Soil Science Society of America Journal 65:1083-1088 (2001)
© 2001 Soil Science Society of America
DIVISION S-1—NOTES
A simple fracture mechanics approach for assessing ductile crack growth in soil
P. D. Hallett* and
T. A. Newson
Dep. of Civil Engineering, Univ. of Dundee, Dundee, DD1 4HN, Scotland
* Corresponding author (p.hallett{at}scri.sari.ac.uk)
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ABSTRACT
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The weak understanding of crack growth mechanisms in ductile soil is addressed by testing a new fracture mechanics approach. Samples are fractured using a deep-notch (3-point) bend test, with data on sample bending, crack growth, and crack mouth opening collected to assess the crack opening angle (COA), the crack tip opening angle (CTOA), and plastic energy dissipation rate (Dpl). The test variables are clay and salinity content, with samples formed from mixtures of kaolinite and fine sand. The CTOA and Dpl detect differences in fracture mechanics due to clay, but not salinity. The energy needed to drive crack extension, Dpl, is one order of magnitude higher for samples containing a ratio of sand to kaolinite of 75:25, as compared with 50:50. However, the CTOA due to plasticity was 0.19 and 0.24 for the same samples respectively, indicating that more strain is needed for crack growth in the specimens with more clay.
Abbreviations: COA, crack opening angle over its entire length • COD, crack opening displacement at its mouth • CTOA, crack tip opening angle
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INTRODUCTION
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MODELING SOIL STRUCTURE and its temporal nature could be improved significantly if the mechanics of crack development were included (Moran and Kirby, 2000). Research in this area, however, in comparison with other aspects of soil physical behavior, has been minimal despite its obvious significance as a basic physical characteristic of soil. Over the past decade, there has been a surge in pore structure modeling research (Perrier et al., 1995). Some of the investment in these studies, however, needs to be redirected so that soil structure modeling can be backed with a firm understanding of the origin of porosity through cracking.
Cracks originate in soil when the strain energy imposed by shrinking and swelling or tillage is sufficient to break interparticle bonds (Raats, 1984). Physically based models of soil cracking describe this phenomenon using techniques based on Griffith's (1920) pioneering work on the fracture mechanics of ideal linear elastic materials (Snyder and Miller, 1985; Lima and Grismer, 1994; Ayad et al., 1997). This approach describes the thermodynamic conditions required for catastrophic fracture. Evaluating the parameters needed for Griffith's theory is complicated for soil, hence most research has used the Irwin-Orowan extension to the model, which is stress rather than energy based (Lima and Grismer, 1994; Morris et al., 1992; Hallett et al., 1995). Either approach may be applicable to dry brittle soil, but they do not account sufficiently for plasticity in wet soil (Hallett, 1996). Like metals and other materials, plasticity can be a dominant sink to imposed strain energy in soil (Hatibu and Hettiaratchi, 1993). These types of materials are classified as ‘ductile’, as opposed to ‘brittle’, because fracture is accompanied by considerable plastic deformation.
Cracking can initiate in wet ductile soil as a means to relieve the strain imposed by shrinking clays (Chertkov and Ravina, 1998). In this condition, the ability of primary soil particles to rearrange in the stress field at the crack tip probably induces material toughness. The term ‘toughness’ is used in materials science to describe the deviation of a material from Griffith's (1920) ideal condition where fracture energy, G, is dependent only on crack surface energy (Lawn, 1993). A tough material may have the same strength as a brittle one, but requires far more energy to fracture. Tillage is an excellent example of this process in practice, although only Chandler (1984) has discussed soil fragmentation using these principles. Wet soil is far weaker than dry soil, but tillage is done at the point where the combination of crack surface energy (i.e., interparticle bonds) and energy dissipation through other processes (i.e., plasticity) are about the lowest. Most fracture mechanics models applied to soil do not account for toughness (Snyder and Miller, 1985; Lima and Grismer, 1994).
A notable exception was Chandler's (1984) research. He used J-integral analysis, which provides a robust description of crack propagation that includes plasticity. It is unfortunate that this excellent research was not pursued. Part of the problem was Chandler's (1984) finding that J-integral analysis produced similar results regardless of the soil treatment. Another problem is that obtaining the data necessary for J-integral analysis is quite difficult, although Chandler (1984) presented the necessary techniques.
