The essential mechanics of capillary crumbling of structured agricultural soils

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1 Soil & Tillage Research 55 (2000) 117±126 The essential mechanics of capillary crumbling of structured agricultural soils O.B. Aluko *, A.J. Koolen Soil Technology Group, Wageningen University, Bomenweg 4, 6703HD Wageningen, Netherlands Received 5 July 1999; received in revised form 28 January 2000; accepted 17 February 2000 Abstract In soil loosening processes like seedbed preparation, signi cant soil crumbling is often desired. A better understanding of the mechanics of crumbling is necessary to optimize crumbling operations, particularly in structured agricultural soils in which capillary bonds are dominant. To this end, a model of the mechanics of capillary crumbling in structured agricultural soils is developed. Four sets of experiments on cylindrical soil samples were carried out to investigate the validity of the model and soil crumbling characteristics as in uenced by freezing, thawing and drying. After preparation and sample pre-treatment, the samples were dried to different moisture contents and then tested to determine soil bonding strength. Soil water suction was also monitored at testing as were sample dimensions at different stages of experimentation. The model was found to account satisfactorily for the mechanics of capillary crumbling in structured agricultural soils. Freezing had the effect of reducing the strength of inter-aggregate bonds whilst preserving the integrity of soil aggregates during crumbling. # 2000 Elsevier Science B.V. All rights reserved. Keywords: Soil bonding strength; Soil structure; Capillary bonds; Soil moisture content; Mechanics of crumbling 1. Introduction When such assessments as traf cability, water in ltrability and workability of an agricultural soil are desired for cultivation and agricultural production purposes, two physical properties are of primary importance: soil textural composition and soil structure. The textural composition of a soil is largely xed with possible but minor changes being made in the organic matter content. Soil structure, on the * Corresponding author. Present address: Department of Agricultural Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria. Tel.: ; fax: address: oaluko@oauife.edu.ng (O.B. Aluko) other hand, is a transient property which can be altered by different processes. Such processes include shrinking and swelling, drying and wetting, freezing and thawing, human, animal and vehicular traf c, mechanical tillage operations and incorporation of organic matter. The importance of soil structure cannot be overemphasized. For example, the state of the soil structure can be a farmer's greatest asset or his greatest risk. The integrity of aggregates, the type and strength of aggregate bonds, the size and distribution of pore spaces within the soil and the moisture content are salient factors that determine the state of structure of an agricultural soil. When aggregate bonding is the decisive factor, Koolen and Kuipers (1989) have suggested that a soil consisting of rm units /00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S (00)

2 118 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117±126 (aggregates) with relatively little bonding between adjacent aggregates at the mutual points (or areas) of contact, can be regarded as having a good structure. The phenomenon of ``crumbling'' in agricultural soils is very important in the development of soil structure. Indeed, agricultural soils can be regarded as having a crumb structure (Russell, 1973). This crumb structure approach has been used by Chandler (1985) to explain the crumbling of agricultural soils as a fracture process which involves the breaking of bonds between soil crumbs (or aggregates). When the crumbs are hard and the bonding between crumbs is weak, the soil cracks in a brittle manner. On the other hand, when the bonding between the crumbs is strong compared with its internal strength, it may become distorted without breaking (i.e. it is ductile). Snyder and Miller (1989) also attributed crumbling in agricultural soils during tillage operations, to failure by cracking due to tensile stresses. For structured agricultural soils where capillary bonds play a dominant role, Koolen (1987) proposed a capillary bonding stress, p w (N/cm 2 ), which is a function of the soil water suction and the degree of pore saturation. This paper reports an experimental-analytical investigation to examine the mechanics of capillary crumbling in a structured agricultural soil. In particular, Koolen's model of capillary bonding was further developed and the effects of freezing, thawing and drying on crumbling as well as the integrity of soil structure, were studied. Fig. 1. Physical model of the contact zone between two aggregates in a structured soil (after Koolen, 1987): p w, capillary bonding stress; u, soil water suction; S, degree of pore saturation. per unit area (i.e. capillary bonding stress), p w (N/ cm 2 ), given by p w ˆ u S (1) where u is the soil water pressure (i.e. soil water suction in units of stress) and S is the degree of pore saturation, which is the fraction of the total pore volume within the bulk soil occupied by water. The range of S is 0±1. In a further analysis of Eq. (1) (see Fig. 2), Koolen considered its implications in a structured clay soil with inter-aggregate pore spaces that fall within one narrow size class. He reasoned that as the soil dries out from an initially wet state, the trend in the capillary bonding stress (p w ) within aggregates would differ 2. Theory In general, soil strength is derived from two sources: inter-aggregate bonds and intra-aggregate bonds. The strength of a soil is a re ection of the force required to break these bonds. Fig. 1 is a physical model of two aggregates in a structured soil, hatched vertically and horizontally, respectively, that are in contact at positions 1, 2 and 3 in plane I. The pores between the aggregates are relatively larger than those within the aggregates. At contact positions 1, 2, and 3 the tension in the soil water ``clenches'' the particles together. Based on this clenching force, Koolen (1987) considered surface-elements within the soil such as planes I and II and derived a capillary bonding force Fig. 2. The p w ±u±s diagram of capillary crumbling proposed by Koolen (1987). Curves: A, pf (ˆlog u) curve within aggregates; B, pf curve between aggregates; C, capillary bonding stress within aggregates; D, capillary bonding stress between aggregates.

