Principle of fracture mechanics pdf

In anisotropic materials Gc may depend on the direction of crack growth. Consider fracture in wood for example, where cracks will prefer to grow along the grain rather than across it, demonstrating that the toughness along the grain is much less than across the grain. For now only materials with isotropic fracture and elastic properties will be considered. Such materials have no preferred direction of crack growth.

Epoxy 0. A selection of typical fracture toughness values for nominally brittle materials is given in table 4. Environmental conditions play an important role in fracture and most other aspects of material behavior. For example the fracture toughness of metals and polymers is generally reduced as the temperature is reduced. In some cases this effect can be drastic. For example, in figure 4. Since energy is the square of KI this implies that the energy needed for fracture drops by almost a factor of 10 and the material could be considered as brittle at high temperatures.

Avoiding brittle fracture at low temperatures is of key importance in modern structural design, be it for railcars traversing North Dakota in winter or ships crossing the North Atlantic. The simple fracture criteria above will let you determine if a crack will grow or not, but it does not tell us anything about how fast, how far, or in what direction the crack will grow. These topics are addressed in subsequent chapters and sections. If as the crack grows, G increases and becomes ever higher than GC the crack will become unstable, rapidly growing until the body is completely fractured.

The prediction, or better yet, prevention of unstable crack growth except when it is desirable as in some manufacturing operations is of primary concern in mechanical and structural design and hence warrants careful attention. The stability of crack growth depends on both the characteristics of the material and on the geometry and nature of the loading.

For example, from the analysis of the DCB specimen it is easily shown that if the applied load is fixed then G grows as a2 eq 3. Thus it is far more likely for fracture to be stable under conditions of fixed displacement loading than under conditions of fixed force. Adapted from [32]. In terms of generalized forces and displacements such a general loading can be represented as a body loaded by a spring with compliance CM , to which a fixed displacement, qT , is applied, as sketched in figure 4.

The energy release rate is unchanged from the previous results, eq 3. How so? The physical sources of an increasing GR curve are numerous; two examples are given here: 1 For elastic-plastic materials, the level of crack tip strain for the same applied stress intensity factor, is less for propagating cracks that for stationary cracks. If a specific strain ahead of the crack is needed to grow the crack, this implies that the applied stress intensity must be increased for the growing crack.

To overcome these closing forces the externally applied stress intensity factor must be increased. For perfectly brittle materials, GR is a constant. For other materials, GR may rise and then reach a steady state value or it may continue to increase. In ductile metals it is generally found that the resistance curve increases at a faster rate for tests done on thin sheets than for tests done on thick sections.

In small scale yielding, KI and G are related by eq 3. Regardless of the source of the resistance curve, a criterion for crack stability can be stated. Generally GR increases with crack extension. It may or may not reach a steady state. For elastic-plastic materials, GR increases much more for plane stress than for plane strain problems, due to the decreased constraint and hence greater ease of plastic flow in plane stress.

In the case of fixed load, G is an increasing function of a, e. For fixed displacement G decreases with a, e. In all cases G increases as Q2 or q 2. The available energy release rates G Q, a for the fixed load case and G q, a for the fixed displacement case are superimposed with the GR resistance curves in figures 4. In the case of fixed displacement loading, the crack growth is always stable, i. To continue to grow the crack the applied displacement must be increased to satisfy equation 4.

In many real applications the loading is somewhere between fixed load and fixed displacement, thus G may increase or decrease with respect to a and the stability will depend on the relative stiffness of the loading and on the slope of the R curve. Loading with fixed force will be unstable after a small amount of crack growth even for a material with a rising R-curve.

Crack growth is always stable under fixed displacement loading. Most applications would involve loading through a compliant system, thus the stability will be between the two extremes above. The prediction not only of when and how far a crack will grow, but of its path is important in the analysis of potential failures.

