Multifractal Scaling Law Research Articles

Anisotropic size effect of 3D printed LC3-based engineered cementitious composites (LC3-ECC)

One of the critical factors affecting the development of 3D concrete printing (3DCP) is the challenging issue of in-situ reinforcement. Engineered Cementitious Composites (ECC) are printable self-reinforced materials, and their application in 3DCP technology can facilitate de-steelisation. However, traditional ECC materials pose certain concerns for sustainable environmental development. The integration of Limestone Calcined Clay Cement (LC3) suitable for 3DCP into ECC (LC3-ECC) presents an ideal solution to the aforementioned challenges. Moreover, conducting comprehensive research into its size effects is an essential step from experimentation to engineering application. A thorough exploration of the size effects of 3D printed LC3-ECC in compression, shear, and flexural fracture aspects is conducted to derive size effect laws. The relationship between the anisotropic mechanical properties and size effects has been established. The mold-cast LC3-ECC precisely follows the Bažant size effect law (SEL). The fiber orientation effect enhances bridging efficiency, leading to the fact that the mechanical properties of certain printed LC3-ECC are hardly affected by the size effect. The mechanical performance of some printed LC3-ECC depends on interlayer strength, making its primary reason for size effect not in the Fracture Process Zone (FPZ) energy release, hence closer to the Multifractal Scaling Law (MFSL). LC3-ECC exhibits weak anisotropy in mechanical properties, showing satisfactory stability. However, under flexural fracture behavior, the anisotropic size effect is higher, necessitating careful consideration in structural design.

Multifractal scaling analyses of the spatial diffusion pattern of COVID-19 pandemic in Chinese mainland

Revealing spatio-temporal evolution regularity in the spatial diffusion of epidemics is helpful for preventing and controlling the spread of epidemics. Based on the real-time COVID-19 datasets by prefecture-level cities, this paper is devoted to exploring the multifractal scaling in spatial diffusion pattern of COVID-19 pandemic and its evolution characteristics in Chinese mainland. The ArcGIS technology and box-counting method are employed to extract spatial data and the least square regression based on rescaling probability (μ-weight method) is used to calculate fractal parameters. The results show multifractal distribution of COVID-19 pandemic in China. The generalized correlation dimension spectrums are inverse S-shaped curves, but the fractal dimension values significantly exceed the Euclidean dimension of embedding space when moment order q«0. The local singularity spectrums are asymmetric unimodal curves, which slant to right. The fractal dimension growth curves are shown as quasi S-shaped curves. From these spectrums and growth curves, the main conclusions can be drawn as follows: First, self-similar patterns developed in the process of COVID-19 pandemic, which seems to be dominated by multifractal scaling law. Second, the spatial pattern of COVID-19 across China can be characterized by global clustering with local disordered diffusion. Third, the spatial diffusion process of COVID-19 in China experienced four stages, i.e., initial stage, the rapid diffusion stage, the hierarchical diffusion stage, and finally the contraction stage. This study suggests that multifractal theory can be utilized to characterize spatio-temporal diffusion of COVID-19 pandemic, and the case analyses may be instructive for further exploring natural laws of spatial diffusion.

Effect of water-cement ratio and size on tensile damage in hardened cement paste: Insight from peridynamic simulations

This paper uses the improved bond-based peridynamics (BB PD) method to predict the crack evolution and macroscopic mechanical properties of the hardened cement paste at the micro-scale. Specifically, the prototype micro-elastic brittle (PMB) model is improved by considering an attenuation kernel function and the surface effect correction. Subsequently, the uniaxial tension simulations of hardened cement microstructures generated by μic software are systemically conducted to investigate the influences of the heterogeneity of microstructure, size effect, and water-cement ratio (w/c) on crack growth and mechanical properties, i.e., tensile strength and Young's modulus. The results show that the heterogeneity of the microstructure inevitably will cause the discreteness of the macroscopic properties of the numerical cement paste. Therefore, a statistical method is considered to analyze the mechanical properties of the hardened cement. Moreover, the tensile strength has a remarkable size effect, with the size increase of microstructure, the tensile strength decreases; the simulation results fit well with the existing analytical size effect model, i.e., Weibull size effect model, multifractal scaling law, and Bažant's size effect law. Besides, the higher ratio of w/c is likely to result in lower tensile strength, Young's modulus and premature damage to the cement paste. The predicted macroscopic mechanical properties are in good agreement with the previous simulation and experimental data, which demonstrates the reliability of the improved bond-based peridynamic method in studying the crack initiation and propagation of hardened cement paste.

