Lognormal Research Articles

Perinatal mortality after Chornobyl in contaminated regions of Ukraine.

Belarus and Ukraine were the countries most affected by the consequences of the Chornobyl nuclear power plant in 1986. A study of perinatal mortality in Belarus found a highly statistically significant increase in the 1990s in the most contaminated oblast Gomel but no increase during this period in the rest of Belarus. As a possible mechanism to interpret this increase as a late Chornobyl effect, it has been suggested that strontium-90 contained in Chornobyl fallout, incorporated during menarche, impairs the immune system of pregnant women which in turn increases perinatal mortality. In the present study, this hypothesis is tested using data from Ukraine. Annual data on perinatal mortality, in the period 1981-2006 were provided by the Ministry of Public Health of Ukraine. Trends in perinatal mortality rates in the most contaminated regions of Ukraine (Kyiv and Zhytomyr oblasts and the city of Kyiv; study region) were compared with rates in the rest of Ukraine (control region). To identify any differences in perinatal mortality trends between the study and control regions, the ratios of perinatal mortality rates in the study region to the rates in the control region were analyzed using the calculated strontium concentration as a predictor. A trend analysis of perinatal mortality rates in Ukraine revealed two bell-shaped deviations from a long-term exponential trend with maxima at the beginning and end of the 1990s. The same pattern was found in the data from the study and control regions, but the deviations were almost three times higher in the study region than in the control region. An analysis of the ratios of perinatal mortality rates in the study region to the rates in the control region (odds ratios) showed an increase and decrease during the 1990s which was approximated by a lognormal density distribution. The calculated strontium concentration, when used as a predictor, also fitted the data well. Thus, the data from Ukraine confirms the results from Belarus. The analysis of the odds ratios revealed about 1000 excess perinatal deaths in the study region in the period 1990-2004. The corresponding figure for Ukraine as a whole was estimated at 3500 perinatal deaths. It is hypothesized that the observed increase in perinatal mortality in the 1990s may be a late effect of incorporated strontium-90 on the immune system of pregnant women. The analysis is based on a theoretical model, as no data on strontium concentrations were available; the results should therefore be interpreted with caution. An updated study for Belarus would be desirable to corroborate the results.

Impact of wave action on the entrainment rate and droplet size of oil

Oil dispersed on the water surface undergoes fragmentation into droplets, subsequently entering the water column under wave action. The oil entrainment rate and the droplet size distribution (DSD) significantly influence the subsequent migration and transformation of oil in the water column. A wave tank system was employed to investigate the impact of wave action on oil entrainment rate and DSD. The experimental results revealed that increased wave height and wave frequency facilitated the entry of oil into the water column and fragmented it into dispersed droplets. Relationships between oil entrainment rate and energy dissipation, as well as DSD, were established and supported by mathematical regression analyses. Non-linear regression analysis indicated that the cumulative frequency distribution of DSD adhered to the Log-Normal and Rosin-Rammler distributions. These findings contribute to a comprehensive understanding of natural oil dispersion and subsea transport, valuable for estimating the origin of oil spills.

Collapse analysis of masonry arches and domes considering finite friction and uncertainties in compressive strength

Within the context of the lower bound theorem of classical limit analysis, the collapse of stone masonry arches and domes characterized by finite friction and uncertain finite compressive strength is addressed. Self-weight is considered, along with a live point load applied at the crown. A multi-constrained funicular approach based on the concept of force densities is implemented to provide the maximum static load multiplier for any assigned value of the friction coefficient and of the compressive strength. The numerical method, which takes into full account the stereotomy of blocky stone structures, is validated by using a novel version of the semi-analytical Durand-Claye’s stability area method. Two different types of brickwork are considered in the investigation, addressing the compressive strength as a random variable with prescribed log-normal distributions. Due to the limited availability of data to elaborate probabilistic models for the angle of friction, the effect of this mechanical parameter on the probability of failure of the blocky structures is assessed by developing a set of fragility curves. For each discrete value in a representative range of friction coefficients, a Monte Carlo investigation is performed by iteratively applying the funicular method considering samples of the random variable that describes the compressive strength. The retrieved load multipliers are processed to estimate the probability of failure for any given magnitude of live load, which depends on the value of the friction coefficient. An insight into the collapse modes occurring in the masonry arches and domes under examination is provided, as well.

