Experimental study is performed on the probabilistic models for the long fatigue crack growth rates (da/dN) of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity f...Experimental study is performed on the probabilistic models for the long fatigue crack growth rates (da/dN) of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity factor range going down to the threshold and the average stress effect. The probabilistic models were presented on the equation. They consist of the probabilistic da/dN-ΔK relations, the confidence-based da/dN-ΔK relations, and the probabilistic- and confidence-based da/dN-ΔK relations. Efforts were made respectively to characterize the effects of probabilistic assessments due to the scattering regularity of test data, the number of sampling, and both of them. These relations can provide wide selections for practice. Analysis on the test data of LZ50 steel indicates that the present models are available and feasible.展开更多
Experimental study is performed on the probabilistic models for the long fatigue crack growth rates （da/dN） of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity...Experimental study is performed on the probabilistic models for the long fatigue crack growth rates （da/dN） of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity factor range going down to the threshold and the average stress effect. The probabilistic models were presented on the equation. They consist of the probabilistic da/dN-△K relations, the confidence-based da/dN-△K relations, and the probabilistic- and confidence-based da/dN-△K relations. Efforts were made respectively to characterize the effects of probabilistic assessments due to the scattering regularity of test data, the number of sampling, and both of them. These relations can provide wide selections for practice. Analysis on the test data of LZ50 steel indicates that the present models are available and feasible.展开更多
As is already known, most of the plasma literature is occupied with the plasma instabilities and the inevitable plasma waves, which remain major obstacles to the thermonuclear fusion process. Many experimental data on...As is already known, most of the plasma literature is occupied with the plasma instabilities and the inevitable plasma waves, which remain major obstacles to the thermonuclear fusion process. Many experimental data on the plasma waves （growth or damping） and their accompanied theoretical interpretations have been published during the last five decades; lots of them have been identified and justified as well, some not yet. One of these is our previous research on plasma waves, which originated in the early 80＇s at the Plasma Physics Laboratory of the NCSR ＂Demokritos＂. As the wave rising is defined by the growth rate （or the damping on the extinguishment）, these important wavy quantities will be studied in detail in the present paper. Three examples are taken from our previous theoretical results, and the first observation reveals that the involved quantities are complicated enough to be studied by themselves. So, the use of suitable approach models, which may interpret the experimental wavy quantities, is the central idea of the present attempt, Furthermore, calculations with a little change of the initial conditions have been repeated in order to determine whether the plasma behaves as a chaotic medium.展开更多
Population is an important strategic resource for national development, a fundamental element of socio-economic development. The coordinated development of population and economy is an effective way to achieve rapid e...Population is an important strategic resource for national development, a fundamental element of socio-economic development. The coordinated development of population and economy is an effective way to achieve rapid economic growth. Based on the population statistics data of counties (districts) in Henan Province, China, from 2006 to 2021. The paper firstly uses the logistic population growth mathematical model to calculate the resident population growth rate of counties (districts), then utilizes the hotspot analysis and spatial semi-variogram analysis, to research the spatial distribution characteristics of the resident population growth rate in Henan Province. The research results show that the evolution of the regional resident population in the province basically conforms to the logistic natural growth model. The resident population growth rate shows the characteristics of high in the north and low in the south, high in the center and low in the surrounding regions. The resident population growth rate is positively correlated with the level of economic development;the urban built-up areas, especially the new regions in urban planning, have a fast growth rate of resident population, which has a significant siphon effect on the population of surrounding regions. The hotspots of resident population growth rate in the province are mainly distributed in the urban built-up areas and surrounding regions of Zhengzhou, Luoyang, and Xinxiang, accounting for about 3.51% of the total area of the province. The cold spots are mainly distributed in the eastern part of the province, forming zonal distribution, which spans across Shangqiu City, Zhoukou City, and Zhumadian City, accounting for about 8.61% of the total area of the province. The area with negative growth of resident population accounts for approximately 53.47% of the total province. The spatial distribution of the growth rate of the resident population in the whole province basically conforms to the spherical model, with a small dispersion degree and a short range. In the range, there is a high degree of variability in resident population growth rate.展开更多