Scaling Laws for Partially Developed Turbulence

We formulate multifractal models for velocity differences and gradients which describe the full range of length scales in turbulent flow, namely: laminar, dissipation, inertial, and stirring ranges. The models subsume existing models of inertial range turbulence. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from their inertial range values. In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which inertial range study can not address. In the ranges of partially developed turbulence, the flow can be far from universal, so that standard inertial range turbulence scaling models become inapplicable. The model proposed here serves as a replacement. Details of the fitting of the parameters for the τp and ζp models in the dissipation range are discussed. Some of the behavior of ζp for larger p is unexplained. The theories are verified by comparing to high resolution simulation data.

Multi-Fractal Scaling Law applied to VHCF with an emphasis on statistical fluctuations

In recent decades, ultrasonic testing machines have allowed to investigate the very-high cycle fatigue (VHCF) range, featuring feasible gigacycle fatigue testing in a very short time, although the influence of frequency is still partly controversial. The fatigue behaviour depends on the random distribution of initial defects, whatever the stress range amplitude is. As a consequence, the structural size comes into play and the so called ”size-effects” are usually observed. On the other hand, the scaling is not uniform when sufficiently large dimensional ranges are analysed. To this aim, the multi-fractal formalism was adopted to explain the observed specimen-size effect in the VHCF regime when a wide dimensional range is investigated. In addition, the statistical dispersion of experimental results was accounted by adopting different probabilistic functions. Among them, the Generalized Extreme Value Distribution Type-I was selected. In this way, it was possible to derive the analytical relationship for probabilistic-scale dependent P-S-N-b curves.Subsequently, the proposed model was used to analyse the experimental data obtained in an experimental campaign carried out at Politecnico di Torino by the present authors. More in detail, ultrasonic fully reversed tension–compression fatigue tests in the very-high cycle fatigue (VHCF) range were conducted on a set of hourglass and dog-bone specimens made of EN AW-6082 aluminium alloy, with a diameter in the middle cross-section ranging from 3 mm up to 30 mm.The comparison between the theoretichal model and the experimental data allowed to demonstrate the ability of Multi-Fractal Scaling Law (MFSL) to provide objective values of the fatigue strength of full-size components subjected to VHCF and to guarantee safe structural design.

Multifractal scaling analyses of urban street network structure: The cases of twelve megacities in China

Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory.

Clebsch confinement and instantons in turbulence

The turbulence in incompressible fluid is represented as a field theory in 3 dimensions. There is no time involved, so this is intended to describe stationary limit of the Hopf functional. The basic fields are Clebsch variables defined modulo gauge transformations (symplectomorphisms). Explicit formulas for gauge invariant Clebsch measure in space of generalized Beltrami flow compatible with steady energy flow are presented. We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the “instantons and intermittency” program we started back in the 1990s.1 These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier–Stokes equation smears this delta function to the Gaussian with width [Formula: see text] at [Formula: see text] with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation [Formula: see text] around fixed loop [Formula: see text] in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations2 including pre-exponential factor [Formula: see text]. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions [Formula: see text] can be fitted at small [Formula: see text] to the parabola, coming from the Liouville theory with two parameters [Formula: see text], [Formula: see text]. At large [Formula: see text] the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of [Formula: see text] in agreement with DNS.

Evaluation of Size-Effect Models for High-Strength Steel Fiber–Reinforced Concrete

Abstract In an effort to mitigate crack growth, nine high-strength steel fiber–reinforced concrete (HSSFRC) mixtures were designed in this study, and specimens with various sizes were cast and cured. First, a total of 135 cylinder specimens were tested under the load- and displacement-control apparatus. The crack patterns and size-effect models (size-effect law [SEL], modified SEL [MSEL], multi-fractal scaling law [MFSL], and Sim et al.) were then investigated. The results confirm that with the addition of steel fiber, the rate of crack propagation dramatically decreases, indicating that specimens absorb more energy to break. Accordingly, the effect of size in cylindrical specimens is significantly improved, characterizing a size-independent state. Furthermore, the efficiency of SEL, MSEL, and MFSL models is considerably enhanced when steel fibers are added to the mixtures. For nonfibrous high-strength concrete specimens with brittle nature, SEL and the Sim et al. models are found to be of higher efficacy. Besides, the SEL outperforms the others since it is not only compatible with HSSFRC but also reflects all the characteristics of the specimens in various sizes. Moreover, the results demonstrate that the effect of steel fiber on reducing the size effect compared to other fiber types obtained from various researches is quite noticeable. It is also important to note that there is very little difference between the results of 10 % and 20 % fiber reinforcement; therefore, using 10 % steel fiber reinforcement in concrete is recommended.

