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61-603: Keepit Dam is a major gated mass concrete gravity dam with an earth fill abutment and a central gated concrete overflow crest and six radial gate spillways across the Namoi River upstream of its junction with the Peel River in the North West Slopes region of New South Wales , Australia . The dam's purpose includes flood mitigation , hydro-power , irrigation , water supply and conservation. The impounded reservoir

122-473: A plastic zone develops at the tip of the crack. As the applied load increases, the plastic zone increases in size until the crack grows and the elastically strained material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat . Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy

183-457: A crack front in a linear elastic solid. This asymptotic expression for the stress field in mode I loading is related to the stress intensity factor K I {\displaystyle K_{I}} following: where σ i j {\displaystyle \sigma _{ij}} are the Cauchy stresses , r {\displaystyle r} is the distance from

244-429: A crack is present in a specimen that undergoes cyclic loading, the specimen will plastically deform at the crack tip and delay the crack growth. In the event of an overload or excursion, this model changes slightly to accommodate the sudden increase in stress from that which the material previously experienced. At a sufficiently high load (overload), the crack grows out of the plastic zone that contained it and leaves behind

305-421: A finished mechanical component. Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause failure of the flawed structure. Despite these inherent flaws, it

366-429: A geometry dependent region of stress concentration replacing the crack-tip singularity. In actuality, the stress concentration at the tip of a crack within real materials has been found to have a finite value but larger than the nominal stress applied to the specimen. Nevertheless, there must be some sort of mechanism or property of the material that prevents such a crack from propagating spontaneously. The assumption is,

427-433: A specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material. To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental glass specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in

488-444: A specimen. The experiments showed that the product of the square root of the flaw length ( a {\displaystyle a} ) and the stress at fracture ( σ f {\displaystyle \sigma _{f}} ) was nearly constant, which is expressed by the equation: An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence

549-411: Is Poisson's ratio , and K I is the stress intensity factor in mode I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I, mode II (sliding mode), and mode III (tearing mode) stress intensity factors for the most general loading conditions. Next, Irwin adopted the additional assumption that the size and shape of

610-462: Is 5,700 square kilometres (2,200 sq mi). The central gated overflow crest and six radial gates of the spillway are capable of discharging 10,480 cubic metres per second (370,000 cu ft/s). An A$ 146.6 million upgrade of facilities commenced in 2009 and resulted in the construction of two spillways and three saddle dams, completed during 2011. A further upgrade is due to commence in 2014 for completion by 2016 that will involve raising

671-418: Is accepted as the defining property in linear elastic fracture mechanics. In theory the stress at the crack tip where the radius is nearly zero, would tend to infinity. This would be considered a stress singularity, which is not possible in real-world applications. For this reason, in numerical studies in the field of fracture mechanics, it is often appropriate to represent cracks as round tipped notches , with

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732-743: Is called Lake Keepit . Commenced in 1939, with construction halted during World War II , and completed in 1960, the Keepit Dam is a major dam on the Namoi River, located approximately 56 kilometres (35 mi) west of Tamworth and 39 kilometres (24 mi) north-east of Gunnedah , upstream of the confluence of the Namoi and Peel rivers. The dam was built by the New South Wales Water Conservation & Irrigation Commission to supply water for irrigation, flood mitigation and potable water for

793-440: Is centered at the crack tip. This equation gives the approximate ideal radius of the plastic zone deformation beyond the crack tip, which is useful to many structural scientists because it gives a good estimate of how the material behaves when subjected to stress. In the above equation, the parameters of the stress intensity factor and indicator of material toughness, K C {\displaystyle K_{C}} , and

854-532: Is considered a material property. The subscript I {\displaystyle I} arises because of the different ways of loading a material to enable a crack to propagate . It refers to so-called "mode I {\displaystyle I} " loading as opposed to mode I I {\displaystyle II} or I I I {\displaystyle III} : The expression for K I {\displaystyle K_{I}} will be different for geometries other than

915-904: Is dimensionless, the stress intensity factor can be expressed in units of MPa m {\displaystyle {\text{MPa}}{\sqrt {\text{m}}}} . Stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface weakness energy. Both of these terms are simply related to the energy terms that Griffith used: and K c = { E G c for plane stress E G c 1 − ν 2 for plane strain {\displaystyle K_{c}={\begin{cases}{\sqrt {EG_{c}}}&{\text{for plane stress}}\\\\{\sqrt {\cfrac {EG_{c}}{1-\nu ^{2}}}}&{\text{for plane strain}}\end{cases}}} where K I {\displaystyle K_{I}}

