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 Internal stresses in soils evolve as a result of the various interactions among soil particles and the pore fluids in response to natural or human-induced activities. Whether attributed to suction, structure, or the various physicochemical forces that may develop, assessing the internal stresses in soils has been an active area of research in soil science and engineering. This paper presents the restrained ring method as an experimental tool for measuring the stresses that develop in a drying soil when it is restrained from shrinking freely. The internal stress that develops can be related to the water content of the soil. This method is validated using results for both silty clay loam and clay loam soils. Tensile stress at failure (estimated from the restrained ring method) was compared to tensile strength ranges of similar soils, and a good correlation was observed. Moreover, internal stresses measured by the restrained ring method were compared with existing empirical relationships (using suction and degree of saturation). This comparison concluded that suction is not the sole contributor to the soil's internal stress.
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 Bridging the gaps between soil's mechanical behavior, its structure, and the soil hydrologic properties as well as establishing the links between the local-scale soil processes and the mesoscopic field scale is a major research challenge yet to be resolved. Achieving these grand challenges requires additional research at the interface of these disciplines particularly related to transdisciplinarity, transfer of scales, global change assessment, and empirical against physically based characterization [Braudeau and Mohtar, 2008; Lin, 2003; Lin et al., 2006; Balser et al., 2006; Ahuja et al., 2006]. Lin et al.  presented the quantification of soil structure and its impact on flow and transport as main challenges despite the various attempts of scaling and representations of heterogeneity of flow and transport problem at the local and landscape scales. This challenge presented a major barrier against the fundamental understanding of soil behavior and led to major empiricism in the equations of soil physical and mechanical properties in the sense that they do not take into account soil structure and its hierarchical organization in the soil-water dynamics and interactions [Braudeau et al., 2004a, 2004b; Braudeau and Mohtar, 2004, 2008].
 In fact, bridging soil physics, hydrology, and mechanics with pedology (discipline dealing with the internal and external soil organization), by clearly mapping and defining the interaction of associated soil processes across the various functional scales would allow for resolving some knowledge gaps in earth and environmental sciences and provide fundamental understanding of the ecological responses to the various biogeochemical processes at the landscape level [Balser et al., 2006; Braudeau and Mohtar, 2008].
 One challenging example of bridging the gaps across the soil's functional scales is to understand and be able to measure the evolution of the soil's internal stresses during the drying starting from saturation and linking them to soil behavior. In fact, the complexity of this problem has widened the gap between research among the soil physics and soil mechanics groups, each driven by their own range of applications and interests. Understanding the interaction among soil particles as well as the structural and physicochemical factors contributing to the internal stress generated in soil owing to environmental loads like drying has always been a major challenge. Owing to its complexity, the soil physics research addressed this problem by assuming a constitutive relationship between soil suction and water content (or degree of saturation) through the soil-water characteristic curve, or developing functional relationships like the shrinkage/swelling curves. Soil mechanics research addressed this problem through the development of the effective stress theory [Terzaghi, 1936] for saturated soils that was extended later by Bishop  into the unsaturated soils assuming an empirical relationship between suction and the effective stress. Advancements in colloid sciences have triggered the interest in extending the effective stress theory to account for physicochemical effects [Bolt, 1956; Lambe, 1953; Low, 1979] with various attempts to modify the effective stress theory to account for physicochemical forces [Lambe, 1960; Sridharan and Vinkatappa Rao, 1971, 1973; Mitchell, 1960; Barbour and Fredlund, 1989]. However, the lack of comprehensive understanding of internal stress evolution in soils with water content and the inability to measure those stresses have caused a disconnect between the two groups with soil mechanics more concerned about the soil behavior in reaction to an external load (from foundations or engineering structures). In fact, the current state of the art in soil experimentation allows for the development of stress-strain relationships in soils only as function of an external pressure (which is the main interest in geotechnical application) through the traditional tools used in soil mechanics. This paper presents a tool to assess internal stresses that develop owing to natural and environmental loads or processes like drying. Such tool will reintroduce the stress-strain concepts to the soil physics community from a relevant perspective and related applications. It will also fill a major gap and establish a link between soil physics and mechanics especially with the evolving interest in the field of unsaturated soil mechanics [Fredlund, 2006]. Once this link is established, scalable and physically based constitutive relationships in soils can be developed and applied in the various fields of geosciences.
