Abstract
The modern methods of materials (including cement matrix materials) design and testing impose the application of an approach appropriate to materials engineering. A quantitative description of the association between the properties of these materials and their structure is a necessity. What remains the scientific aim, however, is the clarification and description of the occurring phenomena by means of models mapping their actual behavior in the closest way possible. The article presents a cracking fractal model based on tests on the morphology of concrete fracture surfaces. The recorded fractal nature of the cracking of cement matrix materials enabled fractal geometry in the model development to be applied. Owing to the application of statistical analysis, together with an extensive base of data on the profile lines separated out of the real fracture surfaces of concrete, it was possible to develop a cracking fractal model. Not only does this model satisfy the condition of the equality of the fractal dimension of the real and model profile lines, it also offers the possibility of introducing an order to the apparently chaotic phenomena, such as the cracking process. An advantage and novelty of the model is that unlike the other authors’ proposals, there is a possibility of reaching an infinitely large number of solutions for model profile lines, which approximates the model to the real-life scenario. The results of fractal tests were supplemented with strength measurements, identifying concrete’s compressive and fracture toughness (determining the critical stress intensity factor KIcS). A connection between the fractal dimension and the investigated properties of concrete was demonstrated. A higher fractal dimension was observed in the profile lines separated out of the fracture surfaces of concretes of higher water–cement ratio. The advantages of the model include the simplicity and applicability in model studies on other materials of the cement matrix.
Highlights
Apart from the traditional approach appropriate for fractography to the analysis of the morphology of cement composites fracture surfaces or the lines separated out of them, attempts were made to use the possibilities offered by fractal geometry developed in the 1980s in research
The scientist regarded as the pioneer of the new branch of mathematics–fractal geometry is Benoit Mandelbrot, a French mathematician born in Warsaw in 1924
This paper presents a proposal of a fractal cracking model, based on the analysis of the statistical results of real-life fractal measurements performed on the profile lines, separated out of metakaolinite modified concrete fracture surfaces
Summary
Apart from the traditional approach appropriate for fractography to the analysis of the morphology of cement composites fracture surfaces or the lines separated out of them, attempts were made to use the possibilities offered by fractal geometry developed in the 1980s in research. Fractal procedures enable the modeling and examination of the growth of bacteria or plants, description of chaotic processes (e.g. weather forecasting) or the analysis of the stock market They are employed in genetic tests for the analysis of DNA chains and in music for the analysis of self-similar harmonic structures. Obtained two values of fractal dimension: at a smaller measurement step ~2.10–2.13, at a bigger one ~2.03 Apart from their multi-fractal character, cement composites fractures exhibit the features of self-similarity. Researchers indicate a correlation between the microstructure of the materials of the cement matrix and their mechanical parameters, as well as the properties connected with water or aggressive media conveyance [21,22] They use fractal geometry in the modeling of the structure of these materials [23,24]. This paper presents a proposal of a fractal cracking model, based on the analysis of the statistical results of real-life fractal measurements performed on the profile lines, separated out of metakaolinite modified concrete fracture surfaces
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