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Agricultural Engineering Division, Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel
* Corresponding author (agvictor{at}techunix.technion.ac.il)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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A general feature of all the above works is the important role of the shrinkage geometry factor, rs (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) in considering the effects of both the crack network geometry and shrink-swell anisotropy. In addition, the rs factor is used for experimental estimation of the total crack volume (Baer and Anderson, 1997).
Important applications of the rs concept are based on Bronswijk's known presentation (Bronswijk, 1988, 1989, 1991a, 1991b) and measurement method (Bronswijk, 1990) of the rs value. According to this presentation, the shrinkage geometry factor determines the relation between the variations of soil subsidence (
z) and those of the matrix volume (V) of a soil layer in field conditions to be
 | [1] |
Aside from the constant rs value, the practical use of the exact Eq. [1] in the frame of Bronswijk's (1988)(1989, 1990, 1991a, 1991b) approach is based on a geometrical schematization of the crack system in a soil layer (Fig. 1) . Recently Chertkov et al. (2004) noted three implicit assumptions of Bronswijk's schematization and formulated them explicitly [below Assumption 1 is given in an equivalent but more visual form than in Chertkov et al. (2004)].
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Chertkov et al. (2004) emphasized that the correcting M factor does not compensate for an inaccuracy of the rs value in Bronswijk's approximation when using Assumption 1 and this inaccuracy should be addressed in a separate paper. In fact, Assumptions 1 and 2 are incorporated together into Bronswijk's approximation. One should therefore consider the corresponding correction that would compensate for the inaccuracy of the rs value in Bronswijk's approximation because of the simultaneous use of Assumptions 1 and 2. The major objective of this work is namely to address this correction and thereby to find the totally corrected rs value. Notation is summarized in Appendix 1.
Before proceeding to the main exposition, it is worth specifying the links between the concepts of cracking and of porosity in clay soils (clay content > 40%) and that we imply following Bronswijk (1988)(1989, 1990, 1991a, 1991b) and Hallaire (1984). The scale of clay soil aggregates or interaggregate (so-called structural) pores is simultaneously the scale of the smallest macrocracks (or simply cracks). We cannot distinguish between these smallest cracks and the interaggregate pores from the viewpoint of volume shrinkage. They coincide, and we say about the interaggregate cracks. Large subvertical cracks in a soil, small interaggregate cracks of different orientations, and all cracks of intermediate size, relate to one category of shrinkage cracks with varying volume. The rs factor by definition (Eq. [1] and Bronswijk 1988, 1989, 1990, 1991a, 1991b) relates namely to the total volume of all these cracks in a clay soil layer or sample (rs in these cases is different, see Chertkov et al. [2004] and below). In turn, the volume V of a clay soil matrix and its variation, V, in Eq. [1] are understood to be the summary volume of the intraaggregate matrix of all aggregates and the volume variation, respectively. The intraaggregate matrix includes solids (a network of clay particles embracing silt and sand grains), interparticle (or matric) pores, and possible microcracks. Scales of clay particles, matric pores, and the microcracks coincide by order of magnitude. The contribution of the microcracks to soil matrix volume is negligible for clay soils with clay content >40% by weight (Fiès and Bruand, 1990, 1998) and that we use in the following. Thus, the volume variation and shrinkage curve of the soil matrix are only determined by matric porosity.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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Figure 2
visually shows the difference among the shrinkage curve of the soil matrix without cracks [
; o is an initial value]; the shrinkage curve of the soil layer including the matrix and cracks in Bronswijk's approximation 
; the true shrinkage curve of the soil layer including the matrix and cracks 
; and the shrinkage curve of the soil sample with cracks 
. The simultaneous replacement, 
s, l'l, and rsr's in exact Eq. [A2.1] (that is equivalent to Eq. [1]) leads to the corresponding equation in Bronswijk's approximation,
 | [2] |
s, 'l, and r's in Eq. [2] differ from , l, and rs in Eq. [A2.1] at a given water content. Equations [A2.1], [A2.2], and [2] lead to the presentation of the totally corrected rs value as
 | [3] |
 | [4] |
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Since l
'lo (see Fig. 2, Curves 1 to 3) the additional logarithmic multiplier, L from Eq. [4] is in the range 0 < L 1. Because the correcting M factor in Eq. [3] has values M 
1 (Chertkov et al., 2004), the L and M factors, in part, mutually compensate each other.
