Fractal domains crack

Hammad, Cement and Concrete Research 24 — Carpinteri and B. Chiaia, Materials and Structures 28 — Davidson, Journal of Materials Science 24 — Carpinteri, Mechanics of Materials 18 89— Chiaia, Materials and Structures , to be published. Mandelbrot, Physica Scripta 32 — Carpinteri, International Journal of Solids and Structures 31 — Chudnowski and B.

Kunin, Journal of Applied Physics 62 — Irwin, Journal of Applied Mechanics 24 — Wilson, Physical Review B4 — Mosolov, Europhysics Letters 24 — Carpinteri, International Journal of Fracture 51 — Gong and Z. Lai, Engineering Fracture Mechanics 44 — Williford, Scripta Metallurgica et Materialia 24 — Wittmann, H. Mihashi and N. Nomura, Engineering Fracture Mechanics 35 — Kim, H. Mihashi, K. Kirikoshi and T.

Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Carpinteri, A. Crack-resistance behavior as a consequence of self-similar fracture topologies. Int J Fract 76, — Download citation. Received : 20 April In practical engineering applications, when the corrosion expansion force reaches the ultimate tensile strength of concrete, it is considered that the inner surface of concrete cover begins to crack.

At first, the crack width is small, and the cracking has little influence on the overall strength and stability of concrete. However, once cracks reach the surface of the concrete cover, they provide access for harmful substances from outside to penetrate the concrete cover. Therefore, exploring the complexity of cracks in the initial stage of cracking can provide a basis for researching the transfer rate of harmful substances.

In this research, the distribution area of corrosion cracks and the complexity of cracking pattern I are studied. Pattern L and T can be thought of as being made up of many cracks I, and their characteristics are based on cracking pattern I. Although the random distribution of coarse aggregate causes varied tortuosity of cracks, the distribution of cracks is always within a certain range, which is the area directly above the steel bar.

Figure 9 shows a simplified crack development. As shown in the figure, l represents the spacing between the leftmost and rightmost end of crack development. Table 7 shows the mean value of l for cracking pattern I. In this test, l ranges from 0. The mean value of all the cracked slices for all the specimens is 0. Since the concrete cover thickness is known, the general range of crack development can be roughly estimated. In practice, by focusing the object region above the steel bar, the area of concrete members that may propagate corrosion-induced cracks can be monitored and maintained.

Under the same corrosion ratio, the complexity of corrosion cracks mainly depends on the distribution of coarse aggregate for cracking pattern I. The coarse aggregate will influence the complexity of corrosion cracks from two aspects. The coarse aggregate firstly affects the transfer of an external harmful substance, which leads to the non-uniform corrosion of the steel bar. Secondly, the random distribution of coarse aggregate will increase the tortuosity of cracking propagation.

In comparison with cement mortar, concrete is prone to cracking along the interfacial zone ITZ on the surface of coarse aggregate. A random distribution of coarse aggregate can induce tortuous cracks.

Therefore, it is necessary to furtherly clarify the influence of coarse aggregate on the formation of the corrosion-induced cracking pattern I.

According to whether the cracks have contact with coarse aggregate or not, the cracks in cracking pattern I are divided into cracks I-1 that have no contact with coarse aggregate and cracks I-2 that have contact. Furtherly, according to the contact position of coarse aggregate, cracks I-2 are furtherly specified as cracks I that are in contact with aggregates deep in the concrete cover, cracks I that are in contact with aggregates near the surface of concrete cover, and cracks I that are in contact with both internal and surficial aggregates.

The detailed cracking patterns of these cracks are shown in Figure As can be seen from Figure 10 , compared with the other three types of cracks, the direction of crack I-1 development is single, and the crack width is smaller.

Since cracks I, I, and I are in contact with aggregates in different positions, their tortuosity is larger than cracks I-1, and they develop along the edge of the aggregates. Because cracks cannot penetrate the aggregates, the complexity of their trajectories depends on how irregular the shape of the aggregates is.