976-403: Is large, which results in a larger plastic radius. This implies that the material can plastically deform, and, therefore, is tough. This estimate of the size of the plastic zone beyond the crack tip can then be used to more accurately analyze how a material will behave in the presence of a crack. The same process as described above for a single event loading also applies and to cyclic loading. If

1037-399: Is low, one knows that the material is more ductile. The ratio of these two parameters is important to the radius of the plastic zone. For instance, if σ Y {\displaystyle \sigma _{Y}} is small, then the squared ratio of K C {\displaystyle K_{C}} to σ Y {\displaystyle \sigma _{Y}}

1098-540: Is needed for crack growth in ductile materials as compared to brittle materials. Irwin's strategy was to partition the energy into two parts: Then the total energy is: where γ {\displaystyle \gamma } is the surface energy and G p {\displaystyle G_{p}} is the plastic dissipation (and dissipation from other sources) per unit area of crack growth. The modified version of Griffith's energy criterion can then be written as For brittle materials such as glass,

1159-420: Is possible to achieve through damage tolerance analysis the safe operation of a structure. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new. Fracture mechanics should attempt to provide quantitative answers to the following questions: Fracture mechanics was developed during World War I by English aeronautical engineer A. A. Griffith – thus

1220-602: Is small scale yielding, the condition that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures. Linear-elastic fracture mechanics is of limited practical use for structural steels and Fracture toughness testing can be expensive. Most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions that involve large loads. In such materials

1281-465: Is termed linear elastic fracture mechanics ( LEFM ) and can be characterised using the stress intensity factor K {\displaystyle K} . Although the load on a crack can be arbitrary, in 1957 G. Irwin found any state could be reduced to a combination of three independent stress intensity factors: When the size of the plastic zone at the crack tip is too large, elastic-plastic fracture mechanics can be used with parameters such as

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1342-474: Is that their large concrete structures are susceptible to destabilising uplift pressures relative to the surrounding soil. Uplift pressures can be reduced by internal and foundation drainage systems. During construction, the exothermic curing of concrete can generate large amounts of heat. The poorly-conductive concrete then traps this heat in the dam structure for decades, expanding the plastic concrete and leaving it susceptible to cracking while cooling. It

1403-408: Is the criterion for which the crack will begin to propagate. For materials highly deformed before crack propagation, the linear elastic fracture mechanics formulation is no longer applicable and an adapted model is necessary to describe the stress and displacement field close to crack tip, such as on fracture of soft materials . Griffith's work was largely ignored by the engineering community until

1464-406: Is the designer's task to ensure this does not occur. Gravity dams are built by first cutting away a large part of the land in one section of a river, allowing water to fill the space and be stored. Once the land has been cut away, the soil has to be tested to make sure it can support the weight of the dam and the water. It is important to make sure the soil will not erode over time, which would allow

1525-565: Is the mode I {\displaystyle I} stress intensity, K c {\displaystyle K_{c}} the fracture toughness, and ν {\displaystyle \nu } is Poisson's ratio. Fracture occurs when K I ≥ K c {\displaystyle K_{I}\geq K_{c}} . For the special case of plane strain deformation, K c {\displaystyle K_{c}} becomes K I c {\displaystyle K_{Ic}} and

1586-555: The J-integral or the crack tip opening displacement . The characterising parameter describes the state of the crack tip which can then be related to experimental conditions to ensure similitude . Crack growth occurs when the parameters typically exceed certain critical values. Corrosion may cause a crack to slowly grow when the stress corrosion stress intensity threshold is exceeded. Similarly, small flaws may result in crack growth when subjected to cyclic loading. Known as fatigue , it

1647-419: The glass transition temperature, we have intermediate values of G {\displaystyle G} between 2 and 1000 J/m 2 {\displaystyle {\text{J/m}}^{2}} . Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around

1708-449: The weight of the material and its resistance against the foundation. Gravity dams are designed so that each section of the dam is stable and independent of any other dam section. Gravity dams generally require stiff rock foundations of high bearing strength (slightly weathered to fresh), although in rare cases, they have been built on soil. Stability of the dam primarily arises from the range of normal force angles viably generated by

1769-408: The assumptions of linear elastic fracture mechanics may not hold, that is, Therefore, a more general theory of crack growth is needed for elastic-plastic materials that can account for: Historically, the first parameter for the determination of fracture toughness in the elasto-plastic region was the crack tip opening displacement (CTOD) or "opening at the apex of the crack" indicated. This parameter

1830-416: The biggest danger to gravity dams and that is why, every year and after every major earthquake, they must be tested for cracks, durability, and strength. Although gravity dams are expected to last anywhere from 50–150 years, they need to be maintained and regularly replaced. Concrete fracture analysis Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe

