Compaction of an Eutric Cambisol under heavy wheel traf®c
in Switzerland Ð ®eld data and modelling
M. Gysi
a,*, G. Klubertanz
b, L. Vulliet
baSwiss Federal Research Station for Agricultural Economics and Engineering (FAT), CH-8356 Taenikon, Switzerland bSwiss Federal Institute of Technology (EPFL), Soil Mechanics Laboratory, CH-1015 Lausanne, Switzerland
Received 16 September 1999; received in revised form 19 May 2000; accepted 2 June 2000
Abstract
Heavy agricultural machinery can cause structural degradation in agricultural subsoils. Severe structural degradation impedes plant growth. Therefore, compaction must be limited to layers that can be structurally reclaimed and remoulded with reasonable effort by tillage. The purpose of this study was to investigate the impact of a single pass with a sugar beet harvester on the soil properties of an unploughed Eutric Cambisol. Field measurements and laboratory testing were carried out in Frauenfeld, Switzerland. In addition 2D calculations of strain, stress and subsequent compaction were conducted using a three-phase (soil skeleton, pore water, and air) model for unsaturated soil incorporating a recently developed constitutive law. Model data were compared to the ®eld measurements. Due to the pass of the machinery, the soil was compacted down to a depth of at least 0.15 m and at most 0.25 m. This compaction was indicated by an increase in soil bulk density and pre-consolidation pressure as well as by a decrease in total porosity and macroporosity. The surface displacement measured in the ®eld was consistent with the calculated model data. The calculated and measured stresses at depths of 0.35 and 0.55 m stand in good accordance with each other, whereas at a depth of 0.15 m the pressure measured in the ®eld exceeded the calculated pressure. In this study, we show the degree of compaction due to heavy wheel traf®c and the suitability of a model approach to describe compaction processes.#2000 Elsevier Science B.V. All rights reserved.
Keywords:Compaction; Soil properties; Model; Laboratory; Field experiment; Eutric Cambisol; Switzerland
1. Introduction
In Swiss agriculture, economic pressure progres-sively favours the use of heavy machinery. The ever increasing weight of machinery and the necessity to work also under unfavourable conditions increase the risk of soil compaction. Compaction may signi®cantly impair the production capacity of a soil. HaÊkansson and Reeder (1994) showed a crop yield loss of 14 per cent the
®rst year after repeated wheel traf®c on agricultural soils in seven different countries in Europe and North Amer-ica. These effects may not immediately become evident. According to Moullart (1998), soil compaction can impede root growth; however, the above-ground parts of the plants often do not show reduced dry matter production. Compaction reduces the penetrability of the soil for roots (Unger and Kaspar, 1994). In nutritional de®ciency situations, changes in the soil pore volume may reduce water and nutrient supply of the root. Furthermore, soil organisms depending on oxygen may be affected. Horn et al. (1995) observed an increase of N2O, CH4and CO2emissions in compacted soils.
*Corresponding author: Tel.:
41-52-368-3354; fax:41-52-365-1190.
E-mail address: michael.gysi@fat.admin.ch (M. Gysi).
Topsoil compaction is not considered to be a serious problem because the regeneration potential of the soil structure is relatively high due to biological (below ground animals, plant roots), climatic (dehydration and frost) and anthropogenic (soil tillage, cultivation) in¯u-ences. In the subsoil these factors are less effective and structural recovery is therefore less intense. The effect of subsequent subsoil compaction is to be considered cumulative (Semmel et al., 1993). In addition, technical measures taken in order to loosen the subsoil are expen-sive and their bene®ts are often controversial.
Subsoil compaction can be predicted by the means of ®nite element model (FEM). To simulate compac-tion under tyres, non-linear elastic (Perumpral et al., 1971; Pollock et al., 1986; Chi et al., 1993), plasticity (Chung and Lee, 1975; Kirby, 1989a) and critical state (Kirby, 1989b) models have been used.
In this paper, we calculate compaction under a tyre using a fully coupled three phase approach recently developed by Klubertanz (1999) and implementing the new constitutive law of Geiser (1999) for unsatu-rated soil behaviour. Further on, we present the results of a compaction experiment carried out in Switzer-land. A sugar beet harvester was used for the experi-mental treatment which took place in October, 1998. A full set of soil mechanical and soil physical para-meters were measured in the plot subjected to wheel traf®c and in an untraf®cked plot nearby. Some of these parameters served as input variables for the model. The output of the model is compared to data recorded during the ®eld experiment.
