Physical modeling of shrink–swell cycles and cracking in a clayey vadose zone

Geomechanics for Energy and the Environment 1 (2015) 16–33

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Physical modeling of shrink–swell cycles and cracking in a clayey vadose zone V.Y. Chertkov ∗ Division of Environmental, Water, and Agricultural Engineering, Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel

highlights • • • • •

Crack origin far from the soil surface in a clayey vadose zone is addressed. Soil structure, layer size, loading, water content range, and cycling are caught. Clay, intra-aggregate matrix, soil without and with cracks are studied. Primary and scanning shrinkage and swelling curves are derived. Crack volume hysteresis for steady shrink–swell cycles is predicted.

article

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Article history: Received 16 September 2014 Received in revised form 28 January 2015 Accepted 26 February 2015 Available online 18 March 2015 Keywords: Physical modeling Clay soil profile Intra-aggregate structure Multifold scanning shrink–swell cycles Overburden pressure Crack volume hysteresis

abstract Physical understanding of the crack origin and quantitative physical prediction of the crack volume variation far from the clay soil surface are necessary to protect the underlying aquifers from pollutants. The basis of this work is an available physical model for predicting the shrink–swell curves in the maximum water content range (the primary curves) and crack volume variation in the range. The objective of the work is to generalize this model to the conditions of the deep layer of a clayey vadose zone with the overburden pressure, multiple shrinkage–swelling, and variation of water content in a small range. We aim to show that the scanning shrinkage and swelling curves, and steady shrink–swell cycles existing in such conditions, lead to the occurrence of cracks and a hysteretic crack volume. The generalization is based on the transition to the increasingly complex soil medium from the contributive clay, through the intra-aggregate matrix and aggregated soil with no cracking, to the soil with cracks. The results indicate the single-valued physical links between the scanning shrink–swell cycles and crack volume variation of the four soil media on the one hand, and primary shrinkage and swelling curves of the media on the other hand. The predicted cycles and crack volume hysteresis can be expressed through the physical properties and conditions of the soil at a given depth. The available observations of the cracks and crack volume variation in the clayey vadose zone give strong qualitative experimental evidence in favor of the feasibility of the model. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Shrinkage cracks occurring close to the soil surface owing to the vertical water content gradient are known.1,2

∗

Tel.: +972 4829 2601. E-mail addresses: [email protected], [email protected].

http://dx.doi.org/10.1016/j.gete.2015.02.001 2352-3808/© 2015 Elsevier Ltd. All rights reserved.

Their characteristics cannot explain the origin and volume variation of the cracks that are directly or indirectly observed at sufficiently large depths of a clayey vadose zone3,4 (among others). Such cracks can essentially increase the hydraulic conductivity of the clayey vadose zone. The physical understanding of the origin of such cracks and quantitative physical prediction of the crack volume variation are important to protect the underly-

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

ing aquifers from different pollutants. The physical understanding and prediction should take into account the major specific limitations existing in a clay soil at large depths: (i) overburden action at a given depth; (ii) limited and frequently small range of water content variation (compared to the maximum range); and (iii) multifold drainage-wetting alternation. The works that consider the explanation and prediction of shrink–swell cycling and accompanying cracking in a clayey vadose zone, based on the physical characteristics of soil texture and structure (with no fitting) and local hydrological characteristics, are absent. The contemporary approaches that combine the microphysics of water–vapor–solid interactions with different variants of transition to a clay soil continuum based on conservation laws and thermodynamics (see5 among others), essentially consider a clay paste with no crack occurrence as well as no inter- and intra-aggregate structure leading to soil cracking at shrink–swell processes. The known approach to overburden effects in swelling soils6 (for previous references see6 ) and subsequent discussion (see7 among others) are based on the equilibrium thermodynamics and the shrinkage curve slope as a function of water content and applied load, with the use of some form of fitted shrinkage curve in the maximum water content range. The physical description (i.e., with no fitting) of overburden effects in connection with soil structure effects as well as cracking, swelling, and multifold drainage-wetting in an arbitrary range of water content are beyond the scope of this approach. The same relates to a recent work8 that modifies the approach6 using still another form of the fitted shrinkage curve. It should be noted that available approaches to soil cracking, based on either a physical description (e.g.,9 ) or using a number of empirical dependences (e.g.,10 ), only relate to shrinkage in the maximum water content range close to the soil surface or in small samples, that is, with no effects of overburden, swelling, and multifold shrink–swell cycling in a small water content range. A natural background for the development that accounts for the above limitations in a clay soil of large depths should include some description of the shrink–swell curves and crack volume variation during a single drainage-wetting cycle in the maximum range of water content and without soil loading. Such a description was recently suggested and experimentally validated.11 The objective of the present work is to extend the approach11 accounting for overburden, arbitrary water content range, and multifold drainage-wetting cycles, and keeping the physical character of the approach. The methodology11 is based on the recently suggested inter- and intra-aggregate structure of inorganic soils12–14 (Fig. 1) and successive consideration of the increasingly complex soil media: clay (paste), intra-aggregate soil matrix (including clay, silt, sand, and lacunar pores), aggregated soil without cracks (small samples), and aggregated soil with cracks (soil layer). The key point is also a link between the swelling curves of these soil media as well as between the swelling and shrinkage curves for each of the media. First, we extend the shrinkage–swelling of contributive clay to conditions of overburden pressure and multifold drainage-wetting with small variations of water content (Section 2). Then, we extend the results to be found for clay

17

to the case of the intra-aggregate matrix (Section 3) and aggregated soil without and with cracks (Section 4). Section 5 sums the theoretical results. In Sections 6 and 7 we analyze the limited available data on shrinkage under loading to check the major model aspects. 2. Modeling shrinkage and swelling of a pure clay 2.1. Modeling clay shrinkage and swelling with no loading 2.1.1. Available modeling of primary shrinkage and swelling curves with no loading We will refer to the shrinkage and swelling curves of a clay paste in the maximum possible range of water content as primary curves. The expressions of the primary shrinkage, v(ζ ) and swelling, vˆ (ζ ) curves of clay in relative coordinates (Fig. 2, curves 1 and 2; v and vˆ are the ratios of the clay volume at shrinkage and swelling to its maximum in the solid state, i.e., at the liquid limit; ζ is the ratio of the clay water content to its maximum in the solid state) have been derived and experimentally validated11 (for previous references see11 ). The derivation was based on the smallness of a number of values as ζn − ζz ≪ 1, vn − vz ≪ 1, ζh ∼ = 0.5 < 1, vh − vz ≪ 1 (Fig. 2; the (ζz , vz ), (ζn , vn ), and (ζh , vh ) points correspond to the clay shrinkage limit, air-entry point, and maximum swelling point, respectively). The v(ζ ) curve (Fig. 2) consists of two linear parts in the 0 ≤ ζ ≤ ζz and ζn ≤ ζ ≤ ζh ranges and a square part in the ζz ≤ ζ ≤ ζn range. vˆ (ζ ) is presented by one square line (Fig. 2). Note also the porosity, P (v(ζ )) and the maximum and minimum internal sizes, rm (v(ζ )) and ro (v(ζ )), respectively, of clay matrix pores at primary shrinkage (excluding pore wall thickness) as well as the corresponding values at primary swelling, Pˆ (ζ ) = P (ˆv (ζ )), rˆm (ζ ) = rm (ˆv (ζ )), and rˆo (ζ ) = ro (ˆv (ζ )).11 The two physical clay parameters, vs (ratio of clay solid volume to clay volume at the liquid limit) and vz determine v(ζ ) and vˆ (ζ )11 (for experimental estimating vs and vz see12 ; note that vh = vs + (1 − vs )ζh ∼ = 0.5(1 + vs )11 ). The coordinates (ζ , v) or (ζ , vˆ ) give the customary (w, ¯ V ) or (w, ¯ Vˆ ) (the specific volume V or Vˆ vs. gravimetric water content, w ¯ of the clay) by V = v/(vs ρs ), Vˆ = vˆ /(vs ρs ), and w ¯ = ((1 − vs )/vs )(ρw /ρs )ζ (ρs and ρw are clay solid and water density). 2.1.2. Extension to scanning swelling and shrinkage curves of clay with no loading The swelling curve, vˆ (ζ , ζo ) (Fig. 2, curve 3) that starts at a point (ζo , v(ζo )) of the primary shrinkage curve, v(ζ ) (Fig. 2, curve 1) will be referred to as the scanning swelling curve. The shrinkage curve, v(ζ , ζo ) (Fig. 3, curve 3) that starts at a point (ζo , vˆ (ζo )) of the primary swelling curve, vˆ (ζ ) (Fig. 3, curve 2) will be referred to as the scanning shrinkage curve. The possible initial ζo values are 0 < ζo < ζh (Figs. 2 and 3). Only one pair of scanning swelling and shrinkage curves passes through any point of the (ζ , v) plane between the primary shrink–swell curves (Figs. 2 and 3). The analytical expressions for the scanning swelling curves (Fig. 2, curve 3) flow out of: (i) the available primary swelling curve (Section 2.1.1); (ii) the same smallness

