Soil parameters in terms of entropy coordinates

. In the ongoing research, an approximate, grading entropy based, advanced interpolation method is applied to establish empirical functions between the grading curves and the model parameters of sands. The space of the grain size distribution curves with N fractions is isomorphic to the space of the unit-sided simplex with dimension N -1. The traditional interpolation over the simplex with dimension N -1 is problematic since the number of the sub-simplexes (and the interpolation points) may increase exponentially with N . To overcome this difficulty, the function is approximately interpolated, which means that the interpolation is made on some 2-dimensional sections of the simplex and is extended to the whole simplex, using the grading entropy map. Two kinds of 2-dimensional sections can be used based on either fractal distributions or partly fractal partly on some maximal gap-graded distributions. In this paper the experiments were made on the artificial mixtures of natural sand grains earlier. Two kinds of functions were approximately interpolated for the minimum dry density using the measured data. One kind of function was determined for the parameters of an SWCC model directly from the level lines previously graphically interpolated from fractal distribution data.


Space of grading curves
To characterise a grading curve, an abstract fraction system is defined. The diameter limits are doubled with the serial number j (j =1, 2. , Table 1): where d0 is the smallest diameter which may be equal to the height of the SiO4 tetrahedron (d0=2 -22 mm).
The relative frequency of xj of grading curves fulfil: (2) where the integer variable N is the number of the fractions between the finest and coarsest non-zero ones.
The relative frequencies xi can be identified with the barycentre coordinates of the points of an N-1 dimensional, closed simplex. The space of the grading curves with N fractions can be identified with the N-1 dimensional, closed simplex. * Corresponding author: imre.emoke@uni-obuda.hu

Grading entropy and diagrams
The grading entropy S is the sum of two parts [1,2]): where S0 is base entropy, S is entropy increment. The base entropy and its normalized form are the weighed (normalised) means of the fraction entropies:  (4) where S0k is the k-th fraction entropy (Table 1), S0max and S0min are the entropies of the largest and the smallest fractions, respectively. The entropy increment S and its normalized form B: , ln 2 ln Any grading curve can be represented as a point in the entropy diagram using the normalised or nonnormalised grading entropy coordinates ( Fig. 1(b)) using the smooth entropy map. The inverse image (Fig.  2) of a regular value is an N-3 -dimensional sphere, the critical values of the map (maximum line of the diagram) is the optimal line. A point of it, the optimal point or optimal grading curve which has fractal distribution, with the following relative frequencies: where parameter a is the root of the equation: The optimal grading curve is a kind of mean grading curve for all the grading curves for given A value.

Interpolation, goal of paper
The direct interpolation of a function over the space of the grading curves ( Fig. 3(a)) needs c(2 N -1) interpolation points, this number is exponentially increasing in terms of the number of the fractions N (see Table 2). To overcome this difficulty, an approximate interpolation method is suggested, based on the grading entropy concept, needing ~c N to ~c N 2 data. The paper briefly presents the method and its application to the minimum dry density and, to the parameters of the Fredlund-Xing soil water characteristic curve equations [3] for sands.

Some definitions
In topology the fibration is defined for topological spaces as follows: where the first map is the inclusion of fiber F into the total space E and the second map π: E → B is a projection of the total space onto the base space B.
The fibration is locally trivial if in small regions of the base space B the total space E behaves just like a projection from corresponding regions of B × F to B.
A subspace I in E is a section of the fibration if there is a continuous map g: B→ I and the map g(π) is the identity of B. A locally trivial fibration is trivial if the base space is contractible. In this case E is diffeomorphic to B x F or I x F, E can be decomposed such that one fiber is attached to each section point.

Interpolation using the entropy map
The simplex (total space) maps to into entropy diagram (base space B). The fiber F is the inverse image (Fig. 2).
We can interpolate on the entropy diagram (or equivalently on a section I of the fibration in the total space E). The function is extended onto the fibers by the constant map point-wise. The extended functionaccording to the continuity of the composite functionsis continuous, but approximate over the total space.
The entropy map is twofold for the triangle diagram (n=2), having two sections as shown in Fig. 3. The optimal line maps into the upper boundary of the entropy diagram and the related grading curves reflect a kind of mean behaviour. It is the part of the boundary of both sections. The other part of the boundary is either the The smin iso-lines on the entropy diagram are different if section 1 or section 2 is mapped (upper/lower part of the triangle diagram, resulting in near linear/curved lines, respectively, Fig. 3).
The measurements were performed on seven sand fractions and their artificial, optimal mixtures with A=2/3 (see Tables 3, 4, [7 -10]). In the suction range of ua-uw ≤ 50 kPa sand boxes were used, in the higher suction range axis translation technique was used. For the model fitting, a nonlinear method was applied [6].
Using the identified parameters, section 1 interpolation was made where not only the fractal gradings related to N=7, but also the fractal gradings N<7 were also used (ie., optimal points of the continuous sub-simplexes were also used, Table 3, [3]). In this work the so determined isolines were approximated by simple parametric functions in terms of the non-normalised grading entropy coordinates.

Dry density measurements
The measurements were performed on five sand fractions and their artificial, optimal mixtures by Lőrincz earlier (see Table 5, [11 -15]).
The minimum dry density level lines were interpolated in two cases, either from the of 50 fractal distributions with various N, or from 9 / 5 maximal gapgraded / maximal fractal distributions ( [11 -15]).
In this work the so determined isolines were approximated by simple parametric functions in terms of the non-normalised grading entropy coordinates. . Table 3. SWCC measuring -A=2/3 optimal (fractal) mixtures [7].