2. Material and methods
2.1. Experimental design and machine properties
The experimental area is located near Frauenfeld (Swiss National Coordinates: 270 000/707 950,
topo-graphical map sheet 1053: Frauenfeld). The soil, situated on a wide valley bottom, is a skeleton-free Eutric Cambisol. The organic C content varies between 510ÿ3and 0.510ÿ3kg kgÿ1. The texture of the samples taken at a depth of 0.12±0.17 and 0.32± 0.37 m, is a sandy loam; at a depth of 0.52±0.57 m texture is a loamy sand (Table 1).
The year prior to the ®eld experiment, winter wheat (Triticumspp.) was grown at the experimental site and harvested with a plot combined harvester. During the last 10 years, the farmer practised a soil preserving cultivation method without ploughing. No plough pan was found with the penetrometer.
The plot was subjected to a single pass with the right front wheel of a self-propelled six-row sugar beet (Beta vulgaris L.) harvester, model Kleine SF10 in October, 1998. Driving speed was 1 m sÿ1. The lifting units of the combine harvester were raised. A Good-year radial tyre 710/70 R 38 was used. This 0.68 m wide tyre had an in¯ation pressure of 220 kPa. The total weight of the sugar beet harvester was 285.2 kN. Total load is unevenly distributed onto the four wheels due to the raising of the lifting units as well as to the laterally extended unloading elevator. The highest load (107 kN) is applied to the right front wheel.
In order to determine the contact area, the tyre print was marked with lime powder, subsequently photo-graphed and the contact area determined by digital image processing. This yielded a value of the contact area of 0.71 m2. The average ground contact pressure was thus 151 kPa as summarised in Table 2. An untraf®cked plot next to the treated plot served as control area.
Both plots were at all times exposed to natural precipitation. This resulted in a matric potential of ÿ26 hPa and a volumetric soil water content of 24% at a depth of 0.12±0.17 m (Fig. 1). At the time of the experiment the ground water level was at a depth of 1 m.
Table 1
Soil organic C content and texture of Eutric Cambisol in the test area in Frauenfeld, Switzerland
Depth (m) Organic C content (kg kgÿ1)
0.12±0.17 0.0053 0.18 0.40 0.42 Sandy loam
0.32±0.37 0.0011 0.17 0.31 0.52 Sandy loam
2.2. Field and laboratory measurements
To measure the soil pressure distribution underneath the contact area, Bolling pressure probes (Bolling, 1987) were placed in the centre of the tyre tracks at depths of 0.15, 0.35 and 0.55 m, six probes at each depth.
The day after thewheel pass, undisturbed soil samples were taken under the tyre track centre at depths of 0.10± 0.13, 0.20±0.23, 0.30±0.33, 0.40±0.43 and 0.50±0.53 m. The original surface of the soil served as O level for the depth measurements. The samples were stored at a temperature of 18C. Eight samples were taken for each parameter (bulk density, macroporosity,
pre-consolida-tion) and each depth. The horizontal distance between the samples were 0.5 m.
The samples for bulk density, total porosity and macroporosity had a length of 45 mm and a diameter of 55 mm. Macroporosity was measured by gravi-metric method. An equivalent pore radius of 100mm corresponds to a matric potential of ÿ30 hPa (Soil Science Society of America, SSSA, 1997).
Cylindrical soil samples of 100 mm in diameter and 30 mm in height (Horn, 1981) were used to measure the pre-consolidation pppressure at an initial matric
potential of ÿ60 hPa. Consolidation pressure was increased in the following sequence: 8, 13, 20, 30, 50, 75, 100, 150, 200, 300, 400, 600, 800, 1200, 1400, 2000 kPa. Each pressure step was applied during 30 min in order to monitor consolidation. Pre-conso-lidation pressureppwas determined by means of the
Casagrande criterion (Casagrande, 1936). In addition, minimum, maximum and most probable pp values
were determined (McBride and Joosse, 1996). The settlement of the soil surface in the tyre track was determined by means of a laser pro®le meter. The laser pro®le meter measures the distance between a reference level and the soil surface in steps of 2 mm.