Researchers in fracture mechanics such as Turner and Kolednik (1994) argue that whilst J-integral analysis may be the best way to describe crack initiation, its use may not be applicable after subsequent stable crack growth. They suggested that crack growth resistance be described by its energy resistance rate, D. This approach is theoretically sound, degrades to the classical J-contour integral, J, at the point of crack initiation, and degrades to G for linear elastic materials. The difficulty with applying the approach to soil is measuring the necessary parameters.
An alternative, and much simpler approach to understand, is the crack opening displacement, COD (Lawn, 1993). If a crack is forced open, it will eventually grow, which is a property commonly observed in drying soil (Raats, 1984). Crack opening displacement was applied to stiff soils by Sture et al. (1999), who found that catastrophic fracture occurs at a consistent amount of crack opening for a specific soil. Crack opening displacement is often criticized on theoretical grounds for its empiricism, and it fails to predict ductile growth adequately (Turner and Kolednik, 1994). Its use is widespread, however, because it is easy to get the necessary parameters and it has given highly successful results in predicting material integrity. Crack opening displacement can also be used in finite element modeling to characterize the strain-dependent fracture of materials.
An ideal testing approach to describe crack growth in soil needs to (i) account for plasticity, (ii) be easy to conduct, and (iii) provide parameters that are theoretically sound. A recent approach presented by Turner and Kolednik (1997) may offer the solution. They show how a simple deep-notch (3-point) bend test can be used to assess D and the CTOA. Unlike COD, which considers global crack properties, CTOA considers the condition at the crack tip required for stable ductile crack growth. This is more applicable to ductile crack growth.
In this note, we adapt this approach to assess the fracture mechanics of ductile soil. Deep-notch bend specimens were formed from mixtures of sand and kaolinite. The experimental treatments investigated were clay content and salinity. The water content of the test specimens was maintained to give a ductile consistency at the 200-kPa postconsolidation level. Video-analysis during fracture was used to provide a measurement of crack opening and elongation at different levels of strain.
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Fracture Test
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A modified version of the standard 3-point bend test was used to obtain the data necessary for the approach that will be presented. With this procedure, a rectangular bar sample is flexed across three load points to provide the strain energy to drive crack elongation. An initial crack of specified length, 
0, is often intentionally inserted into the center of the specimen to act as a zone of failure initiation. Figure 1 illustrates a typical test specimen and the symbols used to refer to its dimensions.
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Fig. 1. Diagram of a Deep-Notch (3-Point) Bend Specimen used to assess ductile crack growth in soil. The specimen consists of a rectangular test piece of soil with a crack of length a intentionally inserted. Ligament length (b), span distance between rollers (S), and width of test piece (W) are also represented.
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Excess plastic deformation at the load-points and failure under self-weight limits the application of the 3-point bend test to ductile soils. Furthermore, unless the crack is sufficiently large at the onset of testing, there is a tendency for the crack to deviate during loading because of the presence of anisotropic stress fields. These problems were overcome by inserting a large preformed crack into the specimen (0/W = 0.5), as suggested by Turner and Kolednik (1997), and by supporting the specimen on glass slides to distribute the load and remove self-weight (Fig. 1). By increasing the size of the crack, the strain field generated at the crack-tip in relation to the applied force becomes larger. This reduces the influence of compressive stress from the upper loading point.
At the onset of testing, the specimen is balanced against self-weight on the two lower glass slides and load-points as illustrated in Fig. 1. During testing, the upper load point is lowered, causing the sample to flex and the crack mouth to open. The lower load points were rollers that allowed for the glass-slides to move freely during loading, thus lowering any influence friction may have.
The 3-point bend tests were conducted using an INSTRON (Canton, MA) mechanical test frame (Model 5544). The cross-head displacement was recorded using Merlin software supplied with the mechanical test frame. Crack opening and elongation were obtained from video images that were captured to a computer at 5-s intervals. For many materials, strain gauges and extensiometers are fixed to the crack to measure opening and elongation. This is not possible with ductile soil because the samples are too weak. Another common practice is to deduce crack opening and elongation from cyclic unloading and reloading. Ductile soil samples have a tendency to rebond during unloading, so the direct analysis using video images is more appropriate. To minimize viscous effects, and to allow for monitoring of crack propagation, the displacement rate was 1 mm min-1, as suggested by Hallett et al. (1998).