Size and Shape Selective Classification of Nanoparticles

As nanoparticle syntheses on a large scale usually yield products with broad size and shape distributions, the properties of nanoparticle-based products need to be tuned after synthesis by narrowing the size and shape distributions or via the removal of undesired fractions. The development of property-selective classification processes requires a universal framework for the quantitative evaluation of multi-dimensional particle fractionation processes. This framework must be applicable to any property and any particle classification process. We extended the well-known one-dimensional methodology commonly used for describing particle size distributions and fractionation processes to the multi-dimensional case to account for the higher complexity of the property distribution and separation functions. In particular, multi-dimensional lognormal distributions are introduced and applied to diameter and length distributions of gold nanorods. The fractionation of nanorods via centrifugation and by orthogonal centrifugal and electric forces is modeled. Moreover, we demonstrate that analytical ultracentrifugation with a multi-wavelength detector (MWL-AUC) is a fast and very accurate method for the measurement of two-dimensional particle size distributions in suspension. The MWL-AUC method is widely applicable to any class of nanoparticles with size-, shape- or composition-dependent optical properties. In addition, we obtained distributions of the lateral diameter and the number of layers of molybdenum disulfide nanosheets via stepwise centrifugation and spectroscopic evaluation of the size fractions.

Analysis of Shift in Nil-Ductility Transition Reference Temperature for RPV Steels Due to Irradiation Embrittlement Using Probability Distributions and Gamma Process

Reactor pressure vessel (RPV) steels are highly susceptible to irradiation embrittlement due to prolonged exposure to high temperature, high pressure, and intense neutron irradiation. This leads to the shift in nil-ductility transition reference temperature—∆RTNDT. The change in ∆RTNDT follows a certain distribution pattern and is impacted by factors including chemical composition, neutron fluence, and irradiation temperature. Existing empirical procedures can estimate ∆RTNDT based on fitting extensive irradiation embrittlement data, but their reliability has not been thoroughly investigated. Probability statistical distributions and the Gamma stochastic process were performed to model material property degradation in RPV steels from a pressurized water reactor due to irradiation embrittlement, with the probability models considered being normal, Weibull, and lognormal distributions. Comparisons with existing empirical procedures showed that the Weibull distribution model and the Gamma stochastic model demonstrate good reliability in predicting ∆RTNDT for RPV steels. This provides a valuable reference for studying irradiation embrittlement in RPV materials.

Time-Dependent Multi-Particle Model Describing the Hydrogen Absorption of Nanocrystalline Magnesium Powders: A Case Study

Classical kinetic models describing the hydrogen absorption of nanocrystalline metallic hydrides generally do not involve any parameter related to the change in the crystallite size during the hydrogenation at constant temperature. In the present investigation, ball-milled nanocrystalline Mg powders exhibiting lognormal crystallite size distribution have been subjected to hydrogen absorption in a Sievert-type apparatus. Partially absorbed states were achieved by interrupting the hydrogenation cycle at different hydrogen content, i.e., when 15%, 50%, and 90% of Mg powder transformed to MgH2. The evolution of the characteristic size of the nucleating MgH2 phase was determined from X-ray diffraction analysis. Considering the crystallite size distribution of the as-milled powder agglomerate as well as the growth during the isothermal hydrogenation process, a time-dependent multi-particle reaction function ∝CV¯t;R(t) was developed. It was shown unambiguously for this case study that the measured hydrogen absorption curve of the ball-milled Mg powder shows the best correlation with this model when it is compared to classical kinetic functions or the previously developed multi-particle reaction function excluding the change in the average crystallite size during hydrogenation.

