自组织临界性、分形及灾变理论研究
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摘要
本文围绕着沙堆模型实验,开展了工程领域的散粒体系统呈现自组织临界性的判据的研究;然后基于自组织临界性的判据对两个典型灾害系统——地震系统和斜坡松散堆积物系统的组构特征和自组织临界性动力学行为之间的关系进行了探讨;最后主要针对影响公路和铁路的斜坡重力作用灾害,在自组织临界性概念的框架下,分别对灾害的形成机理、运动过程、预测预报和防治对策进行了专题研究。研究的主要工作及结论如下:
1、在课题组先前实验的基础上,开展了各种粒径级配的沙堆实验。在大尺度下,粒径均匀的沙堆和粒径级配曲线斜率不连续的非均匀沙堆表现为序列的准周期行为或规模的正态分布,而粒径级配曲线斜率连续的非均匀沙堆呈现出SOC特征。
2、提出用信息熵来描述自组织临界性系统中幂次分布所包含的信息。首次定义了真实沙堆演化中沙崩大小分布的信息熵,并对沙崩信息熵进行了分析。随着沙堆演化的向前推进沙崩信息熵增大,并在临界态基本趋于稳定,达到最大值。沙崩信息熵体现了系统的无序度,表现了沙堆系统演化的复杂性,这对于理解复杂系统的潜在作用机制有着十分重要的意义。同时沙崩信息熵也表明临界态的趋近可以通过熵的变化来体现,沙崩信息熵大小表明了沙堆当前状态与临界态之间的距离,沙崩信息熵越小表明沙堆当前状态距离临界态越远,沙崩信息熵越大表明沙堆当前状态距离临界态越近,当沙崩信息熵达到最大时表明沙堆系统的临界态已经达到。
3、利用筛下沙粒数量构造了关联维数,推导出了关联维数的具体计算公式,计算了单面坡沙堆实验各种粒径级配的分维数。结合沙堆实验得出具有自组织临界性的沙堆系统,其颗粒粒径具有分形分布,提出了具有自组织临界性的系统其组构具有分形特点和系统组构的分形结构可以作为自组织临界现象发生的判据的假说,并利用颗粒空间排列具有分形特点的元胞自动机沙堆模型验证了这一假说。
4、依据G—R定律,分析了中国大陆地区的地震活动性,发现地震活动在空间和能量分布上,当考虑其整体行为时具自组织临界性。利用分形理论中的粗视化网格法,深入研究了中国大陆地区断层系的分形特点和分形结构的跨尺度特征。建立了分形断裂力学简化模型,给出了地震活动性的b值与断层系的分维数D以及断层面的分维数d之间的定量关系,探讨了b的物理机制,以及b和D之间的正相关关系,从理论上和实践上证明了地震的自组织临界现象源于分形几何断层系的自组织临界性的动力学过程,明确了系统组构分形和地震活动性之间的因果关系。
5、研究了国道317都江堰至汶川K74+775～K75+740段7处斜坡松散堆积物颗粒粒度分布的分维规律,研究表明:斜坡松散堆积物颗粒分布具有良好的分形结构,分维数随着斜坡高度的增加而增大,分维数越小,斜坡松散堆积物中粗颗粒成分越多,分维数越大,细颗粒成分越多,分维数在2到3之间。
以实测资料为依据,从颗粒流理论出发,结合大型相对密度仪和直接剪切仪实验,探讨了材料性质相同的颗粒物质其宏观力学性质与颗粒粒径分形分布之间的关系。推导了剪切应力系数、正应力系数、内摩擦角和颗粒浓度与粒径分维数之间的关系。结果表明:粒径分维数D小的斜坡松散堆积物,流动性能差,崩塌流动过程中,斜坡松散堆积物系统与干扰相互作用的效应使松散物沿坡面下落阻碍增强,从而限制了系统各部分敏感性递增的趋势,使得扰动强度不至于总是超过自稳能力,该斜坡松散堆积物系统仍具有自组织临界性动力学特征,进而得出斜坡松散堆积物系统对干扰传播制约的能力是由组构的分形特征所决定的系统自身的性质决定的。
6、分析了岩石破坏过程的自组织临界特征。为了研究这种临界问题,提出了一种利用自组织程度参数α刻画单元之间的应力转移和局部单元上的应力集中的方法,并以单元的破坏强度服从二次Weibull分布为基础,运用重整化群理论建立了计算力学模型,推导了岩石破坏的临界破坏概率p_c。结果表明,当单元的破坏概率p小于临界破坏概率p_c时,单元的破坏是独立的和随机的,系统向稳定态演化;当单元的破坏概率p大于临界破坏概率p_c时,单元的破坏向宏观贯通的张裂面这一吸引域集中;临界破坏概率p_c以及单元承受外载F的能力均与初始条件有关,p_c和F/F_0随自组织程度参数α增大而减小,且F/F_0＜1,表明由于应力的转移使得单元破坏应力降低。
7、通过对区域滑坡的空间规模分布,以及单个滑坡演化过程的位移时间序列分析,证明了滑坡灾害具有自组织临界性。
滑坡一般发生在众多不连续面中强度效应最弱的主滑面上。滑面在演化过程中是一种远离平衡态的自组织临界现象。运用图型动力学模型从图型演化角度分析了滑动面损伤破坏方式的“多米诺效应”和损伤破坏的“链式反应”过程。以确定性的、不可逆的、非线性、非局域和没有特征尺度的动力学规则,描述损伤单元之间的一种自动放大的非局域相互作用的连接过程,引入分布函数φ(p)和ζ(p)刻画断裂态吸引域在相空间的分布。探讨了不同损伤率下,滑面破坏的可能模式。
8、分析了自组织临界性系统的可预测性,以及自组织临界性理论对灾害预测预报和灾害防治工程的启示。
在自组织临界性的概念框架下,将改进的重标极差分析方法即RSH分析方法引入泥石流灾害前兆观测资料的处理和分析中,并在计算赫斯特指数H时,定义一个涵盖序列历史和序列长时记忆信息的时间窗,应用该方法对泥石流灾害进行了预测预报。
基于自组织临界性的频率—规模的幂律分布关系,从可靠性设计角度出发,建立了斜坡防护系统的可靠性分析模型,并应用于实例的计算。
