Technology & Engineering
Linear Elastic Fracture Mechanics
Linear Elastic Fracture Mechanics (LEFM) is a theory used to analyze the behavior of cracks in materials under stress. It focuses on the relationship between the applied load, the material's properties, and the size and shape of the crack. LEFM is based on the assumption that the material behaves elastically, allowing for the prediction of crack growth and failure.
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12 Key excerpts on "Linear Elastic Fracture Mechanics"
- Ashok Saxena(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
1Introduction and Review of Linear Elastic Fracture Mechanics
1.1 Why Nonlinear Fracture Mechanics
Fracture of load-bearing components is always an important consideration for engineers who design, build, operate, and maintain bridges, highways, automobiles, trains, airplanes, power plants, chemical process equipment, and numerous other large pieces of machinery. Everyone understands the catastrophic consequences of structural failure and that sometimes it happens because the factors involved in predicting it are complex and are not well understood at the time the component is designed. Since the late 1950s, the developments in fracture mechanics have contributed immensely to our understanding of fractures that emanate from cracks or crack-like defects which could potentially escape detection.A large fraction of failures in structural components occurs due to preexisting defects or defects that initiate rapidly from clusters of nonmetallic inclusions or from other imperfections such as casting, forging, and welding defects. On the other hand, several defects also remain dormant in the components and pose no threat of fracture. In fracture mechanics, we are interested in both, the defects that can ultimately cause fracture and those that are benign.Several fractures can be analyzed using the principles of Linear Elastic Fracture Mechanics (LEFM). These are typically brittle fractures that are accompanied by limited or small amounts of plastic deformation. When fractures are accompanied by significant amounts of plastic deformation or by creep deformation, LEFM is inadequate in those circumstances. Nonlinear fracture mechanics has been developed as a viable engineering tool since the 1970s to address specifically those classes of problems. Some examples that depend on the use of nonlinear fracture mechanics for explaining the fracture process are considered next to illustrate its need.
eBook - ePubExtended Finite Element Method
Theory and Applications
- Amir R. Khoei(Author)
- 2014(Publication Date)
- Wiley(Publisher)
et al. (2001), in which a discontinuous function based on the Heaviside step function was employed in modeling two-dimensional (2D) linear elastic crack tip displacement fields. A methodology for modeling crack discontinuities was presented by Sukumar and Prévost (2003) with implementation of X-FEM in isotropic and bimaterial media. The technique was implemented into three-dimensional fatigue crack propagation simulation of multiple coplanar cracks by Sukumar, Chopp, and Moran (2003a) and Chopp and Sukumar (2003). They combined the X-FEM to the fast marching method using the PUM to model the entire crack geometry, including one or more cracks, by a single signed distance (level set) function. An enriched FEM with arbitrary discontinuities in space–time was presented by Chessa and Belytschko (2004). They modeled discontinuities by the X-FEM with a local PU enrichment to introduce discontinuities along a moving hyper-surface. In this chapter, the X-FEM technique is presented to model the crack propagation in the Linear Elastic Fracture Mechanics (LEFM). The basis of LEFM is first introduced by defining the stress and displacement distributions around the crack tip and the SIFs for different loading modes. The governing equation of a solid fractured body is then derived in the framework of X-FEM. The procedure of crack growth simulation using the X-FEM technique is illustrated, and the method is employed to model the curved crack with higher order elements, and the crack growth simulation for bimaterial interface cracks in complex composite components.7.2 The Basis of LEFM
Fracture mechanics originally concentrated on the elastic material behavior where Hook’s law was obeyed. A number of experiments and theories were presented by Orowan (1948), Irwin (1957), and Barenblatt (1962) to illustrate the behavior of cracked domain in linear elastic materials. Irwin (1960) and Shih and Hutchinson (1976) extended the concept of LEFM to nonlinear behavior of materials such as plastic solids. A general description of the fractured materials is presented in Figure 7.1 a, where a tension load is applied to an infinite plate that contains a crack with the length of 2a at the center of the plate. The fractured domain causes a singularity in the stress field at the crack tip region for the elastic material. This singularity is depicted in Figure 7.1 b where the stress reaches an infinite value. However, in plastic materials, a plastic zone occurs at the crack tip that causes the stress to reach a finite value equal to yield stress of the material.(a) A crack of length 2a in an infinite plate subjected to a uniform tensile stress σ; (b) the distribution of normal stress ahead of a crackFigure 7.1There are a number of microscopic and macroscopic studies performed by researchers to investigate the fracture behavior of the material. From the microscopic point of view, the crack can be propagated if the potential energy of atoms exceeds the bound energy existed between two adjacent atoms. Consider that x0
eBook - PDF- Brian Cotterell(Author)
- 2010(Publication Date)
- ICP(Publisher)
