A fatigue testing machine (I designed) in which changing temperature and atmosphere as well as mechanical stretching are appied the specimen, the piece of metal in the center.
Fatigue cracks prefer to form along specific weak crystallographic planes or along the interfaces between crystals in the metal. (Metals are polycrystalline.)
The quasi-cleavage fracture surface, common in fatigue, may reflect the different orientations of these favored planes. When the fatigue fracture takes place at a high homologous temperature, surface tension may round the corners between different crystals or crystal surfaces. This surface annealing and other causes tend to obliterate detail.
When fatigue cracks originate at grain boundaries, looking directly into the forming crack with a scanning electron microscope may be possible. In this case the fracture proceeds through voids at the crack tip. The spacing of these voids may reflect the moire effect of the two lattices. Excess lattice vacancies may contribute to forming these voids.
As a crack propagates, it may encounter obstacles and other cracks. Here the crack has propagated from the left and is viewed growing into the material in this transverse section.At the right it has encountered a void evidentally created by an internal crack along one end of a crystal. The void blunted the crack and prevented it from growing further.
Metal Fatigue
One of the most useful properties of metals is that they can be bent or stretched and they will spring back to their original shape. Even if they take a "permanent set"" they continue to be strong. This allows metals to be formed into shapes and is why they rarely shatter in spite of their great strength. The great lesson of time is that no event is truly reversible no matter how it seems. Even though a piece of metal may spring back, it has been altered by bending. In fact metals are altered by most of their experiences. One economically important aspect of this state of affairs is that repeated deformation of metals causes them to crack and break. This cracking process is called fatigue It is not a property of metals alone. Plastic materials behave in a similar way. Brittle materials such as glasses and ceramics undergo even more complex changes when repeatedly bent or loaded. But, their tendency to simply break from overloading obscures fatigue in them.
In spite of massive amounts of money spent on researching metal fatigue, the actual physics of the subject is poorly understood. This is because of its complexity and inaccessibility (Preparing specimens alters the material prepared. By the time a specimen has been prepares for viewing at the atomic level of resolution, it has undergone an enormous amount of experience.) But, it is also because of the extraordinary similarity of fatigue behavior in different metals and under different circumstances. It is hard to build up a good theory when changing what should be significant variables does not produce a systematic change in the experimental result. Furthermore, fatigue data are noisey; experiments are not as repeatable as good physics would require. This contributes to another factor of the problem that is that no deep understanding is really needed in the short term. Empirical formulas, such as the Coffin-Manson relation often as good enough considering the intrinsic variability of the results. One merely has to determine the values of the coefficients of the desired formula experimentally for the particular material under the particular circumstances. This has kept a lot of experimentalists busy and is why the total bill for fatigue research is so high.
Phenomenology
Fatigue phenomena are surprisingly divers in detail and yet highly repetitive in overall form. People tend to look at different aspects such as fatigue life which is the number of cycles to failure e.g. the number of takeoffs and landings of an airplane or the number of times you can bend and straighten those little metal tabs on the backs of picture frames. Another way of looking at fatigue is through the fatigue growth rate of cracks-- referred to as fatigue crack propagation. One observes a crack and asks how fast it will grow, and how long before the part breaks? A salient example is the burner cans in jet engines. Breakage of one of the burner cans is likely to result in the explosion of the engine. (Commercial planes have armor of under the wings for just this event.) Yet, large cracks are permitted at inspection in burner cans because these cracks grow very slowly. Still another way of looking at fatigue is through crack initiation or nucleation. How long before a crack forms at all. The interesting fact here is that it depends on how hard one looks. The stronger the microscope the more quickly cracks are seen to form. With very powerful electron microscopes, features can be observed in "virgin" material that are cracks if you wish to call them such, but they could also be viewed as roughness, intrusions or extrusions. All of these features are capable of growing into full-size cracks. The interested reader should look at
S. Suresh, Fatigue of Materials, in the Cambridge Solid State Science Series published by Cambridge University Press. There are several editions.
For a review of fine points in crack propagation, the reader should look at
L.Lawson, E.Y. Chen and M. Meshii, "Near-threshold fatigue: a review" in International Journal of Fatigue, volume 21 supplement pages S15-S34 (1999).