Based on the sand-pile model test,this article did deep research on the criterion of self-organized criticality(SOC)about granular system in civil engineering field.The relationship between the fabric characteristics and dynamic behavior of two typical disaster system,earthquake system and loose deposits system,was discussed based on the criterion of SOC.Finally,aiming at the earthquake disaster and slope gravitational disaster in highway and railway engineering,this article carries out the special research on the formative mechanism,the process of movement and forecast,prevention measures of disasters in the frame of idea of SOC.In details,the main jobs and conclusions of this dissertation are as follows.
1.On the basis of previous sand-pile model test,a series of sand-pile model tests of different grain size distribution were carried out.Under the large-scale conditions,the avalanches of uniform sand and non-uniform sand,which grain size distribution curve were discontinuity,exhibited quasi-period distribution or normal distribution,however,the avalanches of non-uniform sand which grain size distribution curve were continuity presented the characteristics of self-organized criticality.
2.The concept of information entropy was provided to describe the information contained by the power-law distribution of SOC system.The information entropy of avalanches was defined and analyzed in the sand-pile slopes evolvement for the first time.The information entropy gradually increases with the evolving of sand-pile,and reaches a maximum to stabilization in the critical state.The information entropy of avalanches indicates the disorder degree of systems and represents the complex of sand-pile systems evolvement,which is important to understand the potential action mechanism of complex system. Meanwhile,the information entropy of avalanches indicate also to obtain the critical state by changing the information entropy,the value of entropy show the distance between current state and critical state,and the value is more smaller to show more further critical state,but more greater to more nearer its,when information entropy reach the greatest value,it shows the sand-pile system attach the critical state.
3.Correlation dimension was constructed by using the amount of sands under riddle,and the concrete calculating formula was deduced,therefore,we obtained dimensions of sand-pile with one-side and different grain size distribution,and gained the grain size distribution of sand-pile system bearing SOC present fractal distribution combined with sand-pile test,so we put forward the configuration of SOC system have fractal characteristics and the fractal configuration of system can be considered a hypothesis of SOC,and proved the hypothesis by means of cellular automata sand-pile model,which grains array in space have fractal characteristics.
