Mechanical Vibration and Shock Analysis, Volume 4, Fatigue Damage E-Book

Christian Lalanne

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2014

Seitenzahl: 483

Beschreibung

Fatigue damage in a system with one degree of freedom is one of the two criteria applied when comparing the severity of vibratory environments. The same criterion is also used for a specification representing the effects produced by the set of vibrations imposed in a real environment. In this volume, which is devoted to the calculation of fatigue damage, Christian Lalanne explores the hypotheses adopted to describe the behavior of material affected by fatigue and the laws of fatigue accumulation. The author also considers the methods for counting response peaks, which are used to establish the histogram when it is not possible to use the probability density of the peaks obtained with a Gaussian signal. The expressions for mean damage and its standard deviation are established and other hypotheses are tested.

Leseprobe

Table of Contents

Foreword to Series

Introduction

List of Symbols

Chapter 1 Concepts of Material Fatigue

1.1. Introduction

1.2. Types of dynamic loads (or stresses)

1.3. Damage arising from fatigue

1.4. Characterization of endurance of materials

1.5. Factors of influence

1.6. Other representations of S-N curves

1.7. Prediction of fatigue life of complex structures

1.8. Fatigue in composite materials

Chapter 2 Accumulation of Fatigue Damage

2.1. Evolution of fatigue damage

2.2. Classification of various laws of accumulation

2.3. Miner’s method

2.4. Modified Miner’s theory

2.5. Henry’s method

2.6. Modified Henry’s method

2.7. Corten and Dolan’s method

2.8. Other theories

Chapter 3 Counting Methods for Analyzing Random Time History

3.1. General

3.2. Peak count method

3.3. Peak between mean-crossing count method

3.4. Range count method

3.5. Range-mean count method

3.6. Range-pair count method

3.7. Hayes’ counting method

3.8. Ordered overall range counting method

3.9. Level-crossing count method

3.10. Peak valley peak counting method

3.11. Fatigue-meter counting method

3.12. Rainflow counting method

3.13. NRL (National Luchtvaart Laboratorium) counting method

3.14. Evaluation of time spent at a given level

3.15. Influence of levels of load below fatigue limit on fatigue life

3.16. Test acceleration

3.17. Presentation of fatigue curves determined by random vibration tests

Chapter 4 Fatigue Damage by One-degree-of-freedom Mechanical System

4.1. Introduction

4.2. Calculation of fatigue damage due to signal versus time

4.3. Calculation of fatigue damage due to acceleration spectral density

4.4. Equivalent narrowband noise

4.5. Calculation of damage from the modified Rice distribution of peaks

4.6. Other approaches

4.7. Calculation of fatigue damage from rainflow domains

4.8. Comparison of S-N curves established under sinusoidal and random loads

4.9. Comparison of theory and experiment

4.10. Influence of shape of power spectral density and value of irregularity factor

4.11. Effects of peak truncation

4.12. Truncation of stress peaks

Chapter 5 Standard Deviation of Fatigue Damage

5.1. Calculation of standard deviation of damage: Bendat’s method

5.2. Calculation of standard deviation of damage: Mark’s method

5.3. Comparison of Mark and Bendat’s results

5.4. Standard deviation of the fatigue life

5.5. Statistical S-N curves

Chapter 6 Fatigue Damage using Other Calculation Assumptions

6.1. S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit)

6.2. S-N curve defined by two segments of straight line on log-lin scales

6.3. Hypothesis of non-linear accumulation of damage

6.4. Random vibration with non-zero mean: use of modified Goodman diagram

6.5. Non-Gaussian distribution of instantaneous values of signal

6.6. Non-linear mechanical system

Chapter 7 Low-cycle Fatigue

7.1. Overview

7.2. Definitions

7.3. Behavior of materials experiencing strains in the oligocyclic domain

7.4. Influence of the level application sequence

7.5. Development of the cyclic stress–strain curve

7.6. Total strain

7.7. Fatigue strength curve

7.8. Relation between plastic strain and number of cycles to fracture ….

7.9. Influence of the frequency and temperature in the plastic field

7.10. Laws of damage accumulation

7.11. Influence of an average strain or stress

7.12. Low-cycle fatigue of composite material

Chapter 8 Fracture Mechanics

8.1. Overview

8.2. Fracture mechanism

8.3. Critical size: strength to fracture

8.4. Modes of stress application

8.5. Stress intensity factor

8.6. Fracture toughness: critical K value

8.7. Calculation of the stress intensity factor

8.8. Stress ratio

8.9. Expansion of cracks: Griffith criterion

8.10. Factors affecting the initiation of cracks

8.11. Factors affecting the propagation of cracks

8.12. Speed of propagation of cracks

8.13. Effect of a non-zero mean stress

8.14. Laws of crack propagation

8.15. Stress intensity factor

8.16. Dispersion of results

8.17. Sample tests: extrapolation to a structure

8.18. Determination of the propagation threshold Ks

8.19. Propagation of cracks in the domain of low-cycle fatigue

8.20. Integral J

8.21. Overload effect: fatigue crack retardation

8.22. Fatigue crack closure

8.23. Rules of similarity

8.24. Calculation of a useful lifetime

8.25. Propagation of cracks under random load

Appendix

A1. Gamma function

A2. Incomplete gamma function

A3. Various integrals

Bibliography

Index

Summary of Other Volumes in the Series

First edition published 2002 by Hermes Penton Ltd © Hermes Penton Ltd 2002Second edition published 2009 in Great Britain and the United States by ISTE Ltd and John Wiley &Sons, Inc. © ISTE Ltd 2009Third edition published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd 2014The rights of Christian Lalanne to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014933740

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-643-3 (Set of 5 volumes)ISBN 978-1-84821-647-1 (Volume 4)

Foreword to Series