Typically the life of a crack begins with some discontinuity ( as small as nanometers in size) in the material that produces a discontinuity in the stress field. This discontinuity tends to grow into a microcrack, a crack with a size measured in microns. In areas of stress concentration, microcracks may be abundant. Since they generally cannot be seen without an electron microscope, they are often regarded as non-present. One of the interesting points of contemplation is that such tiny openings are thermodynamically unstable and should heal themselves. But, they don't. Oxidation, adsorption of gasses, surface crystallographic reconstructions, dislocation backstresses and the migration of less soluble impurities to the crack surface have all been used as arguments for why cracks rarely heal. As microcracks grow, they encounter either obstacles that they cannot easily grow through, e.g. voids, particles and grain boundaries,or else they encounter locally enhanced stresses that speed their growth. Microcracks seem to grow by the same rules as large cracks but in a much more variable environment. Eventually microcracks encounter other microcracks and join together. This coalescence is usually the way a large-enough-to-be-seen crack or engineering crack forms. Once a large crack does form, the mechanisms that promoted the rapid growth of microcracks are less important because the large crack must grow in its entirety. If the crack grows a small protuberance, the rest of the crack may not be able to follow. A phenomenon called crack closure becomes important. Originally this term meant peices of dirt that got wedged into the crack blocking it from cyclically opening and closing. A crack that does not cyclically open and close does not grow. Crack closure has been greatly generalized to the point that some experts have denied its existence. In modeling the life expectancy of parts and civil structures, crack closure remains a useful construct if not always good physics. The fact is that many events including sudden overloads will cause cracks to arrest their growth for long period of time. This has been told to me as one of the reasons Harley-Davidson owners are encouraged to every so often really stomp on their bikes through rapid acceleration.
Modeling Fatigue
Fatigue is a statistical process. Fatigue predictions are thus manifestations of statistics. Questions like "What are the chances the Kinzua railroad bridge will collapse in the next year?"(it did) become rooted in summing the conditioned probabilities that a certain loading will interact to produce a certain stress in the vicinity of a certain crack (real or imagined) to produce a certain stress intensity (a driving force for crack growth) resulting in... Obviously such summations are very complicated and most of the details are guesswork.
In the midst of all this uncertainty is one bright star. It is called the Paris Law. But, since noone has proven its legality, it is properly called the Paris Relation: 
This relation says that the crack growth increment per cycle is proportional to the amount the stress intensity changes in a cycle raised to a power. The stress intensity in turn is the product of the stress in which the crack grows times a geometry factor times the square root of the length of the crack (in this case x). This means that the crack's growth rate is proportional to its size raised to a power. Such power law relations have no geometric length scale. Ceteris parabus one could not tell the difference between a movie of a giant crack growing in an oil tanker or a microcrack in a watch spring. This is also a property of fractal geometry and much thought has been spent on it. See for example, M Schroeder, Fractals, Chaos, Power Laws, Freeman, NY (1991). What these things have in common with crack growth is self-similarity, that is, scale independence. At a first glance -- there is not enough space here to tell you how to look -- the case of the fatigue crack seems trivial: One is simply looking at a point in some sort of space and a point looks like a point at any length scale. The problem with this triviality is that stress singularities do not really exist but are fictions of theoretical mechanics. In real life there is plasticity to complicate matters. Yet, like the child who does not know how to ask at Passover, there is a sufficient answer. Detailed knowledge of plasticity is rarely needed. But why? I have some ideas based on my own work, that of Neubers and that of Glinka. You too can guess.
In terms of elementary events, cracks grow, become arrested and join with other cracks. Cracks usually link up on a micro level, but then there was the Aloha incident in which many cracks growing out of rivets suddenly joined together and the fuselage of the Aloha plane burst in mid air. Models of these elementary processes can be incorporated into what is called a Monte Carlo Markov Chain (MCMC) model of a field of many cracks. At the microcrack size level fields of many microcracks are not uncommon. The evolution of these fields amounts to what engineers call crack initiation. The idea is to consider the many sites at which cracks could grow. A computer then tosses dice at each site for each event and tabulates the resulting cracks and their sizes. One could construct a detailed spatial model and track each crack until it grows into another one. An often better approach -- one much less computationally intensive -- is to consider only the density of cracks and their probabilities of intersection based on stochastic geometry. There is a good derivation in stereology that can be used as the basis for this calculation. I have written many papers with my student, Ed Chen, on this type of model. They are listed with my publications. Most are quite readable. They don't even use the term, Monte Carlo Markov Chain, because it had not yet become popular. Now, I believe, there is an entire journal devoted to it.
