n/(n+1)
Therefore, the stress and strain rate fields of non-linear viscous materials are also HRR-type fields. In the next section this description of the stress and strain filed ahead of a creep crack is used to develop a model which can predict crack initiation and growth at elevated temperatures.
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5.3 Predictive Models in High-Temperature Fracture Mechanics At elevated temperatures where creep is dominant, time-dependent crack growth is observed. The rate of this time-dependent crack growth is measurable and parameters are needed to predict it. In order to predict the crack behaviour in such materials, ˙ creep crack growth rate a(= da/dt) must be estimated using appropriate parameters. Several fracture parameters have been applied for this purpose. The most commonly used parameters are stress intensity factor K, the C∗ integral [8, 22–23, 27–33]. Steady-state crack growth rate a˙ s is usually given as follows using these parameters: a˙ s = AK m
(5.28)
a˙ s = DC ∗φ
(5.29)
A suitable parameter to describe crack growth at elevated temperature will depend on material properties, loading condition, size, geometry and the period of time during which crack growth is observed. Any of these variables could affect the stress state at the crack tip of a creeping material, hence making the cracking to behave in a ductile or brittle manner. The testing standard ASTM E1457 [8] goes to a considerable length to quantify these regions with respect to the appropriate parameters.
5.3.1 Derivation of K and C∗ The definitions for the stress intensity factor K are readily available for numerous geometries in the literature [38–41]. For C∗ this analysis is more complicated and the parameter is both non-linear as well as time dependent. There are a number of ways to derive C∗ [22, 23] both numerically and experimentally. However, only the validated methods for laboratory specimens and components adopted in standards and codes of practice are presented here. C∗ is estimated experimentally, using Equation (5.25), from measurements of creep load-line displacement according from tests carried out on compact tension (C(T)) specimens based on the recommendations of ASTM E 1457-07 [8] giving C∗ =
˙c P ·Δ ·F Bn · (W − a)
(5.30)
Values of F as function of n are given for a number of geometries in ASTM E1457-07. The crack initiation and growth rate data obtained from C(T) tests are usually considered as ‘benchmark’ for creep crack growth properties of the materials in the same way as creep strain rate and rupture for uniaxial creep tests. These data can then be used directly in crack initiation and growth prediction models as described in the different codes [3–9] to estimate residual lives in components. For
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components, such as cracked pipes and plated, C∗ must be determined from finite element analysis or reference stress methods described briefly below. In most cases, the reference stress approach of evaluating C∗ is adopted in line with that used in a number of defect assessment codes [3–9]. With this approach C∗ is expressed approximately as Cr∗e f = σr e f .˙εr e f .
K σr e f
2 (5.31)
Where the reference creep strain rate ε˙r e f at an applied stress σr e f can be derived directly from uniaxial data. When described in terms of the Norton creep law in Equations (5.9) and (5.31) can be rewritten as 2 Cr∗e f = A.σrn−1 e f .K
(5.32)
The typical value for n is between 5 and 12 for most metals. In addition, average creep rate, ε˙ A , from Equation (5.10) obtained directly from rupture data, described earlier to account for all three stages of creep as an approximate method can be used for estimating ε˙ A−r e f in conjunction with uniaxial data and σr e f .
5.3.2 Example of CCG Correlation with K and C∗ Figure 5.5a represents a typical relationships for creep crack growth a˙ versus the linear stress intensity factor K in compact tension specimens for an aluminium alloy Al-2519 at 135◦ C. Crack growth might also be characterised to a certain extent by the stress intensity factor K in such ‘creep brittle’ materials using Equation (5.28). It is clear, however, that K does not adequately correlate the data as different test loads 100
Crack Growth Rate, da/dt (mm/h)
100
10–1
10–1
10–2
Al 2519 at 135°C K = 17.07 (B = 22 mm) K = 18.97 (B = 22 mm) K = 18.24 (B = 22 mm) K = 16.37 (B = 22 mm) K = 15.36 (B = 22 mm) K = 21.44 (B = 6.35 mm) K = 20.44 (B = 6.35 mm) K = 19.53 (B = 6.35 mm) K = 18.22 (B = 6.35 mm) K = 18.97 (B = 6.35 mm)
–3
10
10–4
10–5 10
20
30
Stress Intensity Factor, K (MPa
(a)
40
a 10–2 (mm/h)
10–3
10–4 10–3
10–2
10–1
100
101
102
C* (KJ/m2h)
m1/2)
(b)
Fig. 5.5 A typical relationship for creep crack in compact tension specimens growth versus (a) K for an aluminium alloy Al-2519 at 135◦ C and (b) C∗ correlation for a 1CrMoV steel at 550◦ C showing an initial transient ‘Tail’ and a steady crack growth region
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initiate different cracking rates. In Fig. 5.5b, the non-linear C∗ parameter correlating cracking rate in a 1CrMoV steel at 550◦ C is more appropriate. The C∗ correlation in Fig. 5.5b exhibits an initial transient ‘tail’ and a steady crack growth region. If a material is elastic, immediately upon loading, the stress distribution around the crack tip will be elastic. For this situation, the stress and strain at the crack tip are described by linear elastic stresses. The ‘transient’ region will be modelled later in this chapter in order to derive initiation times. Thus, in a material that shows little creep deformation, the stress distribution will remain virtually unchanged by creep and the tail is usually small. On the other hand, in a ‘creep ductile’ material, where large creep strains can be dominant and the ‘tail’ is usually larger. In the extreme where creep deformation is prevalent everywhere in the material, the singularity at the crack tip will be lost and correlation in terms of the reference stress on the uncracked ligament could be obtained. Since the reference stress is a parameter which describes the overall damage across an uncracked ligament and may characterise net section rupture of the ligament rather than creep crack growth. For materials between the two extremes of creep brittle and very creep ductile, substantial creep deformation could accompany fracture, and stress redistribution will occur around a crack tip but a singularity by the crack will still remain. For this situation, creep crack growth will be characterised by C∗ using Equation (5.29). For engineering metals, most experimental evidence suggests that the widest range of correlation is achieved with C∗ [27–33].
5.3.3 Modelling Steady-State Creep Crack Growth Rate Once a steady-state distribution of stress and creep damage has been developed ahead of a crack tip, it is usually found that creep crack growth rate can be described by Equation (5.28). Most often, the constants D and φ are obtained from tests that are carried out on compact tension (C(T)) specimens based on the recommendations of ASTM E 1457-07 and described above. Using the steady-state assumptions at the crack tip model of creep crack growth under steady state conditions, proposed previously [27], is briefly presented in this section. The model called the NSW model is based on stress and strain rate fields characterised by the C∗ integral combined with a creep damage mechanism and ductility exhaustion at the crack tip. For a material which deforms according to the Norton’s creep law described by Equation (5.9), the stress and strain rate distributions ahead of a crack are given by Equations (5.26) and (5.27) respectively. When the creep damage accumulated within the creep zone shown in Fig. 5.6a reaches a critical failure value, the crack is postulated to progress. If the material starts to experience creep damage when it enters the process zone at r = rc and accumulates creep strain εij by the time it reaches a distance r from the crack tip, the condition for crack growth is given using the ductility exhaustion criterion as
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σ Creep Strain Rate
r
.
∫ε dt = εφ
θ .
as
Creep Process Zone rc
Distance r
a)
rc
b)
Fig. 5.6 Concept of the steady-state NSW model, (a) the creep process zone, (b) creep strain rate distribution in the process zone
r εi j =
ε˙ i j dt
(5.33)
r =rc
Using the strain rate distribution in Equation (5.27), the creep strain can be written as
r εi j =
εo r =rc
c∗ In σo ε˙ o r
n/n+1 ε˜ i j
dt dr dr
(5.34)
If a steady (constant) crack growth rate a˙ s is assumed dr/dt = a˙ s
(5.35)
Then Equation (5.34) is analytically integrated as εi j = (n + 1)˙εo
C∗ In σo ε˙ o
n/(n+1)
. 1/(n+1) / rc − r 1/(n+1)
(5.36)
When the creep ductility considering the stress state (multiaxial stress condition) is given by ε∗f , substituting εi j = ε∗f at r = 0 into Equation (5.36) gives (n + 1)˙εo a˙ s = ε∗f
C∗ In σo ε˙ o
n/(n+1)
. / ε˜ i j rc1/(n+1)
(5.37)
The non-dimensional function ε˜ i j is normalised so that its maximum equivalent becomes unity. Hence, assuming this maximum value, the constants D and φ in Equation (5.29) become
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D=
(n + 1)˙εo ε∗f
1 In σo ε˙ o
n/n+1 (5.38)
where n (5.39) n+1 The creep ductility considering the stress state ε∗f is equal to the uniaxial creep ductility ε f in plane stress conditions. This may be estimated from the multiaxial stress factor (MSF), using φ=
ε∗f = MSF. ε f
(5.40)
The value of the MSF may be estimated using appropriate models [42, 43]. In the present study the Cocks and Ashby void growth and coalescence model [40] was used. The calculated MSF using the Cocks and Ashby model is given as 0 n − 1/2 σm 2 n − 1/2 sinh 2 = sinh MSF = εf 3 n + 1/2 n + 1/2 σe ε∗f
(5.41)
This model will make ε∗f dependent on the stress triaxiality, the ratio between the mean (hydrostatic) stress and equivalent (Mises) stress, σ m /σ e , and the creep stress exponent, n. The stress triaxiality ratio, σ m /σ e , may be evaluated using FE for specific geometries or estimated for some geometries. For plane strain conditions, using Equation (5.41), ε∗f could range anywhere between 25 to 80 times smaller than ε f depending on the creep void growth model assumed [44, 45]. The relationships are relatively insensitive to the creep index n. Originally, in the NSW model, ε f /ε∗f = 50 was recommended as a extreme upper-bound value for plane strain conditions [27]. This factor was further refined to ε f /ε∗f = 30 for most relevant engineering materials [46]. Furthermore, if available data exist for a range of sizes and geometries for a specific material, this factor can be further reduced depending on the level of conservatism that is sought. Thus from the relationship between normalised ductility ε f /ε∗f and the ratio of hydrostatic tensile stress over the equivalent stress (σ m /σ e ), it is apparent that an increase in hydrostatic tension causes a large reduction in creep ductility [44, 45]. As an example, experimental data bands in Fig. 5.7 show a range of data for 316H stainless steel. The variation in cracking rate is due to effect of specimen size and geometry and scatter. The cracking rate for the for 316H stainless steel at 550◦ C varies by a factor of ∼7 and in the extreme a factor of 15 for the large specimens. The NSW model from Equation (5.37) predicts a factor of 30 using a failure strain εf = 0.2 using minimum strain rate date to derive the creep index n. The average creep strain rate ε˙ A can also be used from Equation (5.10) to derive n since the material at the crack tip undergoes three stages during its transverse of the creep zone similar to its uniaxial creep behaviour.