232 Chapter 9 Fundamentals of Fracture and Metal Fracture from 1950 to the Present The second half of the twentieth-century saw fracture mechanics emerge as an engineering discipline and come to maturity. The development of Linear Elastic Fracture Mechanics (LEFM) in the 1950s was the first practical move away from a maximum stress fracture criterion in engineering. At the beginning of the period the problem of the brittle fracture of steel had not been solved. The solution lay mainly in determining the effective transition temperature and in metallurgical improvements to produce steels of lower transition temperatures. However, for the assessment of the significance of defects needed elasto-plastic fracture mechanics (EPFM). The growth of computers during this period enabled more realistic models of deformation and fracture to be implemented. At the end of the period computing power had grown enormously, enabling fracture at the atomic scale to be simulated using molecular dynamics which will be discussed in Chapter 12. 9.1 Linear Elastic Fracture Mechanics (LEFM) George Rankin Irwin (1907–1998) was working on armour materials in the in the Ballistics Branch of the Mechanics Division of the Naval Research Laboratory (NRL) during War World II. Evidence of a fracture size effect led him to obtain a research contract for the study of fracture at the University of South Carolina from 1941 to 1948. 1 Irwin was joined at NRL in 1948 by Joe Kies (1906–1975). The initial fracture experiments of Irwin and Kies were on polymethyl methacrylate (PMMA) and cellulose acetate sheets loaded to different tensile stresses and fractured by firing a bullet into them, where the stress necessary to cause complete fracture was size dependent. 2 In 1952 Irwin and Kies used Fundamentals of Fracture and Metal Fracture from 1950 to the Present 233 Griffith’s theory to estimate the brittle fracture strength of a steel plate containing a large crack.
eBook - PDFFracture Mechanics
Fundamentals and Applications
- Surjya Kumar Maiti(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
2 Linear Elastic Fracture Mechanics 2.1 Introduction The foundation for the understanding of brittle fracture originating from a crack in a component was laid by Griffith (1921), who considered the phenomenon to occur within the framework of its global energy balance. He gives the condition for unstable crack extension in terms of a critical strain energy release rate (SERR) per unit crack extension. The next phase of development, which is due to Irwin (1957a and b), is based on the crack-tip local stress–strain field and its characterization in terms of stress intensity factor (SIF). The condition of fracture is given in terms of the SIF reaching a critical value, and the parameter is shown to be related to the critical energy release rate given by Griffith. Later, the scope of the SIF approach was amended to take care of small-scale plastic deformation ahead of the crack-tip. Most of the present applications of the principles of Linear Elastic Fracture Mechanics (LEFM) for design or safety analysis have been based on this SIF. This chapter presents the gradual developments that have taken place to advance the understanding of fracture of brittle materials and other materials that give rise to small-scale plastic deformation before the onset of crack extension. Examples are presented to illustrate the applications of LEFM to design. 2.2 Calculation of Theoretical Strength A fracture occurs at the atomic level when the bonds between atoms are broken across a fracture plane, giving rise to new surfaces. This can occur by breaking the bonds perpendicular to the fracture plane, a process called cleavage, or by shearing bonds along a fracture plane, a process called shear. The theoretical tensile strength of a material will therefore be associated with the cleavage phenomenon (Tetelman and McEvily 1967; Knott 1973).
eBook - ePub- Ashok Saxena(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
3 Theoretical Basis for Linear Elastic Fracture MechanicsDOI: 10.1201/9781003292296-3Before discussing the mechanics of how cracks degrade the ability of structures to withstand loads, it is important to understand the deformation behavior of structural materials. Except in the case of very brittle materials, fracture is preceded by inelastic deformation that leads to concentration of strains in the region where fracture initiates. Microcracks first form in microstructurally weak regions or in regions that have the highest stresses and then fracture spreads by coalescence of microcracks and leads to separation. An example of this phenomenon is the onset of necking during a tensile test in metallic materials where strain concentration occurs in vicinity of nonmetallic inclusions or a cluster of inclusions that are present as impurities at some location along the length of the specimen. Fracture occurs at that location because it is microstructurally weaker than the rest of the material. In the case where macro-cracks are already present, strain concentration occurs at the tip of the crack and fracture emanates from that pre-existing defect.As discussed in Chapter 2 , if cracks exist in a load-bearing component, its resistance to fracture depends on (a) the material's ability to absorb energy through irreversible processes such as plastic deformation in the crack tip region, (b) meandering of the crack, or (c) by fiber pull out in composite materials etc. If these energy absorbing processes are absent, such as during fracture in glasses, all available energy is directed toward forming new surfaces leading to a low energy, brittle fracture.3.1 Engineering Materials and Defects
Engineering materials for structural applications are classified as metals and alloys, ceramics, polymeric materials, and composites. Composites are combinations of two or more types of materials mixed in pre-determined proportions to create tailored properties. Examples of composites include brittle concrete that is reinforced with steel rods/wires to increase the toughness. Because the fibers are aligned in one direction, the properties of composites vary with direction and are therefore anisotropic.
