Water Resources Research

Impact of hydraulic suction history on crack growth mechanics in soil

Authors


Abstract

[1] The mechanics of crack formation and the influence of soil stress history were described using the crack tip opening angle (CTOA) measured with fractography. Two soils were studied: a model soil consisting of 40% Ca-bentonite and 60% fine silica sand and a remolded paddy soil with similar clay content and mineralogy. Fracture testing used deep-notch bend specimens formed by molding soils at the liquid limit into rectangular bars, equilibrating to soil water suction ranging from 5 kPa to 50 kPa (with some 50 kPa specimens wetted to 5 kPa), and inserting a crack 0.4× specimen thickness. Bend tests at a constant displacement rate of 1 mm min−1 provided data on applied force and load point displacement. The growth and geometry of the cracks were quantified from a series of images to determine the CTOA. Modulus of rupture, evaluated from the peak force, increased as water suction increased. However, rewetting did not alter the peak stress from the 50 kPa value, indicating that shrinkage-induced consolidation was more important than the soil water suction at the onset of testing. CTOA measured during stable crack growth decreased with drying. CTOA decreased even further when specimens equilibrated initially to 50 kPa were rewetted to 5 kPa. These results suggested that CTOA was primarily governed by the stiffness, although rewetting probably altered the capillary stresses in advance of the crack tip. Our future work will combine CTOA with a model that couples hydrological and mechanical processes to take into account the dependency of CTOA on the soil water regime so that crack propagation in soil can be predicted.

1. Introduction

[2] The hydrological properties of soil are influenced considerably by changes in pore structure caused by weathering. Of all processes governing changes in hydrology, the formation of cracks probably has the greatest impact. Without taking account of the contribution of cracks, water movement through swelling clay soil is extremely underestimated. Ritchie et al. [1972], for example, revealed that the hydraulic conductivity of swelling clay measured in field basins was 8 times larger on average than those in 73 cm and 21 cm diameter undisturbed cores. A few studies, notably Chertkov and Ravina [2001, 2002] and De Dreuzy et al. [2001] have attempted to relate crack propagation in swelling clay soils with changes in hydraulic conductivity.

[3] There is a strong need in research in water resources to begin to treat the pore structure as a dynamic property. Quantitative models of crack development could be coupled with existing hydrological models that account for cracks, such as those discussed above. This was the goal of research by Renshaw [1996], who simulated the development of cracks in rock on the basis of the available energy for fracture under incremental uniform tensile stress, and related crack development to the hydraulic conductivity. Extending such an approach to soils presents a considerable challenge, however, because the processes driving crack formation are complicated by plasticity.

[4] A major driver of crack propagation in farmlands is water uptake by crop roots, which causes clay shrinkage through decreases in water potential. Under row-planted crops, many early researchers noticed that large linear cracks formed at the center of the interrow [Johnston and Hill, 1944; Johnson, 1962; Fox, 1964]. The processes causing these cracks were described numerically by Yoshida and Adachi [2004], who evaluated the spatial distribution of tensile effective stress in the interrow using a two-dimensional consolidation equation originally developed by Biot [1941]. This approach succeeded in explaining the mechanism of crack initiation by specifying the hydraulic and mechanical boundary conditions, but could not predict the vertical growth of the crack.

[5] Predicting the vertical growth of shrinkage cracks in soil requires the mechanics of crack propagation at the crack tip in addition to the consolidation model developed by Yoshida and Adachi [2004]. Whereas the aforementioned research by Renshaw [1996] on crack growth in rock was able to draw on considerable knowledge of rock fracture mechanics, soil fracture mechanics is poorly understood. Among the studies that have been conducted, Lee et al. [1988] proposed an elastic finite element model for analysis of tensile crack development caused by the construction of embankments or vertical cuts in sloping soil. They assumed that the stress concentration at the crack tip in soil could be described with Linear Elastic Fracture Mechanics (LEFM). Morris et al. [1992] incorporated shear and tensile strength theory for unsaturated soil into LEFM theory to analyze the maximum crack depths in drying clays of tailing ponds. They calculated the vertical distribution of the stress intensity factor, K, caused by the propagation of an unsaturated horizon, and predicted the possible depth of cracks by comparing K and the fracture toughness, Kc (i.e., the stress intensity at fracture). Lima and Grismer [1994] found using an LEFM approach that the elastic strain energy release rate G increased considerably in saline affected soils. Consequently, more energy would be required for cracks to propagate in high-salinity versus low-salinity soil. Nichols and Grismer [1997] measured fracture toughness of clay soil using a three-point bending test and showed that Kc increased with a decrease in water content.