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Sample Properties
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The soils tested were formed from mixtures of silica sand (63- to 125-µm particle diameter) and Kaolinite clay (Supreme Ultrafine China Clay, ECC International, Cornwall, UK). Properties of these two materials can be found in Hallett et al. (1995). The treatments were selected to investigate the effect of clay content and salinity on ductile crack propagation. The mixes used were 75:25 and 50:50 sand to kaolinite. All samples were formed at the 200 kPa postconsolidation water content. This is equivalent to 26.8 and 24.0 g 100 g-1 water content for the 75:25 and 50:50 sand to kaolinite mixes, respectively. The saline treatment was formed using water mixed with 30 mg kg-1 NaCl. Yield stress and Young's Modulus were evaluated from a standard triaxial test. The Poisson's Ratio is assumed to be 0.5 for our samples.
The aggregation of clay to sand was improved by adding water first to the sand, and then mixing in the clay gradually. The mixture was sealed and left for at least 48 h before sample formation to allow for water equilibration. Mechanical test specimens were formed by packing the soil samples into polytetrafluoroethylene-lined molds. The sample dimensions each had a 90-mm length, 20-mm width (W), and 20-mm thickness. There was a vertical slot in the center of the sample mold in which a razor blade was inserted to form an 0 of 
10 mm. The molded samples were sealed and left for a further 48 h, and then the mold was disassembled so that the sample could be tested. The water content after testing was measured for all samples to ensure that it did not deviate from initial conditions. Ten replicate samples were tested for each treatment. Figure 1 illustrates the testing procedure of the samples. The span distance between the loading rollers, S, was 80 mm.
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Statistical Analyses of Data
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Treatment differences for the various mechanical properties presented subsequently were assessed with an ANOVA test (Genstat 5 Committee, 1993) with P referring to the probabilities for the variance ratios.
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ASSESSING THE TEST DATA
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Figure 2 shows a series of video images of a sample being tested. It is apparent that the crack mouth opened under imposed loading, and this translates into strain at the crack tip that drives crack extension. The combination of video analysis and a computer-controlled loading frame allowed for the following data to be collected: (i) cross-head displacement, (ii) COD, and (iii) crack elongation.
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Fig. 2. Video images of a 3-point bend test specimen of the 50:50 sand to kaolinite soil being tested. The images show different levels of strain applied by flexing the sample. The numbers on the scale represent 10 mm increments.
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These data are easy to obtain and can be used directly to characterize ductile crack growth using nonlinear fracture mechanics. Sture et al. (1999) reported on the usefulness of COD analysis for assessing the fracture mechanics of cemented soils. This is a measure of the opening distance of the crack mouth at the initiation of crack growth. Crack opening displacement analysis is suitable for highly cemented soils where the energy input up to crack initiation is sufficient to cause catastrophic fracture. In our tests, crack growth is slow because of the high ductility of the wet clay soils being examined. Determining COD at the point of crack initiation is difficult because a large amount of bending strain is required. It would not be descriptive of ductile fracture because crack extension would cease without the input of additional strain energy.
Ductile fracture requires a continuous source of energy input for the crack to elongate. By combining COD and crack extension data, the COA can be assessed. Its derivation and measurement are simple from the plastic whole crack mouth opening (Vpl), and crack length (a), as
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This measurement considers the global behavior of the entire crack path. It omits the bending nature of the test specimen and how this influences the strain field locally around the crack tip.
A more rigorous analysis of ductile crack growth in materials was provided by Turner and Kolednik (1997). Their approach recognizes that the driving force behind crack growth is the localized strain field near to the crack tip. On the basis of an earlier theoretical description of ductile crack growth (Turner and Kolednik, 1994), they described how a global measure of the CTOA could be assessed using 3-point bend specimens (Turner and Kolednik, 1997). In a 3-point bend specimen, the global CTOA measurement due to plasticity, which will be denoted g,pl, depends on the S, the length of material that is being flexed (b), and the load-point displacement (q). By describing bending as a fully plastic hinge and taking into account the geometry of the test sample, g,pl/rpl is evaluated as
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where rpl defines the distance ahead of the crack tip to the instantaneous center of rotation of the arms of the piece. The negative slope of the graph shown in Fig. 3 provides the ratio g,pl/rpl. This relationship is linear in the plastic regime of displacement which occurs during steady-state crack growth, hence the subscript pl. At the onset of bending, a transient regime may occur. For both previous tests on metals (Turner and Kolednik, 1997) and our tests on soils, however, it has a negligible influence on the slope shown in Fig. 3, and can therefore be included in the analysis. The r2 value for the slope was typically >0.90. The ratio g,pl/rpl for the samples tested are summarized in Table 1.
