Contração Energy Capillary - Baungart

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Modeling Volume Change and Mechanical Properties with Hydraulic Models

Th. Baumgartl* and B. Ko ¨ ck

ABSTRACT Volume Change by Internal andExternal StressesVolume change of soils may be caused either by external (mechani-

cal) or internal (hydraulic) stresses or a combination of both. A com- Processes of volume change in a porous system likeplete description of volume change must therefore include both me- soil can either occur in a two-phase system (solid–waterchanical and hydraulic stresses. By combining theories of mechanical

or solid–air) or a three-phase system (solid–water–air).and hydraulic stress states, a hydraulic function, which predicts the

For reasons of simplification, it is generally assumedchange of water volume as a function of the stress state parameterthat air-filled pores are interconnected to the atmospheresoil water suction (water retention curve), is adopted to model volumeand that the influence of pore air pressure buildup duringchange. The utilization of such a continuous function also enables the

derivationof soil mechanical parameters (e.g., preconsolidationstress, volume changes can therefore be neglected. The stressYoungs modulus) by determining mathematically the point of maxi- state can be defined by two kinds of stresses within themum curvature and inflection point. This information can then be soil. It is a result of external (mechanical) stresses andused to calculate the preconsolidation stress according to the method internal (hydraulic) stresses. Mechanical stresses areof Casagrande. Thepresented calculationhas considerable advantages transmitted via the solid phase. Volume change occurscompared with the graphic method of Casagrande or other methods. when maximum shear resistances between particles areOn the basis of stress–strain relationships of various textured and

exceeded. Hydraulic stresses or internal stresses arestructured soils and soil substrates and various test procedures (oedo-caused by the water phase in the pore system and influ-meter test, triaxial test, shrinkage test), volume change is modeled usingence the shearing resistances. When stresses are applied,the described method. It is shown that modeling volume change by

they will be positive in a two-phase system (water-satu-the van Genuchten equation using the software RETC is possible withhigh accuracy. Soil mechanical parameters are derived using the pa- rated soil), but may also be positive in the short termrameters of the van Genuchten equation. The comparison of results in a three-phase system (water-unsaturated soil), whileof this method with the Casagrande and a statistical method shows that compaction caused by external stresses will redistributethese methods have deficiencies when the data sets have a high vari- water into a pore system with a modified pore-size distri-ability, the samples are not homogeneous, and when the stress–strain bution. At the state of equilibrium of the water phasecurve is flat. The accuracy of the mathematical method in contrast is

within the pore system (equivalent with the pore sizevery high and the calculated preconsolidation stress very reliable.

distribution), hydraulic stresses will be generally nega-tive and can be characterized by tensile stresses. Under

Predicting volume change of soils under stress ap- unsaturated conditions, the change of internal stressesby wetting and drying will cause reduction or expansionplication is important when calculating the stabilityof a soil volume. Depending on the intensityof the stressof soil substrates. This information is commonly usedapplication, a deformation will be elastic or plastic.in engineering or agriculture to derive mechanical pa-

For a complete description of the stress state, bothrameters (e.g., preconsolidation stress), which explain

the external and internal stresses have to be combinedthe stress history of a material. Preconsolidation stress, as it is formulated in the Terzaghi equation (Terzaghifor example, shows which stresses a soil substrate hasand Jelinek, 1954) for a two-phase system and in the ex-been exposed to in its history (mechanical, hydraulic)tended form for a three-phase system, first describedand up to which stress level it can be loaded without fur-by Bishop and Blight (1963). This relationship is ex-ther unrecoverable volume change. Information aboutpressed as follows:the kind of volume change is also important for the

understanding of shrinkage behavior, crack and struc- ( ua) (ua uw). [1]ture formation, as well as resistance to (shearing) stresses

The stresses explained by the matric water potential of (Horn and Baumgartl, 2002). In the literature, variousEq. [1] represent the tensile stresses (Snyder and Miller,methods can be found which describe stress-dependent1985), whereas the effective stresses are represented bystrain and the derivationof mechanical properties. How-the mechanical stress state.ever, many of these basic approaches lack in simplicity