For example if a gear tooth was cracked would the crack propagate across the tooth, breaking the tooth? Or would the crack propagate into the hub of the gear, causing the entire gear to fly apart? Would a crack in the skin of a pressurized aircraft fuselage grow straight and unzip the entire fuselage or would it curve and be contained with one bay a section of the fuselage bounded by the circumferential frames and the longitudinal stringers?

Such problems involve complex, 3D geometries. The cracks in such cases generally have a mix of Mode-I,-II,-III loadings that varies along the crack front and that may vary as well during a cycle of loading. As a start to understanding such problems we will start with 2D mixed-mode loadings.

If as sketched in figure 4. Theories for mixed-mode fracture include: 1 maximum circumferential stress [33], 2 mini- mum strain energy density [34], 3 maximum energy release rate [35] and 4 local symmetry [36]. Only the theory of maximum circumferential stress and of maximum energy release rate will be discussed here. The results are plotted in figure 4. Comparison of this theory with experimental results shows that the maximum hoop stress theory predicts the angle of crack growth well but somewhat underestimates the envelope of failure.

Nonetheless, at least for crack growth angle the maximum hoop stress theory is quite accurate and is easily implemented in fracture simulations. In elementary mechanics of materials one learns that brittle fracture will occur along the plane of maximum tensile stress. This criterion is in fact the same as the maximum hoop stress theory.

The answer is that the presence of the crack disturbs the stress field, hence changing the directions of maximum principal stress. It is interesting to note however, that the path of the crack will evolve to lie along the plane of maximum principal stress. To demonstrate this a numerical finite element calculation was performed of a crack in a plate under pure shear pure Mode-II loading. Using the maximum hoop stress criterion the tip of the crack is moved ahead in small increments.

The result, shown in figure 4. In principal this criterion can be applied in 3D and to crack surfaces of arbitrary shape. Initial crack straight line in figure is under pure Mode-II loading. Crack path evolves along planes of maximum tensile stress. The results show that the kink angle predicted by the energy release rate and hoop stress criteria are quite similar. However what would happen if, for example a crack were to grow in a stress field, with respect to the coordinate system in figure 4.

However, the maximum tensile stress away from the crack is not in the x1 direction, but in the x2 direction. So perhaps the crack will grow in the x2 direction so that the material fractures on the plane of maximum tensile stress.

The above brings up the question of crack-path stability. This problem can be analyzed by considering a semi-infinite crack in a 2D stress field. Cotterell and Rice [37] developed a first order method to calculate the stress intensity factors at the tip of a slightly curved or kinked crack and applied it to predict the stability of crack paths.

From eqs. The stability of a straight crack under Mode-I loading can be deduced from the above analysis. Suppose that the crack has a slight kink or that the loads are not perfectly aligned orthogonal to the crack.

Experimental results on crack paths under biaxial tension [38] are in agreement with this prediction. Note that compression parallel to the crack stabilizes the crack path.

This result is often exploited in experiments where one may wish to constrain the crack path. After [37]. Applying the criterion eq 4. Methods for computing T are reviewed in ref. Note, however that the presence of plastic deformation very close to the crack tip will change the actual stress fields rendering the above analysis an approximation for elastic-plastic materials.

Examples include layered materials such as mica or wood that have distinctly weaker bonding between the layers than along the in-plane directions and rolled metals that have slightly different toughnesses in different directions.

When the toughness is anisotropic the criteria for fracture and for crack path selection must be modified. This function could be smooth as in the case of rolled metals, or could be nearly constant with very low values only on specific planes as in the case of layered materials.

For example, layered materials loaded such that the principal stress is parallel to the layers may nonetheless fracture between the layers. The physical mechanisms for these stages will depend on the material and environmental conditions at hand. However in all cases, stage I will consist of the development of microstructural damage such a microcracks or slip bands.

These will grow and eventually coalesce to form a dominant crack. Such a crack could be on the order of 1 mm long at the smallest.

