Extended Finite Element Method for Crack Propagation E-Book

Sylvie Pommier

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

Seitenzahl: 276

Veröffentlichungsjahr: 2013

Beschreibung

Novel techniques for modeling 3D cracks and their evolution in solids are presented. Cracks are modeled in terms of signed distance functions (level sets). Stress, strain and displacement field are determined using the extended finite elements method (X-FEM). Non-linear constitutive behavior for the crack tip region are developed within this framework to account for non-linear effect in crack propagation. Applications for static or dynamics case are provided.

Leseprobe

Table of Contents

Foreword

Acknowledgements

List of Symbols

Introduction

Chapter 1. Elementary Concepts of Fracture Mechanics

1.1. Introduction

1.2. Superposition principle

1.3. Modes of crack straining

1.4. Singular fields at cracking point

1.5. Crack propagation criteria

Chapter 2. Representation of Fixed and Moving Discontinuities

2.1. Geometric representation of a crack: a scale problem

2.2. Crack representation by level sets

2.3. Simulation of the geometric propagation of a crack

2.4. Prospects of the geometric representation of cracks

Chapter 3. Extended Finite Element Method X-FEM

3.1. Introduction

3.2. Going back to discretization methods

3.3. X-FEM discontinuity modeling

3.4. Technical and mathematical aspects

3.5. Evaluation of the stress intensity factors

Chapter 4. Non-linear Problems, Crack Growth by Fatigue

4.1. Introduction

4.2. Fatigue and non-linear fracture mechanics

4.3. eXtended constitutive law

4.4. Applications

Chapter 5. Applications: Numerical Simulation of Crack Growth

5.1. Energy conservation: an essential ingredient

5.2. Examples of crack growth by fatigue simulations

5.3. Dynamic fracture simulation

5.4. Simulation of ductile fracture

Conclusions and Open Problems

Summary

Bibliography

Index

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from La simulation numérique de la propagation des fissures: milieux tridimensionnels, fonctions de niveau, éléments finis étendus et critères énergétiques published 2009 in France by Hermes Science/Lavoisier © LAVOISIER 2009

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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The rights of Sylvie Pommier, Anthony Gravouil, Alain Combescure, Nicolas Moës to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Microrobotics for micromanipulation/edited by Nicolas Chaillet, Stéphane Régnier.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-209-1

1. Fracture mechanics--Mathematics. 2. Finite element method. I. Pommier, Sylvie.

TA409.E98 2011

620.1′1260151825--dc22

2010048620

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-209-1

Foreword

This book, offered to us by Sylvie Pommier, Anthony Gravouil, Nicolas Moës, and Alain Combescure, is undoubtedly the most awaited book today in the solid mechanics community. It certainly fills a gap in the very important subject of the new “eXtended” Finite Element Method, known only very recently and available only through specialized articles. The fact, that one of the authors of this new “X-FEM” method is also the joint author of this book, makes it all the more valuable.

The raison d’être of this book is to provide a method, certainly the only truly efficient one, to model real crack propagation, from the initiation stage to final fracture.

We know that the conventional Finite Element Methods using fixed meshes can only deal with this type of problem, either if the crack path travels through mesh nodes, or if we remove mesh elements. This is an extremely important limitation in industrial applications.

The new “X-FEM” method no longer requires the mesh to be constantly stuck to discontinuous surfaces or propagating cracks. It introduces discontinuous finite elements. It uses, as in other known methods, the partition of unity and introduces level sets to model the crack surface and the crack front, while maintaining the same mesh.

In addition to the discussion of the “X-FEM” method, in Chapters 2 and 3, there is also a discussion on the non-linear behavior of the cracked body and the related crack growth laws (Chapter 4); brittle, ductile, static, and dynamic fracture (Chapter 5), as well as an investigation of the voluminal forces and thermal loading.

I would also like to add that the numerous illustrations have conferred a particularly enticing feel to this book, which, I do not doubt, will have great success in the solid mechanics community.