I hope I have given the reader some quick insight. The subject is vast.
Publications
Home
The quasi-cleavage fracture surface, common in fatigue, may reflect the different orientations of these favored planes. When the fatigue fracture takes place at a high homologous temperature, surface tension may round the corners between different crystals or crystal surfaces. This surface annealing and other causes tend to obliterate detail.
When fatigue cracks originate at grain boundaries, looking directly into the forming crack with a scanning electron microscope may be possible. In this case the fracture proceeds through voids at the crack tip. The spacing of these voids may reflect the moire effect of the two lattices. Excess lattice vacancies may contribute to forming these voids.
As a crack propagates, it may encounter obstacles and other cracks. Here the crack has propagated from the left and is viewed growing into the material in this transverse section.At the right it has encountered a void evidentally created by an internal crack along one end of a crystal. The void blunted the crack and prevented it from growing further.
Metal Fatigue
One of the most useful properties of metals is that they can be bent or stretched and they will spring back to their original shape. Even if they take a "permanent set"" they continue to be strong. This allows metals to be formed into shapes and is why they rarely shatter in spite of their great strength. The great lesson of time is that no event is truly reversible no matter how it seems. Even though a piece of metal may spring back, it has been altered by bending. In fact metals are altered by most of their experiences. One economically important aspect of this state of affairs is that repeated deformation of metals causes them to crack and break. This cracking process is called fatigue It is not a property of metals alone. Plastic materials behave in a similar way. Brittle materials such as glasses and ceramics undergo even more complex changes when repeatedly bent or loaded. But, their tendency to simply break from overloading obscures fatigue in them.
In spite of massive amounts of money spent on researching metal fatigue, the actual physics of the subject is poorly understood. This is because of its complexity and inaccessibility (Preparing specimens alters the material prepared. By the time a specimen has been prepares for viewing at the atomic level of resolution, it has undergone an enormous amount of experience.) But, it is also because of the extraordinary similarity of fatigue behavior in different metals and under different circumstances. It is hard to build up a good theory when changing what should be significant variables does not produce a systematic change in the experimental result. Furthermore, fatigue data are noisey; experiments are not as repeatable as good physics would require. This contributes to another factor of the problem that is that no deep understanding is really needed in the short term. Empirical formulas, such as the Coffin-Manson relation often as good enough considering the intrinsic variability of the results. One merely has to determine the values of the coefficients of the desired formula experimentally for the particular material under the particular circumstances. This has kept a lot of experimentalists busy and is why the total bill for fatigue research is so high.
Phenomenology
Fatigue phenomena are surprisingly divers in detail and yet highly repetitive in overall form. People tend to look at different aspects such as fatigue life which is the number of cycles to failure e.g. the number of takeoffs and landings of an airplane or the number of times you can bend and straighten those little metal tabs on the backs of picture frames. Another way of looking at fatigue is through the fatigue growth rate of cracks-- referred to as fatigue crack propagation. One observes a crack and asks how fast it will grow, and how long before the part breaks? A salient example is the burner cans in jet engines. Breakage of one of the burner cans is likely to result in the explosion of the engine. (Commercial planes have armor of under the wings for just this event.) Yet, large cracks are permitted at inspection in burner cans because these cracks grow very slowly. Still another way of looking at fatigue is through crack initiation or nucleation. How long before a crack forms at all. The interesting fact here is that it depends on how hard one looks. The stronger the microscope the more quickly cracks are seen to form. With very powerful electron microscopes, features can be observed in "virgin" material that are cracks if you wish to call them such, but they could also be viewed as roughness, intrusions or extrusions. All of these features are capable of growing into full-size cracks. The interested reader should look at
S. Suresh, Fatigue of Materials, in the Cambridge Solid State Science Series published by Cambridge University Press. There are several editions.