A two-dimensional global chemistry model is developed to study the distribution andlong-term trends of methane. The model contains 34 species and 104 chemical andphotochemical reactions. Using the model, the long-term...A two-dimensional global chemistry model is developed to study the distribution andlong-term trends of methane. The model contains 34 species and 104 chemical andphotochemical reactions. Using the model, the long-term trends of CH4, CO and OH in atmosphere are simulated, comparison between the model and observations shows that thesimulation is successful.Experiments are done to investigate the causes of dramatic decrease in the growth rate ofCH4 in 1992 such as OH increase due to stratospheric ozone depletion, decrease of temperature in the troposphere due to Mount Pinatubo eruption and descendent of CH4 sources fluxes.A new explanation is proposed and verified by this model that the decrease of CO emissionplays an important role for the abnormal growth rate of CH4 in 1992. We find that the decreases of CH4 and CO emissions are the main reasons for the sudden decrease of growth rateof CH4 in 1992, which account for 73% and 27% respectively.展开更多
The objective of this study is to analyze a chemostat model of very simple type with the Haldane expression of growth rate and a variable yield coefficient. The proposed modified model is analyzed qualitatively and qu...The objective of this study is to analyze a chemostat model of very simple type with the Haldane expression of growth rate and a variable yield coefficient. The proposed modified model is analyzed qualitatively and quantitatively. Analytic conditions for stability and optimality are determined for washout and no washout equilibrium solutions. One of the main focuses of the study is to determine parameter values for which Hopf Bifurcations occur in a bioreactor. It has been shown that the maximum stable non-washout equilibrium exits at a residence time under suitable parameter values. Hopf bifurcation is observed at three different conditions of the parameters.展开更多
During underground coal gasification （UCG）, whereby coal is converted to syngas in situ, a cavity is formed in the coal seam. The cavity growth rate （CGR） or the moving rate of the gasification face is affected by...During underground coal gasification （UCG）, whereby coal is converted to syngas in situ, a cavity is formed in the coal seam. The cavity growth rate （CGR） or the moving rate of the gasification face is affected by controllable （operation pressure, gasification time, geometry of UCG panel） and uncontrollable （coal seam properties） factors. The CGR is usually predicted by mathematical models and laboratory experiments, which are time consuming, cumbersome and expensive. In this paper, a new simple model for CGR is developed using non-linear regression analysis, based on data from 1 l UCG field trials. The empirical model compares satisfactorily with Perkins model and can reliably predict CGR.展开更多
Depositional growth of ice crystal is one of the major processes for development of precipitation systems and can be represented by depositional growth of cloud ice from cloud water(P_(IDW)) and depositional growth of...Depositional growth of ice crystal is one of the major processes for development of precipitation systems and can be represented by depositional growth of cloud ice from cloud water(P_(IDW)) and depositional growth of snow from cloud ice(P_(SFI)) in cloud-resolving model. Four parameterization schemes are analyzed in the cloud-resolving model simulations of four rainfall cases over the tropics and midlatitudes. The comparison of time and model domain mean data shows that Shen's scheme produces the closest rainfall simulation to the observation. Compared to Zeng's scheme,Shen's scheme improves the mean rain-rate simulation significantly through the dramatic decrease in depositional growth of cloud ice from cloud water. Compared to other schemes, Shen's scheme produces the better rainfall simulation via the reduction in the mean rain rate associated with the enhanced gain of cloud water and ice.展开更多
Bangladesh is a densely populated country than many other countries of the world. The population growth is termed as alarming, however, knowledge of growth in the years to come would be useful in planning for the deve...Bangladesh is a densely populated country than many other countries of the world. The population growth is termed as alarming, however, knowledge of growth in the years to come would be useful in planning for the development of the country. This article is based on the projection of future population growth of the country. The available actual population census data during 1991-2011 of Bangladesh was applied to the application of a non-linear, non-autonomous ordinary differential equation familiar as Verhulst logistic population model with the maximum environmental capability of Bangladesh. Bangladesh will reach its carrying capacity of 245.09 million population in the next 56 years i.e. the year 2067 and then it decreases as S-shaped curve. The article has provided a focus on the changing trends of the growth of the population of Bangladesh.展开更多