 Another application will be to monitor the evolution of stresses in drying soils and model the cracking behavior as the point when internal stress development exceeds the tensile stress in the soil at that specific water content. While this concept is still at the theoretical level owing to the lack of an experimental method to measure the stress evolution in soils as well as the limited understanding of the mechanics of tensile strength in soils, the restrained ring method may provide the experimental basis for the characterization of shrinkage-induced cracks. Such characterization presents a major challenge, as it requires fundamental understanding of the governing processes associated with the cracking behavior of soil. Those processes include shrinkage/swelling, stress-strain relationships, soil tensile strength, and the evolution of soil's internal stresses as a function of water content. Shrinkage/swelling processes were modeled from volume change perspectives where the internal volume change (aggregates, water pools, and micropore and macropore systems) is governed by the shrinkage parameters of the soil structure by linking the soil's specific volume to gravimetric water content [Braudeau et al., 2004a; Braudeau and Mohtar, 2006]. However, coupling the soil's shrinkage/swelling-induced volume changes with the evolution of internal stresses as function of the water content is yet to be established especially with the inability of the effective stress principle to explain volume change behavior in unsaturated soils [Jennings and Burland, 1962; Matyas and Radhakrishna, 1968; Donald, 1960]. Reasons are multiple. The first one is conceptual: the inability to formulate the role of the soil structure in the shrinkage-swelling processes and its thermodynamic properties. This leads to great deal of empiricism in soil physics in an attempt to describe soil behavior. Some aspects of soil behavior were addressed by physically based models including the water potential [Braudeau and Mohtar, 2004], the shrinkage curve [Braudeau et al., 2004b], and finally the thermodynamics of the organized soil medium represented by its pedostructure: structure representative elementary volume (SREV) of the soil medium [Braudeau and Mohtar, 2008]. The second reason is the inability to measure the internal stress development in soil as the soil dries. The work presented in this paper evolved from this background, recognizing the need to improve the current understanding of how internal stresses develop in soils, thus contributing toward bridging the gap between soil physics and pedology. This objective may only be achieved by taking account of the internal soil organization (structure) in all physical equations of hydrophysical processes in soils (as suggested by Braudeau and Mohtar ), including soil mechanics notions like effective stress, overburden pressure transmission, domains of elasticity, friability, soil cracking, etc. Once this link is established, such physical (nonempirical) characterization would allow for physically scaling up the current soil-water interaction formulation to the field level. This scaling would allow for the soil-water flow in the field to be modeled in two distinct pools: across the soil matrix and in preferential flow paths.
 The objective of this paper is to introduce the restrained ring method to the soil science community. The restrained ring method provides a method for measuring the stress that develops in soil when it is restrained from shrinking freely. The restrained ring tests provide an experimental method to improve the current understanding of internal stress evolution in soils as they dry and are restrained from shrinking freely. Specifically, this paper (1) introduces the restrained ring method as an experimental method to quantify the internal stresses that are generated as a restrained soil dries, (2) presents the results of internal stress development for restrained silty clay loam and clay loam soils, (3) compares the results of tensile stresses obtained for the two soils with tensile strength ranges of similar soils from the literature, and (4) compares the results of the internal stress measured by the restrained ring method to those estimated by soil-water potential or suction. The restrained ring method quantifies the resultant internal stresses as a function of soil-water content.
2. Current Understanding of Stress Evolution in Unsaturated Soils
 Internal stresses have been recognized from various soil-related observations particularly from problems associated with slope failures upon soil saturation [Terzaghi, 1950; Rao, 1996]. Such losses of soil's strength (recognized in the literature as loss of apparent cohesion) have triggered the interest in assessing the evolution of soil's internal stresses as a function of soil's water content, suction, or degree of saturation (in the context of its influence on effective stress). Experimental evidence has demonstrated that internal stresses develop as soil loses water and shrinks causing a change in the soil's pore pressure, shear strength [Escario and Saez, 1986; Fredlund et al., 1987], and compressive strength [Aitchison, 1957]. Assessing internal stresses in soil has always been a major challenge in soil physics and mechanics owing to the lack of the experimental means and theoretical framework to quantify them. Experimentally, soil internal stresses have been assessed in the context of shear stress resistance using the direct shear apparatus under controlled suction [Vanapalli et al., 1996; Escario, 1980; Escario and Saez, 1986] or triaxial testing [Fredlund et al., 1987; Aitchison, 1957]. Theoretically, internal stresses have been mostly understood in the context of the effective stress theory (through relationships with suction). Thus, increase in shear strength due to drying was explained using the Coulomb-Hvorslev relationship by linking the increase in shear strength nonlinearly to suction or the apparent cohesion of soil (both varying nonlinearly with water content).
 In the context of shear strength, various studies (Table 1) assumed that soil's internal stresses are mainly a function of suction, or basically the capillary forces that develop in the soil structure owing to the air-water interface generated by the difference between water pressure, uw, and air pressure, ua [Bishop, 1959; Fredlund et al., 1987, 1996; Vanapalli et al., 1996; Oberg and Sällfors, 1997]. Those stresses have been observed as the “contribution to shear stress due to matric suction” (the second term in the noncomprehensive list of shear strength equations of Table 1). However, those correlations between shear strength and suction, especially for cohesive soils, have failed to predict accurate shear strength values when tested for different soil types [Garven and Vanapalli, 2006]. It is our belief that such failure is attributed to the degree of empiricism associated with internal stress understanding and representation. Most of the proposed parameters are empirical variables (χ, Θκ, or S) that change with water content (and its corresponding suction value) having a value ranging between zero and one to reflect the nonlinearity between suction (that reaches infinity in existing models at residual water content) and increase in shear strength. Such parameters ignore other forces attributing to the internal stresses of soils including the various physicochemical and structural forces.