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Fig. 5.7 Size and geometry effects on steady-state CCG rate for C(T) and middle tension M(T) ◦ data versus C∗ for 316 H stainless steel at 550 showing the effect of constraint. NSW plane stress/strain prediction (×30) was derived from Equation (5.37). NSW-MOD prediction (grey band x ∼6) Plane stress/strain was derived from Equation (5.54)
To simplify the predictions for crack rate with respect to C∗ for most steels, n > 1, usually 5–10, so that Equation (5.37) becomes relatively insensitive to the value of rc . Considering most engineering materials, Equation (5.37) has been reduced to [32]: a˙ =
3C ∗0.85 ε∗f
(5.42)
The model therefore indicates that crack growth rate should be inversely proportional to the creep ductility ε∗f appropriate to the state of stress at the crack tip. The bounds on this value for plane stress and plane strain are as described above. Using this simplified prediction is particularly useful when the user does not have material cracking data and needs a conservative initial estimate using the plane strain upper-bound lines for prediction component cracking behaviour.
5.3.4 Transient Creep Crack Growth Modelling A transient creep crack growth model has also been proposed as described schematically in Fig. 5.8 [22, 47]. The differences between the steady crack growth model and the transient crack growth model are that transient creep crack growth starts from the undamaged state in the creep damage process zone, whereas the steadystate model assumes a steady-state distribution of damage in the process zone. Also creep damage accumulates in elements with width dr ahead of a crack tip. Steady
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K.M. Nikbin
.
∫ε dt = εφ
a)
Distance r
b)
rc
Fig. 5.8 Schematic of the transient NSW model process zone showing (a) the creep zone and (b) the creep strain rate field both showing a schematic view of pseudo-uniaxial specimens operating in the creep zone under local uniaxial conditions
state creep crack growth can be predicted analytically but numerical integration is necessary for the transient model. From the transient creep crack growth model, the initial crack growth rate a˙ can be estimated by the following equation assuming secondary creep behaviour: ε˙ o a˙ i = ∗ εf
C∗ In σo ε˙ o
n/n+1 (dr )1/n+1
(5.43)
This can be further simplified [48] to relate the initial transient crack growth a˙ i to the steady state cracking rate a˙ s as (1/n+1) dr 1 a˙ i = a˙ s n + 1 rc
(5.44)
For the same reason described for the steady state model for the creep zone rc in Equation (5.37), the choice of dr as the size of finite creeping region behaving as a pseudo round uniaxial specimens, as shown schematically in Fig. 5.8 appears to be insensitive for predicting the initial cracking rate because it is raised to a small fractional power when n>>1. For element i, the model predicts a crack growth rate a˙ i of ε˙ o a˙ i = ∗ ε f − ε1.i−1
C∗ In σo ε˙ o
n/n+1 (dr )1/n+1
(5.45)
where εi,i –1 is the creep strain accumulated at r = ri = idr when the crack reaches R = (ri –1) = (i –1)dr The ligament dr can be chosen to be a suitable fraction of rc . However since dr/rc is raised to a small power in Equation (5.45) this will give a˙ i =
1 a˙ s n+1
(5.46)
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For most engineering materials, therefore, the initial crack growth rate is expected to be approximately an order of magnitude less than that predicted from the steady-state analysis. The cracking rate will progressively reach the steady-state cracking rate as damage is accumulated. Thus, using the transient and the steady-state NSW models, the extreme regions of the crack growth behaviour is covered adequately within the bounds of the present model. Figure 5.9 shows the cracking rate bounds covering a wide range of a number of alloys taken from [32]. The lighter shade is the band of data for creep ductile materials (such as 316H, P22 steels, 1CMV). Figure 5.9 acts as a material-independent engineering crack growth assessment diagram where cracking rate is multiplied by the uniaxial ductility of the material. Note that most of the grey data band below the NSW plane stress line is in the initiation and transient region and not steady cracking rates. Therefore, the transient region which is usually below the plane stress line, assuming that the initial cracking rate is given as a˙ i ≈ a˙ s /10, can be introduced as an extreme lower-bound band to estimate the initiation times and the transition ‘tail’ region of damage development observed in many test pieces and components. Fig. 5.9 Material-independent engineering assessment diagram showing CCG rate (da/dt. εf ) versus C∗ over the plane stress/strain and transient bounds using the NSW and NSW-MOD prediction lines
Therefore, the most conservative line to choose for life assessment would be the plane strain NSW line. However, for a narrower range of data sets of creep ductile materials (shown in Fig. 5.9) this may be overly conservative. Furthermore, using known specific material properties with standard size and geometries a much lower acceptable safety factor would be needed to predict cracking behaviour and reduce over-conservatism. To deal with this, a modified NSW-MOD model [44, 45] based on the assumption that damage would occur over a wide range of angles at the crack tip was introduced. NSW-MOD in effect reduced the crack growth bounds relevant to ‘creep ductile’ engineering materials. This is shown in the example in Fig. 5.9 where it is found that an approximate factor of 6 predicts the divergence
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between plane stress and plane strain cracking rates. This modification to NSW will be described in detail later in this chapter.
5.3.5 Predictions of Initiation Times ti Prior Onset of Steady Creep Crack Growth In creep crack growth experiments and in real components, an incubation period when the crack seems to be stationary can often be observed. In cases of sensitive and vital component parts an initiation of a crack could mean the end of life. Also sometimes the incubation time occupies most of the life of a cracked body. Therefore, methods to predict this period are important. In addition, in order to reduce conservatism in predicting crack growth life times it may be important to incorporate the incubation time into creep crack growth predictions in practical applications. There are many ways to deal with this problem and as yet no one methodology can cater for all conditions. In the transient analysis described earlier the initiation time prediction was calculated from the NSW model. Lower and upper bounds can be obtained from the steady creep crack growth model and transient model shown above as follows: ε∗f rc = Lower bound ti := a˙ s (n + 1) εo Upper bound ti :
ε∗f rc = a˙ i εo
= (n + 1)
In σo ε˙ o rc C∗
r 1/n+1 c
dr
In σo ε˙ o rc C∗
n/n+1
n/n+1
rc 1/n+1 dr
ti (lower bound)
(5.47)
(5.48)
(5.49)
Further, the upper-bound incubation can be obtained by taking a˙ = a˙ i ti =
ε∗f o εo
In σo ε˙ o rc C∗
ν/n+1 (5.50)
Alternative estimates of an incubation period can also be obtained from the approximate creep crack growth rate Equation (5.42) assuming steady state. The prediction based on an approximate steady-state cracking rate expression will be the lower bound of the incubation period. ti =
rc ε∗f 3C ∗0.85
(5.51)
When the initial crack growth rate is used instead of the steady-state crack growth rate, the upper bound is determined as
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Fig. 5.10 Experimental incubation periods for different geometries for crack extension Δai = 0.2 mm for Type 316H austenitic stainless steel at 550◦ C compared to NSW crack initiation predictions (using dotted lines) based on Equations (5.51) and (5.52) and on uniaxial creep properties for Type 316H austenitic stainless steel at 550◦ C shown in Table 5.1
ti =
(n + 1) rc ε∗f 3C ∗0.85
(5.52)
Figure 5.10 shows the predicted ti from Equations (5.51) and (5.52) versus experimental incubation times for different geometries for crack extension Δai = 0.2 mm for Type 316H austenitic stainless steel at 550◦ C. The scatter is typical of hightemperature data where 0.2-mm crack extension occurring over long times is difficult to measure accurately. Figure 5.10 does not indicate a clear difference in initiation times due to geometry. Whereas the overall inherent scatter of data shows a much larger difference. Using the steady-state model the prediction lines from Equation (5.52) are conservative but using the initiation line from Equation (5.51) the lower bound covers most of the data.
5.3.6 Consideration of Crack Tip Angle in the NSW Model Based on the form of the crack tip fields in Equations (5.26) and (5.27) and using a ductility exhaustion argument as in the NSW model, it has been shown that the ˙ may be predicted over the bounds of plane strain/stress creep crack growth rate, a, conditions. It was also recommended that under plane stress conditions the multiaxial ductility, εf∗ , be taken as the uniaxial failure strain, εf , and εf /30 under plane strain conditions. This recommendation would cover a wide range of materials and conditions which may make it necessarily over conservative.
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In the NSW model, it is implicitly assumed that fracture occurs first at the value of the crack tip angle, θ , at which the equivalent creep strain, quantified by ε˜ e (θ, n) in Equation (5.27), reaches its maximum value. A more general expression can be obtained, which considers the dependence of ε˜ e (θ, n) and εf∗ on angle, θ . For this condition, the NSW model may be extended to give a modified crack growth rate, (hereafter referred to as the ‘NSW-MOD model’) as described below.