Table 2
Wheel load, contact area, tyre in¯ation pressure and average ground contact pressure of the right front wheel of the sugar beet harvester
Parameter Unit Value
Wheel load kN 107.6
Contact area m2 0.71 Tyre inflation pressure kPa 220 Average ground contact pressure kPa 151
2.3. Model approaches
A numerical model developed by Klubertanz et al. (1997a,b) for multiphase ¯ow in deformable porous media was used. A newly developed constitutive law (Geiser, 1999) has been incorporated. The model has been successfully applied in the past to laboratory cases (Klubertanz et al., 1997a). A brief description of the multiphase model is given below. The main equa-tions can be found using an approach based on studies by Hassanizadeh and Gray (1979), Schre¯er (1995), Hutter et al. (1999), and Lewis and Schre¯er (1999). For details of the derivation, the reader is referred to Klubertanz et al. (1997a) or Klubertanz (1999). The principal equations of the model are given in Appen-dix A.
2.4. Modelling details
The calculations were determined using a 6030 4-node quadrilateral, non-uniform grid on a 2D cross-section of 8 m4 m. A load corresponding to the 152 kPa surface pressure under the tyre was applied
in the middle of the cross-section over the width of the tyre, i.e. 0.7 m in the form of a ramp load: the load rose in 0.1 s to its nominal value and was maintained constant for 0.9 s. This load application corresponds to a driving speed of 1 m sÿ1.
The material parameters are given in Table 3. Most of them have been determined by means of laboratory tests on soil samples taken from the experimental site. Some of the parameters have been estimated based on values obtained for similar soils (see the footnote in Table 3).
The initial yield surface at each Gauss point was generated based on the oedometer test data, using linear interpolation (Table 4). Computations were made on an O2(SIG) microstation.
Table 3
Material parameters used for calculation (related equations are given in Appendix A)
Parameter Short description Unit Value
Elastic parameters
E Young's modulus MPa 190
v Poissons ratio ± 0.4
Parameter for plasticity model
g Steady state parameter ± 0.0273
b Steady state parameter ± 0.52
n1 Phase change parameter ± 3.1
a1 Hardening parameter forF1 ± 4.1310ÿ9
Z1 Hardening parameter forF1 ± 3.22
k0 Describing the non-associated behaviour ofF1 ± 8 k/ Describing the non-associated behaviour ofF1 ± 0.6
R Cohesion kPa 15
pce Air entry pressure kPa 5
a2 Hardening for suction-dependent evolution ofF1 ± 0.0025a
a3 Hardening parameter forF2 ± 9.510ÿ9a
Further parameters
N Porosity ± 0.48
rs Grain density g/cm3 2.7
K Geometric permeability m2 210ÿ13
C0 Material parameter ± 4
C1 Material parameter ± 0.06
aParameters estimated prior to calculations.
Table 4
Measured pre-consolidationppin three depths of the control plot
Depth (m) Unit Initial pre-consolidation stress
0.15 kPa 75
0.35 kPa 145
3. Results
3.1. Field and laboratory measurements
Due to a single pass of the wheel, a statistically signi®cant increase (P<0.05) of 0.035 g/cm3in bulk density was recorded at a depth of 0.15 m (Fig. 2). Between 0.25 and 0.55 m of depth, no statistically signi®cant changes were measured.
Corresponding to the increase of bulk density, the total porosity decreased (P<0.05) by 1.9% at 0.15 m of depth. Below this depth, no statistically signi®cant changes were observed (Fig. 3).
Macroporosity (radius >100mm) decreased signi®-cantly (P<0.05) by 3.9 at a depth of 0.15 m (Fig. 4). At depths of 0.25 and 0.35 m, this decrease is still measurable, but not at a statistically signi®cant level. At 0.45 and 0.55 m, no decrease in macroporosity was measured.
A statistically signi®cant (P<0.05) increase in pre-consolidation from 75 to 152 kPa was recorded at a depth of 0.15 m (Fig. 5). At depths of 0.35 and 0.55 m, no changes in pre-consolidation were observed.
3.2. Modelling and comparison with ®eld measurements
Fig. 6 shows the calculated displacement ®eld (after pass). The vertical displacement under the track centre amounted to 0.037, 0.025 and 0.017 m at depths of 0.15, 0.5 and 1 m, respectively. The calculated dis-placement under the track centre is considered to be reasonable. Unfortunately no comparison with ®eld measurements can be done since displacement mea-surement at depth were not available.