[6] Although these LEFM approaches were very useful for showing links between the crack size and fracture criterion, they have limits in predicting crack propagation quantitatively because the LEFM approach cannot cover the large-scale yielding observed in the cracking of wet clayey soil. Chandler [1984] first introduced nonlinear fracture mechanics to soil. He performed a four-point bend test of plastic clay and granular clay stabilized by cement to illustrate the J integral as the fundamental fracture toughness of soil. The J integral was originally defined as a path-independent contour integral around the crack tip. The value of this integral is shown to be equal to the energy release rate in nonlinear elastic materials. However, Chandler [1984] found that J was not sensitive to differences in soil composition, presumably because this energetic approach was affected too much by plastic energy.

[7] Of various fracture criteria applied to materials, the critical crack tip opening angle (CTOA) or displacement (CTOD) at a specified distance from the crack tip can describe large-scale, stable, ductile crack growth. The CTOA criterion has been successfully applied to numerous structural applications, such as aircraft fuselages, complex structures and pipelines [Newman et al., 2003]. Hallett and Newson [2001] developed an experimental method to evaluate CTOA in wet soil based on the work of Turner and Kolednik [1997]. In a further study, Hallett and Newson [2005] found that CTOA was affected by clay content and salinity. Considerably more research is needed to understand CTOA in soil, however, to apply it to practical problems such as the cracking of desiccating clays. Of particular importance is the dependency of CTOA on soil water suction and the shrinkage-induced consolidation history. Moreover, the approach used by Hallett and Newson [2001] is limited as it assumes that the material behaves as a perfect plastic hinge and requires a correction factor to evaluate CTOA close to the crack tip.

[8] The present study aims to quantify the critical CTOA of clayey soil depending on the shrinkage stress history and the soil water suction at testing. We evaluated CTOA directly from crack morphology. The soils investigated were a paddy soil and a model soil of similar texture and mineralogy. The extension of this work to numerical simulations to predict crack propagation is also discussed.

2. Materials and Methods

2.1. Specimen Formation

[9] Two soils were used: a model soil consisting of a mix of 40% Ca-bentonite (Surrey Finest, Laporte Absorbents, Faringdon, United Kingdom) and 60% fine silica sand (Redhill 110, Hepworth Minerals, Sandbach, United Kingdom, average particle diameter of 110 μm) and a remolded paddy soil with similar clay content and mineralogy. The paddy soil was taken at a lowland paddy field located in Niigata, Japan (37.116°N, 138.271°E), air-dried and sieved to <0.42 mm. The particle size composition, plastic limit (PL) and liquid limit (LL) are provided in Table 1. The soils were wetted to about LL, kneaded in a rectangular tray with a spatula for 10 min, and equilibrated for at least 24 h. Bars were formed by packing the soil in aluminum molds that were lined with an acetate insert, and the dimensions were 140 mm long, 30 mm wide (denoted by B), and 35 mm thick (denoted by W). The water content just before packing is shown in Table 1. The water suction of the rectangular bars of the model soil were equilibrated to 5, 10, 20, and 50 kPa on suction plates (EcoTech, Bonn, Germany) that were attached to a vacuum pump. Half of the 50 kPa specimens were returned to 5 kPa to determine the effect of soil water suction in comparison to shrinkage stress history. The paddy soil was tested subsequently. On the basis of the long sample preparation and testing time experienced for the model soil, fewer replicates were tested because the experimental error associated with the various tests was small. The paddy soil was equilibrated to 12.5, 25, and 50 kPa so that it covered the major influence of soil water suction observed for the model soil.

Table 1. General Properties of Soil Samples
 Particle Size Distribution (Gravimetric %)Water Content Before Hydraulic Consolidation (g kg−1)Liquid Limit (g kg−1)Plastic Limit (g kg−1)
<2 μm2 μm < 20 μm20 μm < 200 μm200 μm <
Bentonite and sand mix35.04.355.35.4662683244
Paddy soil40.036.320.43.3655608330

[10] The equilibrated water contents, number of replicates, void ratio, volumetric shrinkage, and degree of saturation are listed in Table 2. Volumetric shrinkage and degree of saturation were determined using 50 mm diameter by 25 mm height cores, with volume following water equilibration determined by coating the soil with wax and using Archimedes principle. Although the measured degree of saturation does not have a clear trend with suction treatments, it shows that the specimens were almost saturated regardless of the suction treatments. The water content of the specimens decreased linearly with the increasing logarithm of soil water suction (Figure 1). Although the relationship between suction and water content reflects pore size distribution and air entry suction, the present curves should be understood as the “compression curve,” which is conventionally called e-log p (void ratio versus mean stress) curve in soil mechanics because the pores shrank and remained saturated. The slope represents “compressibility” of the soil fabric against isotropic consolidation due to soil water suction.