Based on the basic ecological model proposed by Jorgensen and the species composition of lake ecosystem, a new approach to calculate the growth rate of phytoplankton weighted growth rate method was presented. The the...Based on the basic ecological model proposed by Jorgensen and the species composition of lake ecosystem, a new approach to calculate the growth rate of phytoplankton weighted growth rate method was presented. The thermodynamic concept “Exergy” was also introduced into the ecodynamic model for lakes,and the changes in the species composition and the ecological structure of lake ecosystem were reflected by the changing exergy.An improved ecodynamic model for lakes was developed by combining the weighted growth rate with the application of exergy. These improvements comprehensively take the ecological properties of lake ecosystems into account, making the ecodynamic model ecologically more reasonable and practically more flexible. The applications of the improved ecodynamic model in the ecological modelling and prediction of Dianchi Lake, have demonstrated that this model is practicable and satisfactory.展开更多
A self adjusting model was presented on the basis of the effect of temperature gradient on eutectic growth and a curved solid/liquid interface. Finite differential method was adopted to solve the model. The average la...A self adjusting model was presented on the basis of the effect of temperature gradient on eutectic growth and a curved solid/liquid interface. Finite differential method was adopted to solve the model. The average lamellar spacing of the Al Al 3Fe eutectic alloy and the content fields ahead of the solidifying interface under different growth rates were calculated. Directional solidification experiments were carried out in order to prove the modification of the modeling. The experimental results are in relatively good agreement with the calculations.展开更多
Many curves have been proposed and debated to model individual growth of marine invertebrates. Broadly, they fall into two classes, first order (e.g. von Bertalanffy) and sigmoidal (e.g. Gompertz). We provide an innov...Many curves have been proposed and debated to model individual growth of marine invertebrates. Broadly, they fall into two classes, first order (e.g. von Bertalanffy) and sigmoidal (e.g. Gompertz). We provide an innovative approach which demonstrates that the growth curves are not mutually exclusive but that either may arise from a simple three-stage growth model with two steps (k<sub>1 </sub>and k<sub>2</sub>) depending on the ratio of the growth parameters . The new approach predicts sigmoidal growth when is close to 1, but if either growth from stage A to stage B or B to C is fast relative to the other, the slower of the two steps becomes the growth limiting step and the model reduces to first order growth. The resulting curves indicate that there is a substantial difference in the estimated size at time t during the period of active growth. This novel two-step rate model generates a growth surface that allows for changes in the rate parameters over time as reflected in the new parameter n(t) = k<sub>1</sub>(t) - k<sub>2</sub>(t). The added degree of freedom brings about individual growth trajectories across the growth surface that is not easily mapped using conventional growth modeling techniques. This two (or more) stage growth model yields a growth surface that allows for a wide range of growth trajectories, accommodating staged growth, growth lags, as well as indeterminate growth and can help resolve debates as to which growth curves should be used to model animal growth. This flexibility can improve estimates of growth parameters used in population models influencing model outcomes and ultimately management decisions.=展开更多
A better understanding of the mechanisms that control nutrient acquisition in the context of plant and ecosystem responses to climate change is needed. Mechanistic nutrient uptake models provide a means to investigate...A better understanding of the mechanisms that control nutrient acquisition in the context of plant and ecosystem responses to climate change is needed. Mechanistic nutrient uptake models provide a means to investigate some of the impacts of temperature change on soil nutrient supply and root uptake kinetics through the simulation of key soil and plant processes. The NST 3.0 model, in combination with literature values on plant and soil parameters from a red spruce (Picea rubens L.) site in the southern Appalachians, was used to conduct a series of model simulations focused on the combined effects of changes to the maximal rate of nutrient influx at high concentrations (Imax), root growth rate (k), concentration of nutrient occurring in the soil solution (Cli), and the ability of the soil solid phase to buffer changes to the soil solution nutrient concentration (b). Previous research has indicated that these four parameters are responsive to changes in root zone temperature. Simulated uptake of NH4 increased by a factor of up to 2.6 in response to increases in soil temperature of 1°C to 5°C. The model also projected an increase in P uptake coupled with up to an 80% reduction in solution P concentration in response to a 1°C -5°C increase over a 147-d simulation period. These hypothetical changes, if validated, have interesting implications for plant growth and competition and point to a need for additional studies to better define the impacts of soil temperature on soil nutrient supply and root uptake.展开更多
Based on analysis of deformation in an infinite isotropic elastic matrix containing an embedded elliptic crack, subject to far field triaxial compressive stress, the energy release rate and a mixed fracture criterion ...Based on analysis of deformation in an infinite isotropic elastic matrix containing an embedded elliptic crack, subject to far field triaxial compressive stress, the energy release rate and a mixed fracture criterion are obtained by using an energy balance approach. The additional compliance tensor induced by a single closed elliptic microcrack in a representative volume element and its in-plane growth is derived. The additional compliance tensor induced by the kinked growth of the elliptic microcrack is also obtained. The effect of the microcracks, randomly distributed both in geometric characteristics and orientations, is analyzed with the Taylor＇s scheme by introducing an appropriate probability density function. A micromechanical damage model for rocks and concretes under triaxial compression is obtained and experimentally verified.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos.50375130and50323003), the Special Foundation of National Excellent Ph.D.Thesis (No.200234) and thePlanned Itemforthe Outstanding Young Teachers ofMinistry ofEducationofChina (No.2101)