H. D. Bui, Institute Member

Acknowledgements

The authors would first of all like to thank P. Devalan and the “Fondation Cetim” who are behind the management of this book. If this work is to be published today, we owe it particularly to the various public or private partners who supported the research tasks summarized here.

For this reason, we would particularly like to thank ANR, Cetim, DGA, SAFRAN, AREVA, EDF. We ask forgiveness from those who we may have forgotten.

A big thank you also goes to all our colleagues who actively encouraged the drafting of this collective work, and who allowed us to improve its quality. We also direct our gratitude to R. Desmorat, J. Lemaître, A. Benallal and H.D. Bui who all read this document, in part, or as a whole.

Finally, to finish, we would like to send warm thanks to the PhD students and post-doctorals who contributed to this work and to whom we are highly indebted.

List of Symbols

x

Vector position of a point.

u

(

x

), υ(

x

)

Displacement vector, and velocity of a point.

σ

(

x

),

ε

(

x

)

Stress and strain tensors.

C

Hooke tensor.

K

I

,

K

II

,

K

III

Stress intensity factors of Modes I, II, and III.

ψ

and

ϕ

Level set functions associated with the surface and front of the crack.

H

(

x

)

Generalized Heaviside function.

Displacement modes of the enriched Finite Element approximation.

Rate of elastic displacement field intensity factors.

Rate of complementary field intensity factors.

G

, (

G

I

,

G

II

)

Energy release rate vector (and the components).

G

Energy release rate in crack plane.

Introduction

The predicted lifespan of parts and components is a key question regarding the safety of certain machines, such as airplanes, cars, production factories, or with regard to the reliability of micro-electronic components or implants in the human body.

Mechanical parts are designed and controlled in order to guarantee that they do not contain a macroscopic crack, i.e. detectable by standardized test methods such as metal sweating or ultrasonic control. These increasingly sophisticated devices, together with strict and standardized procedures, are used in order to guarantee that at the end of the production cycle the assembled parts are free from detectable cracks. In addition, defects can be implemented during the assembly stage (e.g., during welding). Lastly, a mechanical system, even when completely healthy at the end of manufacturing and assembly stages, may still be damaged when in use due to the encountered stresses, naturally mechanical ones as well as thermal or environmental (chemical or biological attack, scratches or wearing, minor impacts, etc.). Sometimes, synergies exist between these damage mechanisms which can then lead to anticipated failures. Stress-corrosion or fatigue-corrosion are both well-known examples.

Figure 1.Photographs of broken, moored liberty-ships during the winter of 1941

Some examples of failures have made us more aware of the importance of the potential presence of these defects (from manufacturing, assembly, or usage) and of the risk of fracture they induce. These examples can be regarded as “founders” of fracture mechanics. The most spectacular example is undoubtedly the brittle fracture of liberty-ships during the Second World War. These ships were built as a series in order to transport men and materials across the Atlantic Ocean. Some of these boats were found split through by enormous cracks, as seen in Figure 1. The breaks occurred during the night, in the winter of 1941, when the boats were moored and waiting to depart. The role of the subjected temperature was quickly identified: with low-temperatures and close-welding, the material became brittle and the residual stresses introduced during welding were sufficient for the manufacturing defects to propagate and lead to the unstable fracture of the ship.

The development of fracture mechanics began at this time. This discipline, originating at the beginning of the 20th century, tries to predict or rather avoid, the breaking of parts and components of structures.

The initially vague and invisible-to-the-eye damage gradually leads to the appearance of macroscopic cracks. These cracks can then propagate and lead to fracture.

Figure 2.High-speed “ICE” train in Eschede, Germany in 1998. This accident was caused by a crack in the wheel caused by fatigue

The sudden fracture of a component can sometimes have very serious consequences for people’s safety. For example, in Eschede (Germany) in 1998, a dramatic train accident took place (Figure 2) due to crack initiation, and then fatigue crack propagation in a wheel. This was followed by a brutal breaking of the wheel at full speed. A fracture can also have important financial consequences for the operators, with direct costs (replacement of broken or damaged parts) and indirect costs coming from the unforeseen unavailability of the systems (airplanes, trains, electricity power stations, etc.).