eBook - PDFDislocation Modelling of Physical Systems
Proceedings of the International Conference, Gainesville, Florida, USA, June 22-27, 1980
- M.F. Ashby, R. Bullough, C.S. Hartley, M.F. Ashby, R. Bullough, C.S. Hartley(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
In this paper we review the above mentioned applications of dislocation theory to fracture mechanics and the micromechanisms of cracking. Several cases studied by the present authors will be presented in more d e t a i l . First the use of dislocations in stress analysis is shown with the emphasis on methods of determining stress intensity factors for various configurations of cracks. The application of the dislocation models in elastic-plastic fracture mechanics (EPFM) is then presented. The physical meaning of both the c r i t i c a l crack tip opening displace-ment (CTOD) and J-integral (J ), and the contribution of dislocation models towards their interpretation will be discussed. A specific example is an interpretation of the phenomenon of warm pre-stressing. Applications of the dislocation models to fatigue and creep fracture will also be briefly outlined. In the part of the paper which deals with micromechanisms the dislo-cation models of crack nucleation, in particular the pile-up models, will be discussed. A model of microcrack propagation which is concomitant with dislocation emission from i t s tip will be presented and it will be shown how a relationship between the plastic work, , and the ideal work to fracture, , can be established. p 2. Linear Elastic Fracture Mechanics (LEFM) Fracture mechanics is concerned with macroscopically sized crack-like defects and the effects these have on the load bearing capacity of structures. In LEFM the fracture parameter of interest is the stress intesity factor, K, which determines the magnitude of the stress singularity at the tip of a loaded sharp crack in an elastic material (19) . This singularity goes ar r ^ where r is the distance from the t i p . I t is postulated that fracture ensues when the value of K reaches a c r i t i c a l value, K , called the fracture toughness. For engineer-ing purposes i t is regarded as a fundamental material parameter and i t has been measured for many structural materials.
eBook - PDF- François Henri Cornet(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
7 Elements of linear fracture mechanics As pointed out in chapter 2, an elastic material may be modeled by a spring, and it is quite intuitive that a spring cannot be extended indefinitely: some maximum force, and correl- atively some maximum spring extension, must exist that defines an elastic limit. When the force, or the spring extension, becomes larger than this maximum value the spring either breaks or extends further but with a nonreversible component, so that some residual deformation exists when the loading is relaxed. In a crude way, this defines two possible post-elastic behaviors: either the spring breaks, in which case the material used to make the spring is said to be brittle, or the spring deforms in a nonelastic manner, so that a permanent deformation is observed when the load is relaxed, in which case the material is said to be ductile. While a spring may help illustrate elastic phenomena in one dimension, we saw in chapters 3–5 that for three-dimensional problems of elasticity we must introduce three- dimensional stresses and strains. These quantities are second-order tensors, and elasticity implies a linear relationship between the stress and small-strain components. By com- parison with the spring’s behavior, we may anticipate that when the principal stress (or principal strain) magnitudes, or some function of them, become larger than some criti- cal values, either the material breaks into pieces or it deforms according to a nonreversible process. As for the case of a spring, we call the post-elastic behavior brittle when the mater- ial fails because of the extension of fractures. We call the post-elastic behavior ductile when the material deforms continuously with occurrence of a nonreversible component. The object of the present chapter is to discuss brittle failure; ductility is discussed in chapters 8 and 9.