Random and time-persistent depositional processes in turbidite successions: an example from the marine deep-water Aoshima Formation (Neogene, Kyushu Island, southwest Japan)

ABSTRACT The deposits of flood- and earthquake-derived subaqueous sediment gravity flows represent a significant fraction of lacustrine and deep-sea sedimentary successions, thus providing a valuable record of such natural disasters. The magnitude of these events and the thickness of the associated deposits are considered to follow a lognormal or power-law frequency distribution, whilst that of time intervals between subsequent events appear to be best approximated by a Poisson model, indicative of a random, time-independent phenomenon. However, the debate on whether the sedimentary record of these natural disasters is governed by randomness alone or whether there is some underlying stratigraphic ordering is still unsettled and requires detailed time-series analysis. This study consists of a time-series analysis of mudstone- and sandstone-dominated turbidite successions offshore a fan-delta system in the Neogene Aoshima Formation that belongs to the sedimentary fill of the forearc basin of southwest Japan. The formation consists of a monotonous alternation of very fine- to medium-grained sandstones capped by hemipelagic mudstones and, more rarely, by turbidite mudstones. The results show that the autocorrelation function of the time series suggests quasi-periodic variability in the upper sandstone-dominated part, whereas the lower mudstone-dominated part shows a white-noise-like pattern. Rescaled range analysis shows that the number of events per unit time in the lower part is characterized by a random time series, such as Brownian noise with a Hurst exponent of 0.5. In contrast, the thickness of event beds of the lower part and the thickness and the number of events of the upper part are persistent time series with a Hurst exponent > 0.5. These results suggest that the number of turbidite depositional events in the mudstone-dominated part indicates random timing, whereas its thickness time series and the sandstone-dominant part are not governed by simple stochastic processes but are affected by sea-level changes, sediment transport dynamics, and other factors such as, for example, seafloor topography.

Which distribution to choose for deriving a species sensitivity distribution? Implications from analysis of acute and chronic ecotoxicity data

Species sensitivity distributions (SSDs) estimated by fitting a statistical distribution to ecotoxicity data are indispensable tools used to derive the hazardous concentration for 5 % of species (HC5) and thereby a predicted no-effect concentration in environmental risk assessment. Whereas various statistical distributions are available for SSD estimation, the fundamental question of which statistical distribution should be used has received limited systematic analysis. We aimed to address this knowledge gap by applying four frequently used statistical distributions (log-normal, log-logistic, Burr type III, and Weibull distributions) to acute and chronic SSD estimation using aquatic toxicity data for 191 and 31 chemicals, respectively. Based on the differences in the corrected Akaike’s information criterion (AICc) as well as visual inspection of the fitting of the lower tails of SSD curves, the log-normal SSD was generally better or equally good for the majority of chemicals examined. Together with the fact that the ratios of HC5 values of other alternative SSDs to those of log-normal SSDs generally fell within the range 0.1–10, our findings indicate that the log-normal distribution can be a reasonable first candidate for SSD derivation, which does not contest the existing widespread use of log-normal SSDs.

An iterative algorithm to simultaneously retrieve aerosol extinction and effective radius profiles using CALIOP

Abstract. The Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) on board the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite has been widely used in climate and environment studies to obtain the vertical profiles of atmospheric aerosols. To retrieve the vertical profile of aerosol extinction, the CALIOP algorithm assumes column-averaged lidar ratios based on a clustering of aerosol optical properties measured at surface stations. On one hand, these lidar ratio assumptions may not be appropriate or representative at certain locations. One the other hand, the two-wavelength design of CALIOP has the potential to constrain aerosol size information, which has not been considered in the operational algorithm. In this study, we present a modified inversion algorithm to simultaneously retrieve aerosol extinction and effective radius profiles using two-wavelength elastic lidars such as CALIOP. Specifically, a lookup table is built to relate the lidar ratio with the Ångström exponent calculated using aerosol extinction at the two wavelengths, and the lidar ratio is then determined iteratively without a priori assumptions. The retrieved two-wavelength extinction at each layer is then converted to the particle effective radius assuming a lognormal distribution. The algorithm is tested on synthetic data, Raman lidar measurements and then finally the real CALIOP backscatter measurements. Results show improvements over the CALIPSO operational algorithm by comparing with ground-based Raman lidar profiles.

