Fracture Of Plain Concrete Beams Via Fractals

Abstract

The quantitative description of rough fracture surfaces of concrete has been an important challenge for many years. Looking at the fracture surface of a concrete specimen, one realizes that the self-affine geometry of crack faces results from the stochastic nature of the crack growth. This is due to the heterogeneous nature of concrete that makes the crack tortuous leading its way through weak bonds, voids, mortar and getting arrested on encountering a hard aggregate forming crack face bridges. These mechanisms contribute to the tendency of the crack to follow a tortuous path. The self-similarity contained in the tortuous fracture surface of concrete makes it an ideal candidate to be considered as a fractal. Further, the softening response itself has been treated as a singular fractal function by earlier investigators. The very process of cracking and microcracking, could be considered very close to the stick and slip process and therefore as a fractal. Therefore modeling a crack as a fractal and characterizing it by a fractal dimension have become the focus of research in recent years. Due to randomly distributed discontinuous flaws and high heterogeneity of the internal structure of concrete, mechanical properties also randomly vary. Under the effect of the same external force, the stress intensity factors to which different points in the concrete are subjected are different. Hence the microcracks induced by the external force are distributed discontinuously and randomly. Therefore in the present study the effect of the random nature of the microcracks in the fracture process zone of concrete is investigated using both fractal and probabilistic approach. The most probable fractal dimension of a network of micro cracks is obtained as a function of the branching angle ‘α’ of the microcracks, considered as a random variable. Further, an ensemble of cracks is synthetically generated using Monte Carlo technique imposing a constraint that the random deviations do not exceed the maximum size of the aggregate. Such tortuous cracks are analyzed by extending Fictitious Crack Model (FCM) proposed by Hillerborg et al [37]. A numerical study is carried out to examine the influence of certain important fracture parameters on the beam response of plain concrete beams. The contents of this thesis are organized in seven chapters with references at the end. Chapter-1 summarizes the historical development of fracture mechanics. A brief review of the basic concepts of fracture mechanics theory is presented. In chapter-2 a brief review of literature on fracture mechanics of concrete is presented. An overview of the analytical models, numerical models and fractal models till date has been presented in a systematic way. In chapter-3 the fracture processs zone has been modeled as a fractal following the work of Ji et al [118]. The contribution here has been to improve the work of Ji et al [118] (which considers the region of microcracks as a fractal tree) by considering the branching angle as a random variable. Mean fractal dimension thus obtained is found to match well with the experimental results available in the literature. In chapter-4 FCM, as proposed by Hillerborg et al [37] has been modified to be applicable to cracks with varying inclined faces by considering both horizontal and vertical components of the closing forces. The theoretical aspects of the modified FCM have been described in detail. The procedure for the determination of influence co- efficient matrices for a random tortuous crack in mode-I and mixed-mode along with a fractal crack has been explained. In the subsequent chapters the study has been taken up in two parts. In the first part only one generator of the fractal tree considered by Ji et al [118] has been analyzed by FCM to obtain load-deformation responses and fracture energy. In part two, a random tortuous crack, as already defined earlier has been analyzed both in mode-I and mixed mode using FCM. In chapter-5 plain concrete beams with one generator of fractal tree has been analyzed. The influence of the branching angle on the post-peak response of (P-δ) curves and fracture energy has been obtained. In chapter-6 a random tortuous crack has been analyzed in mode-I by FCM. The analysis reveals the influence of maximum aggregate size upon the pre and post-peak behaviour in support of the experimental findings. The nominal stress at peak is found to depend on the characteristic dimension of the structure thereby confirming the size effect. Further fracture energy values have been obtained by the work of fracture method and the results show good agreement with the results obtained in the literature. In chapter-7 a random tortuous crack has been analyzed in mixed mode by FCM. While modeling, symmetry has been assumed only to facilitate computational work though it is known that loss of symmetry affects the peak load. However analysis of the whole beam can be handled by the code developed in the thesis In chapter-8 a summary of the research work is presented along with a list of major observations and references at the end.