The more irregular the shape of aggregate is, the greater the tortuosity of the crack is, and the more difficult it is for harmful substances to penetrate deep into the concrete cover. Because the cracks do not develop along the entire edge of the aggregates, it is not represented to study the shape and size of a single aggregate. In other words, the edge of the aggregates only partly affects the cracking patterns. Cracks I are in contact with aggregates deep in the concrete cover, so their widths are larger than cracks I Thus, the presence of aggregate will increase the degree of cracking.

Due to the low strength of interface transition zone ITZ , concrete cover cracks along ITZ under the corrosion expansion force.

For cracks I in contact with aggregates near the concrete cover surface, they develop a high tortuosity, and their widths near the surface are smaller than that close to the steel bar. This suggests that the cracking rate accelerates significantly when the crack reaches ITZ, which further proves that the aggregate makes the cracking worse.

For cracks I in contact with both internal and surficial aggregates, their widths are larger than cracks I, and their development patterns are more complex. In summary, the development complexity of cracks I are greater than cracks I, and that of cracks I are greater than cracks I The development patterns of cracks I-1 are the simplest of the four types.

For cracks with a single development trend, the presence of coarse aggregate and its surface irregularity greatly affect the development of cracks. The development pattern of corrosion-induced cracks in reinforced concrete has fractal characteristics and obvious self-similarity, which can be described by fractal geometry. The self-similarity of the cracking pattern can be described by fractal dimension D , the value is mainly between 1.

The complexity of the cracking pattern can be described by scale coefficient C , and the value is mainly between 20 and The greater the complexity of the crack, the greater the value of C. For cracks in general, the influence of cracking patterns on the complexity of crack distribution is greater than the coarse aggregate.

Therefore, the cracking pattern is the main factor affecting the complexity of crack distribution in concrete. The presence of aggregate will increase the degree of cracking. The more irregular the aggregate surface is, the greater the tortuosity of the crack is, and the more difficult it is for harmful substances to penetrate deep into concrete cover.

Conceptualization, H. Haodong Ji , H. Haoyu Jiang , and Y. Haodong Ji and H. Haoyu Jiang ; validation, H. Haoyu Jiang and R. Haodong Ji and Y. Haodong Ji ; data curation, H. Haodong Ji and R. Haoyu Jiang ; writing—review and editing, N. All authors have read and agreed to the published version of the manuscript. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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Find articles by Nanguo Jin. Find articles by Jing Tong. Author information Article notes Copyright and License information Disclaimer. Received Aug 4; Accepted Aug Abstract Based on the fractal geometry, a quantitative index describing the development degree of the internal corrosion expansion of reinforced concrete was proposed.

Keywords: reinforced concrete structure, corrosion-induced cracks, fractal geometry, characteristic contour parameter, cracking pattern. Introduction Corrosion cracking is one of the manifestations of durability failure of reinforced concrete structures [ 1 , 2 ].

Experimental Investigation 2. Materials The mix proportion of concrete is listed in Table 1. Open in a separate window. Figure 1. Table 1 Mix proportions of concrete. Element C Mn Si Proportion 0. HPBhot-rolled plain steel bars Table 4 Parameters of the specimens.

Experimental Procedure Accelerated corrosion can greatly shorten the test period on the premise of achieving the target corrosion effect. Figure 2. Analysis of Corrosion-Induced Cracking Pattern in Concrete Corrosion-induced cracks form from the steel-concrete interface.

Figure 3. Table 5 The number of cracking patterns in each specimen. Crack Distribution Described by Fractal Geometry At present, some scholars apply fractal geometry to the study of the fracture section of metal materials [ 38 ] and the research of brittle materials, such as ceramics [ 39 , 40 ] and geotechnical materials [ 41 ]. Figure 4. Analysis of Fractal Characteristics of the Specimens Each specimen was cut into 16 slices.

Figure 5. Figure 6. Figure 7. Analysis of Characteristic Contour Parameter Xu et al. Figure 8. Table 6 Mean values of the parameters. Analysis of Distribution and Types of Corrosion-Induced Cracks In practical engineering applications, when the corrosion expansion force reaches the ultimate tensile strength of concrete, it is considered that the inner surface of concrete cover begins to crack.