For a review of fine points in crack propagation, the reader should look at
L.Lawson, E.Y. Chen and M. Meshii, "Near-threshold fatigue: a review" in International Journal of Fatigue, volume 21 supplement pages S15-S34 (1999).
Typically the life of a crack begins with some discontinuity ( as small as nanometers in size) in the material that produces a discontinuity in the stress field. This discontinuity tends to grow into a microcrack, a crack with a size measured in microns. In areas of stress concentration, microcracks may be abundant. Since they generally cannot be seen without an electron microscope, they are often regarded as non-present. One of the interesting points of contemplation is that such tiny openings are thermodynamically unstable and should heal themselves. But, they don't. Oxidation, adsorption of gasses, surface crystallographic reconstructions, dislocation backstresses and the migration of less soluble impurities to the crack surface have all been used as arguments for why cracks rarely heal. As microcracks grow, they encounter either obstacles that they cannot easily grow through, e.g. voids, particles and grain boundaries,or else they encounter locally enhanced stresses that speed their growth. Microcracks seem to grow by the same rules as large cracks but in a much more variable environment. Eventually microcracks encounter other microcracks and join together. This coalescence is usually the way a large-enough-to-be-seen crack or engineering crack forms. Once a large crack does form, the mechanisms that promoted the rapid growth of microcracks are less important because the large crack must grow in its entirety. If the crack grows a small protuberance, the rest of the crack may not be able to follow. A phenomenon called crack closure becomes important. Originally this term meant peices of dirt that got wedged into the crack blocking it from cyclically opening and closing. A crack that does not cyclically open and close does not grow. Crack closure has been greatly generalized to the point that some experts have denied its existence. In modeling the life expectancy of parts and civil structures, crack closure remains a useful construct if not always good physics. The fact is that many events including sudden overloads will cause cracks to arrest their growth for long period of time. This has been told to me as one of the reasons Harley-Davidson owners are encouraged to every so often really stomp on their bikes through rapid acceleration.
Modeling Fatigue
Fatigue is a statistical process. Fatigue predictions are thus manifestations of statistics. Questions like "What are the chances the Kinzua railroad bridge will collapse in the next year?"(it did) become rooted in summing the conditioned probabilities that a certain loading will interact to produce a certain stress in the vicinity of a certain crack (real or imagined) to produce a certain stress intensity (a driving force for crack growth) resulting in... Obviously such summations are very complicated and most of the details are guesswork.
In the midst of all this uncertainty is one bright star. It is called the Paris Law. But, since noone has proven its legality, it is properly called the Paris Relation:
This relation says that the crack growth increment per cycle is proportional to the amount the stress intensity changes in a cycle raised to a power. The stress intensity in turn is the product of the stress in which the crack grows times a geometry factor times the square root of the length of the crack (in this case x). This means that the crack's growth rate is proportional to its size raised to a power. Such power law relations have no geometric length scale. Ceteris parabus one could not tell the difference between a movie of a giant crack growing in an oil tanker or a microcrack in a watch spring. This is also a property of fractal geometry and much thought has been spent on it. See for example, M Schroeder, Fractals, Chaos, Power Laws, Freeman, NY (1991). What these things have in common with crack growth is self-similarity, that is, scale independence. At a first glance -- there is not enough space here to tell you how to look -- the case of the fatigue crack seems trivial: One is simply looking at a point in some sort of space and a point looks like a point at any length scale. The problem with this triviality is that stress singularities do not really exist but are fictions of theoretical mechanics. In real life there is plasticity to complicate matters. Yet, like the child who does not know how to ask at Passover, there is a sufficient answer. Detailed knowledge of plasticity is rarely needed. But why? I have some ideas based on my own work, that of Neubers and that of Glinka. You too can guess.
