## Publications

Updated 30 June 2017

**Selected Publications by Les Pook**

It has been truly said that there is no agony like the agony of literary composition.

P G Wodehouse, *Jill the Reckless*.

**Frost N E, Marsh K J and Pook L P. Metal fatigue.** Oxford: Clarendon Press, 1974. ISBN 0 19 856114 8. Reprinted with minor corrections. Mineola, NY: Dover Publications Inc., 1999. ISBN 0-486-40927-9.

When parts fail due to metal fatigue, results can be catastrophic. This book, the definitive text on the subject, addresses all aspects of the fatigue of metals, components and structures. It begins with the historical development of the understanding of metal fatigue, and leads up to the vital concepts of the cyclic stress that causes a crack to grow, and of the macrocrack propagation characteristics of metals.

An in depth description of the mechanism of fatigue crack initiation is followed by the interpretation of a comprehensive collection of data on factors affecting the fatigue strength of materials. The effect of the stress concentrations on notches and the stress necessary for macrocrack growth follow, concluding with specific aspects of fatigue that warrant a more detailed treatment. Seven appendixes describe many elementary concepts, among them stress-strain relationships of a material, repeated loading and fracture, plastic deformation in ductile metals, and fatigue testing machines.

This was the first major book on metal fatigue that used a fracture mechanics approach. It was the also first book in English to use SI units.

**Pook L P. The role of crack growth in metal fatigue.** London: Metals Society, 1983. ISBN 0 904357 63 5.

**Pook L P. Linear elastic fracture mechanics for engineers. Theory and applications.** Southampton: WIT Press, 2000. ISBN 1 85312 7035.

The book fulfils the need for a short, modern, introductory text on linear elastic fracture mechanics and its engineering applications. It is suitable for use by engineering undergraduates, and also by other newcomers to the subject. The main ideas underlying present day linear elastic fracture mechanics are explained, and also how these ideas have been developed. The book shows how these ideas can be used to carry out calculations answering the question ‘Does this crack matter?’ from the viewpoint of an engineering designer. It provides an explanation of the basis of standard methods and of software employed to carry out calculations. Additional, more advanced material is included, where this will increase understanding of the sometimes formidable mathematics involved, and of the various simplifications and approximations used in practical applications. The author includes all the material central to an undergraduate introductory course and ends each chapter with an overview of the material covered to aid accessibility. Familiarity with the mechanical properties of metallic materials, and with the linear elastic stress analysis of untracked bodies is assumed.

**Pook L P. Crack paths.** Southampton: WIT Press, 2002. ISBN 1-85312-927-5.

Many engineering components and structures contain cracks or crack-like flaws, and it is widely recognised that crack growth must be considered both in design and the analysis of failures. The complete solution of a crack growth problem includes determination of the crack path. At present the factors controlling the path taken by a propagating crack are not completely understood. In general crack paths are difficult to predict, while in practice their development in structures is often determined by large scale structural tests. In introductory texts on fracture mechanics it is usually assumed that the crack path is known, either from theoretical considerations, or from the results of laboratory tests. A great amount of published information on crack paths exists, but this is scattered and is often incomplete.. This book provides an accessible state-of-the-art survey, with major themes presented in as unified a manner as possible.

This monograph was the first book to be devoted to crack paths. A spin off has been the organisation of the following two international conferences on crack paths.

**Carpinteri A and Pook L P (Editors). Proceedings of the International Conference on Fatigue Crack Paths (FCP 2003)** Parma (Italy), 18-20 September 2003. Parma: University of Parma, 2003.

**Pook L P. Flexagons inside out.** Cambridge: Cambridge University Press, 2003. ISBN 0 521 81970 9 hardback 0 521 52574 8 paperback.

Flexagons are hinged polygons that have the intriguing property of displaying different pairs of faces when they are flexed. Workable paper models of flexagons are easy to make and entertaining to manipulate. Flexagons have a surprisingly complex mathematical structure, and just how a flexagon works is not obvious on casual examination of a paper model Flexagons may be appreciated at three different levels: firstly as toys or puzzles: secondly as a recreational mathematics topic, and finally as the subject of serious mathematical study.. This book is written for anyone interested in puzzles or recreational mathematics. No previous knowledge of flexagons is assumed, and the only prerequisite is some knowledge of elementary geometry. An attractive feature of the book is a collections of nets, with assembly instructions, for a wide range of paper models of flexagons. These are printed full size and laid out so they can be photocopied.