5.3.7 The New NSW-MOD Model In the NSW model the value of the non-dimensional equivalent stress function, used to describe the Mises equivalent stress and strain rate fields in Equations (5.26) and (5.27), was taken to be its maximum value of unity. This is considered to be a conservative measure and implicitly assumes that failure will occur first at the angle, θ , where σ˜ e attains its maximum value [44, 45]. Figure 5.11 shows the dependence of crack-tip stress field and ε f /ε∗f on θ under plane stress and strain. Since both σ m and σ e depend on n and θ , then the MSF is also a function of n and θ which can be then estimated by considering all angles. 1
1
Plane strain n=5
Plane stress 0.1
σ~e (θ , n) 0.1
εf*/εf
n
n = 20
Plane strain
Plane stress 0.01
n=5 n = 10 n = 20
n = 10 0.01
a)
0
45
90
θ
135
180
0.001
b)
0
45
90
θ
135
180
Fig. 5.11 Dependence of (a) σ˜ en on angle θ and n and (b) ε∗f on angle θ and n (note that the lines in this case are relatively insensitive to n)
Figure 5.11 shows the stress distributions σ˜ e (θ, n) and ε f /ε∗f against θ for n = 5, 10 and 20 under plane stress and strain. It is seen that the maximum value of [σ˜ e (θ, n)]n is unity at θ ≈ 0◦ and 90◦ under plane stress and plane strain conditions, respectively. At θ = 0◦ , which is the condition assumed in the NSW model, the difference in the value of [σ˜ e (θ, n)]n under plane stress and plane strain conditions can be up to a factor of 50–100 depending on the value of n. It is also seen in Fig. 5.11(b) that εf∗ increases with angle, both under plane stress and plane strain conditions. At θ = 00 the difference of εf∗ between plane stress and plane strain conditions is well above a factor of 100. This variation will also affect the value of εf∗ in Equation (5.41) which itself will vary for a range of stress states. In the NSW-MOD1model, therefore, failure is considered to occur first where the angular function σ˜ en MSF attains its maximum value. The resulting crack growth rate equations are
5 Predicting Creep and Creep/Fatigue Crack Initiation and Growth
1
a˙ N SW −M O D = (n + 1) (Arc ) n+1
C∗ In
n n+1
127
1 σ˜ en (θ, n) ε f MSF (θ, n) max
(5.53)
Giving a˙ N SW −M O D
ε˙ 0 = (n + 1) ∗ ε f (θ, n)
C∗ ε˙ 0 σ0 In
n/(n+1) rc1/(n+1) ε˜ e (θ, n)
(5.54)
.
.
10
10
8
8
aPE/aPS for θ = 0°
aPEmax / aPSmax
Comparing the maximum value of CCG rate (see Fig. 5.12(a)), the CCG rate for plane strain conditions is about 3–7 times faster than that for plane strain condition, although the ratio depends on the value of n. However, partly because of side grooves, experimentally creep crack growth in C(T) specimens is contained in the direction of θ = 0◦ . Hence, comparing the value of CCG rate at θ = 0◦ , the CCG rate under plane strain conditions is about 0.5–7 times faster than that under plane stress conditions, depending on the values of n (the ratio is decreasing with increasing of the values of n).
6 4
.
2
4
2
0
0
5
a)
6
.
10
15
20
n
5
b)
10
15
20
n
Fig. 5.12 Comparison of (a) the maximum value of CCG rate (a˙ max ) and (b) CCG rate at the angle of 0◦ between plane stress and plane strain conditions
From this analysis the indications are that in Fig. 5.9, as shown in the lighter shade data band, the upper bound can be reduced and modified for most creep ductile alloys such as the 316H stainless steel to a factor of ∼6–7. Clearly, this is an improvement from the factor of 30 predicted in Fig. 5.9 for the NSW model. It should also be noted if pedigree data were available for a particular batch the predicted bounds could be further reduced.
5.3.8 Finite Element Framework The NSW model using the ductility exhaustion and crack-tip constraint effects on damage has been independently verified using a finite element framework.
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Numerical predictions, using basic materials elastic–plastic–creep properties, may be developed to predict CCG rate in fracture mechanics geometries [44]. The elastic, plastic and creep strains are taken to be independent of strain rate giving the total strain as ε = εel + ε p + εc
(5.55)
Power-law creep behavior as in Equation (5.9) is assumed or using the materials average creep strain rate properties as in Equation (5.10). As an example of the modelling procedure the material properties for 316H stainless steel at 550◦ C are shown in Table 5.1. The plastic response of the material is assumed to be governed by a Mises flow rule with isotropic strain hardening, which was obtained by fitting to uniaxial tensile test data at 550◦ C. In the FE model, the yield strength of the material is taken to be its 0.2% proof stress, and the post-yield strain hardening response is treated as piece-wise linear up to the point relating to the ultimate tensile stress, σ UTS , and corresponding strain, εUTS , beyond which no strain hardening occurs. Table 5.1 Uniaxial tensile and creep properties for 316H at 550◦ C used Young’s modulus, E (GPa) σ 0.2 (MPa) σ UTS (MPa) εUTS (%) A (h–1 MPa–n ) n 140
170
415
33
–32
7.24×10
εf (%)
10.6 21
Figure 5.13 gives an example for a C(T) mesh with fine element sized in the expected crack growth line to allow for debonding on damage accumulation.
5.3.9 Damage Accumulation at the Crack Tip For identifying a criterion for damage development leading to a finite crack extension, the principle of the ductility exhaustion NSW model proposed above is used. An uncoupled damage approach is employed, where the rate of damage accumulation, ω, ˙ is controlled by the equivalent creep strain rate, ε˙ c , and determined from the relation ω˙ =
ε˙ c ε˙ c = ε∗f MSF ε f
(5.56)
where ε∗f is calculated from the uniaxial creep failure strain (creep ductility) of the material, εf , using the multiaxial strain factor (MSF) given in Equation (5.41). It should be noted that within an FE analysis the value of ε∗f changes for any fixed material point since it depends on the derived triaxiality through Equation (5.41). Therefore, ε∗f changes with time and crack length and at every angular position ahead of the crack tip as the stress redistributes locally.
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a W Fig. 5.13 Example of finite element mesh for a C(T) specimen with the centre line region refined to allow for virtual crack growth simulation by element debonding at the nodes based on the MSF criterion
(a)
(b)
Fig. 5.14 (a) Damage contour plot for an elastic–plastic–creep analysis of a CT specimen using the node-release technique. (b) Contours of plastic strain at the same amount of crack growth
A creep damage parameter, ω, can be defined such that 0 ≤ ω ≤ 1 is introduced to quantify the extent of creep damage in a body at any instant, which is evaluated from the time integral of ω˙ in Equation (5.56), i.e.
t
ω= 0
t ω˙ dt = 0
ε˙ c dt ε∗f
(5.57)
Initially, at t = 0, it is assumed that there is no damage anywhere in the material, thus ω = 0 and failure occurs at a material point when ω = 1. Note that the evolution of damage is not coupled to creep deformation, such that the creep rate of the material is not enhanced due to the accumulation of damage.
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In a numerical model of, for example, a C(T) specimen, the parameter C∗ can be calculated from the calculated FE creep load line displacement rate as in Equation (5.30). Hence, independently, a virtual crack growth rate, da/dt, can be predicted using Equation (5.57) incrementally allowing crack-tip element debond when ω reaches a value of unity and with the current crack-tip position taken to be at the position of the most recently released node. Figure 5.15 show a graphic example of damage buildup and plastic deformation at the crack tip for the virtual cracking method described. Thus, by modelling the crack tip and using the MSF method used in the NSW and NSW-MOD models it has been shown that the results compare well [44]. Clearly there are possibilities that can be explored to develop robust virtual creep crack growth methods both to reduce and optimize the number of test and to predict component crack growth behaviour at elevated temperatures. Fig. 5.15 Influence of frequency on the crack profile of a corner crack tension (CCT) specimen of AP1 nickel-base superalloy tested ◦ at 700 C [49] showing the specimen cross section and the region of creep and fatigue domination
2 mm x-section
Fatigue creep
Fatigue
5.3.10 Elevated Temperature Cyclic Crack Growth With temperature increase, time-dependent processes become more significant even under fatigue loading but are dependent on the loading frequency. Creep and environmentally assisted crack growth can take place more readily since they are aided by diffusion and rates of diffusion increase with rise in temperature. A graphic example of creep and fatigue mechanisms on crack growth is shown in Fig. 5.15 [49]. The regions of high-frequency loading (10 Hz) on a square section corner crack tension specimen shows no effect of stress state as the crack extension is the same at the centre and at the surface of the specimen. However, when static creep is applied there is a clear state of stress effect. The creep regime in this case has been dealt with the modelling described above. For the fatigue component, the analysis is much simpler and has been described by the Paris Law described below. The summation of the two in the manner that the codes of practice [3, 4] treat creep/fatigue is described below.