4.According as the laws of G-R,the earthquake activities were analyzed in the tectonic regions of the Chinese mainland,we found the earthquake activities in space and energies distribution show SOC as a whole.The fractal characteristics and trans-scale features of fractal structural of faults in Chinese mainland and tectonic regions were researched comprehensively with the macroscopic grid method in fractal theory.A fractal fracture mechanics simple model was established,the quantitative relationship between earthquake the activities value,b, the dimensions of faults,D and the dimensions of fault-face,d was presented, furthermore,the physical mechanism of the b and the positive correlation between the b and the D were discussed.From the theory and practice,these simple models proved the SOC phenomenon of earthquake have its origin in the process of dynamics of SOC and its fractal faults;made clear the consequence between the configuration fractal and earthquake activities.
5.The dimensions law of grain size distribution of loose deposits along the national road 317 from Doujiangyan to Wenchuan(7-site in road section K74+775～K75+740)were studied,the results showed that the grain size distribution of loose deposits have fractal structure,and the dimensions increase with slope height,when the value of dimensions was littler,the more components of coarse grain to loose deposits on slope,but the value was greater,the more components of smaller grain,in a word,the value of dimensions varied from two to three.
From the theory of particle flow,combined with the test results of large-scale relative density meter and direct shear apparatus,the relation between macroscopically mechanical character of the identical particles and the fractals distribution of grain size was discussed based on the data of experiments. Furthermore,we also deduced the relation between coefficients of stress,φ,and grains density of loose deposits and dimensions of grain size distribution.From these results,we found the fluidness of loose deposits was not flowing,when the value of dimension D of grains was little,during the landslip process,the effect of disturbances upon the systems of loose deposits on a slope strengthened the loose deposits fall,accordingly,it limited the increasing trend of systems sensitivity,and the intensities of disturbances does not always exceed the capability of self-stability,ultimately,the system of loose slope had the character of SOC.So we deduced the capability of a system of loose slope restriction against disturbances spreading was decided by the fractal character of system itself.
6.This article analyzed a characteristic of Self-organized criticality of rock failure process.A method which use Self-organized extent parameterαto express stress transfer among units and stress concentration in local units was put forward. Based on the theory that the failure strength of rock units obey second power Weibull distribution,a mechanical model was established with renormalization group theory,and the critical probability of rock failure,pc,was derived.Results indicate that the failure of units is independent and stochastic,and the system evolves to a stable state when the failure probability of unit,p,is less than the critical failure probability pc.When p is higher than pc,the failure of units focus on the macroscopic-transfixion tension split plane,which is a dynamic attractor of Self-organized criticality.The critical failure probability pc and the unit capacity of bearing extemal load F is related to initial condition,pc as well as F/F_0,decrease with the increase of Self-organized extent parameterα,and furthermore,F/F_0＜1. The failure stress of units decreases due to stress transfer.
7.We testified the disaster of landslips have the character of SOC using the space distribution of landslips in a region,and analysis on sequence of velocity and time to a single landslip evolvement process.
Landslides generally occur on the main slide surface whose strength is the lowest among multitudinous discontinuities.Traditional stability analysis methods for slide surface are mostly a kind of deterministic analysis based on the limit equilibrium theory of rigid body or improved limit equilibrium theory.Actually, slide surface is a kind of extended dissipative dynamical complex systems which vary in the time and space dimension,it exchanges substance,energy and information with outside environment.The evolvement process of slide surface is a self-organized criticality phenomenon which is far away from the equilibrium state.SOC is the dynamical attractor in the course of slide surface running through. By using pattern dynamic model,the "domino-effect" mode and the "chain reaction" process of slide surface failure was analyzed in this paper.A sort of auto-amplifying interactional joining process among damage units was described with deterministic,irreversible,non-linear,non-local and non-characteristic scale dynamic regulation.Distribution functions ofφ(p)andξ(p)were introduced to depict the distribution of rupture attractable area in phase space.At length,the possible destruction patterns of slide surfaces with different damage ratio were also discussed.
8.Finally,this article analyzed the predictability of SOC and the apocalypse of the theory of SOC to forecast and prevention of disasters.