Spectral decomposition of ground motions in New Zealand using the generalized inversion technique

SUMMARY To gain new insights into ground-motion phenomena in New Zealand (NZ), we apply the non-parametric generalized inversion technique (GIT) in the Fourier domain to isolate the systematic source, path, and site effects from 20 813 seismograms, recorded by 693 sensors at 439 unique locations, from 1200 shallow crustal events (Mw > 3) during the period 2000–2021. From the inverted source spectra, we derive Brune's stress parameter, ∆σ, which is found to follow a lognormal distribution with a log10 standard deviation of 0.36 or equivalently 0.83 in natural log unit. ∆σ slightly increases with focal depth and is practically independent of earthquake size (i.e. self-similar), but displays a statistically significant spatial clustering. Based on the inverted attenuation, a trilinear geometric-spreading function, and a distance-dependent quality-factor Q(f) model are found to well describe the attenuation in NZ; though a single $Q( f )$ model is also obtained for the whole distance range: $Q( f ) = 149.1{f}^{0.62}$. Using the site response decomposed from GIT, we find that the soil classification scheme specified in NZ seismic code, NZS1170.5, has a limited capability in discerning the site-specific frequency-dependent amplification functions in comparison to a non-parametric clustering with the same number of discrete classes. The potential use of the spatial variation in source parameters from this GIT analysis in region-specific physics-based simulations is discussed.

Application of deep learning and inline holography to estimate the droplet size distribution

We examine five machine learning-based architectures to estimate the droplet size distributions obtained using digital inline holography. The architectures, namely, U-Net, R2 U-Net, Attention U-Net, V-Net, and Residual U-Net, are trained using synthetic holographic images. Our assessment focuses on evaluating the training, validation, and prediction performance of these architectures. Our results indicate that U-Net and R2 U-Net are the most proficient architectures, consistently demonstrating superior performance trends and achieving the highest Intersection Over Union (IOU) scores among the five architectures studied. We employ additional training using experimental holographic images for the two top-performing architectures to validate their efficacy further. Subsequently, they are employed to segment an experimental dataset illustrating the bag breakup phenomenon, facilitating the extraction of size distribution. The extracted size distribution from U-Net and R2 U-Net segmentation is then compared with the analytical model proposed by Jackiw and Ashgriz (2022) by employing the gamma and log-normal distributions. Our findings indicate that the gamma distribution provides a more accurate prediction of the multi-modal size distribution than the log-normal distribution owing to its long exponential tail. The present study offers valuable insights into the effectiveness of machine learning architectures in estimating particle/droplet sizes, highlighting their practical application in real-world experimental scenarios.

Characterizations of continuous log-symmetric distributions based on properties of order statistics

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function h ( X ) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions.

A new approach to estimate parameters of the two-parameter Weibull distribution

This study proposes a new approach to estimate the parameters of the two-parameter Weibull distribution in the case of complete (uncensored) or doubly Type II censored samples. The proposed estimators have closed forms and can be adapted to Type II right or Type II left censored samples. The new estimators are strongly consistent and the new shape estimator follows asymptotically a normal and the new scale estimator a lognormal distribution. Bias, mean square and efficiency performance of the new estimators are investigated through simulations and compared to some known estimation methods such as maximum likelihood estimation, method of moments, quantile estimation, maximum goodness of fit estimation, L-moments, U-statistics and bias-corrected maximum likelihood. According to the simulations, the newly proposed estimators seem to have good efficiency performance in comparison to the other methods, particularly in small samples. Furthermore, two real data sets are used to illustrate the new estimators.

Research on the Equivalent Criterion for Ratios of Safety Margin in Terms of the Factor of Safety and the Reliability Index Under Lognormal Distribution

The ratio of safety margin is the bridge between deterministic design and reliability design. However, the equivalence criterion between the ratios of safety margin in terms of the factor of safety (FOS) and the reliability index is ambiguous. This paper considers the traditional and reliability-based design method at the same safety level. A general form of the FOS following lognormal distribution is adopted. It was found that the variability of FOS is affected by the normalized gradient and the coefficient of variation of random variable. The ratios of safety margin in terms of the FOS and the reliability index are equivalent in the specific project if the normalized gradient does not vary with the design parameters. The feasibility of applying the ratio of safety margin may be increased in engineering practice.

Elucidating the influence of the lower and upper microbiological limits: When is a 3-class sampling plan useful to test for pathogens in food?