In the course of their lifetime simple items in everyday use such as mobile telephones, wristwatches, electronic components in cars or more specific items such as satellite equipment or flight systems in aircraft, can be subjected to various conditions of temperature and humidity, and more particularly to mechanical shock and vibrations, which form the subject of this work. They must therefore be designed in such a way that they can withstand the effects of the environmental conditions to which they are exposed without being damaged. Their design must be verified using a prototype or by calculations and/or significant laboratory testing.

Sizing, and later, testing are performed on the basis of specifications taken from national or international standards. The initial standards, drawn up in the 1940s, were blanket specifications, often extremely stringent, consisting of a sinusoidal vibration, the frequency of which was set to the resonance of the equipment. They were essentially designed to demonstrate a certain standard resistance of the equipment, with the implicit hypothesis that if the equipment survived the particular environment it would withstand, undamaged, the vibrations to which it would be subjected in service. Sometimes with a delay due to a certain conservatism, the evolution of these standards followed that of the testing facilities: the possibility of producing swept sine tests, the production of narrowband random vibrations swept over a wide range and finally the generation of wideband random vibrations. At the end of the 1970s, it was felt that there was a basic need to reduce the weight and cost of on-board equipment and to produce specifications closer to the real conditions of use. This evolution was taken into account between 1980 and 1985 concerning American standards (MIL-STD 810), French standards (GAM EG 13) or international standards (NATO), which all recommended the tailoring of tests. Current preference is to talk of the tailoring of the product to its environment in order to assert more clearly that the environment must be taken into account from the very start of the project, rather than to check the behavior of the material a posteriori. These concepts, originating with the military, are currently being increasingly echoed in the civil field.

Tailoring is based on an analysis of the life profile of the equipment, on the measurement of the environmental conditions associated with each condition of use and on the synthesis of all the data into a simple specification, which should be of the same severity as the actual environment.

This approach presupposes a proper understanding of the mechanical systems subjected to dynamic loads and knowledge of the most frequent failure modes.

Generally speaking, a good assessment of the stresses in a system subjected to vibration is possible only on the basis of a finite element model and relatively complex calculations. Such calculations can only be undertaken at a relatively advanced stage of the project once the structure has been sufficiently defined for such a model to be established.

Considerable work on the environment must be performed independently of the equipment concerned either at the very beginning of the project, at a time where there are no drawings available, or at the qualification stage, in order to define the test conditions.

In the absence of a precise and validated model of the structure, the simplest possible mechanical system is frequently used consisting of mass, stiffness and damping (a linear system with one degree of freedom), especially for:

– the comparison of the severity of several shocks (shock response spectrum) or of several vibrations (extreme response and fatigue damage spectra);

– the drafting of specifications: determining a vibration which produces the same effects on the model as the real environment, with the underlying hypothesis that the equivalent value will remain valid on the real, more complex structure;

– the calculations for pre-sizing at the start of the project;

– the establishment of rules for analysis of the vibrations (choice of the number of calculation points of a power spectral density) or for the definition of the tests (choice of the sweep rate of a swept sine test).

This explains the importance given to this simple model in this work of five volumes on “Mechanical Vibration and Shock Analysis”.

Volume 1 of this series is devoted to sinusoidal vibration. After several reminders about the main vibratory environments which can affect materials during their working life and also about the methods used to take them into account, following several fundamental mechanical concepts, the responses (relative and absolute) of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms are defined. By placing the properties of sinusoidal vibrations in the contexts of the real environment and of laboratory tests, the transitory and steady state response of a single-degree-of-freedom system with viscous and then with non-linear damping is evolved. The various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of sweep rate are shown and a rule for choice of this rate is deduced from it.

Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be made with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution.

Volume 3 examines the analysis of random vibration which encompasses the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process, enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed, as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing, overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). The relationships that provide the response of a one-degree-of-freedom linear system to a random vibration are established.

Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behavior of a material subjected to fatigue, the laws of damage accumulation and the methods for counting the peaks of the response (used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal). The expressions of mean damage and its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non-linear accumulation law, etc.). The main laws governing low-cycle fatigue and fracture mechanics are also presented.