This was the first book on flexagons to be written at the recreational mathematics level.

**Carpinteri A and Pook L P (Editors). Proceedings of the International Conference on Crack Paths (CP 2006)** Parma (Italy), 14-16 September 2006. Parma: University of Parma, 2006.

**Pook L P. Metal fatigue. What it is, why it matters**. Dordrecht: Springer, 2007. ISBN 1-4020-5596-6. Click the following link for the flyer productFlyer_978-1-4020-5596-6[1]** **The book can be read online. Go to www.springer.com and search for Pook.

There do not appear to be any recent books on metal fatigue which are presented in a format that appeals to engineers, and which can be recommended to newcomers to the topic. This book aims to present the important ideas in metal fatigue in as straightforward a manner as possible for the benefit of readers who need to be able understand more advanced documents on a wide range of metal fatigue topics. Indications on how metal fatigue problems are solved in engineering practice are included. The prerequisite knowledge required for readers is a basic understanding of stress analysis and mathematics covered in engineering undergraduate courses. No prior knowledge of metal fatigue is assumed.

The book has several objectives. Firstly, to explain the terminology used in metal fatigue. Secondly, to provide s brief description of the mechanisms of metal fatigue. Thirdly, to describe the basis of design against the fatigue failure of components and structures. Fourthly, to provide examples illustrating various points made. Finally, the overall objective is to provide a firm foundation for readers who wish to study various aspects of metal fatigue in more detail, including finding solutions to specific fatigue problems.

**Pook L P. Serious fun with flexagons. A compendium and guide.** Dordrecht: Springer, 2009. ISBN 978-90-481-2502-9. Click the following link for the flyer productFlyer_978-90-481-2502-9[1]** **The book can be read online. Go to www.springer.com and search for Pook.

A flexagon is a motion structure that has the appearance of a ring of hinged polygons. It can be flexed to display different pairs of faces, usually in cyclic order. Flexagons can be appreciated as toys or puzzles, as a recreational mathematics topic, and as the subject of serious mathematical study. Workable paper models of flexagons are easy to make and entertaining to manipulate. The mathematics of flexagons is complex, and how a flexagon works is not immediately obvious on examination of a paper model. Recent geometric analysis, included in the book, has improved theoretical understanding of flexagons, especially relationships between different types.

This profusely illustrated book is arranged in a logical order appropriate for a textbook on the geometry of flexagons. It is written so that it can be enjoyed at both the recreational mathematics level, and at the serious mathematics level. The only prerequisite is some knowledge of elementary geometry, including properties of polygons. A feature of the book is a compendium of over 700 nets for making paper models of some of the more interesting flexagons, chosen to complement the text. These are accurately drawn and reproduced at half full size. Many of the nets have not previously been published. Instructions for assembling and manipulating the flexagons are included.

**Pook L P. Fatigue failure and replacement of a mainspring.** Horological Science Newsletter, 2009, (2009-4), 26-33. Click the following link for the text. Mainspring 7

**Atzori B, Carpinteri A, Lazzarin P and Pook L P (Editors). Proceedings of the International Conference on Crack Paths (CP 2009)** Vicenza (Italy), 23-25 September 2009. University of Padua, Vicenza,. 2009. Click the following link for the flyer **CP 2009 Flyer FINAL**

**Pook L P Fifty years of crack path research.** In: Atzori B, Carpinteri A, Lazzarin P and Pook L P (Editors). Proceedings of the International Conference on Crack Paths (CP 2009) Vicenza (Italy), 23-25 September 2009. University of Padua, Vicenza, 2009. Proceedings on CD (15 pp.). Abstract in abstract booklet, p. 26. Click the following link for the text. Fifty years 3

**Pook L P. A serviceman’s traveling clock.** Watch & Clock Bulletin, 2010, **52**(385), 160.

The serviceman referred to is my father in law, who was a mechanic in the Royal Air Force during the Second World War.