A great deal of the fatigue life of a component could be spent in stage I. In stage II the dominant crack grows stably under the application of repeated loads. Fatigue life of structures is determined using total life or damage tolerant approaches. The total life approach predicts the fatigue life of a component as a total of the initiation and propagation time until failure.

The damage tolerant design assumes that structures have imperfections and flaws from the beginning. Results of fatigue fracture tests are generally plotted on a log-log scale and will have a form similar to that sketched in figure 4. The data can be separated into three general regions. Below the threshold value at the left of region 1 there is no crack growth. Just above this threshold the crack grows very slowly. In region 3, the final stage of growth is marked by accelerating crack growth on the way to instability.

Actual data, figure 4. Adapted from [43]. Crack closure can arise from many sources including plasticity in the wake of the growing crack, roughness of the fracture surface, oxidation of the new fracture surfaces and other effects, see [3], [46] and [47]. To understand the mechanics of crack closure, assume for now that the minimum of the cyclic load is zero.

When of the load is removed from the component, the crack faces close and can go into compressive contact with each other, partially holding the crack open.

This maintains a non-zero KI at the crack tip, as shown in figure 4. As R increases to above approximately 0. Note as well that environmental conditions can play an important part in fatigue crack growth. For example fatigue tests on aluminum alloys show that the crack growth rate is faster in humid air than in inert environments.

Examples include cracking of aluminum alloys in the presence of salt water, steels in the presence of chlorides or hydrogen and glass in the presence of water.

In metals, stress corrosion cracks will typically grow between the grains, but may also grow across grains. Stress corrosion cracking is time dependent with the rate of growth depending on both the stress intensity factor and the corrosive environments. Adapted from [48]. The rate of crack growth in high strength AISI V steel in the presence of salt water, hydrogen and hydrogen sulfide is shown in figure 4.

Typically below a threshold level of KI , no cracking occurs, and as the stress intensity factor reaches KIC the crack grows rapidly. In the middle region the rate of growth depends on the availability of hydrogen. However, such plateau regions are not found in all materials. Stress corrosion cracking is a significant issue in structures of many types.

Much more information on this topic can be found in [48] and [49]. Compare the predicted kink angle to the results of the maximum hoop stress criterion, figure 4. What does the disagreement or agreement between these results tell you? Derive the equations 4. Furthermore the use of stress intensity factors and energy release rate as criteria for fracture have been introduced.

However for only a small number of cases have solutions for the stress intensity factors been given. Thus in this chapter analytical, look-up and computational methods for the determination of the stress intensity factors and energy release rate since there is a one-to-one correspondence between the two in linear elastic fracture will be described.

With such methods the solutions to a great number of fracture problems have been found. A brief outline of the method is given in section 2. Further examples a description of the method can be found in [17] and [50]. Analytical solutions are useful not only to calculate stress intensity factors for physical problems that can be approximated by these idealizations, but as building blocks for more complex solutions and as examples against which to test computational methods for calculating stress intensity factors.

Finite Crack in an Infinite Body The case of an anti-plane shear crack under uniform, remote stress is given in figure 2. The stress intensity factors for a finite crack under uniform remote tensile or shear stress are given in section 2. Summarizing the results given in section 2.

In such cases if the energy or stiffness of the structure can be determined as a function of crack length, or area, then the energy release rate and if the loading is Mode-I stress intensity factor can be computed.

A few examples are given here. The energy release rate for this sample can easily be computed, however, since the cracks are subject to mixed- mode loading, further analysis is needed to determine the individual stress intensity factors [51]. The portion of the test sample between the inner loading pins is loaded in pure bending with a moment M. Treating this as a fixed force problem the energy release rate can be calculated from equation 3.

In pure bending the strain energy per unit length is 2EI , where I is 3 the moment of inertia. In an effort spanning decades these results have been tabulated in easy to use, well organized handbooks [25, 52]. Generally these handbooks provide equations for stress intensity factors as a function of the geometry and dimensions of the crack and of the object containing the crack. The results are given a graphs, equations and tables of coefficients. A sampling of stress intensity factor solutions for common fracture test specimens is given in table 5.