The potential advantages of using 3-class sampling plans in the management of pathogens in food has been discussed for years. Using a 3-class plan, some lots would be accepted if, among a total number of samples n, the pathogen was detected in no more than a predefined number of samples c (such as c = 1) exceeding a microbiological limit m, and no sample was at a level above a predefined maximal concentration M. We built upon and extended beyond published methodologies and developed a web application to evaluate the performance of 2-class and 3-class sampling plans. We showed that there is little benefit of a 3-class plan over a 2-class plan with equal values for m, n and c under the classical assumption of a lognormal distribution of bacteria concentration in the lot with a standard deviation of 0.8 log cfu/g, for m value set to “absence in 25g” and M set to values > 1 cfu/g. In this situation, the lot will almost always be rejected based on the criterion “>c detected samples” and will almost never (or seldom) be rejected based on only the criterion “at least one sample > M cfu/g”. However, the benefit of a 3-class plan is realized when the distribution of pathogens is not a simple lognormal but includes a proportion of samples contaminated beyond the lognormal distribution, at levels > M. When this is the case, the performance of a 3-class plan relative to the 2-class plan with equal values of m, n, and c improves, then becomes essentially equivalent, as the proportion of highly contaminated samples increases. The behavior of 3-class plan may be useful in such situations. While such distributions of pathogens have been observed across lots in the food supply, better characterization of within-lot contamination distribution in food is needed to better evaluate the performance of sampling plans.

Evolution of the magnetic field rotation distributions in the inner heliosphere

The nature and evolution of the solar wind magnetic field rotations is studied in data from the Parker Solar Probe We investigated the magnetic field deflections in the inner heliosphere below 0.5 au in a distance- and scale-dependent manner to shed some light on the mechanism behind their evolution. We used the magnetic field data from the FIELDS instrument suite to study the evolution of the magnetic field vector increment and rotation distributions that contain switchbacks. We find that the rotation distributions evolve in a scale-dependent fashion. They have the same shape at small scales regardless of the radial distance, in contrast to larger scales, where the shape evolves with distance. The increments are shown to evolve towards a log-normal shape with increasing radial distance, even though the log-normal fit works quite well at all distances, especially at small scales. The rotation distributions are shown to evolve towards a previously developed rotation model moving away from the Sun. Our results suggest a scenario in which the evolution of the rotation distributions is primarily the result of the expansion-driven growth of the fluctuations, which are reshaped into a log-normal distribution by the solar wind turbulence.

Intrinsic Mass–Richness Relation of Clusters from THE THREE HUNDRED Hydrodynamic Simulations

The main systematics in cluster cosmology is the uncertainty in the mass–observable relation. In this paper, we focus on the most direct cluster observable in optical surveys, i.e., richness, and constrain the intrinsic mass–richness (MR) relation of clusters in THE THREE HUNDRED hydrodynamic simulations with two runs: GIZMO-SIMBA and GADGET-X. We find that modeling the richness at a fixed halo mass with a skewed Gaussian distribution yields a simpler and smaller scatter compared to the commonly used lognormal distribution. Additionally, we observe that baryon models have a significant impact on the scatter while exhibiting no influence on the mass dependence and a slight effect on the amplitude in the MR relation. We select member galaxies based on both stellar mass M ⋆ and absolute magnitude M . We demonstrate that the MR relations obtained from these two selections can be converted to each other by using the M⋆–M relation. Finally, we provide a seven-parameter fitting result comprehensively capturing the dependence of the MR relation on both stellar mass cutoff and redshift.

Quantification of grain boundary mobilities in natural olivine by annealing experiments and full-field modelling

We investigate olivine grain boundary (GB) migration in natural peridotites experimentally annealed at high pressure and high temperature, and couple the experimental observations to full-field grain growth models to provide the distribution of GB mobilities in natural olivine polycrystals. A stack of four slices of natural mylonitic peridotite (Oman ophiolite) was annealed at 1473 K for 5 h under a confining pressure of 300 MPa of argon in a Paterson press. The three sintered interfaces of the stack are characterized using scanning electron microscopy and electron backscatter diffraction (EBSD) to extract a distribution of apparent (2D) GB displacements. Full-field simulations of numerically sintered interfaces are then used to infer the GB mean mobility and distribution in olivine polycrystal allowing forward modelling exercise to reproduce the experimental GB displacement distribution.This yields widely dispersed mobilities, which can be approximated by a log-normal distribution of 10−15.85±2.58m4.J−1.s−1. Both the average and the dispersion of GB mobilities can be explained by silicon grain boundary diffusion. Finally, we demonstrate that the high variability of GB mobilities in olivine implies a decrease of the mean growth rate with time. This elucidates the difficulties of extrapolating experimental grain growth rates to geological timescales and observed microstructures in peridotites.