In terms of elementary events, cracks grow, become arrested and join with other cracks. Cracks usually link up on a micro level, but then there was the Aloha incident in which many cracks growing out of rivets suddenly joined together and the fuselage of the Aloha plane burst in mid air. Models of these elementary processes can be incorporated into what is called a Monte Carlo Markov Chain (MCMC) model of a field of many cracks. At the microcrack size level fields of many microcracks are not uncommon. The evolution of these fields amounts to what engineers call crack initiation. The idea is to consider the many sites at which cracks could grow. A computer then tosses dice at each site for each event and tabulates the resulting cracks and their sizes. One could construct a detailed spatial model and track each crack until it grows into another one. An often better approach -- one much less computationally intensive -- is to consider only the density of cracks and their probabilities of intersection based on stochastic geometry. There is a good derivation in stereology that can be used as the basis for this calculation. I have written many papers with my student, Ed Chen, on this type of model. They are listed with my publications. Most are quite readable. They don't even use the term, Monte Carlo Markov Chain, because it had not yet become popular. Now, I believe, there is an entire journal devoted to it.
I hope I have given the reader some quick insight. The subject is vast.
Publications
Home
As a crack propagates, it may encounter obstacles and other cracks. Here the crack has propagated from the left and is viewed growing into the material in this transverse section.At the right it has encountered a void evidentally created by an internal crack along one end of a crystal. The void blunted the crack and prevented it from growing further.
Metal Fatigue
One of the most useful properties of metals is that they can be bent or stretched and they will spring back to their original shape. Even if they take a "permanent set"" they continue to be strong. This allows metals to be formed into shapes and is why they rarely shatter in spite of their great strength. The great lesson of time is that no event is truly reversible no matter how it seems. Even though a piece of metal may spring back, it has been altered by bending. In fact metals are altered by most of their experiences. One economically important aspect of this state of affairs is that repeated deformation of metals causes them to crack and break. This cracking process is called fatigue It is not a property of metals alone. Plastic materials behave in a similar way. Brittle materials such as glasses and ceramics undergo even more complex changes when repeatedly bent or loaded. But, their tendency to simply break from overloading obscures fatigue in them.
In spite of massive amounts of money spent on researching metal fatigue, the actual physics of the subject is poorly understood. This is because of its complexity and inaccessibility (Preparing specimens alters the material prepared. By the time a specimen has been prepares for viewing at the atomic level of resolution, it has undergone an enormous amount of experience.) But, it is also because of the extraordinary similarity of fatigue behavior in different metals and under different circumstances. It is hard to build up a good theory when changing what should be significant variables does not produce a systematic change in the experimental result. Furthermore, fatigue data are noisey; experiments are not as repeatable as good physics would require. This contributes to another factor of the problem that is that no deep understanding is really needed in the short term. Empirical formulas, such as the Coffin-Manson relation often as good enough considering the intrinsic variability of the results. One merely has to determine the values of the coefficients of the desired formula experimentally for the particular material under the particular circumstances. This has kept a lot of experimentalists busy and is why the total bill for fatigue research is so high.
Phenomenology
Fatigue phenomena are surprisingly divers in detail and yet highly repetitive in overall form. People tend to look at different aspects such as fatigue life which is the number of cycles to failure e.g. the number of takeoffs and landings of an airplane or the number of times you can bend and straighten those little metal tabs on the backs of picture frames. Another way of looking at fatigue is through the fatigue growth rate of cracks-- referred to as fatigue crack propagation. One observes a crack and asks how fast it will grow, and how long before the part breaks? A salient example is the burner cans in jet engines. Breakage of one of the burner cans is likely to result in the explosion of the engine. (Commercial planes have armor of under the wings for just this event.) Yet, large cracks are permitted at inspection in burner cans because these cracks grow very slowly. Still another way of looking at fatigue is through crack initiation or nucleation. How long before a crack forms at all. The interesting fact here is that it depends on how hard one looks. The stronger the microscope the more quickly cracks are seen to form. With very powerful electron microscopes, features can be observed in "virgin" material that are cracks if you wish to call them such, but they could also be viewed as roughness, intrusions or extrusions. All of these features are capable of growing into full-size cracks. The interested reader should look at
S. Suresh, Fatigue of Materials, in the Cambridge Solid State Science Series published by Cambridge University Press. There are several editions.
For a review of fine points in crack propagation, the reader should look at
L.Lawson, E.Y. Chen and M. Meshii, "Near-threshold fatigue: a review" in International Journal of Fatigue, volume 21 supplement pages S15-S34 (1999).