Volume 5 is dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification as from the lifecycle profile of the equipment is then detailed taking into account the uncertainty factor (uncertainties related to the dispersion of the real environment and of the mechanical strength) and the test factor (function of the number of tests performed to demonstrate the resistance of the equipment).

First and foremost, this work is intended for engineers and technicians working in design teams responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance following the choice of the most suitable simulation means.

Introduction

Fatigue damage to a system with one degree of freedom is one of the two criteria adopted for comparing the severity of different vibratory environments, the second being the maximum response of the system.

This criterion is also used to create a specification reproducing the same effects on the equipment as all the vibrations to which it will be subjected in its useful lifetime. This book is not intended as a treatise on material fatigue. Instead, it is meant to provide the elements necessary for understanding the behavior of components or materials going through fatigue and to describe the methods that can be used specifically for calculating damage caused by random vibration.

This requires the following items:

– Knowledge of the fatigue behavior of the materials, characterized by the S-N curve (stress versus number of cycles), yields the number of cycles to failure of a specimen depending on the amplitude of the stress applied. The main laws used to represent the curve are quoted in Chapter 1, emphasizing the random nature of fatigue phenomena. This is followed by some measured values of the variation coefficients of the numbers of cycles to failure.

– The law of accumulation of the damage caused by all the stress cycles must be selected. The most common laws with their limitations are described in Chapter 2.

– The histogram of the peaks of the response stress, assumed here to be proportional to the relative displacement, is determined. When the signal is Gaussian stationary, as was seen in Volume 3, the probability density of its peaks can easily be obtained from only the power spectral density (PSD) of the signal. When this is not the case, the response of the given one-degree-of-freedom system must be calculated digitally and the peaks then counted directly. Numerous methods, ranging from the simplest (counting of the peaks) to the most complex (rainflow) have been proposed and are presented, with their disadvantages, in Chapter 3.

All these data are used to estimate the damage – characterized statistically if the probability density of the peaks is available and deterministically otherwise (Chapter 4) – and its standard deviation (Chapter 5).

A few elements for damage estimation from other hypotheses are provided in Chapter 6. These concern the shape of the S-N curve, the existence of an endurance limit, the non-linear accumulation of damage, the law of distribution of peaks and the existence of a non-zero mean value.

The Wöhler curve describes three fields based on the level of stress: with unlimited endurance in which the useful lifetime is very long, or even infinite; limited endurance (considered in the first chapters of this book); and for when stress is close to yield stress (low-cycle fatigue). Chapter 7 shows how the S-N curve can be characterized in this context by a strain – number of cycles to failure relation, and how calculation of fatigue damage can then be calculated.

All these approaches are “black box”, with no analysis of physical phenomena leading to failure. Experience shows that a crack will eventually appear in a part submitted to alternating stresses. This crack grows until the part fails. Several studies were carried out to understand and model propagation mechanisms in order to evaluate the remaining useful lifetime of cracked parts and to introduce an inspection and maintenance strategy, particularly in the aeronautics field. Chapter 8 discusses the major laws proposed to describe these phenomena and to evaluate a useful lifetime from these criteria.

The elements necessary for calculating the Gamma function and the different integrals involved in the relations established in this book are provided in the Appendix.

List of Symbols

The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have locally another meaning which will be defined in the text to avoid any confusion.

Chapter 1

Concepts of Material Fatigue

1.1. Introduction

1.1.1. Reminders on the strength of materials

1.1.1.1. Hooke’s law

We accept that the strain at a point of a mechanical part is proportional to the elastic force acting on this point. This law assumes that the strains remain very small (elastic phase of the material). It enables us to establish a linear relationship between the forces and the deformation or between the stresses and the strains. In particular, if we consider the normal stress and the shear stress, we can write successively

[1.1]

[1.2]

where

The following table gives several values of the Young’s modulus E:

Table 1.1.Some values of Young’s modulus

Material

Young’s modulus E (Pa)

Steel

2 to 2.2 10

11

Brass

1 to 1.2 10

11

Copper

1.1 10

11

Zinc

9.5 10

10

Lead

5 10

9

Wood

7 to 11 10

9

NOTE.– Hooke’s law is only an approximation of the real relationship between stress and strain, even for small stresses [FEL 59]. If Hooke’s law is perfectly respected, the stress strain process would thus be, below the elastic limit, thermodynamically reversible, with complete restitution of the energy stored in the material. Experience shows that this is not the case and that, even at very low levels of stress, a hysteresis exists. The process is never perfectly reversible.

1.1.1.2. Stress–strain curve

Engineering stress–strain curve

The stress strain curve thus obtained, traced in the axes (σ, ε), has an identical shape since the changing of the variable corresponds to a proportional transformation (S cross-section, l useful length of the bar). This dimensionless diagram is characteristic of the material here, and not of the sample considered (Figure 1.1).