Fig. 7 shows the pore-water pressure at the end of the loading. Calculated pore-water pressures of 80 and 10 kPa were obtained at depths of 0.4 and 1 m, respectively. This important pore-water pressure gen-eration strongly in¯uences the displacement pattern; in fact, it contributes to wider and more regular distri-bution of strains. Moreover, the extension of the saturated zone has a negative in¯uence on soil strength and signi®cantly favours plastic deformations.
In Fig. 8, the simulated porosity change is compared to the measured porosity change underneath the track centre. In the model, porosity change is calculated on
Fig. 3. Total porosity of the single pass plot (white circles) and the control plot (black triangles). The error bars areS.D.
the basis of the strain under the track centre. The fact that the calculated changes in porosity shows exten-sion over the whole depth is most likely due to the overall volumetric behaviour of the Geiser-Model, which tends to overestimate dilative soil behaviour (Geiser, 1999; Klubertanz, 1999).
In Fig. 9, calculated and measured surface settle-ment pro®les are compared. Two calculated pro®les are
shown: The ®rst one (circles) represents the maximum displacement during the pass (including elastic and plastic components); the second one (triange) repre-sents the residual displacement (plastic component only) after the pass. The measured surface displace-ment pro®le is the average of four measuredisplace-ments, with the indication of the standard deviation. The calculated residual displacement at the surface after wheel traf®c
Fig. 5. Pre-consolidationppof the single pass plot (white circles) and the control plot (black triangles). In addition, soil pressure (black
quadrangles) measured by means of bolling probes is indicated. The error bars areS.D.
Fig. 7. Pore water pressure ®eld in the soil pro®le at the end of loading. The centre line of the track is at the surface (depth 0) and at a horizontal distance of 4 m.
is consistent with the displacement observed in the ®eld. The measured displacement of the surface lies between the elastic/plastic component displacement and the residual plastic displacement. In the ®eld, the shape of the track is more pronounced than in the model. The cross-section of the tracks in the ®eld had a more or less ¯at bottom with vertical side walls whereas the calculated cross-section has a V-shape. This could be a consequence of the heterogeneous stress distribu-tion under the tyre (Wood and Burt, 1987; Kirby et al., 1991; Blunden et al., 1992; DoÈll, 1998).
In a separate ®nite element calculation, we veri®ed the relation between the ®rst invariant of the stress tensor ((sxsysz)/3) and the pressure in a Bolling
probe (sb). For that reason, a simple cross-section
with a water ®lled tube in the middle was simulated. On the top of the cross-section, we applied an increas-ing load rangincreas-ing from 0 to 200 kPa. The cross-section consisting of a highly permeable sand with Mohr± Coulomb as constitutive law was overlaid by a non-uniform mesh. An undrained material with a low Young's modulus was used for the tube. The compar-ison of the pressure in the tube with the ®rst invariant of the stress tensor showed a linear relationship also in
the plastic range of the deformation. The ratio of the pressure in the tube to the ®rst invariant of the stress tensor (sb/((sxsysz)/3)) amounted to 0.89. This
proves that the pressure in the Bolling probe is a good indicator for the ®rst invariant of the stress tensor.
Fig. 10 shows a comparison of the maximal calcu-lated and measured total stresses during pass. The calculated stress is the ®rst invariant of the total stress tensor. The calculated total stress and the measured maximal stress underneath the track centre coincide quite well at 0.35 and 0.55 m of depth. The measured peak value at a depth of 0.15 m is much higher than the value calculated in the model. A similar phenomenon was observed using a critical state model by Kirby et al. (1997).
4. Discussion
A total load of 285.2 kN is currently amongst the highest for off-road vehicles in Swiss agriculture. According to the manufacturer, the wheel load of 107 kN is at the limit of the bearing capacity of this tyre.
The matric potential and water content measured at the time of the pass indicated that the soil was wet. The matric potential was far higher than ®eld capacity. Such a wet soil condition is common for sugar beet harvest in autumn in Switzerland.
A simple FE model shows that the pressure in a Bolling probe corresponds well with the ®rst invariant of the stress tensor. The disturbance of the surrounding soil is relatively low when installing the probe. For these reasons, Bolling probes are a good tool to quantify soil pressure in the soil.
The main features of the used constitutive model may be summarised as follows (Geiser et al., 1999):
Pre-consolidation pressure increases with matrix suction. A dry season would have the same effect on the soil as load induced pre-consolidation. Stiffness increases with suction. A dry soil is less
deformable than a wet soil.