The first way to experimentally estimate an L value using Eq. [4] is to combine separate core sample measurements to find the specific volume of a soil layer in Bronswijk's approximation and separate measurements in situ to estimate the true specific volume of a soil layer . We could only find the available data on similar measurements (simultaneously on samples and in situ) permitting one to obtain independent estimates of 'l and l, in the work of Hallaire (1984). A corresponding illustrative example will be given below.
The second way is to suggest some sound relation between 'l and l and use the relation to exclude l or 'l from Eq. [4]. The following theoretical subsection is devoted to the derivation of such a simple relation based on two assumptions that have repeatedly and successfully been applied in soil science literature. One can find 'l from core sample measurements and then l using the relation (a corresponding illustrative example with data from Chertkov et al. [2004] will be given below), or one can find l from in situ measurements and then 'l using the relation (this way is not illustrated because of the lack of data).
The Relation between the True Specific Volume of a Soil Layer and Specific Volume in Bronswijk's Approximation
In this subsection we graphically introduce the soil matrix displacements that are needed to derive the relation between l and 'l. Then we accept two assumptions known in literature, which lead to the relation. After that we give the relation (Eq. [5]) and its connection to Poisson's ratio of the soil matrix. The interested reader can find all details of the derivation in Appendix 4.
Figures 1 and 3
schematically illustrate the vertical and horizontal displacements in the matrix of a soil layer and cube with a water content decrease. Because of the tensile stresses the layer subsidence, z is no less than the subsidence z' of an isolated soil cube (Chertkov et al., 2004), that is, z z' (Fig. 1). For the same reason, after shrinkage, the horizontal size x of a stretched layer matrix without cracks (per one initial imaginary cube of size z) is no less than the horizontal size x' of an isolated cube (Chertkov et al., 2004), that is, x x' (Fig. 3). Figure 4
visually shows these results.
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V (per one initial soil cube) entering Eq. [1] is considered to be the same for both the isolated cube of Bronswijk's approximation and the cube that is mentally outlined in a real connected layer. For this reason both the volume of an isolated-cube matrix, x'2 (z – z') (see Fig. 4), and that of the layer matrix without cracks (per one initial soil cube), x2 (z – z), (Fig. 4) are equal to the same value of (z3 – V) at a given water content, w < wo and correspondingly at a given V. Following Bronswijk (1988)(1989, 1990, 1991a, 1991b), we also accept this assumption. The second assumption is the elastic model. The concepts from the theory of elasticity apply to soil only in an approximate way. However, many results of the consideration of shrinkage crack development in clay soils, especially in water saturated states, in a number of works (Haberfield and Johnston, 1990; Morris et al., 1992; Murdoch, 1993; Harison et al., 1994; Lima and Grismer, 1994; Konrad and Ayad, 1997a, 1997b; Ayad et al., 1997; Chertkov, 2002) in the frame of this simplest model are quite reasonable and give a sufficiently sound basis. Following these works we also assume the model in this work. That is, the deformations under the action of tensile stresses in the matrix of an unlimited shrinking soil layer (field conditions) are considered to be longitudinal deformations of a thin elastic plate.
The final relation between l and 'l is as follows (see Appendix 4):
 | [5] |
 | [6] |
 | [7] |
is Poisson's ratio of the soil matrix.
Replacing l(w) in Eq. [4] from Eq. [5] leads to the final expression for the value of the correcting L factor,
 | [8] |
(w), s(w), and 'l(w) (Fig. 2) one can estimate r's(w) in Bronswijk's approximation from Eq. [2], M(w) from Eq. [A2.2], L(w) from Eq. [8], and the corrected rs(w) value from Eq. [3], if in Eq. [8] the evolution of the 
parameter with water content is also known for a given soil. As indicated above the parameter is a function of Poisson's ratio () of the soil matrix (see Eq. [6] and [7]).
