Fracture Mechanics

Front matter

Fracture Mechanics An Engineering Primer M. Janssen, J. Zuidema, R.J.H. Wanhill

Fracture Mechanics An Engineering Primer

Fracture Mechanics An Engineering Primer 2nd Edition M. Janssen, J. Zuidema and R. J. H. Wanhill

Part I Introduction

An Overview 1.1 About this Course This course is intended as a basic grounding in fracture mechanics for engineering use. In order to compile the course we have consulted several textbooks and numerous research articles. In particular, the following books have been most informative and are recommended for additional reading: • D. Broek, “Elementary Engineering Fracture Mechanics”, Martinus Nijhoff (1986) The Hague; • J.F. Knott, “Fundamentals of Fracture Mechanics”, Butterworths (1973) London; • Richard W. Hertzberg, “Deformation and Fracture Mechanics of Engineering Materials”, John Wiley and Sons (1988) New York; • T.L. Anderson, “Fracture Mechanics, Fundamentals and Applications”, CRC Press (1991) Boston.

Four international journals are also recommended: • Fatigue and Fracture of Engineering Materials and Structures; • International Journal of Fatigue; • International Journal of Fracture; • Engineering Fracture Mechanics.

As indicated in the table of contents the course has been divided into five parts. Part I, consisting of this chapter, is introductory. In Part II the well established subject of Linear Elastic Fracture Mechanics (LEFM) is treated, and this is followed in Part III by the more recent and still evolving topic of Elastic-Plastic Fracture Mechanics (EPFM). In Part IV the applicability of fracture mechanics concepts to crack growth behaviour is discussed: namely subcritical, stable crack growth under cyclic loading (fatigue) or sustained load, and dynamic crack growth beyond instability. Finally, in Part V the mechanisms of fracture in actual materials are described together with the influence of material behaviour on fracture mechanics-related properties. 1.2 Historical Review Strength failures of load bearing structures can be either of the yielding-dominant or fracture-dominant types. Defects are important for both types of failure, but those of primary importance to fracture differ in an extreme way from those influencing yielding and the resistance to plastic flow. These differences are illustrated schematically in figure 1.1.

4 Introduction For yielding-dominant failures the significant defects are those which tend to warp and interrupt the crystal lattice planes, thus in terfering with dislocation glide and providing a resistance to plastic deformation that is essential to the strength of high strength metals. Examples of such defects are inte rstitial and out-of-size substitutional atoms, grain boundaries, coherent precipitates and dislocation networks. Larger defects like inclusions, porosity, surface scratches and small cracks may influence the effective net section bearing the load, but otherwise have little effect on resistance to yielding. Figure 1.1. Types of structural failure.

For fracture-dominant failures, i.e. fracture before general yielding of the net section, the size scale of the defects which are of maj or significance is essentially macroscopic, since general plasticity is not involved but only the local stress-strain fields associated with the defects. The minute lattice-related defects which control resistance to plastic flow are not of direct concern. They are important insofar as the resistance to plastic flow is related to the material’s susceptibility to fracture. Fracture mechanics, which is the subject of this course, is concerned almost entirely with fracture-dominant failure. The commonly accepted first successful analysis of a fracture-dominant problem was that of Griffith in 1920, who considered the propagation of brittle cracks in glass. Griffith formulated the now well-known concept that an existing crack will propagate if thereby the total energy of the system is lowered, and he assumed that there is a simple energy balance, consisting of a decrease in elastic strain energy within the stressed body as the crack extends, counteracted by the energy needed to create the new crack surfaces. His theory allows the estimation of the theoretical strength of brittle solids and also gives the correct relationship between fracture strength and defect size.

1. An overview 5 The Griffith concept was first related to brittle fracture of metallic materials by Zener and Hollomon in 1944. Soon after, Irwin pointed out that the Griffith-type energy balance must be between (i) the stored strain energy and (ii) the surface energy plus the work done in plastic deformation. Irwin defined the ‘energy release rate’ or ‘crack driving force’, G, as the total energy that is released during cracking per unit increase in crack size. He also recognised that for relatively ductile materials the energy required to form new crack surfaces is generally insignificant compared to the work done in plastic deformation.