Under the frame of theory of SOC,The precursor monitoring data of the debris flow disaster was processed and analyzed with a method of improved rescaled range analysis(RSH algorithm).And,a time window which contains sequences history and longtime memory information was defined when calculating Hurst index H.The method can predict debris flow disaster.
From the angle of dependable design,we built a model of dependable analysis of the system of safeguard to a loose slope,and the model was applied to practice engineering.
1、在课题组先前实验的基础上,开展了各种粒径级配的沙堆实验。在大尺度下,粒径均匀的沙堆和粒径级配曲线斜率不连续的非均匀沙堆表现为序列的准周期行为或规模的正态分布,而粒径级配曲线斜率连续的非均匀沙堆呈现出SOC特征。
2、提出用信息熵来描述自组织临界性系统中幂次分布所包含的信息。首次定义了真实沙堆演化中沙崩大小分布的信息熵,并对沙崩信息熵进行了分析。随着沙堆演化的向前推进沙崩信息熵增大,并在临界态基本趋于稳定,达到最大值。沙崩信息熵体现了系统的无序度,表现了沙堆系统演化的复杂性,这对于理解复杂系统的潜在作用机制有着十分重要的意义。同时沙崩信息熵也表明临界态的趋近可以通过熵的变化来体现,沙崩信息熵大小表明了沙堆当前状态与临界态之间的距离,沙崩信息熵越小表明沙堆当前状态距离临界态越远,沙崩信息熵越大表明沙堆当前状态距离临界态越近,当沙崩信息熵达到最大时表明沙堆系统的临界态已经达到。
3、利用筛下沙粒数量构造了关联维数,推导出了关联维数的具体计算公式,计算了单面坡沙堆实验各种粒径级配的分维数。结合沙堆实验得出具有自组织临界性的沙堆系统,其颗粒粒径具有分形分布,提出了具有自组织临界性的系统其组构具有分形特点和系统组构的分形结构可以作为自组织临界现象发生的判据的假说,并利用颗粒空间排列具有分形特点的元胞自动机沙堆模型验证了这一假说。
4、依据G—R定律,分析了中国大陆地区的地震活动性,发现地震活动在空间和能量分布上,当考虑其整体行为时具自组织临界性。利用分形理论中的粗视化网格法,深入研究了中国大陆地区断层系的分形特点和分形结构的跨尺度特征。建立了分形断裂力学简化模型,给出了地震活动性的b值与断层系的分维数D以及断层面的分维数d之间的定量关系,探讨了b的物理机制,以及b和D之间的正相关关系,从理论上和实践上证明了地震的自组织临界现象源于分形几何断层系的自组织临界性的动力学过程,明确了系统组构分形和地震活动性之间的因果关系。
5、研究了国道317都江堰至汶川K74+775～K75+740段7处斜坡松散堆积物颗粒粒度分布的分维规律,研究表明:斜坡松散堆积物颗粒分布具有良好的分形结构,分维数随着斜坡高度的增加而增大,分维数越小,斜坡松散堆积物中粗颗粒成分越多,分维数越大,细颗粒成分越多,分维数在2到3之间。
以实测资料为依据,从颗粒流理论出发,结合大型相对密度仪和直接剪切仪实验,探讨了材料性质相同的颗粒物质其宏观力学性质与颗粒粒径分形分布之间的关系。推导了剪切应力系数、正应力系数、内摩擦角和颗粒浓度与粒径分维数之间的关系。结果表明:粒径分维数D小的斜坡松散堆积物,流动性能差,崩塌流动过程中,斜坡松散堆积物系统与干扰相互作用的效应使松散物沿坡面下落阻碍增强,从而限制了系统各部分敏感性递增的趋势,使得扰动强度不至于总是超过自稳能力,该斜坡松散堆积物系统仍具有自组织临界性动力学特征,进而得出斜坡松散堆积物系统对干扰传播制约的能力是由组构的分形特征所决定的系统自身的性质决定的。
6、分析了岩石破坏过程的自组织临界特征。为了研究这种临界问题,提出了一种利用自组织程度参数α刻画单元之间的应力转移和局部单元上的应力集中的方法,并以单元的破坏强度服从二次Weibull分布为基础,运用重整化群理论建立了计算力学模型,推导了岩石破坏的临界破坏概率p_c。结果表明,当单元的破坏概率p小于临界破坏概率p_c时,单元的破坏是独立的和随机的,系统向稳定态演化;当单元的破坏概率p大于临界破坏概率p_c时,单元的破坏向宏观贯通的张裂面这一吸引域集中;临界破坏概率p_c以及单元承受外载F的能力均与初始条件有关,p_c和F/F_0随自组织程度参数α增大而减小,且F/F_0＜1,表明由于应力的转移使得单元破坏应力降低。
7、通过对区域滑坡的空间规模分布,以及单个滑坡演化过程的位移时间序列分析,证明了滑坡灾害具有自组织临界性。
滑坡一般发生在众多不连续面中强度效应最弱的主滑面上。滑面在演化过程中是一种远离平衡态的自组织临界现象。运用图型动力学模型从图型演化角度分析了滑动面损伤破坏方式的“多米诺效应”和损伤破坏的“链式反应”过程。以确定性的、不可逆的、非线性、非局域和没有特征尺度的动力学规则,描述损伤单元之间的一种自动放大的非局域相互作用的连接过程,引入分布函数φ(p)和ζ(p)刻画断裂态吸引域在相空间的分布。探讨了不同损伤率下,滑面破坏的可能模式。
8、分析了自组织临界性系统的可预测性,以及自组织临界性理论对灾害预测预报和灾害防治工程的启示。
在自组织临界性的概念框架下,将改进的重标极差分析方法即RSH分析方法引入泥石流灾害前兆观测资料的处理和分析中,并在计算赫斯特指数H时,定义一个涵盖序列历史和序列长时记忆信息的时间窗,应用该方法对泥石流灾害进行了预测预报。
基于自组织临界性的频率—规模的幂律分布关系,从可靠性设计角度出发,建立了斜坡防护系统的可靠性分析模型,并应用于实例的计算。