**Carpinteri A, Pook L P and Sonsino C M (Editors). Proceedings (on CD) of the 9 ^{th} International Conference on Multiaxial Fatigue & Fracture (ICMFF9)** Parma (Italy), 7-9 June 2010. University of Parma, Parma.

**Pook L P. A brief introduction to coiled springs as a power source.** In: Carpinteri A, Pook L P and Sonsino C M (Editors). Proceedings (on CD) of the 9^{th} International Conference on Multiaxial Fatigue & Fracture (ICMFF9) Parma (Italy), 7-9 June 2010. University of Parma, Parma. pp. 147-194. Click the following link for the text. Coiled 4B

**Pook L P. Five decades of crack path research.** Eng. Fract. Mech. 2010, **77**(11), 1619-1630. Erratum. Ibid. 2011, **78**(2), 450.

This is an extended version of ‘Fifty years of crack path research.’.

**Pook L P. An introduction to coiled springs (mainsprings) as a power source.** Int. J.Fatigue, 2011, **33**(8), 1017-1024.

This is an extended version of ‘A brief introduction to coiled springs as a power source.’.

**Pook L P. Understanding pendulums. A brief introduction.** Dordrecht: Springer, 2011. ISBN978-94-007-1414-4.

Despite their apparent simplicity, the behaviour of pendulums can be remarkably complicated. Historically, pendulums for specific purposes have been developed using a combination of simplified theory and trial and error. There do not appear to be any introductory books on pendulums, written at an intermediate level, and covering a wide range of topics. This book aims to fill the gap. It is written for readers with some background in elementary geometry, algebra, trigonometry and calculus. Historical information, where available and useful for the understanding of various types of pendulum and their applications, is included.

Perhaps the best known use of pendulums is as the basis of clocks in which a pendulum controls the rate at which the clock runs. Interest in theoretical and practical aspects of pendulums, as applied to clocks, goes back more than four centuries. The concept of simple pendulums, which are idealised versions of real pendulums is introduced. The application of pendulums to clocks is described, with detailed discussion of the effect of inevitable differences between real pendulums and simple pendulums. In a clock, the objective is to ensure that the pendulum controls the timekeeping. However, pendulums are sometimes driven, and how this affects their behaviour is described. Pendulums are sometimes used for occult purposes. It is possible to explain some apparently occult results by using modern pendulum theory. For example, why a ring suspended inside a wine glass, by a thread from a finger, eventually strikes the glass. Pendulums have a wide range of uses in scientific instruments, engineering, and entertainment. Some examples are given as case studies.

**Pook L P. From hexaflexagons to edge flexagons to point flexagons.** The College Mathematics Journal, 2012, **43**(1), 11-14.Also in HENLE M and HOPKINS B (Ed). Martin Gardner in the Twenty-First Century. The Mathematical Association of America, 2012, pp.109-112. Click the following link for the text. Gardner 13-cmj011-014-pook

This is one of three flexagon papers in the Martin Gardner Special Issue of ‘The College Mathematics Journal’.

Flexagons, introduced to a wide audience 50 years ago by Martin Gardner, now have an extensive literature and are an active area of research. This paper describes two kinds: edge flexagons and point flexagons, and gives an example of one means of classification.

**Berto F, Lazzarin P, Kotousov A and Pook L P. Induced out-of-plane mode at the tip of blunt notches and holes under in-plane shear loading.** Fatigue Fract. Engng. Mater. Struct. 2002, **35**(6), 538-555. Published online 16 Jan 2012.

As it is well known the Poisson’s effect in a cracked plate subjected to anti-symmetric plane loading leads to the generation of a coupled out-of-plane singular mode. Recent theoretical and numerical analyses have shown that this effect is present also in plates weakened by sharp V-notches and might play a role in failure initiation phenomena in plates under Mode II loading, especially in the presence of a large opening angle.