The -factor of Eq. [1] accounts for the amount of and uniqueness. A method is presented here which al-water-filled pores which is, when defined as stress states,lows the description of volume change as a function of dependent on the matric water potential. Under satu-stresses based on hydraulic and mechanical stress staterated conditions, the parameter 1 and Eq. [1] re-models and a well-defined derivation of mechanicalduces to the general Terzaghi equation. It could beproperties using this theory.shown that this factor is explained within certain limitsby the water retention curve (Baumgartl, 2002). TheTh. Baumgartl, Inst. for Plant Nutrition and Soil Sci., Univ. of Kiel,calculation of the tensile stresses is possible as long asOlshausenstrasse 40, 24118 Kiel, Germany;B. Köck, School of Mathe-

matics, Univ. of Southampton, Southampton, SO17 1BJ, U.K. Re- the pore size distribution for each state of water poten-ceived 19 Sept. 2002. *Corresponding author ([email protected] tial is known (in fact, it is often assumed that the porekiel.de). size distribution is constant). In principle, a calculation

is possible even in soils with volume change (shrinkagePublished in Soil Sci. Soc. Am. J. 68:57–65 (2004).or swelling, compaction), when the deformation behav-© Soil Science Society of America

677 S. Segoe Rd., Madison, WI 53711 USA ior is known (Katou et al., 1987; O’Sullivan and Ball,

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58 SOIL SCI. SOC. AM. J., VOL. 68, JANUARY–FEBRUARY 2004

1993). Because volume change as a combination of com- stress. For reasons of normalization in the following,paction and shrinkage is very complex, deformation due pore volume is defined as the void ratio e (pore volume/to mechanical or hydraulic stresses are in the following volume of solids) and water content as the moistureviewed and modeled separately. ratio (water volume/volume of solids). Figure 2 shows

several possible scenarios.Adoption of the Stress–Strain Relationship The application of mechanical stress on a water-satu-

with a Hydraulic Model rated substrate (with 0 at drained conditions) may

result in curve II (stress–strain relationship). The slopeIn mechanical and agricultural engineering, the rela- is defined by de/d. A steep slope at higher stressestionship of volume change with stress application is a(virgin compression curve) can be distinguished from acommon feature to describe the mechanical characteris-less steep slope at low mechanical stresses (preconsoli-tics of soil substrates. With a semilogarithmic plot of dation curve).normal stress vs. strain, a recompression curve at a lower

Taking the equivalent relation for hydraulic stressstress range can be distinguished from a virgin compres-() vs. void ratio (e), the void ratio will decrease lesssion curve at higher stresses (Casagrande, 1936; Terzaghiwith decreasing matric potential due to the emptying of and Jelinek, 1954). Both curves are separated by thepore space and hence reduced tensile stress. A possiblepreconsolidation stress (Fig. 1). Likewise, volume changerelationship may be curve I in Fig. 2 and can be charac-caused by increased internal stresses results in a rela-terized as shrinkage-strain relationship. The slope attionship which behaves similarly (Fig. 1). At lower capil-each point is defined by de/d and has low values atlary stresses (i.e., less negative matric potentials), thevery low water potentials (preshrinkage curve) and veryshrinkage is smaller, whereas after exceeding a certainhigh water potentials (residual shrinkage curve). Thedegree of dryness, the shrinkage rate increases with lesscurve in between these two ranges of water potential

negative matric water potentials. Within the range of depict the virgin shrinkage curve.small matric potentials, this relationship can be definedReferring the parameter moisture ratio to mechani-as the preshrinkage line. It is separated from the virgin

cal stress and assuming saturated conditions (withshrinkage line by the preshrinkage stress (Baumgartl, 0 at drained conditions), then this relationship is2002). The preshrinkage line at high matric water poten-equivalent to curve II because e. The slope is de-tials is equivalent to the structural shrinkage of the well-fined by d/d. When referring matric water potentialknown shrinkage curve derived from the void ratio- to moisture ratio , the result may be curve III andmoisture ratio relationship (Yong and Warkentin, 1975).describes the water retention curve (after Fredlund andThe above considerations show that processes andRahardjo, 1993).the characteristic of volume change can be expressed

Similar considerations have been conducted by Tollsimilarly. Additionally, when both stresses occur, the me-(1995). He compared the shrinkage behavior of nor-chanical and hydraulic stresses have to be coupled tomally consolidated soil and overconsolidated samplesdefine the complete stress state (Groenevelt and Bolt,with the moisture ratio and void ratio. He states that1972; Richards and Smettem, 1992; Richards et al.,as long as a soil sample is saturated, the e– and –2000). Fredlund and Rahardjo (1993) point out that

relationships are identical. Once the sample starts toseveral coefficients can be defined which formulate thevolumetric deformation with hydraulic and mechanical be desiccated, is reduced with further desaturation

Fig. 1. Stress–strain relationship as a consequence of compaction and shrinkage.