 More recent studies [Alonso et al., 1990; Gens and Alonso, 1992; Santamarina, 2001; Lu and Likos, 2006] considered the soil's structural hierarchy, aggregation, and the microscale or macroscale interaction in an attempt to better understand the physical phenomena governing stresses in soils. They observed the interaction between “a capillary-controlled macrostructure and a microstructure where physicochemical and other phenomena occurring at particle level take place” [Gens and Alonso, 1992, p. 1030]. They realized the importance of discontinuities of soil internal forces across its different scales as well as the spatial variability of its pore and particle size distribution, considering them key attributes contributing to the complexity of the problem. This understanding of the interaction between the two distinct scales is providing an explanation for the nonlinearity between soil strength and its corresponding suction value (no more being the sole contributor to internal stresses) as Alonso et al. [1990, p. 408] stated that “the increase in strength cannot continue indefinitely with applied suction” especially for very low water contents.
 This new interest in the effect of the soil's “microstructure” on internal stress evolution, especially with the introduction of concepts like the electrical double layer theory to soils [Bolt, 1956; Lambe, 1953; Low, 1979], has triggered the need to update the current stress evolution formulation. It is now clear that understanding internal stresses in soils cannot be established through simple correlation with soil suction (with their clear implications on shear strength formulation as summarized in Table 1) and that other forces and interaction in the soil structure need to be considered. According to Lu and Likos , the soil's internal stresses include (1) Van der Waals forces, (2) cementation between particles due to cementing agents, (3) electrical double layer forces, and (4) capillary interparticle forces due to pore water pressure and surface tension of water.
 Currently, limited understanding of the contribution and interaction among the various force components inside the soil structure exists. The restrained ring method provides a “resultant” estimate for the internal stress evolution in a soil restrained from drying. This value would constitute an estimate of the cumulative effect of the internal stress built up, being the summation of the various contributions of all “sources” and types of forces that interact in the soil structure.
 In the restrained ring method, a sample (i.e., soil) is placed around a ring (i.e., core) of known material that will restrain the shrinkage. As the soil is allowed to dry, it attempts to shrink (Figure 1a). The core material prevents this shrinkage resulting in the development of a pressure between the soil and the core. This pressure can be used to describe the internal stresses that develop in the soil. The following sections describe the theoretical framework and the experimental procedure for the restrained ring method.
3.1. Estimation of Soil Internal Stresses
 The analytical stress formulation of a restrained material undergoing shrinkage using only the measured strain in the restraining ring was developed by Weiss and Furgeson  (Figure 2a). This is done by separating the problem into two components: (1) a ring of restraining material pressurized at its outer surface (Figure 2b) and (2) a soil cylinder under pressure at its inner surface (Figure 2c).
 The pressure on the restraining ring (i.e., core) can be determined by finding the pressure that is required to generate a strain that is equivalent to the strain measured on the inner circumference of the core. Assuming uniform shrinkage throughout the soil and the elasticity of the restraining ring material, the pressure on the restraining ring can be calculated:
where PInternal,ISoil(t) is the internal compressive pressure acting on the restraining ring, ε(t)Ring is the time-dependent strain measured at the inner surface of the restraining ring, RORing is the outside radius of the restraining ring, RIRing is the inside radius of the restraining ring, and ERing is the modulus of elasticity of the restraining ring.
 Once the interface pressure in the restraining ring is known, the average circumferential tensile stresses in the vertical direction (σθ) as well as the radial compressive stresses (σr) can be computed (Figure 3) in the soil (at any point along the radial direction) using the solution for an elastic stress in pressurized cylinders [Timoshenko and Goodier, 1987; Weiss, 1997]:
where ROSoil is the outside radius of the restrained soil specimen and r is the radius at any point along the restrained soil specimen.
Figure 3 illustrates that the circumferential stress is highest at the restraining core and decreases as a function of 1/r2 as the radius is increased to the outside of the soil ring. The radial stress is equal to the pressure at the core but decreases to zero at the outer edge. From equation (2), the maximum tensile stress at r = RORing (Figure 3) is obtained:
The resultant stress, σRing,element, of any soil element (Figure 4) in the restrained ring method setup is obtained by
The average resultant internal stress across the soil (σInternal) can be determined by integrating σRing,element from the inner radius of soil (RISoil) to the outer radius of soil (ROSoil) and dividing the result by (ROSoil − RISoil):
σInternal can be obtained numerically from equation (6). It is proposed to represent the average resultant internal stress across the soil. In other words, σInternal represents the resultant of all the internal stresses discussed in the previous sections and is therefore an integral part of the effective stress of the soil. Such “macro” resultant force becomes very practical in the application of the effective stress theory in the absence of a comprehensive understanding of the individual effect of various stress components (suction, repulsive, attractive, surface hydration, structural, osmotic, etc.).