In the middle 1950s Irwin contributed another major advance by showing that the energy approach is equivalent to a stress intensity ( K) approach, according to which fracture occurs when a critical stress distribu tion ahead of the crack tip is reached. The material property governing fracture may therefore be stated as a critical stress intensity, Kc, or in terms of energy as a critical value Gc.

Demonstration of the equivalence of G and K provided the basis for development of the discipline of Linear Elastic Fracture Mechanics (LEFM). This is because the form of the stress distribution around and close to a crack tip is always the same. Thus tests on suitably shaped and loaded specimens to determine Kc make it possible to determine what cracks or crack-like flaws are tolerable in an actual structure under given conditions. Furthermore, materials can be compared as to their utility in situations where fracture is possible. It has also been found that the sensitivity of structures to subcritical cracking such as fatigue crack growth and stress corrosion can, to some extent, be predicted on the basis of tests using the stress intensity approach. The beginnings of Elastic-Plastic Fracture Mechanics (EPFM) can be traced to fairly early in the development of LEFM, notably Wells’ work on Crack Opening Displacement (COD), which was published in 1961. In 1968 Rice introduced an elastic-plastic fracture parameter with a more theoretical basis: the J integral. Although both COD and J are now well established concepts, EPFM is still very much an evolving discipline. The reason is the greater complexity of elastic-plastic analyses. Important topics are: • the description of stable ductile crack growth (tearing), • the development of failure assessment methods that combine the effects of plasticity and fracture.

As opposed to using the above-mentioned global fracture mechanics parameters, fracture problems are also increasingly being tackled by means of local fracture criteria. Here the mechanical conditions that actually exist in the crack tip region are being determined and are being related to the material properties. 1.3 The Significance of Fracture Mechanics In the nineteenth century the Industrial Revolution resulted in an enormous increase in the use of metals (mainly irons and steels) for structural applications. Unfortunately, there also occurred many accidents, with loss of life, owing to failure of these structures. In particular, there were numerous accidents involving steam boiler explosions and railway equipment.

6 Introduction Some of these accidents were due to poor design, but it was also gradually discovered that material deficiencies in the form of pre-existing flaws could initiate cracking and fracture. Prevention of such flaws by better production methods reduced the number of failures to more acceptable levels.

A new era of accident-prone structures was ushered in by the advent of all-welded designs, notably the Liberty ships and T-2 tankers of World War II. Out of 2500 Liberty ships built during the war, 145 broke in two and almost 700 experienced serious failures. Many bridges and other structures also failed. The failures often occurred under very low stresses, for example even when a ship was docked, and this anomaly led to extensive investigations which revealed that the fractures were brittle and that flaws and stress concentrations were responsible. It was also discovered that brittle fracture in the types of steel used was promoted by low temperatures. This is depicted in figure 1.2: above a certain transition temperature the steels behave in a ductile manner and the energy required for fracture increases greatly.

Figure 1.2. Schematic of the general effect of temperature on the fracture energy of structural metals.

Current manufacturing and design procedures can prevent the intrinsically brittle fracture of welded steel structures by ensuring that the material has a suitably low transition temperature and that the welding process does not raise it. Nevertheless, serviceinduced embrittlement, for example hydrogen embrittlement in the petrochemical industries, irradiation effects in nuclear pressure vessels and corrosion fatigue in offshore platforms, remains a cause for concern.

Looking at the present situation it may be seen from figure 1.3 that since World War II the use of high strength materials for structural applications has greatly increased. These materials are often selected to obtain weight savings — aircraft structures are an obvious example. Additional weight savings have come from refinements in stress analysis, which have enabled design allowables to be raised. However, it was not recognised until towards the end of the 1950s that although these materials are not intrinsi-

1. An overview 7 cally brittle, the energy required for fracture is comparatively low, as figure 1.2 shows. The possibility, and indeed occurrence, of this low energy fracture in high strength materials stimulated the modern development of fracture mechanics. The object of fracture mechanics is to provide quantitative answers to specific problems concerning cracks in structures. As an illustration, consider a structure containing pre-existing flaws and/or in which cracks initiate in service. The cracks may grow with time owing to various causes (for example fatigue, stress corrosion, creep) and will generally grow progressively faster, figure 1.4.a. The residual strength of the structure, which is the failure strength as a function of crack size, decreases with increasing crack size, as shown in figure 1.4.b. After a time the residual strength becomes so low that the structure may fail in service.

