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FATIGUE FRACTURE, AND MECHANICAL RELIABILITY
PART I
An Introduction to Fracture Mechanics
by
James W. Provan
Spring 2008 Department of Mechanical Engineering University of Victoria Victoria, BC, Canada
Part I-1
1.
INTRODUCTION
Materials used in the construction of most engineering components are chosen according to their mechanical (as well as their chemical and electrical) properties so that they may sustain their applied design loads. Many components, however, fail under either fracture or fatigue loading patterns that are either well below these design loads or well before their life expectancy. These types of loads initiate and propagate cracks that eventually lead to the failure of the component. For well over a century, researchers have studied the fracture and fatigue properties of engineering materials, the main objective being to be able to predict the fracture/failure characteristics or the fatigue life of a component. The fracture properties of different materials undergoing different load patterns associated with different environments are generally obtained by experimental testing. Knowing these fracture properties, it is possible for engineers and designers to assess, with the help of fracture mechanics, just how components are likely to behave while being loaded. Hence, the ability to predict the life of a whole engineering structure, e.g., a vessel, an aircraft, an engine, etc., by studying every component is, today, an obligation of all design engineers. Over the decades, designers have been made aware of the very detrimental effects of mechanical structural failure. Most textbooks report some spectacular examples in their introductions and case studies, such as those reported by Rich and Cartwright [1], have a definite educational value and should be consulted during the development stage of a particularly complicated design. In addition to these catastrophic and unfortunately often newsworthy failures, there are countless "minor" failures that result in delays, inconveniences, expensive repairs and lawsuits. Furthermore, as designs become more complex, high-strength alloys become more common, minimum weight and cost become factors, and stresses increase while safety factors lower, the possibility of failures in expensive/complex structures must be considered and the designer must become aware of the methods available for preventing failures. There are two important items to be considered in the design of engineering components and structures. The first is the determination of the loads and the stress analysis of the problem aimed at finding the magnitude and direction of the stresses and strains as they vary with time and position within a mechanical component. The second item is the selection of a failure criterion, which determines both the material and the final dimensions of the component. The traditional failure criterion is to design the cross-sectional area of a component such that the applied stresses are kept below the yield strength of the material by means of a safety factor. It is now common knowledge that this conventional design criterion cannot adequately describe the failure of materials, which are sensitive to the presence of inherent flaws or mechanical defects. Indeed, in most design situations where stresses are high it is now accepted practice to design the element on the assumption that flaws and/or microcracks exist in the neighborhood of a region of high stress. The primary objective of fracture, fatigue and mechanical reliability is to expose the engineer to the germane design concepts that are presently finding their way into all practical aspects of industrial design.
Part I-2
References: [1] Rich, T.P. and Cartwright, D.J., Case Studies in Fracture Mechanics, Dept. of the Army, Army Mtls. and Mechs. Res. Cen., Watertown, MA, (1977).
Part I-3
2.
2.1
LEFM
Preamble:
The science of fracture mechanics has become the primary approach of controlling the brittle fracture and fatigue failure of engineering structures. It may be used to quantitatively describe the tradeoffs between stress, material toughness, and flaw size so that the designer may determine the relative importance of each during the design stage rather than during a failure analysis. Design codes, initially adopted by various engineering professions in the 1970s and 80s, now include material specifications that are based upon fracture mechanics concepts. Although a vast amount of research on the characteristics of fracture has shown that numerous factors may contribute to these failures, fracture mechanics itself has shown that there are three primary factors that control the susceptibility of a structure to brittle failure. 1. Material Fracture Toughness. Material fracture toughness may be defined as the ability to carry loads or deform plastically in the presence of a notch. It may be described in terms of the critical stress intensity factor, K Ic , under a variety of conditions. (These terms and conditions are fully discussed in the following chapters.) Crack Size. Fractures initiate from discontinuities that can vary from extremely small cracks to much larger weld or fatigue cracks. Furthermore, although good fabrication practice and inspection can minimize the size and number of cracks, most complex mechanical components cannot be fabricated without discontinuities of one type or another. Stress Level. For the most part, tensile stresses are necessary for brittle fracture to occur. These stresses are determined by a stress analysis of the particular component.
2.
3.