Typically the life of a crack begins with some discontinuity ( as small as nanometers in size) in the material that produces a discontinuity in the stress field. This discontinuity tends to grow into a microcrack, a crack with a size measured in microns. In areas of stress concentration, microcracks may be abundant. Since they generally cannot be seen without an electron microscope, they are often regarded as non-present. One of the interesting points of contemplation is that such tiny openings are thermodynamically unstable and should heal themselves. But, they don't. Oxidation, adsorption of gasses, surface crystallographic reconstructions, dislocation backstresses and the migration of less soluble impurities to the crack surface have all been used as arguments for why cracks rarely heal. As microcracks grow, they encounter either obstacles that they cannot easily grow through, e.g. voids, particles and grain boundaries,or else they encounter locally enhanced stresses that speed their growth. Microcracks seem to grow by the same rules as large cracks but in a much more variable environment. Eventually microcracks encounter other microcracks and join together. This coalescence is usually the way a large-enough-to-be-seen crack or engineering crack forms. Once a large crack does form, the mechanisms that promoted the rapid growth of microcracks are less important because the large crack must grow in its entirety. If the crack grows a small protuberance, the rest of the crack may not be able to follow. A phenomenon called crack closure becomes important. Originally this term meant peices of dirt that got wedged into the crack blocking it from cyclically opening and closing. A crack that does not cyclically open and close does not grow. Crack closure has been greatly generalized to the point that some experts have denied its existence. In modeling the life expectancy of parts and civil structures, crack closure remains a useful construct if not always good physics. The fact is that many events including sudden overloads will cause cracks to arrest their growth for long period of time. This has been told to me as one of the reasons Harley-Davidson owners are encouraged to every so often really stomp on their bikes through rapid acceleration.
Modeling Fatigue
Fatigue is a statistical process. Fatigue predictions are thus manifestations of statistics. Questions like "What are the chances the Kinzua railroad bridge will collapse in the next year?"(it did) become rooted in summing the conditioned probabilities that a certain loading will interact to produce a certain stress in the vicinity of a certain crack (real or imagined) to produce a certain stress intensity (a driving force for crack growth) resulting in... Obviously such summations are very complicated and most of the details are guesswork.
In the midst of all this uncertainty is one bright star. It is called the Paris Law. But, since noone has proven its legality, it is properly called the Paris Relation:
This relation says that the crack growth increment per cycle is proportional to the amount the stress intensity changes in a cycle raised to a power. The stress intensity in turn is the product of the stress in which the crack grows times a geometry factor times the square root of the length of the crack (in this case x). This means that the crack's growth rate is proportional to its size raised to a power. Such power law relations have no geometric length scale. Ceteris parabus one could not tell the difference between a movie of a giant crack growing in an oil tanker or a microcrack in a watch spring. This is also a property of fractal geometry and much thought has been spent on it. See for example, M Schroeder, Fractals, Chaos, Power Laws, Freeman, NY (1991). What these things have in common with crack growth is self-similarity, that is, scale independence. At a first glance -- there is not enough space here to tell you how to look -- the case of the fatigue crack seems trivial: One is simply looking at a point in some sort of space and a point looks like a point at any length scale. The problem with this triviality is that stress singularities do not really exist but are fictions of theoretical mechanics. In real life there is plasticity to complicate matters. Yet, like the child who does not know how to ask at Passover, there is a sufficient answer. Detailed knowledge of plasticity is rarely needed. But why? I have some ideas based on my own work, that of Neubers and that of Glinka. You too can guess.
In terms of elementary events, cracks grow, become arrested and join with other cracks. Cracks usually link up on a micro level, but then there was the Aloha incident in which many cracks growing out of rivets suddenly joined together and the fuselage of the Aloha plane burst in mid air. Models of these elementary processes can be incorporated into what is called a Monte Carlo Markov Chain (MCMC) model of a field of many cracks. At the microcrack size level fields of many microcracks are not uncommon. The evolution of these fields amounts to what engineers call crack initiation. The idea is to consider the many sites at which cracks could grow. A computer then tosses dice at each site for each event and tabulates the resulting cracks and their sizes. One could construct a detailed spatial model and track each crack until it grows into another one. An often better approach -- one much less computationally intensive -- is to consider only the density of cracks and their probabilities of intersection based on stochastic geometry. There is a good derivation in stereology that can be used as the basis for this calculation. I have written many papers with my student, Ed Chen, on this type of model. They are listed with my publications. Most are quite readable. They don't even use the term, Monte Carlo Markov Chain, because it had not yet become popular. Now, I believe, there is an entire journal devoted to it.