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V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Fig. 1. Schematic illustration of the accepted soil structure.11–14 Shown are (1) an assembly of many soil aggregates and inter-aggregate pores contributing to the specific soil volume, Y ; (2) an aggregate, as a whole, contributing to the specific volume Ua = Ui + U ′ ; (3) an aggregate with two parts indicated: (3a) an interface layer contributing to the specific volume Ui and (3b) an intra-aggregate matrix contributing to the specific volumes U and U ′ = U /K ; (4) an aggregate with indicated intra-aggregate structure: (4a) clay, (4b) silt and sand grains, and (4c) lacunar pores; and (5) an inter-aggregate pore leading at shrinkage to an inter-aggregate crack contributing to the specific volume Ucr . U is the specific volume of an intra-aggregate matrix (per unit mass of the oven-dried matrix itself). U ′ is the specific volume of an intra-aggregate matrix (per unit mass of the oven-dried soil). Ui is the specific volume of the interface layer (per unit mass of the oven-dried soil). Ucr is the specific volume of cracks (per unit mass of the oven-dried soil). Ua is the specific volume of aggregates (per unit mass of the oven-dried soil). K is the aggregate/intra-aggregate mass ratio.

of a number of values (Section 2.1.1); (iii) the square approximation of a scanning swelling curve; (iv) transition continuity of the scanning swelling curves to the primary that at ζo → 0 (Fig. 2); and (v) condition, vˆ (ζo , ζo ) = v(ζo ) (from the definition of the scanning swelling curves at 0 < ζo < ζh ; Fig. 2). The scanning swelling curves, vˆ (ζ , ζo ) at 0 < ζo < ζh (Fig. 2, curve 3) can be divided into two subfamilies of usual scanning swelling curves, at 0 < ζo < ζn and specific ones, at ζn < ζo < ζh (Fig. 2). The former curves keep the general shape of the primary swelling curve, vˆ (ζ ) (Fig. 2, curve 3) and are written as

vˆ (ζ , ζo ) = vh + aˆ (ζo )(ζ − ζh ) − bˆ (ζo )(ζ − ζh )2 , ζo < ζ < ζh , 0 < ζo < ζn , aˆ (ζo ) = (1 − vs )ζo /ζn ,

0 < ζo < ζn ,

(1a) (1b)

bˆ (ζo ) = [vh − v(ζo ) + aˆ (ζo )(ζo − ζh )]/(ζo − ζh )2 , 0 < ζo < ζn .

(1c)

The latter coincide with the primary shrinkage curve, v(ζ ) at ζn < ζo < ζ < ζh (Fig. 2) as

vˆ (ζ , ζo ) = v(ζ ) = vh − (1 − vs )(ζh − ζ ),

ζn < ζo < ζ < ζh .

(2)

The specific curves (Eq. (2)) differ from the usual curves by their linear shape. Note that the vˆ (ζ , ζn ) curve plays the part of the boundary one between the two subfamilies. The analytical expressions for the scanning shrinkage curves (Fig. 3, curve 3) flow out of: (i) the available primary shrinkage curve (Section 2.1.1); (ii) the same smallness of a number of values (Section 2.1.1); (iii) the square approximation of a scanning shrinkage curve; (iv) transition continuity of the scanning shrinkage curves to the primary that at ζo → ζh (Fig. 3); and (v) condition, v(ζo , ζo ) = vˆ (ζo ) (from the definition of the scanning shrinkage curves at 0 < ζo < ζh ; Fig. 3). Conditions (iii) and (iv) mean that in all the range 0 < ζo < ζh (Fig. 3) the scanning shrinkage curves, v(ζ , ζo ) keep the same general shape as the primary curve, v(ζ ) (Section 2.1.1), i.e., consist of two linear parts at 0 < ζ ≤ ζ (ζo ) and at ζ (ζo ) ≤ z n ζ ≤ ζo and square part between them at ζ z (ζo ) < ζ < ζ n (ζo ). Thus one has two additional conditions of the smooth transition between the linear and square parts of the v(ζ , ζo ) curve at ζ = ζ (ζo ) and ζ = ζ (ζo ). As a result z

n

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Fig. 2. The qualitative view of the clay scanning swelling curve, vˆ (ζ , ζo ) (curve 3) that goes between the primary shrinkage, v(ζ ) (curve 1) and swelling, vˆ (ζ ) (curve 2) curves and starts at ζ = ζo on the primary shrinkage curve. The initial water content, ζo can vary in the 0 < ζo < ζh range. At ζn < ζo < ζh vˆ (ζ , ζo ) = v(ζ ), i.e., curves 3 and 1 coincide.

Fig. 3. The qualitative view of the clay scanning shrinkage curve, v(ζ , ζo ) (curve 3) that goes between the primary shrinkage, v(ζ ) (curve 1) and swelling, vˆ (ζ ) (curve 2) curves and starts at ζ = ζo on the primary swelling curve. ζo can vary in the 0 < ζo < ζh range. The point, ζ = ζ n (ζo ) is a boundary of the initial linear range (ζ (ζo ) < ζ < ζo ) of the v(ζ , ζo ) curve. The point, ζ = ζ (ζo ) is a boundary of the final linear range (0 < ζ < ζ (ζo )) of the v(ζ , ζo ) n z z curve. ζ (ζo ) → ζz and ζ (ζo ) → ζn at ζo → ζh . z n

the scanning shrinkage curves (Fig. 3, curve 3) are written as

v(ζ , ζo ) = vz + a(ζo )ζ ,

0 < ζo < ζh .

v(ζ , ζo ) = d(ζo ) + e(ζo )ζ ,

(3b)

ζ n (ζo ) ≤ ζ ≤ ζo ,

0 < ζo < ζh ,

0 < ζ ≤ ζ (ζo ), 0 < ζo < ζh , z

a(ζo ) = 2λζh (1 − ζo /ζh )

ζ z (ζo ) = ζz ζo /ζh ,

(λ ≡ (vh − vz )/ζh2 ),

(3a)

(4a)

d(ζo ) = vz + [λ − (1 − vs )/ζh ]ζ , 2 o

0 < ζo < ζh ,

e(ζo ) = 2λζh − [2λζh − (1 − vs )]ζo /ζh ,

(4b)

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V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Fig. 4. The qualitative view of the two initial consecutive transitive scanning shrink–swell cycles of a cycle set leading to steady shrink–swell cycle in the ζ1 < ζ < ζ2 range (Fig. 5). The first cycle consists of the swelling branch, vˆ 1 (ζ , ζ1 ) at ζo = ζ1 ≡ ζo1 (curve 1) and shrinkage branch, v 1 (ζ , ζo2 ) at ζo = ζo2 (curve 2). The second cycle consists of the swelling branch, vˆ 2 (ζ , ζo3 ) at ζo = ζo3 (curve 3) and shrinkage branch, v 2 (ζ , ζo4 ) at ζo = ζo4 (curve 4). Curve 5 shows the start of the following transitive cycle.

0 < ζo < ζh .

(4c)

v(ζ , ζo ) = v z (ζo ) + a(ζo )(ζ − ζ z (ζo )) + b(ζo ) × (ζ − ζ z (ζo ))2 , ζ z (ζo ) < ζ < ζ n (ζo ), 0 < ζo < ζh (5a) , v z (ζo ) = vz + 2λ(1 − ζo /ζh )ζz ζo , 0 < ζo < ζh ,   b(ζo ) = [e(ζo ) − a(ζo )]2 / 4 v z (ζo ) − d(ζo )  − e(ζo )ζ z (ζo ) , 0 < ζo < ζh ,

(5b)

(5c)

ζ n (ζo ) = ζ z (ζo ) + 2[v z (ζo ) − d(ζo ) − e(ζo )ζ z (ζo )]/ [e(ζo ) − a(ζo )],

0 < ζo < ζh .

(5d)

The extended expressions for rm (ζ ), ro (ζ ), rˆm (ζ ), rˆo (ζ ), P (ζ ), and Pˆ (ζ ) along the scanning shrinkage (v(ζ , ζo )) and swelling (ˆv (ζ , ζo )) curves are obtained from the available those (Section 2.1.1) after v(ζ ) → v(ζ , ζo ) and vˆ (ζ ) → vˆ (ζ , ζo ). The latter replacements, plus V (w) ¯ → V (w, ¯ w ¯ o ) and Vˆ (w) ¯ → Vˆ (w, ¯ w ¯ o ) where w ¯ o = ((1 − vs )/vs )(ρw /ρs )ζo , determine the transition to customary coordinates as applied to scanning curves based on the same relations from the end of Section 2.1.1. 2.1.3. Extension to transitive and steady scanning shrink–swell cycles with no loading Fig. 4 shows the two initial scanning shrink–swell cycles that start at the primary shrinkage curve at ζ = ζ1 . It is clear from Fig. 4 that eventually we come to a steady multifold shrink–swell cycle (Fig. 5) with repeating shrinkage and swelling branches in the ζ1 < ζ < ζ2 range. The previous cycles (Fig. 4) can be referred to as transitive ones. Starting at the ζ = ζ2 point of the primary swelling curve, vˆ (ζ ) (Fig. 4) and conducting the similar consideration we come