Figure 1.4. The engineering problem of a crack in a structure. With respect to figure 1.4 fracture mechanics should attempt to provide quantitative answers to the following questions: 1. What is the residual strength as a function of crack size? 2. What crack size can be tolerated under service loading, i.e. what is the maximum permissible crack size? 3. How long does it take for a crack to grow from a certain initial size, for example the minimum detectable crack size, to the maximum permissible crack size? Figure 1.3. Introduction of high strength materials for structural applications.

8 Introduction 4. What is the service life of a structure when a crack-like flaw ( e.g. a manufacturing defect) with a certain size is assumed to exist? 5. During the period available for crack detection how often should the structure be inspected for cracks?

This course is intended to show how fracture mechanics concepts can be applied so that these questions can be answered.

In the remaining sections 1.4 1.11 of this introductory chapter an overview of the basic concepts and applications of LEFM a nd EPFM are given in preparation for more detailed treatment in subsequent chapters.

Figure 1.5. A through-thickness crack in a loaded infinite plate. 1.4 The Griffith Energy Balance Approach Consider an infinite plate that is subjected to a uniform tensile stress, V, applied at infinity (see figure 1.5). Suppose that we introduce a through-thickness crack of length 2a. In the area directly above and below the crack the stress (in the loading direction) will decrease significantly and will even become zero along the crack flanks. Hence introduction of the crack changes the elastic strain energy stored in the plate. We can roughly estimate this change by assuming that in a circle-shaped area of radius a around the crack the stress has become zero, while the remainder of the plate experiences the same stress as before. In this case the elastic en ergy in the plate has decreased by an amount equal to the volume of the stress-free material times the original elastic energy per unit volume, i.e. ½×stress×strain. Assuming linear elastic material behaviour, i.e. a Young’s modulus E, the elastic energy change would be1: Sa2·V22E = 12 SV2a2E . (1.1) Obviously, this is only an approximation because the stress field becomes non- 1 In this section we consider two-dimensional geomet ries only and all energies and forces are defined per unit thickness.

1. An overview 9 homogeneous near the crack, as will be shown in chapter 2. Griffith used a stress analysis developed by Inglis to show that for an infinite plate the elastic energy change is actually given by Ua = SV2a2E , (1.2) where Ua =c hange in the elastic strain energy of the plate caused by introducing a crack with length 2 a. The minus sign shows this change is a decrease in elastic energy.

The introduction of a crack will require a certain amount of energy. Griffith assumed that for ideally brittle materials this is in the form of surface energy. A crack with length 2a in a plate involves the creation of a crack surface area (defined per unit thickness) equal to 2·(2a) = 4a, leading to an increase in surface energy of UJ = 4a·Je , (1.3) where UJ = change in surface energy of the plate due to introduction of a crack with length 2a, Je = surface energy per unit area, i.e. the surface tension. Griffith postulated that a crack will extend when the potential energy decreases. He considered the surface energy as a part of this potential energy. In practice the energy involved in creating crack surfaces will not be reversible due to several reasons (oxidation etc.) and strictly speaking is not part of the potential energy. However, as long as only growing cracks are considered, the irreversibility of the surface energy is not relevant. Here, the potential energy according to Griffith will be referred to as the total energy.

For a real plate, i.e. one with finite dimensions, the total energy U is that of the plate and its loading system. When a crack is present the total energy U is composed of U = Uo + Ua + UJ – F , (1.4) where Uo = total energy of the plate and its loading system before introducing a crack (a constant), F = work performed by the loading system during the introduction of the crack = load × displacement.

The combination of plate and loading system is assumed to be isolated from its surroundings, i.e. no work is performed on the plate or on the loading system from outside. This explains why F must be subtracted in equation (1.4): if the loading system performs work it goes at the expense of the energy content of the loading system and therefore lowers the total energy U. A more extensive treatment will be given in section 4.2. In this introductory chapter we will conveniently assume that no work is done by the loading system. This is the case if the specimen is loaded by a constant displacement, a