Figure 1.

Relationship between imposed suction and water content of the specimens. Error bars show the range of standard deviation.

Table 2. Properties of Specimens for Deep-Notch Bending Testsa
 TreatmentNumber of ReplicatesWater Content by Weight at the End of the Tests (g kg−1)Void Ratio (m3 m−3)Degree of Saturation (m3 m−3)Volumetric Shrinkage (m3 m−3)
  • a

    Mean ± standard error.

Bentonite and sand mix5 kPa14606.6 ± 1.11.58 ± 0.030.976 ± 0.0170.070 ± 0.013
10 kPa13582.8 ± 1.01.46 ± 0.010.971 ± 0.0100.113 ± 0.006
20 kPa13533.4 ± 1.41.33 ± 0.010.991 ± 0.0120.163 ± 0.004
50 kPa12464.1 ± 1.71.22 ± 0.010.978 ± 0.0040.190 ± 0.006
5 kPa rewetted.12484.7 ± 1.21.23 ± 0.020.995 ± 0.0130.195 ± 0.012
Paddy soil12.5 kPa6516.9 ± 1.21.32 ± 0.010.993 ± 0.0020.162 ± 0.004
25 kPa5482.7 ± 0.81.29 ± 0.010.951 ± 0.0080.171 ± 0.005
50 kPa5444.5 ± 1.91.16 ± 0.000.993 ± 0.0030.220 ± 0.001

2.2. Deep-Notch Bend Test

[11] The rectangular soil bars were fractured using the three-point bend test, which is one of the most common methods used to determine fracture mechanics parameters of materials. Bending tests were conducted with a mechanical test frame (Model 5544, INSTRON, Massachusetts, United States) fitted with a 5 N load cell accurate to 0.2 mN at maximum load. Just prior to the bending tests, a predefined notch approximately 0.4 times the specimen thickness was inserted at the center of the specimens using a razor blade mounted in a vice to ensure the crack was perpendicular and to minimize differences in crack length between specimens. The apparatus and specimen are shown schematically in Figure 2. The specimens were set on two supports spaced 70 mm apart and subject to bending at a constant compression rate of 1 mm min−1 until the applied force reached 2 N, then unloaded to 0.5 N and then loaded again until the crack propagated through the entire specimen. The applied force and load point displacement were recorded using INSTRON Merlin software. Images of the propagating crack were taken every 3 s with a digital camera (Nikon D100) fitted with a Nikkor 60 mm macro lens, with pictures downloaded automatically using controlling software (Nikon Capture 4). The resolution of the images was 50 pixels per millimeter. The growth and geometry of the cracks were quantified from the series of images. Binary crack images (Figure 3b) were extracted from gray-scale raw images (Figure 3a) through a thresholding process using image analysis software (Image J ver.1.33u, National Institute of Health, United States). The thresholding process judges whether the pixel is on the crack or out of the crack by its gray-scale value. Because the lighting conditions in the laboratory cannot be fixed, the threshold value was required to be adjusted objectively by eye. Furthermore, the existence of small holes or roughness of the surface of the specimens hindered the automatic recognition of the crack. Such obvious “noise” in the images was removed manually.

Figure 2.

Schematic representation of the deep-notch three-point bend test showing the specimen length, L, width, W, thickness, B, crack length, a, and ligament length, b. The specimen is supported on rollers spaced at distance S (S = L/2) and bent by downward displacement at the load point where force is measured.

Figure 3.

Procedure of image analysis. (a) Original raw image of a crack tip. (b) Binary image of the crack tip.

2.3. CTOA Evaluation

[12] The crack opening angle (COA) is an important nonlinear fracture mechanics parameter. The COA measured at the crack tip is the crack tip opening angle (CTOA), which describes crack propagation under large-scale yielding. However, there are various definitions of CTOA, usually based on the methods of experimental measurement used [Lloyd and McClintock, 2003]. The simplest definition of the CTOA αt is

equation image

where d is the width of the crack and x is the distance from the crack tip. The most widely accepted definition measures the opening displacement between the crack walls at a distance of 1 mm behind the instantaneous crack tip location [Lloyd and McClintock, 2003]. Applying this definition to the evaluation of CTOA from the images of crack tips in ideal materials like metals is easy, but the crack walls in soil are rough and this complicates a visual measurement. As a reliable value of d from a single measurement at 1 mm behind the tip was found to be difficult to measure, d was also measured every 5 pixels (= 0.1 mm) to determine how it changed as a function of x. The relationship between d and x within 3.0 mm or 5.0 mm from the most probable tip location was subject to linear regression, and the “averaged” d values were estimated at x = 1.0 mm and 5.0 mm, respectively.