 The restrained ring method is currently used on disturbed laboratory samples. It is anticipated, however, that this method can be extended to field nondisturbed soil samples. The procedure that was followed in our experiments (Figure 5) was as follows.
3.2.1. Specimen Base
 A 4-inch diameter, randomly perforated (with 3/16-inch holes) acrylic sheet was used as the base for the soil material. This sheet was overlaid with randomly cut small geotextile sheets (Figure 5) to prevent the loss of soil during the testing process, yet allow water to infiltrate or evaporate from the bottom of the specimen.
3.2.2. Soil Specimen Preparation
 Specimen base was soaked in a water bath (around 5 mm deep) after placing a first layer of unsaturated soil on the base to protect the geotextile from floating. Saturated soil sample was prepared by placing soil in similar layers (six layers) and compacting them uniformly with a 12-cm-long stainless steel rod at 100 strikes per layer (from a uniform height). Each layer was allowed to saturate by capillarity from the bottom and by adding some water from the top, especially in the last layers. Soil was placed outside the restrained ring, and inside another confining ring that is used only to help in holding the soil. A thin layer of oil was placed on the outside of the inner restrained ring, and the inside of the confining ring to eliminate friction between the ring and the soil.
3.2.3. Beginning of the Test
 The specimen was placed in a temperature chamber at 23°C (with specimen temperature fluctuation of less than 2°C during the whole period of experimentation) over a scale (0.1 g accuracy) that is connected to a data acquisition system. For the gravimetric water content, change in specimen weight was taken every minute using the WinWedge software, and gravimetric water content was obtained from the scale readings by back calculation. As to strain and specimen temperature readings, the strain gauges were connected to the StrainSmart data acquisition system. The StrainSmart system was used for reading the strain gauges connected to the inner ring and the thermocouples connected to the soil specimen. Readings were also taken every minute for each of the strain gauges and thermocouples. Strain, as measured from the strain gauges (ε(t)Ring), was used for the calculation of soil pressures and tensile stresses (equations (1)–(4)).
3.2.4. Experiment Closure
 The specimen was left to dry naturally until cracking took place. After that, the experiment was stopped and the soil sample was removed and the soil was taken to the oven to determine its oven dry weight.
 Two soils, silty clay loam and clay loam, were obtained from the Agronomy Center for Research and Education (ACRE) of Purdue University. Soil properties are listed in Table 2. Sample preparation was conducted by sieving the soil with sieve 4. These criteria were followed with the objective of trying to maintain part of the soil structure by keeping the soil's small aggregates while ensuring that the soil tested includes no large or bulk material (stones) that might bias the result. This method was favored over the soil crushing technique that totally destroys the soil's microstructure.
Table 2. Properties of the Two Soils Used in the Analysis
 As to the experimental apparatus, the restraining ring material was carefully selected to fit within the range of material (soil) strength. If the ring is too stiff, the ring will not deform and strain will not be sufficient to be measured. Although the effect of geometry (ring dimensions and material) is not covered in this paper, work was performed to understand the effect of geometries, moisture distribution, and material properties on the experimental results [Moon and Weiss, 2006]. A brief description of the data acquisition systems and equipment used in the experiment as well as a list of the material and dimension properties of the restraining ring is presented in Tables 3a and 3b.
Table 3a. Equipment Used in Data Acquisition System
0.1 g accuracy (Ohaus Scout Pro digital scale SP2001)
Scale reading will be required to back calculate the water content as a function of time.
Data acquisition for scales
WinWedge 32 STD v3.0 software with RS232 connection
Digital scale is connected to the computer with the RS232 connector, and data are acquired using WinWedge software.
Vishay's CEA-09-250UW-120/P2 strain gauges
Gauges are installed on the inside of the restraining ring to measure ε(t)Ring
wire thermocouple (type T)
Thermocouple is used to compensate for temperature error in the strain gauge readings on the PVC restraining ring.
Data acquisition for the strain and temperature
Vishay's StrainSmart System 5000 Model 5100B
System is used to store strain and temperature data as a function of time. Readings were taken every 1 min.
Table 3b. Material Properties of the Restraining Ring
PVC, schedule 21
Modulus of elasticity, E (GPa)
Soil specimen height (cm)
3.4. Ring Geometry
 Selection of proper geometric dimensions in the restrained ring method is key for the success of the experiment. A sample geometry was desired that was a reasonable size, capable of producing one crack in the radial direction. Determining the geometry started by testing different geometries including different ring radii, materials, and thicknesses. Figure 6a shows how soil 1 behaved under relatively large geometries (RISoil = 15 cm, height = 10 cm) thus producing multiple cracks. The large size of the sample makes it more difficult to insure uniformity in the sample preparation and the analysis can only be performed until the development of a single crack. Figure 6b shows how the use of smaller geometries (RISoil = 2.11 cm, height = 3.81 cm) produced the desirable stress distribution for soil 1 by producing a single crack in the desired direction.