The same two above assumptions eventually allow one to express the parameter through the sample measurement data. Detailed derivation of the expression can be found in Appendix 5. Here we only give the final result as
 | [9] |
 | [10] |
(w) is the specific volume of the intraaggregate soil matrix that can be measured as the shrinkage curve of soil aggregates (e.g., Bronswijk and Evers-Vermeer, 1990). Possibility of the theoretical prediction of (w) for clay soils (clay content > 40% by weight) will be briefly considered below in illustrating the approach. Using values one can estimate the corresponding µ (Eq. [6]), (Eq. [6] and [7]), l (Eq. [5]), L (Eq. [8]), and rs (Eq. [3]) values. |
MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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Additional Analysis of Data from Chertkov et al. (2004)
In this subsection we indicate parameters that were measured by Chertkov et al. (2004) and values that were estimated by these authors from the data. After that we consider an additional use of the parameter data and found values to estimate and l values that were introduced above.
The data relate to clay (>95% of montmorillonite) extracted by standard methods from samples of the 0- to 30-cm layer of soil in Sarid, Israel. The clay paste water saturated nearly to the liquid limit was placed in small cylindrical containers of approximately 1.5-cm height and approximately 3.5-cm diam. Three parameters of the clay itself and three parameters of each clay sample were measured. The former included the clay particle density, 
s, the specific volume of oven-dried clay matrix, z, and the liquid limit, wL (Table 2 from Chertkov et al., 2004). The latter include the diameter, d, height, h, and weight, m of a sample (Table 3 from Chertkov et al., 2004). The measurements were conducted once a day for 8 d with subsequent oven drying for four samples.
The clay properties, s, z, and wL enabled the calculation of the specific volume of a clay matrix as a function of w from Chertkov (2000b)(2003). This dependence is reproduced in Fig. 5
as the solid line. It is worth noting that Chertkov's (2000b)(2003) model relating to pure-clay pastes fits pretty well with data from Bruand and Prost (1987) based on a clay-silt-sand mixture of high clay content, but not pure clay [see Chertkov (2003)(p.78)]. Except for that it can be shown that Chertkov's (2000b)(2003) model, after natural and simple generalization by introducing the content of non-clay solids as an additional parameter, describes (without fitting) the shrinkage curve data of aggregates relating to 21 clay soils (with clay content > 40% by weight) from Bronswijk and Evers-Vermeer (1990). In fact, the model can have rather broader applicability for (w) prediction than only for pure clay.
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'l and the specific volume of cracked clay paste samples, s for a number of w values. These estimates are also reproduced in Fig. 5 by asterisks ('l) and circles (s). In turn, , 'l, and s values allowed the estimation of the rs', M, and rsM for a number of water content values. These estimates are reproduced in Table 1.
The experimental data on md[ = m(0)], h(w), and do[= d(0)] as well as the values of (w) from Chertkov et al. (2004) permit us to additionally estimate the (w) function from Eq. [9] and [10]. Then using the values and the specific volume 'l we estimated the true specific volume of the soil layer, l (Eq. [5]), the correcting L factor (Eq. [8]), and the totally corrected rs value (Eq. [3]).
Data from Hallaire (1984)
In this subsection, we describe Hallaire's data as applied to the following analysis in the frame of the model under consideration.
The data relate to an aggregated clay soil. Clay content, mainly montmorillonite and chlorite, is 52 to 56% by weight. Saturated undisturbed core samples (15-cm diam., 7.2 cm height) were collected during the winter in the depth range from 20 to 120 cm. In the course of laboratory measurements, the water content of the samples decreased from w 
0.3 g g–1 (the field capacity). Figure 6
reproduces the data of three shrinkage curves from Hallaire's (1984) Fig. 2 in coordinates of void ratio versus gravimetric water content. The layer void ratio 
was measured by the vertical shrinkage of core samples, that is, in Bronswijk's approximation in terms of Chertkov et al. (2004). The sample void ratio (es corresponding to s) was measured by the vertical and diameter shrinkage of core samples. The void ratio of aggregates (e corresponding to ) was also measured.