These three factors control the susceptibility of an engineering component to fracture. All other factors such as temperature, loading rate, stress concentrations, residual stresses, etc., influence these three primary factors. Although designers have known about them for many years and have reduced the probability of elements fracturing by controlling the above factors in an empirical manner, they did not have specific design guidelines to evaluate the relative performance and the economic tradeoffs among design, fabrication and materials until the advent of LINEAR ELASTIC FRACTURE MECHANICS (LEFM). LEFM presents these design guidelines and is based on an analytic procedure that relates the stress field magnitude and distribution in the neighborhood of a crack to: i) ii) iii) the nominal stress applied to the structure, the size, shape and orientation of the crack, and to the material properties.
In this latter context, one of the underlying principles of LEFM is that unstable fracture occurs when the stress intensity factor at the crack tip reaches a critical value; the fracture toughness of the material in question. Knowing the critical stress intensity factor, a designer can either determine flaw sizes that possibly may be tolerated at a given stress level or find the design
Part I-4
stress level that may safely be used for a crack that may or may not be present in a component. Griffith [2, 3], who is considered as the founder of fracture mechanics theory, made the first attempt at solving fracture problems in 1921. His approach, which is now known as the Griffith energy criterion, involves a plate of unit thickness with a central, transverse crack of length 2a, whose ends are fixed while inducing a remote or nominal stress of . The load/displacement diagram shown in figure 2.1 illustrates what occurs when the crack extends by an amount da.
Figure 2.1:
Load-displacement diagram for a cracked plate with fixed end conditions
The elastic energy contained in the original plate is represented by the area OAC. When the crack extends by da, the stiffness of the plate decreases (or the compliance increases) and the elastic energy content drops to a value represented by the area ODC. Hence, the crack extension results in an elastic energy release equal in magnitude to the area OAD. Alternatively, Griffith stated that crack propagation will occur if the energy released upon crack growth is sufficient to provide all the energy that is required for the growth of the crack. The condition for crack growth is, then, whenever:
G=
dU dW = =R , da da
(2.1)
where U is the elastic energy contained in the plate and W is the energy required for crack growth (fracture energy). According to work performed in 1913 by Inglis [4], Griffith calculated dU / da per unit thickness and per 1/2 the total crack length, 2a, as: dU 2 a = =G , da E (2.2)
where E is Young's modulus. In eqn(2.2), dU / da is replaced by G, the energy release rate; rate, in this context, referring to: per unit of crack extension. G is also called the crack driving
Part I-5
force; its dimensions being energy per unit plate thickness, per unit of crack extension.
On the other hand, the energy spent in crack propagation is denoted by R = dW / da , the crack resistance of the material. As a first approximation, it is assumed that the energy required to produce a crack is the same for each increment da which implies that R is a material constant. Hence, the crack growth energy condition, eqn(2.1), states that G must be at least equal to R before crack propagation can occur. If R is a constant, G must exceed a critical value, GIc , which is related to the fracture toughness of the material. Griffith derived his equation for glass, which is a very brittle material. Therefore, he assumed that R was related to surface energies of the two exposed crack faces only. In ductile materials, however, plastic deformation occurs at the crack tip and most of the energy introduced into the ductile material is used for the formation of this plastic zone. The energy associated with the production of the plastic zone may then be considered as the energy needed for crack propagation. This implies that, in the case of metals where R is mainly plastic energy, the surface energy remains relatively small so that it may be neglected [5, 6]. As a result, LEFM is invalidated by the elastic-plastic behavior and the formation of large plastic zones in engineering materials. The extensions to LEFM, which are discussed in Chapter 4, are: 1. the R-curve analysis, which is a procedure used to characterize fracture during incremental crack slow-stable extension; 2. the Crack Tip Opening Displacement, CTOD, measures the pre-fracture deformation at the tip of a sharp crack under conditions of inelastic behavior; and 3. the J-integral analysis, which is a path independent integral measuring the elastic-plastic stress-strain field ahead of a crack. All three techniques are used for specifying material toughness in terms of allowable stress or defect size since they are all extensions of LEFM. Finally, fracture mechanics may be used to determine useful information concerning crack propagation. For example, the critical stress intensity factor K Ic represents the terminal conditions on the propagation life of a component, while the stress intensity factor K I may serve as a single term parameter characterizing the rate of crack propagation. Hence, the total useful life of an element is determined by the time necessary to initiate a crack (often an impossible task to assess) and to propagate the crack from its sub-critical dimensions to its critical size. This approach will be fully discussed in subsequent chapters.