3.5. Moisture Gradient Considerations
 The presence of significant vertical moisture gradient was considered at the early stages of designing the experimental procedure of the restrained ring method. The use of perforated acrylic and geotextile as the base of the soil specimen allows for moisture loss from the bottom through infiltration and evaporation. Moreover, previous work on concrete demonstrated that despite the gradient of moisture that develops the average strain and stress in a restrained sample can be determined since unless the retraining ring is very flexible in the z direction the restraining ring will deform uniformly.
 Ring and soil specimen geometry were carefully selected to achieve moisture content distribution within acceptable range. The water contents of the upper and lower parts of the soil specimen (after the soil cracks) were checked and results confirmed they are within a margin of ±0.5% gravimetric water content. Similar results were obtained when testing for a moisture gradient in the radial direction. In fact, achieving close water content values across the soil specimen is an integral part in maintaining the simplicity of this experiment. With such experimental checks, it is safe to assume uniform moisture distribution across the soil specimen.
 Moisture gradients however could be a problem in a case of soil ring with very large depth dimension. In this case, the use of multiple rings and multiple moisture content sensors to capture the change in the water profile is highly recommended. For all the experiments conducted in this paper, the depth of the restraining ring and thus the soil sample was 3.81 cm (1.5 inches).
 It is important to note that different soils are expected to react differently in the same way that every soil has a different soil-water characteristic curve. Internal stresses in soils (as they generate owing to drying) are function of soil type and are expected to be different and thus different soils may require different geometries.
4. Experimental Results
Figures 7 and 8show typical experimental results obtained by a restrained ring test (using a PVC (schedule 21) ring material with RIRing = 1.95 cm, RORing = 2.11 cm, ROSoil = 5.30 cm, specimen height = 3.81 cm) for the silty clay loam soil (soil 1). Figure 7 shows the variation of strain (not corrected for temperature) as a function of time, whereas Figure 8 shows the change in strain as a function of the gravimetric water content (w) for the same test. The abrupt change in strain detected at water content of around 17% as shown in Figure 8 (or after 1,150 min as illustrated in Figure 7) is referred to as the point of cracking. It is also important to note that the strain and stress levels drop considerably owing to cracking. However, this does not mean that the specimen is not shrinking anymore, nor does it mean that the specimen is not developing local internal stresses.
 The methodology section explained the analytical solution that was used to calculate (1) the internal compressive pressure exerted by the drying soil on the ring, PInternal,ISoil (equation (1)), (2) the maximum tensile stress that is developed in the soil, σθmax, (equation (4)), and (3) the resultant or averaged internal stress, σInternal (equation (6)). Results of the internal pressure at the ring and tensile stresses obtained for the silty clay loam and the clay loam soils are presented in sections 4.1 and 4.2.
4.1. Internal Pressure
 The pressures on the restraining ring (exerted by the drying soil on the outside of the ring) obtained using equation (1) are shown in Figure 9 for soil 1 and soil 2. As expected, internal stresses for soil 1 increases with the decrease of water content, reaching a range of 20–35 KPa at gravimetric water contents around 17%. This value was zeroed at water contents near the saturation level. This approach assumes linear elasticity in a continuum, as such the strain values measured after the point of cracking (i.e., for water contents lower than 17% in this case for soil 1) cannot be used to determine a meaningful value of stress.
 If the stress evolution of various soil specimens is examined (Figure 9), a few items should be noted. Stress release after the point of cracking (from the direction of the ring) should not provide the wrong impression about the general behavior of the soil: that the soil reaches a point of maximum internal stress at the water content when cracking took place. Moreover, the stress that develops depends on the degree of restraint provided by the restraining ring (i.e., ring thickness or stiffness). A thinner ring or ring with a lower stiffness will cause cracking to take place later owing to the change in the degree of restraint on the soil specimen [e.g., Hossain et al., 2003].
Figure 9 also provides a comparison between PInternal,ISoil obtained for soil 1 and soil 2 using the restrained ring method. It is clear that soil 2 starts to develop higher stresses at higher water contents than those of soil 1. The reason may be due to the higher clay content and lower sand content of soil 2. Another interesting observation is that for the same restraining ring, soil 2 always tends to crack at higher water contents. This may be attributed to the fact that soil 2 has larger shrinkage range (with specific volume of around 0.85 cm3/g of soil near saturation and around 0.64 cm3/g for gravimetric water content of 24%) than soil 1 (with specific volume of around 0.75 cm3/g of soil near saturation and around 0.61 cm3/g for gravimetric water content of 17%) and thus its capacity to shrink toward the core (the restraining ring) is opposed by the much higher stiffness of the ring. It is thus expected that soil 2 cracks at water contents higher than those of soil 1.
4.2. Tensile Stress Development
 The maximum tensile stress is calculated from equation (4) (located at r = RORing in equation (2)). Figure 10 shows the results obtained for the maximum tensile stresses ranging from zero at saturation to around 30–50 KPa at gravimetric water contents of around 17% (for soil 1). Similar to the earlier discussion, tensile stresses increased with time, that is with the decrease in water content.