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l). These data with large standard deviations were obtained in situ by the thickness of three horizons (25–50, 50–80, 80–120 cm) using reference marks in the ground. Using these 13 points and their standard deviations we estimated the corresponding points of the true layer void ratio, el (Fig. 6, squares) as follows. Knowing the e values in Fig. 6 one can find el values from the differences (el – e). With that the large standard deviations relate to el values. Note that according to its physical meaning a l value at a given water content should be between corresponding s and 'l values (see Fig. 2, Curves 3, 2, and 4, respectively). Therefore, the el points in Fig. 6 should also meet the inequality, es el e'l. Hence, irrespective of their standard deviations, the four el points (from Hallaire's [1984] Fig. 5) that are in the range 0.26 g g–1 < w < 0.29 g g–1 (Fig. 6, squares), where es = el', lie on the trend line of el' and es points. The six experimental el points (from Hallaire's [1984] Fig. 5) at 0.17 g g–1 < w < 0.26 g g–1 (Fig. 6, squares) are close to el = 
/2 values, but have standard deviations so large that the upper and lower boundaries of the possible ranges of el values are higher than the corresponding el' and lower than the corresponding es values. However, accounting for the condition es el e'l that determines the maximum real range of standard deviations of el we take for real standard deviations of these six points the range between es and el'. For the three experimental el points (from Hallaire's [1984] Fig. 5) that are nearest to w = 0.15 g g–1 (Fig. 6, squares) the upper boundary that is determined by the observed standard deviations exceeds the el' values, but the corresponding lower boundary is between the el' and es values. These three experimental points are also close to the average between el' and the observed lower boundary of standard deviations (Fig. 6). Thus, the inequality es el e'l determines a real range of standard deviations 
el, but not the experimental el values from Hallaire's (1984) Fig. 5 that were obtained by in situ measurements.
For numerical estimates we can directly use the experimental points el' (asterisks), es (circles), and e (dots) in Fig. 6. The set of experimental el points (squares) only differ from the el', es, and e point sets by the number of points and larger standard deviations. For this reason we need some fitted curve to approximate el data. Using the ten experimental el points at w < 0.26 g g–1 (Fig. 6, squares) when el 
e'l es and the least-squares criterion, we found a squared el approximation and extrapolation to small water content values to be
 | [11] |
Hallaire (1984) noted that at maximum water content the interaggregate-crack void ratio, 
0. We estimated the corresponding w and e values (Fig. 6) continuing the e'l and e trend lines up to their intersection at wo 0.335 g g–1 and eo 1.07. A possible inaccuracy of this point that is connected to the o value is very slightly reflected in the following estimates.
In general, aggregate water content and soil water content are different. However, as judged by the data presentation in Hallaire's (1984) Fig. 2 (sample measurements of e'l and es, aggregate measurements of e) and Hallaire's (1984) Fig. 5 (in situ measurements of el using reference marks in the ground), he implicitly accepted the approximate coincidence of the aggregate and field water contents. A possible reason for that are the sufficiently small soil, core, and aggregate water content that are below the field capacity. Thus, in the case of these particular data we, following Hallaire (1984), also accepted the equality of aggregate and soil water content. However, in general, the connection between field, sample, and aggregate water content is essential in estimating the rs value in the frame of the approach that uses a number of different shrinkage curves (as in Fig. 2). This connection should be addressed in the future.
Analysis of Hallaire's (1984) Data
The specific volume and a void ratio e are connected by 
= 
/s where s is the density of solids. Therefore r's, M, L, ML, and rs factors are obtained from the data in Fig. 6 using the same Eq. [2], [A2.2], [4], and [A2.1] after replacements: o (eo + 1), (e + 1), s (es + 1), l (el + 1), and 'l 
. In addition, rsM = Mrs'.
Similarly, we have = 
1/2 from Eq. [5]. Note that, unlike the above case of a clay paste, for the aggregated clay soil is immediately obtained from el and el' data and does not serve to find l through 'l. Finally, is found from by Eq. [6] and [7].
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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(for a sufficiently exact and sound approach for calculation based on two known assumptions see Appendices 4 and 5).