Size-dependent Behaviour of Weak Intact Rocks

Characterising the size-dependent behaviour of rock has been a significant challenge in rock engineering particularly during the design of structures on or within a rock mass. Generally, the mechanical characterisation of rock starts at the laboratory scale where the intact rock is tested and then the resulting data is extrapolated to the field conditions using a suitable size effect model. Despite extensive research on the size effect of intact rock, very few have included the weak rocks with low uniaxial compressive strength (UCS). Thus, in this study, the size-dependent behaviour of two different weak rocks namely, Gambier limestone and artificial rock with uniaxial compressive strength of less than 10 MPa were investigated experimentally and analytically. The diameter of the cylindrical Gambier limestone samples varied from 26 to 285 mm while the diameter of artificial rock samples ranged between 26 and 139 mm. For Gambier limestone, the uniaxial compressive, Brazilian and point load experiments were carried out while for artificial rock, only the uniaxial compressive tests were conducted. In both rock types, the ascending and then descending size effect trend was a pronounced behaviour for UCS and Young’s modulus data while the size effect behaviour of Poisson’s ratio was inconclusive. Also, the tensile strength and point load index data obtained from Gambier limestone revealed only descending size effect trends. The unified size effect law and its improved version were fitted to the UCS and Young’s modulus data leading to a very good agreement between the data and the model predictions. It was confirmed that an improved version of unified size effect law can predict a more realistic strength value for a sample with an infinite size. Finally, the applicability of two descending size effect models to the tensile strength and point load data was assessed and concluded that the multifractal scaling law is a suitable model for the point load data while the size effect law can better predict the tensile strength data.

Influence of joint anisotropy on the fracturing behavior of a sedimentary rock

Fracture toughness (FT) and tensile strength (TS) indicate a rock's susceptibility to fracturing. Past studies showed that mechanical properties and FT of homogeneous rocks can be correlated well. In the present study, this capability has been extended to the jointed sedimentary rocks. Sandstone with a wide combination of analogue joints were tested in the laboratory, and the control of joint geometrical properties on the FT, TS, and development of fracture process zone (FPZ) were investigated. A FT prediction model was developed based on the TS of the jointed specimens. Multifractal scaling law (MFSL) was utilized to extend the laboratory results to the field scale. Further, the interaction of propagating crack with the already existing joints were investigated and post-peak behavior were studied. Results show that FT and TS decrease with decreasing joint spacing, but they are more sensitive at higher joint spacing. It is also possible to construct a linear prediction model between the FT and TS of rock. FPZ of jointed rock is less sensitive to joint orientation and more related to joint spacing. A negative correlation between the FPZ size and joint spacing is established in this study. Post-peak behavior of the jointed specimens indicate that crack-joint interaction increases the frictional resistance and dissipated fracture energy of the specimens. Further, comparison of mixed-mode fracture criteria with the experimental results show that Maximum Tangential Stress (MTS) criterion can successfully predict the mixed-mode fracture behavior of jointed sandstone.

Mechanical properties of Graphene: Molecular dynamics simulations correlated to continuum based scaling laws

In this paper, the combined effect of domain size, lattice orientation and crack length on the mechanical properties of Graphene, namely the yield strength and strain, are studied extensively based on molecular dynamics simulations. Numerical predictions are compared with the continuum-based laws of size effect and multifractal scaling. The yield strength is found to vary with the specimen size as ≈L−1/3, which is in agreement with the multifractal scaling law, and with the inverse square of the initial crack length as ≈a0-1/2, according to the Griffith’s energy criterion for fracture.