Taking the use of tabulated solutions further, software packages such as NASCRAC [53] and NASGRO [54] integrate stress intensity factor solutions, material property databases and a graph- ical user interface to provide tools for the estimation of allowable loads, fatigue life and other calculations of interest in practical applications. Adapted from [25] and [55]. The section between the inner loading pins is in pure bending with moment M.

The use of computational methods such as the finite element method, boundary element method and dislocation based methods is invaluable for studying fracture in real-world problems. As computational fracture is itself a vast field, here we will only study some basic but important aspects and we will focus on the finite element method due to its wide ranging use in engineering design and since it is a very flexible method, extendable to nonlinear and dynamic problems.

The emphasis will be on methods that can be used with standard finite element packages and on methods relatively simple minimal post-processing. For further details see [56, 57]. This is the idea behind the stress and displacement correlation methods. Advantages of this method are that it is quite simple, it can be used with any finite element program, no special postprocessing is needed, only one analysis is needed, different modes of stress intensity factors are easily computed and stress intensity factors can be computed along a 3D crack front by taking stresses on lines normal to the crack front at different positions.

The accuracy of the method will depend on mesh refinement and the ability of the mesh to capture the crack tip stress singularity. Displacement Correlation A similar approach can be taken based on the displacements of the crack face. This has the advantage over stress correlation that the displacements are primary solution variables in the finite element method.

Similarly to stress correlation different modes of stress intensity factor can be calculated and stress intensity factors along a 3D crack front can be calculated. Accuracy issues are similar to stress correlation.

Thus if the problem involves only a single mode of loading, then the stress intensity factor can be extracted by finding the energy release rate. Advantages of these global approaches are that they are not as sensitive as stress and displace- ment correlation to the crack tip meshing since they deal with global quantities and no special finite element code or postprocessing is needed. Disadvantages are that two calculations are needed, in- dividual stress intensity factor modes cannot be determined and stress intensity factor variations along a crack front cannot be determined.

Consider the mesh shown in figure 5. Consider the two crack tip meshes shown in figure 5. The crack tip mesh must be sufficiently refined so that for a crack growing over the distance of one element the crack growth is self-similar, i. Consider first the 4 node element with crack tip node j. If the crack were to advance by one node, then the crack opening displacement for the new crack would be the displacement at the original crack tip, i. If the crack growth is self-similar, this is equal to the displacement at two nodes behind the original crack tip, i.

In addition, as with NR, only very simple post-processing of the data, namely extraction of the nodal forces and displacements are needed. Note that in both the NR and MCCI methods if the crack is not a line of symmetry, then the elements above and below the line ahead of the crack must have separate nodes. These nodes can then be constrained to have equal displacement so that nodal reaction forces can be extracted from the analysis.

Note as well that if the crack did not grow straight ahead then the equations above for G would not be valid. Generalizations of MCCI have been developed for use with quarter point and other elements [63, 64] where the simple relations above will not work. Consider the closed contour C with outward unit normal vector m as shown in figure 5. The area enclosed by C is A. In principle q can be any function as long as it is sufficiently smooth.

Recall from equation 3. Thus J has been converted into an integral over a closed contour. The domain integral approach is generally very accurate even with modest mesh refinement since it does not rely on capturing the exact crack tip singular stress field, rather on correctly computing the strain energy in the region surrounding the crack tip.

Since all of the methods use information from a small distance away from the crack tip they are somewhat forgiving of errors induced by not capturing the exact crack tip singular stress field. However, more accurate results could be obtained by capturing the crack tip singular stress field. A number of methods to produce singular crack tip stresses have been developed, some of which require special elements and some of which can be used with standard elements.

We will focus on quarter-point elements that can be implemented using standard elements [68, 69]. In the finite element method the displacement field and the coordinates are interpolated using shape functions. Consider the 2D, 8 node isoparametric element shown in figure 5.