Figure 7
shows the values (squares) corresponding to the measured gravimentic water contents of the clay and gives the quantitative illustration of the qualitative (w) dependence in Fig. A5.1.
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l (squares) of the specific volume of the cracked clay paste layer. Estimates of the L, ML, and rs factors and their standard deviations for this clay at the measured gravimetric water contents, are given in Table 1. These estimates and a comparison between them and estimates from Chertkov et al. (2004) permit one to note the following results.l(w) (squares) and 'l(w) (asterisks) shows that the difference, ('l – l) is beyond the limits of standard deviations 
, l is appreciably less than 'l, and l
/'l
0.8. That is, Assumption 1 is violated. L). ML). w wL beyond the limits of standard deviations (rs). That is, Assumption 3 is also violated.
The corresponding experimental dependence of Poisson's ratio versus water content, (w) from Eq. [6] and [7] looks quite similar to that in Fig. 7 (squares), but with ordinate value range from 0.45 to 0.5. According to available data (Briones and Uehara, 1977; Haberfield and Johnston, 1990; Murdoch, 1993; Harison, 1994; Ayad et al., 1997; Lade, 2001) the Poisson's ratio of the clays changes with water content in saturated states only weakly, if at all, and decreases with water content decrease in the unsaturated states from 0.5 to 0.3. Thus, the available data in general correspond with the obtained data on (w).
Results Flowing out of Hallaire's (1984) Data
The aim of this subsection is to formulate and discuss relevant results based on immediate data on el', el, es, and e from Fig. 6.
The rs', rsM, and rs values that were estimated using data from Fig. 6 are presented in Fig. 8
. The step between the values of the gravimetric water content was w = 0.01 g g–1. The corresponding M, L, and ML estimates are shown in Fig. 9
. The obtained dependences are not quite smooth because numerical values el', el, es, and e were directly taken from the trend lines in Fig. 6. However, this roughness does not influence the major results for the aggregated soil following Fig. 8 and 9.
2.84 and, hence, relatively close to Bronswijk's (1990) experimental estimate, rs' = 3, for his aggregated-clay-soil samples after oven drying. 2.9 at drying (Fig. 8) corresponds to the rapid increase of the total crack volume at the first stage with the appearance of a dense network of small and thin cracks. Further lowering of rs at drying to approximately 2 (Fig. 8) corresponds to a gradual decrease of the total crack volume at the second stage to an approximately stable value, in spite of the appearance of large and wide, but more seldom cracks, because of the closing of many small cracks.
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Based on Fig. 9, one can note that the essential variation of M, L, and ML factors with water content and their differences of unity in the overwhelming part of the w range also confirm violation of Assumptions 1 to 3 for the aggregated clay soil under consideration. In physical terms the difference between M and unity means the essential impact of cracks in the samples on the estimate of the rsM factor relating to the samples and determining the total crack volume inside them. The difference between L and ML and unity means the essential impact of soil stretching in a layer, under the action of shrinkage tensile stresses, on the estimate of the rs factor relating to the layer and determining the total crack volume in it.
Finally, Fig. 10
shows a slight change of Poisson's ratio for Hallaire's (1984) aggregated clay soil with water content decrease as in the above case of the clay paste where 0.45 < 0.5. The small variation of Poisson's ratio is in agreement with available data from the abovementioned works (Briones and Uehara, 1977; Haberfield and Johnston, 1990; Murdoch, 1993; Harison et al., 1994; Ayad et al., 1997; Lade, 2001). Dependence (w) (immediately found from experimental el' and el) for the clay soil looks quite similar to that in Fig. 10 with the ordinate range being from 0.96 to 1.
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'l(w), s(w), and (w) (Fig. 2) are sufficient to estimate rs', M, and rsM. Then, using the M, and rsM values we estimate three separate contributions to the matrix volume change of the sample—the contribution of cracks developing in the cores, that of core height decrease, and that of core diameter decrease. For a cracked soil layer, to estimate the exact rs value and exact contributions of cracks and subsidence into the matrix volume change, one should know either the (w) dependence and in situ l(w) (Fig. 2) or the 'l(w) and s(w) dependencies for samples (along with (w)) (Fig. 2) as well as the parameter as a function of water content. The parameter is estimated as suggested above (for details see Appendices 4 and 5).