2.2
Linear Elastic Fracture Mechanics:
The fundamental principle of fracture mechanics is that the stress field surround a crack in a mechanical element may be characterized by the stress intensity factor, K I , which is related to both the stress and the size of the flaw. The analytic development of the stress intensity factor is described in Subsection 2.2.2, prior to a presentation of the stress intensity factors for a number
Part I-6
of common specimen and crack geometries. With a knowledge of these relationships, fracture mechanics may be better understood and design engineers better equipped to anticipate and thus prevent component deficiencies. In Section 2.4, the relationship between stress intensity factors and material fracture toughness is briefly examined prior to an introduction to fracture mechanic design concepts.
2.2.1 The three modes of fracture
Before analyzing to a variety of crack problems, it is importaand a constant at the origin (crack tip). This is due to the singularity given by
the term 1/ z . The parameter KI is the stress intensity factor (SIF) for opening mode I. It is a second example of an energy release rate and crack driving force. For the problem, the stresses in the vicinity of the crack tip may be deduced from eqn(2.8) and eqn(2.11), which, when expressed in polar coordinates with z = ei , result in:
Part I-10
x y xy
Or, in general terms:
= = =
KI 3 cos 1 - sin sin + 2 r 2 2 2 KI 3 cos 1 + sin sin + 2 r 2 2 2 KI 3 sin cos cos 2 r 2 2 2 +
, , ,
(2.13)
ij =
KI fij ( ) . 2 r
(2.14)
Sih [10] and Eftis and Liebowitz [11] showed for the uniaxial tension case, that the addition of a stress in the y direction, namely - , had no effect on the singular terms in the x solution and a for the purely that the Westegaard's solution is only valid for small values of r , i.e., r elastic response of the material. Since the stresses are elastic, they are proportional to the external load. Hence, K I must be proportional to and the square root of a length in order to obtain the proper dimension for the stresses, i.e., K I must take the form:
KI = Y
a ,
(2.15)
where Y depends on the crack length and the component geometry. From Hooke's law, for any elastic material, the displacement field becomes:
u = v = 2 (1- ) r k -1 KI + sin 2 + cos E 2 2 2 2 2 (1- ) E KI r k +1 + cos 2 + sin 2 2 2 2 ,
(2.16)
,
where:
u and v are displacements in the x and y directions respectively, is Poisson's ratio, E is Young's modulus, k = (3 - 4 ) for a PSN case and k = (3 - ) / (1 + ) in the case where PSS conditions prevail.
Similar equations could be derived for K II and K III (see Broek [12].)
2.3
Stress intensity factors:
While the general case studied above is of considerable importance, cracks in plates and other structural components of finite size are of much greater practical interest to engineers. The
Part I-11
Relationships between the stress intensity factor, KI, and various body configurations, crack sizes, orientations and shapes, and loading conditions have been investigated and published. Those of Paris and Sih [7], Tada, et al. [13], Sih [14, 15] and Rooke and Cartwright [16] all deserve careful review. Furthermore, a large number of computational engineering design aids incorporating various aspects of these LEFM concepts have been and are being developed throughout the world, e.g., NASA-FLAGRO [17].
2.3.1 Through thickness cracks
As previously discussed, the stress intensity factor for an infinite plate subjected to a uniform stress, , that contains a through thickness crack of length, 2a , is given by eqn(2.12). The SIF for a plate of finite width W under the same conditions is:
a W K I = a tan W a
1/ 2
, a <
W . 4
(2.17)
2.3.2
Double edge cracks
The SFI for symmetric double edge cracked plates subjected to a uniform tensile stress is:
a 2 a W K I = a tan + 0.1 sin W W a
1/ 2
.
(2.18)
2.3.3
The single edge notch
The KI for a semi-infinite single edge notched specimen is:
K I = 1.12 a (MPa m or ksi in ) ,
(2.19)
which is also the form that the double edge notch reduces to as W becomes large. The correction factors necessary for accounting for the finite width and non-symmetry of loading in the case of a single edge crack may be summarized by setting: 2a a Y W where Y (2a / W ) is given by table 2.1.
