(1. Beijing Urban Construction Group Co., Ltd. Beijing 100044
2. Institute of Geology and Geophysics, China Academy of Sciences, Beijing 100029)
Soil-rock mixture is a heterogeneous discontinuous body. According to the structural characteristics of soil-rock mixture, the size effect of soil-rock mixture is analyzed, and the stochastic structure model of soil-rock mixture is put forward. In this paper, the measured statistics, distribution function and generation method of three random variables, namely spatial position, size and orientation of gravel block, are deeply studied. By using stochastic simulation method, a set of automatic generation technology of stochastic structure model is proposed, and finally two examples of stochastic structure model are given.
Automatic generation technology of random structure model of soil-rock mixture
The important sign of the transformation from qualitative to quantitative in geotechnical engineering is the establishment of geotechnical medium model, and the rationality and effectiveness of the model is the basic premise to ensure the accuracy and reliability of the quantitative analysis results of engineering. Soil-rock mixture is a heterogeneous discontinuous body, and its mechanical properties are controlled by its internal structure [1 ~ 4]. The uncertainty, irregularity and fuzziness of the mechanical behavior of soil-rock mixture are the concrete manifestations of its structural complexity. Therefore, reflecting and determining the nonlinear characteristics of geotechnical structures has become the key to break through the constraints of continuum mechanics model and develop new ideas of geotechnical mechanics theory and technology. Whether a geotechnical analysis method can successfully solve practical geotechnical engineering problems depends largely on how it truly reflects these characteristics of engineering rock mass. Based on the measured profile of soil-rock mixture, this paper deeply studies the measured statistics, distribution functions and generation methods of three random variables, such as spatial position, size and orientation of gravel blocks, by using image processing technology and finite difference mesh automatic generation technology, and puts forward a set of techniques for automatically generating random structural models by using random simulation method. Finally, several examples of stochastic structural models are given.
Analysis of mechanical properties of 1 soil-rock mixture
Soil-rock mixture has typical heterogeneity and discontinuity because it contains gravel blocks of different sizes, types and quantities, and shows strong anisotropy in mechanical properties.
In practical engineering, the inhomogeneity of rock and soil is not only manifested in the inhomogeneity of material composition distribution, but also mainly in the inhomogeneity of rock and soil structure [5, 6]. Soil-rock mixture can be considered as a composite structure, which can contain clay or sand filler with relatively low strength, or gravel block with relatively high strength, which determines its heterogeneity in material composition. At the same time, the stones contained can have various spatial structures and directions, and their sizes are also very different, which determines its structural heterogeneity.
It is precisely because of the material heterogeneity and structural heterogeneity of soil-rock mixture that its physical and mechanical properties show obvious heterogeneity, anisotropy and discontinuity and the complexity of stress redistribution [7].
The traditional methods to determine the mechanical properties of this material have little effect: the usual field and indoor tests can not provide representative research conditions compared with large-scale rock and soil bodies. In order to reduce these shortcomings, the method adopted in this paper is to give different material strength parameters to gravel blocks with higher strength and clay sand fillers with lower strength respectively to solve the non-uniformity of material composition; At the same time, the percentage content, size, shape and direction of gravel inclusions are simulated in the form of random variables to solve the structural heterogeneity of soil-rock mixture.
In the soil-rock mixture, because it contains two or more substances with different hardness, the stress and displacement are discontinuous during deformation, which is mainly reflected in the interface between the two different substances, and the gravel block and soil unit can be pulled and slid. In this paper, the interface element is used to simulate this interface, and the results show that it can well reflect the discontinuity of displacement and stress.
The uniformity and continuity of geotechnical materials are related to the research scale and investigation scope [8, 9]. Blocks with severe fractures at one scale may be a combination of large blocks at another scale. Changing the research scale will change the heterogeneity, discontinuity and anisotropy of rock mass, so the mechanical methods adopted will also change.
Gravel blocks contained in soil-rock mixture are large and small, and their distribution is extremely uneven. However, in practical research, it is necessary to ensure that the maximum particle size is not greater than 1/5 of the sample size. For special large particles, we should analyze them separately and study the influence of their existence on the whole stress field. Particularly small particles are not considered.
From the above analysis, we can see that the following principles must be adhered to when establishing the model:
(1) In the soil-rock mixture model, it is assumed that there are only two materials with different strengths, which are homogeneous and continuous respectively.
(2) In numerical simulation, the contact interface between "inclusion-filler" should be considered as much as possible to simulate the discontinuity of soil-rock mixture.
(3) The size of the study area should be at least 4 ~ 5 times larger than the characteristic size of the largest inclusion, so as to obtain the representative properties of the soil-rock mixture.