3. Results and Discussion

3.1. Strength and Stiffness of the Soil

[13] Figure 4 illustrates representative loading diagrams at different soil water suctions. The specimens strain hardened once the yield force was exceeded. After the peak force was reached, crack growth initiated and the force declined to a stable level where the crack almost passed through the ligament. The summary of the diagrams for each replicate is shown in Table 3. With increasing soil water suction, the peak force (Pmax) increased while the displacement at the peak force qpeak decreased. The modulus of rupture (MR) is a common approach to describe the resistance of materials to failure [Lima and Grismer, 1994] and was evaluated by

equation image

where Pmax is the peak applied force, S is the span between the supports, a is the length of the predefined notch, and B and W are the width and thickness of the specimen, respectively. MRel represents the maximum tensile stress generated at the outer surface of a linear elastic beam with a thickness of Wa subject to three-point bending. MR can be represented by a more general form by taking the plasticity into account:

equation image

where the work-hardening exponent, n, defines the amount of strain hardening and ranges between zero and one. A rigid-perfectly plastic beam can be represented by setting n as zero, which gives the minimum MR (MRpl). The evaluated minimum (rigid-perfectly plastic) and maximum (linear elastic) MRs are presented in Table 3. If we ignore the stress concentration at the tip of the predefined notch, the real maximum stress ranged between MRpl and MRel.

Figure 4.

Representative loading diagrams for each treatment (clay sand mix).

Table 3. Summary of Crack Resistance Curve (JR Curve) for Each Testa
SampleTreatmentPmax (N)qpeak (mm)MRel (kPa)MRpl (kPa)E′ (MPa)
  • a

    Mean ± standard error.

  • b

    Asterisks ** show significance level of 1%.

  • c

    Letters beside the data show the results of multiple comparisons by Holm's methods. Different letters represents significant difference between the treatments.

Bentonite and sand mix **b********
−5 kPa3.047 ± 0.077 dc0.982 ± 0.049 a22.96 ± 0.72 d15.30 ± 0.48 d2.38 ± 0.07 e
−10 kPa3.245 ± 0.188 d0.684 ± 0.048 b24.17 ± 1.23 c16.11 ± 0.82 c3.32 ± 0.08 d
−20 kPa4.643 ± 0.114 c0.636 ± 0.024 bc37.55 ± 0.89 b25.04 ± 0.59 b5.63 ± 0.19 c
−50 kPa6.086 ± 0.128 a0.524 ± 0.026 cd47.11 ± 0.76 a31.41 ± 0.51a7.06 ± 0.31 a
−5 kPa rewetted5.595 ± 0.185 b0.480 ± 0.021 d46.30 ± 1.54 a30.87 ± 1.02 a6.45 ± 0.25 b
Paddy soil **ns******
−12.5 kPa2.379 ± 0.068 c0.697 ± 0.06519.67 ± 0.86 c13.11 ± 0.58 b4.79 ± 0.33 a
−25 kPa4.063 ± 0.115 b0.556 ± 0.03937.32 ± 1.05 b24.88 ± 0.70 b8.06 ± 0.31 b
−50 kPa6.092 ± 0.254 a0.524 ± 0.03060.71 ± 1.20 a40.47 ± 0.80 a10.96 ± 0.68 c

[14] MR appeared to be influenced by water suction, suggesting that traditional approaches from unsaturated soil mechanics could prove useful in describing critical crack growth conditions. In their development of unsaturated soil mechanics, Fredlund and Rahardjo [1993] described the stress state of unsaturated soil using the combination of two state variables. These were the difference between total stress and pore air pressure (σua) and the difference between pore air pressure and pore water pressure (uauw). They presented a rough estimation of crack depth by assuming constitutive relationships for the soil structures in elasticity form. Although Fredlund and Rahardjo's [1993] approach is relevant and theoretically robust, the critical conditions for tensile failure cannot be described by a single parameter. Effective stress is a single-valued stress state variable which governs all mechanical aspects of saturated soil. Bishop [1959] gained widespread reference among the studies on extension of effective stress in unsaturated soil:

equation image

where σef is effective stress which represents interparticle stress governing any deformation of soil fabric and χ is a nondimensional parameter that accounts for influence of air-filled pore space. In soil mechanics it is generally assumed that soil is unable to withstand tensile (effective) stress and this requires that the minor principal stress cannot be less than zero [Hettiaratchi and O'Callaghan, 1980]. This gives a simple critical condition for tensile failure. According to Yoshida and Adachi [2001], the critical minor principal stress in saturated clay soil for cracking was estimated as small as −0.1 kPa (tensile). Thus it is plausible that interparticle bonding contributes little to crack extension resistance. On the basis of these observations, σef and ua would be zero, with equation (4) reducing to