Applying equations 5. With the collapsed node element the element edges must be straight in order to to obtain accurate solutions [70]. Recommended element. Now let us move the position of node 5 to the quarter-point, i. Other elements can also be used. A better choice might be the natural triangle quarter point element shown in figure 5. With any of these elements accuracy per unit computational time should be significantly better than with the use of non singular elements. For example, Banks-Sills and Sherman [71] compared displacement extrapolation, J integral and total energy approaches using singular and non-singular elements.

For a central cracked plate under tension using 8 node elements displacement ex- trapolation had an error in stress intensity factor of 5. For the same problem, using the total energy method with a mesh of 8 node elements the error was 2. The mode I stress intensity factor will be computed using displacement correlation, global energy, domain integral and MCCI methods.

All quantities in N and mm units. The mesh refinement is characterized by the radial length of the elements at the crack tip. The coarse, medium and fine meshes had element lengths of 5. The computational results for KI and J are summarized in table 5.

Note that the theoretical results are accurate to 0. It is likely that the correct value of KI is In any case, it is seen that both methods provide sufficient accuracy even with the coarsest mesh. However, in every case the domain integral result is more accurate. Accurate results with the domain integral method could be achieved with even coarser meshes. Global Energy Using the medium mesh shown in figure 5. The total energies in each case were 3.

Note that a symmetry model was used, so the energies must be multiplied by two. A medium and a fine mesh, figure 5. The results are summarized in tabletable-mcci. The fine mesh 4 node model and the medium mesh 8 node model have the same number of degrees of freedom, yet KI calculated using the second order method with the 8 node elements in the medium mesh is much more accurate than that calculated using the fine mesh with 4 node elements.

The results show that reasonably good accuracy can be obtained with the MCCI method even for meshes of modest refinement. Figure 5. Symmetry BC at crack line used. Results in N and mm units. Linear 4 node and quadratic 8 node elements were used. For example, perhaps the loading is not known, or is dynamic, or information about parts of the structure that would be needed for a FEM analysis are missing. In such cases one may wish to determine the stress intensity factor experimentally, based on local measurements of stress, strain and displacement.

A number of optical methods such as photoelasticity, the shadow spot caustics method, in- terferometry and digital image correlation have been developed for fracture research [72] and could in principle be applied to determine stress intensity factors in a component. However for practical applications all but the digital image correlation method will be quite difficult or impossible to im- plement. For example the photoelastic methods rely on either the use of photoelastic models built of a birefringent polymer or the placing of photoelastic coatings on the component.

The method of caustics requires either a transparent component or that the surface be optically flat and polished before cracking.

Digital image correlation requires little special surface preparation. However a great deal of experience not to mention specialized and expensive equipment is needed to use this method. However, most engineers and technicians are familiar with resistance strain gauges. Furthermore strain gauges can be bonded to many materials and are relatively inexpensive as is the signal conditioning and other equipment needed.

Thus I will focus on the strain gauge method [72, 73] as an example of a method for experimentally determining stress intensity factor in real-world applications. Let us assume that a strain gauge, or strain gauges can be placed in a region near the crack tip where the stresses are accurately described by the first three terms of the Williams expansion, equation 2. Dally and Sanford [73]show that the strain parallel to the gauge, i. There are three parameters, but only one measurement.

What do you do? Thus careful mounting of the gauges is critical to the accuracy of the method. Note also that the gauge should be small relative to the crack length or component size so that the approximation implicit in the above that the strain is constant over the area of the strain gauge can be realized. Show that for the case of uniform p2 , the stress intensity factor from equation 5. Calculate the stress intensity factor for a finite crack subjected to uniform tractions over a small region near the crack tips, i.

When a body has more than one crack the stress fields from each crack will affect the other. However, it is reasonable as well to expect that when the cracks are widely spaced that their interaction will be minimal.

Calculate the energy release rate for the 4 point bending specimen, figure 5. Extend the calculation of energy release rate for the 4 point bending specimen, figure 5.