to the same steady multifold shrink–swell cycle (Fig. 5). Thus, the ζ1 < ζ < ζ2 range in the single-valued manner determines the corresponding steady shrink–swell cycle of a clay (Fig. 5). Its two branches, v(ζ , ζo1 ) and vˆ (ζ , ζo2 ), i.e., the corresponding ζo1 and ζo2 values that determine the branches (Section 2.1.2) are found from the two evident conditions at ζ = ζ1 and ζ = ζ2 (Fig. 5) as v(ζ1 , ζo1 ) = vˆ (ζ1 , ζo2 ), and v(ζ2 , ζo1 ) = vˆ (ζ2 , ζo2 ). 2.2. Extension to clay shrinkage and swelling under loading 2.2.1. Extension of the primary clay shrink–swell curves to the case of loading The loading, L is considered to be approximately constant at a given depth. The primary clay shrink–swell curves at L = 0 are only expressed through two clay characteristics, the relative volume of clay solids, vs and relative minimum clay volume, vz (Section 2.1.1). By its physical meaning vs cannot change with L. Unlike that, vz , in general, should depend on L. In addition, at L > 0 ζh and vh (Fig. 6) also become functions of L. Thus the primary clay shrink–swell curves depend on L through vz (L) and vh (L) (or ζh (L)) (Fig. 6). The (ζh , vh ) point is displaced under loading to (ζh (L), vh (L)) along the saturation line (Fig. 6) since ζh (L) > ζn (L). 2.2.2. The minimum clay volume (vz ) as a function of loading (L) The volume, vz corresponds to a rigid clay particle network between the oven-dried state (ζ = 0) and shrinkage limit (ζ = ζz ) (Fig. 6). Therefore to describe the vz variation under loading one can take advantage of the elastic model and characterize a dry clay matrix by Young’s

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Fig. 5. The qualitative view of the steady shrink–swell cycle (curves 1 and 2) of a clay in the ζ1 < ζ < ζ2 range. The cycle consists of the shrinkage branch, v(ζ , ζo1 ) (curve 1) and swelling branch, vˆ (ζ , ζo2 ) (curve 2) where ζo1 = ζo1 (ζ1 , ζ2 ) and ζo2 = ζo2 (ζ1 , ζ2 ) are a solution of equations at the end of Section 2.1.3.

Fig. 6. The general qualitative view of transformation of the clay maximum shrink–swell cycle under loading. The indicated values are connected by the following relations: (i) v(ζ ) ≡ v(ζ , L = 0), vˆ (ζ ) ≡ vˆ (ζ , L = 0); (ii) ζh = ζh (L = 0), ζn = ζn (L = 0), ζz = ζz (L = 0); (iii) vh = v(ζ = ζh , L = 0), vn = v(ζ = ζn , L = 0), vz = v(ζ = ζz , L = 0); (iv) vh (L) ≡ v(ζ = ζh (L), L), vn (L) ≡ v(ζ = ζn (L), L), vz (L) ≡ v(ζ = ζz (L), L).

modulus, E and Poisson’s ratio, σ . We consider the state of the dry clay under loading, L as the result of homogeneous compression of a thick isotropic elastic horizontal layer by pressure L that is applied to its surfaces. The deformation of the layer is written as uzz = ∆vz /vz (z is a vertical axis). vz is the initial value (Fig. 6). By definition and from the elasticity theory15 : ∆vz ≡ vz − vz (L) and ∆vz = [L/(α E )]vz where α = (1 − σ )/[(1 + σ )(1 − 2σ )]. As a result we have vz (L) as

vz (L) = vz [1 − L/(α E )].

(6)

For different clays the L/(α E ) factor in Eq. (6) is always small. Indeed, available data on E and σ of dry clays that were obtained by different methods (see,16 among others) are characterized by the spread, approximately, in the ranges, E ∼ 102 –103 MPa and σ ∼ 0.1–0.3 (i.e., α ∼ 1–1.4). However, in any case at L < Lmax ∼ 1–1.5 MPa (for the maximum soil depth ∼50 m) one has the estimate, L/(α E ) ≤ 10−3 –10−2 . The smallness of L/(α E ) justifies the

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Fig. 7. The qualitative view of the ζh (L/L∗ ) dependence. Quantitatively, ζh (L/L∗ ) is described by Eq. (7) and L∗ (see the end of Section 2.2.3).

use of the elastic model in estimating the vz (L) dependence of clay. At sufficiently small loading, L < \ ∼ = 100 kPa Eq. (6) gives vz (L) ∼ = vz since L/(α E ) ≤ 10−4 –10−3 . The corresponding available data are analyzed in Section 7. In spite of the always small change of the dry clay volume, vz (L) with the L growth (Eq. (6)), the volume variation of a soil, that includes this clay, can be quite appreciable at small water contents even at L < 10 kPa when vz (L) ∼ = vz = const. The reason for this is the non-clay porosity variation with L (Section 7). 2.2.3. The maximum swelling point (ζh , vh ) variation as a function of loading (L) ζh (L) varies between the maximum, ζh = 0.5 at L = 0 (Section 2.1.1) and minimum, ζh (L → ∞) = (vz − vs )/(1 − vs ) (Fig. 7). Indeed, (vz − vs )/(1 − vs ) is the ratio of the minimum to the potential maximum (at the liquid limit) pore volumes of the clay, and this physical meaning is in the agreement with the physical meaning of ζh (L → ∞). L* in Fig. 7 is some characteristic loading of the clay that determines the velocity of ζh decrease when L increases. The negative increment, dζh per unit increment of loading, L, i.e., dζh /dL, should be proportional to the part of the relative maximum water content, ζh (L/L∗ ) that can change at a given L, i.e., to the difference [ζh (L/L∗ ) − (vz − vs )/(1 − vs )] (see Fig. 7). Therefore (1/L∗ is a proportionality coefficient) dζh = −[ζh (L/L∗ ) − (vz − vs )/(1 −vs )]dL/L∗ . Integration with condition, ζh (L = 0) = 0.5 (Fig. 7) gives ζh (L/L∗ ) as

ζh (L/L∗ ) = (vz − vs )/(1 − vs ) + [0.5 − (vz − vs )/(1 − vs )] exp(−L/L∗ ), (7) and vh (L) = vs + (1 − vs )ζh (L) (cf. vh = vs + (1 − vs )ζh in Section 2.1.1) gives (at vh = 0.5(1 + vs )):

vh (L/L∗ ) = vz + (vh − vz ) exp(−L/L∗ ).

(8)

The characteristic loading, L* can only depend on the ratio,

(vz −vs )/(1 −vs ) (Fig. 7). With that (vz −vs )/(1 −vs ) ≪ 1,

but L*, in general, is not small. Thus, L∗ is the decreasing (by its physical meaning) and non-small function of the small variable when it increases. The simplest such function is

L∗ = Lu /[(vz − vs )/(1 − vs )] or L∗ = Lu /[(vz − vs )/(1 − vs )]2 where Lu is a constant (with dimension of loading) that does not depend on vs and vz and in this meaning is universal. To estimate Lu we use available data (Section 7). 2.2.4. Extension of different clay shrink–swell curves to the case of loading Eqs. (6)–(8) enable the physical quantitative prediction of clay shrink–swell curves under loading (primary, scanning ones, and cycles) based on their expressions for the case with no loading in Section 2.1. To this end in all equations of Section 2.1 one should just replace vz , vh , and ζh = 0.5 with above vz (L), vh (L), and ζh (L). 3. Modeling shrinkage and swelling of the intraaggregate matrix of a soil 3.1. Modeling shrinkage and swelling of intra-aggregate matrix with no loading 3.1.1. Available primary shrink–swell curves with no loading The primary curves, u(ζ ) and uˆ (ζ ) of the intra-aggregate matrix (Fig. 1) are linked with the similar curves of the contributive clay, v(ζ ) and vˆ (ζ ) as11–13 u(ζ ) = ulpz + k(1 − uS )vz + uS + (1 − k)(1 − uS )v(ζ ), 0 < ζ ≤ ζh ,

(9a)

uˆ (ζ ) = ulpz + k(1 − uS )vz + uS + (1 − k)(1 − uS )ˆv (ζ ), 0 < ζ ≤ ζh

(9b)

where v(ζ ) and vˆ (ζ ) are from Section 2.1.1; u and uˆ are the ratios of the matrix volume at shrinkage and swelling to its maximum in the solid state; ζ is the ratio of the matrix water content to its maximum in the solid state and is equal to ζ for clay; the lacunar factor, k is a function of clay content, c, clay type, and soil texture as17 k(c /c ∗ ) = [1 − (c /c ∗ )3 ]1/3 , k(c /c ) = 0, ∗

0 < c /c ∗ < 1,

1 < c /c < 1/c ∗

∗

(10a) (10b)

where c ∗ = [1 + (vz /vs )(1/p − 1)]−1 is the critical clay content and p is the porosity of the contributive silt

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Fig. 8. The qualitative view of the transformation of the maximum shrink–swell cycle for the intra-aggregate matrix under loading (at c < c ∗ and 0 < k < 1). The u(ζ ) curve with the initial slope, (1 − k)(1 − us ) starts at the (ζh , uh ) point on the pseudo saturation (dash–dot) line (with the slope, (1 − us )) that corresponds to L = 0. The displacement, ζh∗ − ζh = ulph /(1 − us ) of the line relative to the true saturation (dashed) line is connected with the lacunar pores that are non-filled in water. The u(ζ , L) curve with the same initial slope, (1 − k)(1 − us ) starts at the (ζh (L), uh (L)) point on another pseudo saturation (dotted) line that corresponds to the L loading. The smaller displacement of the line relative to the true saturation line corresponds to the smaller lacunar pore volume at loading, L.

and sand grains when they are in the state of imagined contact12 ; uS = vs (1 − c )[c + vs (1 − c )]−113 is the relative volume of non-clay solids of intra-aggregate matrix; and ulpz is the relative volume of lacunar pores in soil intraaggregate matrix at ζ = ζz . ulpz is connected with ulph (similar value, but at ζ = ζh ) as13 ulph = ulpz − k(1 − uS )(vh − vz ).