equation image

[15] For saturated soil, χ is 1. Thus tensile stress is expected to be equal to soil water suction at tensile failure. Because MRs in equations (2) and (3) are total tensile stress at failure, we can replace σ in equation (5) by −MR:

equation image

where ψ denotes soil water suction (−uw). Table 3 shows equation (6) does not predict MR. It appears that the bentonite-sand mix specimens equilibrated to −50 kPa failed at a smaller value than equation (6) would predict, probably because of a stress concentration at the crack tip. In other specimens, the soil water suction at the crack tip probably changed during bending, causing the unexpectedly higher MR values than equation (6) would predict. Crack propagation was likely influenced by the following processes, which would vary in importance depending on the soil stress history and stiffness: (1) change in matric potential at the crack tip from the equilibrated value during crack propagation, (2) variation in σef with matric potential, and (3) stress concentration at the crack tip. MR cannot be applied in the quantitative estimation of the critical condition because it does not account for changes in the internal pore structure and the propagation of already existing cracks [Nichols and Grismer, 1997]. MR also depends on the geometry of the specimen, such as the initial crack length or testing methods. A more rigorous approach that describes the fracture mechanics of a material appropriately should be used.

[16] Young's modulus E is a measure of the stiffness of a material and can be evaluated from the unloading line of the tests, measured before crack propagation:

equation image

Δqunload and ΔPunload denote the load point displacement and decrease in applied force due to unloading, respectively. ZLL denotes the nondimensional load line compliance, which is a function of S, W and a. The formula to evaluate ZLL is given by Anderson [2005]. The ν represents the Poisson ratio, which we assumed to be 0.5, because soil volume does not change during undrained deformation as the present tests. Young's modulus increased significantly as soil water suction increased, but decreased a little by rewetting.

[17] These results indicated that the degree of shrinkage-induced consolidation rather than the current soil water suction was a major factor determining the strength and stiffness of the soil. In other words, porosity was the major determinant of the mechanical properties of the specimens prior to the onset of crack propagation.

3.2. Energy Dissipation by Stable Crack Growth

[18] Crack growth becomes stable or unstable depending on the balance between the crack driving force and the crack growth resistance. The driving force, or energy rate available for dissipation that is stored in the material, C, was defined by Turner and Kolednik [1997] as

equation image

where U is work and wel is recoverable strain energy. The critical condition for stable crack growth occurs when C equals and remains at the energy required for crack extension D. The parameter D is the global energy dissipation rate and includes the energy dissipated because of both crack formation and plastic processes per unit area of crack extension.

[19] Turner and Kolednik [1997] argued persuasively of the merits of D as a physically meaningful parameter. However, its application requires quantifying dwel, which is complicated using loading versus crack extension data alone. A more general description of crack growth, which does not segregate between elastic and plastic energy, is the J integral. In its simplest definition it is computed from the area under the loading diagram, U, as

equation image

where B is the thickness of the specimen, η is a dimensionless constant, and b is the current ligament length. For pure bending in specimens with deep cracks, η = 2. J evaluated by equation (9) is the energy release rate of elastic plastic materials up to the critical condition when the energy release from crack extension causes unloading. When this critical condition exists, equation (9) is adapted to calculate the energy absorbed by the specimen as

equation image

where P is the force, q is the load point displacement, and subscript i denotes the value of the ith incremental step. The subscript R pertains to resistance to crack growth after initiation by the material. Figure 5a provides an example for soil of the change in JR with crack extension, which is commonly referred to as a J-R curve in materials science. The curve is characterized by an initial steep rise followed by a linear slope and final upward deviation. The first steep rise represents the blunting stage of the predefined notch. The force applied to the specimen was still rising in this stage (Figure 5b). The second linear rise corresponds to stable crack growth, where the force rapidly decreased. The third part represents the crack almost reaching the loading point, which is the limit of sample compliance for testing.

Figure 5.

Relationship between (a) JR (•) and crack tip opening angle (CTOA) (×) with crack extension and (b) the corresponding points on loading diagram. CTOA labeled as (+) is not in the region of stable-ductile behavior.