eBook - PDFConcrete Fracture
A Multiscale Approach
- Jan G.M. van Mier(Author)
- 2012(Publication Date)
- CRC Press(Publisher)
11 2 Classical Fracture Mechanics Approaches 2.1 Stress Concentrations Linear Elastic Fracture Mechanics (LEFM) dates back to 1920–1921 when Griffith proposed his energy approach for the brittle fracture of glass. Any material, including a very smooth homogeneous material such as glass, con-tains imperfections. These imperfections are the source of stress concentra-tions, which may lead to failure of the material well below its theoretical strength. Based on a sinusoidal approximation of the atomic bond potential, σ = σ ⋅ π - r x r sin ( ) max 0 (2.1) where σ max is the peak stress in the atomic bond stress-spacing diagram and r is the increase of the original lattice spacing r 0 of the atoms, it is possible to calculate the theoretical strength of crystalline solids, which leads to (Kelly and MacMillan 1986): σ = π E max (2.2) The Young’s modulus E relates stress with strain following σ = E. ε = E . x/r 0 . For example, for alkali-resistant glass fiber, with a Young’s modulus E = 70 GPa (Gupta 2002), the predicted theoretical strength according to Equation (2.2) would be σ max = 23 GPa, whereas in reality about 70 MPa is measured on a sin-gle fiber. The strength of the fiber is very much affected by its diameter. Surface defects result in premature failure at stress levels quite below the maximum attainable value. In the glass rod of Figure 2.1a, the crack seems to have nucle-ated from the small white line at the bottom of the mirror area. The rod, which was a simple off-the-shelf product, was highly polluted on the outside as can be seen in Figure 2.1b. The crack nucleated from an imperfection, and appeared to have started symmetrically in the beginning. After the rather smooth mirror-zone, surface roughness gradually increased into the mist- and hackle-zones. 12 Concrete Fracture: A Multiscale Approach Stress concentrations in real materials are, for instance, caused by pores, inclusions, interfaces between distinct material phases, and the like.
eBook - PDF- Richard A. Schultz(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
For example, driving stress leading to fracture-surface displacement within a linearly elastic material produces singular (infinite) stress mag- nitudes at the fracture tip, and steep near-tip displacement gradients, because the material is defined implicitly as indefinitely strong (Lawn, 1993, pp. 52–53). The small area over which the amplified near-tip stresses are pre- dicted to exceed the rock’s yield strength defines the size of the Irwin plastic Terms for elastic–plastic (non-LEFM) process zone models: ♦ End-zone model ♦ Cohesive end zone ♦ Strip yield model ♦ Slip-weakening model ♦ Fictitious crack model ♦ Tension-softening model ♦ Dugdale–Barenblatt model ♦ Post-yield fracture Paris Law for fatigue fracture propagation 9.3 Better Terminations: Yield Strength and the End-Zone Concept 415 zone (Broek, 1986, pp. 13–14, 99–102; Anderson, 1995, pp. 72–75); within this small area the predicted level of elastic stress would become so large that its magnitude would not have any physical meaning. By setting a separate criterion for yield strength, the predicted stresses within this area are reduced and limited to the yield strength value appropriate to the near-tip stress state (e.g., tension, shear, or compaction). Using an elastic material rheology sets up physically unrealistic conditions at the fracture tip that drive the near-tip stresses to infinitely large values and the near-tip displacement profile to sim- ilarly elliptical or parabolic shapes. Here are two geologic examples that demonstrate (or at least imply) that LEFM analyses might not be correct and must therefore be supplemented by a more comprehensive approach. ● Many geologic discontinuities do not reflect conditions of small-scale yielding and LEFM (Cowie and Scholz, 1992b; Rubin, 1993a; Scholz, 2002, p. 116; Yao, 2012) due to a combination of factors such as finite (small) rock strength at the tipline and relatively large ratios of driving stress to yield strength (e.g., Rubin, 1993a).