Structural collapse is imbedded. Awetting of the soil with the related decrease in matrix suction may produce a structural collapse (reduction of porosity). Peak strength increases with suction, together with increasing brittleness. A non-saturated soil will
exhibit more strength (increasing bearing capacity) but may lead to brittle failure under sufficient displacement.
Concerning the complete multiphase model (Klu-bertanz, 1999) the following capabilities will play an important role in the problem:
The phases are fully coupled. As a consequence, any change in the water content will affect not only the mechanical properties (volume change, sheer deformation) but also the hydraulic properties (per-meability).
Both hydraulic and dynamic loading are possible. Deformation could be caused by wheel load as well as by climatic factors (¯uctuation of the water level, rain, etc.).
5. Conclusions
Soil compaction was observed down to a depth of 0.15±0.25 m, as demonstrated by an increase in bulk density and pre-consolidation pressure as well as by a
decrease in total porosity and macroporosity. Below this depth, no change in any parameter was evident. This result con®rms the observations of Gysi et al. (1999). These ®ndings are quite surprising because other researchers reported much deeper compaction with lower wheel loads (HaÊkansson and Reeder, 1994; Alakukku, 1996).
Generally, calculated values are in good agreement with the ®eld measurements. Calculations seem to con®rm that notable effects of compaction are limited to upper soil layers. No effects can be expected in deeper layers of this unploughed soil.
With the developed model, it is possible to predict the impact of wheel traf®c on agricultural soil under various soil conditions. The advantage of the model is to directly take into consideration the soil water con-tent. Since this variable is known to vary largely in autumn when heavy ®eld traf®c is applied, it will result in more or less severe effects on the soil compaction.
The proposed model makes use of relatively large computational power and extensive laboratory tests. Some simpli®cations may be introduced to allow its use as standard procedure for the prediction of subsoil compaction.
The compaction caused by a single pass with a wheel load of 107.6 kN was restricted to the upper 25 cm of the soil.
The used three-phase model is suitable to predict the depth and the degree of compaction of unsa-turated soils.
Other authors also report that in upper layers the calculated soil pressure is significantly lower than the soil pressure measured in the field.
Appendix A.
The momentum balance of the three phase mixture reads
r s0ijÿwpwdijÿ 1ÿwpadij rgi0 (A.1)
wheres0ijis the effective stress tensor, p
w
andpathe water and the air pressure, gi the gravity vector
and r the density of the mixture as r(1ÿn)rs
nSwrwnSara. Note that here rs,rw andra are the densities of solid, water and air, respectively.vsis the
(Lagrangien) velocity of the solid skeleton. pcis the matric potential (pcpaÿpw). Sw and Sa are the degrees of water and air saturation, respectively, de®ned as the volume ratio of water and air, respec-tively, divided by the total pore volume. Furthermore,
ndenotes the porosity,dijthe Kronecker delta andr
the divergence operator. Bishop's parameterwis cho-sen aswSwaccording to Hutter et al. (1999). Quan-tities related to the solid, water or air phase are denoted by the superscripts s, w, and a, respectively. Similarly, the combined mass balance equations for water/solid or air/solid read with some assumptions (Klubertanz et al., 1997a)
for the water/solid balance and
n
The relative permeabilitykrwof water is assumed to
be a function of the degree of water saturation and porosity (Seker, 1983), the relative permeability of air
kra being a function of the porosity only. K is the
intrinsic permeability andm the dynamic viscosity. A non-linear capillary pressure Ð saturation rela-tion obtained by tests on different materials was used (Seker, 1983):
Sw 1
1=C0log ÿ cdpc=gzrw1=C11
(A.4)
whereC0,C1andcdare material parameters andgz
represents the vertical component of the gravity vec-tor. Eqs. (A.1)±(A.3) represent a set of ®ve equations for the ®ve unknowns solid displacement vectorui, air
pressure scalar pa and water pressure scalar pw (ui
being linked tos0ijin (1) via a constitutive equation and tovsvia derivation in time). Other choices for the set of unknowns are possible.