10 Introduction so-called fixed grip condition. Then the term F in equation (1.4) will vanish. Introducing a crack now leads to a decrease in elastic strain energy of the plate, i.e. Ua is negative, because the plate loses stiffness and the load applied by the fixed grips will drop. A plate with finite dimensions resembles an infinite plate when 2 a << W, the plate width. Consequently, the total energy U of a finite plate loaded with fixed grips and containing a small crack is approximately U = Uo + Ua + UJ = Uo – SV2a2E + 4aJe . (1.5) Following Griffith, crack extension will occur when U decreases. In order to formulate a criterion for crack extension, we consider an increase of the crack length by d(2a). Since Uo is constant, it will not change and dUo/d(2a) is zero. Also, since no work is done by the loading system, the driving force for crack extension can be delivered only by the decrease in elastic energy d Ua due to the crack length increase d(2 a). The crack will extend when the available energy dUa is larger than the energy required dUJ. Thus the criterion for crack extension is dU d(2a) = dd(2a) (Ua + UJ) < 0 or dd(2a) ©¨§ ¹¸·–SV2a2E + 4aJe < 0 . (1.6) This is illustrated in figure 1.6. Figure 1.6.a schematically represents the two energy terms in equation (1.6) and their sum as functions of the introduced crack length, 2 a. Figure 1.6.b represents the derivative, dU/d(2a). When the elastic energy release due to a potential increment of crack growth, d(2 a), outweighs the demand for surface energy for the same crack growth, the introduction of a crack will lead to its unstable propagation.

From the criterion for crack extension, equation (1.6), one obtains SV2a E > 2Je , (1.7) which can be rearranged to V a > 2EJeS . (1.8) Equation (1.8) indicates that crack extension in ideally brittle materials is governed by the product of the remotely applied stress and the square root of the crack length and by material properties. Because E and Je are material properties the right-hand side of equation (1.8) is equal to a constant value characteristic of a given ideally brittle material. Consequently, equation (1.8) indicates that crack extension in such materials occurs when the product V a attains a certain critical value.

1. An overview 11 Figure 1.6. Energy balance for a small crack in a large plate loaded under fixed grip conditions. 1.5 Irwin’s Modification to the Griffith Theory Irwin designated the left-hand side of equation (1.7) as the energy release rate , G, representing the energy per unit new crack area that is available for infinitesimal crack extension.2 The right-hand side of equation (1.7) represents the surface energy increase per unit new crack area that would occur owing to infinitesimal crack extension and is designated the crack resistance, R. It follows that G must be larger than R before crack growth occurs. If R is a constant, this means that G must exceed a critical value Gc = R = constant. Thus fracture occurs when G = SV2aE > Gc = R = 2Je . (1.9) The critical value Gc can be determined by measuring the critical stress Vc required to fracture a plate with a crack of size 2a or by measuring the critical crack size 2ac needed 2 The crack area is defined as the projected area, normal to the crack plane, of the newly formed surfaces.

12 Introduction to fracture a plate loaded by a stress V.

In 1948 Irwin suggested that the Griffith theory for ideally brittle materials could be modified and applied to both brittle materials and metals that exhibit plastic deformation. A similar modification was proposed by Orowan. The modification recognised that a material’s resistance to crack extension is determined by the sum of the surface energy Je and the plastic strain work Jp (both per unit crack surface area) that accompany crack extension. Consequently, in this case the crack resistance is R = 2(Je + Jp) . (1.10) For relatively ductile materials Jp >> Je, i.e. R is mainly plastic energy and the surface energy can be neglected.

Although Irwin’s modification includes a plastic energy term, the energy balance approach to crack extension is still limited to defining the conditions required for instability of an ideally sharp crack. Also, the energy balance approach presents insuperable problems for many practical situations, especially slow stable crack growth, as for example in fatigue and stress corrosion cracking.

The energy balance concept will be treated in more detail in chapter 4. 1.6 The Stress Intensity Approach Owing to the practical difficulties of the energy approach a major advance was made by Irwin in the 1950s when he developed the stress intensity approach. First, from linear elastic theory Irwin showed that the stresses in the vicinity of a crack tip take the form Vij = K2Sr fij(T) + … , (1.11) where r, T are the cylindrical polar co-ordinates of a point with respect to the crack tip, figure 1.7.

Figure 1.7. Stresses at a point ahead of a crack tip.