(11)

The cycles, (v(ζ ), vˆ (ζ )) (Fig. 2) and (u(ζ ), uˆ (ζ )) (Fig. 8) are qualitatively similar. However unlike the clay case (Fig. 2), the slope, du/dζ at ζn < ζ ≤ ζh can be less than the slope of the saturation line (Fig. 8), and the (ζh , uh ) point is on the pseudo saturation line (Fig. 8)13 because of the presence of lacunar pores (Fig. 1). Then 0 < k < 1 and in Eqs. (9a) and (9b) (1 − k) < 1.13 To transit to (w, U ) and (w, Uˆ ) (specific volume vs. gravimetric water content of the intra-aggregate matrix at shrinkage and swelling) one can use U = u/(us ρs ), Uˆ = uˆ /(us ρs ), w = ((1 − us )/us )(ρw /ρs )ζ , 0 < w = c w ¯ ≤ wh = w(ζh ).11 3.1.2. Extension to scanning curves and shrink–swell cycles with no loading Eqs. (9a) and (9b) reflect a particular case of the more general link between a point of (ζ , u) plane (Fig. 8) and the corresponding point of (ζ , v ) plane (Fig. 2). It means the similar relations should also be fulfilled at possible water contents, ζ for the volumes of the intra-aggregate matrix and contributive clay along any scanning shrinkage (v(ζ , ζo ) and u(ζ , ζo )) and swelling (vˆ (ζ , ζo ) and uˆ (ζ , ζo ))

curves as u(ζ , ζo ) = ulpz + k(1 − uS )vz + uS

+ (1 − k)(1 − uS )v(ζ , ζo ),

0 < ζ < ζo ≤ ζh

(12a)

uˆ (ζ , ζo ) = ulpz + k(1 − uS )vz + uS

+ (1 − k)(1 − uS )ˆv (ζ , ζo ),

0 < ζo < ζ ≤ ζh . (12b)

The expressions for u(ζ , ζo ) and uˆ (ζ , ζo ) show that the scanning curves of the intra-aggregate matrix are arranged inside the primary shrink–swell cycle, (u(ζ ), uˆ (ζ )) (Eqs. (9a) and (9b)) similar to the arrangement of the clay scanning curves, v(ζ , ζo ) and vˆ (ζ , ζo ) inside the primary cycle, (v(ζ ), vˆ (ζ )) (Fig. 2). Using the u(ζ , ζo ) and uˆ (ζ , ζo ) scanning curves instead of v(ζ , ζo ) and vˆ (ζ , ζo ) one can construct the transitive and steady shrink–swell cycles for the intra-aggregate matrix without loading in the total analogy to the case of the contributive clay (Section 2.1.3; Figs. 4 and 5). The transition to customary coordinates is realized as in Section 3.1.1 by the replacements, e.g., uˆ (ζ ) → uˆ (ζ , ζo ), Uˆ (w) → Uˆ (w, wo ), and wo = ((1 − us )/us )(ρw /ρs )ζo . 3.2. Extension to shrinkage–swelling of intra-aggregate matrix under loading Eqs. (9a), (9b), (12a) and (12b) transform under loading, L to u(ζ , L) = ulpz (L) + k(L)(1 − uS )vz (L) + uS + (1 − k(L))(1 − uS )v(ζ , L), 0 < ζ ≤ ζh (L), L ≥ 0,

(13a)

24

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Fig. 9. The qualitative view of the reference primary and quasi-primary shrink–swell cycles of a soil. 1—the reference primary shrinkage curve (Yr (W )) of

ˆ

ˆ )) of the first cycle; 2’—the reference quasi-primary swelling curve (Yˆ r (W ˆ )) of the first cycle; the firstcycle; 2—the reference primary swelling curve (Yˆr (W 3—the reference primary shrinkage curve of the second cycle; 4—the swelling branch of the second cycle. Yrh and Wh correspond to maximum swelling ˆ h correspond to maximum swelling before the first cycle; Yrz corresponds to maximum shrinkage in the course of the first and following cycles; Yˆrh and W after the first cycle. The inclined (dashed) line is the saturation or quasi-saturation one.

uˆ (ζ , L) = ulpz (L) + k(L)(1 − uS )vz (L) + uS + (1 − k(L))(1 − uS )ˆv (ζ , L), 0 < ζ ≤ ζh (L), L ≥ 0,

(13b)

u(ζ , ζo , L) = ulpz (L) + k(L)(1 − uS )vz (L) + uS + (1 − k(L))(1 − uS )v(ζ , ζo , L), 0 < ζ < ζo ≤ ζh (L),

(14a)

uˆ (ζ , ζo , L) = ulpz (L) + k(L)(1 − uS )vz (L) + uS + (1 − k(L))(1 − uS )ˆv (ζ , ζo , L), 0 < ζo < ζ ≤ ζh (L),

(14b)

respectively. v(ζ , L), vˆ (ζ , L), v(ζ , ζo , L), and vˆ (ζ , ζo , L) as dependences on loading, L were considered in Section 2.2. k(c /c ∗ (L)) as dependence on loading, L is determined by substitution for vz in c* (see after Eq. (10b)) the vz (L) dependence (Eq. (6)). To estimate ulpz (L), first we estimate ulph (L). At ζ = ζh the intra-aggregate matrix is close to the visco-plastic state, and the pore volumes of different type approximately change with loading, L proportionally to each other. In other words, the lacunar pore volume, ulph (L) varies with loading, L proportionally to the volume, (vh (L) − vs ) of the clay matrix pores (in relative units) in the intra-aggregate matrix at ζ = ζh (Fig. 1). Then, taking ulph (L = 0) ≡ ulph and vh (L = 0) ≡ vh we estimate the ulph (L) dependence as ulph (L) = ulph (vh (L) − vs )/(vh − vs ),

L>0

(15a)

(vh (L) being from Eq. (8)), and ulpz (L) follows from Eq. (11) as ulpz (L) = ulph (L) + k(c /c ∗ (L))(1 − uS )

× (vh (L) − vz (L)),

L > 0.

(15b)

Thus the quantitative prediction of the primary shrinkage (u(ζ , L)) and swelling (uˆ (ζ , L)) curves of an intraaggregate matrix under loading follows from Eqs. (13a)

and (13b) with the preliminarily calculated basic dependences, vz (L) (Eq. (6)), ζh (L) (Eq. (7)), and vh (L) (Eq. (8)). Fig. 8 shows the transformation geometry of the cycle, (u(ζ ), uˆ (ζ )) without loading to the cycle, (u(ζ , L), uˆ (ζ , L)) under loading (cf. Fig. 6). The scanning (u(ζ , ζo , L)) and (ˆu(ζ , ζo , L)) curves under loading follow from Eqs. (14a) and (14b) with the preliminarily calculated vz (L), ζh (L), vh (L). At the found scanning curves, u(ζ , ζo , L) and uˆ (ζ , ζo , L) the determination of the transitive and steady shrink–swell cycles in the intra-aggregate matrix under loading in the range, 0 < ζ1 < ζ < ζ2 < ζh (L) (Fig. 8; ζ1 and ζ2 are not shown) is totally similar to the analogous issue in the case of contributive clay under loading (Section 2.2.4; Figs. 4 and 5 with replacements, v(ζ , ζo ) → u(ζ , ζo , L) and vˆ (ζ , ζo ) → uˆ (ζ , ζo , L)). 4. Modeling shrinkage–swelling and cracking of a soil 4.1. Modeling reference shrinkage and swelling of a soil Reference shrinkage and swelling occurs without cracking and loading.11,12 4.1.1. Available modeling of reference primary shrinkage and swelling of a soil The primary shrink–swell cycle leads to aggregate destruction (see18,11 among others). Soil volume and water content at the new maximum swelling point decrease (Fig. 9)11 unlike the volume and water content of clay paste or intra-aggregate matrix (see Figs. 2 and 8). The ˆ )) (Fig. 9, first reference primary cycle of a soil (Yr (W ), Yˆr (W curves 1 and 2; the soil specific volume vs. total water content) has been derived and validated.11 At shrinkage

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

25

ˆ

ˆ )) (curves 1 and 2’, respectively) and reference scanning Fig. 10. The qualitative view of the reference quasi-primary shrink–swell cycle, (Yr (W ), Yˆ r (W shrinkage curve of a soil, Y r (W , W o ) (curve 3). The inclined (dashed) line is the saturation or quasi-saturation one.