Inspections are also carried out on safety components by devices which also carry risks. The vital areas of the parts are periodically tested to detect the possible appearance of cracks and to estimate their size (by non-destructive testing). Then, procedures are applied to decide whether the part must be replaced or not, as a preventive measure. This is a classic procedure in aviation, railways, or the nuclear industry. As for such preventative maintenance, the decisions are made by relying primarily on statistical results from the systematic and organized monitoring of aircraft fleets in service.

Thus, all the progress of crack prediction or non-destructive test methods bears great economic interest. Certainly, to reduce the costs of preventive maintenance (replacements or periodic part testing) without reducing the safety, it is necessary to reduce uncertainty over the potential lifespan of parts, which requires an improvement in the methods used to predict lifespan.

Normally, the lifespan of mechanical components is divided into two stages: a stage known as “initiation” during which the defects are developed, becoming detectable macro-cracks. Then, there is a propagation stage during which these cracks begin to propagate. Each stage has a specific time duration: incubation time Ti, for the first stage, and propagation time Tp for the second stage. The lifespan is the sum of these two durations. When we can guarantee that the incubation time is much higher than the propagation time , there is little point in simulating crack propagation. In the opposite case, we will adhere to calculating Tp. in order to predict the evolution of the crack, from the detection threshold until the critical dimension where the break occurs.

Thus, during each periodic inspection of a safety component, the following questions are raised:

– Considering the loading applied in service, and the safety coefficients of this loading, are the (detected or potential) cracks critical?

– If yes: the part must be replaced or the device be stopped.

– If no: how much time remains before it does become critical?

– If this time is lower than an inter-inspection interval, considering the safety margins over the calculated lifespan, the part must be replaced or the device stopped.

– Otherwise, the device is put back in to service until the next periodic test.

This process is determined by experiment, the possibilities of simulation, and common-sense and it is commonly accepted in literature on this subject. It allows the risk of fracture to be minimized in operating conditions, by swooping down to pessimistic (or conservative) assumptions. However, this conservative procedure is expensive and can sometimes involve ineffectual or even harmful interventions. In the aviation industry, for example, the probability of a defect creation during repair or replacement of a part is taken into account for certain components.

It is therefore useful to limit the amount of interventions and to spread out the time between inspections, i.e. to increase the calculated lifespan for the part. For a material, a load, and a given geometry, considering the process, the lifespan can be increased mainly by three ways:

– by lowering the crack detection threshold, which requires improving the non-destructive test methods;

– by lowering the safety coefficients on the load, which requires an improved loading knowledge, which can be obtained, inter alia, by the instrumentation of parts in operating conditions (or by health monitoring);

– by lowering the safety margins on the calculated lifespans, which requires the improvement of lifespan calculation methods, which is the objective of this book.

Many books have already been devoted to fracture mechanics. The reader may wonder what a new book might bring to this already well-known subject. What prompted the writing of this work was that the recent scientific developments make it possible to raise two strong hypotheses, which are usually put forward in considering residual lifespan and which are sometimes debatable:

– any crack that is propagated will propagate until breakage;

– the stress state stays the same during propagation.

These hypotheses result from the difficulty of taking into account the effect of spatial and temporal variations of the crack propagation path. For example, a crack that started for a certain a stress concentrator may stop, bifurcate, or propagate in an even more critical plane when its tip moves away. The crack then becomes a curved surface of complex form.

This book aims to present the important recent advances in research that makes it possible to raise the restrictive hypotheses usually used for the remaining lifespan prediction of the cracked parts. This new progress stems from two conceptual jumps in modeling which arrive almost simultaneously:

–The appearance of new numerical methods that allow modeling of complex shaped cracks in three-dimensional media (independently of the mesh), and therefore, to consider spatial variations of crack loading.