- Josip Brnic(Author)
- 2018(Publication Date)
- Wiley(Publisher)
. The existence of this zone is important for the application of the LEFM.Crack tip plasticity (elastic and inelastic stress distribution around the crack tip).Figure 10.15The normal stress, σy, on the crack plane is given by Equation (10.6) . The first assessment of the size of the plastic zone at the crack tip was made by Irwin (Irwin’s approach) in his first approach on the basis that the boundary between the elastic and the plastic zone occurs when stress σy, given by Equation (10.6) satisfies a yield criterion. As mentioned above and shown in Figure 10.11 , the stress σyon the crack plane ( ) at the crack tip is [1, 8–9, 11–18]:(10.14)If we consider that, in a case involving plane stress conditions, yielding occurs when stress σyreaches yield stress level σYof uniaxial stress, that is, then, from Equation (10.14) , the plastic zone ( ) may be estimated (see Figure 10.15 ) as:(10.15)In this case, the stress distribution in the area can be represented by a horizontal line ( ), as illustrated in Figure 10.15 .This analysis does not seem to be entirely correct. It is based on an elastic crack tip. After yielding occurs, in order to satisfy equilibrium, the stress must be redistributed. However, the stress distribution in the area above the estimated plastic region (the area above the horizontal line σYand between the vertical axis, σy, and the curve representing the elastic stress distribution, σy
eBook - ePub- Soheil Mohammadi(Author)
- 2012(Publication Date)
- Wiley(Publisher)
et al. (2004). More recent studies include the study of early cracking of Girth-Gear of an industrial Ball-Mill (up to 12 meters in diameter and over 90 tonnes in weight, with a manufacturing cost of about half a million dollars), which was expected to have a fatigue life of 20 years and more, but cracks initiated and propagated within the first two years of operation (Mirzaei, Razmjoo and Pourkamali, 2001). More or less similar trends were reported in the failure of exploded gas cylinders containing hydrogen (Mirzaei, 2008).This chapter only briefly reviews the basic theoretical concepts of fracture mechanics for linear and nonlinear analyses of isotropic materials. Concepts that are related to specific subjects such as orthotropic materials, dynamic materials and the effect of inhomogeneity will be dealt with in the corresponding chapters. Little or no originality is claimed for this chapter, nor is there any claim of completeness. The sections are selected, organized and presented with regard to the needs of subsequent sections and chapters as a basis of LEFM for isotropic materials, and a precursor to the main subject of the book, XFEM fracture analysis of composites.2.2 Basics of Elasticity 2.2.1 Stress–Strain RelationsBeginning with the definition of stress and strain tensors, and , respectively,(2.1)(2.2)or in a vector (array) notation,(2.3)(2.4)where is the engineering shear strain component. The generalized Hooke's law for linear elastic materials can be defined as:(2.5)where and are the second order stress and strain tensors, respectively, and is the fourth order elasticity modulus tensor with Cartesian components . For general three-dimensional problems, the fourth order elasticity tensor can be represented by a two-dimensional matrix with components
eBook - ePubExtended Finite Element Method
Tsinghua University Press Computational Mechanics Series
- Zhuo Zhuang, Zhanli Liu, Binbin Cheng, Jianhui Liao(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
Chapter 2Fundamental Linear Elastic Fracture Mechanics
Abstract
Fundamental Linear Elastic Fracture Mechanics are described in this chapter. The concept of energy release and balance is introduced during crack growth. Stress fields are provided for mode I, II, and III cracks at the crack tip location. The concepts of stress intensity factor and material fracture toughness are introduced. An analytical method is given to determine the stress intensity factor and energy release rate. Three kinds of complex fracture theories are discussed: the maximum circumference tension stress intensity factor theory, the minimum strain energy density stress intensity factor theory and the maximum energy release rate theory. The crack orientation angle, fracture criterion, and stress intensity factor of complex mode cracks are found using the analytical method. The interaction integral method is described to solve the stress intensity factor under quasi-steady-state conditions.Keywords
Stress intensity factor ; energy release rate ; material fracture toughness ; complex crack ; interaction integralChapter Outline2.1 Introduction 132.2 Two-Dimensional Linear Elastic Fracture Mechanics 152.3 Material Fracture Toughness 192.4 Fracture Criterion of Linear Elastic Material 202.5 Complex Fracture Criterion 222.5.1 Maximum Circumference Tension Stress Intensity Factor Theory 222.5.2 Minimum Strain Energy Density Stress Intensity Factor Theory 242.5.3 Maximum Energy Release Rate Theory 272.6 Interaction Integral 292.7 Summary 312.1 .IntroductionOccurrences of fracture in solids arise predominantly from discontinuous surface displacement in the materials. The fracture problem is usually divided into three fundamental modes. The first of these is mode I opening cracking. The displacements are opposite and orientations are perpendicular to each other between two crack surfaces, which is a common form of crack in engineering practice, as shown in Figure 2.1 (a). The second category is mode II sliding cracking, also called in-plane shear cracking. The displacements are also opposite between two crack surfaces; however, one displacement moves along and the other deviates from the crack growth direction, as shown in Figure 2.1 (b). The third category is mode III antiplane shear cracking. The antiplane displacements occur between two crack surfaces, as shown in Figure 2.1
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