Based on the sand-pile model test,this article did deep research on the criterion of self-organized criticality(SOC)about granular system in civil engineering field.The relationship between the fabric characteristics and dynamic behavior of two typical disaster system,earthquake system and loose deposits system,was discussed based on the criterion of SOC.Finally,aiming at the earthquake disaster and slope gravitational disaster in highway and railway engineering,this article carries out the special research on the formative mechanism,the process of movement and forecast,prevention measures of disasters in the frame of idea of SOC.In details,the main jobs and conclusions of this dissertation are as follows.
1.On the basis of previous sand-pile model test,a series of sand-pile model tests of different grain size distribution were carried out.Under the large-scale conditions,the avalanches of uniform sand and non-uniform sand,which grain size distribution curve were discontinuity,exhibited quasi-period distribution or normal distribution,however,the avalanches of non-uniform sand which grain size distribution curve were continuity presented the characteristics of self-organized criticality.
2.The concept of information entropy was provided to describe the information contained by the power-law distribution of SOC system.The information entropy of avalanches was defined and analyzed in the sand-pile slopes evolvement for the first time.The information entropy gradually increases with the evolving of sand-pile,and reaches a maximum to stabilization in the critical state.The information entropy of avalanches indicates the disorder degree of systems and represents the complex of sand-pile systems evolvement,which is important to understand the potential action mechanism of complex system. Meanwhile,the information entropy of avalanches indicate also to obtain the critical state by changing the information entropy,the value of entropy show the distance between current state and critical state,and the value is more smaller to show more further critical state,but more greater to more nearer its,when information entropy reach the greatest value,it shows the sand-pile system attach the critical state.
3.Correlation dimension was constructed by using the amount of sands under riddle,and the concrete calculating formula was deduced,therefore,we obtained dimensions of sand-pile with one-side and different grain size distribution,and gained the grain size distribution of sand-pile system bearing SOC present fractal distribution combined with sand-pile test,so we put forward the configuration of SOC system have fractal characteristics and the fractal configuration of system can be considered a hypothesis of SOC,and proved the hypothesis by means of cellular automata sand-pile model,which grains array in space have fractal characteristics.