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Fig. 3. The relationship between the plastic load point displacement, qpl and ligament length, ln(b) for different sand:clay deep-notch bend specimens. The lines are from linear regression analysis and the slopes are used to assess parameters in Eq. [2]. The data shown are from a representative sample for each mix.
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The rpl is evaluated from the relationship between Vpl and plastic load-point displacement (qpl) as
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where b0 is the initial value of the ligament lengths. Figure 4 illustrates the linear relationship between Vpl and qpl that was observed for the soil samples that were tested. The lowest r2 value for the slope was 0.90. By combining the results of Eq. [2] and [3], g,pl can be evaluated. Table 1 lists all of the fracture mechanics parameters obtained. The g,pl parameter is evaluated independently for each sample tested rather than using the average values of g,pl/rpl and rpl presented in Table 1. As a result, the values for g,pl are slightly different than what would be expected from multiplying g,pl/rpl by rpl from the average values.
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Fig. 4. The relationship between the plastic whole crack mouth opening, Vpl, and plastic load-point displacement qpl for the different sand:clay deep-notch bend specimens. The lines are from linear regression analysis and the slopes are used to assess parameters in Eq. [3]. The data shown are from a representative sample for each mix.
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These fracture mechanics parameters are useful for describing crack growth in ductile soil. With COA analysis, crack propagation is assumed to occur because of the strain-field induced by opening the crack mouth. On the basis of an ANOVA test, COA is similar for our tests regardless of clay content (P = 0.30) or salinity (P = 0.87). This measurement, however, does not consider how the imposed bending strain translates into crack extension and crack tip opening.
Equations [2] and [3] use measurements of crack growth, crack mouth opening, and cross-head displacement (i.e. bending) to evaluate a number of descriptive parameters. For a fully plastic sample, the parameter g,pl defines the angle at which the crack tip must be open due to plastic deformation for crack extension to occur. Our results suggest that the samples containing a lower clay content will crack at lower levels of crack tip opening angles (P <0.05). This would be expected, since Young's Modulus measurements indicate that these samples are more rigid (Table 1). Salinity does not affect g,pl (P = 0.75).
This study is the first to report g,pl values for soil. From a practical standpoint, g,pl defines the level of plastic strain at the crack tip that is required for crack extension. Under a specific level of mechanical strain, a soil with a lower g,pl would be more susceptible to cracking. A major benefit in obtaining this measurement is that with further computational analysis, it can be used in standard finite element modeling packages (e.g., ABAQUS). It has also been used to investigate critical conditions of crack propagation from spectrophotogrammetric measurements of fractured specimens (Kolednik and Stüwe, 1985). This may offer a powerful tool for evaluating the genesis of soil structure from observations of crack tip properties.
The analysis method presented also provides useful data for assessing the energy criterion for ductile fracture in soil. The ratio g,pl/rpl is determined by the amount of ductile crack growth that occurs for a given amount of mechanical strain, in this case caused by the bending of the test specimen. It can be combined with other material properties to assess the Dpl by the relationship
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(Turner and Kolednik, 1997). The parameter 
is the uniaxial yield stress, and L is the classical plastic constraint factor (1.57 for a/W 0.5 in plane strain, W = width of test piece). The results for the samples with different clay content are listed in Table 1. They are evaluated independently for each sample rather than using the average values of the other parameters listed in Table 1. They are reflective of the lower yield stress required to impart mechanical strain to the samples containing higher clay concentrations.
Equation [4] considers only the plastic energy dissipation. For a highly ductile sample like wet soil, elastic energy dissipation probably has minimal influence on the total D, so DDpl. Drier soil will behave as an elastic–plastic material, with D evaluated by
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where the elastic energy dissipation per unit crack growth, Del, is the same as the mechanical energy release rate, G, defined by Griffith's (1920) for linear elastic materials. Eq. 5 could potentially describe the fracture mechanics of soil over a wide range of water contents. The same testing procedure presented in this paper can be used to assess the necessary parameters for Eq. 5, providing that the applied load is also recorded. This may require improvements to the approach used to remove the effects of sample self-weight. The true measure of CTOA also needs to include crack opening due to elasticity, which we assume is negligible for our samples because of the dominant plastic behavior.