Dealing with blunt notches with a large notch radius, and not just with sharp notches, the presence or not of an out-of-plane mode does not appear to have been systematically investigated in the past. The main aim of this work is to confirm the existence of Mode O and to describe its main features in the presence of a notch radius significantly different from zero. The analyses include U-notches, as well as circular and elliptic holes. The strain energy density in a three-dimensional control volume is used to identify the most critical zone through the plate thickness at the notch tip.

**Pook L P. A double pendulum.** The International Meccanoman, 2012, (67), 17.

**Berto F, Kotousov A, Lazzarin P and Pook L P. On scale effect in plates weakened by rounded V-notches and subjected to shear loading.** Int. J. Fract. 2013, **180**(1), 111-118.

**Pook L P. Book review. Sun C T and Lin Z-H. Fracture mechanics.** Amsterdam: Academic Press, 2012. Materials World. 2012, **20**(6), 48. Erratum. Ibid. 2012, **20**(7), 52. Also http://www.iom3.org/book-review/fracture-mechanics?c=574

**Pook L P. Book review. Lines M A. Ferranti synchronous electric clocks.** Hardback. Milton Keynes: Zazzo Media, 2012. Antiquarian Horology, 2013, **34**(1), 111.

**Afshar R, Berto F. Lazzarin P and Pook L P. Analytical expressions for the notch stress intensity factors of periodic V-notches under tension by using the strain energy density approach.** J Strain Analysis, 2013, **48**(5), 291-305.

**Pook L P. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches.** Fatigue Fract. Engng. Mater. Struct. 2013, **36**(8), 699-723.

This review is a brief survey of three dimensional effects at cracks and sharp notches. The overall aim is to review developments over the past fifty years leading up to the current state of the art. The review is restricted to linear elastic, homogeneous, isotropic materials, with any yielding confined to a small region at a crack or notch tip. It is also restricted to static loading and to constant amplitude fatigue loading. An enormous amount of theoretical and experimental information relevant to three dimensional effects has been published in the past five decades so the review is, of necessity, highly selective. Theoretical topics covered are: linear elastic fracture mechanics, including Volterra distorsioni, stress intensity factors, corner point singularities, crack front line tension, displacement analysis of cracks and notches, and through thickness distributions of stresses and stress intensity factors. Crack path topics covered are: fatigue crack path constraints, determination of fatigue crack paths, oscillating crack fronts in thin sheets, and the transition to slant crack propagation in thin sheets. Plane strain fracture toughness testing, including standards, is covered. Overall, it can be concluded that the existence of three dimensional effects at cracks and sharp notches has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three dimensional effects numerically in detail. Despite increased understanding three dimensional effects are sometimes ignored in situations where they may be important.

**Pook L P. Temco Art Deco domestic synchronous clocks.** Watch & Clock Bulletin, 2014, **56/1**(407), 47-58.

Manufacture of synchronous electric clocks became widespread in the 1930s, as did the Art Deco style, so it is not surprising that this style was sometimes used for synchronous clocks. The Telephone Manufacturing Company were primarily telephone manufacturers, but also made electric master clocks, impulse wall clocks and associated equipment, and a wide range of synchronous electric clocks using the trademark Temco. Some Temco movements and Temco Art Deco domestic synchronous clocks are described in this article.

**POOK L P. A seven-roll roller leveler.** International Meccanoman, 2014, (71), 14-17.

Many modellers use plate rollers to form accurate curves in plates and strips. Les Pook’s device offers a solution to a problem faced by many modellers – distorted plates. Les demonstrated this useful tool at a recent meeting of the Holy Trinity Meccano club where it attracted considerable attention, and a request in the club’s magazine for the plans to be published. Here, for the benefit of a wider readership, is a detailed description of the roller.

**Pook L P, Berto F, Campagnolo A and Lazzarin P. Coupled fracture mode of a cracked disc under anti-plane loading**. Eng. Fract. Mech. September 2014, **128**, 22-36.