SWCC
The entropy coordinates were computed here by assuming that the fraction d = 0.03 to 0.06 mm has a base entropy of S0 = 12. The iso-lines of the parameters a, n, m of the SWCC model of Fredlund-Xing were possible to be graphically interpolated on the entropy diagram, see e.g., in [7 to 10].
In this work the so determined isolines were approximated by simple parametric functions in terms of the non-normalised grading entropy coordinates (Figs  4 to 6). The m-equation was approximated in two parts, with the same parametric function: Different parameter values were valid in the range of S0 > 15 or S0 < 15, respectively, as shown in Table 6. The equations suggested for parameters a and n:

Minimum dry density
The entropy coordinates of the data base of Lőrincz ([1, 11 to 15]) were computed here by assuming that fraction d = 0.06 to 0.125 mm has a base entropy of S0 = 13. Two sections were used. In the first case, fractal gradings of the continuous sub-simplexes of the 4dimensional simplex were used for the measurement. The isolines shown in Fig.7 were determined directly from the following regression equation: min 0 0.0195 0.0318 0.2482 s S S (13) In the second case, the 4-dimensional version of section 2 (see Fig. 3) was used, the interpolation points were selected on the optimal line and on the gap-graded edge 1-N. The iso-lines shown in Fig. 7(b) were computed from the following regression equation: Eq. 14 can be applied in the "lower half" of the simplex where the largest fraction is dominant (see Fig. 3 and Table 4).

Preciseness by the inverse image
To test the variability of the SWCC or the minimum dry density related to a fixed entropy diagram point, a set of grading curves were generated as the inverse image of the entropy diagram point. Concerning the variability of the SWCC-s related to a fixed entropy diagram point, results can be seen in Fig. 8. The drying and wetting curves are similar, when the fine content is larger, the SWCC is keeping more water, the difference is about 0.2 of total saturation volumetric water content at 10 kPa.
The variability of the dry density related to two fixed entropy diagram points is presented in [13]. According to the results, the preciseness is determined by the size of the inverse image of the entropy diagram point.  Fig. 7. N=5, smin iso-lines (a) constructed from optimal (fractal, mean) mixture data (Eq. (13), (b) constructed from maximal gap-graded and maximal optimal mixture data (Eq. 14).

Dry density
Comparing the linear level lines for section 1 interpolation in case of N=3 and 5 (see Fig. 3. and Fig.  7(a)), the similarity is apparent. Comparing the curved level lines for section 2 interpolation in case of N=3 and 5 (see Fig. 3. and Fig. 7(b)), the similarity is also apparent. The relationship seems to be similar for any N, to be practically independent of the value of N.
Concerning section 1 interpolation, linear relation is found for not only the tested fractal mixtures with (R 2 > 0.9) but also for two additional data bases for continuous mixtures (see [17], where it was mentioned that the linear relation supports the idea raised in Edwards statistical mechanics approach [16] that the dry density and the entropy are closely related).
The linear relation is not valid for gap-graded mixtures. However, it is known from earlier study [15] that gap-graded mixtures are slightly denser in the loosest state than the optimal ones at a given A.

Minimum number of interpolation points
If the interpolation is based on mean (fractal) gradings of the whole simplex, and on some 2-fraction mixtures, then the minimum number of data used for the interpolation is equal to ~5+5N and ~ 5+5 for section 1 and 2 interpolation, respectively ( [15]).
The similarity of the level lines for section 1 or section 2 interpolation may explain that the minimum number of the interpolation points is hardly changing with N which has some significance since for a unitsided simplex, representing the space of the grading curves, the ratio of surface/volume increases with N and, the number of sub-simplexes increases exponentially with N (see Tables 8, 2, [15]). In case of section 1 interpolation, according to the experiences, optimal points of the continuous sub-simplexes can also be used as ~c N 2 interpolation point. Table 7. Definition of lower half of the simplex (see e.g., [9,15]) where Eq. 14 can be applied.

Type of the entropy diagram
The normalised grading entropy coordinates are not unique, zero fractions may change their value, but this is not the case for the non-normalised grading entropy coordinates, which are unique. The normalised and nonnormalised diagram representations are equivalent in case of applying zero fractions up to fixed maximum and minimum diameters [4,17,18]. The representation is simpler and theoretically more precise in the nonnormalized diagram since no zero fractions are needed.

Conclusions
In this paper some analytical functions were elaborated to describe the non-linear parameters of the Fredlund-Xing model [4] in terms of non-normalised grading entropy coordinates, based on the previously interpolated level lines. In further research, the preciseness will be analysed and some regression equations will be elaborated directly from measured data.
In this paper it was found that the dry density level lines are similar in dimensions 2 and 4, indicating that level lines interpolated in the triangle above and below the optimal line may qualitatively characterize the level lines of the larger dimensional simplexes determined by section 1 (continuous mixtures) and section 2 (near maximal gap-graded) interpolations.
Especially, linear level lines were found for fractal and continuous mixtures with R2>0.9. This linear relationship supported the idea that dry density and entropy are closely related, as indicated by Edwards' statistical mechanics, the link will be further investigated for fractal gradings.
A quadratic function was elaborated to describe the minimum dry density from near maximal gapgraded and fractal mixtures, the preciseness will be investigated.
The suggested interpolation method can be used for the approximation of any parameter of the complicated critical state models of sands, this and the use for silts and plastic soils will be investigated.