3 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117± from that between aggregates. Thus in Fig. 2, curves A and C illustrate the pf (ˆlog u) curve (where u is expressed as a negative pressure head in cm water column) and the trend of p w within aggregates, respectively, while curves B and D describe the pf curve and the trend of p w between aggregates, respectively. The horizontal portion of curve B refers to the emptying of inter-aggregate pores within a particular size class with little or no increase in soil water suction. Koolen's model suggests that, corresponding to this emptying, the capillary bonding stress between aggregates decreases (see curve D) and then begins to rise again as soil water suction again increases. In an investigation into the mechanics of cracking of remoulded clay bars, Towner (1987) stated that the tensile strength of a soil is a material property that depends in general on both the soil water suction and the water content. In another study of the tensile strength of unsaturated soils, Snyder and Miller (1985) used a modi ed effective stress equation to estimate tensile strength, s t. Their equation can be expressed in the form (Snyder and Miller, 1989): s t ˆ w y u w (2) where the factor w(y) is a function of the degree of pore saturation of the soil and u w is the negative hydrostatic pore water pressure. It is interesting to note that the terms on the right-hand side of Eq. (2) are similar to those in Eq. (1). Now, Mullins and Panayiotopolous (1984) have pointed out that in structured soils, only the fraction of water in the inter-aggregate pore space will contribute to the cohesion between adjacent aggregates which ultimately determines the strength of a structured soil. In view of this, Snyder and Miller (1989) suggested that the use of an ``effective'' degree of pore saturation, ``y e '', based only on the interaggregate porosity of soil, might give a better estimate of the tensile strength of soil in Eq. (2). Assuming that the foregoing arguments are valid, an expression, similar to Eq. (1), can be written for the soil tensile strength s t (N/cm 2 ) as follows: s t ˆ u S i (3) where S i is the degree of inter-aggregate pore saturation, which is de ned as the fraction of the interaggregate pore space occupied by water. Like S, S i ranges from 0 to 1. In conjunction with Koolen's p w ±u±s diagram (see Fig. 2), Eqs. (1) and (3) constitute the proposed capillary crumbling model for structured agricultural soils in which capillary bonds are decisive. In comparison with s t, p w is an index of the bonding strength of the bulk soil whereas s t is an index of the bonding strength at inter-aggregate locations. Series of experiments, in which soil tensile strength, soil water suction and degree of pore saturation were measured, were carried out to investigate the validity of the proposed capillary crumbling model. The materials used and experimental procedure will now be described. 3. Experimental work 3.1. Soil preparation The soil used for this investigation was a Wageningen silty clay loam (36% of minerals <2 mm, 2.3% of soil is humus, water content at pf 2 is 0.27 g/g, CaCO 3 content is 3.3%, ph KCl is 7.4), a full description of which is given by Koolen (1972). Natural (undisturbed) top soil was carefully collected from an arable layer (0±200 mm) and frozen in trays, at a temperature of 188C, for a period of 72 h. The soil was then allowed to air-dry to a moisture content of g/g in order to allow mechanical sieving without internal destruction of aggregates. In the course of this investigation, soil moisture contents were determined by the gravimetric method. The size fraction between 0.6 and 5.0 mm was then obtained by mechanical sieving. This was allowed to equilibrate for 48 h in a sealed plastic container. The soil was then moistened to a moisture content of 0.27 g/g, using water spray jets, and subsequently allowed to equilibrate for 24 h in sealed plastic containers. Cylindrical samples, each having a diameter of 50 mm and a height of 30 mm were prepared using cylindrical core samplers and a tension/compression testing machine. The preparation of each sample entailed weighing a pre-determined quantity of the soil and then compressing it, inside a cylindrical core sampler, to the diameter and height speci ed above. Compression was quick and was stopped before the expulsion of water could start. Each sample was thus prepared to the initial dry density and porosity values of 1.45 g/cm 3 and 0.46 cm 3 /cm 3, respectively.