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BAUMGARTL & KO ¨ CK: HYDRAULIC MODELS FOR VOLUME CHANGE 59

following a virgin desiccation line with a steeper slope e eres er

[1 (e)ne]me [4](curve III in Fig. 2). The slope of the e/ relationship

(curve I in Fig. 2) will remain constant as long as theThe parameters s and es describe moisture ratio atshrinkage limit has not been reached. Beyond this point,

the slope will reduce to zero. water-saturated conditions and void ratio before stressWith the above given hypotheses, the similarity of application, respectively. The parameter r (residual

volume reduction and change of water content due to moisture ratio) is dependent on the texture of the mate-hydraulic stress can be referred to volume change

rial, but is often defined as r 0. The minimum (resid-due to mechanical stress . Additionally, it has been ual) void ratio er is dependent on the maximum packingdescribed (Katou et al., 1987; Bruand and Cousin, 1995) density (i.e., minimum void ratio) of the material andthat mechanical stress will first decrease the size of less will be generally around er 0.27. The van Genuchtenstable coarse pores, which is comparable with draining parameters , n, and m have specific values (indicatedcoarse pores of a water-saturated soil first by increasing by indices) depending on being related to void ratiohydraulic stress. Furthermore, changes of the pore size or moisture ratio and mechanical stress or hydraulicdistribution by mechanical stresses can be compared

stress .with the change of the pore size distribution by tensile

The advantage of this hypothesis is that it enables thestresses, although different in magnitude (see Eq. [1]).definition of volume change by a simple and continuousCombining these concepts of describing volume changefunction. Also, volume change due to external stressesby hydraulic and mechanical stresses provides a basiscan be expressed by the same model as volume changeto model void ratio using an hydraulic model, which hasdue to hydraulic stresses. In the case of shrinkage orbeen adapted to de/d and de/d other than d/de.swelling by internal stresses, the stress state parameterAs a model, the van Genuchten-equation (van Gen- and the functional relationship with moisture ratiouchten, 1980) for determining of the water retentionand void ratio are the same.curve was chosen. The following expressions and param-

eters are used for different void ratio and moisture ra-tio relationships: Derivation of Soil Mechanical Properties

(i) Moisture ratio vs. matrix potential (waterStress–strain relationships provide the basis to describe

retention curve)soil mechanical behavior. Such parameters can be, forexample, preconsolidation stress or Youngs modulus. r

s r [1 ()

n]m [2]The preconsolidationstress is determined either graph-

ically or statistically. The graphic determination after(ii) Void ratio e vs. matric potential (shrinkage-Casagrande (1936) defines this stress by use of the pointstrain relationship)of maximum curvature of the semilogarithmic stress–strain relationship. The preconsolidation stress is the

e er

es er [1 (e)

ne]me [3]point of intersection of two lines. The first line is thebisector line of the angle made by a horizontal and(iii) Void ratio e vs. total stress (stress–strain rela-

tionship) tangent line drawn at the point of maximum. The second

Fig. 2. Void ratio e and moisture ratio as a function of the dependent variable mechanical stress or hydraulic stress .


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line is the extended straight line of the virgin compres- The mechanical parameter preconsolidation stresscan be determined using the second and third derivationsion curve in direction to smaller stresses (Fig. 3). Al-

though this method is commonly used, the determina- of this function. The virgin compression line is deter-mined by the tangent through the inflection point (zero-tion of the point of maximum curvature and the decision

of which range of stresses determine the virgin compres- point of the 2nd derivation) of this function. The pointof maximum curvature (1st zero-point of the 3rd deriva-sion line are often subjective.