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When alternating loads are applied to high-temperature structures, the crack growth in the structures will be subject to both creep and fatigue. The Paris law is usually adequate to describe fatigue at high frequencies given as da/dN = C ΔK c
(5.58)
Some interaction between creep and fatigue is expected under cyclic loading. Some of the causes of creep–fatigue interaction might be the enhancement of fatigue crack growth due to embrittlement of grain boundaries or weakening of the matrix in grains and enhancement of creep crack growth due to acceleration of precipitation or cavitation by cyclic loading. The importance of creep–fatigue interaction effects is largely dependent on the material and loading conditions and detailed analysis for every cycle will reveal a complex interaction of creep and fatigue mechanisms. Nevertheless, for the macro-cracking behaviour of a crack under creep/fatigue, the simple linear summation rule for creep–fatigue crack growth has been successfully applied to predict the crack growth in several engineering metals. This is given as da/dN = (da/dN )c + (da/dN ) f = (1/3600 f )(da/dN )c + (da/dN ) f
(5.59)
where da/dN is crack growth per cycle in mm/cycle, da/dt is crack growth rate in mm/hour, and f is frequency in Hz. There is clear experimental evidence for this general rule [4]. Figure 5.16 shows the fracture mode changes going from transgranular at high frequency to intergranular at lower frequencies. At the intermediate frequencies, there is a mixture of cracking mode indicating that the two mechanisms can run in parallel. Therefore, using the predictive model in creep cracking and fatigue in this chapter, the combined cracking response under creep/fatigue interaction can be predicted in a conservative manner by simple summation depending on the loading history.
5.4 Conclusions This chapter has covered topics relating to the prediction of creep and creep/fatigue crack initiation and growth, under static and cyclic loading, in engineering materials at high temperatures. A short overview of the present standards and codes of practice for fracture-based methods suggest that trends towards a fracture-based design and remaining life prediction is an important methodology for the future. Following a brief description of engineering creep parameters of basic elastic– plastic fracture mechanics methods, high-temperature fracture mechanics parameters are derived from plasticity concepts. Techniques are shown for determining the
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a) 10 Hz
b) 0.1 Hz c) .01 Hz
d) 0.001 Hz Transgranular (Fatigue) >> Intergranular (Creep)
Fig. 5.16 Effects of frequency on mode of failure for AP1 Astroloy Nickel-base superalloy tested at 700◦ C [49]
creep and fatigue fracture mechanics parameters K and C∗ using experimental crack growth data. Models to predict crack initiation, growth and failure under creep conditions are also presented. An analytical failure strain constraint based model called the NSW model has been shown to cover the range of plane stress/strain behaviour found in creep crack growth. The model based on a ductility exhaustion argument and constraint level at the crack tip is able to predict, in a range of as much as a factor of 30 in crack initiation and growth at high temperatures, over the plane stress to plane strain regimes. By taking into account angular damage distribution around the crack tip, the model is further refined and named as the NSW-MOD model is able predict a much improved upper/lower bound of a factor of at most ∼0.5–7 (depending on the value of n) in steady-state crack initiation and growth over the plane stress/strain region in components containing defects. The presence of cyclic load is assumed to introduce a cycle-dependent fatigue component with a linearly summed cumulative effect with the creep. In order to verify the models, a virtual crack extension numerical model was developed which made use of the development of damage at the crack tip based on crack-tip constraint and ductility exhaustion. As advanced computational capabilities hardware and soft-ware improve and new expert systems for software are developed, the task to use predictive methods are made easier. Where there is known to be scatter in the data, probabilistic methods rather than deterministic analysis will need to be used. The models presented in this chapter can be used in predictive techniques in fracture mechanics-based creep crack initiation and growth rate analysis. In conclusion, the creep/fatigue modelling presented can be used in component design of metallic parts, as well as in life assessment of cracked components at elevated temperatures and in predicting
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virtual cracking behaviour in fracture mechanics specimens in order to reduce overconservatism in the analysis.
5.5 Nomenclatures and Abbreviations
˙c Δ θ ε˙ 1c , ε˙ 2c , ε˙ 3c ε˙¯ c ε˙ , ε˙ min , ε˙ A φ i, , Di a˙ i , a˙ s ε˙ , ε˙r e f , ε˙ A−r e f ε˙ A , ε˙ sc , ε f , ε∗f o u˙ i = du i /dt a, ai, af A, n As , ns AA , nA σ˜i j and ε˜ i j B, Bn , W C∗ c’, C’ D, H, α, β, m, φ da/dt or a˙ dr F G, K, ΔK In J N, da/dN N’,n n1 , n2 , n3 N1 , N2 , N3 nj P, Q R rc T, Tm
load line creep displacement rate in a cracked geometry the crack tip angle principal creep strain rates, corresponding equivalent creep strain rate creep strain rate, minimum and average creep strain rates constant for CCI versus C∗ in ti = Di C∗φi initial transient, steady state CCG rate (mm/h) creep rate, creep strain rate at reference stress, average creep strain rate at reference stress average creep rate, secondary creep rate, failure strain, appropriate failure strain relevant to crack-tip constraint displacement rate crack length (mm), initial crack length (mm), final crack length (mm) material constant, Norton’s creep index in ε˙ = Aσ n material constant, creep index value for secondary creep rate material constant, creep index value for average creep rate equivalent stress and strain functions of angle θ and n thickness (mm), net thickness (of a side-grooved geometry) (mm), width (mm) Steady-state creep correlating parameter (MJ/m2 .h) fatigue crack growth (FCG) rate index and material constant in da/dN= C’ΔKc’ material constants dependent on temperature and stress state. creep crack growth rate (mm/h) an increment of distance within creep zone rc factor which depends on geometry and creep index, n the elastic√energy release rate, the stress intensity factor √ (MPa m), stress intensity factor range Δ(MPa m) the non-dimensional factor of n or N’, J-integral Number of loading cycles, FCG/ cycle (mm/cycle) the hardening exponent, the creep index number of cycles spent at each condition the endurance or critical cycle at these conditions. the normal unit vector outside from the path Γ applied load (MN) activation energy Gas constant Creep zone size (mm) Absolute temperature, Absolute melting temperature
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Ti t, ti , tr U∗ W, W∗ εel , εp , εc σ 1, σ 2 and σ 3 θ 1, θ 2, θ 3 θ 4 Δai , Δaf Γ, ds α, β, γ , φ, D σ, σy σ m , σ e, σ ref σ , ε, r ω, ω˙ MSF CCG CCI FCG C(T), SEN(T), M(T), SEN(B), DEN(T), CS(T)
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the traction force test time (h), time to initiation (h), uniaxial rupture time (h). potential energy rate per crack extension Strain energy density change, strain energy density change rate elastic, plastic and creep strain components principal stresses stress and temperature dependent material parameters. initial crack extension (mm), total crack extension (mm) the line integration path is the length along the path Γ stress and temperature dependent material parameters. power index and material constant in da/dt = DC∗φ stress and normalized yield stress (MPa) hydrostatic stress, equivalent stress, reference stress (MPa) material constants as the stress (MPa) and strain (1/h) at a distance r from crack tip, creep damage, creep damage rate parameters Multiaxial strain factor Creep Crack Growth Creep Crack Initiation Fatigue Crack Growth compact tension, single-edge notch tension, middle tension, single-edge notch bend, c-ring tension specimens
Acknowledgement The author would like to express his gratitude to colleagues at Imperial College, British Energy and VAMAS work group and EU project partners in numerous projects undertaken to develop the findings in this chapter.