Improved compressive fracture models for self-consolidating concrete (SCC)

This paper investigates the size effect phenomena in compressive strength of self-consolidating concrete (SCC) with different water to cement ratios, and attempts to model the experimental data coming from a series of laboratory unconfined compression tests. To this aim, several concrete cylinders having a constant slenderness ratio of two and diameters ranging from 50mm to 200mm were prepared, and tested under uniaxial compression. In addition, four theoretical models (i.e. size effect law (SEL), modified size effect law (MSEL), Sim et al.’s model, and multifractal scaling law (MFSL)) were investigated on their ability to catch this size effect by regression analysis using the nonlinear least squares method. The experiments indicated a pronounced strength reduction as the specimen size increases. The comparison of the goodness of fit between the models was quantified by using correlation coefficient R which provides that the models can interpret satisfactory the observed size effect.

Multi-fractal scaling law for split strength of concrete cubes

Experiments on concrete members indicate that the nominal strength of a specimen decreases with increasing specimen size for the same specimen geometry. This phenomenon is known as the size effect in the fracture mechanics of plain and reinforced concrete. Although nominal strength is also highly affected by the width of the distributed load in split-tension cylinder and cube specimens, this effect can be negligible within the practical range of the load-distributed width in diagonal cubes. However, the number of reported theoretical and experimental studies on cube and especially diagonal cube split-tension specimens is much more limited than those on cylindrical specimens. Therefore, in this study, six series of cube and diagonal cube specimens, with three different sizes and with two different maximum aggregate sizes (4 mm and 16 mm) but similar geometries, were tested under different load-distributed widths. The peak loads obtained from the test results were analysed using the multi-fractal scaling law. Subsequently, two prediction formulae are proposed for the design of torsional concrete/reinforced concrete members.

Influence of intermittency on the anisotropy of magnetic structure functions of solar wind turbulence

AbstractIntermittency appears to be connected with the spectral anisotropy of solar wind turbulence. We use the Local Intermittency Measure to identify and remove intermittency from the magnetic field data measured by the Ulysses spacecraft in fast solar wind. Structure functions are calculated based on the time sequences as obtained before and after removing intermittency and arranged by time scale (τ) and ΘRB (the angle between local mean magnetic field B0 and radial direction R). Thus, the scaling exponent (ξ(p, ΘRB)) of every structure function of order (p) is obtained for different angles. Before removing intermittency, ξ(p, ΘRB) shows a distinctive dependence on ΘRB: from monofractal scaling law at ΘRB ~0° to multifractal scaling law at ΘRB ~90°. In contrast after eliminating the intermittency, ξ(p, ΘRB) is found to be more monofractal for all ΘRB. The extended structure‐function model is applied to ξ(p, ΘRB), revealing differences of its fitting parameters α (a proxy of the power spectral index) and P1 (fragmentation fraction) for the cases with and without intermittency. Parameter α shows an evident angular trend falling from 1.9 to 1.6 for the case with intermittency but has a relatively flat profile around 1.8 for the case without intermittency. Parameter P1 rises from around 0.5 to above 0.8 with increasing ΘRB for the intermittency case and is located between 0.5 and 0.8 for the case lacking intermittency. Therefore, we may infer that it is the anisotropy of intermittency that causes the scaling anisotropy of energy spectra and the unequal fragmentation of energy cascading.

Uniaxial compression of rotationally symmetric Norway spruce samples: surface deformation and size effect

The influence of size on strength is well known for various materials and testing methods, but not enough data and knowledge of the effect on wood under compressive strength can be found. Therefore, the deformation and compressive strength of hyperboloid Norway spruce samples were tested for different dimensions. To determine the influence of size on the strength, three different approaches were used: “weakest link theory”, “size effect law” and “multifractal scaling law”. Of the three scaling theories, the “size effect law” and the “weakest link theory” predicted the effect well. Additionally, to determine the surface deformation of various samples, DIC3D was used on the largest sample geometry. The experiments show the existence of a possible size effect during compression, while digital image correlation illustrates the expected differences in deformation due to the shape and dependence of the strain distribution on the structure of wood. Additional experiments are proposed to further verify the observed effects and expand the knowledge of the size effect.

A fractal interpretation of size effects on the strength of metallic glasses

Abstract Yielding strength of metallic glasses in the uniaxial tensile and compressive tests is scale-dependent, which is attributed to the self-similar distribution of atomic cluster and free volume in the work. In contrast with the Weibull statistical theory previously employed in scaling phenomena of metallic glasses, fractal scaling laws are for the first time applied to describe the size effect inherent to the material disorder. Especially, the Multifractal Scaling Law (MFSL) originally proposed for quasi-brittle materials is used to interpret some experimental data in the literature. The best-fitted parameters ( f y and lch) from the MFSL are in good consistency with the bulk yielding strength and the shear band size of metallic glasses observed in the alternative approaches or experiments. The fractal size effect laws provide insight into not only the scaling phenomena, but also further engineering strength predictions and designs.