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. The purpose of the present investigation is to study by means of accurate 3D finite element (FE) models a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs. The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked discs under anti-plane loading. The influence of plate bending is increasingly important as disc thickness decreases. Bažant and Estenssoro’s analysis works well for the symmetric mode (mode I), but it is incomplete for the asymmetric mode (a combination of modes II and III). It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the disc surfaces. Discussion on whether *K*_{III} tends to zero or infinity as a corner point is approached is futile because *K*_{III} is meaningless at a corner point. Calculation of the strain energy density (SED) in a control volume at the crack tip shows that the position of the maximum SED is a function of disc thickness.

**Pook L P. British domestic synchronous clocks 1930 to 1980. The rise and fall of a technology. Dordrecht: Springer, 2015.**

This book complements available one-make books on domestic synchronous clocks. It is also a history of science book that sets British domestic synchronous clocks, their manufacturers and technology in their social context.

Part I covers the historical background, British domestic synchronous clock manufacturers and brands, how synchronous clocks work, domestic synchronous clock cases, practical advice on the servicing of domestic synchronous clocks, and analysis of the marketing and reliability of British domestic synchronous clocks. This analysis provides an explanation of the rise and eventual fall of their technology. Part II contains galleries of a selection of British domestic synchronous clocks, and of the movements with which they are fitted. There is a front and back view of each clock, together with a brief description. Views of each movement include views with the movement partly dismantled, together with a brief technical description of the movement.

This profusely illustrated book is primarily for fellow enthusiasts, and is based on an extensive archive of information on domestic synchronous clocks, their movements, and their manufacturers. Current electrical regulations mean that professional clockmakers are reluctant to repair synchronous clocks. In fact, provided that they have not been mistreated, synchronous clocks are usually reliable, and quite easy to maintain.

**Pook L P. Crack paths and the linear elastic analysis of cracked bodies.** Frattura ed Integrità Strutturale, 2015, 34, 171-180.

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907, but it was not until the introduction of the stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. Linear elastic fracture mechanics (LEFM) developed rapidly in the 1960s, with application to brittle fracture and fatigue crack growth. The first application of finite elements to the calculation of stress intensity factors for two dimensional cases was in 1969. Finite element analysis had a significant influence on the development of LEFM. Corner point singularities were investigated in the late 1970s. It was soon found that the existence of corner point effects made interpretation of calculated stress intensity factors difficult and their validity questionable. In 1998 it was shown that the assumption that crack growth is in mode I leads to geometric constraints on permissible fatigue crack paths. Current open questions are. The need for a new field parameter, probably a singularity, to describe the stresses at surfaces. How best to allow for the influence of corner point singularities in three dimensional numerical predictions of fatigue crack paths. Adequate description of fatigue crack path stability.

**Pook L P. The linear elastic analysis of cracked bodies and crack paths.** Theoretical and Applied Fracture Mechanics, 2015, 79, 34-50.

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907, but it was not until the introduction of the stress intensity factor concept in 1957 that widespread application to practical engineering problems became possible. Linear elastic fracture mechanics (LEFM) developed rapidly in the 1960s, with application to brittle fracture and fatigue crack growth. The first application of finite elements to the calculation of stress intensity factors for two dimensional cases was in 1969. Finite element analysis had a significant influence on the development of LEFM. Corner point singularities were investigated in the late 1970s. It was soon found that the existence of corner point effects made interpretation of calculated stress intensity factors difficult and their validity questionable. In 1998 it was shown that the assumption that crack growth is in mode I leads to geometric constraints on permissible fatigue crack paths. Current open questions are. The need for a new field parameter, probably a singularity, to describe the stresses at surfaces. The use of strain energy density is promising. Adequate description of fatigue crack path stability. How best to allow for the influence of corner point singularities in three dimensional numerical predictions of fatigue crack paths. Development of an adequate plane strain fracture toughness testing standard. Some background information is given in appendices.

**Campagno A, Berto F and Pook L P**. Three-dimensional effects on cracked discs and plates under nominal Mode III loading. Fracture and Structural Integrity. 2015, **34**, 190-199.