4 120 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117±126 Immediately after preparation, each sample (still inside the core sampler) was placed and wrapped in a sealed, transparent plastic bag and then subjected to further treatment(s) prior to testing Tests and measurements On the basis of sample treatment(s) prior to tensile strength testing, four different sets of experiments were carried out. These have been coded EI, EII, EIII and EIV, respectively. These sample treatments consisted of equilibration prior to freezing (in sealed, transparent plastic bags) for 24 h, freezing at a temperature of 188C for 24 h (EI, EII and EIII only), thawing for 24 h (EI, EII and EIII only), drying by passing a warm air stream through the sample for selected periods (minutes and/or hours), equilibration after drying (for 24 h) in sealed, transparent plastic bags and then tensile strength testing. Thus, only samples of experiment set EIV were not subjected to the freezing and thawing pre-treatments prior to drying. The conduct of an experiment set entailed the drying of each sample to a progressively lower soil moisture content than the preceding sample. Therefore, only one sample was tested at each moisture content within a particular experiment set. Experiment sets EI, EII and EIII are replicates of the same investigation, carried out primarily for reasons other than statistical reasons. Experiment set EII replicates EI with a very important difference: the average drying interval between successive tensile strength tests is reduced. Experiment set EIII was carried out for further veri cation of the results obtained in EII. Thus, in these replicates (EI, EII and EIII), the emphasis is on the trends over approximately the same range of soil moisture content and not on the replication of tests at speci c, individual moisture contents. After drying and subsequent equilibration, the diameter and height of each sample, as illustrated in Fig. 3(b) and (c), were measured before further testing to determine its tensile strength. This was accomplished using a pair of vernier calipers. Soil tensile strength was determined by subjecting each sample to the Brazilian test. The Brazilian test consists of diametrically loading a cylindrical soil sample until it fails in tension as shown in Fig. 3(d). The applied load, Fig. 3. Typical sample particulars at different stages of experimentation viz. (a) original dimensions at preparation; (b) and (c) dimensions after drying; (d) particulars at failure in the Brazilian test: H 0, sample height at preparation; h 1, sample height after drying; Dh, change in sample height (h 1 h 0 ); D 0, sample diameter at preparation; D 1, sample diameter after drying; D f, sample diameter at failure; DD, change in sample diameter (D 1 D f ); F, loading force at failure; s t, soil tensile strength. F (N), and the strain during testing were recorded and later used to determine the tensile strength, s t (N/cm 2 ), of the soil. Several of these tests were carried out on soil samples at different moisture contents using a tension and compression-testing machine. The soil water suction, u (mbar), of the sample was measured immediately after the Brazilian test by inserting a small tensiometer needle to prevent undue sample disturbance. The tensiometer needle had a diameter of 2 mm. One measurement per sample was suf cient because the water tension in such samples is very uniform. The tensiometer itself used a very small stress-transducer and the highest water tension value that was measured was 830 mbar. The sample

5 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117± moisture content at testing was then determined by the gravimetric method. For each experiment set, values of capillary bonding stress p w (N/cm 2 ) were calculated using Eq. (1). Sample dimensions and loading force F (N) at failure, recorded prior to and during Brazilian tests respectively (see Fig. 3(d)), were used to estimate tensile strength s t (N/cm 2 ) according to the relationship (Kirkham et al., 1959): s t ˆ 2F (4) pd 1 h 1 where h 1 (cm) and D 1 (cm) are the sample height and diameter, respectively, after drying. Values of S i were subsequently calculated using Eq. (3). The diametric strain-to-failure (e D ) during the Brazilian test was also determined as follows: e D ˆ D1 D f D 1 ˆ DD D 1 (5) where D f (cm) is the sample diameter at failure. An estimate of the size of pores being emptied during drying, can be obtained by considering the capillary rise relationship 0:15 cos a u ˆ (6) r where r (mm) is the radius of the capillary and a (8) the contact angle of water. Assuming that complete wetting takes place (i.e. aˆ08) then Eq. (6) can be expressed as 2r ˆ 0:30 (7) u Eq. (7) has been used in the literature (Kutilek and Nielsen, 1994), and is used in this study to estimate the equivalent diameter of pores (2r) being emptied at different soil water suctions. 