Analysis of Distribution Area of Corrosion-Induced Cracks Although the random distribution of coarse aggregate causes varied tortuosity of cracks, the distribution of cracks is always within a certain range, which is the area directly above the steel bar. Figure 9. Table 7 The distribution range of corrosion-induced cracks. Analysis of the Patterns of Corrosion-Induced Cracks Under the same corrosion ratio, the complexity of corrosion cracks mainly depends on the distribution of coarse aggregate for cracking pattern I.

Figure Conclusions The development pattern of corrosion-induced cracks in reinforced concrete has fractal characteristics and obvious self-similarity, which can be described by fractal geometry. Table A2 The scale coefficient C. Author Contributions Conceptualization, H.

Conflicts of Interest The authors declare no conflict of interest. References 1. Michel A. Propagation of steel corrosion in concrete: Experimental and numerical investigations. Alhozaimy A. Investigation of severe corrosion observed at intersection points of steel rebar mesh in reinforced concrete construction.

Wang L. Investigation on chloride penetration into unsaturated concrete under short-term sustained tensile loading. Laurens S. Steady-state polarization response of chloride-induced macrocell corrosion systems in steel reinforced concrete—Numerical and experimental investigations. Zhao Y. Damage analysis and cracking model of reinforced concrete structures with rebar corrosion. Jin W. Science Press; Beijing, China: Steel Corrosion-Induced Concrete Cracking.

Kwon S. Service life prediction of concrete wharves with early-aged crack: Probabilistic approach for chloride diffusion. Before the hammer collides with specimen, the gravitational potential energy of the hammer is converted into kinetic energy completely, and then works on the specimen by colliding with the specimen. One part of the energy is absorbed by the hammer, the aluminium gasket and the force sensor, and the other part is transformed into the impact force doing work on the specimen.

Figure 4 shows the relationship between gravitational potential energy and work done by impact force, where the hammer weight is 4 kg, the falling height changes and the falling height remains unchanged, the hammer weight changes. From Fig. For impact tests of the same height and different mass, the energy consumed by heavy hammer, aluminium gasket and force sensor is relatively less, accounting for about The ultra-dynamic strain gauge records the stress-time signal and strain—time signal of the specimen.

In order to study the failure mechanism of sandstone with different fissure dips and different fissure number under impact loading, according to the principle of drop hammer impact test, the stress-time signal and strain—time signal can be converted into the stress—strain signal of the specimen under impact compression.

The following takes the first impact as an example to illustrate the stress—strain relationship of the specimen under impact loading.

Similar to static loading, the dynamic compression deformation of the sandstone specimen has gone through the typical pore compaction stage, elastic stage, rapid crack development stage and descending section after fracture. Under dynamic impact, the compaction stage of the rock is extremely weak. It can be seen from Fig. The compaction stage of the other samples is not obvious, and they directly enter the elastic deformation stage at the beginning of loading.

Then enter the rapid development stage of the cracks, the stress—strain curve appears to a certain degree of depression, which is caused by the secondary collapse of the pores in the rock when the loaded stress exceeds the yield limit of most pores.

After the elastic stage, a large number of pores collapse, and then arise a stress relaxation platform section. When loaded to the peak strength of the specimen, the specimen begins to fail, the failure process is flexible, and the stress drops slowly. The stress drop rate of the specimen depends on the integrity of the rock mass. When the sandstone contains more fissures, the internal micro-cracks are more likely to converge and nucleate, so it is easier to form penetrating cracks, resulting in the reduction of peak stress.

After unloading, the specimen appears rebound deformation and unrecoverable residual deformation, which is mainly caused by the closing, slipping and dislocation of the structural surface during compression.

The peak stress of the rock can reflect the ability of the rock to resist damage. There are many factors that affect the peak stress of the rock. On the one hand, it is the factor of the rock itself, and on the other hand, it is related to the relative size of the rock specimen, the processing condition and the loading rate.

In order to study the ability of sandstone to resist damage under different fissure forms, the dynamic growth factor is used to reflect the change of the stress growth amplitude of the test block, that is, the dynamic growth factor formula is defined as follows:.