The crack front is modeled by a continuous function of the three-dimensional medium (level set), which gives the signed distance to the crack plane at each spatial point. A second level set makes it possible to define the crack front.

This modeling is associated with a calculation method by enriched finite elements (X-FEM) based on the partition of unity, which makes it possible for the elements to be completely or partially intersected by the crack. Jump-type discontinuous functions are added to normal displacement interpolation functions for the completely intersected elements, while the asymptotic displacements fields resulting from the linear fracture mechanics are added for the elements where the crack front exists.

Thus we can easily simulate crack propagation in a three-dimensional medium by finite elements: it is useless to remesh when the crack propagates. It is sufficient to update the level sets and to modify the base displacement field of X-FEM elements related to the new crack position.

– Appearance of incremental crack growth laws which integrate effects of the confined plastic strain at the crack tip under mixed mode and variable amplitude loading conditions.

The approach is based on a projection of the velocity field around the crack tip on a base of reference fields (space functions only). The intensity factor of each of these fields constitutes a condensed measure of elastic and plastic strain rates for each mode in the crack tip region.

Determining the evolution laws of these intensity factors makes it possible to equip eXtended finite elements (X-FEM) with cyclic and multiaxial elastic-plastic extended behavior laws. They are used to predict the growth rate of the cracked surface area during loading paths including non-proportional mixed mode and variable amplitude loadings schemes.

These two advances make it possible to demonstrate effective numerical simulations in three-dimensions, with reasonable meshes and calculation times.

Here, we could question the relevance of these “global” methods, which are based on neighboring fields at the crack tip with respect to “local” approaches, which are based on local stress values of strains or strain velocities.

Two brief answers can be put forward: first of all, the efficiency of the numerical solutions is much higher with the “global” methods, a very fine mesh around the crack front not being required as the field shape was given a priori. Secondly, if the “local” approaches are well adapted to the fracture initiation, they are less so for the propagation simulation. Certainly, in practice, the local methods, which use the continuous finite elements also represent, a priori, the displacement fields selected for the elements (linear or quadratic) and can somewhat inadequately apprehend the presence of inherent discontinuities due to the presence of a crack.

The first chapter of this book is devoted to recalling the elementary concepts of the fracture process.

In Chapter 2, the numerical modeling of fixed or moving discontinuities is explained. Here, the discontinuities are cracks, but the same methods apply to represent other interfaces, such as a fluid-solid front in a foundry simulation.

The third chapter is devoted to the presentation of eXtended finite elements X-FEM.

The fourth chapter concerns non-linear constitutive laws for a cracked body and the strategies used to identify them for a given material.

Chapter 5 shows fracture applications through fatigue, brittle, ductile, static, and dynamic means. Some three-dimensional cases are also compared with experimental results.

Chapter 1

Elementary Concepts of Fracture Mechanics

1.1. Introduction

Fracture mechanics is a relatively new discipline, introduced at the beginning of the 20th century, and it actually took off in the 1960s.

With regard to describing the mechanical fields of a fractured medium, the first elements were put into place by In-glis [Inglis 13] who determined stress concentration around an elliptic hole in an infinite plate. Then, Westergaard [Westergaard 39], Muskhelisvili [Muskhelisvili 42], and Williams [Williams 57] gave analytical expressions of the asymptotic mechanical fields near the crack tip in linear elasticity. Lastly, Hutchinson, Rice, and Rosengren [Hutchinson 68] [Rice 68] proposed solutions for asymptotic fields within a restricted non-linear framework (non-linear elasticity).

Concerning fracture criteria, the very first theoretical elements were set up by Griffith [Griffith 20] who proposed, in 1920, an energy criterion for fracture at the atomic scale. This first approach was then expanded by Irwin and Orowan [Orowan 48], who realised that even confined plasticity could significantly change fracture energy.