Dias Junior and Pierce (1995) present a statistical tion) is the starting point for the calculation of the bi-

sector line of the angle made by a horizontal and a tan-procedure with which the difficulties of exact and objec-tive determination of the Casagrande values can be ex- gent line of the stress–strain relationship. When the

zero-point of the 2nd and 3rd derivation is known, thecluded. They also compare this method with a numberof different approaches. The method of Dias Junior point of intersection of the bisector and the extended

straight line of the virgin compression curve can beand Pierce (1995) fits the recompression line and virgincompression line by a regression line starting with the calculated easily using a spread sheet. In the following,

the derivation of the van Genuchten equation is de-smallest and highest applied stress, respectively. Step-by-step, the adjoined stresses for calculation of both scribed in detail.

To simplify notations, the van Genuchten equationregression lines are included and statistically compared.One combination of stresses, which determines the re- (see Eq. [2], [3], and [4]) is rewritten in the following

uniform way:compression line and the virgin compression line, willresult in the greatest statistical difference of these two

f (h) (s r)[1 (h)n]m r, [5]lines. Dias Junior and Pierce (1995) state a good agree-

ment of this statistical method with graphic determina- with s and r being the water content at saturation and

tions. The preconsolidation stress is then defined as the residual water content, respectively, and , n, and mpoint of intersection of these two lines. In practice, like being the van Genuchten parameters. The variable hthe graphic method, this method has disadvantages when represents the matric water potential or mechanicalthe number of observations is small and when the curva- stress . To take account for the logarithmic stress, f (h)ture of the log(stress)-strain relationship is small. Addi- is replaced by the following function:tionally, this method will only be an approximation de-

g( x) f (10 x) (s r)[1 (10 x)n]m r . [6]pendent on the quality and number of data.

With use of a continuous function, the soil mechanical Furthermore, the notation k: k( x): 10 x will beparameters can be determined purely mathematically. used. With use of k( x) k( x)ln(10) and the chain rule,On the basis of this hydraulic model of van Genuchten, the first derivative g( x) of g( x) can be calculated asthe hydraulic/mechanical stress–strain relationship can follows:be modeled with the above given boundary values. Vol-

g( x) (s r)(m)(1 kn)m1nkn1k ln(10)ume change should always be defined by void ratio rather

than relative strain. It simplifies calculations because ckn(1 kn)m1, [7]

residual void ratio can be defined by a constant value where c denotes the constant mn(s r) ln(10). Withas explained above, whereas when using strain a specificthe addition of the product rule, the second derivativebulk density has to be related to each soil and hence a

g″ ( x) and third derivative g( x) of g( x) can be calculatedspecific minimum strain estimated. As the derived param-as follows:eters of the van Genuchten curve respond very sensitively

to the shape of the curve, it is quite important to model g″ ( x ) cnkn1k ln(10)(1 kn)m1with a reliable residual pore volume (void ratio).

ckn(m 1)(1 kn)m2nkn1k ln(10)

cn ln(10)kn(1 kn)m2[(1 kn) (m 1)kn]

c(kn mk2n)(1 kn)m2, [8]

where c denotes the constant cn ln(10).

g( x ) c[nkn1k ln(10) m2nk2n1k ln(10)](1 kn)m2

c(kn mk2n)( m 2)(1 kn)m3nkn1k ln(10)

cn ln(10)kn(1 kn)m3[(1 2mkn)(1 kn)

(m 2)(kn mk2n)]

cn ln(10)kn(1 kn)m3[(1 2mkn kn 2mk2n

(m 2)kn m(m 2)k2n]

cn ln(10)kn(1 kn)m3[1 (3m 1)kn m2k2n]

[9]Fig. 3. Determination of the preconsolidation stress according to themethod of Casagrande (1936). The zero


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Table 3. Summary of results of modeling the stress–strain relationships with the van Genuchten equation and derivation of mechani-cal properties.

Parameter es† er‡ § n§ RMSE NO¶ hmc# hi†† p‡‡

1/kPa kPa

Soil 0.705 0.270§§ 0.0314 1.6419 0.0118 5 20 56 25Spoil 0.941 0.270§§ 0.0255 1.9137 0.0056 5 25 58 29Glacial till 0.481 0.270§§ 0.0074 1.8116 0.0020 8 87 211 103Palsa 5.364 0.270§§ 0.1013 1.3241 0.0004 7 7 29 9

