Fatigue problem is still very much an empirical art rather than a science, despite being a relatively old subject having nearly 160 years of studying history since (Wöhler, 1860). Many problems in the area of fatigue still remain unsolved. (Cui, 2002) pointed out six important examples, such as the fatigue life prediction under variable amplitude; the corrosion fatigue problem; the combination of various environment fatigue; multiaxial fatigue stress in variable amplitude; and the distribution of fatigue life. Among these unsolved problems, let’s point out an important issue as the main work in this thesis, that is, the transferability of fatigue data from small specimens to actual components or structures, and also from short cracks to long cracks. Many tests have shown that the damage sums to failure of actual components is much lower than those of specimens (size effect). This problem is, intentionally or unintentionally, passed over in silence (Schütz, 1996) for a long time. Some concepts, such as the local-strain approach blindly take transferability for granted, and LEFM ignored the disability in the short crack regime. In these cases, they cannot be used to study scale effects. To begin with this complicate problem, it is worth to mention the essence behind this phenomenon--the scaling laws. To discover the scaling law, as we shall see, that involves two key steps: (a) Identify a set of physically-relevant dimensionless groups, and (b) Determine the scaling exponent for each one. Scaling laws are intimately connected to dimensional analysis. The reason of this relationship is that the scaling laws reveal a deep feature of process: self-similarity. Establishing scaling laws and self-similarity was always considered as an important, sometimes crucially important step in construction of engineering and physical theories. The crucial importance of this remarkable study is the preliminary application of dimensional analysis to obtain the result without solving complicated equations. Dimensional analysis cannot tell us everything we need to know. It will be helpful with step (a), but it cannot possibly be helpful with step (b). In fact, it is less than half the battle. Nevertheless, dimensional analysis has shown some simple examples of the invariance of the intermediate asymptotes with respect to the renormalization group. Fractal Geometry has been widely used for the description of irregular phenomena in various scientific fields recently. In the subjects concerning fracture system characterization, fractals represent the fracture surfaces in 2 or 3D problems. In this study, fatigue limit model is proposed through fractal geometry and proved to be valid by statistical analysis of the experimental data. What’s more, with the aid of recently development of acoustic emission, the fractal theory can also be proposed in rock specimens for compression fatigue tests. In Chapter 2, the basic concepts and development of fatigue are reviewed. Based upon the key characteristics with respect to fracture mechanics, the well-know fatigue crack propagation (FCP) approach and cumulative fatigue damage (CFD) approach are presented in details. On the other hand, a special attention is given to the short crack problem because of the limitation of the linear elastic fracture mechanics (LEFM) to describe the phenomenon. In Chapter 3, a review of the dimensional analysis is presented, with regard to the works by Carpinteri and his group at Politecnico di Torino. The implementation and the development of dimensional analysis have shown to be able to give an indication of the relevant effects on the Paris law and Wöhler's curve. In Chapter 4, firstly, the implementation of the fractal geometry theory succeeded in capturing the size-scale effects in fatigue, which show fractal patterns in the failure process. Then, integrate and extend the results in these works with emphasis on the important engineering issue of fatigue limit, a new proposed fractal model is proposed, which coincides with the existing empirical fatigue limit criteria. This approach also leads to a scale-invariant fractal fatigue limit model which is able to predict the crack-size effect in the Wöhler's curve. Furthermore, based on this, the Paris law and Wöhler's curve are intimately connected by this fractal model. In Chapter 5，statistical analysis of the experimental data for fatigue limit with different crack-size is performed. Firstly, a data analysis is performed to eliminate possible outliers. Next, systematic effects are detected, which is a good evidence of the presence of systematic effects in the most accepted model. Then, regression models are proposed to find a better description of the experimental data, and the analysis of the two families indicates some systematic should be considered to propose the model. Finally, the regression results and the analysis can be a validation of the proposed model in Chapter 4. What’s more, the exponent different from 0.5 can be explained by fractality of fracture surface. In Chapter 6, a fractal theory for the prediction of the damage evolution in fatigue process, by means of the Acoustic Emission (AE) technique is proposed. Fractal theory can be applied not only for tensional fatigue in metals, but also for compression fatigue in brittle materials. According to this method, the damage level of a structure can be estimated from AE data of a reference specimen extracted from the structure and tested up to failure. By AE monitoring of the fracture propagation, it is therefore possible to evaluate the damage level of a structure as well as the time corresponding to final collapse. In Chapter 7, the main conclusions from this research are briefly summarized, and the potential developments are discussed.

Scaling Laws and Fractality in Fatigue Crack Growth / Zhou, Lili. - (2015).