Subsequently, Dugdale [Dugdale 60] and Barrenblat [Barrenblat 62] developed an energy approach to fracture which integrates the non-linear aspects within a linear-elastic fracture mechanics framework. Considering the chosen hypotheses, these theoretical approaches are confined to brittle elastic material fracture, and to cases where the plasticity remains confined to a small area close to the crack tip.

In the 1960s, Paris ([Paris 62] [Paris 64]) also proposed an empirical law of crack growth under fatigue within the linear-elastic fracture mechanics framework. These “global” approaches of fracture all define the fracture threshold according to non-local scalar parameters, solely characterizing the loading imposed in the crack tip area (stress intensity or energy release rate). Of course, these parameters are established under certain hypotheses which restrict the scope of these approaches.

More recently, a local fracture mechanics school was developed in France (Pineau [Pineau 93], Lemaitre [Lemaitre 85], and Mazars [Mazars 96] for concrete), which uses the concept of “critical” stress at the crack tip to predict fracture. Considering the presence of the crack and resultant stress concentrations, the exercise is difficult. The concept of critical stress has the advantage of being applicable to ductile fracture, i.e. in cases where the plasticity or creep are no longer confined, allowing a more realistic modeling of materials and their damage.

We must also mention Shih [Shih 83] as well as Bui [Bui 78] who have equally contributed essential components to the comprehension of this field: in particular, they have made it possible to establish the link between local and energy global approaches to fracture mechanics.

We do not wish to provide here a course on fracture mechanics, but simply to point out the useful ideas to enable a better comprehension of the following.

1.2. Superposition principle

We will take up a linear fracture mechanics framework in linear elasticity, which enables the application of the superposition principle. The stress field in the fractured part under external loading can thus be calculated as the stress field superposition in the part without crack growth, under boundary conditions and an auxiliary field, whose boundary conditions make it possible to restore the condition of the free surfaces on the crack faces.

The basic problem corresponds to a crack where only the faces are loaded with a stress vector, equal and opposite in sign to that which controls the “healthy” part where the crack should be.

Figure 1.1 illustrates this principle in the case of a cracked planar plate under the boundary conditions of uniform uniaxial tension.

Figure 1.1.Principle of superposition

Moreover, in linear elasticity, we can always divide the problem into a superposition of elementary problems, which relate to different modes of crack loading.

1.3. Modes of crack straining

Let us consider a crack plane with a rectilinear front in an infinite three-dimensional medium. We will denote this crack as Γ. We estimate that the crack plane is (O, x, z). The crack front will be the straight line (O, z). Each point M of Γ (of coordinates (x, y, z)) belonging to the crack will therefore verify:

[1.1]

Now we will discuss the relative displacement of crack sides in the immediate vicinity of any point N − Γ of the crack front at coordinates (0, 0, z). This relative displacement is a vector [u] whose components in the local base attached to the plane and the crack front make it possible to define the elementary modes of crack stress.

The crack stress is therefore driven by the three modes which relate to the three components of the vector displacement jump [u] along the crack Γ.

The first, called Mode I or opening mode, is characterized by a magnitude which we will call KI, and corresponds to the component normal to the crack front of vector [u].

The second, called Mode II or in-plane shear mode, is characterized by KII and corresponds to its component in the crack plane and normal to its front.

Lastly, the third, called Mode III or out-of-plane shear mode or tearing mode, is characterized by KIII and corresponds to the component tangential to the crack front of vector [u].

Figure 1.2 illustrates these three stress modes. The three denominations Ki will be called stress intensity factors.

More generally, Mode I corresponds to the symmetrical section with respect to the crack plane of the elastic problem, while Mode II corresponds to its anti-symmetric section with respect to (O, z), and Mode III to its anti-symmetric section with respect to (O, x).