文摘Experimental study is performed on the probabilistic models for the long fatigue crack growth rates (da/dN) of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity factor range going down to the threshold and the average stress effect. The probabilistic models were presented on the equation. They consist of the probabilistic da/dN-ΔK relations, the confidence-based da/dN-ΔK relations, and the probabilistic- and confidence-based da/dN-ΔK relations. Efforts were made respectively to characterize the effects of probabilistic assessments due to the scattering regularity of test data, the number of sampling, and both of them. These relations can provide wide selections for practice. Analysis on the test data of LZ50 steel indicates that the present models are available and feasible.
基金国家自然科学基金，Special Foundation of National Excellent Ph.D.Thesis，Outstanding Young Teachers of Ministry of Education of China
文摘Experimental study is performed on the probabilistic models for the long fatigue crack growth rates （da/dN） of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity factor range going down to the threshold and the average stress effect. The probabilistic models were presented on the equation. They consist of the probabilistic da/dN-△K relations, the confidence-based da/dN-△K relations, and the probabilistic- and confidence-based da/dN-△K relations. Efforts were made respectively to characterize the effects of probabilistic assessments due to the scattering regularity of test data, the number of sampling, and both of them. These relations can provide wide selections for practice. Analysis on the test data of LZ50 steel indicates that the present models are available and feasible.
文摘As is already known, most of the plasma literature is occupied with the plasma instabilities and the inevitable plasma waves, which remain major obstacles to the thermonuclear fusion process. Many experimental data on the plasma waves （growth or damping） and their accompanied theoretical interpretations have been published during the last five decades; lots of them have been identified and justified as well, some not yet. One of these is our previous research on plasma waves, which originated in the early 80＇s at the Plasma Physics Laboratory of the NCSR ＂Demokritos＂. As the wave rising is defined by the growth rate （or the damping on the extinguishment）, these important wavy quantities will be studied in detail in the present paper. Three examples are taken from our previous theoretical results, and the first observation reveals that the involved quantities are complicated enough to be studied by themselves. So, the use of suitable approach models, which may interpret the experimental wavy quantities, is the central idea of the present attempt, Furthermore, calculations with a little change of the initial conditions have been repeated in order to determine whether the plasma behaves as a chaotic medium.
文摘Population is an important strategic resource for national development, a fundamental element of socio-economic development. The coordinated development of population and economy is an effective way to achieve rapid economic growth. Based on the population statistics data of counties (districts) in Henan Province, China, from 2006 to 2021. The paper firstly uses the logistic population growth mathematical model to calculate the resident population growth rate of counties (districts), then utilizes the hotspot analysis and spatial semi-variogram analysis, to research the spatial distribution characteristics of the resident population growth rate in Henan Province. The research results show that the evolution of the regional resident population in the province basically conforms to the logistic natural growth model. The resident population growth rate shows the characteristics of high in the north and low in the south, high in the center and low in the surrounding regions. The resident population growth rate is positively correlated with the level of economic development;the urban built-up areas, especially the new regions in urban planning, have a fast growth rate of resident population, which has a significant siphon effect on the population of surrounding regions. The hotspots of resident population growth rate in the province are mainly distributed in the urban built-up areas and surrounding regions of Zhengzhou, Luoyang, and Xinxiang, accounting for about 3.51% of the total area of the province. The cold spots are mainly distributed in the eastern part of the province, forming zonal distribution, which spans across Shangqiu City, Zhoukou City, and Zhumadian City, accounting for about 8.61% of the total area of the province. The area with negative growth of resident population accounts for approximately 53.47% of the total province. The spatial distribution of the growth rate of the resident population in the whole province basically conforms to the spherical model, with a small dispersion degree and a short range. In the range, there is a high degree of variability in resident population growth rate.
