Response to discussion on "the fractal effect of irrergularity of crack branching on the fracture toughness of brittle materials"

Xie, Heping

doi:10.1007/bf00012381

Cited by 7 publications

(5 citation statements)

References 5 publications

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“…This model also shows the sufficiently different effects of fractal crack propagation on the local (or fractal) static stress intensity factor for various 'structure parameters' (d/Aa and the kinking angle, see Table 2): the monotonous relative 'unloading' in a case of d/Aa = 0.1 and non-monotonous dependence for d/Aa = 0.01 with the relative 'overloading' starting from kinking angles more than 60 ° . These results explain well the experimental phenomena that for different materials the static toughness (Kc) may increase or decrease during crack branching and kinking [34].…”

Section: Discussionsupporting

confidence: 72%

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Xie

Sanderson

1996

Int J Fract

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Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity Vo, microstructure parameter d/Aa (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path O. For fractal crack propagation, the apparent (or measured) crack velocity Vo, cannot approach the Rayleigh wave speed Cr. Why Vo is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstmcture parameter (d/Aa), fractal dimension D, and fractal kinking angle of crack extension path O. This is in good agreement with experimental findings. I n t r o d u c t i o nStudies of crack propagation in a material under dynamic and static loading are of considerable importance, but are very difficult. Much of the previous research on crack propagation has been reviewed by Freund [1]. A number of critical issues, particularly some of the discrepancies between experimental findings and theoretical predictions, however, remain to be fully resolved [2]. For example, observed maximum crack velocities, Vo, in most materials are significantly lower than (typically about half of), the theoretically predicted terminal crack velocity (the Rayleigh wave speed Cr). Under high rates of loading with the large stresses, the dynamic stress intensity factor, K ( L ( D , t), V), predicted from theory agrees with the experimentally measured result only up to the time of crack propagation. A probable reason for the differences between observed and theoretical results is that current theories of dynamic fracture assume crack extension along a straight line, whereas, crack propagation is usually along an irregular trace, producing rough fracture surfaces [2][3][4][5][6][7]. Watanabe [6], Goldstein and Lewandowski [7], Guo [2] and Xie and Sanderson [23] have considered the effects of the roughness of crack surfaces on crack extension velocity and on dynamic stress intensity. Guo [2]proposed a wave-crack model to describe the effects of curved crack propagation. Such results indicate that it is difficult to obtain the quantitative agreement between theoretical and experimental results using current theories of dynamic fracture.The theory of fractals has been introduced in many sciences since Mandelbrot [8]. Whereas classical geometry provides an approximation to the structure of physical objects, fractal geometry is an extension of it, and it can be used to describe precisel...

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Section: Discussionsupporting

confidence: 72%

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Xie

Sanderson

1996

Int J Fract

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“…This relationship was validated by Gao et al [14] through their investigation on the performance of concrete containing carbon nanotubes at high temperatures. As shown in Figure 26, the   lg ln r T M M r  curves of plain concrete (PC) exhibited a linear growth relationship under different working conditions.The high correlation coefficients (R 2 > 0.9) signify that the crushing distribution of the sample conforms to the fractal principles[156]. Furthermore, as shown in Figure27, the fractal dimension f D of modified concrete (MC) is lower than that of PC.…”

mentioning

confidence: 77%

“…Evidently, the degree of concrete material cracking after exposure to high temperatures is closely related to the size and shape of the specimen. However, in contrast to other crack parameters, the fractal dimension c The high correlation coefficients (R 2 > 0.9) signify that the crushing distribution of the sample conforms to the fractal principles [156]. Furthermore, as shown in Figure 27, the fractal dimension f D of modified concrete (MC) is lower than that of PC.…”

Section: Correlation Between Pore Structure and High-temperature Resi...mentioning

confidence: 94%

Fractal Analysis of Cement-Based Composite Microstructure and Its Application in Evaluation of Macroscopic Performance of Cement-Based Composites: A Review

