an introduction to fracture mechanics for engineers

An Introduction to Fracture Mechanics for Engineers

Supplement to the FESI Bulletin Summer 2021, no 23

FESI Publishing

Roderick A Smith

2 Origins

This is a collection of three papers I wrote

more than 40 years ago and forms the basic

reading for courses, also called Introduction

to fracture mechanics for engineers, which I

have given over the intervening years at the

Universities of Cambridge, Sheffield, and

Imperial College London.

In the years that followed their original

publication, I was pleasantly surprised by

the large number of requests I received from

many practitioners of engineering who want an

easy introduction to fracture mechanics from

first principles. It was clear that many people

were put off fracture mechanics because of

the complicated mathematical frameworks

erected by many authors, sometimes so com-

plicated that they hide the physical fundamen-

tals concerning the separation of material. I

have therefore relegated any formal mathe-

matics to an appendix. I deliberately used the

word engineers in the title because they are

the people who shape our world, make things

and create wealth—often relying on judge-

ments based on imperfect information. And I

was also writing from an engineering depart-

ment, and not a metallurgy or materials or

indeed a materials engineering department.

Over the intervening years I have acted

as an investigator of accidents and as an

expert witness in many legal cases arising from

failures. Two things have struck me as being

particularly important: it is extremely important

to see a failure, as far as possible, for real, and

in the place it occurred. It is quite remarkable

how this clarifies one’s thinking and enables

one to ask the right questions. Secondly, I have

been surprised, time after time, how, in many

applications, the loads and therefore the

stresses and strains to which components and

structures are subjected in service, are largely

unknown. Surprised, because materials fail

when there is an imbalance between the prop-

erties of a material and the loads to which they

are subjected, and rates of failure are gener-

ally proportional to strong power laws of the

applied stresses. The reader might imagine

the balance of a set of scales: if they are tilted

to the side of the properties, then the materi-

al’s resistance is greater than the load, so all is

well. Trouble arises when the converse is true,

and the balance is such that the applied loads

are greater than the ability of the materials to

resist them. The measurement, calculation

and verification of service loads are thus of

vital importance.

In order to limit the size and scope of

this work, it is almost exclusively concerned

with metals. Recent years have seen a huge

increase in the use of ceramics, polymers,

and composites. With some modifications the

methods of fracture mechanics can be applied

to all these materials. Manufacturing tech-

niques have also been introduced which need

special consideration. Sintering and additive

manufacture spring readily to mind, whilst

joining techniques, particularly welding, can

change material properties in the vicinity of

the weld, in the so-called heat affected zone.

Extensive discussions of all these topics can

be found in the literature.

Although the concept of a loads versus

property balance is an important simple con-

cept, it should be realised that this is a balance

which can change with time and exposure

to service. A bridge, for example, might be

exposed to many more and heavier vehicles

than those for which it was originally designed.

Wear might make a section thinner and the

stresses therefore higher, as might wastage

from corrosion. At higher temperatures, met-

als may creep, whilst at lower temperatures,

the propensity to brittle fracture may increase.

Metals may become embrittled by the pres-

ence of hydrogen, or properties may become

degraded by exposure long to radiation. These

and similar factors must be taken into account

when decisions have to be made on how long

a structure or component might remain in

service and we need to know if the effects

are independent or reinforce each other. On

occasions, competing mechanisms can be

helpful. If the wear rate of a railway line is such

that it effectively rubs out initiating fatigue

cracks, then the rail will not fail by fatigue.

Although condition monitoring is only men-

tioned in passing in this elementary work, it is

becoming increasing important for informing

operators of changing conditions, changing

stresses and as an input to calculation to esti-

mate residual safe life.

An Introduction to Fracture Mechanics for Engineers 3Roderick A Smith

It is also worth noting that we are now

waking up to the fact that more weather

events and more severe weather events are

features of the effect of man-made climate

change, and monitoring to inform evi-

dence-based calculations are now vital tools

in our attempts to improve resilience and

mitigate the effects of climate change.

I have not made any changes to what I

wrote many years ago. I consider that the text

has withstood the test of time and can still be

regarded as doing ‘what it says on the can’.