Clay-I 0.980§§ 0.270§§ 0.0212 1.5397 0.0236 8 30 93 38Clay-II 1.084 0.270§§ 0.0669 1.3831 0.0003 7 10 38 13Clay-III 0.655 0.270§§ 0.0191 2.0921 0.0239 7 34 71 39Clay-IV-triax 0.839 0.270§§ 0.0076 1.2302 0.0013 26 93 514 130Kaolin K2 1.579 0.839 0.0094 1.6318 0.0003 16 67 189 82Kaolin K3 1.408 0.732 0.0083 1.5395 0.0002 14 77 238 96

† es, modeled void ratio at stress 0 kPa.‡ er, minimum void ratio.§ and n, van Genuchten parameters.¶ NO, number of observations.# hmc, stress at maximum curvature.†† hi , stress at inflection point.‡‡ p, preconsolidation stress or preshrinkage stress.§§ fixed value during modeling.

RESULTS AND DISCUSSION samples can be modeled with high accuracy as indicatedby the RMSE value. Only the soil samples clay-I and

The principal results of the modeling procedure with clay-III have a higher deviation between data and fittedRETC and the calculation of the preconsolidation stressesvalues. This is because of the relatively small numberusing the point of highest curvature and the inflectionof replicates, but also because of the fact that eachpoint of the van Genuchten equation are listed in Ta-observation is represented by an individual sample. Theble 3. The results show that the volume change of allheterogeneity between the replicates is high, particu-larly in structured samples, and the accuracy is thereforereduced. Regarding the oedometer tests with increasingstress on the same-sample (multistep) and the shrinkagetest, the accuracy is very high. The results show thateven when the heterogeneity of the data set is high, itis possible to find a well-fitted curve.

Figures 4a, 4b, 4c, and 5 show graphically the modeledresults. In Fig. 5, which shows the volume change of a shrinking soil, the calculated mechanical parameters(preshrinkage stresses) and the Youngs modules are in-cluded.

The results of the tests depict very well the parametersstructure and water potential, which influence soil me-chanical properties and stability (Baumgartl and Horn,1991). The combination of homogenized soil and low

Fig. 5. Data,model, and Youngs modulus of thestress–strain relation-Fig. 4. Data and model of thestress–strain relationship(compression) ship (shrinkage) of the substrates Kaolin K2 and Kaolin K3 and

preshrinkage stresses.of the investigated substrates.


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Table 4. Comparison of preconsolidation stresses using differ-water potentials at test conditions result in small precon-ent methods.solidation stresses (soil, spoil, clay-I, clay-II, clay-III;

Graphical Statistical MathematicalFig. 4a, 4c). The organic material of the substrate Palsa(Mohr-Coulomb) (four-parameter fit) (van Genuchten)(Fig. 4b) with very high initial void ratio also presented

kPaa very low preconsolidation stress.Soil 20 21 25The triaxial test was performed on homogenized ma-Spoil 18 17 29

terial under isotropic conditions (Fig. 4c). This kind of Glacial till 80 73† 103Palsa 18 6 9compaction reduces the volume to a lesser extent. The Clay-I 68 46† 38

steepness of the curve is lower and the calculated pre- Clay-II 32 71† 13Clay-III 22 112† 39consolidation stress higher.Clay-IV-triax 145 49 130The test on the structured glacial till (Fig. 4c) was Kaolin K2 105 35† 82Kaolin K3 85 37† 96performed at water conditions drier than field capacity.

The sample originated from a site which was exposed † Not statistically significant from second best fit.to high stresses resulting from heavy agricultural ma-chinery. The preconsolidation stress as a result of boththe higher mechanical and hydraulic stresses, has to be sion line are similar as a cause of either a small numberclassified as very high. of points (e.g., clay-II, clay-III) or a shallow stress–strain

The influence of both the mechanical compaction as relationship in case of mechanical stress applicationwell as the hydraulic impact of shrinkage can be ob- (glacial till, clay-IV-triax) or reduced volume change inserved in the samples Kaolin K2 and Kaolin K3 (Fig. 5; case of shrinkage (Kaolin K2, Kaolin K3).

data taken from Toll, 1995, after Biarez et al., 1988). The comparison shows that the quality of the precon-The sample Kaolin K2 was exposed to about 50% of the solidation stresses of the graphic and the statisticalmechanical stress of Kaolin K3 (55 kPa). Both samples method are very much dependent on the range of thewere then water saturated and dried. The preshrinkage applied stresses, which later define the recompressionstress which can be calculated from the shrinkage char- and virgin compression line. The accuracy also improvesacteristic is higher than the mechanical preconsolidation with greater homogeneity of the samples, that is, thestress. The shrinkage capacity of the kaolinitic substrate more the slopes differ from each other. Regarding shal-exceeds the mechanical compressing stresses when the low stress–strain relationships with only little differencematerial is dried and therefore increases the preconsoli- between the slopes of the recompression and the virgin

compression line, the risk of small accuracy will alwaysdation/shrinkage stress to higher values. Nevertheless,a memory effect can be recognized as the sample Kaolin be high.