HISS-family of models (Desai et al., 1991).J1is the
®rst invariant of the stress tensor andJ2Dthe second
invariant of the deviatoric effective stress tensor. The two yield surfaces read, respectively
F1J2D
with the plastic potentialGfor surfaceF1 (mechanical yielding is non-associated)
whileF2 is associated. In (5)±(7),gandnare material constants,pais a normalising constant taken as some
constant value of the atmospheric pressure andaand aQare hardening parameters. The second yield surface
allows to cover the creation of plastic deformation due to hydric loading, i.e. changes in capillary pressure. The generation of plastic deformation is limited to values of capillary pressure lower than the air entry valuepce.Fsreads
Desai et al. (1991) proposed several forms for the hardening functiona1,2g1,2(x,xv,xD,rv,rD). In this
paper, we shall use the expression (Geiser, 1999):
a1
a1
xZ1 (A.9)
for the hardening parameter inF1 and
a2
orxDits volumetric or deviatoric part, respectively.
Further holdsrvxv/xandrDxD/x.a1,a3andZ1are
material parameters.
The hardening parameter for the potentialaQreads
aQak a0ÿa 1ÿrv (A.11)
witha0, the value ofabefore the onset of loading and
k a material parameter.
The dependency of the hardening and cohesion on the capillary pressure is modelled by the following relations:
with the material parametersa2andras well as the air
entry valuepce.
For further details of the model and appropriate ways to determine the values of the parameters and initial values, the reader is referred to Geiser (1999). The implementation in a FEM model is discussed in detail in Klubertanz (1999).
References
Alakukku, L., 1996. Persistence of soil compaction due to high axle load traf®c. II. Short-term effects on the properties of clay and organic soils. Soil Till. Res. 37, 211±222.
Blunden, B.G., McLachlan, C.B., Kirby, J.M., 1992. A recording system for measuring in situ stresses due to traf®c. Soil Till. Res. 25, 35±42.
Bolling, I., 1987. Bodenverdichtung und Triebverhalten bei Reifen Ð neue Mess Ð und Rechentechniken. Forschungsbericht Agrartechnik des Arbeitskreises Forschung und Lehre der Max-Eyth-Gesellschaft (MEG), Dissertation UniversitaÈt MuÈnchen. Casagrande, A., 1936. Determination of the pre-consolidation load
and its practical signi®cance. In: Proceedings of the Interna-tional Conference on Soil Mechanics and Foundation Engi-neering, Vol. III, Cambridge, pp. 60±64.
Chi, L., Tessier, S., McKyes, E., Lague, C., 1993. Modelling mechanical behaviour of agricultural soils. Trans. Am. Soc. Agric. Eng. 36, 1563±1570.
Chung, T.L., Lee, J.K., 1975. Dynamics of a viscoelastoplastic soil under a moving wheel. J. Terramech. 12, 15±31.
Desai, C.S., Sharna, K.G., Wataguala, G.W., Rigby, P.B., 1991. Implementation of hierarchical single surface d0 and d1 models in ®nite element procedure. Int. J. Num. Anal. Meth. Geomech. 15, 649±680.
Kriterien. In: Van den Akker, J.J.H., Arvidsson, J., Horn, R. (Eds.), Experience with the Impact and Prevention of Subsoil Compaction in the European Community, Part 1. Proceedings of the First Workshop of the Concerted Action on Subsoil Compaction, Wageningen, the Netherlands, May 28±30, pp. 73±82.
Geiser, F., 1999. Comportement meÂcanique d'un limon non satureÂ. TheÁse de Doctorat. Lausanne, Ecole Polytechnique FeÂdeÂrale de Lausanne.
Geiser, F., Laloui, L., Vuilliet, L., 1999. Unsaturated soil modelling with special emphasis on undrained conditions. In: Pande, G.N., Pietruszczak, S., Schweiger, H.F. (Eds.), Numerical Models in Geomechanics. Proceedings of the Seventh Interna-tional Symposium on Numerical Models in Geomechanics-NUMOG VII, Graz, Austria, September 1±3, pp. 9±14. Gysi, M., Ott, A., FluÈhler, H., 1999. In¯uence of single passes with
high wheel load on a structured, unploughed sandy loam soil. Soil Till. Res. 52, 141±151.
HaÊkansson, I., Reeder, R.C., 1994. Subsoil compaction by vehicles with high axle load Ð extent, persistence and crop response. Soil Till. Res. 29, 277±304.
Hassanizadeh, M., Gray, W.G., 1979. General conservation equations for multi-phase systems: 2. Mass, momenta, energy and entropy equations. Adv. Water Res. 2, 191±203. Horn, R., 1981. Eine Methode zur Ermittlung der Druckbelastung
von BoÈden anhand von Drucksetzungskurven. Z. f. Kulturtech-nik und Flurbereinigung 22, 20±26.