### Scaling Laws and Fractality in Fatigue Crack Growth

#####
*ZHOU, LILI*

##### 2015

#### Abstract

Fatigue problem is still very much an empirical art rather than a science, despite being a relatively old subject having nearly 160 years of studying history since (Wöhler, 1860). Many problems in the area of fatigue still remain unsolved. (Cui, 2002) pointed out six important examples, such as the fatigue life prediction under variable amplitude; the corrosion fatigue problem; the combination of various environment fatigue; multiaxial fatigue stress in variable amplitude; and the distribution of fatigue life. Among these unsolved problems, let’s point out an important issue as the main work in this thesis, that is, the transferability of fatigue data from small specimens to actual components or structures, and also from short cracks to long cracks. Many tests have shown that the damage sums to failure of actual components is much lower than those of specimens (size effect). This problem is, intentionally or unintentionally, passed over in silence (Schütz, 1996) for a long time. Some concepts, such as the local-strain approach blindly take transferability for granted, and LEFM ignored the disability in the short crack regime. In these cases, they cannot be used to study scale effects. To begin with this complicate problem, it is worth to mention the essence behind this phenomenon--the scaling laws. To discover the scaling law, as we shall see, that involves two key steps: (a) Identify a set of physically-relevant dimensionless groups, and (b) Determine the scaling exponent for each one. Scaling laws are intimately connected to dimensional analysis. The reason of this relationship is that the scaling laws reveal a deep feature of process: self-similarity. Establishing scaling laws and self-similarity was always considered as an important, sometimes crucially important step in construction of engineering and physical theories. The crucial importance of this remarkable study is the preliminary application of dimensional analysis to obtain the result without solving complicated equations. Dimensional analysis cannot tell us everything we need to know. It will be helpful with step (a), but it cannot possibly be helpful with step (b). In fact, it is less than half the battle. Nevertheless, dimensional analysis has shown some simple examples of the invariance of the intermediate asymptotes with respect to the renormalization group. Fractal Geometry has been widely used for the description of irregular phenomena in various scientific fields recently. In the subjects concerning fracture system characterization, fractals represent the fracture surfaces in 2 or 3D problems. In this study, fatigue limit model is proposed through fractal geometry and proved to be valid by statistical analysis of the experimental data. What’s more, with the aid of recently development of acoustic emission, the fractal theory can also be proposed in rock specimens for compression fatigue tests. In Chapter 2, the basic concepts and development of fatigue are reviewed. Based upon the key characteristics with respect to fracture mechanics, the well-know fatigue crack propagation (FCP) approach and cumulative fatigue damage (CFD) approach are presented in details. On the other hand, a special attention is given to the short crack problem because of the limitation of the linear elastic fracture mechanics (LEFM) to describe the phenomenon. In Chapter 3, a review of the dimensional analysis is presented, with regard to the works by Carpinteri and his group at Politecnico di Torino. The implementation and the development of dimensional analysis have shown to be able to give an indication of the relevant effects on the Paris law and Wöhler's curve. In Chapter 4, firstly, the implementation of the fractal geometry theory succeeded in capturing the size-scale effects in fatigue, which show fractal patterns in the failure process. Then, integrate and extend the results in these works with emphasis on the important engineering issue of fatigue limit, a new proposed fractal model is proposed, which coincides with the existing empirical fatigue limit criteria. This approach also leads to a scale-invariant fractal fatigue limit model which is able to predict the crack-size effect in the Wöhler's curve. Furthermore, based on this, the Paris law and Wöhler's curve are intimately connected by this fractal model. In Chapter 5，statistical analysis of the experimental data for fatigue limit with different crack-size is performed. Firstly, a data analysis is performed to eliminate possible outliers. Next, systematic effects are detected, which is a good evidence of the presence of systematic effects in the most accepted model. Then, regression models are proposed to find a better description of the experimental data, and the analysis of the two families indicates some systematic should be considered to propose the model. Finally, the regression results and the analysis can be a validation of the proposed model in Chapter 4. What’s more, the exponent different from 0.5 can be explained by fractality of fracture surface. In Chapter 6, a fractal theory for the prediction of the damage evolution in fatigue process, by means of the Acoustic Emission (AE) technique is proposed. Fractal theory can be applied not only for tensional fatigue in metals, but also for compression fatigue in brittle materials. According to this method, the damage level of a structure can be estimated from AE data of a reference specimen extracted from the structure and tested up to failure. By AE monitoring of the fracture propagation, it is therefore possible to evaluate the damage level of a structure as well as the time corresponding to final collapse. In Chapter 7, the main conclusions from this research are briefly summarized, and the potential developments are discussed.##### Pubblicazioni consigliate

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`https://hdl.handle.net/11583/2606171`

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