The mathematical method allows a continual fit of K3 with higher mechanical preconsolidation stress,shows as well higher preshrinkage stresses after drying. the stress–strain relationship including the boundary

conditions initial and residual void ratio and improvesThe difference in the pretreatment is also identifiable

in the elastic properties. The course of the Youngs mod- the accuracy of the derived preconsolidation stress (ac-cording to the Casagrande method).ulus shows a higher width at lower water suctions for

the substrate Kaolin K2 (Fig. 5). The shape of the stress–strain relationship modeledwith the van Genuchten equation is based on the param-To value the mathematical method of deriving me-

chanical parameters from a functional relationship, the eters and n. A sensitivity analysis shows that generallythe preconsolidation stress is mainly influenced by theherewith calculated preconsolidation stresses are com-

pared with the commonly used graphic method (Casa- -value of the van Genuchten equation. For n values 1.6, the influence of n on the preconsolidation stress isgrande) and statistical method (Dias Junior and Pierce,

1995). The preconsolidation stress according to the Ca- negligible (Fig. 6). Only for stress–strain relationshipswith a small change in void ratio with increasing me-sagrande method was determined by a manual line of

best fit. The statistical method consists of a four-parame- chanical or hydraulic stress, that is, a small steepness of the van Genuchten equation, will n values have an influ-ter fit. As described earlier, two regression lines are

calculated using an increasing number of measured data ence on the value of the preconsolidation stress. As the

shape of the curve has a sensitive influence on the calcu-for the first regression line and a equivalently decreasingnumber of observed data for the second regression line. lated value of the preconsolidation stress, void ratioshould be preferably used instead of, for example, anThe regressions with the highest statistical difference

were used to calculate the preconsolidation stress. absolute or relative strain value.In the hydraulic use of the van Genuchten model, theThe preconsolidation stress of the three methods are

summarized in Table 4. The resultsof the methods differ -value represents the air-entry value, which may be de-fined as the elastic property of thepore system in respectpartly to a great extent. The variation between the physi-

cal and mathematical method is higher with greater het- to emptying of water filled pores (Hillel, 1998). Hence,for high n values in very deformable substrates, the pre-erogeneity of the data set; that is, the less a true line

could be drawn through the points (e.g., clay-I, clay-II). consolidation stress is equivalent to the air entry-value.The results of the modeling procedure show that itThe statistical method has a high deviation when the

slopes of the recompression line and the virgin compres- is possible to predict stress–strain relationships by a hy-


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APPENDIX

List of Parameters

effective stress

total stress

water potential

ua pore air pressureuw pore water pressure water suction [ (matric

water potential)]

parameter related to the degree of saturation of thesoil (0 1)

water content

s water content at saturation

r residual water content

moisture ratio

r residual moisture ratio

s moisture ratio at saturation

e void ratioer residual (minimum) void ratio (e 0.27)

Fig. 6. Influence of the van Genuchten parameters and n on the es maximum void ratiovalue of the preconsolidation stress (isobars in kPa; in 1/kPa).

van Genuchten parameter (subscripts relate to thedependent variables e, and , )

n van Genuchten parameter (subscripts relate to thedraulic function (van Genuchten-equation) and to derive dependent variables e, and , )mechanical parameters like preconsolidation stress and

m van Genuchten parameter (subscripts relate to theYoungs modulus from derivatives of that equation.

dependent variables e, and , )

h variable representing or

CONCLUSION E Youngs modulus

ε strainThe combination of the theories which describe stress

state functions of mechanical and hydraulic stresses and p preconsolidation stressthe utilization of a model, which describes the behavior s preshrinkage stressof volume change due to either mechanical or hydraulicstresses, bears great advantages when modeling volume REFERENCESchange. The application of a continuous function en-

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Contração Energy Capillary - Baungart

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