DIGITAL MODELING IN PROBLEMS OF RADIATION MONITORING OF THE DENSITY OF BODIES WITH A HETEROGENEOUS STRUCTURE

V. A. VOROB'EV, G. N. PARVATOV, V. I. SES'

(Presented by the scientific seminar of the sector of defectoscopy of building materials, Scientific Research Institute of Reinforced Concrete)

The main type of monitoring that makes it possible to obtain the most complete and objective information in determining the texture–structural parameters of composite materials is radiography [1]. However, the effectiveness of the method depends strongly on the quality of the film itself and on the procedure used for its processing. Therefore, the method proposed in [2] for estimating the heterogeneity of structures by means of a radiographic coefficient, representing the change in the relative density of blackening of gamma images on a radiographic photograph, cannot be effective for studying the density distribution over the volume of structures.

In the present work, for studying the density distribution along the length of a specimen, a method of digital modeling is proposed [3].

The structure of concrete as a multicomponent body can be represented by a model of a bounded space filled with convex bodies with a uniform distribution over the volume. In most cases, with a sufficient degree of approximation, convex bodies can be replaced by spheres with distributed effective radii [4]; moreover, the number of different radii can be limited to three, four, etc., depending on the specific conditions of the problem. For engineering calculations of the parameters of the granulometric composition, one usually confines oneself to three sizes [5]. Thus, the uniform distribution of the sizes constituting the components can be replaced by a discrete one, and their distribution in the volume by frequencies. Then the mathematical model can be formulated, in the general case, by specifying the sizes of the spheres—\(r_i\), their frequencies \(f_i\), and the packing conditions:

\[ a \leq q_{ji} \leq b; \tag{1} \]

\[ \sum_i (q_{ji}^{0} - q_{ji})^2 \geq l_{ik}. \tag{2} \]

Here \(i = 1, 2 \ldots k \ldots\) is the number of the size being packed; \(q_j\) are the coordinates of the center of the sphere being packed in \(S\)-dimensional space; \(j = 1, 2, 3 \ldots s\); the index “0” refers to a packed sphere, and “1” to the one being packed; \(l_{ik}\) is the distance between the centers of the spheres being packed.

Conditions (1) and (2) determine the boundary conditions of packing, that is, the sizes of the boundaries and the smallest possible distance between granules, respectively.

The mathematical model described above was implemented on the M-20 digital computer for packing a unit cube with spheres of three sizes, specified in the form of the ratio \(1:2,\;17:3.85\) with frequencies \(f_1, f_2, f_3\). The implementation algorithm is as follows.

The coordinates of the centers of the packed spheres of radius \(r_i\) were represented by triples of numbers \((x,y,z)\) with a uniform distribution in the interval \([01]\). Uniformity of the distribution was ensured by special pseudorandom-number generators [6], with a separate generator for each coordinate in order to reduce their correlation.

Numbers are chosen at random from the interval \([01]\) in such a way that conditions (1) and (2) are satisfied, with \(a=0.5,\; b=0.95\). The frequency ratio \(f_1:f_2:f_3\) is varied by a separate size-selection generator, which must be called after each packed sphere.

The sign that packing is complete is the number of unsuccessful attempts in succession to pack the next sphere—“\(n\).” As soon as \(n=1000\), packing is considered complete and the results are output.

For convenience, the mathematical description of the packing and the algorithm for its implementation on a digital computer will be called a digital model. As a result of a statistical experiment with the digital model on the M-20 digital computer, mean density values were obtained as a function of the frequencies for fixed values of \(r_i\). The results are presented in Table 1.

Using the digital model, let us consider the density characteristics of a cement-concrete block as random functions of its sections. As the density characteristic of the concrete, we use the homogeneity of the filling, which is the differential density of the section [4]. Then, on a previously superimposed grid of points (Fig. 1), it is not difficult to calculate the values of homogeneity “\(\nu\)” in each section by the formula:

\[ \nu=\frac{1}{H}\sum_i l_i, \]

where
\[ H \]
is the thickness of the block;

\[ l_i \]
is the length of the chord along the ray,

\[ i=1\ldots n \]
is the number of spheres that intersect the ray.

Let \(\nu_i(L_k)\) be the value of homogeneity along the \(i\)-th horizontal in the \(k\)-th section; then the mathematical expectation of \(\nu_i\) and the variance will be, respectively:

\[ \bar{\nu}_x(L_k)=\frac{1}{n}\sum_i \nu_i(L_k); \]

\[ \bar{D}_x(L_k)=\frac{n}{n-1} \left[ \frac{\sum_i [\nu_i(L_k)]^2}{n} - [\bar{\nu}(L_k)]^2 \right]. \]

Thus, the found values \(\nu_i(L_k)\) of the random function according to the scheme of Fig. 1, with mean value \(\bar{\nu}_x\) and variance \(\bar{D}_x\), form a random field whose internal structure is described by the correlation function [8]:

\[ \bar{K}_x(L_k L_j)=\frac{n}{n-1} \left[ \frac{\sum_i \nu_i(L_k)\nu_i(L_j)}{n} - \bar{\nu}(L_k)\bar{\nu}(L_j) \right], \]

passing to the normalized correlation function, we obtain:

\[ \bar{\gamma}_x(L_k L_j)= \frac{\bar{K}_x(L_k L_j)} {\bar{\sigma}_x(L_k)\bar{\sigma}_x(L_j)}, \tag{5} \]

where \(\bar{\sigma}_x\) is the root-mean-square deviation in the \(k\)-section.