Horn, R., Domzal, H., Slowinska-Jurkiewicz, A., van Ouwerkerk, C., 1995. Soil compaction processes and their effects on the structure of arable soils and the environment. Soil Till. Res. 35, 23±36.
Hutter, K., Laloui, L., Vulliet, L., 1999. Thermodynamically based mixture models of saturated and unsaturated. J. Coh. Frict. Mat. 4, 295±338, in press.
Kirby, J.M., 1989a. Shear damage beneath agricultural tyres Ð a theoretical study. J. Agri. Eng. Res. 44, 217±230.
Kirby, J.M., 1989b. Critical soil mechanics for unsaturated soils Ð measuring the parameters and predicting deformations. In: Larson, W.E., Blake, G.R., Allmaras, R.R., Voorhees, W.B., Gupta, S.C. (Eds.), NATO Advanced Workshop on Mechanics and Related Processes of Structured Agricultural Soils. Kluwer Academic Publishers, Dordrecht, 224 pp.
Kirby, J.M., Blunden, B.G., McLachlan, C.B., 1991. Soil stresses beneath tracked and tyred vehicles. The Aust. Cotton Grower 12, 46±49.
Kirby, J.M., Blunden, B.G., Trein, C.R., 1997. Simulating soil
deformation using a critical-state model: II. Soil compaction beneath the tyres and tracks. Eur. J. Soil Sci. 48, 59±70. Klubertanz, G., 1999. Zur hydromechanischen Kopplung in
dreiphasigen poroÈsen Medien. TheÁse de doctorat. Lausanne, Ecole Polytechnique FeÂdeÂrale de Lausanne.
Klubertanz, G., Laloui, L., Vulliet, L., 1997a. Numerical modeling of the hydro-mechanical behaviour of unsaturated porous media. In: Proceedings of the NAFEMS World Congress 1997, Stuttgart.
Klubertanz, G., Laloui, L., Vulliet, L., 1997b. Numerical modeling of unsaturated porous media as a two and three phase medium: a comparison. IACMAG, Wuhan, China, pp. 1302±1314. Lewis, R.W., Schre¯er, B.A., 1999. The ®nite element method in
the static and dynamic deformation and consolidation of porous media. Wiley, Chichester, UK, pp. 1159±1164.
McBride, R.A., Joosse, P.J., 1996. Overconsolidation in agricultur-al soils: II. Pedotransfer functions for estimating pre-consolida-tion stress. Soil Sci. Soc. Am. J. 60, 373±380.
Moullart, J., 1998. Factors in¯uencing soil and subsoil compaction and impact of compaction on yield of different plants. In: Van den Akker, J.J.H., Arvidsson, J., Horn, R. (Eds.), Experience with the Impact and Prevention of Subsoil Compaction in the European Community, Part 2. Proceedings of the First Work-shop of the Concerted Action on Subsoil Compaction, Wageningen, the Netherlands, May 28±30, pp. 145±154. Perumpral, J.V., Liljedahl, J.B., Perloff, W.H., 1971. A numerical
method for predicting the stress distribution and soil deforma-tion under tractor wheel. J. Terramech. 8, 9±22.
Pollock, D., Perumpral, J.V., Kuppusamy, T., 1986. Finite element analysis of multipass effects of vehicles on soil compaction. Trans. Am. Soc. Agric. Eng. 29, 45±50.
Schre¯er, B.A., 1995. Thermo-hydromechanical coupling in geomaterials. In: Chambon, R. (Ed.), Proceedings of the Seventh European Autumn School on Nonlinear Modelling of Geomaterials with the Finite Element Method, p. 54. Semmel, H., Horn, R., Steinkampf, H., Schwanghart, H., Neukam,
M., 1993. Druckfortp¯anzung im Boden. Landtechnik 48, 1±2. Seker, E., 1983. Etude de la deÂformation d'un massiv de sol non satureÂ. TheÁse de Doctorat. Lausanne, Ecole Polytechnique FeÂdeÂrale de Lausanne.
Soil Science Society of America, 1997. Glossary of Soil Science Terms. American Society of Agronomy, Madison, WI, 83 pp. Unger, P.W., Kaspar, T.C., 1994. Soil compaction and root growth:
a review. Agron. J. 86, 759±766.