References 1. ASME Boiler and pressure vessel code, section XI: Rules for in-service inspection of nuclear power plant components, American Society of Mechanical Engineers, 1998 2. AFCEN, Design and construction rules for mechanical components of FBR nuclear islands, RCC-MR, Appendix A16, AFCEN, Paris, 1985 3. BS7910, Guide on methods for assessing the acceptability of flaws in fusion welded structures, London, BSI, 2007 4. R5, Defect assessment code of practice for high temperature metallic components, British Energy Generation Ltd., 2007 5. R6, Assessment of the integrity of structures containing defects, Revision 3, British Energy Generation Ltd., 2007 6. P.J. Budden, “Validation of the High-Temperature Structural In-tegrity Procedure R5 by Component Testing,” R5, Document, Vol. 8–9, July, 2003 7. B. Drubay, D. Moulin, and S. Chapuliot. A16: Guide for Defect Assessment and Leak Before Break, Third Draft, Commissariat a l’Energie Atomique (CEA), DMT 96.096, 1995 8. ASTM, “E 1457-07: Standard Test Method for Measurement of Creep Crack Growth Rates in Metals,” Annual Book of ASTM Standards, Vol. 3, 2007, pp. 936 –950 9. API RP-579-Recommended Practice (RP) “Standardized Fitness-for-Service Assessment Techniques for Pressurized Equipment Used in the Petrochemical Industry” American Petroleum Inst. 2004
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10. J.E. Dorn (Ed.), Mechanical Behaviour of Materials at Elevated Temperatures, McGraw-Hill, Inc., New York, 1961 11. A.J. Kennedy, Processes of Creep and Fatigue in Metals, Wiley, New York, 1962 12. F. Garofalo, Fundamentals of Creep and Creep-Rupture in Metals, MacMillan, New York, 1965 13. G.A. Webster, “A Widely Applicable Dislocation Model for Creep,” Phil. Mag., Vol. 14, 1966, pp. 775–783 14. J. Gittus, Creep, Viscoelasticity and Creep Fracture in Solids, Applied Science, London, 1975 15. F.A. Mclintock and A.S. Argon, Mechanical Behaviour of Materials, Addison-Wesley, Massachusetts, 1966 16. J.D. Lubahn and R.P. Felgar, Plasticity and Creep of Metals, Wiley, New York, 1961 17. A.E. Johnson, J. Henderson, and B. Khan, Complex Stress Creep, Relaxation and Fracture of Metallic Alloys, HMSO, London, 1962 18. I. Finnie and W.R. Heller, Creep of Engineering Materials, McGraw-Hill, New York, 1959 19. Y.N. Rabotnov, Creep Problems in Structural Members, F.A. Leckie, Ed., North Holland, Amsterdam, 1969 20. R. Viswanathan, Damage Mechanisms and Life Assessment of High-Temperature Components, ASM International, Metals Park, Ohio, 1989 21. L.M. Kachanov, Introduction to Continuum Damage Mechanics, Kluwer Academic Publishers, Dordrecht, 1986 22. G.A. Webster and R.A. Ainsworth, High Temperature Component Life Assessment, Chapman and Hall, London, 1994 23. A. Saxena, “Evaluation of C∗ for the Characterization of Creep Crack Growth Behavior in 304 Stainless Steel,” Fracture Mechanics: Twelfth Conference, 1980 ASTM STP 700, ASTM, pp. 131–151 24. H. Riedel and J.R. Rice, “Tensile Cracks in Creeping Solids,” Fracture Mechanics: Twelfth Conference, ASTM STP 700, ASTM, 1980, pp. 112–130 25. J.W. Hutchinson, “Singular Behavior at the End of a Ten-sile Crack in a Hardening Material,” J. Mech. Phys. Solids, Vol. 16, 1968, pp. 13–31 26. C.F. Shih, “Tables of Hutchinson-Rice-Rosengren Singular Field Quantities,” Brown University Report MRL E-147, Providence, RI. 1983 27. K.M. Nikbin, D.J. Smith and G.A. Webster, “Pre-diction of Creep Crack Growth from Uniaxial Data,” Proc. R. Soc. Lond. A, Vol. 396, 1984, pp. 183–197 28. A. Saxena, “Evaluation of Crack Tip Parameters for Characterizing Crack Growth: Results of the ASTM Round-Robin Program,” Mater. High Temp., Vol. 10, 1992, pp. 79–91 29. K.M. Nikbin, “Foreward: Creep Crack Growth in Components,” Guest Editor,’ Int. J. Pres. Ves. Pip. Elsevier Ltd., Vol. 80, No. 7–8, 2003, pp. 415–595 (July–August 2003) 30. K.H. Schwalbe, R.H. Ainsworth, A. Saxena, and T. Yoko-bori, “Recommendations for Modifications of ASTM E1457 to Include Creep-Brittle Materials,” Eng. Fract. Mech., Vol. 62, 1999, pp. 123–142 31. M. Tabuchi, T. Adachi, A.T. Yokobori, Jr., A. Fuji, J.C. Ha, and T. Yokobori, “Evaluation of Creep Crack Growth Properties Using Circular Notched Specimens,” Int. J. Pres. Ves. Pip., Vol. 80, 2003, PP. 417–425 32. K.M. Nikbin, D.J. Smith, and G.A. Webster, “An Engineering Approach to the Prediction of Creep Crack Growth,” J. Eng. Mater. Technol. Trans. ASME, Vol. 108, 1986, pp. 186–191 33. B. Dogan, K.M. Nikbin, and B. Petrovski, “Development of European Creep Crack Growth Testing Code of Practice for Industrial Specimens,” Proc. EPRI Int. Conf. on Materials and corrosion experience for fossil power plants, Isle of Palm, SC, USA Nov. 18–21, 2003 34. L.F. Coffin, “Fatigue at high temperature” in “Fatigue at elevated temperatures,” ASTM STP 520, 1973, pp. 5–34 35. S.S. Manson, “A Challenge to Unify Treatment of High Temperature Fatigue,” ASTM STP 520, 1973, pp. 744–775 36. J. Bressers, (Ed.) Creep and Fatigue in High Temperature Alloys, Applied Science, Barking, UK, 1981
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37. R.A. Ainsworth, G.C. Chell, M.C. Coleman, I.W. Goodall, D.J. Gooch, J.R. Haigh, S.T. Kimmins, and G.J. Neate, CEGB Assessment Procedure for Defects in Plant Operating in the Creep Range, Fatigue and Fracture of Engineering Materials and Structures, Vol. 10(2), 1987, pp. 115–127 38. H. Tada, Stress Analysis of Cracks Handbook. Paris Productions Incorpo-rated, 2nd edition, 1985 39. J.C. Newman and I.S. Raju, “An Empirical Stress In-tensity Factor Equation for the Surface Crack”. Eng. Fract. Mech., Vol. 15, No. 1–2, 1981, pp. 185–192 40. D.P. Rooke and D.J. Cartwright, Compendium of Stress Inten-sity Factors by. HMSO, London, 1976 41. C. Davies, K.M. Nikbin, and N.P. O’ dowd, “Experimen-tal Evaluation of the J or C∗ Parameter for a Range of Cracked Geometries,” ASTM STP 1480, 2007 42. J.R. Rice and D.M. Tracey, “On the Ductile Enlarge-ment of Voids in Triaxial Stress Fields,” J. Mech. Phys. Solids, Vol. 17, 1969, pp. 201–217 43. A.C.F. Cocks and M.F. Ashby, “Intergranular Fracture During Power-Law Creep Under Multiaxial Stress,” Met. Sci., Vol. 14, 1980, pp. 395–402. 44. M. Yatomi, N.P. O’ Dowd, and K.M. Nikbin, “Computational Modelling of High Temperature Steady State Crack Growth Using a Damage-Based Approach,” in PVP-Vol. 462, Application of Fracture Mechanics in Failure Assessment Computational Fracture Mechanics, ASME 2003, P. -S. Lam, Ed., ASME New York, NY 10016, 2003, pp. 5–12 45. M. Yatomi, K.M. Nikbin, and N.P. O’ Dowd, “Creep Crack Growth Prediction Using a Creep Damage Based Approach,” Int. J. Pres. Ves. Pip., Vol. 80, 2003, pp. 573–583 46. M. Tan, N.J.C. Celard, K.M. Nikbin, and G.A. Webster, “Comparison of Creep Crack Initiation and Growth in Four Steels Tested in HIDA,” Int. J. Pres. Ves. Pip., Vol. 78, 2001, pp. 737–747 47. K.M. Nikbin, “Transient Effects on the Cyclic Crack Growth of Engineering Materials,” Published in ‘3rd Conference on low cycle fatigue and elasto-plastic behaviour of materials III, LCF3’, K.T. Rie, Ed., Elsevier Publishers, 1992, pp. 558–664 48. C.M. Davies, N.P. O’Dowd, K.M. Nikbin, and G.A. Webster, Prediction of Creep Crack Initiation under Transient Stress Conditions, American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP, 2006, Vancouver, BC, Canada, 2006, American Society of Mechanical Engineers, New York, NY 10016-5990, United States, p. 9 49. M.R. Winstone, K.M. Nikbin, and G.A. Webster, “Modes of Failure Under Creep/Fatigue Loading of a Nickel-Based Superalloy,” J. Mater. Sci., Vol. 20, 1985, pp. 2471–2476
Chapter 6
Computational Approach Toward Advanced Composite Material Qualification and Structural Certification Frank Abdi, J. Surdenas, Nasir Munir, Jerry Housner, and Raju Keshavanarayana
Abstract The objective of this chapter is to perform accurate simulation of physical tests using multi-scale progressive failure analysis (PFA) and to simulate the scatter in the physical test results by using probabilistic analysis. The multi-scale analysis is based on a hierarchical analysis, where a combination of macro-mechanics and micro-mechanics is used to analyze material and structures in great detail. To calculate the correct micro-mechanical constituent properties for the multi-scale analysis, a three-step process is used: (1) calibration step, (2) verification step, and (3) probabilistic analysis step. The discussion in this chapter will focus mainly on the use of FAA composite material certification requirements and estimation of mechanical and fracture properties of composites; A-basis and B-basis allowable properties generation that are recognized as statistical in nature; and categories of damage tracking for composite structure under service.
6.1 Overview The use of advanced composites in product design is becoming increasingly more attractive due to advantageous weight-to-stiffness and weight-to-strength ratios. Increasingly, composite structures are being subjected to severe combined environments and are expected to survive for long periods of time. There is neither an adequate test database for composite structures nor significant long-life service experience to aid in risk assessment. To ensure safe designs, it is estimated that aerospace companies spend an estimated $350 million per year on testing. Due to the difficulty and cost in assessing and managing risk for new and untried systems, the general method of risk mitigation consists of applying multiple conservative factors of safety and significant inspection requirements to already conservative designs in lieu of costly full system tests. Unfortunately, this approach can lead to excessively conservative designs and the full potential of composite systems is often not fully realized. F. Abdi (B) Alpha STAR Corporation, Long Beach, CA, USA e-mail:
[email protected] B. Farahmand (ed.), Virtual Testing and Predictive Modeling, C Springer Science+Business Media, LLC 2009 DOI 10.1007/978-0-387-95924-5 6,
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This chapter primarily focuses on the use of FAA advanced composite material and structural components certification requirements with a view to obtaining efficient designs. The following items are covered: • Treating FAA categories of damage tracking for composite structures under service. • Estimation of mechanical and fracture properties of composites. • Generation of A-basis and B-basis allowable properties that are statistical in nature. The objective is to perform a virtual testing (VT) process which involves an accurate simulation of physical tests using multi-scale progressive failure analysis (PFA), including the scatter in physical tests by using probabilistic analysis. The multi-scale analysis is based on a hierarchical analysis, where a combination of micro-mechanics and macro-mechanics is used to analyze material and structures in great detail. Certification required predictions are important for reducing risk in structural designs. Moreover, determination of allowable properties is a time-consuming and expensive process, since a large amount of testing is required. In order to reduce costs and product lead time, VT can be used to reduce necessary physical tests both for certification and for determining allowables. In summary, whereas the aerospace industry tends to rely on expensive testintensive empirical methods to establish design allowables for sizing advanced composite structures, the developed VT methodology relies on physics-based failure criteria to reduce its dependence on such empirical-based procedures. This is more than a simple mix of analysis and test because: (1) the root cause of failure at the micro-scale is modeled for accurate failure and life prediction, (2) VT is incorporated into each stage of the FAA building-block process and the FAA categories of damage tolerance, and (3) natural material and manufacturing data scatter is created giving rise to the unique capability to estimate A- and B-basis allowables.