[20] For all soil treatments investigated, J-R curves had the first and second part. The slope of the second linear part and its intercept to the vertical axis for all tests are summarized in Table 4. The slopes reflect the energy absorbed during the stable extension of a crack by unit length, and the intercepts represent rough estimates of critical J for initiation of crack growth. For the model soil, neither parameter showed clear trends to the increasing suction treatment. The paddy soil showed higher intercepts and slopes of JR at greater suction. The dJR/da decreased significantly following rewetting from 50 kPa to 5 kPa suction. These results generally showed that energy based parameters suffered from the high plasticity. Chandler [1984] also found J-R analysis to be insensitive to compaction energy, which like suction alters particle packing and effective stress conditions considerably.

Table 4. Summary of Crack Resistance Curve (JR Curve) for Each Testa
SampleTreatmentdJR/da (J m−3)Jinter (J m−2)
  • a

    Mean ± standard error.

  • b

    Asterisks * and ** show significance level of 5% and 1%, respectively.

  • c

    Letters beside the data show the results of multiple comparisons by Holm's methods. Different letters represents significant difference between the treatments.

Bentonite and sand mix **b**
−5 kPa0.153 ± 0.014 bcc6.362 ± 0.366 a
−10 kPa0.110 ± 0.009 c4.402 ± 0.554 b
−20 kPa0.207 ± 0.010 a5.355 ± 0.325 ab
−50 kPa0.191 ± 0.012 ab4.900 ± 0.285 ab
−5 kPa rewetted0.134 ± 0.010 c4.234 ± 0.226 b
Paddy soil **
−12.5 kPa0.272 ± 0.041 b3.952 ± 0.457 b
−25 kPa0.363 ± 0.012 ab4.081 ± 1.007 b
−50 kPa0.396 ± 0.029 a6.910 ± 0.505 a

[21] Samples rewetted to 5 kPa from 50 kPa exhibited interesting behavior (Table 4). The crack initiation condition was similar to the 50 kPa specimens, but the energy absorption with crack extension was similar to the 5 kPa specimens. Reaching the initiation conditions depends heavily on material stiffness, which was similar between the 50 kPa and rewetted specimens. The dJR/da depends on plasticity and bond energy, with the latter influenced by suction, which was the same for rewetted and 5 kPa specimens. However, there were no clear trends at other suctions and differences between elastic and plastic energy during loading would be required to assess these relationships properly. This requires further study as does decreases in compliance during crack growth.

3.3. CTOA as the Toughness Parameter of Soil With Propagating Cracks

[22] Global measurements of crack growth, such as J described previously, could have limited application in wet soil because large-scale plasticity will dominate the result. Measurements more local to the crack tip might provide a better description of the critical conditions required for crack extension. CTOA concentrates on crack tip processes and has been shown by Turner and Kolednik [1997] to be related to D.

[23] Earlier studies on metals showed that CTOA was nearly constant from crack initiation, but two-dimensional FEM analyses showed that CTOA at crack initiation was larger than the value needed for stable crack growth [Newman et al., 2003]. Rice et al. [1980] postulated that crack growth occurs at a critical opening displacement d, at a distance x behind the crack tip.

equation image

where α, β and R are constant. E and σ0 denote Young's modulus and yield strength, respectively. Equation (11) predicts that the CTOA should be constant with constant dJ/da. Figure 5a shows a correspondence of JR curve to measured CTOA values. Variation in CTOA values within this range was small enough to regard them as constant. It is plausible that the second linear part of JR corresponds to the stable crack growth stage. Thus we regard the CTOA values averaged through the second linear section as the CTOA of stable crack growth.

[24] Table 5 provides CTOA averaged through the stable crack growth stage. The CTOA at 5 mm from the crack tip, represented by αt5, decreased significantly with increasing soil water suction. However, the differences between −20 kPa and −50 kpa in the bentonite-sand mix and between −25 kPa and −50 kPa in the paddy soil were too small to be statistically significant. The CTOA measured at 1 mm, represented by αt1, decreased significantly from −10 kPa to −20 kPa. As stated in the previous section, MR (strength) and E′ (stiffness) decreased slightly by rewetting. These decreases can be attributed to the small recovery of water content and void ratio. However, it should be noted that the rewetting treatment caused a further significant reduction in CTOA regardless of the measurement location.

Table 5. Crack Tip Opening Angle at 5 mm, αt5 and 1 mm, αt1 From the Crack Tip at Steady State Crack Growth Bend Testsa
SampleTreatmentαt5 radαt1 radκ15b
  • a

    Mean ± standard error.

  • b

    The coefficient κ15 is the ratio of αt5 to αt1.

  • c

    Asterisks ** show significance level of 1%.

  • d

    Letters beside the data show the results of multiple comparisons by Holm's methods. Different letters represents significant difference between the treatments.