However, I can’t recall that this last phrase was

in use when the text was written! So, although

times, customs, speech and fashions change,

fundamentals don’t. There has been an explo-

sion of published material on fatigue, fracture

and fracture mechanics. Valuable data has

been generated, but huge swathes of this new

material consist of dotting the i’s and crossing

the t’s. Sweeping generalisations are made as

are wild extrapolations, and the reader is often

left bemused about the relevance. It is hoped

that this present modest publication will assist

readers to sort the relevant from the forgetta-

ble and to constantly question the fundamen-

tals. I have added references to particularly

valuable, practical and relevant material which

has emerged in the last few decades. It is my

intention to follow this work with a collec-

tion of case studies based on my own direct

experience and on other key incidents which

have occurred since Introduction to fracture

mechanics for engineers was first written.

Roderick A Smith, Oxford

Emeritus Professor, Imperial College London

[email protected]

November 2020

mailto:[email protected]

4 Biography

Professor Roderick Smith, ScD, FREng is

currently Emeritus Professor of Railway

Engineering, Imperial College London and

Chair of the Future Railway Research Centre.

He was Chief Scientific Advisor to the UK

Department for Transport from 2012 to May

2014 and the 126th President of the Institution

of Mechanical Engineers (2011–12). He was

previously a lecturer in the Cambridge

University Engineering Department

(1980–88) and Professor of Mechanical and

Process Engineering, University of Sheffield

(1988–2000), including a period as Head

of Department (1992–95), and Head of

Mechanical Engineering at Imperial College

(2000–05). He was a consultant to the Board

of British Rail (1992–96) for which he chaired

a major investigation into the crashworthi-

ness of trains, the outputs of which have been

incorporated to standards internationally.

In 2019 he became Distinguished Visiting

Chair Professor at Hong Kong Polytechnic

University.

He was Chair of Heathrow Airport

Consultative Committee (2015–18) and is

currently an infrastructure commissioner for

Wales. He served eight years as a Trustee of

the National Museum of Science and Industry,

which includes the Science Museum, and the

National Railway Museum in York to which

he arranged the gift of an original 0 Series

Shinkansen from Japan. He has also worked in

Japan with many Japanese railway organisa-

tions, manufacturers, and universities.

Rod Smith has published extensively on

structural integrity, railway engineering and

energy and is frequently invited to address

international meetings and conferences. He

is rapporteur and discussion chair for the

Tokyo-based International High-Speed Rail

Association (IRHA).

He is frequently called as an expert wit-

ness in legal cases and has advised on many

accident investigations. Amongst the

major railway cases in which he has been

involved are: the 1998 Deutsche Bahn

Eschede accident; the 2000 Hatfield

derailment as Chair of the Investigation

Committee for Railtrack, later as expert

for Balfour Beatty; the 1998 Sandy Hook

derailment in Washington County, USA;

the 1997 Ladbrook Grove accident; the

2012 failure in the Singapore Mass Rapid

Transit system, as Government advi-

sor, later as member of the Singapore

MRT maintenance advisory committee

(2013–16); the 1997 Southall collision; the

2002 Southall derailment; and the Pandrol rail-

clips patent case in the Philippines.

Rod has been involved in many failure

investigations of ships, and he played a major

role in the UK Health & Safety Executive’s

investigations of the 1989 Hillsborough

Stadium disaster and the UK’s 2010 response

to the emergency resulting from the effects of

volcanic ash on airplane jet engines.

He is President of the Japanese Railway

Society, President of Engineering Integrity

Society (EIS), and a director of the UK Forum

for Structural Integrity (FESI).

Rod is a keen mountaineer, a member

of the Alpine Club, and of the Fell & Rock

Climbing Club (FRCC) in the English Lake

District.

http://rodericksmith.synthasite.com

The

aut

hor

http://rodericksmith.synthasite.com

6 Supplement to the FESI Bulletin, no 23


I Suggested additional resources

I.1 Stress intensity factors handbook

Murakami Y, ed (2005)

The Stress intensity factors handbook was first published in 1987 as a two-volume edition

by the Committee on Fracture Mechanics of the Society of Materials Science, Japan. In 1992 the

information contained in the first and second volumes was updated and published as a single vol-

ume edition. Since then, a large number of additional stress intensity factors have been developed,

and by 1999 the Committee on Fracture Mechanics felt it was time to publish a third edition in order

to include this new information. The handbook is considered by engineers and investigators in the

field of fracture mechanics as the most comprehensive and reliable source of information available

on stress intensity factors. This new edition provides this information in two volumes.