and swelling it is determined by the volume of the intraaggregate matrix, U and Uˆ (Section 3), the specific initial volume of inter-aggregate pores, Us (Fig. 1); the interface layer (Fig. 1) contribution to the specific soil volume, Ui and Uˆ i ; aggregate/intra-aggregate mass ratio, K and Kˆ ; and the interface layer (Fig. 1) contribution to the total water content, ω and ω ˆ. 4.1.2. Extension to the reference scanning shrink–swell curves of a soil The primary cycles with aggregate destruction can only be realized in the upper part of the soil profile. At sufficiently large depths the water content usually varies in a relatively small range, and shrinkage–swelling does not lead to aggregate destruction. It means that Uˆ i = Ui , ˆ h (Fig. 9). Thus the reference Kˆ = K , Yrh = Yˆrh , and Wh = W scanning shrink–swell curves (that are in a range ∆W < ˆ
ˆ

ˆ) with the reference quasi primary swelling curve, Yˆ r (W ˆ ˆ corresponding to Ui = Ui and K = K (Fig. 9, dashed line 2’). The extension of the major relations from11 (Section 4.1.1) to the case of the reference scanning shrink–swell curves is realized by a number of value replacements. In the case of a scanning shrinkage curve: Yr (w ′ ) = Yr (W ) → Y r (w ′ , wo′ ) = Y r (W , W o ) (W is the total soil water content along the curve of scanning shrinkage; W o is the W value at the start of scanning shrinkage; w ′ = w/K is the contribution of the intra-aggregate matrix to W ; wo′ corresponds to W o in terms of w ′ ); U (w) → U (w, wo ); U ′ (w ′ ) → U ′ (w ′ , wo′ ) (U ′ = U /K ; U ′ = U /K ); W (w ′ ) → W (w ′ , wo′ ); ω(w ′ ) → ω(w ′ , wo′ ). Characteristics of the contributive clay, R(ζ ) (the maximum internal size of water-filled clay pores at primary shrinkage), v(ζ ), and P (ζ ) are replaced with R(w′, wo′ ) = R(ζ , ζo ) (the maximum internal size of water-filled clay pores at scanning shrinkage), v(ζ , ζo ),

P (ζ , ζo ). Finally two additional conditions that flow out of the definition of scanning shrinkage should be fulfilled:

ˆ Y r (W o , W o ) = Yˆ r (W o ) (Fig. 10) and R(ζo , ζo ) = Rˆ (ζo ).

In the case of a scanning swelling curve: Yˆr (w ′ ) = ˆ ) → Yˆ r (w ′ , wo′ ) = Yˆ r (W ˆ ,W ˆ o ) (W ˆ is the total Yˆr (W soil water content along the curve of scanning swelling; ˆ value at the start of scanning swelling); ˆ o is the W W Uˆ (w)

→ Uˆ (w, wo ); Uˆ ′ (w ′ ) → Uˆ ′ (w ′ , wo′ ) (Uˆ ′ = ˆ (w ′ ) → W ˆ (w ′ , wo′ ); ω(w Uˆ /K ; U = Uˆ /K ); W ˆ ′) → ′ ′ , wo ). Characteristics of the contributive clay, Rˆ (ζ ) ω(w ˆ ˆ′

(the maximum internal size of water-filled clay pores at primary swelling), vˆ (ζ ), and Pˆ (ζ ) are replaced with Rˆ (w′, wo′ ) = Rˆ (ζ , ζo ) (the maximum internal size of water-

filled clay pores at scanning swelling), vˆ (ζ , ζo ), Pˆ (ζ , ζo ). Two conditions that flow out of the definition of scanning ˆ o, W ˆ o ) = Yr (W ˆ o ) (Fig. 11) swelling should be met: Yˆ r (W

and Rˆ (ζo , ζo ) = R(ζo ).

4.1.3. Extension to the reference transitive and steady shrink–swell cycles of a soil The found reference scanning shrinkage (Y r (W , W o ))

ˆ ,W ˆ o )) curves of a soil inside the quasi and swelling (Yˆ r (W primary shrink–swell cycle (Figs. 10 and 11) enable one to construct the reference transitive and steady shrink–swell cycles of the soil in a given water content range, 0 < W1 < W < W2 < Wh in the total analogy to the consideration for a clay paste in Section 2.1.3 (Figs. 4 and 5). Fig. 12 shows the qualitative view of the soil reference steady shrink–swell cycle. The conditions for clay (at the end of Section 2.1.3) turn into Y r (W1 , Wo1 ) = Yˆ r (W1 , Wo2 ) and Y r (W2 , Wo1 ) = Yˆ r (W2 , Wo2 ) to determine the starting points, W o = Wo1

ˆ o = Wo2 of of the shrinkage branch (Fig. 12, curve 3) and W the swelling branch (Fig. 12, curve 4) of the steady cycle.

26

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

ˆ

ˆ )) (curves 1 and 2’, respectively) and reference scanning Fig. 11. The qualitative view of the reference quasi-primary shrink–swell cycle, (Yr (W ), Yˆ r (W ˆ ,W ˆ o ) (curve 3). The inclined (dashed) line is the saturation or quasi-saturation one. swelling curve of a soil, Yˆ r (W

Fig. 12. The qualitative view of the reference steady shrink–swell cycle (curves 3 and 4) of a soil in the W1 < W < W2 range. The cycle consists ˆ , Wo2 ) (curve 4) [which are inside the reference quasi-primary shrink–swell cycle, of the shrinkage branch, Y r (W , Wo1 ) (curve 3) and swelling one, Yˆ r (W

ˆ

ˆ )) (curves 1 and 2’, respectively)] where Wo1 = Wo1 (W1 , W2 ) and Wo2 = Wo2 (W1 , W2 ) are a solution of equations at the end of Section 4.1.3. (Yr (W ), Yˆ r (W The inclined (dashed) line is the saturation or quasi-saturation one.

4.2. Extension to shrinkage–swelling and cracking of a soil layer with no loading 4.2.1. Available modeling of the first primary cycle of a cracked layer with no loading The primary shrinkage curve, Y (W ) and crack volume, Ucr (W ) of the soil layer in the range, 0 < W < Wh , ˆ ) (Fig. 13) as well as the primary swelling curve, Yˆ (W

ˆ

ˆ ) in the case under consideration (that coincides with Yˆ (W

when Uˆ i = Ui , Kˆ = K , Xˆ m = Xm ; see Section 4.1.2) ˆ ) in the range 0 < W ˆ < and crack volume, Uˆ cr (W ˆ h , without loading, are expressed through the reference W primary curves as11,14 Y (W , h/h∗ ) = (1 − q(h/h∗ ))Yr (W ) + q(h/h∗ )Yrh , 0 < W < Wh ,

(16)

Ucr (W , h/h ) = q(h/h )(Yrh − Yr (W )) + Us , ∗

0 < W < Wh ,

∗

(17)

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

27

ˆ )) is the reference primary shrink–swell cycle. Fig. 13. The qualitative view of the four different primary shrink–swell cycles of a soil. (Yr (W ), Yˆr (W ˆ

ˆ

ˆ )) is the reference quasi primary shrink–swell cycle. (Y (W ), Yˆ (W ˆ )) is the primary shrink–swell cycle with a crack contribution. (Y (W ), Yˆ (W ˆ )) (Yr (W ), Yˆ r (W is the quasi primary shrink–swell cycle with a crack contribution. The differences between the soil volume with and without cracks give the corresponding ˆ ) − Yˆr (W ˆ ) = Uˆ cr (W ˆ ) − Us ; crack volume at a given water content minus the volume of the structural pores: Y (W ) − Yr (W ) = Ucr (W ) − Us ; Yˆ (W ˆ

ˆ

ˆ

ˆ ) − Yˆ r (W ˆ ) = Uˆ cr (W ˆ ) − Us . W ˆ h∗ is the W value at which the residual cracks after the (Y (W ), Yˆ (W ˆ )) shrink–swell cycle would be water-filled11 . Yˆ (W

ˆ , h/h∗ ) = (1 − q(h/h∗ ))Yˆr (W ˆ) Yˆ (W

(18)) should coincide at W

+ q(h/h∗ )(Uh − Uz )/K + q(h/h∗ )Yrz , ˆ
(18)

ˆ , h/h∗ ) = q(h/h∗ )(Uh − Uz )/K + Us Uˆ cr (W ˆ ) − Yrz ), − q(h/h∗ )(Yˆr (W

ˆ
(19)

where the crack factor, q(h/h∗ ) in the layer case depends on the initial layer thickness, h (i.e., at W = Wh ) and characteristics of the aggregate size distribution at W = Wh (the maximum and minimum aggregate sizes and inter-aggregate porosity).11,14 The critical layer thickness, h∗ ∼ = 2–5 cm is determined through Xm and the mean distances, lmin and lm between the aggregates of the minimum, Xmin and maximum, Xm size at W = Wh .11,14 4.2.2. Extension to scanning shrinkage–swelling of a cracked layer with no loading The scanning shrinkage curves with no cracking, Y r (W , W o ) (Section 4.1.2; curve 3 in Fig. 10) and with cracks, Y (W , W o ) differ by the crack contribution as Y (W , W o ) = Y r (W , W o ) + U cr (W , W o ) − Us , 0 < W < W o < Wh

(20)

where U cr (W , W o ) − Us is the crack volume (minus its initial value, Us ) along the Y (W , W o ) curve that starts at W = W o . Using the definition of the crack factor, q(h/h∗ ) (Section 4.2.1) we have the differential relation for U cr as dU cr (W , W o ) = −q(h/h )dY r (W , W o ), ∗

0 < W < W o < Wh .