Figure 1.2.Elementary modes of three-dimensional crack stress

1.4. Singular fields at cracking point

In linear elasticity the stresses and strains in the vicinity of point N of the crack front are singular, i.e. we find that they tend toward infinity when the distance r between N and a point P close to the crack tip tends toward zero (Figure 1.3). Williams [Williams 57] has shown that the stresses in the vicinity of the singularity could be expressed in the form of an infinite expansion with respect to r, where each term of the expansion is a product of an intensity factor Kp, a spatial distribution function fp, and distance r at the power of λp.

[1.2]

Figure 1.3.Definition of the coordinates r and θ

Stress and displacement expressions of the first order are known as Westergaard’s asymptotic solutions [Westergaard 39] and are given for a point P of coordinates (r, θ) in a plane normal to the crack front. The stresses at point P near crack tip N are usually given in Polar coordinates (r, θ) (see Figure 1.3). The first terms of power expansion of the stress tensor are a linear combination of three solutions which correspond to each elementary mode, characterized by their respective intensities KI, KII, and KIII:

[1.3]

These stress fields show a singularity of type . The stresses tend toward infinity when r → 0. Consequently, the first term of the expansion dominates when r is small. We can use the solution of the first degree and distinguish three areas, illustrated in Figure 1.4, in the vicinity of the crack tip:

– when r is large, the term of the first degree of the asymptotic solutions is no longer dominant. It is also necessary to consider the terms of higher degree and the loading applied in the non-cracked medium by applying the superposition principle [1.1],

– when r gets smaller, we define the K-dominant zone in which the asymptotic development terms of the first degree are much higher than the other terms. Intensity factors KI, KII, and KIII are sufficient to characterize the stresses in this zone,

– these asymptotic solutions were established in a linear elasticity framework, which is generally no longer applicable when the stresses are very high. Also, within the K-dominant zone, there is a small zone around the crack tip in which the material expresses non-linear behavior (plastic, for example). This zone is then called the plastic zone,

– finally, in the immediate vicinity of the crack tip, the material is damaged, which allows the crack to propagate. This last zone is usually called the process zone. Its dimensions can be compared on a micro-structural material scale, a dimension to which the continuous theories of plasticity are no longer applicable. This area is generally disregarded in linear or nonlinear fracture mechanics but it comes into picture when we try to understand the local mechanisms that allow the crack propagation.

Figure 1.4.Definition of the zones of interest around the crack tip

We also know the analytical shape of the displacement fields in the vicinity of the crack tip. They are given for any point P by equation [1.4] below:

[1.4]

We can see that displacements at the mean crack tip are in proportion to . The relative displacement of the crack lips is therefore parabolic.

There are similar expressions in the case of anisotropic materials or for cracks where the end is located at the interface between two different isotropic materials. But, these are of a more complex nature and will not be discussed in this book. Such examples can be found in [Hills 96].

1.4.1. Asymptotic solutions in Mode I

The stress field is given by:

[1.5]

The displacement field is given by:

[1.6]

1.4.2. Asymptotic solutions in Mode II

The stress field is given by:

[1.7]

The displacement field is given by:

[1.8]

1.4.3. Asymptotic solutions in Mode III

The stress field is given by:

[1.9]

The displacement field is given by:

[1.10]

1.4.4. Conclusions

We therefore have all the analytical expressions needed to represent the displacement field at the crack tip for each of the three basic stress loading Modes I, II and III. We will stay thereafter under the assumption of plane strain along the direction tangent to the crack front as the basis for the displacement field.

This hypothesis is valid everywhere along the crack front except in the region very close to the skin of the body. In this region neither of the two hypotheses are generally valid (plane stress or plane strain). Hence for general 3D cracks we shall keep only the plane strain fields knowing that their validity is debatable where the crack approaches the free surface of the body.

1.5. Crack propagation criteria

In the first section, we saw how to characterize crack loading. The subsequent question is to find out if this loading will lead to the propagation of the crack, and if so then how. The answer to this question will be taken up again later in more detail but we will provide here the most common fracture criteria. We will consider this issue with two approaches: the “local” fracture vision and the “global” energetic vision.