I hope I have given the reader some quick insight. The subject is vast.
Publications
Home
Metal Fatigue
One of the most useful properties of metals is that they can be bent or stretched and they will spring back to their original shape. Even if they take a "permanent set"" they continue to be strong. This allows metals to be formed into shapes and is why they rarely shatter in spite of their great strength. The great lesson of time is that no event is truly reversible no matter how it seems. Even though a piece of metal may spring back, it has been altered by bending. In fact metals are altered by most of their experiences. One economically important aspect of this state of affairs is that repeated deformation of metals causes them to crack and break. This cracking process is called fatigue It is not a property of metals alone. Plastic materials behave in a similar way. Brittle materials such as glasses and ceramics undergo even more complex changes when repeatedly bent or loaded. But, their tendency to simply break from overloading obscures fatigue in them.
In spite of massive amounts of money spent on researching metal fatigue, the actual physics of the subject is poorly understood. This is because of its complexity and inaccessibility (Preparing specimens alters the material prepared. By the time a specimen has been prepares for viewing at the atomic level of resolution, it has undergone an enormous amount of experience.) But, it is also because of the extraordinary similarity of fatigue behavior in different metals and under different circumstances. It is hard to build up a good theory when changing what should be significant variables does not produce a systematic change in the experimental result. Furthermore, fatigue data are noisey; experiments are not as repeatable as good physics would require. This contributes to another factor of the problem that is that no deep understanding is really needed in the short term. Empirical formulas, such as the Coffin-Manson relation often as good enough considering the intrinsic variability of the results. One merely has to determine the values of the coefficients of the desired formula experimentally for the particular material under the particular circumstances. This has kept a lot of experimentalists busy and is why the total bill for fatigue research is so high.
Phenomenology
Fatigue phenomena are surprisingly divers in detail and yet highly repetitive in overall form. People tend to look at different aspects such as fatigue life which is the number of cycles to failure e.g. the number of takeoffs and landings of an airplane or the number of times you can bend and straighten those little metal tabs on the backs of picture frames. Another way of looking at fatigue is through the fatigue growth rate of cracks-- referred to as fatigue crack propagation. One observes a crack and asks how fast it will grow, and how long before the part breaks? A salient example is the burner cans in jet engines. Breakage of one of the burner cans is likely to result in the explosion of the engine. (Commercial planes have armor of under the wings for just this event.) Yet, large cracks are permitted at inspection in burner cans because these cracks grow very slowly. Still another way of looking at fatigue is through crack initiation or nucleation. How long before a crack forms at all. The interesting fact here is that it depends on how hard one looks. The stronger the microscope the more quickly cracks are seen to form. With very powerful electron microscopes, features can be observed in "virgin" material that are cracks if you wish to call them such, but they could also be viewed as roughness, intrusions or extrusions. All of these features are capable of growing into full-size cracks. The interested reader should look at
S. Suresh, Fatigue of Materials, in the Cambridge Solid State Science Series published by Cambridge University Press. There are several editions.
For a review of fine points in crack propagation, the reader should look at
L.Lawson, E.Y. Chen and M. Meshii, "Near-threshold fatigue: a review" in International Journal of Fatigue, volume 21 supplement pages S15-S34 (1999).