Bentonite and sand mix **c** 
−5 kPa0.0422 ± 0.0014 ad0.0748 ± 0.0028 a0.56
−10 kPa0.0324 ± 0.0019 b0.0737 ± 0.0038 a0.43
−20 kPa0.0238 ± 0.0006 c0.0529 ± 0.0028 b0.45
−50 kPa0.0244 ± 0.0010 c0.0553 ± 0.0029 b0.44
−5 kPa rewetted.0.0180 ± 0.0015 d0.0393 ± 0.0024 c0.46
Paddy soil **** 
−12.5 kPa0.0468 ± 0.0045 a0.1052 ± 0.0083 a0.44
−25 kPa0.0319 ± 0.0002 b0.0585 ± 0.0016 b0.55
−50 kPa0.0260 ± 0.0015 b0.0545 ± 0.0054 b0.48

[25] Within small-scale yielding, the elastic stress field around the crack tip determines the crack propagation, which is the basis of linear elastic fracture mechanics [Rice et al., 1980]. In small-scale yielding, J controls fracture and equation (11) enables us to use both J and CTOA as equivalent parameters. However, in fully plastic conditions, J does not describe fracture well, and the J-CTOA relationship (Figure 5b) may depend on the size and on configuration of the test.

[26] Under conditions of large-scale yielding, such as crack propagation in wet soil, fracture is determined not only by the stress field but by the strain concentration. The latter property is influenced considerably by the stress history, which causes irreversible global plastic deformation of soil. Therefore, the criterion for a ductile fracture should be represented by the combination of the stress and the strain state. Although the application of CTOA to the large-scale yielding case does not have a perfect theoretical basis [Turner and Kolednik, 1997], CTOA dexterously represents both the stress and strain field around the crack tip. On the one hand, as the soil experiences greater water suction than it ever has had previously, stiffness increases irreversibly because soil water suction causes irreversible consolidation of the soil fabric and a less deformable soil structure. On the other hand, soil water suction reinforces the soil fabric against shear or tensile stress during the bending test.

[27] The CTOA should be smaller for a stiff material, which exhibits small strain against the stress, and greater for a strong material, which bears large stress. Because the stiffness and strength simultaneously increase with increasing soil water suction, the dependency of CTOA on soil water suction cannot be easily predicted without performing actual tests. The comparison between the −50 kPa specimen and specimens rewetted to −5 kPa showed that the soil fabric was reinforced with soil water suction against rupture at the crack tip because differences in void ratio and water content between these treatments were minimal. It should be noted, however, that the water suction at the crack tip, which probably controlled fracture, is not necessarily same as the global or preimposed soil water suction.

[28] Other properties of soil have been shown to influence CTOA considerably. Hallett and Newson [2005] indicated that the addition of 20% sand to pure kaolinite led to a 40% reduction in CTOA. They also showed that CTOA is much affected by composition of clay mineral. Thus, the texture of soil and mineralogy must have a substantial effect on critical CTOA. In our tests, two soils were investigated with similar clay content and mineralogy. CTOA was similar between these two soils when tested at the same soil water suction, whereas other mechanical properties such as MR and E′ were very different. It appears that the critical CTOA was not very sensitive to small differences in mechanical properties when the clay content and major clay mineral were the same. Therefore, the collection of CTOA data for the representative specimens is expected to contribute to the practical application of CTOA for soil in which CTOA has not actually been measured.

3.4. Technical Issues for Future Applications

[29] CTOA can vary across the specimen width because the crack tip propagates faster in the center at an early stage of crack growth because of plastic constraint [Lloyd and McClintock, 2003]. This phenomenon is called tunneling. All of the aforementioned methods rely on measurement at the outer surface of the test specimen to infer some through-thickness average value of CTOA. To avoid tunneling, side grooving is often used in fracture tests of metals, but side grooving was found to be difficult to apply to wet soil so it was not used in our study. This technical problem is one of the difficult subjects that must be addressed in a future study on accurate measurement of the fracture mechanics parameters of soil.

[30] Turner and Kolednik [1997] relate “global CTOA, αgpl” with the load point displacement and advancing location of the crack tip assuming a rigid plastic hinge model. Although Hallett and Newson [2001, 2005] introduced this indirect method to evaluate the CTOA of soil, it requires adequate correction, especially for very wet specimens because the measured load point displacement includes considerable error due to wedging in of the loading bar into the specimen. This problem may also affect estimation of the J integral from the loading diagram. Because the dJR/da is evaluated in the latter stage of bending, the influence of associated overestimation due to “wedging in” on the judgment of crack growth stability was negligible. However, if the J integral is applied as the critical condition for rupture, another testing approach or correction will be required. Recent work by Nishimura [2006] on a direct tensile device for clay specimens is among the alternative methods being developed to measure the fracture mechanics parameter of very soft soil.