KI =
(MPa m or ksi in ) ,
(2.20)
2a/W Y(2a/W) Table 2.1
0.1 1.15
0.2 1.2
0.3 1.29
0.4 1.37
0.5 1.51
0.6 1.68
0.7 1.89
0.8 2.14
0.9 2.46
1.0 2.86
The Geometrical Factor Y for Single Edge Notches
Part I-12
2.3.4
A single crack in a bent beam
The stress intensity factor for a beam height W with a crack of length a being bent by a moment M , is given by:
KI =
6M 2 a Y , 3/ 2 (W - a) W
(2.21)
where the values of Y (2a / W ) are presented in table 2.2.
a/W Y(a/W) Table 2.2
0.05 0.36
0.1 0.49
0.2 0.6
0.3 0.66
0.4 0.69
0.5 0.72
0.6 0.73
The Geometrical Factor Y for a Bent Beam
Finally, other expressions for various crack geometries are introduced during the subsequent developments of further LEFM concepts.
2.4
The Stress Intensity Factor, Fracture Toughness and Design:
Briefly, as this topic is covered in depth in later sections, one of the fundamental principles of fracture mechanics is that unstable fracture occurs when the stress intensity factor at the crack tip reaches a critical value. The critical stress intensity or fracture toughness describes the ability of a particular material to carry loads or to deform plastically in the presence of a notch or crack. For mode I deformation and for small crack tip plastic deformation (PSN conditions), the critical stress intensity factor for fracture instability is designated by K Ic . Thus K Ic is also a measure of the crack resistance of a material under PSN conditions and has the units of MPam (ksiin). The distinction between the analytic crack driving force K I and the material crack resistance property K Ic is analogous to the distinction between stress, , and yield strength, ys . With the knowledge of fracture toughness values for a given material of a particular thickness and at a specific temperature and loading rate, the design engineer can design for the tolerable flaw size for a given stress level. Conversely, the safe stress level can be determined with an existing crack that may be present in a structure. As discussed in the Introduction, the aspects of design that are of interest are; the selection of materials, the choice of appropriate stress design levels, and the determination of the tolerable crack size for quality control or inspection. In this sense, fracture mechanics design assumes that the engineer has already established the following information; the type and overall dimensions of the mechanical components in the assemblage, the required performance criteria and the stress and stress ranges being experienced by these components. With this basic information, the
Part I-13
engineer can incorporate K Ic values at the appropriate service temperature and loading rate into the successful design of a fracture resistant structure. Accordingly, after the engineer has selected the K Ic and the yield stress ys of the material being considered, the designer should then select the most probable type of flaw prior to consulting a listing of the stress intensity factor equations. In this manner the designer can determine the stress-flaw-size relation at various possible stress levels. As an example, consider a through thickness crack with a SIF given by K I = a . For a material whose K Ic = 97.4 MPam, the critical flaw size 2acr for a maximum tensile stress of 100 MPa is 0.604 m. Suppose, on the other hand, that both the K Ic = 97.4 MPam and the maximum allowable flaw size of 2a = 0.6 m are specified, then for this chosen crack configuration a maximum permissible nominal stress of just over 100 MPa is easily deduced. Finally, suppose that 2a = 0.6 m and = 100 MPa are specified, then the question arises as to which materials have a fracture toughness greater than 97.4 MPam to operate successfully under these conditions. From a table similar to that given in Chapter 10 of Hertzberg [18] or from the literature surveyed by Hudson and Seward [19], the materials which meet this requirement may be ascertained. This problem illustrates one of the most dramatic advances in safe design due solely to the incorporation of fracture mechanic concepts into normal engineering practice.
References:
[2]
[3]
[4]
[5]
[6]
Griffith, A.A., The Phenomena of Rupture and Flow in Solids, Phil. Trans. Royal Society of London, vol. A221, pp. 163-198 (1921). Reproduced in: Selected Papers on Foundations of Linear Elastic Fracture Mechanics, R.J. Sanford, ed., SEM classic papers; v. CP 1: SPIC milestone series; v. MS 137, (1997). Griffith, A.A., The Theory of Rupture, Proc. First Int. Congress of Applied Mechanics, Biezeno and Burgers, eds., Waltman Press, pp. 55-63, (1925). Reproduced in: Selected Papers on Foun