(4) The numerical simulation can be carried out according to the plane deformation mode, ignoring the body stress state of the specimen. Experiments show that the results are actually equal to or slightly less than the strength and deformation characteristics in the experiment under bulk stress. The compound theory put forward by Christensen in 1982 confirms this idea.
2. Automatic generation technology of random structure model based on random distribution.
Monte Carlo method, also known as random simulation method or statistical calculation method, is a numerical calculation method of computer simulation of random variables determined by statistical sampling theory [10 ~ 12]. It is widely used. In geotechnical engineering, Monte Carlo simulation of joints and cracks has attracted wide attention and been applied in many projects. Stochastic finite element method and stochastic boundary element method, which are developed by combining Monte Carlo method with finite element method, are further developed.
In this paper, Monte Carlo method is used to simulate the size, orientation and spatial distribution of gravel in soil-rock mixture, so as to establish a random structure model of soil-rock mixture based on random distribution. The basic idea is: according to the field geological investigation of the soil-rock mixture, the particle group analysis of typical samples and the statistics of stone content, for the convenience of research, it is assumed that the spatial position of gravel in the soil-rock mixture obeys the uniform distribution, and the size and orientation of gravel obey the lognormal distribution. Firstly, a location point is generated uniformly in a certain research area; Then, given two random geological parameters, size and orientation, the random structure model of soil-rock mixture can be established by using these attribute parameters and the drawing function of AutoCAD or Ansys.
When establishing the stochastic structure model of soil-rock mixture, three random variables, namely the spatial position, size and orientation of gravel blocks, are needed.
2. Spatial position of1block
Generally speaking, the blocks are evenly and randomly distributed in the sample, as shown in figure 1. Because it is a plane problem, it is necessary to generate a randomly distributed number pair (x, y). This is a two-dimensional random variable, where x and y are the x and y coordinates of a point respectively. According to the actual situation, the random variables X and Y are statistically independent, and their joint probability density function is:
Soil-rock mixture
Where: fx(x) is the edge distribution density function of X; Fy(y) is the edge distribution density function of y.
At this time, the random number of each random variable can be generated independently by using a single random variable. In practical application, two random sequences belonging to different intervals can be generated by inverse function method, that is
Soil-rock mixture
Soil-rock mixture
From this, a random number pair (,yi) can be obtained, that is, a point is generated in the area shown in figure 1
Figure 1 Spatial Arrangement of Uniform and Random Distribution
2.2 Block size
The shape and size of gravel blocks in soil-rock mixture vary according to their types. For example, the blocks in the soil-rock mixture formed by landslide accumulation are mostly round or sub-round with different sizes, while the blocks in the soil-rock mixture formed by collapse accumulation are mostly angular. Regarding the shape, this paper considers two limit cases: triangle and circle. As for the size of the block, the author counts the particle size of gravel blocks in two samples in Baiyian landslide area, and also counts the area size and average particle size of the blocks in the samples through digital photography. The histogram is shown in Figure 2 to Figure 5.
Fig. 2 Histogram of manual statistics 1 sample particle size distribution
Fig. 3 Histogram of Particle Size Distribution of Sample 2 by Manual Statistics
The abscissa in the figure represents different particle size groups. For example, in Figure 2, the group 1: is less than1cm; Group 2:1~ 2 cm; The third group: 2 ~ 4cm；; The fourth group: 4 ~ 6cm；; Group 5: 6 ~ 8cm；; Group 6: 8 ~10 cm; Group 7:10 ~12 cm; Group 8: greater than 12cm. As can be seen from the figure, the particle size basically conforms to the lognormal distribution. The probability density function of lognormal distribution is:
Fig. 4 Average particle size histogram of automatically counted numerical samples.
Histogram of numerical sample area distribution with automatic counting.
Soil-rock mixture
The density function f(x) of lognormal distribution is partial normal, as shown in Figure 6. As can be seen from the figure, the value of the lognormal random variable X is in the positive domain, so a considerable number of random variables in engineering, such as soil cohesion and internal friction angle, obey lognormal distribution [12].
Fig. 6 Probability density function of lognormal distribution
For this kind of random variable, coordinate transformation is generally used to generate it.
Let r 1 and r2 be two independent random numbers evenly distributed on [0, 1], and convert them into
Soil-rock mixture
η 1 and η2 are two independent random numbers, which obey the standard normal distribution. From the relationship between normal distribution and standard normal distribution, we can easily find the random number of nonstandard normal distribution with mean value μx and standard deviation σx x 1, x2:
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From the relationship between normal distribution and lognormal distribution, the lognormal random number is obtained:
Soil-rock mixture
2.3 the direction of the block
The orientation of the block refers to the dominant direction of the block, which does not exist for circular blocks; For triangles, especially planar polygons, it is of great significance, because different orientations have different deformation and failure mechanisms, and the strength properties of samples are also very different. In this calculation, only the equilateral triangle is considered, and its direction can be defined as shown in Figure 7.