(21)

The crack volumes, U cr (W , W o , h/h∗ ) along the scanning

ˆ , h/h∗ ) (Eq. shrinkage curve, Y (W , W o , h/h∗ ) and Uˆ cr (W

ˆ , h/h∗ ) (Eq. (19)) along the primary swelling curve, Yˆ (W

=

ˆ W

=

W o as

U cr (W o , W o , h/h∗ ) = Uˆ cr (W o , h/h∗ ) (at 0 < W o < Wh ). Integrating Eq. (21) with this condition as well as the similar condition, Y r (W o , W o ) = Yˆr (W o ) for the soil volumes with no cracks, we obtain the crack volume, U cr (W , W o , h/h∗ ) at 0 < W < W o < Wh to be U cr (W , W o , h/h∗ ) = q(h/h∗ )(Yˆr (W o ) − Y r (W , W o ))

+ Uˆ cr (W o , h/h∗ ).

(22)

Here Uˆ cr (W o , h/h∗ ) is the crack volume that has existed in the starting point, W = W o of the scanning shrinkage curve, Y (W , W o , h/h∗ ). The term with q is the addition to the crack volume, U cr (W , W o , h/h∗ ) at scanning shrinkage. Substitution for U cr (W , W o , h/h∗ ) in Eq. (20) from Eq. (22) leads to Y (W , W o , h/h∗ ) at 0 < W < W o < Wh as (at q → 0 when there are no cracks Uˆ cr → Us , and Y → Y r )

Y (W , W o , h/h∗ ) = (1 − q(h/h∗ ))Y r (W , W o )

+ q(h/h∗ )Yˆr (W o ) + Uˆ cr (W o , h/h∗ ) − Us .

(23)

Similar considerations for scanning swelling lead to the ˆ ,W ˆ o , h/h∗ ) with cracks at scanning swelling curve, Yˆ (W

ˆ < Wh as (at q → 0 Ucr → Us , and Yˆ → Yˆ r ) ˆo
ˆ ,W ˆ o , h/h∗ ) = (1 − q(h/h∗ ))Yˆ r (W ˆ ,W ˆ o) Yˆ (W ˆ o ) + Ucr (W ˆ o , h/h∗ ) − Us , + q(h/h∗ )Yr (W

(24)

ˆ ,W ˆ o , h/h∗ ) at 0 < W ˆo < and the crack volume, Uˆ cr (W

ˆ < Wh as W

ˆ ,W ˆ o , h/h∗ ) = −q(h/h∗ )(Yˆ r (W ˆ ,W ˆ o ) − Yr (W ˆ o )) Uˆ cr (W ˆ o , h/h∗ ). + Ucr (W

(25)

28

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Fig. 14. The qualitative view of the hysteretic steady shrink–swell cycle of the specific crack volume (at a given soil depth). Curve 3 presents the crack volume evolution, U cr (W , Wo1 ) Eq. (26a) at the shrinkage stage and corresponds to the reference scanning shrinkage curve, Y r (W , Wo1 ) (curve 3 in Fig. 12) in the W1 < W < W2 range. Curve 4 presents the crack volume evolution, Uˆ cr (W , Wo2 ) Eq. (26b) at the swelling stage and corresponds to the reference scanning swelling curve, Yˆ r (W , Wo2 ) (curve 4 in Fig. 12) in the W1 < W < W2 range. Wo1 and Wo2 are the starting points of the Y r (W , Wo1 ) and Yˆ r (W , Wo2 ) scanning curves (see Fig. 12). Wm is the maximum point of the hysteretic variation of the crack volume, ∆U cr (W ) (Eq. (28)). U cr min (Eq. (27a)) and U cr max (Eq. (27b)) indicate the minimum and maximum crack volume of the hysteretic crack volume cycle in the W1 < W < W2 range.

ˆ o , h/h∗ ) is the crack volume that has existed in the Ucr (W ˆ o of the scanning swelling curve, ˆ = W starting point, W ˆY (W ˆ ,W ˆ o , h/h∗ ). The term with q is the decrease of the ˆ o , h/h∗ ) at scanning swelling. ˆ ,W volume, Uˆ cr (W 4.2.3. Extension to the steady cycle and crack volume hysteresis with no loading Eq. (22) (at W ≡ W and W o ≡ Wo1 ) and Eq. (25) (at

ˆ ≡ W and W ˆ o ≡ Wo2 ) determine the shrinkage and W swelling stages of the hysteretic steady shrink–swell cycle of the specific crack volume (Fig. 14, curves 3 and 4) as U cr (W , Wo1 , h/h∗ ) = C − q(h/h∗ )Y r (W , Wo1 ), 0 < W1 < W < W2 < Wo1 < Wh , ˆ U cr (W , Wo2 , h/h∗ ) = C − q(h/h∗ )Yˆ r (W , Wo2 ), 0 < Wo2 < W1 < W < W2 < Wh

(26a)

(26b)

where C ≡ q(h/h )Yr (Wo2 ) + Ucr (Wo2 , h/h ) does not depend on W . The minimum and maximum crack volume follows from Eqs. (26a) or (26b) at W = W2 and W1 (Fig. 14) as ∗

U cr

U cr

∗

min

= U cr (W2 , Wo1 , h/h∗ ) (27a)

max

= Uˆ cr (W2 , Wo2 , h/h∗ ), = U cr (W1 , Wo1 , h/h∗ ) = Uˆ cr (W1 , Wo2 , h/h∗ ).

(27b)

The hysteretic variation of the crack volume, ∆U cr (W ) at W1 < W < W2 (Fig. 14) is as

∆U cr (W ) = U cr (W , Wo1 , h/h∗ ) − Uˆ cr (W , Wo2 , h/h∗ )

= q(h/h∗ )(Yˆ r (W , Wo2 ) − Y r (W , Wo1 )).

(28)

The maximum hysteretic variation of the crack volume, ∆U cr (Wm ) (Eq. (28); Fig. 14) essentially influences the soil hydraulic conductivity. The steady cycle of a cracked soil layer, (Y (W , Wo1 , h/h∗ ), Yˆ (W , Wo2 , h/h∗ )) is written from Eqs. (23) and (24) as Y (W , Wo1 , h/h∗ ) = C − Us + (1 − q(h/h∗ ))

× Y r (W , Wo1 ), 0 < W < Wo1 < Wh , ˆY (W , Wo2 , h/h∗ ) = C − Us + (1 − q(h/h∗ )) × Yˆ r (W , Wo2 ),

0 < Wo2 < W < Wh .

(29a)

(29b)

Thus, the steady cycle of a cracked layer (Y (W , Wo1 , h/h∗ ), Yˆ (W , Wo2 , h/h∗ )) in the W1 < W < W2 range qualitatively repeats the reference steady cycle (Y r , Yˆ r ) in Fig. 12. In the course of the steady cycle of a cracked soil layer the crack volume (Fig. 14): (i) is always more than zero; (ii) varies between the minimum at W = W2 and maximum at W = W1 ; and (iii) at drying is always higher than at wetting (Fig. 14, curves 3 and 4). 4.2.4. Estimating the initial thickness (h) of a cracked layer with no loading Usually one only knows the current layer thickness, z < h at W < Wh , but not the h thickness. The h value can be estimated from the known z at the known water content, W and corresponding specific volume, Y with cracks. Y and z are simply connected with the initial (maximum) thickness, h of the layer and the maximum specific volume, Yh of the soil at W = Wh as z /h = Y /Yh (even though the curve under consideration is a scanning one and does not include the point W = Wh ). We are interested in the substitution for Y the two scanning curves that correspond to the shrinkage and swelling stages of the steady cycle in the cracked layer (Eqs. (29a) and (29b)). To estimate h as

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

29

Fig. 15. The experimental points8 and predicted shrinkage curves at five loadings (see Tables 1–4). θh , θs , θn , and θz correspond to the maximum swelling point, end point of structural shrinkage, end point of normal (basic) shrinkage, and shrinkage limit, respectively. The ranges of θh , θs , θn , and θz values with loading variation are indicated by two arrows.

a function of z /h∗ and W we can solve equation, z /h = Y (W , Wo1 , h/h∗ )/Yh at 0 < W < Wo1 < Wh (or z /h = Yˆ (W , Wo2 , h/h∗ )/Yh at 0 < Wo2 < W < Wh ) with respect to h/h∗ (Wo1 and Wo2 are the functions of W1 and W2 from Section 4.1.3). 4.3. Extension to shrinkage–swelling and cracking of a soil under loading The variation of shrink–swell behavior of the intraaggregate matrix (of a soil) with increase in L is determined by the vz (L) and vh (L) (or ζh (L)) dependences of the contributive clay (Sections 2 and 3). The third basic value that determines soil shrink–swell behavior with the L increase (including crack volume behavior) can be the maximum aggregate size (at the shrinkage limit), Xmz or (at maximum swelling) Xm . In the beginning of swelling with no loading (at W ∼ = 0) Xmz decreases to Xˆ mz < Xmz .18,11 However, loading during swelling should, in part or totally, offset the internal pressure in aggregates and prevent their destruction (see Section 7). In addition, at least at L < 0.1–1 MPa (see Section 2.2.2) the maximum aggregate size, Xmz should not change under loading (see Section 7). The scanning shrink–swell curves (Section 4.2.2), steady cycle, andcrack volume hysteresis (Section 4.2.3) are in the single-valued manner connected with the primary ones (Sections 4.1 and 4.2). The latter are determined by Us , Ui , K , ω, and q (Sections 4.1.1 and 4.2.1). Expressions for Ui , K , ω, and q through vz , vh , and Xmz are known11,14,19 and can be extended to the loading case using vz (L), vh (L), and Xmz (L). The Us (L) dependence is derived similarly to that of ulph (L) (Eq. (15a)). The inter-aggregate pore volume at maximum swelling, Us (L) (Fig. 1) varies with loading, L proportionally to the volume, Uh (L) − 1/ρs of the intraaggregate pores (lacunar and clay matrix pores; Fig. 1) at W = Wh . Then, accounting for Us (L = 0) ≡ Us and Uh (L = 0) ≡ Uh (uh (L = 0) ≡ uh ), we estimate Us (L) to be Us (L) = Us (Uh (L) − 1/ρs )/(Uh − 1/ρs ) = Us (uh (L) − us )/(uh − us ) (L > 0) where us = vs [c +vs (1 − c )]−1 .13 uh (L) is estimated from Eq. (13a) at ζ = ζh (L) and depends on L through vz (L) and vh (L) (or ζh (L)).