I.2 ASM handbook, volume 19—fatigue and fracture

Lampman SP (1996)

This book is the first reference book of its kind to put critical information on both fatigue

and fracture mechanics in one convenient volume. It provides comprehensive data on a broad

spectrum of engineering structural materials and alloys. You get coverage on mechanisms,

testing, analysis, and characterization. You’ll learn about fatigue and fracture from both the fun-

damental and practical standpoint. It’s the essential data necessary to make informed decisions

on alloy design and material selection. It also gives an invaluable insight into fracture control, life

assessment, and failure analysis. Providing a working knowledge of fatigue and fracture proper-

ties in actual engineering practice, this handbook is especially useful in evaluating test data and

helping to understand the key variables that affect results.

I.3 Knovel—engineering technical reference information

https://app.knovel.com/kn

The Knovel website integrates validated content, optimized search, and data analysis

tools (including our unit converter) enabling engineers to solve problems quickly.

I.4 Materials: engineering, science, processing and design

Ashby MF, Shercliff H, Cebon D (2018)

Materials: engineering, science, processing and design is an essential materials engineer-

ing text and resource for students developing skills and understanding of materials properties

and selection for engineering applications. Taking a unique design-led approach that is broader

in scope than other texts, Materials meets the curriculum needs of a wide variety of courses

in the materials and design field, including introduction to materials science and engineering,

engineering materials, materials selection and processing, and behaviour of materials. This new

edition retains its design-led focus and strong emphasis on visual communication while expand-

ing its coverage of the physical basis of material properties, and process selection.

Mike Ashby and his collaborators have produced software to characterise and prioritise

material properties over the whole range of engineering materials. This software is a learning

and educational aid, but can also be used by professionals to explore alternative and optimise

designs. See, for example: https://www.grantadesign.com/

The CES Selector provides an unrivalled combination of materials data and tools to plot,

analyse, and compare materials data, and enable systematic materials selection.

https://app.knovel.com/kn

https://www.grantadesign.com/


II Additional introductions to fatigue and fracture

I hope that the reader’s curiosity is by now sufficiently aroused to seek alternative and more

comprehensive treatments. I have found the books listed below clear and useful:

II.1 Fatigue of structures and materials

Schijve J (2nd edition, 2008)

https://b-ok.cc/book/601448/8569fa

This book is primarily a textbook written for people working on fatigue problems of

engineering structures and materials associated with design, predictions, load spectra and

experimental verifications. Many different fields of interest are involved, as the block diagram

on the front cover indicates. It is of great importance that all these aspects are well recognized

and understood. The author explains the various topics in a number of chapters. Understanding

of the fatigue phenomena covering both crack initiation and crack growth is emphasized in

view of possible influences of design variables, material selection, production technology and

load spectra on the fatigue performance of a structure. Prediction methods for fatigue lives and

fatigue crack growth are discussed as well as verification by experiments. Load spectra analysis

and statistical aspects are also addressed.

II.2 Fatigue of materials

Suresh S (2nd edition, 1998)

Written by a leading researcher in the field, this revised and updated second edition of a

highly successful book provides an authoritative, comprehensive and unified treatment of the

mechanics and micromechanisms of fatigue in metals, non-metals and composites. The author

discusses the principles of cyclic deformation, crack initiation and crack growth by fatigue,

covering both microscopic and continuum aspects. The book begins with discussions of cyclic

deformation and fatigue crack initiation in monocrystalline and polycrystalline ductile alloys as

well as in brittle and semi-/non-crystalline solids. Total life and damage-tolerant approaches

are then introduced in metals, non-metals and composites followed by more advanced topics.

The book includes an extensive bibliography and a problem set for each chapter, together with

worked-out example problems and case studies. This will be an important reference for anyone

studying fracture and fatigue in materials science and engineering, mechanical, civil, nuclear

and aerospace engineering, and biomechanics.