1891-460: The case of plane strain should be divided by the plate stiffness factor ( 1 − ν 2 ) {\displaystyle (1-\nu ^{2})} . The strain energy release rate can physically be understood as: the rate at which energy is absorbed by growth of the crack . However, we also have that: If G {\displaystyle G} ≥ G c {\displaystyle G_{c}} , this

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1952-399: The center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a dimensionless correction factor , Y {\displaystyle Y} , in order to characterize the geometry. This correction factor, also often referred to as the geometric shape factor , is given by empirically determined series and accounts for

2013-469: The change in elastic strain energy per unit area of crack growth, i.e., where U is the elastic energy of the system and a is the crack length. Either the load P or the displacement u are constant while evaluating the above expressions. Irwin showed that for a mode I crack (opening mode) the strain energy release rate and the stress intensity factor are related by: where E is the Young's modulus , ν

2074-455: The crack tip, θ {\displaystyle \theta } is the angle with respect to the plane of the crack, and f i j {\displaystyle f_{ij}} are functions that depend on the crack geometry and loading conditions. Irwin called the quantity K {\displaystyle K} the stress intensity factor. Since the quantity f i j {\displaystyle f_{ij}}

2135-607: The dam can begin. Usually gravity dams are built out of a strong material such as concrete or stone blocks, and are built into a triangular shape to provide the most support. The most common classification of gravity dams is by the materials composing the structure: Composite dams are a combination of concrete and embankment dams . Construction materials of composite dams are the same used for concrete and embankment dams. Gravity dams can be classified by plan (shape): Gravity dams can be classified with respect to their structural height: Gravity dams are built to withstand some of

2196-584: The early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic. Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. For ductile materials such as steel , although

2257-415: The energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new material property was given the name fracture toughness and designated G Ic . Today, it is the critical stress intensity factor K Ic , found in the plane strain condition, which

2318-428: The flow of the water leaving Keepit Dam with an average output of 10.2 gigawatt-hours (37 TJ) per annum. The station was completed in 1960 and upgraded in 1983. The facility is managed by Meridian Energy . The name Keepit originates after a riverside property called Keypit or Keepit , resumed for part of the storage area. The word probably means 'keep it', a derogatory remark about the apparent worthlessness of

2379-434: The foundation. Also, the stiff nature of a gravity dam structure endures differential foundation settlement poorly, as it can crack the dam structure. The main advantage to gravity dams over embankments is the scour -resistance of concrete, which protects against damage from minor over-topping flows. Unexpected large over-topping flows are still a problem, as they can scour dam foundations. A disadvantage of gravity dams

2440-550: The height of the main dam wall by 3.4 metres (11 ft) and enhancing post tension in the concrete section of the wall. Keepit Dam is operated in conjunction with Split Rock Dam . The two dams supply water requirements along much of the Namoi Valley, used for irrigation including cotton , cereal and wheat crops, lucerne, fodder and pasture, vegetables, vines, orchards and oil seeds. A hydro-electric power station generates up to 7.2 megawatts (9,700 hp) of electricity from

2501-414: The material to the far-field stresses of the y-direction along the crack (x direction) and solved for the effective radius. From this relationship, and assuming that the crack is loaded to the critical stress intensity factor, Irwin developed the following expression for the idealized radius of the zone of plastic deformation at the crack tip: Models of ideal materials have shown that this zone of plasticity

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2562-478: The pastoral run. The area surrounding Lake Keepit is used for local recreation including camping, picnics, swimming, boating, sailing, water skiing and fishing. The lakeshore is the location of Lake Keepit Soaring Club, the second largest gliding club by membership in Australia. Gravity dam A gravity dam is a dam constructed from concrete or stone masonry and designed to hold back water by using only

2623-406: The plastic deformation at the crack tip effectively blunts the crack tip. This deformation depends primarily on the applied stress in the applicable direction (in most cases, this is the y-direction of a regular Cartesian coordinate system), the crack length, and the geometry of the specimen. To estimate how this plastic deformation zone extended from the crack tip, Irwin equated the yield strength of

2684-422: The plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress (the plastic zone) at the crack tip. In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture. The energy release rate for crack growth or strain energy release rate may then be calculated as

2745-412: The pocket of the original plastic deformation. Now, assuming that the overload stress is not sufficiently high as to completely fracture the specimen, the crack will undergo further plastic deformation around the new crack tip, enlarging the zone of residual plastic stresses. This process further toughens and prolongs the life of the material because the new plastic zone is larger than what it would be under