5. Theoretical results and discussion (1) The primary shrink–swell curves with no loading determine the corresponding scanning curves in a single-valued manner (the physical input that is necessary in the modeling of the primary curves has been considered11–14 ). (2) The single-valued links between the primary shrink–swell curves of clay, intra-aggregate matrix, aggregated soil with no cracks, and the soil with cracks for no loading case11,14 were extended under loading to similar links. (3) Results (1) and (2) permit the single-valued prediction of the different scanning curves under loading, based on the primary curves and loading conditions. (4) Result (3) allows the single-valued quantitative prediction of the transitive and steady shrink–swell cycles in a given layer of a clayey vadose zone at a given range of the water content variation. Such cycles were recently qualitatively observed.20 (5) The model explains the origin of the crack volume and quantitatively predicts the crack volume hysteresis in a given layer of a clayey vadose zone. The crack volume was recently qualitatively observed.3 6. Analysis of available data to check the basic model relations The major functions of L to be checked are vz (L) (Eq. (6)), vh (L) (Eq. (8)), ζh (L) (Eq. (7)), and Xm (L) ∼ = const (see Section 4.3). The available data for soil layers (large depths) only indicate the fact itself of the crack volume occurrence.3 The quantitative data are only available as applied to the soil sample case. In this case the crack factor, q(h/h∗ ) is modified14 (h is the sample height of the same order of magnitude as the sample diameter). The data that are obtained in an oedometer and a triaxial test apparatus (see21 among others) are oriented to estimating the swelling pressure and do not usually contain the physical soil characteristics that are necessary for checking the model. The only data on the primary shrinkage curves of soil samples under loading we could use were obtained in8,22 (the data8 on the soils that were compacted by different loads before shrinkage are not of interest in this work).

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V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Table 1 Primary dataa and input physical soil parameters for model predictionb . Data source

Load L (kPa)

ϑh

Primary data ez

Peng et al.,8 Fig. 12 (Dystric Gleysol)

0 1.8 3.8 5.5 7.3

1.6350 1.5211 1.4298 1.3802 1.3200

0.7541 0.7601 0.7324 0.7489 0.7429

Input parameters ρs c (kg dm−3 ) 2.65 0.649 2.65 0.649 2.65 0.649 2.65 0.649 2.65 0.649

Talsma,22 Fig. 1 (Black Earth from NSW, Australia)

0c 0.14 6.3 11.2

1.9123 1.9027 1.6955 1.5594

1.3042 1.2523 1.1700 1.0573

2.69 2.69 2.69 2.69

0.560 0.560 0.560 0.560

s 0.151 0.151 0.151 0.151 0.151

Us (dm3 kg−1 ) 0 0 0 0 0

Ulph (dm3 kg−1 ) 0 0 0 0 0

Wh (kg kg−1 ) 0.6170 0.5740 0.5395 0.5208 0.4981

Yz (dm3 kg−1 ) 0.6619 0.6642 0.6537 0.6600 0.6577

h (mm) 41 41 41 41 41

0.150 0.150 0.150 0.150

0 0 0 0

0 0 0 0

0.7109 0.7073 0.6303 0.5797

0.8566 0.8373 0.8067 0.7648

20 20 20 20

Soil moisture ratio at maximum swelling state before shrinkage (ϑh ), void ratio at shrinkage limit (ez ). Mean solid density (ρs ), clay content (c ), silt content (s), specific volume of structural pores at maximum swelling (Us ), specific lacunar pore volume at maximum swelling before shrinkage (Ulph ), soil water content at maximum swelling before shrinkage (Wh ), specific soil volume at the shrinkage limit (Yz ), sample height at maximum swelling before shrinkage (h). c Actual minimum load from22 , 0.02 kPa was taken to be zero. a

b

Peng et al.8 observed shrinkage of the soil samples 4.1 cm in height under five loads including L = 0 (Table 1; Fig. 15). Each constant load was imposed at the beginning of the previous swelling and acted up to the maximum swelling and then up to the end of shrinkage. Ulph = 0 for different L (Table 1) since the experimental points at θ = θh (Fig. 15) are on the saturated line. Us = 0 for different L (Table 1) since the experimental curves (Fig. 15) do not have the horizontal part close to θ = θh . Talsma22 observed shrinkage of samples 2 cm in height from Black Earth (NSW, Australia) under four loads (Table 1; Fig. 16). Each constant load was imposed at the beginning of shrinkage and acted up to its end. We took the c and s values (Table 1) based on the available data on Black Earth (see23 among others). Talsma22 speaks about the samples of clay soil paste, but they show behavior (Fig. 16) that is similar to that of aggregated soils with the structural shrinkage range. Therefore we tried to estimate the xm and Xm values for the samples. Ulph = 0 and Us = 0 were taken for22 (Table 1) from the same considerations as for.8 The data8,22 do not include the maximum sand grain size, xm and maximum aggregate size, Xmz that are needed for the prediction of a shrinkage curve.11,14 By its physical meaning xm is constant when L increases. We assume that at the L values from Table 1, Xmz also does not change with L. We find xm and Xmz only using data8 at L = 0 and 1.8 kPa and22 at L = 0 based on the approach.11,14 Then, xm , Xmz and data from Table 1 participate in the prediction of the shrinkage curves for8,22 at other L. After finding us (that does not depend on L) for case L = 0 we can estimate ζh (L) at L > 0 knowing Wh (L) (Table 1) as ζh (L) = ρs Wh (L)/(1/us − 1). The characteristics of the soil, intra-aggregate matrix, and contributive clay are given in Tables 2–4. Fig. 15 shows the predicted curves at L > 1.8 kPa for.8 Tables 2–4 show the characteristics at L > 1.8 kPa. Similar predictions for22 at L > 0 are in Fig. 16 and Tables 2–4. We also estimated the xm and Xmz values for8 at L > 1.8 kPa and22 at L > 0 using Xmz as the only fitting parameter. All the characteristics at these L (Tables 2–4) were kept. The fitted and predicted curves also coincide (Figs. 15 and 16). We fitted the theoretical curve, ζh (L/L∗ ) (Eq. (7)) to the five points, ζh (Li ) (i = 1, . . . , 5) from8 (Table 4) using L*

as the only fitting parameter. For the results see Fig. 17 and its caption. Using vs and the mean vz value by the five loading values (Table 4; the differences between vz at different L are only stipulated by the spread of the input Yz values in Table 1) we also estimated the characteristic universal loading, Lu (see the end of Section 2.2.3; Fig. 17). A similar procedure was also utilized to the four points, ζh (Li ) (i = 1, . . . , 4) from22 (Table 4; Fig. 17). 7. Data analysis results and discussion I. Dependences of the contributive-clay characteristics on L. (1) The theoretical curve ζh (L/L∗ ) (Eq. (7)) is in agreement with the ζh values (Table 4; Fig. 17; see rζ2 and σζ in the figure caption). This also justifies vh (L/L∗ ) (Eq. (8)). (2) L* (Fig. 17), vs , and vz for two soils (Table 4) only lead to universal Lu (Fig. 17) if L∗ = Lu /[(vz − vs )/(1 − vs )]2 (Section 2.2.3). (3) The vz and ζz constancy (Table 4) for each soil (within the limits of a spread) confirms Eq. (6) at sufficiently small L (the end of Section 2.2.2). II. Dependences of the characteristics of the intraaggregate matrix on L. (1) p (Table 3) is constant since xm (Table 2), c, and s (Table 1) are constant.14 (2) The spread of c* (Table 3) is connected with that of vz . (3) k > 0 leads to Ulpz > 0 (Table 3) albeit Ulph = 0 for both soils (Table 1). (4) By their physical meaning us and uS (Table 3) do not vary with L. (5) Uh decreases with L growth for both soils (Table 3). For22 Uz , Un , Ulpz > 0, and Ulpn > 0 also decrease with L growth at k > 0 (Table 3), but Uz − Ulpz and Un − Ulpn are constant, i.e., Uz (L) and Un (L) are linked with Ulpz (L) > 0 and Ulpn (L) > 0. For8 Uz and Un are constant, Ulpz = 0, and Ulpn = 0 at k = 0 (Table 3). III. Dependences of the soil characteristics and shrinkage curves on L. (1) re2 and σe (Table 2) show that the predicted curves, e(θ ) (Figs. 15 and 16) are in agreement with the data8,22 at all L. (2) xm and xn (Table 2) have reasonable values for clay soils. (3) Preliminary sample swelling at L = 0 in8 leads to some destruction of aggregates, but at L > 0 there are not the aggregate destruction11 (Section 4.3). It explains the difference between Xmz (Table 2) at L = 0 and Xmz at L > 0 (Table 2) for.8 In22 the samples were loaded only at shrinkage and Xmz were similar at all L ≥ 0 (Table 2). (4) Large Xmz and Xm for22 (Table 2) agree with