Figure 10 also compares the results of maximum tensile stresses for soil 1 and soil 2. Similar to the results discussed in section 4.1, soil 2 tends to develop higher tensile stress at higher water contents than soil 1. This is most likely attributed to the higher clay content. Soil 2 also has the tendency to crack at higher water contents than soil 1 under the same restraining ring.
5.1. Implication on the Understanding of the Effective Stress Theory
 Soil's effective internal stresses constitute a cornerstone in the effective stress approach developed by Bishop for unsaturated soils [Bishop, 1959; Bishop and Blight, 1963] after Terzaghi's effective stress for saturated soils [Terzaghi, 1936]. Bishop assumed an empirical relationship between suction (ua − uw) and internal stress, χ(ua − uw), which develops as soil gains or loses water through the effective stress parameter, χ:Terzaghi's effective stress
Bishop's effective stress
where σ is the exerted stress, σ′ is the effective stress, ua is the pore air pressure, uw is the pore soil-water potential (positive for saturated and negative for unsaturated soils), χ is the effective stress parameter depending on the soil structure and the degree of water saturation (in fact the shrinkage curve).
 Accounting for the various physicochemical, structural, and osmotic forces that develop in soil owing to change in moisture content have been widely addressed in the unsaturated soil mechanics literature [Bolt, 1956; Lambe, 1953, 1960; Low, 1979; Sridharan, 1968; Barbour and Fredlund, 1989]. Unfortunately, those studies could not get the deserving attention owing to the lack of experimental tools to quantify those contributions (forces), especially the role of the soil structure, which is represented by the empirical function χ in equation (8). This limitation has caused the empirical modeling of the internal soil stress (or effective stress) as function of suction or water content.
 The relationship between stresses measured by the restrained ring method (tensile and compressive) and the soil-water potential is under current investigation. The full capabilities of the restrained ring method as tool to assess the internal stress evolution with decreasing water content in soils are not completely exploited at this point. Different ring material, cross sections, and thicknesses (which control the degree of restraint) need to be explored in order to capture the effect of restraint on soil as well as to obtain the maximum range of water content that can be investigated by the restrained ring method.
 The state-of-the-art understanding of the internal forces and stresses in soils has not yet matured into a comprehensive understanding (based on the pedostructure organization) of the various stress components and their dynamic effect and contribution to the resultant total internal stress measured using the restrained ring method. This new approach may contribute toward better understanding of the current empirical and mathematical forms that link the soil-water potential uw to effective stress through parameters like Bishop's effective stress parameter (χ).
5.2. Comparing Maximum Tensile Stress and Tensile Strength
 The tensile strength, ft, is defined as “the stress, or force per unit area, required to cause soil to fail in tension” [Imhoff et al., 2002, p. 1656]. Another widely used indicator of soil strength in soil science is soil friability (F) redefined by Watts and Dexter [1998, p. 73] as “the coefficient of variation of soil tensile strength.” Lu et al.  suggested a range of tens of pascals for coarse sand, several kilopascals for fine sands, and several tens of kilopascals for silts. They recorded a maximum tensile strength of 1,448, 1,416, and 890 Pa for silty sand, fine sand, and medium sand, respectively. Vomocil et al.  reported tensile strengths of 8.8–30 KPa for the Yolo loam (35% sand and 19% clay) with water contents ranging from 24.9% to 10.8%. They also reported tensile strength of 4.7–19.1 KPa for the San Joaquin sandy soil (54% sand and 12% clay) with water contents ranging from 14.2% to 4.2%. Tensile strength values of 2.2–122.8 KPa were reported for a internal basaltic clayey soil with water contents ranging from 122% to 21.5% [Nahlawi et al., 2004]. For a 50:50 mixture of sand and Na-rich bentonite, tensile strengths of 228–318 KPa were reported for water contents ranging from 19.8% to 17.4% and suction levels of 4–10 MPa [Tang and Graham, 2000].
 Conceptually, tensile strength is governed by the weakest continuous resultant of the macro–interparticle forces and active skeletal forces in the soil body. Obviously, this tensile strength value is highly dependent on scale. Watts and Dexter  stressed the importance of hierarchy in soil structures, moving up in the scale from particles to micro-aggregates, to aggregates then clods. They stressed that each scale is weaker than its preceding owing to the presence of microcracks and flaws. This argument is in line with results obtained by Barzegar et al. [1994, 1995], who reported tensile strengths of a variety of Australian soils with different clay contents. They reported a range of tensile strength values from 147 to 1,439 KPa for oven dry remolded small soil aggregates (discs of 20-mm diameter and 10-mm thickness) with clay content range of 9.2% to 55.3%. Obviously, results obtained by Barzegar et al. [1994, 1995] are at least 1 order of magnitude higher than the referenced similar soils tested at a macroscale (although with higher water contents).