Zhang,

Ding,

Guo

et al. 2024

Fractal Fract

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Cement-based composites’, as the most widely used building material, macroscopic performance significantly influences the safety of engineering structures. Meanwhile, the macroscopic properties of cement-based composites are tightly related to their microscopic structure. The complexity of cement-based composites’ microscopic structure is challenging to describe geometrically, so fractal theory is extensively applied to quantify the microscopic structure of cement-based composites. However, existing studies have not clearly defined the quantification methods for various microscopic structures in CCs, nor have they provided a comprehensive evaluation of the correlation between the fractal dimensions of different microscopic structures and macroscopic performance. So, this study categorizes the commonly used testing methods in fractal theory into three categories: particle distribution (laser granulometry, etc.), pore structure (mercury intrusion porosity, etc.), and fracture (computed tomography, etc.). It systematically establishes a detailed process for the application of testing methods, the processing of test results, model building, and fractal dimension calculation. The applicability of different fractal dimension calculation models and the range of the same fractal dimension established by different models are compared and discussed, and the advantages and disadvantages of different models are analyzed. Finally, the research delves into an in-depth analysis of the relationship between the fractal dimension of cement-based composites’ microscopic structure and its macroscopic properties, such as compressive strength, corrosion resistance, impermeability, and high-temperature resistance. The principle that affects the positive and negative correlation between fractal dimension and macroscopic performance is discussed and revealed in this study. The comprehensive review in this paper provides scholars with methods and models for quantitative research on the microscopic structural parameters of cement-based composites and offers a pathway for the non-destructive assessment of the macroscopic performance of cement-based composites.

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“…So there must be a minimum value of measurement scale δ c . At present, a standard of minimum value of measurement scale δ c is given by Xie [41,42] that the grain size in the rock should be regarded as the smallest scale. Here, δ c should be 0.1 mm, which is determined by the average grain size of black sandstone applied in the experiment.…”

Section: Fractal Methodsmentioning

confidence: 99%

Rock Dynamic Crack Propagation Behaviour and Determination Method with Improved Single Cleavage Semi-circle Specimen Under Impact Loads

Wang

Zhu

et al. 2020

Acta Mech. Solida Sin.

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This paper studied the rock dynamic fracture propagation under impact loads elaborately with a determination method proposed to calculate crack propagation dynamic stress intensity factor (DSIF). By utilizing the split-Hopkinson pressure bar, the impact experiments with an improved single cleavage semicircle (ISCSC) specimen were conducted to illuminate the dynamic crack propagation behaviour. Meanwhile, the fracture characteristics and crack propagation velocity were obtained by the crack propagation gauges. Coordinating experiments with a numerical approach, the crack propagation dynamic stress intensity factors were calculated by an experimental-numerical method with fractal theory. Then, a finite difference model was developed based on the tensile fracture softening damage criterion. With the analysis of numerical and experimental results, the crack propagation behaviour and mechanism of crack arrest were discussed sophisticatedly. The results demonstrate that the novel ISCSC specimen shows a definite advantage in determining crack propagation and arrest DSIF. Additionally, the crack arrest DSIF is larger than the average propagation DSIF with a sharp increase. Meanwhile, the numerical simulation results which agree well with the actual crack propagation illustrate that the crack arrest should be dominated by the compressive stress perpendicular to the crack path, and there were several arrest pauses existing in the transitory crack arrest process.

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Response to discussion on "the fractal effect of irrergularity of crack branching on the fracture toughness of brittle materials"

Cited by 7 publications

References 5 publications

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Fractal Analysis of Cement-Based Composite Microstructure and Its Application in Evaluation of Macroscopic Performance of Cement-Based Composites: A Review

Rock Dynamic Crack Propagation Behaviour and Determination Method with Improved Single Cleavage Semi-circle Specimen Under Impact Loads

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