Differences Between the Two above Experimental Examples
Besides using an aggregated clay soil unlike a pure-clay paste, Hallaire's (1984) data differ from Chertkov et al.'s (2004) data in two relations. First, Hallaire's (1984) data on the true void ratio of a soil layer, el (connected to l) were obtained based on in situ measurements. Unlike that, Chertkov et al.'s (2004) data do not immediately contain the values of the true specific volume, l of a layer. We estimated the l values for a clay paste layer using the parameter that was in turn estimated based on Chertkov et al.'s (2004) data of sample height and diameter evolution with drying. Second, the data on an aggregate void ratio, e (connected to ) from Hallaire (1984) were also immediately measured using the aggregates. Unlike Hallaire (1984), Chertkov et al. (2004) estimated using the data of clay paste properties (s, z, mL) and a model from Chertkov (2000b)(2003).
Estimating Poisson's Ratio
Comparison between the estimates (Fig. 7, square points) that were obtained for the clay paste using the quasi-elastic model and those (0.96 < 1) obtained for the aggregated clay soil directly from the el and el' data (from Eq. [5] 2 = 
/
), speaks in favor of the feasibility of the quasi-elastic model of a soil layer as applied to the connection between its true specific volume (l) and the specific volume in Bronswijk's approximation ('l).
Poisson's ratio is directly connected to the parameter (Eq. [6] and [7]). For this reason the feasibility of the quasi-elastic model as well as similarity between estimates of Poisson's ratio for the clay paste (0.45 < 0.5) and in Fig. 10 for the aggregated clay soil, speaks in favor of the possibility of estimating the Poisson's ratio of a swell-shrink soil either from the l/'l ratio as it was made for the aggregated clay soil or from the sample diameter and height measurements as was made for the clay paste. Thus, the approach for rs estimation simultaneously gives a method for experimentally estimating Poisson's ratio for swelling soils at different moisture contents.
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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'l(w), the shrinkage curve of a sample with cracks, s(w), and the shrinkage curve of a soil matrix without cracks, (w).
In the present work, we consider correction of the rs factor value accounting for the stretching of a shrinking soil layer. This correction leads to changes in the soil subsidence and crack volume compared with Bronswijk's approximation. To estimate the corrected rs value, starting from the initial estimate, rs' (Bronswijk, 1990) we introduce, along with (w), s(w), and 'l(w), the shrinkage curve of a stretched layer with cracks, l(w), based on two known physical conditions:
Two different experimental examples are considered to illustrate the corrected rs value estimations. The data of the first example (Chertkov et al., 2004) relate to a clay paste. The specific volumes l(w) and (w) are estimated from data on the evolution of the height and diameter of clay samples, and on clay properties. The data of the second example (Hallaire, 1984) relate to an aggregated clay soil. The specific volumes l(w) and (w) are directly measured in situ and on the aggregates, respectively. Results show that the shrinkage geometry factor in Bronswijk's approximation (rs'), that for a sample (rsM), and that for a layer (rs), significantly differ. That is, impacts of cracks in soil samples and a soil layer stretching at shrinkage should be taken into account when estimating the rs factor value. The results also show that rs', rsM, and rs significantly change with water content. The results can be useful in discussing the possibility of transferring the hydraulic properties and flow features of a swelling soil that are observed on or calculated for soil samples, to the case of the soil in field conditions.
In conclusion, it is worth stressing the following considerations. Introducing corrections to the rs value to be obtained in the traditional way (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) is, without doubt, a necessity (Chertkov et al., 2004). Chertkov et al. (2004) and this work suggest an approach to estimate the corrections. The approach is theoretically sound. However, the experimental estimates of the corrected rs value are considered for the first time and we have no material for comparison with different soils. Data that we could use to illustrate the calculation of the totally corrected rs value are limited (this is usual when dealing with new things). They relate to clay paste samples (Chertkov et al., 2004) and an aggregated clay soil (Hallaire, 1984). Nevertheless, in our opinion this illustration is sufficiently clear and convincing, although the use of more extensive data in the future is desirable.