6.2 Background 6.2.1 FAA Durability and Damage Tolerance Certification Strategy The generally accepted strategy for verifying an aircraft structural design for FAA certification is a building-block testing approach consisting of coupon, subelement, and full-scale prototype experimental testing. Building a comprehensive VT database of building blocks that conforms to FAA requirements will put at designers’ disposal a readily available compendium of certified designs that can be beneficially interrogated relative to the FAA certification potential of a newly proposed design.
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To insure advanced composite aircraft flightworthiness, the Federal Aviation Administration (FAA) requires that the aircraft builder/user address the damage levels for primary structures. Categories of damage and defect considerations for primary composite aircraft structures are outlined in Table 6.1. Table 6.1 FAA Categories of damage and defect considerations primary composite aircraft structures (Courtesy of FAA) Category
Examples
Category 1: Damage that may go undetected by field inspection methods (or allowable defects) Category 2: Damage detected by field inspection methods @ specified intervals (repair scenario) Category 3: Obvious damage detected within a few filights by operations focal (repair scenario) Category 4: Discrete source damage known by pilot to limit flight maneuvers (repair scenario) Category 5: Servere damage created by anomalous ground or flight events (repair scenario)
BVID, minor environmental degradation ,scratches, gouges, allowable mfg defects VID (ranging small to large), mfg defects/mistakes, major environmental degradation Damage obvious to operations in a “walk-around” inspection or due to loss of form/fit/function Damage in flight from events that are obvious to pilot (rotor burst, bird-strike, lighithing) Damage occurring due to rare service events or to an extent beyond that considered in design
Safety Considerations (Substantiation, Managment) Demonsrate reliable service life Retain Ultimate Load capability Design-driven safety Demonstrate reliable inspection Retain Limit Load capability Design, maintenance, mfg Demonstrate quick detection Retain Limit Load capability Design, maintenance, operations Defined discrete-source events Retain “Get Home” Capability Design, operations, maintenance Requires new substantiation Requires operations awareness for safety (immediate reporting)
6.2.2 Damage Categories and Comparison of Analysis Methods and Test Results Five damage categories are identified by the FAA, ranging from minor to severe. This section describes the damage and the corresponding analysis methods that can be employed to simulate the damage events of each category. Category 1: Damage That May Go Undetected by Field Inspection Methods Barely visible damage can occur due to matrix transverse cracking and microcrack density formation during manufacturing and service (e.g., static loading, fatigue loading). Quantifying and characterizing the micro-cracking “transverse matrix crack” response during the composite cool down process and subsequent
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in-service fatigue life is important because the micro-cracks can form continuous paths through the thickness of the laminates resulting in lower stiffness, and leakage (Fig. 6.1). Fig. 6.1 Typical micro-cracks in the polymer matrix [7]
Methods for estimating stress fields in cracked laminates are the variational approach [1], the shear lag method [2–3], approximate elasticity [4] and internal variable models [5]. The shear lag method is an efficient and simple means of calculating stresses in laminate fibers and matrix. It has also been extended to micro-crack density prediction in laminates. Zhang et al. [6, 7] developed an equivalent constraint model (ECM), which predicts the reduction in stiffness properties due to transverse ply cracks as well as the initiation and growth of matrix cracking with increasing mechanical load. In their study, an improved two-dimensional (2-D) shear lag analysis is used to determine the stress distribution in the cracked laminates. Though crack density prediction using this method shows consistency with test results, it can only be applied to simple lay-ups subjected to uniaxial tension. In order to predict micro-crack densities in structures, the ECM was incorporated into a VT progressive failure finite element analyzer (GENOA) [8]. Stress and strain fields calculated using the finite element analyzer are transferred into the ECM. Within the ECM, the average stress and strain fields in each constituent are used to calculate the micro-crack formation and growth as described by the magnitudes of the micro-crack density. This method assumes that the micro-crack spacing is uniform in the ECM. Only transverse cracking is considered in the ECM. Longitudinal cracking involves fiber failure, which is the macro-fracture of the entire ECM. The degradation of composite properties due to the existence of cracks in a ply is determined by the crack density, stress level and constraints supplied by the adjacent plies. This may be determined using an iterative process as introduced into the VT ECM method that determines the stress redistribution resulting from matrix damage [7]. In each iteration, crack densities and the corresponding degradation of
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composite properties are re-calculated. The iterative process reaches convergence when crack densities throughout the structure reach saturation under the current load level. New cracks may occur when loads are increased to the next higher load level. The cool-down thermal stresses in the ply transverse direction are often large enough to cause damage in laminate matrices. For such conditions, crack density verification was performed by comparing analytic predictions with the experimental test observations reported in the literature [7, 8] for T300/934 uniaxial laminates under tension (Figs. 6.2a and b). Considering the nature of test data scatter, it is safe to say that the crack density simulation agreed reasonably well with the test results. Crack density development in the angled plies of the laminate in the case of fatigue are illustrated in Fig. 6.3 Micro-cracks in the matrix at cryogenic temperature multiplied more rapidly than those at room temperature as shown in both simulation and test observations. Fig. 6.3b compares simulation and test fatigue lives of the laminate at various stress levels. It can be seen that the numerical results fell within scattered test data at each stress level. Both test and analysis show that the laminate fatigue S–N curve stabilizes at a higher stress level at cryogenic (liquid nitrogen) temperature than at room temperature. As shown in Figs. 6.2 and 6.3 the simulation agrees with test results for most cases. Crack counting in tests is sensitive to sampling locations, mechanical test processes and other factors. Thus the discrepancies shown are acceptable. (b) Laminates are T300/934 [0/902]sand [25/-25/904]s 12 Simulation - [0/90]s Test - [0/90]s Simulation - [25/-25/90/90]s Test - [25/-25/90/90]s
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Fig. 6.2 Comparison of crack density in 90 plies – predicted vs. test [1]
Category 2–3: Damage Detected by Field Inspection Methods Visible damage may be observed during manufacturing such as wrinkling and fiber waviness and void distribution in thick laminates (Category 2). In addition, obvious damage may be detected within a few flights by flight operations and maintenance personnel (Category 3). Low-speed impact, tool drop and part buckling are representative events of these categories. Low-speed impact and static indentation on composite laminates may cause interior damage. As a result, its residual strength during its service loading is reduced: (a) compression after impact (CAI) and (b) tension after impact (TAI).
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Fig. 6.3 Comparison between simulation and test: (a) Crack density development in the laminate under tension, (b) Cyclic loading of the laminate at room and cryogenic temperatures. Laminate lay-up is [0/45/90/-45]s
Impactor Fixture plate
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Panel Size: 5 inch by 5 inch panel Impact Speed: 3.01 ft/sec Impact Energy: 7.58 ft-lbs Total Time: 19.75 msec Test Fixture: Clamped boundary condition on all four sides Impactor: Spherical Steel (diameter 1 in) Material: G30-500/45 R6367: Ply Orientation: /-45/0/90/0/90/0/90/0/90/45/45ly Orientation: /
Fig. 6.4 Schematic of impacting panels and fixtures
Figure 6.4 shows an example of low-velocity impact [9]. Figure 6.5 shows the experimentally observed damaged plies and prediction by the progressive failure dynamic analysis. Figure 6.6 shows the impact of a steel ball on an advanced composite plate, the impact event load versus time, and the TAI comparison between test and simulation [10]. Figure 6.7 shows the impact of a steel ball on a composite foam sandwich plate and CAI [11–13]. Comparisons between test and simulation include: (a) impact energy, (b) impact peak load, (c) CAI residual strength, and (d) failure mechanisms. Analyses such as these may be used to assess damage tolerance of composite components to impact events in Categories 2 and 3 of Table 6.1. Category 4: Discrete Source Damage Discrete source damage (DSD) can limit an aircraft’s flight maneuver envelope. Herein, discrete source damage is defined as a through-penetration of a structure with an area of collateral (non-visible) damage emanating from where high-density (>0.1 lb/in.3 ) projectiles impact the structure at velocities sufficient for penetration. A DSD event that the FAA is concerned with is the disintegration of a mechanical
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a) Experimental
b) Simulated Damaged Area
Fig. 6.5 Comparison of the fracture pattern between test and simulation
(a) Impact of steel ball on composite plate
(b) Load vs. time
(c) Tension after impact (TAI) stress vs. damage size
Fig. 6.6 Tension after impact, residual strength – test vs. simulation
(a) Residual impact foot print (Compression load = 0 lb)
(b) Crack path (Compression load = 24,680 lb)
Fig. 6.7 Compression after impact
(c) Test vs. Simulation
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Shallow Angle Impact Fan Shrapnel + 1/3 Disc Sector 1/3 Disc Sector 3-Blade Sector Small Shrapnel
ACT96-051
Fig. 6.8 Potential engine fragment paths [14]
component, such as an engine failure that ejects items such as a blade disk, blades, and/or blade fragments (Fig. 6.8). The components and fragments (usually made of titanium or steel) can strike and penetrate the surface of the lower wing skin and possibly penetrate through the upper wing skin. The FAA describes such events in AC20-128 [14] to include ejection of: • One-third of a fan disk ejected at the critical rotation speed. (This usually occurs at take-off.) • A turbine blade fragment of 3 pounds at a velocity of 900 ft/s. • Small blade fragments weighing 0.6 lb at velocities up to 400 ft/s Federal Aviation Regulation Part-25 is the principal certification guideline [15]. A discrete source damage event is described in Section 25.571 of this document as one facet of a damage-tolerant design that must be considered for certification. A DSD event is an incident that is immediately apparent to an aircraft’s flight crew or ground personnel at the time of its occurrence, and the post-event residual strength of the structure must be 70% of its design limit load [16]. As stated, this type of damage is often caused by projectiles emanating from disintegrating engine components with variations in projectile mass, velocity, and impact location [14]. These variables create a complex problem for engineers designing structures that are DSD tolerant. To reduce this design complexity, DSD evaluations are typically simplified by evaluating the residual strength of multi-stringer test panels containing a two-bay crack or slot across a principal stiffening element [17, 18]. Certification guidelines also state that any future application of composite materials in primary structures will be required to demonstrate a level of damage tolerance after the occurrence of such an event [19]. Therefore, the successful development and verification of a DSD
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analysis methodology for laminated composites [20] would reduce future certification costs of any advanced composite structures fabricated with these materials. DSD Geometry: An initial demonstration of discrete source damage tolerance of composite materials applicable to primary wing structures used machined slots (saw cuts) to simulate DSD sites [21]. Also, prior discussions with Federal Aviation Administration and B-2 bomber durability and damage tolerance personnel strongly suggested the use of a two-bay slot to demonstrate DSD residual strength of composite structures [17, 22]. This geometry provides a low-cost alternative to ballistic tests and produces a repeatable damage site at a precise location that offers consistent test comparisons between sets of data. All the slots are positioned in the center of each specimen, both in the vertical and horizontal directions. The machined slot geometries used for the compression and tension test specimens are shown in Fig. 6.9. Both slots were 18 cm (7.0 inches) long with a 0.24 cm (0.094 inches) tip radii. The compression slot was modified into a diamond configuration to reduce the possibility of the upper and lower surfaces coming into contact during testing. Straight slots were used in the initial DSD evaluation of composite structures. Post-failure analysis of the compression specimen raised concerns that the slot had closed under load during the test and loaded the machined surfaces. Minimizing the potential for the machined surfaces to contact each other during a test was considered an experimental refinement over the prior DSD compression test approach.