II.3 Deformation and fracture mechanics of engineering materials

Hertzberg RW, Vinci RP, Hertzberg JL (5th edition, 2012)

Deformation and fracture mechanics of engineering materials provides a combined

fracture mechanics/materials approach to the fracture of engineering solids with comprehensive

treatment and detailed explanations and references, making it the perfect resource for senior

and graduate engineering students, and practicing engineers alike. The fifth edition includes

new end of chapter homework problems, examples, illustrations, and a new chapter on prod-

ucts liability and recall addressing the associated social consequences of product failure. The

new edition continues to discuss actual failure case histories and includes new discussion of

the fracture behaviour and fractography of ceramics, glasses, and composite materials, and a

section on natural materials including bone and seashells. New co-authors Richard P Vinci and

Jason L Hertzberg add their talent and expertise to broaden the books perspective, while main-

taining a balance between the continuum mechanics understanding of the failure of solids and

https://b-ok.cc/book/601448/8569fa


the roles of the materials nano and microstructure as they influence the mechanical properties

of materials.

II.4 Fundamental of fracture mechanics

Knott J (1973)

This was one of the first texts to appear and was something of a bible when I was a

research student. Sadly, John has now passed away, but his book may be downloaded gratis

from the UK Forum for Engineering Structural Integrity (FESI) website: https://www.fesi.org.uk/

fesipublishing/download/

II.5 Fatigue analysis on the web

https://www.efatigue.com

This is a particularly valuable web resource which has been developed over many years

by a long-time colleague Darell Socie. The eFatigue website gives you easy access to modern

fatigue analysis tools and technology from any web browser— everything you need for comput-

ing the fatigue lives of metallic machine components and structures, including fatigue calcu-

lators, material databases, and stress concentration factors. Registration is free, but with an

eFatigue subscription, you’ll also have access to our state-of-the-art web-based fatigue analysis

software to help you solve more complex fatigue and durability problems.

II.6 Fracture toughness of engineering materials: estimation and application

Wallin KRW (2011)

Professor K. R. W. Wallin is a recognised expert in the field of fracture mechanics. Indeed

his work has provided a major input to the currently accepted ASTM Standard on the Master

Curve Method , E1921. This method facilitates characterisation of ductile to brittle fracture for

ferritic steels.

This publication should provide an aid to both fracture mechanics experts and those engi-

neers and scientists who use fracture mechanics in their daily work. It also offers an insight for

the standards that need to be developed in the area of structural integrity methodologies. The

intention is also to challenge and inspire the scientific experts in the field to develop possibly

competing and improved fracture mechanics solutions, because fracture mechanics is still, by

and large, a maturing discipline.

Available to purchase at https://www.fesi.org.uk/fesipublishing/bookstore/

II.7 Modern metal fatigue analysis

Draper J (2008)

Modern metal fatigue analysis is a concise introduction to modern methods of fatigue

analysis as well as the more traditional methods. It introduces the concepts of strain-based

fatigue analysis and the traditional S-N curve methods. Modern theories of multiaxial fatigue

are described, together with their application to strain gauge measurements and fatigue anal-

ysis of finite element models. There are chapters on statistical analysis, crack propagation, and

recent advances in fatigue analysis of welded steel joints. The final chapters discuss the merits

and disadvantages of different types of fatigue tests and aspects of practical fatigue analysis

and its application to real-world problems. Throughout the book the emphasis is on practical

application.

Available to purchase at https://www.fesi.org.uk/fesipublishing/bookstore/

https://www.fesi.org.uk/fesipublishing/download/

https://www.fesi.org.uk/fesipublishing/download/

https://www.efatigue.com

https://www.fesi.org.uk/fesipublishing/bookstore/

https://www.fesi.org.uk/fesipublishing/bookstore/


MPa√m ksi √in N/mm−³/² kp/mm−³/²

MPa√m 1 0.9101 31.623 3.2235

ksi √in 1.0988 1 34.747 3.5420

N/mm−³/² 0.031623 0.028780 1 0.10194

kp/mm−³/² 0.31022 0.28233 9.8067 1

MPa ksi N/mm² kp/mm²

MPa 1 0.1449 1 0.1019

ksi 6.904 1 6.9033 0.7037

N/mm² 1 0.1449 1 0.1019

kp/mm² 9.8067 1.4211 9.8067 1

Table 1 Stress intensity factor.