4.According as the laws of G-R,the earthquake activities were analyzed in the tectonic regions of the Chinese mainland,we found the earthquake activities in space and energies distribution show SOC as a whole.The fractal characteristics and trans-scale features of fractal structural of faults in Chinese mainland and tectonic regions were researched comprehensively with the macroscopic grid method in fractal theory.A fractal fracture mechanics simple model was established,the quantitative relationship between earthquake the activities value,b, the dimensions of faults,D and the dimensions of fault-face,d was presented, furthermore,the physical mechanism of the b and the positive correlation between the b and the D were discussed.From the theory and practice,these simple models proved the SOC phenomenon of earthquake have its origin in the process of dynamics of SOC and its fractal faults;made clear the consequence between the configuration fractal and earthquake activities.
5.The dimensions law of grain size distribution of loose deposits along the national road 317 from Doujiangyan to Wenchuan(7-site in road section K74+775～K75+740)were studied,the results showed that the grain size distribution of loose deposits have fractal structure,and the dimensions increase with slope height,when the value of dimensions was littler,the more components of coarse grain to loose deposits on slope,but the value was greater,the more components of smaller grain,in a word,the value of dimensions varied from two to three.
From the theory of particle flow,combined with the test results of large-scale relative density meter and direct shear apparatus,the relation between macroscopically mechanical character of the identical particles and the fractals distribution of grain size was discussed based on the data of experiments. Furthermore,we also deduced the relation between coefficients of stress,φ,and grains density of loose deposits and dimensions of grain size distribution.From these results,we found the fluidness of loose deposits was not flowing,when the value of dimension D of grains was little,during the landslip process,the effect of disturbances upon the systems of loose deposits on a slope strengthened the loose deposits fall,accordingly,it limited the increasing trend of systems sensitivity,and the intensities of disturbances does not always exceed the capability of self-stability,ultimately,the system of loose slope had the character of SOC.So we deduced the capability of a system of loose slope restriction against disturbances spreading was decided by the fractal character of system itself.
6.This article analyzed a characteristic of Self-organized criticality of rock failure process.A method which use Self-organized extent parameterαto express stress transfer among units and stress concentration in local units was put forward. Based on the theory that the failure strength of rock units obey second power Weibull distribution,a mechanical model was established with renormalization group theory,and the critical probability of rock failure,pc,was derived.Results indicate that the failure of units is independent and stochastic,and the system evolves to a stable state when the failure probability of unit,p,is less than the critical failure probability pc.When p is higher than pc,the failure of units focus on the macroscopic-transfixion tension split plane,which is a dynamic attractor of Self-organized criticality.The critical failure probability pc and the unit capacity of bearing extemal load F is related to initial condition,pc as well as F/F_0,decrease with the increase of Self-organized extent parameterα,and furthermore,F/F_0＜1. The failure stress of units decreases due to stress transfer.
7.We testified the disaster of landslips have the character of SOC using the space distribution of landslips in a region,and analysis on sequence of velocity and time to a single landslip evolvement process.
Landslides generally occur on the main slide surface whose strength is the lowest among multitudinous discontinuities.Traditional stability analysis methods for slide surface are mostly a kind of deterministic analysis based on the limit equilibrium theory of rigid body or improved limit equilibrium theory.Actually, slide surface is a kind of extended dissipative dynamical complex systems which vary in the time and space dimension,it exchanges substance,energy and information with outside environment.The evolvement process of slide surface is a self-organized criticality phenomenon which is far away from the equilibrium state.SOC is the dynamical attractor in the course of slide surface running through. By using pattern dynamic model,the "domino-effect" mode and the "chain reaction" process of slide surface failure was analyzed in this paper.A sort of auto-amplifying interactional joining process among damage units was described with deterministic,irreversible,non-linear,non-local and non-characteristic scale dynamic regulation.Distribution functions ofφ(p)andξ(p)were introduced to depict the distribution of rupture attractable area in phase space.At length,the possible destruction patterns of slide surfaces with different damage ratio were also discussed.
8.Finally,this article analyzed the predictability of SOC and the apocalypse of the theory of SOC to forecast and prevention of disasters.
Under the frame of theory of SOC,The precursor monitoring data of the debris flow disaster was processed and analyzed with a method of improved rescaled range analysis(RSH algorithm).And,a time window which contains sequences history and longtime memory information was defined when calculating Hurst index H.The method can predict debris flow disaster.
From the angle of dependable design,we built a model of dependable analysis of the system of safeguard to a loose slope,and the model was applied to practice engineering.
引文
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