 In the restrained ring method, the maximum tensile stress at cracking for each soil specimen can be compared to the tensile strength. Figure 11 shows the range of tensile strength (maximum tensile stress at cracking) values obtained for soil 1 and soil 2 in comparison to the maximum tensile stress from one experiment for each type of soil before cracking. Results demonstrate an increasing tensile strength of soils as a function of the soil's moisture content. Variability in results for the same soil is attributed to the heterogeneity of soil particle distribution and structure as well as the procedure and sample preparation (resulting in different starting water content values as well as some variation in the overall soil specimen's dry weight).
 A comprehensive literature review was conducted to compare the values of the maximum tensile stress measured by the restrained ring method to tensile strength of similar soils reported in the literature. Table 4 presents a summary of those soils' properties and compares them with soil properties of soil 1 and soil 2. Figure 12 shows the ranges of tensile strengths reported in the literature for the various soil types (Table 4) ranging from sand to silty sand to loam to clay loam to clays. Logarithmic scale was used so that the whole range of soils types can be presented. The maximum tensile stresses for soil 1 (silty clay loam) and soil 2 (clay loam) obtained using the restrained ring method are also plotted on the same graph. The maximum tensile stress determined using the restrained ring method is comparable to tensile strengths reported for similar soils. Soil 1 and soil 2 (silty clay loam and clay loam) had stress values below the tensile strengths reported for kaolinite, the 50:50 sand/Na-rich bentonite mix, and the internal basaltic soil. Stresses of both soils were within the same order of magnitude of silty clay loams and loams tensile strength, and higher than the tensile strengths reported for the sand group of soils (Figure 12). A rough delineation of soil textural classes and tensile strength is drawn on the basis of the literature data provided in this paper. Such delineation remains a rough estimate of the range of tensile strength based on the soil's textural class neglecting any effect from the soil structure, level of compaction, or clay type.
Table 4. Characteristics of Soils Used for Tensile Strength Comparisons
5.3. Relationship Between Internal Stress and Pore Pressure
 For the first time, the internal stress of restrained soil is experimentally detected rather than theoretically or heuristically estimated on the basis of tensiometric measurements of the pore water matric potential. This independent quantification of internal stress may be used in an attempt to understand its relationship with pore water matric potential (measured by tensiometers). Here, it is very important to stress the difference between the tensiometer and the restrained ring method readings. Tensiometers measure the equilibrium pressure between the last layer of the water in the soil (at the interface with the tensiometer cap) and the water inside the tensiometer and relate this to suction. On the other hand, the Restrain Ring Method measures the stress at the interface between the restrained soil and the ring thus representing the whole surface rather than just the interface between wetting and nonwetting fluids.
 Results from two experiments of internal stress (soil 2) were compared to the water potential curve and to the traditional effective stress defined in equation (8) and assuming for χ(θ) the empirical equation (9) [Vanapalli et al., 1996]. Four points on the soil-water characteristics curve (SWCC) for soil 2 were obtained using the pressure plate method, at suction values 0, 33, 100, and 1500 KPa (assuming θ1500KPa is the residual volumetric water content), and the three parameters of the van Genuchten model [van Genuchten, 1980] were derived (Table 2) using the Retention Curve (RETC) code [van Genuchten et al., 1991]:
Figure 13 shows a plot of (ua − uw), χ(ua − uw), and σInternal as function of the gravimetric water content for soil 2. Since internal stress is the collective contribution of solid rearrangement due to shrinkage (to remain at equilibrium of potential) as well as the potential (measurable by tensiometer) of the water films, the internal stress is expected to be similar to water potential at and near saturation (since solids can rearrange freely and are surrounded by water molecules). Figure 13 shows that the plots of χ(ua − uw) and internal stress are very close near saturation. As the water content continues to decrease and the slope of the shrinkage curve deviates considerably from 1, the ability of particle to rearrange within the soil structure decreases leading to an increase in the effective stress relative to the soil-water suction. The effective stresses here refer to the internal pressure on the inner cylinder. Thus, internal stress measured by the restrained ring method (σInternal) is expected to be lower than the pore pressure at lower water contents. Figure 13 shows a great discrepancy between σInternal and χ(ua − uw) toward the low water contents (left-hand side of Figure 13), indicating that equation (9) of the χ function is absolutely not appropriate. In fact, χ(θ) must be related to the soil structure behavior with the water content [Yong and Warkentin, 1975; Khalili et al., 2004].