Calculate the energy release rate for the sample shown in figure 5. As such a day is likely to be long or perhaps infinite in coming, some acquaintance with physical testing is required to understand and to apply fracture mechanics. A complete description of experimental methods in fracture would require several long books. Thus in this section the bare outlines of the equipment, measurements, basic test and sample types, standards and interpretation of data will be described, focusing on the fracture of materials for which linear elastic fracture mechanics is a good approximation.

Both ASTM standard and several non-standard but useful methods will be outlined here. Physical tests related to elastic-plastic fracture will be briefly discussed in a separate chapter. The discussion here will focus on testing at ambient temperatures and in laboratory air environ- ments. Although the principles remain essentially the same, elevated and low temperature tests require special equipment and considerations. The environment, for example the presence of corro- sive agents, or of hydrogen, or water has a significant impact on crack propagation.

Experimental methods for environmentally assisted and high temperature fracture deserve an entire chapter themselves. To ensure that the toughness values used in the application are correct, great care must be taken in the tests used to determine the toughness.

The method is based on ensuring that sufficient constraint exists to provide plane- strain conditions at the crack. The views expressed here are those of the authors, and unless otherwise noted, do not necessarily reflect the views of the Brookside Associates, Ltd. All writings, discussions, and publications on this website are unclassified.

All rights reserved. Other Brookside Products. Advertising on this Site. Nursing Care Related to the Musculoskeletal System In treating a fracture, the objectives of the treatment are as follows: 1 To regain and maintain the normal alignment of the injured part.

This website is privately-held and not connected to any governmental agency. The views expressed here are those of the authors, and unless otherwise noted, do not necessarily reflect the views of the Brookside Associates, Ltd.

All writings, discussions, and publications on this website are unclassified. All rights reserved. Other Brookside Products. Advertising on this Site. Nursing Care Related to the Musculoskeletal System The book also tackles the crack tip plasticity and covers crack growth. The last chapter in the text discusses some applications in fracture mechanics. The book will be of great use to engineers who want to get acquainted with fracture mechanics.

The book will be of great use to engineers who want to The book outlines analytical and experimental methods for determining the fracture resistance of mechanical and structural components, also demonstrating the use of fracture mechanics in failure analysis, reinforcement of cracked structures, and remaining life estimation.

The characteristics of crack propagation induced by fatigue, stress-corrosion, creep, and absorbed hydrogen are also discussed. The book concludes with a chapter on the structural integrity analysis of cracked components alongside a real integrity assessment.

This book will be especially useful for students in mechanical, civil, industrial, metallurgical, aeronautical and chemical engineering, and for professional engineers looking for a refresher on core principles. Concisely outlines the underlying fundamentals of fracture mechanics, making physical concepts clear and simple and providing easily-understood applied examples Includes solved problems of the most common calculations, along with step-by-step procedures to perform widely-used methods in fracture mechanics Demonstrates how to determine stress intensity factors and fracture toughness, estimate crack growth rate, calculate failure load, and other methods and techniques.

Fracture Mechanics Author : Emmanuel E. Gdoutos Publisher : Springer Nature Release Date : Genre: Electronic books Pages : ISBN 10 : GET BOOK Fracture Mechanics Book Description : This book discusses the basic principles and traditional applications of fracture mechanics, as well as the cutting-edge research in the field over the last three decades in current topics like composites, thin films, nanoindentation, and cementitious materials.

Experimental methods play a major role in the study of fracture mechanics problems and are used for the determination of the major fracture mechanics quantities such as stress intensity factors, crack tip opening displacements, strain energy release rates, crack paths, crack velocities in static and dynamic problems.

Furthermore, numerical methods are often used for the determination of fracture mechanics parameters. They include finite and boundary element methods, Greens function and weight functions, boundary collocation, alternating methods, and integral transforms continuous dislocations.

This third edition of the book covers the basic principles and traditional applications, as well as the latest developments of fracture mechanics.