2806-498: The relation σ f a = C {\displaystyle \sigma _{f}{\sqrt {a}}=C} still holds, the surface energy ( γ ) predicted by Griffith's theory is usually unrealistically high. A group working under G. R. Irwin at the U.S. Naval Research Laboratory (NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials. In ductile materials (and even in materials that appear to be brittle ),

2867-414: The simple case of a thin rectangular plate with a crack perpendicular to the load, the energy release rate, G {\displaystyle G} , becomes: where σ {\displaystyle \sigma } is the applied stress, a {\displaystyle a} is half the crack length, and E {\displaystyle E} is the Young's modulus , which for

2928-414: The state around a crack. Fracture mechanics is used to characterise the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of different parameters have been developed. When the plastic zone at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of elastic forces within the material and

2989-427: The strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed. The growth of a crack, the extension of the surfaces on either side of the crack, requires an increase in the surface energy . Griffith found an expression for the constant C {\displaystyle C} in terms of

3050-485: The strongest earthquakes . Even though the foundation of a gravity dam is built to support the weight of the dam and all the water, it is quite flexible in that it absorbs a large amount of energy and sends it into the Earth's crust. It needs to be able to absorb the energy from an earthquake because, if the dam were to break, it would send a mass amount of water rushing downstream and destroy everything in its way. Earthquakes are

3111-724: The surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was: where E {\displaystyle E} is the Young's modulus of the material and γ {\displaystyle \gamma } is the surface energy density of the material. Assuming E = 62   GPa {\displaystyle E=62\ {\text{GPa}}} and γ = 1   J/m 2 {\displaystyle \gamma =1\ {\text{J/m}}^{2}} gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass. For

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3172-454: The surface energy term dominates and G ≈ 2 γ = 2 J/m 2 {\displaystyle G\approx 2\gamma =2\,\,{\text{J/m}}^{2}} . For ductile materials such as steel, the plastic dissipation term dominates and G ≈ G p = 1000 J/m 2 {\displaystyle G\approx G_{p}=1000\,\,{\text{J/m}}^{2}} . For polymers close to

3233-469: The term Griffith crack – to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts: A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be

3294-399: The town of Walgett . The dam wall height is 55 metres (180 ft) and is 533 metres (1,749 ft) long. The maximum water depth is 48 metres (157 ft) and at 100% capacity the dam wall holds back 425,510 megalitres (15,027 × 10 ^  cu ft) of water at 329.6 metres (1,081 ft) AHD . The surface area of Lake Keepit is 4,370 hectares (10,800 acres) and the catchment area

3355-403: The type and geometry of the crack or notch. We thus have: where Y {\displaystyle Y} is a function of the crack length and width of sheet given, for a sheet of finite width W {\displaystyle W} containing a through-thickness crack of length 2 a {\displaystyle 2a} , by: Irwin was the first to observe that if the size of

3416-555: The usual stress conditions. This allows the material to undergo more cycles of loading. This idea can be illustrated further by the graph of Aluminum with a center crack undergoing overloading events. But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics

3477-493: The water to cut a way around or under the dam. Sometimes the soil is sufficient to achieve these goals; however, other times it requires conditioning by adding support rocks which will bolster the weight of the dam and water. There are three different tests that can be done to determine the foundation's support strength: the Westergaard, Eulerian, and Lagrangian approaches. Once the foundation is suitable to build on, construction of

3538-443: The yield stress, σ Y {\displaystyle \sigma _{Y}} , are of importance because they illustrate many things about the material and its properties, as well as about the plastic zone size. For example, if K c {\displaystyle K_{c}} is high, then it can be deduced that the material is tough, and if σ Y {\displaystyle \sigma _{Y}}

3599-399: Was determined by Wells during the studies of structural steels, which due to the high toughness could not be characterized with the linear elastic fracture mechanics model. He noted that, before the fracture happened, the walls of the crack were leaving and that the crack tip, after fracture, ranged from acute to rounded off due to plastic deformation. In addition, the rounding of the crack tip

3660-529: Was found that for long cracks, the rate of growth is largely governed by the range of the stress intensity Δ K {\displaystyle \Delta K} experienced by the crack due to the applied loading. Fast fracture will occur when the stress intensity exceeds the fracture toughness of the material. The prediction of crack growth is at the heart of the damage tolerance mechanical design discipline. The processes of material manufacture, processing, machining, and forming may introduce flaws in

3721-414: Was more pronounced in steels with superior toughness. There are a number of alternative definitions of CTOD. In the two most common definitions, CTOD is the displacement at the original crack tip and the 90 degree intercept. The latter definition was suggested by Rice and is commonly used to infer CTOD in finite element models of such. Note that these two definitions are equivalent if the crack tip blunts in

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