Typically the life of a crack begins with some discontinuity ( as small as nanometers in size) in the material that produces a discontinuity in the stress field. This discontinuity tends to grow into a microcrack, a crack with a size measured in microns. In areas of stress concentration, microcracks may be abundant. Since they generally cannot be seen without an electron microscope, they are often regarded as non-present. One of the interesting points of contemplation is that such tiny openings are thermodynamically unstable and should heal themselves. But, they don't. Oxidation, adsorption of gasses, surface crystallographic reconstructions, dislocation backstresses and the migration of less soluble impurities to the crack surface have all been used as arguments for why cracks rarely heal. As microcracks grow, they encounter either obstacles that they cannot easily grow through, e.g. voids, particles and grain boundaries,or else they encounter locally enhanced stresses that speed their growth. Microcracks seem to grow by the same rules as large cracks but in a much more variable environment. Eventually microcracks encounter other microcracks and join together. This coalescence is usually the way a large-enough-to-be-seen crack or engineering crack forms. Once a large crack does form, the mechanisms that promoted the rapid growth of microcracks are less important because the large crack must grow in its entirety. If the crack grows a small protuberance, the rest of the crack may not be able to follow. A phenomenon called crack closure becomes important. Originally this term meant peices of dirt that got wedged into the crack blocking it from cyclically opening and closing. A crack that does not cyclically open and close does not grow. Crack closure has been greatly generalized to the point that some experts have denied its existence. In modeling the life expectancy of parts and civil structures, crack closure remains a useful construct if not always good physics. The fact is that many events including sudden overloads will cause cracks to arrest their growth for long period of time. This has been told to me as one of the reasons Harley-Davidson owners are encouraged to every so often really stomp on their bikes through rapid acceleration.
Modeling Fatigue
Fatigue is a statistical process. Fatigue predictions are thus manifestations of statistics. Questions like "What are the chances the Kinzua railroad bridge will collapse in the next year?"(it did) become rooted in summing the conditioned probabilities that a certain loading will interact to produce a certain stress in the vicinity of a certain crack (real or imagined) to produce a certain stress intensity (a driving force for crack growth) resulting in... Obviously such summations are very complicated and most of the details are guesswork.
In the midst of all this uncertainty is one bright star. It is called the Paris Law. But, since noone has proven its legality, it is properly called the Paris Relation:
This relation says that the crack growth increment per cycle is proportional to the amount the stress intensity changes in a cycle raised to a power. The stress intensity in turn is the product of the stress in which the crack grows times a geometry factor times the square root of the length of the crack (in this case x). This means that the crack's growth rate is proportional to its size raised to a power. Such power law relations have no geometric length scale. Ceteris parabus one could not tell the difference between a movie of a giant crack growing in an oil tanker or a microcrack in a watch spring. This is also a property of fractal geometry and much thought has been spent on it. See for example, M Schroeder, Fractals, Chaos, Power Laws, Freeman, NY (1991). What these things have in common with crack growth is self-similarity, that is, scale independence. At a first glance -- there is not enough space here to tell you how to look -- the case of the fatigue crack seems trivial: One is simply looking at a point in some sort of space and a point looks like a point at any length scale. The problem with this triviality is that stress singularities do not really exist but are fictions of theoretical mechanics. In real life there is plasticity to complicate matters. Yet, like the child who does not know how to ask at Passover, there is a sufficient answer. Detailed knowledge of plasticity is rarely needed. But why? I have some ideas based on my own work, that of Neubers and that of Glinka. You too can guess.
In terms of elementary events, cracks grow, become arrested and join with other cracks. Cracks usually link up on a micro level, but then there was the Aloha incident in which many cracks growing out of rivets suddenly joined together and the fuselage of the Aloha plane burst in mid air. Models of these elementary processes can be incorporated into what is called a Monte Carlo Markov Chain (MCMC) model of a field of many cracks. At the microcrack size level fields of many microcracks are not uncommon. The evolution of these fields amounts to what engineers call crack initiation. The idea is to consider the many sites at which cracks could grow. A computer then tosses dice at each site for each event and tabulates the resulting cracks and their sizes. One could construct a detailed spatial model and track each crack until it grows into another one. An often better approach -- one much less computationally intensive -- is to consider only the density of cracks and their probabilities of intersection based on stochastic geometry. There is a good derivation in stereology that can be used as the basis for this calculation. I have written many papers with my student, Ed Chen, on this type of model. They are listed with my publications. Most are quite readable. They don't even use the term, Monte Carlo Markov Chain, because it had not yet become popular. Now, I believe, there is an entire journal devoted to it.
I hope I have given the reader some quick insight. The subject is vast.