Fig. 7 Orientation Map of Triangle Block
The three vertices of a triangle are 1, 2 and 3 respectively. Taking the point 1 as an example, the angle between the line connecting the point 1 and the origin and the X axis is defined as the azimuth angle α of the triangle. Because it is an equilateral triangle, 0 < α < 120.
For the orientation of the block, a section can be measured on the spot, or it can be counted automatically by digital photography. This part of the work has not been done for the time being, but according to previous experience [8], the azimuth of known blocks generally obeys lognormal distribution. For a certain type or region of soil-rock mixture, its block azimuth has a dominant value, that is, the average value. Then, given a variance, a random variable α that obeys lognormal distribution can be generated according to the method in the previous section.
3 the generation of stochastic structure model
With the above three random variables, the random structure model of soil-rock mixture can be generated by programming. Firstly, a randomly distributed random point is randomly generated in a certain area as the center of the circle or the center of mass of the triangle (it is also the center of mass because it is an equilateral triangle). Then give this point two random parameter variables, li and αi, of course, if it is a circle, there is only one parameter li. Finally, a triangle or circle can be drawn from these two parameters. The specific method is as follows:
(1) For a circle, because it has a center and a radius (a random variable li representing the size), you can use Fortran language or drawing commands in ANSYS or AutoCAD to generate a circle, as shown in Figure 8.
(2) For a triangle, point 1 is generated from the centroid, size and orientation, and point 2 is generated according to the difference between point 1 and point 2 120. Similarly, point 3 is generated. Connect 1 and 2,2 and 3,3 and 1 to get a randomly generated equilateral triangle, as shown in Figure 9.
Fig. 8 is a schematic diagram of generating a random circle
Fig. 9 schematic diagram of random triangle generation
After a circle or triangle is generated, a series of circles or triangles can be generated in the same way, because the three random variables that generate these entities are independent of each other, so they are also independent of each other. In this process, one thing needs to be pointed out in particular, that is, when generating an entity, we must first judge whether this entity intersects with other already generated entities, and if it does, we will discard it and regenerate the next one. For a circle, it is enough to judge whether the distance between the centers of two circles is greater than the sum of the radii of two circles (Figure 10). For triangles, there are three situations: points are opposite (figure 1 1), points are opposite (figure 12) and edges are opposite (figure 13). For simplicity, only the most unfavorable first case (Figure 1 1) is considered, that is, whether the distance between two centroids is greater than or equal to the sum of the radii of two triangles. In addition, when a triangle and a circle are mixed, only the unfavorable way is judged, that is, the vertices of the circle and the triangle are opposite, so that it can be judged that the distance from the center of mass of the triangle to the center of the circle is greater than or equal to the sum of the radius of the circumscribed triangle and the radius of the circle.
Figure 10 schematic diagram for judging the intersection of two circles
Figure 1 1 Schematic diagram for judging the relative intersection of triangle points
Figure 12 schematic diagram for judging the relative intersection of triangle points and edges
Figure 13 schematic diagram for judging the relative intersection of triangle edges
The random structure model of soil-rock mixture generated by the above method can be imported into Ansys to extract a lot of useful information: surface information, line information and point information. From these information, the following quantitative indicators can be counted: the area of gravel, the perimeter of gravel, the stone content of soil-rock mixture, the area-perimeter ratio of gravel, and the slenderness ratio of gravel. In this way, the stochastic structure model of soil-rock mixture can be established for numerical calculation. Using this model, the effects of stone content, block shape and block distribution on the strength characteristics and deformation and failure forms of soil-rock mixture can be systematically studied, as shown in reference [4].
An example of random structure model of soil-rock mixture
On the basis of the above analysis, the author compiled a Fortran language program to generate different random structure models. Two more complex models are given: ① circular and triangular blocks are mixed, and their size and orientation obey lognormal distribution (Figure14); ② Circular, triangular and quadrilateral blocks are mixed, and their size and orientation obey lognormal distribution (Figure 15).
Thank you for the great assistance from many fields and scientific research institutions in the process of writing this article. I would like to express my heartfelt thanks to them, especially chief engineer Yin Yueping of China Institute of Hydrogeology and Engineering Geology, researcher Zhang Nianxue and Qu Yongxin of Institute of Geology and Geophysics of China Academy of Sciences, and professor He Changjun of Beijing Urban Construction Group.
Figure 14 stochastic structure model (1)
Figure 15 Stochastic Structure Model (2)
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