4. Results and discussion 4.1. Effect of moisture content and drying interval on bonding strengths The capillary bonding stress (p w ) and the measured tensile strength (s t ) are plotted over a range of moisture content in Fig. 4(a) for the rst set of experiments, Fig. 4. Bonding strength plotted against soil moisture content for experiment sets EI (a) and EII (b): p w, capillary bonding stress; s t, soil tensile strength. EI. As the soil dries out, both p w and s t initially increase. This is in agreement with Causarano (1993), who showed that tensile strength of aggregates depends mainly on water content. It appears that s t increases at a continuously increasing rate as the soil dries out. The recorded increase in s t in the relatively drier moisture range (between 0.20 and 0.15 g/g) is more than double the increase in the wetter moisture range (between 0.25 and 0.20 g/g). The trend of p w on the other hand, shows little or no increase in the drier moisture range (0.20±0.15 g/g), its increase being almost entirely limited to the wetter moisture range (0.25±0.20 g/g). The gure also shows that the values of s t are generally lower than the corresponding values of p w. This is in agreement with Panayiotopoulos

6 122 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117±126 (1996) who reported that theoretical tensile strength values (calculated with a formula similar to Eq. (1)) were 10 or more times greater than measured values. According to the present theory, in the wetter moisture range especially when the degree of pore saturation S is near 1, it is expected that both p w and s t should approach the same value because both S and S i approach the value 1. At a moisture content of g/g in Fig. 4(a) however, p w is approximately three times the value of s t. This suggests that appreciable emptying of inter-aggregate pores had already taken place between the initial moisture content of 0.27 g/g and the moisture content g/g at testing, rendering S i at g/g moisture content much lower than S. The trends of p w and s t are plotted as a function of the equivalent pore size (diameter) being emptied in Fig. 5(a). It can be seen that both p w and s t increase with decreasing pore size. In Figs. 4(a) and 5(a), there is no evidence of the expected fall in tensile strength which, as predicted by the theory, should coincide with the emptying of pores of a certain size class at an approximately constant soil water suction. On close examination, it appears that the relatively large drying interval between successive tests in experiment set EI, might have had an obscuring effect on this phenomenon. These observations, whilst precluding a satisfactory analysis of the p w ±u±s diagram from these initial results, informed the decisions to begin sample testing at higher initial moisture contents (i.e. S closer to 1) and reduce the average drying interval between successive tests in the conduct of subsequent experiment sets. Fig. 4(b) shows the values of p w and s t for experiment set EII plotted as a function of moisture content. As drying proceeds, both s t and p w increase in a manner similar to that shown in Fig. 4(a). As discussed earlier, the capillary crumbling model postulates that s t p w at all times and that the condition s tˆp w occurs when S i Sˆ1. It is therefore interesting to note that in Fig. 4(a) and (b), the magnitude of s t is always lower than that of p w. Furthermore, at a moisture content of g/g (see Fig. 4(b)), which corresponds to a degree of pore saturation S of 0.899, the values of p w and s t are almost the same Ð the difference being of a negligible order of magnitude. These results indicate that non-capillary bonds were not involved in soil crumbling and that capillary bonds Fig. 5. Bonding strength plotted against equivalent pore diameter for experiment sets EI (a) and EII (b): p w, capillary bonding stress; s t, soil tensile strength. indeed constituted the dominant bonding force in this soil The capillary crumbling model The fall in tensile strength, predicted by the theory, appears to have occurred in Fig. 4(b) in the narrow moisture range between and g/g. The size range (i.e. equivalent pore diameter) of interaggregate pore spaces emptied within this narrow moisture range is 5.56±6.00 mm (see Fig. 5(b)). The range of soil water suction at which these pores were emptied is 500±540 mbar.