文摘A two-dimensional global chemistry model is developed to study the distribution andlong-term trends of methane. The model contains 34 species and 104 chemical andphotochemical reactions. Using the model, the long-term trends of CH4, CO and OH in atmosphere are simulated, comparison between the model and observations shows that thesimulation is successful.Experiments are done to investigate the causes of dramatic decrease in the growth rate ofCH4 in 1992 such as OH increase due to stratospheric ozone depletion, decrease of temperature in the troposphere due to Mount Pinatubo eruption and descendent of CH4 sources fluxes.A new explanation is proposed and verified by this model that the decrease of CO emissionplays an important role for the abnormal growth rate of CH4 in 1992. We find that the decreases of CH4 and CO emissions are the main reasons for the sudden decrease of growth rateof CH4 in 1992, which account for 73% and 27% respectively.
文摘The objective of this study is to analyze a chemostat model of very simple type with the Haldane expression of growth rate and a variable yield coefficient. The proposed modified model is analyzed qualitatively and quantitatively. Analytic conditions for stability and optimality are determined for washout and no washout equilibrium solutions. One of the main focuses of the study is to determine parameter values for which Hopf Bifurcations occur in a bioreactor. It has been shown that the maximum stable non-washout equilibrium exits at a residence time under suitable parameter values. Hopf bifurcation is observed at three different conditions of the parameters.
文摘During underground coal gasification （UCG）, whereby coal is converted to syngas in situ, a cavity is formed in the coal seam. The cavity growth rate （CGR） or the moving rate of the gasification face is affected by controllable （operation pressure, gasification time, geometry of UCG panel） and uncontrollable （coal seam properties） factors. The CGR is usually predicted by mathematical models and laboratory experiments, which are time consuming, cumbersome and expensive. In this paper, a new simple model for CGR is developed using non-linear regression analysis, based on data from 1 l UCG field trials. The empirical model compares satisfactorily with Perkins model and can reliably predict CGR.
基金National Natural Science Foundation of China(41475039)National Key Basic Research and Development Project of China(2015CB953601)
文摘Depositional growth of ice crystal is one of the major processes for development of precipitation systems and can be represented by depositional growth of cloud ice from cloud water(P_(IDW)) and depositional growth of snow from cloud ice(P_(SFI)) in cloud-resolving model. Four parameterization schemes are analyzed in the cloud-resolving model simulations of four rainfall cases over the tropics and midlatitudes. The comparison of time and model domain mean data shows that Shen's scheme produces the closest rainfall simulation to the observation. Compared to Zeng's scheme,Shen's scheme improves the mean rain-rate simulation significantly through the dramatic decrease in depositional growth of cloud ice from cloud water. Compared to other schemes, Shen's scheme produces the better rainfall simulation via the reduction in the mean rain rate associated with the enhanced gain of cloud water and ice.
文摘Bangladesh is a densely populated country than many other countries of the world. The population growth is termed as alarming, however, knowledge of growth in the years to come would be useful in planning for the development of the country. This article is based on the projection of future population growth of the country. The available actual population census data during 1991-2011 of Bangladesh was applied to the application of a non-linear, non-autonomous ordinary differential equation familiar as Verhulst logistic population model with the maximum environmental capability of Bangladesh. Bangladesh will reach its carrying capacity of 245.09 million population in the next 56 years i.e. the year 2067 and then it decreases as S-shaped curve. The article has provided a focus on the changing trends of the growth of the population of Bangladesh.