Cost-effectiveness of Kang Ai injection plus chemotherapy vs. Shenqi Fuzheng injection plus chemotherapy in the first-line treatment of advanced non-small cell lung cancer.

This study aimed to evaluate the cost-effectiveness of two Chinese patent medicines, including Kang Ai injection and Shenqi Fuzheng injection with each combined with platinum-based chemotherapy as the first-line treatment for patients with advanced non-small cell lung cancer (NSCLC) in China. From Chinese healthcare system perspective, a three state Markov model with a cycle of 3 weeks and a 10-year horizon was constructed to derive the incremental cost-effectiveness ratio (ICER). Since only individual patient data of progression-free survival (PFS) of Kang Ai injection group can be obtained, we extrapolated median overall survival (mOS) of Kang Ai injection group and median progression-free survival (mPFS) and mOS of Shenqi Fuzheng injection group based on published literature and methods. Then survival curves were estimated by the method of declining exponential approximation of life expectancy (DEALE), which is based on the assumption that survival follows a declining exponential function. We performed one-way sensitivity analysis and probabilistic sensitivity analysis to test the robustness. Additionally, a scenario analysis was adopted to investigate the impact of using best-fitting distribution for PFS curve of Kang Ai injection group on the economic conclusion. The base-case result indicated that Kang Ai injection group provided 0.217 incremental quality-adjusted life years (QALYs) at an incremental cost of $103.38 compared with Shenqi Fuzheng injection group. The ICER was $476.41/QALY, which was much lower than the willingness to pay threshold of one time the GDP per capita of China in 2022 ($12,070/QALY). Deterministic sensitivity analysis result showed that ICER was most sensitive to the changes in odds ratio (OR) value. The probabilistic sensitivity analysis confirmed the robustness of base-case analysis results. The scenario analysis result showed that by using Log-Normal distribution to fit the PFS curve of Kang Ai injection group and shortening the time horizon to 5 years, the ICER was $4,081.83/QALY, which was still much lower than the willingness to pay threshold. Kang Ai injection combined with platinum-based chemotherapy appeared to be more cost-effective for the treatment of advanced NSCLC than Shenqi Fuzheng injection combined with platinum-based chemotherapy.

Energy evolution and strain localization in fractured sandstone under freeze-thaw cycling and uniaxial loading-unloading

In cold regions, combined cyclic freeze-thaw (F-T) and loading-unloading can cause rock slope instability and sliding. The macro-failure of rock masses originates from the expansion of strain localization areas driven by energy. Therefore, a deep understanding of the energy evolution and strain distribution can reveal the instability process. Along these lines, in this work, intact and intermittently jointed sandstones were subjected to a cyclic F-T environment and subsequently carried out uniaxial graded loading-unloading to obtain their mechanical parameters and X-ray computed tomography images. By investigating the energy density, it was found that the input, elastic, dissipated, and damage energies all increased with loading according to a quadratic polynomial function, with the energies obeying a linear allocation law. Meanwhile, F-T changed the energy storage capacity and energy dissipation coefficient of the rock by causing meso-damage. To measure the meso-scale strain localization characteristics, the advanced digital volume correlation technique was also used to obtain three-dimensional strain fields. In terms of local strain, the failure behavior of intact specimens under the application of low and high numbers of F-T cycles was controlled by potential fracture bands and frost-heave cracks, respectively, while fractured specimens were mainly dependent on the macro-fracture arrangement. However, both frost heaving and thawing transformed sandstone from brittle to ductile. In terms of statistical strain, the equivalent strain (εeq) showed a log-normal distribution and continued to deviate to the right due to loading-unloading. Meanwhile, the normal strain (εxx, εyy, εzz) exhibited a normal distribution, where fractured sandstone mainly developed along with the slip direction. Finally, the intrinsic correlations among the energy parameters, meso-damage, and macro-failure were discussed. From our analysis, three stages of energy allocation and transformation were revealed, namely the uniform allocation of damage energy, banded accumulation of damage energy, and fracture caused by the elastic energy release.