CALCULATION OF OPTIMAL STRUCTURES OF COMPOSITE MATERIALS BY THE METHOD OF DIGITAL MODELING

I. E. Naats, V. I. Ses’

(Presented by the scientific seminar of the Department of Higher Mathematics)

In studying the structure of composite materials made up of an inert component (aggregate) and a binder, it is often necessary to choose the optimal composition of the aggregate in order to obtain the greatest density of the material. The most widespread composite material is concrete, for which the choice of the optimal granulometric composition of the aggregate makes it possible to reduce the consumption of cement per unit volume at a prescribed strength, which lowers the cost of concrete, increases its durability, and facilitates automation in the molding of products [1].

In the present work a method is proposed for the numerical calculation of the optimal granulometric composition of a composite material, based on the construction of a probabilistic-geometric model of the spatial arrangement of the component, followed by the computation of a series of statistical tests on a digital computer.

As the geometric model, a packing of rigid spheres of different radii in random tetrahedra was proposed. Experimental data substantiating the probabilistic-geometric model are given in [2].

The specific surface, the number of contacts, the geometric density, and certain other parameters of sphere packing were estimated by a method conventionally called the tetrahedron method, described in [3, 4]. In this method, a space filled with tetrahedra is considered, the vertices of which are formed by the centers of any four mutually tangent spheres. The geometric density for each tetrahedron is determined by the ratio of the sum of the volumes cut out from the spheres to the volume of the given tetrahedron. The mean number of contacts is determined by the values of the angles cut out by the tetrahedra from spheres of a given radius. In addition to the parameters listed, the tetrahedron method makes it possible to estimate the mean number of tetrahedra for a central sphere of a given radius, the mean number of spheres located around a pair of tangent spheres, and the frequencies of tetrahedra containing spheres with given radii. The estimate of the density and of the number of contacts obtained when considering a set of spheres by the tetrahedron method is overestimated, since only tetrahedra are considered as the structural cell of the packing; however, the space of a dense packing of spheres may be completely replaced by a set of tetrahedra and quadrangular pyramids. The relative volume occupied by the spheres for a quadrangular ...

pyramids are smaller than for tetrahedra, and therefore excluding pyramids from consideration shifts the values of the parameters being determined.

The tetrahedron method is applicable to contacting spheres. In order to make it possible to use it for studying non-dense packings, it is proposed to introduce the parameter \(\delta\)—the average gap between spheres along the lines connecting their centers—and to replace the initial set of spheres by a new one, in which the radii of the spheres are increased by \(\delta/2\). Thus, knowing the average gap between spheres, one can estimate the density, the specific surface area, and the number of neighbors for loose packings of spheres, and also establish the dependence of these parameters on the value of \(\delta/2\).

This method was implemented on the M-20 computer for calculating a set of spheres with radii \(3:6:19\) and \(3:10:19\). For comparison with the experimental data given in [1], the gradations of the granulometric composition and the corresponding values of density and specific surface area were chosen in the amount necessary for constructing diagrams similar to the experimental ones [1].

Fig. 1, a shows an experimental diagram constructed from the data of [1] for a concrete aggregate whose mean fractional grain sizes are in the ratio \(1:3:12\). Fig. 1, b shows a diagram calculated according to the proposed method. As is clear from Fig. 1, the results obtained for the model agree qualitatively with the experimental ones. The difference in the fractional composition at which the greatest aggregate density is attained (50–60% of the coarsest fraction for the experimental diagram and 70–80% for the diagram constructed from the data obtained from the model) is explained by the difference in the relative mean sizes of the grains of the fractions. Thus, the mean fractional grain sizes for the experimental diagram in [1] are in the ratio \(1:3:12\), whereas the radii for the model are in the ratio \(1:2:6\); therefore the shift occurred toward an increase in the coarse fraction. To illustrate the influence of changing the mean grain sizes on the character of the isolines, Fig. 2, a gives a diagram constructed for sphere sizes \(1:3:6\) (\(3:10:19\)), i.e., the mean radius is increased. The isolines in diagram 2, a are more extended toward an increase in the middle fraction than the isolines in Fig. 1, b.

The isolines in the diagrams considered were constructed for dense structures, i.e., for the case in which there is no gap at the contact points of neighboring grains (\(\delta = 0\)). However, if there is a gap between grains (and in concrete it is always present, and its magnitude depends on the mobility of the concrete and the viscosity of the cement paste), then, in order to obtain the optimal granulometric composition of the aggregate, it is necessary to take into account the specific surface area of the structure. The influence of the specific surface area consists in the fact that, as the gap between grains increases, the granulometric composition corresponding to the greatest density shifts toward a decrease in specific surface area, as is confirmed by the diagram in Fig. 2, b. This diagram gives isolines for the packing of spheres with radii \(3:6:19\) and an average gap between spheres of \(5 \cdot 10^{-1}\). As is clear from Fig. 2, b, the optimal granulometric composition for loose structures in the sense of geometric density does not correspond to the optimal composition for dense structures.

The fact that the results obtained on the model agree with experiment indicates the possibility of using the method set forth above for determining the optimal granulometric composition of an aggregate. The construction of the above composition–property diagrams was carried out at 21 points. Calculation of the compositions corresponding to these points took about 2 min of machine time on the M-20 computer. However, the optimal granulometric composition of the structure can

a)

b)

Fig. 1. Diagrams of the dependence of the density and specific surface of the filler on its granulometric composition at \(\delta = 0\)

a) experimental,
b) calculated on the model

a)

b)

Fig. 2. Diagrams of the dependence of density on granulometric composition, calculated on the model
a) for spheres with radii \(3:10:19\);
b) for spheres with radii \(3:6:19\) and \(\delta = 5 \cdot 10^{-1}\).

to estimate as a result of carrying out an active machine experiment on the proposed model. In this case the number of compositions (points) to be calculated is reduced, and there is no need to construct the diagram. The computer prints out the optimal granulometric composition for the specified radii and the mean value of the gap between granules at points of contact. The tetrahedron method makes it possible to estimate the density of packings specified by any number of radii, but as the number of radii increases, the required computer time rapidly increases: thus, for three radii the calculation of one packing takes 7 sec, while for 12 radii about 10 min is required. Significant limitations of the method are that the largest radius must not exceed the smallest by more than a factor of 6.46; therefore it does not allow coarse and fine filler to be covered at once, and the calculation must be carried out in parts.

REFERENCES

1. V. I. Soroker, V. I. Galaktionov. Selection of optimal mixtures of fractionated fillers for concrete at reinforced-concrete-products plants. Izv. vuzov, “Construction and Architecture,” 5, Novosibirsk, 1966.

2. I. E. Naats, V. I. Ses, V. M. Goncharov. Toward an experimental substantiation of the probabilistic-geometric model of the structures of inhomogeneous materials of the concrete type. Reports at the Second Interuniversity Conference on Radiation Methods and Means of Nondestructive Testing of Materials and Products, Tomsk, TSU Press, 1968. (In press).

3. I. E. Naats, V. I. Ses. Application of digital modeling to the study of the geometric characteristics of random packings. Reports to the Interuniversity Conference on Radiation Physics. Tomsk, TSU Press, 1968. (In press).

4. M. J. Hogendijk. Random dense packing of Spheres with a discrete distribution of the radii. Philips Res. Repts., 18, 1963.