The calculations carried out by formula (5) showed that the random function under study, \(v(L)\), is not stationary, since the values of the normalized correlation function are not constant. Eliminating these values and passing to a stationary function, we obtain the values of the normalized correlation function \(\bar v_x(h)\), where \(h\) is an arbitrary interval between sections. The graph of this function is shown in Fig. 2. The variation of the function \(\bar v_x(h)\) can be approximated by expression (6), where the coefficients \(A\), \(a\), \(B\), and \(b\) are determined in each particular case by the least-squares method. The dotted line in Fig. 2 shows the curve smoothed according to (6), with coefficients \(A=0.65\), \(a=0.35\), \(B=0.2\), \(b=0.46\), for the digital model.

Thus, since “\(v\)” is the differential coefficient of the packing density of the coarse aggregate, one can write the mathematical model of the density distribution of cement concrete in the form:

Fig. 1. Arrangement scheme of computation points; \(I_0\)—incident radiation flux; \(k=1, 2\ldots\) points along the horizontal; \(i=1, 2\ldots\) sections along the beam

Fig. 2. Graph of the normalized correlation function. 1. Digital model; 2. Approximation according to 6; 3. Cement-concrete specimen

\[ \bar v_x(h)=A\exp(-a^2h^2)+B\exp(-b^2h^2); \tag{6} \]

\[ \left. \begin{aligned} \bar\rho_x(L)&=\rho_0;\\ \bar\rho_x(h)&=A\exp(-a^2h^2)+B\exp(-b^2h^2). \end{aligned} \right\} \tag{7} \]

As an experiment, we carried out physical modeling of packings in a cube \((10\ \text{cm}\times10\ \text{cm}\times10\ \text{cm})\) using spheres and gravel, with the same ratio of sizes and frequencies as in the digital models. The results are summarized in Table 1. The gravel and sphere packings were obtained under identical conditions. However, the density values of the gravel packings are somewhat higher than in the sphere packings. Nevertheless, from Table 1 one can see a regular change in density values as a function of frequencies in all the models—an increase in densities with an increase in the frequency of the first size. This confirms the possibility of using digital models to study structures of the concrete type.

From the gravel packings, cement-concrete specimens were then obtained, followed by radiography with X-rays \((E=210\ \text{keV})\) and decoding of the gamma images on an MF-4 microphotometer. The results of decoding the images were calculated by formula (5). For illustration, Fig. 2 gives the distribution function \(v(h)\) for a cement-concrete specimen with a packing density by coarse aggregate of 0.68. From a comparison of the curves of the digital model and the cement-concrete specimens, it can be concluded that in real concretes the distribution of the coarse aggregate in the specimen obeys approximation (6), obtained on the digital model. Thus, the use of the digital modeling method—

… makes it possible to apply the theory of random functions to the study of heterogeneous structures and to provide a mathematical model of the density distribution of coarse aggregate in cement concrete.

Table 1

Size ratio 1 : 2.17 : 3.85

REFERENCES

1. Yu. D. Gavkalov. Candidate dissertation, TPI, Tomsk, 1968.

2. V. G. Firstov. Features of radiography of structures with an anisotropic structure. Defektoskopiya, No. 6, 83, 1969.

3. V. A. Vorob’ev, V. K. Kivran, I. E. Naats. Study of the distribution of bremsstrahlung fluxes beyond heterogeneous barriers by methods of mathematical modeling. Izvestiya TPI, 193 (1969). Tomsk State University Press, Tomsk.

4. V. A. Vorob’ev, I. E. Naats. On the relation between density of aggregate and the linear coefficient of homogeneity of concrete-type structures. Radiation Methods and Means of Nondestructive Testing of the Quality of Materials and Products. Proceedings of the First Interuniversity Scientific-Technical Conference. ONTI PRIBOR, Moscow, 1969.

5. A. D. Gvozdev, S. M. Krylov. Features of deformations of concrete and reinforced concrete and the use of computers to assess their influence on the behavior of structures. “Gosstroi SSSR,” Moscow, 1969.

6. V. F. Lyashenko. Programming for Digital Computing Machines. Sovetskoe Radio Publishing House, Moscow, 1967.

7. A. A. Sveshnikov. Applied Methods of the Theory of Random Functions. Sudpromgiz, Leningrad, 1961.