Fig. 6.9 Compression and tension slot geometries simulating DSD
Compression DSD Failure Behavior: Damage development in the compression specimens progressed in a stable manner from the slot radii toward the outer stringer flanges. The development of this type of failure mode from a flaw site is consistent with behavior observed in compression-dominated post-impact fatigue tests performed on composite materials [23]. A close-up view of one edge of a failed compression test article is shown in Fig. 6.10b. This is typical of the failures observed in compression test articles. The compression failures displayed classic transverse shear surfaces typically observed in stitched composites [23]. Another feature of the failure mode of these compression test articles included no observed stringer pull-off problem.
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Shear Cracks
(b) Close-up view of compression test article failure zone
(a) DSD test article
(c) Tension test Article translaminar shear damage zones
Fig. 6.10 A DSD test article – compression and tension
Tension DSD Failure Behavior: Generally, the three-stringer tension specimen failures were characterized by initiation of damage at the radii of the slots. Increasing the applied load on each panel caused the tension failure damage to rapidly propagate transversely to the loading direction until it reached the inner flanges of the outer stringer region of each specimen. Once this tension damage reached the outer stringer flanges, the damage then propagated parallel to the flanges in the loading direction. This vertically oriented translaminar shear failure mode was observed in all four tension test articles. The stable propagation of this shear failure mode continued until the damage neared the loading tabs and caused catastrophic panel failure. A post-test view of a tension panel is shown in Fig. 6.10c [24]. DSD Residual Strengths: Results of the residual strength tests and analytical predictions for the three-stringer stitched resin film infusion (S/RFI) panels containing DSD sites are shown in Table 6.2. Table 6.2 Summary of virtual test predictions with experimental results Compression Results
Tension Results
Sealed Post-test Panel Geometry envelope re-analysis
Experimental Sealed Post-test Experimental value envelope re-analysis value
6-Stack skin 2.3 Inch blades 6-Stack skin 1.8 Inch blades 4-Stack skin 2.3 Inch blades 4-Stack skin 1.8 Inch blades
291
314
294.0
421
422
427
270
292
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364
336
382
238
247
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294
341
309
199
230
206.8
294
278
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The catastrophic failure behavior of the compression test articles allowed the ultimate load a panel carried to be used as its failure load. However, the more complex behavior of the tension panels required closer reviews of strain gage and video data to determine when each panel reached its defined ultimate failure load. Tension panels were considered to have failed when translaminar damage started to propagate along the inner flanges of the outer stringers. In Table 6.2, the columns labeled, “Sealed Envelope” refer to pre-test computer predictions provided independently to the aircraft company overseeing the test program. The predictions provided in Table 6.2 indicate that the analysis methodology shows promise for predicting the residual strength of S/RFI structural panels containing DSD sites. The pretest predictions for the compression panels were within 5% of the experimental failure loads. Analytical predictions of tension residual strengths were also reasonably close to the experimental values. The pretest tension predictions were conservatively below the test results by no more than 5%. After the experimental evaluations were completed, a unified S/RFI material database was developed based on coupon compression, tension, and shear test data. The columns labeled “Post-Test Re-analysis” in Table 6.2 refer to analyses carried out using the unified material database. Using this database, the compression and tension panel failure load predictions were within 11% and 12% of the test results, respectively. Category 5: Severe Damage Created by Anomalous Ground or Flight Events In-flight fire is an example of a scenario that requires a half-hour emergency mission abort. Fire gives rise to high temperatures, which can cause epoxy resins to soften or burn, thus effectively undermining the strength of a composite part. The effects of fire are particularly serious in the confined space of vehicles, aircraft, boats, and trains. Therefore, innovative designs should not only be cost-effective but fire resistant. Developing and using composite life prediction analytical procedures to study any composite experiencing extreme fire conditions is important since the reduced analysis and/or experimental effort can represent a significant cost savings. This applies to both the design and redesign of a composite structure. This composite analytical procedure must predict the response of a composite structure accurately. 6.2.2.1 Example of Simulation of Composite Deck under Fire The fire resistance of simply supported deck sandwich (E-Glass DKNE face sheet, and balsa wood core) panel and deck-bulkhead assembly structures (Fig. 6.11) subject to service mechanical loads of 300 psf and the heat load due to the action of fire with the maximum flame temperature of 2,000◦ F was considered [25]. The mid-span deflection of the deck panel was chosen as a criterion for the fire-resistance assessment of the both structures. Two simulation approaches of coupled and un-coupled thermal structural analysis were employed to determine the mid-span deflections of the deck panel [26–27]. First, the PFA was performed based on the equivalent
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a) quarter model
b) Mid-span deflection of the bottom facesheet of the deck sandwich panel
Fig. 6.11 Deck-bulkhead assembly subject to service mechanical loads and surface heat fluxes
temperature load profile, which was obtained from the given fire profile using the NASA CSTEM computer program. In the second case, the coupled CSTEM– progressive failure simulation was performed where the PFA was invoked immediately after the CSTEM temperature redistribution had been calculated for each time interval. In both cases, the effect of fire on the bottom part of the deck panel was simulated through the application of surface heat fluxes to the bottom of the panel and using CSTEM to calculate the equivalent temperature load required for the PFA [28–31]. The calculated mid-span deflection of the simply supported deck sandwich panel after one-hour exposure to fire correlated well with the experiment. Both the simulation and experimental results show that the mid-span deflection was significantly below the 2-foot deflection limit. In addition, the initiation and propagation of damage and fracture patterns over the deck structures were computed. Figure 6.11a shows the finite element model of the deck-bulkhead assembly. In this case, the equivalent uniform service load on the deck panel was 300 psf. The finite element model of the deck-bulkhead assembly consisted of the 7,761 threedimensional isoparametric solid elements and 16,579 nodes. Figure 6.11b shows the predicted deflection after thirty minutes of exposure was about 1.75 and 2.75 inches according to coupled and uncoupled analysis, respectively. The coupled simulation was much closer to the measured 2.5 inches. Both analyses predicted that the panel design was adequately fire-resistant according to the prescribed limit of 2 feet, which also agreed with the experimental observations. Figure 6.12 shows delamination due to the transverse and normal tensile damages throughout the deck-bulkhead structure after 10 minutes of fire exposure.