7 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117± Fig. 6. Experimentally determined p w ±u±s diagrams for Wageningen silty clay loam (a) EII and (b) EIII: p w, capillary bonding stress; s t, soil tensile strength; u, soil water suction (mbar); S, degree of pore saturation; S i, degree of inter-aggregate pore saturation. The p w ±u±s diagram for experiment sets EII and EIII are shown in Fig. 6(a) and (b), respectively. The general forms of curves A, B, C and D in Fig. 2 can be recognized in the trends for S, S i, p w and s t data, respectively, in both diagrams. Pertaining to a decrease in bonding strength, two observations are evident. Firstly, the fall in bonding strength is re ected not only in the trend for s t but also in that for p w.in Fig. 6(a) and (b), the data points in the region where this phenomenon occurred have been encircled for emphasis. This observation is better appreciated by carrying out a closer examination of these data points. The encircled data points of Fig. 6(a) are therefore shown blown-up in Fig. 7 with lines and arrows included to show the actual path of succession between the data points. Secondly, the fall in tensile strength is not as pronounced as that predicted by the theory (see curve D of Fig. 2). This apparent difference between the theory and the experimental results can be explained by considering the distribution and

8 124 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117±126 Fig. 7. The encircled data points of Fig. 6(a) blown-up. The lines and arrows in the diagram show the actual path of succession between the data points: p w, capillary bonding stress; s t, soil tensile strength. relative proportion of inter-aggregate pores of different sizes. The condition for curve D in Fig. 2 is that inter-aggregate pores of a particular size class must occur in a relatively overwhelming proportion within the soil. Conversely, when inter-aggregate pores of different sizes are uniformly distributed within a soil, the fall in tensile strength illustrated in curve D in Fig. 2 will disappear. The results illustrated in Fig. 6(a) and (b) indicate that such inter-aggregate pores of a particular size class, did occur in appreciable but not overwhelming proportion within the soil tested. As mentioned earlier, the size class of these pores in experiment set EII, for example, is given by the equivalent pore diameter range 5.56±6.00 mm. When compared with curve A in Fig. 2, the trend of S data with increasing soil water suction u in Fig. 6(a) and (b), is consistent with what is expected. A de nite trend can also be seen in the plotted S i data as the soil water suction u increases. Bearing in mind that interaggregate pores of a particular size class did not occur in overwhelming proportion as explained above, this trend compares reasonably well with curve B in Fig. 2. However, at the highest levels of u (720 and 830 mbar in Fig. 6(a) and (b), respectively), S i seems to increase Ð a deviation from the general trend. Considering that S i data are calculated from measured values of s t and u, a possible explanation for this apparent deviation is that the measured values of u at those levels were inaccurate. This means that relatively high levels of soil water suction (u720 mbar) are dif cult to measure accurately using a small tensiometer Effect of freezing on bonding strengths The effects of freezing on the capillary bonding strengths p w and s t are illustrated in Fig. 8. In the gure, p w and s t data for experiment sets EII and EIV, are plotted as a function of the soil moisture content. It will be recalled that whilst the samples of experiment set EII were subjected to freezing, those of experiment set EIV were not. It can be seen that the values of both p w and s t for experiment set EIV are consistently higher than the corresponding values for experiment set EII at all moisture contents within the range studied. Furthermore, the relative magnitude of the difference increases with decreasing moisture content. For experiment set EIV, the fall in tensile strength predicted by the theory appears to have occurred between the moisture contents of and g/g. Indeed, this moisture content range though slightly higher, corresponds closely with that observed for experiment set EII (i.e ± g/g). On close examination of the equivalent pore diameter, the results show that the same size range of inter-aggregate pores were emptied in both cases the range being 5.56±6.00 mm for EII and 5.45±5.77 mm for EIV. These results show that freezing depresses the strength of inter-aggregate bonds. This means that further reductions in the energy required to crumble structured agricultural soils, can be achieved by performing crumbling operations on such soils as they dry from a frozen condition Effect of freezing on brittleness of soil The diametric strain-to-failure, e D, determined using Eq. (7), provides an index of the brittleness of the soil. Values of e D are plotted as a function of moisture content for both experiment sets EII and EIV in Fig. 9. In spite of some degree of scatter in the results, it is evident from the gure that at any

9 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117± Fig. 8. Bonding strength plotted against soil moisture content for experiment sets EII and EIV. p w, capillary bonding stress; s t, soil tensile strength. moisture content, the values of e D for EII are lower than the corresponding values for EIV. The magnitude of the difference between the two appears to reduce as the soil further dries out. The immediate deduction here is that freezing increases brittleness of the soil. Perhaps a more interesting implication of these results is that, due to increased brittleness, the integrity of soil aggregates as well as overall soil structure will be Fig. 9. Diametric strain-to-failure, e D, plotted against soil moisture content for experiment sets EII and EIV.