[31] To describe fracture mechanics, the CTOA should be measured as close to the tip as possible, because the local state of deformation can be the determinant of crack propagation. The CTOA evaluated from the crack opening displacement at 5 mm is a global rather than local measure. However, these microscale measurements close to the crack tip require much greater image resolution. The expected difference in the crack width near the tip due to the suction treatment is of the order of 10−2 mm, which requires at least 100 × 100 pixel/mm of resolution for the image. Moreover, the shape of the crack tip in soil is much rougher than metals because of the larger dimension of the particle structure. Therefore, microscopic observation does not necessarily result in an accurate estimation of the crack tip geometry. Turner and Kolednik [1997] related the above mentioned global CTOA αgpl with the local CTOA αt as follows:

equation image

where κ denotes the undetermined coefficient relating the global to local CTOA. Turner [1995] and Turner and Kolednik [1997] explained that the discrepancy between the local and global CTOA was mainly caused by the elastic component of angular deformation. By replacing αgpl and αt with αt5 and αt1, we can define κ15 as a coefficient bridging αt5 and αt1. The κ15s evaluated in this definition are provided in Table 5. The coefficient κ15 ranged between 0.45 and 0.64, without much variability due to treatment except for −5 kPa in the bentonite-sand mix. The nearly constant κ15 suggests that CTOA measured slightly distant from the tip can still reflect the degree of local deformation at the crack tip. Any error due to the quality of the crack tip images does not depend on the length to be measured, so the error is greater if the measured opening is smaller. Therefore, αt5 is much less susceptible to the quality of the images than αt1.

[32] In the simulation of crack propagation using a conventional finite element model with zero tip opening, the current CTOA value was evaluated by the crack opening displacement:

equation image

where δt−1 denotes the opening stretch as represented at one element spacing before the actual tip, and h denotes the element spacing. It is obvious that the location at which the critical CTOA is determined should be consistent with h. Turner [1995] used a 0.4 mm element spacing for analysis of the ductile crack growth in metal taking into account the continual crack growth step due to the microvoid coalescence. Lee et al. [1988] constructed a finite element model using 0.6 m to 3.0 m meshed elements for analysis of tensile crack development in the embankment on the soft clay. Because Lee et al. [1988] did not use CTOA but the stress intensity factor as the criterion of fracture, the element size was not required to be in the scale of the local crack tip deformation. If CTOA is applied in such a large-scale model, the element spacing along the propagating crack would be too small, and therefore whole meshes could not be simulated without any complicated treatment. When CTOA is measured slightly distant from the tip, this measurement can possibly be used as the criterion for ductile crack propagation in a field-scale model.

[33] An advantage of CTOA over the J integral is the transferability to a coupled hydraulic and mechanical model. The conventional deformation models use a displacement vector as the primary variable [Yoshida and Adachi, 2004]. The judgment of crack growth can be done directly by comparing the current and critical opening angle close to the crack tip without referring to a complicated stress-strain relationship. Our future work will combine CTOA with coupled hydrological and mechanical models of wet soil to predict vertical crack growth. The dependency of CTOA on the water regime will be taken into account.

4. Conclusions

[34] The crack tip opening angle (CTOA), which is a promising nonlinear fracture mechanics parameter for predicting stable crack growth in wet soil, can be determined by direct measurements of crack images. The crack opening angle measured at 5 mm behind the crack tip provides the state of local deformation at the crack tip without being disturbed by the roughness of the crack walls.

[35] CTOA during stable crack growth in saturated clayey soil depends both on the highest imposed soil water suction and the current soil water suction. The preconsolidating soil water suction makes a stiffer soil structure, resulting in lower CTOA for stable crack propagation. The soil water suction also reinforces the soil fabric against tensile stress. A decrease in soil water suction due to rewetting reduces CTOA for stable crack growth. The simulation model of crack propagation should take into account this dependency of CTOA on the soil water regime. J integral analysis, which is used commonly to describe the fracture of elastic-plastic materials, was not sensitive to the large differences in soil suction used in this study.

Acknowledgments

[36] We thank Dennis Gordon for his help with the specimen preparation. Roy Neilson and Charlie Scrimgeour are thanked for their helpful comments on the manuscript. This international collaborative research study was funded by fellowships from the National Agriculture and Food Research Organization, Japan, a grant-in-aid for scientific research from the Japan Society for the Promotion of Science, and a grant-in-aid from the Scottish Government to the Scottish Crop Research Institute.

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