6.2.3 FAA Building-Block Approach Within the composite engineering community, the structural substation process, which uses testing and analysis at increasingly complex levels, has become known as the “building-block approach.” Such an approach has traditionally been used to
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Fig. 6.12 Transverse tensile damage in the deck-bulkhead assembly after 10 minutes of exposure to fire in the: (a) balsa wood core; and (b) facesheets (damaged areas are shaded)
address durability and damage tolerance as well as static strength for both metal and composite aircraft structure. The virtual and experimental testing building-block approaches are interactive. Experimental test results are used to validate methods for analytical predictions and reduce uncertainties in VT results. VT provides assistance to planning and reduction of experimental testing at coupon and large component levels. With experimental verification, VT of composite structure can be performed to understand: (1) crack initiation at multiple sites; (2) uncertainties in material properties; (3) effects of barely visible, visible, and discrete source damage; (4) means of predicting damage growth and residual strength; and (5) how to demonstrate durability and robustness to assist in the FAA certification process. Figure 6.13 provides a conceptual schematic of test included in a building-block approach for wing structures. Lower levels of testing are more generic and likely to be applicable to other parts of the airplane and other products. In order to perform these analyses, the material stress–strain curve needs to be established to failure (or
Fig. 6.13 FAA Schematic diagram of building-block tests (Courtesy of FAA)
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a strain cutoff in the test methods) for each composite material used in the design. Analysis has proven reliable to minimize the numbers of tests needed to define this characteristic for laminated composite material forms. There are generally more repetitions at lower levels, such as needed to provide a statistical basis for material performance. Since some lower levels of building-block tests can be considered generic, the concept of databases shared between programs is reasonable. Engineering protocol for base material qualification and the equivalency testing to use shared composite databases has been published previously. Each certification project will have its own certification plan and methods approved by the local aircraft certification office. The integration of the design and manufacturing process becomes evident in larger studies. The larger scales of testing are needed to address the effects of more complex loads and geometry. Fewer tests are performed at larger scales. The relevance of these tests is to address specific structural details. Analysis validation is an important part of the building-block process because it provides a basis to expand beyond the specific tests performed in development and certification. Such validation starts with prediction of the structural stiffness, internal load paths, and stability. Verification of internal load paths may require additional building-block tests, which are designed to evaluate load share between bonded and mechanically attached elements of a design. This is particularly difficult analytically as failure is approached, where some nonlinear behavior can be expected. Combined load effects can further complicate the problem of analytical predictions. 6.2.3.1 Damage, Defects, Repeated Load, and Environmental Effects Prediction of the effect of multiple influences (environment, repeated loads, damage, and manufacturing defects) on the failure modes that affect structural strength traditionally relies on the building-block tests. Often, semi-empirical analyses have been adopted for composite strength. In such analyses, special considerations are given to structural discontinuity (for example, joints, cutouts or other stress risers) and the other design or process-specific details. One of the most important parts of the building-block analysis and test development comes in providing engineering databases to deal with manufacturing defects, field damage, and repairs likely to occur in production and service. Traditionally, not enough attention was given to these issues during composite product development and certification. This has caused significant work slowdowns and increased costs for subsequent product manufacturing and maintenance. Without sufficient analysis and a test database to cover commonly allowed manufacturing defects, damages, and repairs, engineers are often forced to either adopt conservative assumptions (part rejections or expensive repairs) or generate the data as it is needed (leading to down time and associated cost or lost revenue). Unfortunately, production and service experiences with new technologies such as composite materials are often needed to completely define the problems and to plan for unanticipated defects and damage. Nevertheless, an awareness of the likely
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production and service issues will help define practical levels of building-block tests and analyses to be performed as part of the structural substantiation. MIL-HDBK-17 (DoD Composite Material Handbook) provides some detailed background on the engineering practices that have been successfully applied with composite materials used in airplane structures. Chapter 2.1 from Volume 1 – Polymer Matrix Composite (PMC), provides some introduction to this subject, including a synopsis of test levels and the data uses. Many of the engineering practices outlined in MIL-HDBK-17 were derived from composite applications to military and commercial transport structures. The composite material types, structural design details, and associated manufacturing processes selected for such applications may have significant differences from those used for small airplanes.
6.2.4 Test Reduction Process The building-block strategy builds from the lowest configuration level fiber/matrix to the full-scale assembly. To perform test reduction, failure mechanism predictions (such as longitudinal tensile, longitudinal compressive, transverse tensile, transverse compressive, delamination, in-plane shear, out-of-plane shear, peel-off stress, etc.) are determined and then used to identify which tests to perform. The following process (Fig. 6.14) is proposed for characterizing a new material: 1. Material and Manufacturing Development: This step is done by the material vendor where the chemistry, physical form (sheet, plate, cast, etc), and preliminary static and fatigue properties are obtained. 2. Unnotched Static and Fatigue Testing: The basic static material properties and unnotched fatigue material behavior are determined for different stress ratios.
Material and Mfg. Characterization Unnotched Static & Fatigue Testing
Probabilistic Analysis
Deterministic Analysis
Reduced Testing
Fig. 6.14 Test reduction process
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3. Probabilistic Analysis: Treating the material and manufacturing as variables, statistical distributions are applied to simulate test coupons. These simulated test coupons will have different properties, and hence the scatter that is seen in traditional test plans will be reproduced. 4. Deterministic Analysis: Using the test coupons generated in Step 3, a VT matrix in concert with traditional testing is completed to characterize the material completely. To perform the test reduction process, three analytic processes are used. 1. Material Property Characterization Analysis including the effects of manufacturing and service conditions such as hygral (thermal, moisture) effects. Material characterization is used to predict mechanical properties for temperatures where no test data is available including: 1) mechanical properties at the laminate, lamina, and constituent (fiber/matrix) levels under room, cryogenic, and high-temperature testing conditions. Material characterization also validates the simulation results against experimental observation 2. Material Property Uncertainty Analysis of failure mechanisms and the percent contribution of mechanical properties to these failures are used to determine their sensitivity to variations over a temperature range and to generate mechanical properties for temperatures where no test data was available. 3. Probabilistic Analysis. In order to properly characterize a material, several factors must be taken into consideration. These factors, depending on the severity and distribution, have different effects on the fatigue life of the material. The following list provides some of the larger variables: 1. Lot-to-Lot Material Variation (large effect on scatter) a. Porosity (size, density, shape, location) b. Inclusions (same as above) 2. Surface Quality Variation (moderate-to-large effect on scatter) a. Surface finish b. Machining marks, scratches 3. Residual stresses imparted or removed due to cutters, drill bits, etc. 4. Deburring techniques used on edges, holes 5. Loading Variation (moderate effect on scatter) a. Test machine/specimen alignment b. Accuracy of loads/strains applied during test c. Crack measurement/detection capability (for small cracks or to determine crack initiation) 6. Geometry Variation (small effect on scatter) 7. Geometric tolerances causing variation in Kt ∗ from specimen to specimen
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Each these variables will have a statistical distribution depending on how these values change from one specimen to another. Once these distributions have been defined, probabilistic analysis will then come up with a specified number of specimens with a distribution of properties that have the same test scatter.
6.3 Computational Process for Implementing Building-Block Verification The building-block approach focuses on hierarchical progressive failure analyses at each step of the design process to verify basic material constituents, joints, builtup substructures (e.g., spars and bulkheads), and the final product (Fig. 6.15). At each verification stage, materials and structures require evaluation of their mechanical properties and the corresponding uncertainties to determine the adequacy of the structure’s durability and reliability. (PFA) (Fig. 6.16) implements the basic concept that a structure will fail when defects and flaws, that may initially be microscopic, grow and/or coalescence to a critical size at which the structure no longer has an adequate safety margin to avoid catastrophic global fracture (Fig. 6.17). Damage is considered to progress through five stages: (1) initiation, (2) growth, (3) accumulation and coalescence of propagating flaws, (4) stable propagation (up to critical dimensions), and (5) unstable or very rapid propagation to catastrophic failure. Computational PFA involves a formal procedure for identifying the five different stages of damage, quantifying the amount of damage at each stage, and relating the damage to the overall behavior of the deteriorating structure.
Material Constituent Verification 1. 2. 3. 4. 5. 6. 7. 8. 9.
Strength Stiffness Material Uncertainty Fiber/Matrix Interface Micro Mechanics–Failure Void Fiber Volume Fraction Margin of Safety Multi-Factor/Degradation Fracture Toughness
Coupon Verification Strength due to Notch Effects Calibration Basis Waviness Material Uncertainties Load Redistribution ASTM 3039 ASTM 5766 ASTM 3410 ASTM 4255 ASTM 5379
Sub-Element Verification Virtual Testing Ply Failure Waviness Ply Schedule Residual Strength Margin of Safety Residual Strain Load Redistribution Energy Release Rate Design Guidelines Joints
Full Scale Verification Global Multi Site Crack Initiation Multi Site Crack propagation Residual Strength Contribution of Failure Photo Elasticity Deflectometer & Strain Gauge Far Field Strain Inspection Predictions for Non destructive Testing Sensitivity of Failure Modes
Fig. 6.15 Virtual testing building-block verification strategy for FAA FAR-025 certification
Utilization of PFA design has already been demonstrated by successfully predicting damage tolerance limits and failure criteria of significant aerospace structural
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Large Scale Structural Component Multi Site Damage (Initiation, Progression, Residual) Linear/ non linear Structural Analysis Failure Mechanism Contribution Part Inspection Guidance Life Prediction, Final Failure Loads Fracture crack growth Analysis Probabilistic Assessment Reliability Based Optimization Building Block Verification Strategy Certification Process, Conforms to FE Standards
Fig. 6.16 Functionality of virtual testing (GENOA) software modules
Fig. 6.17 Virtual testing multi-scale hierarchical progressive failure analysis process
designs. Results such as these would support satisfying Federal Aviation Regulations for certification of critical aerospace structures.
6.3.1 Multiple Failure Criteria Even though many failure criteria have been developed, there is a lack of evidence to show whether any of the criteria can provide accurate and meaningful predictions of failure for other than a very limited range of circumstances. This conclusion may be surprising to many. After all, there is a large body of composite materials research to draw upon, spanning at least 50 years, along with numerous examples (aircraft, boat hulls, etc.) where composite materials have been used widely and successfully for primary load-bearing structures. One might therefore logically conclude that design procedures (including strength prediction) for composite structures are fully mature. Closer examination reveals that current commercial design practices place little or no reliance on the ability to predict the ultimate strength of the structure with substantial accuracy. Failure theories are often used in the initial calculations to “size” a component (i.e. to establish the approximate dimensions, such as panel thickness, width, etc.). Beyond that, experimental tests of coupons or structural elements are used to determine the global design allowables. These are typically set at levels which are less than 30% of the ultimate load carrying capability, thereby providing
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a wide safety margin to accommodate loss in performance due to fatigue, operating environments, impact and any other possible aspect. The coupon/structural-element testing approach is widespread in the aerospace industry leading to establishment of large databases at great expense. Small-to-medium-sized companies tend to follow a broadly similar path, though on a much smaller scale. A “make and test” approach combined with generous safety factors is commonplace. There are several well-known failure criteria such as Puck, Tsai-Wu, and TsaiHill that are able to predict strength values for some composites extremely well, but differ when the ply lay-up or material system is switched. Using these failure criteria judicially can eliminate the complexity involved, while predicting the failure design envelop without deviating seriously (