Table 2 Stress.

III Dimensions, units, significant figures, calculators, and computers

A brief reminder of the importance of these topics is worthwhile.

Physical quantities can be reduced to their dimensions, and it is always useful to check

dimensional consistency.

For example, stress is force divided by area, that is (mass × acceleration)/area.

So has dimensions (ML/T²)/L² or M/(T²L).

Similarly, stress intensity factor is stress × length¹/² and therefore has dimensions

M/(T²L¹/²).

And stress concentration factor is merely a ratio of stresses and is therefore dimension-

less!

To ensure consistence of units, the above may be worked to convert from first principles.

For convenience unit conversions are given (Tables 1 and 2).

As well as remining us of the multipliers used in the SI system, Table 3 reminds us of the

enormous range of length scales encountered in fracture theory.

We should never think that the accuracy of our computations extends beyond the accu-

racy of the input data.

If we use Table 3, we see that 1 MPm¹/² is equal to 0.9101 ksi in¹/². If we had converted

1.000 MPm¹/², then we could accept 0.9101 ksi in¹/². But 1 converts to 0.9.

significant figures matter ! How often have you seen that the Poisson

ratio is given as 0.3 and the stresses are calculated and quoted to four, five, or six significant

figures? When you look down a microscope is the crack length you measure 5.1 mm or 5.12 mm

or 5.123 mm and was the number of fatigue cycles applied at that instant 10,000 cycles or

10,020 cycles? From a table of such data, we differentiate to find the growth rate of the crack

against the number of loading cycles. We plot the data on a log/log graph and, of course,

the scatter looks almost vanishingly small. We fit a straight line to the data and produce the


Factor Prefix Symbol Example

1,000,000,000 = 10⁹ giga G 1 gigameter (Gm) = 10⁹ m

1,000,000 = 10⁶ mega M 1 megameter (Mm) = 10⁶ m

1,000 = 10³ kilo k 1 kilogram (kg) = 10³ g

100 = 10² hecto h 1 hectogram (hg) = 100 g

10 = 10¹ deka da 1 dekagram (dag) = 10 g

0.1 = 10−¹ deci d 1 decimeter (dm) = 0.1 m

0.01 = 10−² centi c 1 centimeter (cm) = 0.01 m

0.001 = 10−³ milli m 1 milligram (mg) = 0.001 g

*0.000 001 = 10−⁶ micro μ 1 micrometer (μm) = 10−⁶ m

*0.000 000 001 = 10−⁹ nano n 1 nanosecond (ns) = 10−⁹ s

*0.000 000 000 001 = 10−¹² pico p 1 picosecond (ps) = 10−¹² s

Table 3 Length scales used in fracture theory.

Typical sizes An atom = ~ 100 pm, 10−¹⁰ m

Grain size = 50-200 μm, 50–200 10−⁶ m

Crack visible to naked eye = 1 mm, 10−³ m

Weld toe = 5–10 mm, 5–10 10−³ m

Bicycle, person = ~ 1 m

Aircraft wingspan = 50 m

Wind turbine tower = 100 m

Large ship length = 300 m

Bridge = 1 km, 10³ m

Floating ice = 100 km, 10⁵ m

so-called constants in a Paris law relating the growth rate of the crack

to the stress intensity factor. If we are unwise, we quote these contacts

with far too many significant figures. If we are even more unwise we

extrapolate outside the range of our experimental data. In other words,

reader beware, don’t believe all that is published, and in particular, be

very sceptical about the accuracy of the derived data.

In the very first paper I ever had published, the data was plotted

by hand, a curve was fitted by eye, growth rates replotted on log-log

paper and a straight line fitted by eye. This process ensured I was very

aware of the accuracy of the results. How different if all this is done by

a computer! (Or on a hand calculator). We can get results apparently

accurate to many significant figures! Magic, or is it? Think on, as they

say in my native Lancashire!

All this applies to stress analysis by finite element methods.

The truth is that the computations are no better than the quality

of the input.

fesipublishing.org.uk

http://fesipublishing.org.uk

an introduction to fracture mechanics for engineers

Documents