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APPENDIX 1 |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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do initial cylindrical-sample diameter, cm
dm equivalent diameter of a soil sample matrix without cracks, cm
e aggregate void ratio, dimensionless
el true layer void ratio, dimensionless
el' layer void ratio in Bronswijk's approximation, dimensionless
es sample void ratio, dimensionless
h current sample height, cm
L multiplicative correction to the rsM value accounting for the violation of Assumption 1, dimensionless
M multiplicative correction to the rs' value accounting for the violation of Assumption 2, dimensionless
ML multiplicative correction to the rs' value accounting for the violation of Assumptions 1 and 2 simultaneously, dimensionless
m weight of the clay paste sample, g
md oven-dried weight of the clay paste sample, g
rs shrinkage geometry factor of a soil layer, dimensionless
rsM shrinkage geometry factor of a soil sample or rs value corrected, in part by M multiplier, dimensionless
rs' rs factor in Bronswijk's approximation, dimensionless
uxx longitudinal deformation of a soil layer, dimensionless
uyy deformation of soil layer along a normal to it, dimensionless
V initial layer volume in Bronswijk's model (per one mentally separated cube with free boundaries), cm3
specific volume of a soil matrix without cracks, cm3 g–1
o initial specific volume of a soil matrix, cm3 g–1
cr.l specific crack volume in a soil layer, cm3 g–1
cr.s specific volume of cracks in a soil sample, cm3 g–1
l specific volume of the soil layer with cracks, cm3 g–1
'l specific volume of the soil layer with cracks in Bronswijk's approximation, cm3 g–1
lz specific volume of the soil layer with cracks at w = 0, cm3 g–1
s specific volume of soil sample with cracks, cm3 g–1
sz specific volume of the oven-dried soil sample with cracks, cm3 g–1
z specific volume of the oven-dried soil matrix without cracks, cm3 g–1
w gravimetric water content of a soil, g g–1
wo initial water content of a clay sample close to the liquid limit, g g–1
wL liquid limit of the clay paste, g g–1
X variable from Eq. [A4.5], dimensionless
x horizontal size of a stretched soil layer matrix (per one initial imaginary cube of size z), cm
x' horizontal size of a cube of initial z3 volume in Bronswijk's model, cm
z initial soil layer thickness, cm
V matrix volume decrease of a soil layer for drying in Bronswijk's model (per one mentally separated cube with free boundaries), cm3
z subsidence of soil layer, cm
z' subsidence of isolated soil cube in Bronswijk's model, cm
L standard deviation of the averaged L value, dimensionless
M standard deviation of the averaged M value, dimensionless
ML standard deviation of the averaged ML value, dimensionless
rs' standard deviation of the averaged rs' value, dimensionless
rs standard deviation of the averaged rs value, dimensionless
l standard deviation of the averaged l value, cm3 g–1
'l standard deviation of the averaged 'l value, cm3 g–1
maximum error in value, dimensionless
µ parameter of a soil layer from Eq. [A4.1] and [A4.2], dimensionless
Poisson's ratio of soil matrix, dimensionless
s clay particle density, g cm–3
parameter of the soil layer from Eq. [5]–[7], dimensionless
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APPENDIX 2 |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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 | [A2.1] |
o and are initial (at the maximum possible water content) and current values of the specific volume of the soil matrix without cracks; l 
+ cr.l is the specific volume of the layer including the matrix () and cracks (cr.l). The presentation itself given by Eq. [A2.1] shows that rs is a function of water content, w because the (w) and l(w) functions are the corresponding shrinkage curves.
By analogy to Eq. [A2.1] relating to an unlimited layer with cracks one can introduce an equation for the shrinking sample of an arbitrary shape, in particular a cylinder, and with developing cracks as
 | [A2.2] |
(w) of the sample matrix: the contribution of the change in external sample sizes and contribution of crack appearance and development within the sample (see Fig. A2.1)
. s + cr.s is the specific sample volume (as a function of w); cr.s is the specific volume of cracks in the sample.
