Study of Statistical Properties of Linear Inhomogeneity in Concrete-Type Structures Using Mathematical Modeling

STUDY OF THE STATISTICAL PROPERTIES OF LINEAR INHOMOGENEITY IN CONCRETE-TYPE STRUCTURES BY METHODS OF MATHEMATICAL MODELING

V. A. VOROBYEV, V. K. KIVRAN, I. E. NAATS

(Presented by the joint scientific seminar of the sectors DSM and MRD)

In radiation flaw detection of concrete products, inhomogeneity of the shadow image is observed, reducing the detectability of defects [1]. Therefore, when developing particular methods for recording defects, it is necessary to take into account this negative significance of inhomogeneity; for this it is necessary to know the magnitude of the inhomogeneity, its dependence on the density of the concrete, its granulometric composition, and the thickness of the product.

The inhomogeneity of the shadow image is caused by the fact that the radiation flux, in different areas of the body being inspected, travels paths of different length in the aggregate and in the binding medium. Since these components have different linear attenuation coefficients, the flux undergoes different attenuation and acquires an inhomogeneous character. Consequently, the inhomogeneity of the flux is caused by the linear inhomogeneity of the structure; all properties of the flux inhomogeneity are determined by the corresponding properties of the linear inhomogeneity of the structure and by the geometry of the structure.

The linear inhomogeneity of the structure can be quantified by the inhomogeneity of the ratio of the part of the flux path along a ray in the aggregate to the entire path in the product being inspected. When this ratio is averaged over the whole volume of the product, we obtain a certain mean quantity representing the packing density [2], and then the value of the indicated ratio may be interpreted as the linear geometric differential density.

Thus, the inhomogeneity of the flux, as already noted above, is determined by the inhomogeneity of the linear geometric differential density. Here it should be borne in mind that the aggregate in concrete has particles of different size, shape, and orientation in the structure, and the spatial distribution of aggregate particles is random in character. Consequently, the magnitude of the linear differential density is also random in character; moreover, in a plane perpendicular to the direction of propagation of the flux, this quantity forms a random function, or more precisely, a random field. The main statistical parameters of this function are the mean density, or packing density, the variance, and the autocorrelation dependence.

It is very convenient to study the properties of these characteristics by the method of mathematical modeling on digital computers. Such modeling has well-known advantages in comparison—

...compared with physical modeling; moreover, in mathematical modeling it is very easy to abstract away from extraneous factors.

In the concrete model, the aggregate particles are replaced by spheres of the same sizes, and the structure of concrete is represented by a random packing of spheres.

Using models of random packings of spheres, a dependence was revealed between the dispersion of the linear differential density and the aggregate density. To identify this dependence, packing was modeled by different algorithms: with packing density from 0.05 to 0.25 by one algorithm, and at 0.45 and 0.54 by two different dense-packing algorithms. The algorithm with low aggregate density is based on random sampling of the coordinates of the centers of spheres with rejection of intersecting spheres. The dense-packing algorithms model the pouring of balls into a packed container without displacement of previously packed balls [3].

Thus, with these three random-packing algorithms the range of representation of aggregate density is from 0.05 to 0.54.

Figure 1 presents the dependence, calculated from the indicated packings, of the dispersion of the linear differential density \(\delta_z\) on the value of the aggregate density \(\rho\) for a ratio of the sphere diameter to the total thickness of the packing \(H\) equal to 0.1. From the graph one can see a slight deviation from a linear dependence; moreover, as the aggregate density increases, the dispersion of the linear differential density increases. With a change in the ratio of the sphere radius to the thickness of the packing, the shape of the curve does not change; the curve is only shifted upward as this ratio increases and downward as it decreases.

Fig. 1. Dependence of the dispersion of the linear inhomogeneity of the structure on the aggregate density

The magnitude of the dispersion could be calculated analytically if the spatial distribution of the centers of the spheres did not depend on possible intersections of spheres, i.e., when the packing of spheres could be represented as a set of three-dimensional random numbers that are the coordinates of the centers of the spheres. In this case, the probability distribution of the number of points (spheres) in some closed volume obeys the Poisson law. The calculation of the dispersion is then quite simple; we give the final result:

\[ \varepsilon^2=\frac{1}{2}\rho\frac{1-r}{H}. \tag{1} \]

As can be seen, here the dispersion is directly proportional to the value of the aggregate density and to the value of the ratio of the sphere dimensions to the thickness of the packing. A qualitatively analogous dependence can also be observed in the graph in Fig. 1, only here estimate (1) is the lower bound of the curve. Indeed, at low aggregate densities the deviation of the dispersion value from estimate (1) is significant, and in the positive direction; however, the higher the aggregate density, the closer the value...

value of the variance to estimate (1), and at a density of 0.54 this deviation is only 30%. If it is taken into account that usually the packing density in concrete is 0.5–0.6, then estimate (1) can also be used to determine the linear inhomogeneity of a structure so far with particles more or less uniform in size. To reveal the regularity in the change of the variance of the linear differential density as a function of packing density and granulometric composition for distributed particle sizes, additional studies should be carried out on packings of spheres with the corresponding size distribution.

For the above packings of spheres, an autocorrelation dependence of the value of the linear differential density on the distance in a plane perpendicular to the direction of propagation of the radiation flux was obtained. Figure 2 shows a typical form of the autocorrelation dependence. It is evident from the graph that after the transition of the zero mark near the coordinate \(\tau = r\), the absolute value of the amplitude of the first half-wave is approximately 8% of the variance value \((\delta^2 = f(0))\). The further autocorrelation dependence may be neglected (beginning with \(\tau = 2r\)). When the variance changes, the shape of the curve, its periods, and the relative amplitudes of the half-waves change very little. Thus, the principal characteristic determining the structure is the variance of the linear differential density at a given packing density and given sphere sizes. In this sense, when studying the possibility of detecting defects by methods of mathematical modeling, one may in general dispense with modeling the structure, replacing it by a random function (random plane) of the linear differential density with an autocorrelation dependence of the form shown in Fig. 2, with prescribed variance and particle sizes.

Fig. 2. Autocorrelation dependence of the linear inhomogeneity of a concrete-type structure

LITERATURE

1. V. A. Vorob’ev. Radiation flaw detection of building materials and structures using bremsstrahlung radiation. Defektoskopiya, 5, 33, 1965.

2. L. F. Tot. Arrangements on the Plane, on the Sphere, and in Space. Fizmatgiz, Moscow, 1958.

3. V. K. Kivran, I. E. Naats, V. I. Ses’, G. I. Parvatov. Digital modeling in problems of radiation flaw detection of bodies with an inhomogeneous structure. In: Proceedings of the II Interuniversity Scientific-Technical Conference on Radiation Means and Methods of Nondestructive Testing. Tomsk, 1968.