1.5.1. Local criterion

This criterion is not generally applied to a linear-elastic fracture mechanics (LEFM) framework. In elasticity, the stresses tend toward infinity when the distance from the crack tip tends toward zero, a stress or strain criterion would have no meaning. Thus, we have a more complex constitutive law for material; in general, an elastic-plastic damageable stressstrain law.

In this case, the stresses do not tend toward infinity at the crack tip, but remain limited due to the non-linear property of the material. The propagation of the crack then follows from the damage law for the chosen material. The damage rate in the material may be, for example, akin to local plastic rate, or of the stress level reached or strain from the level of damage [Lemaitre 96]. When the critical damage is reached the material point is broken; the “damaged” Young’s modulus is brought back to zero. This process makes it possible to “naturally” propagate cracks.

This method of simulation is very appealing. We can define laws of local damage, which are akin to physics, identified on the basis of various tests, which include the priming and the propagation of the cracks for various fracture mechanisms (ductile fracture, creep, propagation of a primary crack, or the appearance of multiple microscopic cracks, etc.). Also, this approach has had a strong repercussion at its appearance but leads to important difficulties in its numerical implementation.

First of all, as the crack criteria relies on the mechanical fields calculated in the plastic zones and in crack development, we can easily understand that it is necessary to have a fine-enough mesh in order to correctly describe the evolution of these fields. This leads to very heavy calculations when dealing with three-dimensional problems. On the other hand, the local approach is perfectly suitable when the plastic and development zones become large, and in particular when the K-dominant zone no longer exists.

It is also necessary to define and characterize the damage laws which will be used in calculations, and which require a large effort with regard to the experiment, and must be adapted to each stress type: tear, fatigue, creep, and a combination of these fracture modes.

Then, when the elements become closer to their broken state, the strain tends to localize and the results then become mesh dependent: the more we refine the mesh, the earlier the “numerical” fracture occurs! To overcome this difficulty, two main approaches exist. The first is pragmatic and consists of identifying the damage laws by simulating the experimental database with a mesh of fixed-size. Then, during the application of the model on real industrial problems, simulations will be carried out with the same mesh size. The second approach consists of inserting a material characteristic length Lcar. We then have a “non-local” damage criteria defined from the average (and possibly from the gradients) of the useful mechanical quantities, in a sphere of radius Lcar. If these approaches have made it possible to solve the problem of the mesh dependency, they do not lighten the calculations, since the Lcar is often very small (about 10 micrometers for steel). And then finally, the fracture criterion is no longer a local criterion! In fact, the first approach reverts “implicitly” to obtaining a characteristic length Lcar which is the same as the mesh size.

In fatigue, the time calculation problem is important. Firstly, as the size of the plastic zone is much smaller in fatigue than in ductile fracture or in creep, the mesh must be even finer. Moreover, this approach requires a calculation of all the cycles, which can quickly become inaccessible, particularly for a three-dimensional problem in elastic-plasticity.

Lastly, the use of these methods shows that they are relatively suitable for the initiation and the first stages of cracking but function rather poorly (the calculations do not converge) for significant propagations. Indeed, the “broken” finite elements are not really broken but it is simply their behavior that is modified. However, they can be greatly distorted if the relative displacement of the crack lips is large, which leads to major numerical difficulties. We can remove the broken elements, but it is then the processing of the contact and closing again which becomes complex.

We will not return to these local methods in the remainder of the book.

1.5.2. Energy criterion

1.5.2.1. Energy release rate G

This criterion is based on the intuitive idea that a certain quantity of the energy Gc must be provided in order to break the material, i.e. to create a cracked surface area dA. The irreversible amount of energy used up during the creation of a surface dA is written as:

[1.11]

Let us consider the two-dimensional elementary case of Figure 1.5 and look at which conditions are needed for the crack length a(t) to propagate. From an energy point of view, this question can be formulated in the following way: is the energy available in the part, which is released when the cracked surface area extends from a quantity dA