文摘Based on the basic ecological model proposed by Jorgensen and the species composition of lake ecosystem, a new approach to calculate the growth rate of phytoplankton weighted growth rate method was presented. The thermodynamic concept “Exergy” was also introduced into the ecodynamic model for lakes,and the changes in the species composition and the ecological structure of lake ecosystem were reflected by the changing exergy.An improved ecodynamic model for lakes was developed by combining the weighted growth rate with the application of exergy. These improvements comprehensively take the ecological properties of lake ecosystems into account, making the ecodynamic model ecologically more reasonable and practically more flexible. The applications of the improved ecodynamic model in the ecological modelling and prediction of Dianchi Lake, have demonstrated that this model is practicable and satisfactory.
文摘A self adjusting model was presented on the basis of the effect of temperature gradient on eutectic growth and a curved solid/liquid interface. Finite differential method was adopted to solve the model. The average lamellar spacing of the Al Al 3Fe eutectic alloy and the content fields ahead of the solidifying interface under different growth rates were calculated. Directional solidification experiments were carried out in order to prove the modification of the modeling. The experimental results are in relatively good agreement with the calculations.
文摘Many curves have been proposed and debated to model individual growth of marine invertebrates. Broadly, they fall into two classes, first order (e.g. von Bertalanffy) and sigmoidal (e.g. Gompertz). We provide an innovative approach which demonstrates that the growth curves are not mutually exclusive but that either may arise from a simple three-stage growth model with two steps (k<sub>1 </sub>and k<sub>2</sub>) depending on the ratio of the growth parameters . The new approach predicts sigmoidal growth when is close to 1, but if either growth from stage A to stage B or B to C is fast relative to the other, the slower of the two steps becomes the growth limiting step and the model reduces to first order growth. The resulting curves indicate that there is a substantial difference in the estimated size at time t during the period of active growth. This novel two-step rate model generates a growth surface that allows for changes in the rate parameters over time as reflected in the new parameter n(t) = k<sub>1</sub>(t) - k<sub>2</sub>(t). The added degree of freedom brings about individual growth trajectories across the growth surface that is not easily mapped using conventional growth modeling techniques. This two (or more) stage growth model yields a growth surface that allows for a wide range of growth trajectories, accommodating staged growth, growth lags, as well as indeterminate growth and can help resolve debates as to which growth curves should be used to model animal growth. This flexibility can improve estimates of growth parameters used in population models influencing model outcomes and ultimately management decisions.=
文摘A better understanding of the mechanisms that control nutrient acquisition in the context of plant and ecosystem responses to climate change is needed. Mechanistic nutrient uptake models provide a means to investigate some of the impacts of temperature change on soil nutrient supply and root uptake kinetics through the simulation of key soil and plant processes. The NST 3.0 model, in combination with literature values on plant and soil parameters from a red spruce (Picea rubens L.) site in the southern Appalachians, was used to conduct a series of model simulations focused on the combined effects of changes to the maximal rate of nutrient influx at high concentrations (Imax), root growth rate (k), concentration of nutrient occurring in the soil solution (Cli), and the ability of the soil solid phase to buffer changes to the soil solution nutrient concentration (b). Previous research has indicated that these four parameters are responsive to changes in root zone temperature. Simulated uptake of NH4 increased by a factor of up to 2.6 in response to increases in soil temperature of 1°C to 5°C. The model also projected an increase in P uptake coupled with up to an 80% reduction in solution P concentration in response to a 1°C -5°C increase over a 147-d simulation period. These hypothetical changes, if validated, have interesting implications for plant growth and competition and point to a need for additional studies to better define the impacts of soil temperature on soil nutrient supply and root uptake.
基金supported by the National Natural Science Foundation of China (Nos. 10872220 and 50725414)Japan Society for the Promotion of Science JSPS (No. L08538)
文摘Based on analysis of deformation in an infinite isotropic elastic matrix containing an embedded elliptic crack, subject to far field triaxial compressive stress, the energy release rate and a mixed fracture criterion are obtained by using an energy balance approach. The additional compliance tensor induced by a single closed elliptic microcrack in a representative volume element and its in-plane growth is derived. The additional compliance tensor induced by the kinked growth of the elliptic microcrack is also obtained. The effect of the microcracks, randomly distributed both in geometric characteristics and orientations, is analyzed with the Taylor＇s scheme by introducing an appropriate probability density function. A micromechanical damage model for rocks and concretes under triaxial compression is obtained and experimentally verified.