For samples with different fissure dips and different number of fissures, the dynamic growth factors under different working conditions are obtained, as shown in Table 2 , and then plotted in Fig.

The analysis of Table 2 and Fig. From the above analysis, it can be seen that the dynamic peak stress of the yellow sandstone specimens gradually decreases with the increase of the dip between the fissure and the loading section, and with the increase of the fissure dip, the peak stress decreases faster.

In Fig. It can be seen that the more the number of fissures in the rock specimen, the lower the strength, which is consistent with the theory. The dynamic elastic modulus is the tangent modulus of the stress—strain curve, that is, the slope of the straight or close to the middle of the stress—strain curve, which can be used to characterize the deformation properties of the rock.

The decrease of the dynamic elastic modulus indicates that the strength of the rock is attenuating, and the degree of microcracks and damage is gradually increasing. Figure 7 shows the relationship between the dynamic elastic modulus and the fissure dip and the number of fissures.

The dynamic elastic modulus decreases with the increase of the fissure dip and the number of fissures, indicating that with the increase of the fissure dip and the number of fissures, the more prone the sandstone sample is to produce microcracks and fracture behavior under the load.

Compared with the dip of the fissures, the number of fissures has a greater impact on the dynamic elastic modulus of the rock. When there are a lot of micro-cracks, the dynamic mechanical properties of the rock will be greatly attenuated. In actual engineering, especially when there is dynamic disturbance, special attention should be paid to the impact of cracks on engineering safety.

Natural rock materials usually contain different forms of initial defects, which often change the mechanical properties and failure mechanism of rocks. When they are subjected to external loads of different degree, the internal defects of rock will continue to expand and evolve, which will lead to the deterioration of the bearing capacity of rock mass. Damage and fracture of rocks are the fundamental reasons of rock mass instability and various geological hazards. It is of great theoretical and engineering significance to study the damage and fracture process of rock materials under different fissure forms for predicting and evaluating the stability of engineering rock mass scientifically and accurately, and preventing the occurrence of major engineering geological hazards.

Therefore, in order to study the effect of fissure morphology on the dynamic mechanical properties and crack propagation modes of rock materials, specimens with different fissure forms were processed by sandstone, and impact loading tests were carried out on drop hammer impact testing machine.

In the tests, a hammer weight of 5 kg with a falling height of 2 m was selected, and the dynamic failure process of sandstone surface cracks during the test was recorded, as shown in Fig. Crack propagation process of specimens with different fissure numbers in Fig. From Table 3 , the longitudinal splitting cracks of intact specimens and one-fissure specimens start from the loading surface, with the initiation stress of 7.

And the longitudinal splitting cracks of two-fissure specimens and three-fissure specimens start from the middle or near the middle of the prefabricated fissures, with the initiation stress of 2. It indicates that the dynamic compressive strength of the specimens with one fissure is lower than that of the complete specimens, but the reduction is small. The dynamic compressive strength of the specimens with two and three fissures is significantly lower than that of the complete specimen, and the dynamic compressive strength of the specimens with three fissures is smaller than that of the specimens with two fissures, but the difference is not significant.

With the increase of impact loading, shear wing cracks appear at or near the tip of the splitting crack, following far-field cracks occur. Splitting cracks and shear cracks continue to expand and extend, and their widths also increase. When the stress—strain curve reaches the post-peak stage, energy concentrates around the cracks and gradually releases, resulting in internal crack failure and a large number of rock debris.

Longitudinal splitting cracks penetrate along the loading direction, while or later transverse shear cracks penetrate perpendicularly to the loading direction.

Then the specimen slightly distorts along the crack surface, which eventually leads to instability and failure. The crack initiation direction is parallel to the loading direction for both intact and jointed specimens, and the penetration failure of specimens is caused by tension-shear composite cracks.

The more the number of prefabricated fissures is, the denser the cracks on the surface of the specimen are when failed, and the earlier the failure time is. From the above analysis, the crack propagation process is closely related to the number of prefabricated fissures.