1. An overview 13 K is a quantity which gives the magnitude of the elastic stress field. It is called the stress intensity factor.3 Dimensional analysis shows that K must be linearly related to stress and directly related to the square root of a characteristic length. Equation (1.8) from Griffith’s analysis indicates that this characteristic length is the crack length, and it turns out that the general form of the stress intensity factor is given by K = V Sa · f(a/W) , (1.12) where f(a/W) is a dimensionless parameter that depends on the geometries of the specimen and crack, and V is the (remotely) applied stress. For an infinite plate with a central crack with length 2 a, f(a/W) = 1 and thus K = V Sa. For this case we also have G = SV2a/E, see equation (1.9). Combining the two formulae for K and G yields the relation: G = K2E , (1.13) which Irwin showed to be valid for any geometry.

Since K = V Sa for a central crack in an infinite plate, it follows from the result of Griffith’s energy balance approach, equation (1.8), that crack extension will occur when K reaches a certain critical value. This value, Kc, is equal to 2EJe or, after applying Irwin’s modification, 2E(Je + Jp). The criterion for crack extension in terms of K is K = V Sa > Kc . (1.14) The parameter governing fracture may therefore be stated as either a critical energy release rate, Gc, or a critical stress intensity, Kc. For tensile loading the relationships between Gc and Kc are Gc = Kc2E . (1.15) The value of the critical stress intensity Kc can be determined experimentally by measuring the fracture stress for a large plate that contains a through-thickness crack of known length. This value can also be measured by using other specimen geometries, or else can be used to predict critical combinations of stress and crack length in these other geometries. This is what makes the stress in tensity approach to fracture so powerful, since values of K for different specimen geometries can be determined from conventional elastic stress analyses: there are several handbooks giving relationships between the stress intensity factor and many types of cracked bodies with different crack sizes, orientations and shapes, and loading conditions. Furthermore, the stress intensity factor, K, is applicable to stable crack extension and does to some extent characterize processes of subcritical cracking like fatigue and stress corrosion, as will be mentioned in section 1.10 of this chapter and in greater detail in chapters 9 and 10. 3 The stress intensity factor is essentially different from the well-known stress co ncentration factor. The latter is a dimensionless ratio that describes the increase in stress level relative to the nominal stress.

14 Introduction It is the use of the stress intensity factor as the characterizing parameter for crack extension that is the fundamental principle of Linear Elastic Fracture Mechanics (LEFM). The theory of Linear Elastic Fracture Mechanics is well developed and will be discussed in chapter 2. 1.7 Crack Tip Plasticity The elastic stress distribution in the vicinity of a crack tip, equation (1.11), shows that as r tends to zero the stresses become infinite, i.e. there is a stress singularity at the crack tip. Since structural materials deform plastically above the yield stress, there will in reality be a plastic zone surrounding the crack tip. Thus the elastic solution is not unconditionally applicable.

Irwin considered a circular plastic zone to exist at the crack tip under tensile loading. As will be discussed in chapter 3, he showed that such a circular plastic zone has a diameter 2ry, figure 1.8, with ry = 12S ©¨§ ¹¸·KVys 2 , (1.16) where Vys is the yield stress.

Irwin argued that the occurrence of plasticity makes the crack behave as if it were longer than its physical size — the displacements are larger and the stiffness is lower than in the elastic case. He showed that the crack may be viewed as having a notional tip at a distance ry ahead of the real tip, i.e. in the centre of the circular plastic zone (see figure 1.8). Beyond the plastic zone the elastic stress distribution is described by the K corresponding to the notional crack size. As shown in figure 1.8, this elastic stress distribution takes over from the yield stress at a distance 2ry from the actual crack tip. Since the same K always gives the same plastic zone size for materials with the same Figure 1.8. The crack tip plastic zone according to Irwin.

1. An overview 15 yield stress, equation (1.16), the stresses and strains both within and outside the plastic zone will be determined by K and the stress intensity approach can still be used. In short, the effect of crack tip plasticity corresponds to an apparent increase of the elastic crack length by an increment equal to ry.