V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

31

Table 2 Predicted major physical characteristics of soil as a whole at different loading values and handling parameters. Load L (kPa)

Soil characteristicsa xm (mm) xn (= Xmin ) (mm)

Xmz (mm)

Xm (mm)

K

Dystric Gleysol from8

0 1.8 3.8 5.5 7.3

0.120 0.120 0.120 0.120 0.120

0.0216 0.0216 0.0216 0.0216 0.0216

0.7325 0.9615 0.9615 0.9615 0.9615

0.8913 1.1273 1.1166 1.1016 1.0906

Black Earth from22

0 0.14 6.3 11.2

0.118 0.118 0.118 0.118

0.0288 0.0288 0.0288 0.0288

7.715 7.715 7.715 7.715

8.3744 8.4318 8.3236 8.3280

Soil

h* (mm)

q

Handling parametersb re2 σe

1.3570 1.2770 1.2799 1.2840 1.2870

Ui (dm3 kg−1 ) 0.2616 0.2064 0.2005 0.1987 0.1953

53.4193 52.9015 52.9198 52.9462 52.9659

0 0 0 0 0

0.9982 0.9965 0.9904 0.9865 0.9827

0.0135 0.0195 0.0283 0.0307 0.0318

1.0472 1.0468 1.0475 1.0474

0.0488 0.0483 0.0455 0.0431

44.5376 44.5358 44.5392 44.5391

0 0 0 0

0.9884 0.9767 0.9940 0.9956

0.0213 0.0376 0.0152 0.0130

a Maximum sand grain size (xm ), mean size of soil solids (xn = 0.001c + 0.026s + (0.025 + xm /2)(1 − c − s)), minimum aggregate size (Xmin ), maximum aggregate size at shrinkage limit (Xmz ), maximum aggregate size at maximum swelling before shrinkage (Xm ), aggregate/intra-aggregate mass ratio at shrinkage (K ), contribution of the interface aggregate layer to the specific volume of soil aggregates at shrinkage (Ui ), critical sample size at shrinkage (h*), sample crack factor at shrinkage (q). b Goodness of fit of e(ϑ) (Xmz is the only fitting parameter) to shrinkage curve data (re2 ), estimated standard errors of shrinkage curve data8,22 in Figs. 15 and 16 (σe ).

Table 3 Predicted major physical characteristics of intra-aggregate matrix at different loading values. Load L (kPa)

Characteristics of intra-aggregate matrixa p c* k us

uS

Ulpz

Ulpn

Dystric Gleysol from8

0 1.8 3.8 5.5 7.3

0.2875 0.2875 0.2875 0.2875 0.2875

0.1940 0.1794 0.1809 0.1769 0.1760

0 0 0 0 0

0.2342 0.2342 0.2342 0.2342 0.2342

0.0822 0.0822 0.0822 0.0822 0.0822

0 0 0 0 0

0 0 0 0 0

Black Earth from22

0 0.14 6.3 11.2

0.7235 0.7235 0.7235 0.7235

0.6007 0.5908 0.5991 0.5892

0.5746 0.5295 0.5682 0.5208

0.2073 0.2073 0.2073 0.2073

0.0912 0.0912 0.0912 0.0912

0.3201 0.2851 0.2694 0.2125

0.2524 0.2202 0.2020 0.1484

Soil

Uz (dm3 kg−1 ) 0.5431 0.5844 0.5799 0.5921 0.5660 0.8458 0.8258 0.7973 0.7558

Un

Uh (= Yh )

0.6365 0.6758 0.6687 0.6794 0.6812

0.9943 0.9514 0.9169 0.8982 0.8755

0.8982 0.8861 0.8496 0.8153

1.0826 1.0791 1.0020 0.9514

a Porosity of contributive silt and sand grains in the state of imagined contact (p), critical value of soil clay content (c*), lacunar factor (k), relative volume of solid phase of intra-aggregate matrix (us ), relative volume of non-clay solids of intra-aggregate matrix (uS ), specific volume of lacunar pores at ζ = ζz and ζ = ζn (Ulpz and Ulpn , respectively), specific volume of intra-aggregate matrix at ζ = ζz , ζ = ζn , and ζ = ζh (Uz , Un , and Uh , respectively).

Table 4 Predicted major physical characteristics of contributive clay (see Fig. 6) at different loading values. Soil

Dystric Gleysol from8

Black Earth from22

Load L (kPa) 0 1.8 3.8 5.5 7.3 0 0.14 6.3 11.2

Clay characteristicsa

vs

vz

vh

ζz

ζh

0.1656 0.1656 0.1656 0.1656 0.1656 0.1277 0.1277 0.1277 0.1277

0.2777 0.3056 0.3025 0.3108 0.3128 0.2222 0.2314 0.2236 0.2330

0.5828 0.5538 0.5305 0.5178 0.5024 0.5639 0.5617 0.5145 0.4834

0.0612 0.0982 0.0938 0.1057 0.1086 0.0254 0.0327 0.0265 0.0341

0.5 0.4652 0.4373 0.4221 0.4037 0.5 0.4976 0.4434 0.4078

a Relative clay solid volume (vs ), relative clay volume at ζ = ζz (vz ), relative clay volume at ζ = ζh (vh ), shrinkage limit on the ζ axis of relative clay water content (ζz ), maximum clay swelling point on the ζ axis (ζh ).

the fact that the soil is close to the paste-like state. (5) The differences of the Xm and Xmz at a given L illustrate the variation of the maximum aggregate size at shrinkage. (6) The similar Xmz at L > 0 for8 and at L ≥ 0 for22 confirms the constancy of Xmz at shrinkage under sufficiently small loading (Section 4.3). (7) h* (Table 2) are in agreement with the estimates from11,14 (several centimeters). (8) Data8,22 and predicted curves in Figs. 15 and 16 show two types of shrinkage behavior under loading. The behavior in Fig. 15 is connected with the absence of cracks (q = 0, Table 2) because of the small sample size, and lacunar pores (Ulph = 0, Table 1; k = 0, Table 3) since c > c* (Tables 1 and 3) at all

L and water contents. The contributive-clay pores only exist in samples.8 With drying the clay pore volume decreases, and the curves of the different L in Fig. 15 converge. The behavior in Fig. 16 is connected with the development of lacunar pores (k > 0, Table 3) (cracks is not the case of22 because of the small sample size). The contributiveclay pores and lacunar pores exist in samples.22 Again, with drying the clay pore volume decreases, but the lacunar pore volume at different L increases, and the gaps between the curves in Fig. 16 are retained. The large samples or consideration of a layer for soil8 should lead to an

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V.Y. Chertkov / Geomechanics for Energy and the Environment 1 (2015) 16–33

Fig. 16. As in Fig. 15, but for the experimental points22 and predicted shrinkage curves at four loadings (see Tables 1–4).

Fig. 17. The white squares and solid line indicate the five estimated ζh (L/L∗ ) points from Table 2 and fitted ζh (L/L∗ ) curve (Eq. (7)), respectively, for Peng et al.’s8 soil and fitted characteristic loading (see the found L* and Lu in the figure). Goodness of fit is rζ2 = 0.9848. The estimated standard error of the ζh (L/L∗ ) points is σζ = 0.005. The white circles and dashed line indicate the four estimated ζh (L/L∗ ) points from Table 2 and fitted ζh (L/L∗ ) curve (Eq. (7)), respectively, for Talsma’s22 soil and fitted characteristic loading (see the found L* and Lu in the figure). Goodness of fit is rζ2 = 0.9994. The estimated standard error of the ζh (L/L∗ ) points is σζ = 0.001.

crack development11,14 and to the shrinkage behavior with loading increase as in Fig. 16.

soil characteristics (including crack volume) with loading increase.

8. Conclusion

References

We propose an approach to the physical quantitative prediction of the crack volume within the limits of a soil layer in a clayey vadose zone accounting for: (i) overburden; (ii) multifold shrinkage–swelling; and (iii) small water content range. The approach relies on the available models of: (1) the inter- and intra-aggregate soil structure; and (2) the soil shrink–swell curves in the maximum water content range (primary curves) without loading. The major new theoretical results are (1) the extension of the primary shrink–swell curves (with cracks) with no loading to the loading case; and (2) the single-valued links between the primary shrink–swell curves on the one hand and scanning curves (i.e., in a small water content range), shrink–swell cycling, and crack volume hysteresis on the other hand. The major new analysis results of available data testify in favor of the basic model relations that reflect the variation of the

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