Investigation of the Indentation‐Size Effect (ISE) in a Commercial SiAlON: Multifractal Scaling Analysis and Underlying Mechanisms

The indentation‐size effect (ISE) in a commercial SiAlON was investigated by conducting Knoop indentation between 0.1 and 20 kg. The resulting ISE was analyzed utilizing Meyer's Law, Proportional Specimen Resistance (PSR) model, and a Multifractal Scaling Law (MFSL). Meyer's law and the PSR model fits to the hardness–load data were not excellent. Further analysis based on the PSR model and MFSL revealed three piecewise linear fits corresponding to load regimes 0.1–0.3, 0.5–2, and 5–20 kg. Physical inference of MFSL fit parameters suggested that these three load regimes correspond to where indentation behavior is governed by deformation mechanisms limited to single grains, grain boundaries, and multiple grains, respectively. Independent of the ISE analysis results, comprehensive examination of indents by scanning electron microscopy revealed that changes in deformation mechanisms could also be grouped into these three load regimes. Corresponding changes in deformation mechanisms were microcleavage cracking, grain‐boundary cracking, and macrocracking, respectively. These observations are consistent with the findings of both the PSR model and MFSL with respect to the physical aspects of the governing mechanisms. It is concluded that these mechanisms are responsible for the observed ISE in this commercial SiAlON.

Nanoindentation hardness of hot-pressed boron suboxide

Abstract The existence of the indentation size effect implies the absence of a single hardness value for the material under investigation especially at low applied loads. In this paper we present an investigation of the indentation size dependence behaviour of nanoindentation hardness in boron suboxide ceramic compacts prepared by uniaxial hot-pressing. Berkovich nanohardness indentations were conducted and analyzed accordingly. In addition to the ordinary Oliver and Pharr method of nanoindentation data analysis, a quantitative approach for the loading curve analysis is proposed. Using the proposed approach, the description and characterization of the observed indentation size effect through the application of the Meyer’s law, and the classical and the modified proportional specimen resistance models as well as the multi-fractal scaling law was conducted and is reported. The load-independent hardness values deduced from our quantitative approach are comparable to the results calculated with conventional methods, especially with the multi-fractal scaling law.

Analysis of the Indentation Size Effect in the Microhardness Measurements inB6O

The Vickers microhardness measurements of boron suboxide (B6O) ceramics prepared by uniaxial hot-pressing was investigated at indentation test loads in the range from 0.10 to 2.0 kgf. Results from the investigation indicate that the measured microhardness exhibits an indentation load dependence. Based on the results, we present a comprehensive model intercomparison study of indentation size effects (ISEs) in the microhardness measurements of hot-pressed B6O discussed using existing models, that is, the classical Meyer's law, Li and Bradt's proportional specimen resistance model (PSR), the modified proportional specimen resistance model (MPSR), and Carpinteri's multifractal scaling law (MFSL). The best correlation between literature-cited load-independent Vickers microhardness values, the measured values, and applied models was achieved in the case of the MPSR and the MFSL models.

Multifractal analysis of sunspot time series: the effects of the 11-year cycle and Fouriertruncation

Multifractal theory provides an elegant statistical characterization of many complex dynamicalvariations in Nature and engineering. It is conceivable that it may enrich characterizationof the sun’s magnetic activity and its dynamical modeling. Recently, on applying Fouriertruncation to remove the 11-year cycle and carrying out multifractal detrended fluctuationanalysis of the filtered sunspot time series, Movahed et al reported that sunspot data arecharacterized by multifractal scaling laws with the exponent for the second-order moment,h(2), being 1.12. Moreover, they think the filtered sunspot data are like a fractional Brownian motionprocess with anti-persistent long-range correlations characterized by the Hurst parameterH = h(2)−1 = 0.12. By designing an adaptive detrending algorithm and critically assessing the effectiveness ofFourier truncation in eliminating the 11-year cycle, we show that the values of the fractalscaling exponents obtained by Movahed et al are artifacts of the filtering algorithm thatthey used. Instead, sunspot data with the 11-year cycle properly filtered are characterizedby a different type of multifractal with persistent long-range correlations characterized byH≈0.74.