A plastic zone at the tip of a through-thickness crack will inevitably tend to contract in the thickness direction along the crack front. If the plate thickness is of the order of the plastic zone size or smaller, this contraction can occur freely and a plane stress state will prevail. On the other hand, if the plate thickness is much larger than the plastic zone size, contraction is constrained by the elastic material surrounding the plastic zone. The strain in the thickness direction will then be small, meaning that a plane strain state is present.4 The occurrence at the crack tip of either a plane stress or plane strain state has a large effect on the plastic behaviour of the material. In plane strain the plastic deformation occurs only when the stresses amply exceed the yield stress. Actually, equation (1.16) is valid for a plane stress state only. For plane strain ry = 12S ©¨§ ¹¸·KCVys 2 , (1.17) where C is usually estimated to be about 1.7. Thus in plane strain the plastic zone size is considerably smaller. 1.8 Fracture Toughness From sections 1.6 and 1.7 it follows that under conditions of limited crack tip plasticity the parameter governing tensile fracture can be stated as a critical stress intensity, Kc. The value of Kc at a particular temperature depends on the amount of thickness constraint and thus on specimen thickness. It is customary to write the limiting value of Kc for maximum constraint (plane strain) as KIc.5 KIc can be considered a material property ch aracterizing the crack resistance, and is therefore called the plane strain fracture toughness. Thus the same value of KIc should be found by testing specimens of the same material with different geometries and with critical combinations of crack size and shape and fracture stress. Within certain limits this is indeed the case, and so a knowledge of KIc obtained under standard conditions can be used to predict failure for different combinations of stress and crack size and for different geometries.

Kc can also be determined under standard conditions, and the value thus found may also be used to predict failure, but only for situations with the same material thickness and constraint. 4 In all formulae up to this point a plane stress state was implicitly assumed. 5 The subscript I refers to the loading mode where the crack flanks are pulled straight apart (see section 2.1). In fracture mechanics it is customary to include th is subscript in expressions that contain the stress intensity factor as a variable, i.e. KI. However, in this introductory chapter this is not yet done.

16 Introduction As an introductory numerical example of the design application of LEFM, consider the equation for a through-thickness crack in a wide plate, i.e.

K = V Sa . (1.18) Assume that the test results show that for a particular steel the Kc is 66 MPa m for the plate thickness and temperature in service. Using equation (1.18) a residual strength curve for this steel can be constructed relating Kc and nominal stress and crack size. This is shown in figure 1.9. Al so assume that the design stress is 138 MPa. It follows from equation (1 .18) and figure 1.9 that the tolerable crack size would be about 145 mm. For a design stress of 310 MP a the same material could tolerate a crack size of only about 28 mm. Note from figure 1.9 that if a steel with a higher fracture toughness is used, for example one with a Kc of 132 MPa m, the permissible design stress for a given crack size is significantly increased. Thus a material with a higher fracture toughnes s permits a longer crack at a given stress or a higher stress at a given crack length.

Figure 1.9. Residual strength curves for two steels. 1.9 Elastic-Plastic Fracture Mechanics Linear Elastic Fracture Mechanics can deal with only limited crack tip plasticity, i.e. the plastic zone must remain small compared to the crack size and the cracked body as a whole must still behave in an approximately elastic manner. If this is not the case then the problem has to be treated elasto-plastic ally. Due to its complexity the concepts of Elastic-Plastic Fracture Mechanics (EPFM) are not so well developed as LEFM theory, a fact that is reflected in the approximate nature of the eventual solutions. In 1961 Wells introduced the crack opening displacement (COD) approach. This approach focuses on the strains in the crack tip region instead of the stresses, unlike the stress intensity approach. In the presence of plasticity a crack tip will blunt when it is loaded in tension. Wells proposed to use the crack flank displacement at the tip of a blunting crack, the so-called crack tip opening displacement (CTOD) as a fracture parameter (see figure 1.10).

Even for tougher materials exhibiting considerable plasticity critical CTOD values could be defined corresponding to the onset of fracture. Such critical CTOD values

1. An overview 17 could then be used to qualify the mater ials concerned for a given application. However, initially it proved difficult to determine the required CTOD for a given load and geometry or alternatively to calculate critical crack lengths or loads for a given material. In 1968 Rice considered the potential energy changes involved in crack growth in non-linear elastic material. Such non-linear elastic behaviour is a realistic approximation for plastic behaviour provided no unloading occurs in any part of the material. Rice derived a fracture parameter called J, a contour integral that can be evaluated along any arbitrary path enclosing the crack tip, as illustrated in figure 1.11. He showed J to be equal to the energy release rate for a crack in non-linear elastic material, analogous to G for linear elastic material.