10 126 O.B. Aluko, A.J. Koolen / Soil & Tillage Research 55 (2000) 117±126 better preserved if the soil is crumbled (say by a tillage operation) in the condition represented by EII than in that represented by EIV in Fig Conclusions The proposed capillary crumbling model accounts satisfactorily for the mechanics of crumbling in structured agricultural soils where capillary bonding forces are predominant. The fall in inter-aggregate bonding strength (i.e. tensile strength) corresponding to the emptying of pores of a particular size class at a narrow soil suction range occurred as predicted by the model. This took place over a narrow range of soil moisture content. The greater the relative proportion of these pores within the soil, the more prominent will be the fall in the tensile strength of the soil. When tillage and cultivation operations are carried out on relatively wet soils large plastic deformations occur and both the soil structure and integrity of aggregates are undermined. On the other hand, when these operations are carried out on rather dry, baked or cemented soils, much energy is expended and equipment and implements are exposed to high risks of damage. The proposed capillary crumbling model can be exploited in the determination of friable and/or workable conditions in structured agricultural soils. Certainly, efforts to crumble the soil would be optimized by taking advantage of the fall in inter-aggregate bonding strength at a moisture content and soil suction where the risk of damage to the integrity of aggregates is relatively small. In temperate regions, this risk can be further minimized by performing crumbling operations on these soils as they dry from a frozen condition. This should lead to an optimal combination of the energy required to crumble the soil as well as the quality of tilth (soil structure) produced by the crumbling operation. Acknowledgements The rst author wishes to thank Wageningen University for the award of a Postdoctoral Research Fellowship. References Causarano, H., Factors affecting the tensile strength of soil aggregates. Soil Tillage Res. 28, 15±25. Chandler, H.W., Topsoils and cultivations. SPAN 28 (3), 106±107. Kirkham, D., De Boodt, M.F., De Leenheer, L., Modulus of rupture determination on undisturbed soil core samples. Soil Sci. 87, 141±144. Koolen, A.J., Mechanical behaviour of soil by treatment with a curved blade having a small angle of approach. J. Agric. Eng. Res. 17, 355±367. Koolen, A.J., Workability Limits and their Soil Physical Background. PAGV-Verslag No. 64 Lelystadm, The Netherlands, pp. 21±41(in Dutch). Koolen, A.J., Kuipers, H., Soil deformation under compressive forces. In: Larson, W.E., et al. (Eds.), Mechanics and Related Processes in Structured Agricultural Soils. Kluwer Academic Publishers, New York, pp. 37±52. Kutilek, M., Nielsen, D.R., Soil Hydrology. Catena Verlag, Germany. Mullins, C.E., Panayiotopolous, K.P., The strength of unsaturated mixtures of sand and kaolin and the concept of effective stress. J. Soil Sci. 35, 459±468. Panayiotopoulos, K.P., The effect of matric suction on stress± strain relation and strength of three Al sols. Soil Tillage Res. 39, 45±59. Russell, E.W., Soil Conditions and Plant Growth. Longman, London, pp. 479±491. Snyder, V.A., Miller, R.D., Tensile strength of unsaturated soils. Soil Sci. Soc. Am. J. 49, 58±65. Snyder, V.A., Miller, R.D., Soil deformation and fracture under tensile forces. In: Larson, W.E., Blake, G.R., Allmaras, R.R., Voorhees, W.B., Gupta, S.C. (Eds.), Mechanics and Related Processes in Structured Agricultural Soils, NATO ASI Series E, Vol Kluwer Academic Publishers, Dordrecht, pp. 23±35. Towner, G.D., The mechanics of cracking of drying clay. J. Agric. Eng. Res. 36, 115±124.