The more the number of prefabricated fissures is, the easier the initial cracks occur, the more dense the cracks are, and the easier the specimen is to destabilize and destroy. In addition, crack propagation process of specimens with different fissure dips in Fig. It was found that the abnormal phenomenon may be related to the sampling position. Whether splitting cracks or shear cracks, they are expanding and extending, and the crack width is also increasing.

Longitudinal splitting cracks penetrate along the loading direction, while or later shear cracks penetrate staggeringly, resulting in instability and failure of the specimens. This is because the main failure mode of specimens under impact loading is splitting failure. In summary, the crack propagation process is closely related to the prefabricated fissure dip.

With the increase of the dip, the main crack gradually transits from splitting-shear crack to tension splitting crack. Fractal dimension is an important parameter for describing fractal, which can reflect the basic characteristics of fractal.

Fractal characteristics of crack distribution can be obtained by fractal calculation of crack propagation and evolution process on sandstone surface. It is very helpful for further understanding the failure mechanism of rock under impact loading and putting forward reasonable precursor criterion of rock failure.

With the different application of fractal, there are many definitions and calculation methods of fractal dimension. Similar dimension, Houston dimension, capacity dimension and box-counting dimension are common Deng et al.

This paper mainly calculates the fractal dimension of the surface crack image of the sample taken. Therefore, it mainly introduces the calculation method of the box-counting dimension of the two-dimensional digital image. Using the image processing and numerical calculation function of MATLAB, firstly, the image of surface crack of the sample is processed by gray level and binarization, and the related data is stored.

Then the binary image is covered by a square box with the size of edge length r. The number of square boxes N r in the destroyed area of rock samples is counted, and the relevant data are saved.

Among them, the relationship between the square edge length r and the number of square blocks N r is shown in Eq. Fractal dimension D can be expressed as Eq. In this section, we mainly study the variation of fractal dimension of crack propagation process on sandstone surface under different fissure number and fissure dip. And the fractal growth model of crack growth process is intended to building-up. Firstly, the surface cracks of sandstone samples are segmented and extracted by using digital image processing technology and MATLAB software.

Figure 9 shows an example of an intact sample. As shown in Fig. It is proved that the selection of appropriate box size is very important in the calculation of box dimension. Among them, the opposite of the slope of the fitting curve is the box dimension. The box dimensions of surface cracks of specimens with different fissure number and different fissure dip are calculated in Tables 5 and 6 respectively.

From Tables 5 , 6 and Fig. Under impact loading, the surface cracks of sandstone have good fractal characteristics, and the fractal dimensions of crack evolution tend to increase with time as a whole. The change of fractal dimension is closely related to energy. In the early stage of loading, the impact energy is larger, and the damage rock subjected is accordingly greater. With the extension of time, the energy is constantly attenuating, and the damage caused by impact loading is smaller.

Beyond a certain time range, the damage of rock caused by impact loading gradually disappears. In summary, under impact loading, the damage of rock is more serious in the early stage than in the later stage, which indicates that the fractal dimension increasing range is more obvious in the later stage than in the earlier stage.

Therefore, fractal dimension can be used as a parameter to indicate the damage degree of rock. The impact process of the test satisfies the momentum-impulse conservation relationship. According to the energy conservation relationship, under the impact test conditions of the same mass and different heights, the energy absorbed by the sandstone sample only accounts for about The energy not absorbed by the sample is consumed by the weight, aluminum gasket and force sensor.

Under the test conditions where the number of fissures is changed, the crack initiation direction is parallel to the loading direction, and the penetration failure of the specimen is caused by the tensile-shear compound crack. Under the test conditions where fissure dip changes, as the dip increases, the main crack gradually transitions from a split-shear crack to a tensile split crack.

In the fractal analysis of crack propagation, according to the idea of box dimension algorithm and the principle of digital image storage, an algorithm of box dimension of digital image based on MATLAB software is designed.

The box dimension of rock surface crack under different test conditions is obtained, and the fractal growth model of crack with time is established. The fractal dimension can be used as a parameter to express the rock damage degree. Liu, Y. Numerical investigation of the dynamic properties of intermittent jointed rock models subjected to cyclic uniaxial compression.

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