The testing and data analyses procedures presented in this note provide a simple method for evaluating the fracture mechanics of ductile soil specimens. Shrinking soils may begin to crack when soil is ductile, so assessing this physical process is extremely important in understanding the genesis of soil structure and developing predictive transport models. We have tested the approach using the ideal case of a fully plastic soil. Turner and Kolednik (1997) provide the necessary theory, however, to apply the same testing procedure to elastic–plastic and linear elastic soil specimens. There are many variables that could be investigated using this approach. Among these are biological, mineralogical, and chemical influences on soil fracture mechanics. The elegance of the approach is the derivation of CTOA, which can be used directly in Finite Element Modeling. This could be the route towards including the influence of hydraulic transport and heterogeneity on the development of cracking patterns in soil.
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ACKNOWLEDGMENTS
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We thank Brendan Gillard for his assistance with the experiments. SCRI receives grant-in-aid support from the Scottish Executive Rural Affairs Department.
Received for publication October 12, 2000.
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REFERENCES
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- Ayad, R., J.M. Konrad, and M. Soulie. 1997. Desiccation of a sensitive clay: application of the model CRACK. Can. Geotech. J. 34:943–951.
- Chandler, H.W. 1984. The use of non-linear fracture mechanics to study the fracture properties of soils. J. Agric. Eng. Res. 29:321–327.
- Chertkov, V.Y., and I. Ravina. 1998. Modeling the crack network of swelling clay soils. Soil Sci. Soc. Am. J. 62:1162–1171.[Abstract/Free Full Text]
- Genstat 5 Committee. 1993. Genstat 5, Release 3, Reference manual. Clarendon Press, Oxford, UK.
- Griffith, A.A. 1920. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. London, Ser. B. A221:163–198.
- Hallett, P.D. 1996. Fracture mechanics of soil and agglomerated solids in relation to microstructure. PhD thesis. University of Birmingham, UK.
- Hallett, P.D., A.R. Dexter, T. Baumgartl, J.P.K Seville, and R. Horn. 1998. Changes to pore water pressure caused by indirect and direct tensile loading of unsaturated soil aggregates [WCSS CD-ROM]. 16th World Congress on Soil Science, Montpellier, France. 20–26 Aug. 1998. CIRAD, Montpellier, France.
- Hallett, P.D., A.R. Dexter, and J.P.K. Seville. 1995. The application of fracture mechanics to crack propagation in dry soil. Eur. J. Soil Sci. 46:591–599.
- Hatibu, N., and D.R.P. Hettiaratchi. 1993. The transition from ductile flow to brittle fracture in unsaturated soils. J. Agric. Eng. Res. 54:319–328.
- Kolednik, O., and H.P. Stüwe. 1985. The stereophotogrammetric determination of the critical crack tip opening displacement. Eng. Fract. Mech. 21:145–155.
- Lawn, B.R. 1993. Fracture of brittle solids—Second Edition. Cambridge University Press, Cambridge, MA.
- Lima, L.A., and M.E. Grismer. 1994. Application of fracture mechanics to cracking of saline soils. Soil Sci. 158:86–96.
- Moran, C.J., and J.M. Kirby. 2000. In Discussion of: Horgan, G.W., Young, I.M., 2000. An empirical stochastic model for the geometry of two-dimensional crack growth in soil. Geoderma. 96:277–289.
- Morris, P.H., J. Graham, and D.J. Williams. 1992. Cracking in drying soils. Can. Geotech. J. 29:263–277.
- Perrier, E., C. Mullon, M. Rieu, and G. de Marsily. 1995. Computer construction of fractal soil structure: simulation of their hydraulic and shrinkage properties. Water Resour. Res. 31:2927–2943.
- Raats, P.A.C. 1984. Mechanics of cracking soils. p. 23–38. In J. Bouma and P.A.C. Raats (ed.) ISSS Symp. Water and Solute Movement in Heavy Clay Soils. 27–31 Aug. 1984. Int. Inst. for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands.
- Snyder, V.A., and R.D. Miller. 1985. Tensile strength of unsaturated soils. Soil Sci. Soc. Am. J. 49:58–65.[Abstract/Free Full Text]
- Sture, S., A. Alqasabi, and M. Ayari. 1999. Fracture and size effect characters of cemented sand. Int. J. Fract. 95:405–433.
- Turner, C.E., and O. Kolednik. 1994. Application of energy-dissipation rate arguments to stable crack-growth. Fatigue Fract. Eng. Mater. Struct. 17:1109–1127.
- Turner, C.E., and O. Kolednik. 1997. A simple test method for energy dissipation rate, CTOA, and the study of size and transferability effects for large amounts of ductile crack growth. Fatigue Fract. Eng. Mater. Struct. 20:1507–1528.
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ABSTRACT
INTRODUCTION