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APPENDIX 3 |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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s + cr.s > . However, in Bronswijk's approximation the specific volume of the soil matrix without cracks, entering the left part of Eq. [A2.1], is replaced by the specific volume, s of a cylindrical sample (Bronswijk, 1990) with possible cracks. So, if we consider that Assumption 1 is fulfilled (i.e., stretching a soil layer does not change l), but Assumption 2 is not fulfilled (i.e., cr.s 0), the replacement of by s in Eq. [A2.1] as
 | [A3.1] |
According to Eq. [A2.1], [A2.2], and [A3.1] the rs value accounting for cracks in samples is obtained by rs' (i.e., rs in Bronswijk's approximation) as
 | [A3.2] |
1 of cylindrical samples with cracks plays the part of a multiplicative correction to the inexact estimate of rs' from Eq. [A3.1]. If cracks in the sample are absent cr.s = 0, s = and in Eq. [A2.2] and [A3.2] M = 1. |
APPENDIX 4 |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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 | [A4.1] |
 | [A4.2] |
is Poisson's ratio of the soil matrix without cracks. According to the physical meaning of z and z' values as well as x and x' values (see Fig. 4) one can write
 | [A4.3] |
 | [A4.4] |
 | [A4.5] |
 | [A4.6] |
z') = x2(z – z) (see Fig. 4). That is, x/x' ratio can be written as
 | [A4.7] |
Replacing in Eq. [A4.4] x/x' from Eq. [A4.7] and using the X variable from Eq. [A4.5] one can rewrite Eq. [A4.4] as
 | [A4.8] |
 | [A4.9] |
 | [A4.10] |
z and z' as well as between x and x' as
 | [A4.11] |
Finally, the relation, that we need (Eq. [5]), between the specific volume l 
z2 of the real layer with cracks (Fig. 1c and 4) and the specific volume 'l 
z2 of the soil layer in Bronswijk's approximation (Fig. 1d and 4), follows Eq. [A4.11].
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APPENDIX 5 |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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x(w) z is fulfilled for the horizontal size x(w) (see Fig. 4). Therefore, the value x(w) = xmin(w) x'(w) according to Eq. [A4.11] corresponds to max(w) = 1, and the value x(w) = xmax(w) z corresponds to min(w) = x'(w)/z. Hence, at a given water content the value of the parameter is in the range
 | [A5.1] |
|
(w) dependence (Fig. A5.1) as
 | [A5.2] |
(w) value as
 | [A5.3] |
(0)/(0) 0.1, and (w)/(w) quickly decreases with water content increase. For this reason, in the experimental illustrative estimation of the corrected rs values we find (w) in Eq. [8] using the average dependence, (w) from Eq. [A5.2].
In the practical use of core sample measurements one can take x'(w) d(w), z do, and x'/z = d/do where d is the current sample diameter and do is the initial d value. However, the x' size by definition relates to a soil matrix without cracks (see Fig. 1d, 3, and 4) while the soil samples in the course of drying can contain cracks. That is, the relation x' d is inaccurate. Therefore the corresponding estimate from Eq. [A5.2],
 | [A5.4] |
/ < 0.1 and quickly decreases with water content increase. Accounting for the cracks in the samples one can introduce an equivalent diameter, dm of the soil matrix without cracks in a sample and estimate dm from the equality of two different expressions for the soil matrix volume of the sample as md = h d2m 
/4 where md is the oven-dried weight of the sample, h is the current sample height; and is the current specific volume of the soil matrix. This estimate gives dm from Eq. [10] and together with the exact relation x'/z = dm/do and Eq. [A5.2] gives the (w) estimate from Eq. [9]. Equation [A5.3] and x'/z = dm/do give the maximum error in (w) value as
 | [A5.5] |
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ACKNOWLEDGMENTS |
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Received for publication October 25, 2004.
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4APPENDIX 5REFERENCES |
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, the correcting M factor, and the corrected in part rs factor value (rsM) (w, r's, M, rsM from 
of a layer with dotted initial upper surface (corresponding to 