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations still is. Understanding improved when the existence of corner point singularities and their implications became known. Increasingly powerful computers made it possible to investigate three-dimensional effects numerically in detail. Despite increased understanding, three-dimensional effects are sometimes ignored in situations where they may be important. The purpose of the present contribution is to review the study carried out by the same authors in some recent investigations, in which a coupled fracture mode generated by anti-plane loading of a straight through-the-thickness crack in linear elastic discs and plates has been analysed by means of accurate 3D finite element (FE) models.

The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading. The influence of plate bending is increasingly important as the thickness decreases. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the free surfaces. Discussion on whether *K*_{III} tends to zero or infinity as a corner point is approached is futile because *K*_{III} is meaningless at a corner point.

The intensity of the local stress and strain state through the thickness of the cracked components has been evaluated by using the strain energy density (SED) averaged over a control volume embracing the crack tip. The SED has been considered as a parameter able to control fracture in some previous contributions and can easily take into account also coupled three-dimensional effects. Calculation of the SED shows that the position of the maximum SED in the discs case is a function of the thickness. In the plates case instead the position of the maximum SED is independent of plate thickness, contrary to disc results.

**Pook L P. The linear elastic analysis of cracked bodies, crack paths and some practical crack path examples**. Eng. Fract. Mech. 2016, 167(Nov), 2-19.

The linear elastic analysis of cracked bodies is a Twentieth Century development, with the first papers appearing in 1907. The stress intensity factor concept, introduced in 1957, developed rapidly in the 1960s, with widespread application to practical problems. Finite element analysis had a significant influence on the development of linear elastic fracture mechanics. Corner point singularities were investigated in the late 1970s, but a new field parameter is still needed to describe the stresses at surfaces. There are geometric constraints on permissible fatigue crack paths. Some case studies of fatigue crack paths in metallic materials are included.

**Pook L P, Berto F and Campagnolo A. Coupled fracture modes of discs and plates under anti-plane loading and a disc under in-plane shear loading.** Fatigue Fract. Engng. Mat. Struct. 2016, **39**(8), 924-938.

Three-dimensional effects near crack tips have been deeply investigated in the past, but the topic is still ongoing and under investigation. However, the understanding is in some cases limited, and the problem requires further attention. Three-dimensional effects are in fact neglected in situations where they may play an important role. The main aim of this review paper is to summarise some recent results of a study carried out on the coupled in-plane fracture mode induced by a nominal anti-plane (mode III) loading applied to plates and discs weakened by a straight crack. Only linear elastic conditions are considered herein. The results derived by a large bulk of finite element models allow a better understanding of coupled modes generated by anti-plane loading. The influence of plate/disc bending on three-dimensional stress fields is considered in the investigation, showing that it becomes non-negligible as the thickness decreases. Some considerations on the trend of K_{III} as the corner point is approached are also part of this paper. In addition, the trend of the strain energy density (SED) in a volume surrounding the crack tip is shown and commented on. This parameter allows the precise determination of the weakest point of the component through the thickness becoming a useful tool. Some new results for a disc under in-plane shear loading are included. These results clarify the influence of disc bending under anti-plane loading.

**Pook L P, Berto F and Campagnolo A. State of the art of corner point singularities under in-plane and out-of-plane loading**. Eng. Fract. Mech. 2017, 174, 2-9

The linear elastic analysis of homogeneous, isotropic cracked bodies started in the 1900s. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s. An approximate solution by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. Results derived from finite element models showed that the analysis is incomplete. The stress field in the vicinity of a corner point appears to be the sum of two singularities.

**Pook L P. Berto F and Campagnolo. Some implications of corner point singularities revisited.** 14th International Conference on Fracture (ICF 14) 18-23 June 2017, Rhodes, Greece.

The linear elastic analysis of homogeneous, isotropic cracked bodies started in the 1900s. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s. An approximate solution by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. Results derived from finite element models showed that the analysis is incomplete. The stress field in the vicinity of a corner point appears to be the sum of two singularities. Under mode I loading Bažant and Estenssoro’s solution correctly predicts the angle at which a crack front intersects a free surface. This leads to bowed crack fronts in sheets and plates of constant thickness. These bowed crack fronts mean that two dimensional analyses are inherently approximations. Three examples are given.

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