5.4. Implication on the Understanding of the Cracking Behavior of Soil
 Experimental results (Figures 7–10) show that strain (shrinkage) begins to develop in the ring (at ROSoil and RIRing) immediately after the start of the experiment (with soil drying from ROSoil toward the core). The age of cracking varies with the ring size and stiffness, and is smallest for the largest restraining ring thickness and stiffness. Thus it is very important not to misinterpret the point of cracking as the point at which the soil cracks in the field or the point of maximum strength after which the soil becomes weaker. Cracking rather occurs at the point when the tensile stresses generated owing to restraint are equal to the tensile strength of the soil. This is a very important consideration in the upscaling of this method to the field scale. It is widely known that shrinkage cracks take place at much higher water contents at the field [Chertkov, 2002]. This behavior is captured in the restrained ring method by strain observed at the outer radius of the specimen, ROSoil. On the other hand, stress generation is observed by strain detected by the strain gauges at RIRing indicating that the soil is drying toward the core. Thus, one can think of the soil specimen in the restrained ring method as one soil unit at the field shrinking toward its core and losing volume by shrinkage at the boundary (similar to shrinkage from ROSoil to RISoil in the restrained ring method) until it develops enough internal stress higher than its tensile strength so that it cracks rather than simply shrinks.
 In fact, current investigation showed that using digital imagery to quantify volume change (thus strain), a linear relationship between the strains generated by the drying soil and the logarithm of the resultant internal stress developed by the soil (as measured by the RRM) was obtained experimentally. This relationship obtained is similar to the soil compression curves obtained by measuring the strain developed in soils as function of an external load on the soil. The uniqueness of the results obtained using the restrained ring method is that stress-strain relationships measured are not due to external loads but due to changes in soil's internal stresses due to shrinkage (although it can be argued that drying is an external load).
 In summary, the restrained ring method may contribute toward understanding the cracking behavior of soils by (1) providing the experimental means to understand and quantify the stress evolution of soils as result of shrinkage, (2) allowing for measurement of the tensile stresses and strength in soils (in a simple model, the point of cracking occurs when the internal stresses generated by soil owing to shrinkage–function of water content or saturation–exceeds the tensile strength of the soil at the same water content), and (3) allowing for the development of stress-strain relationships in soils, where the stress is the internal stresses that generate owing to shrinkage and strains are due to the soil's loss of moisture during the shrinkage process. In fact, the restrained ring method may be the only method capable of generating stress-strain relationships due to natural or environmental “loads” like shrinkage. The triaxial method (also compression and oedometer tests) used in geotechnical engineering develops stress-strain relationships owing to the effect of an external (usually vertical) stress and a confining stress.
6. Limitations and Assumptions
 The limitations of the restrained ring method for soils should be discussed. For soils that are extremely weak in tension (like coarse sand) require the use of very thin rings with material that has relatively low modulus of elasticity. If the ring is too thin the inner ring may bend resulting in non uniform behavior in the vertical direction. If the restraining ring is made of a material like PVC, though they have lower stiffness than metals, they usually have higher coefficients of thermal expansion and thus become more sensitive to temperature variation. Another limitation is to be able to capture the stress evolution from saturation to zero water content. So far, and with the equations we have, we can only make use of data before the point of cracking, which is at low water contents usually.
 Accuracy of the results depends on a variety of effects including the following: (1) sample preparation and the ability to produce uniform properties all across the sample given the high heterogeneity of soils, (2) precision of measurement equipment (the resolution of results cannot be higher than the level of accuracy and noise level of the measurement equipment), and (3) selection of ring material and dimensions (material stiffness, diameter, and thickness) based on the work of Moon and Weiss . If the material and dimensions provided a very stiff combination, the experiment will not be able to capture system internal stresses with an acceptable accuracy. On the other hand, if the ring used is too flexible, the stresses will develop but soil may not crack; therefore it will not be clear from the failure mode whether soil was shrinking and generating stresses toward the core.
 The restrained ring method is performed on circulate cross sections which are convenient in terms of understanding the stress distribution in the soil and the restraining ring. The axisymmetric property of the circular cross section allows for obtaining analytical solutions for the stress distribution. The dimensions of the rings can be changed by increasing/decreasing the radius, thickness, and depth of the restraining ring and soil around it. The following assumptions were made during our analysis:
 1. The soil sample is uniform. It is assumed to have uniform density and moisture content in the radial direction. Although preparing a sample with uniform density and moisture content is a challenge and may not be achievable, special care in sample preparation is made to have the sample be as homogeneous as possible.
 2. Stress gradient in the vertical direction is not implicitly considered, and an average section stress in the vertical direction is determined (given the assumption of uniform water content and thus the assumption of no vertical moisture gradient).
 3. The soil is assumed to act like an elastic continuum prior to cracking. As such the effect of nonlinear material behavior and microcracking are not considered. This assumption requires further considerations as specimen that had considerable amounts of microcracks have experimentally showed considerable amounts of strain release. Other behaviors can be considered; however, they are beyond the scope of this paper.
 4. The experiment is within the elastic range of the restraining ring material. This allowed us to make use of the elastic solutions provided in the literature. This is a valid assumption since the strain values detected on the ring material are small, usually not exceeding 400 microstrain.
 5. Saturation point is the point of zero stress (or negligible stress). Such an assumption is required to offer a starting point, since the restrained ring method measures strains starting with any water content for the soil sample. At that point, a “relative” zero-stress state is assumed, and stresses calculated are only those that develop between the initial soil's water content (at the point when the experiment started) and the point of analysis.
 6. Soil properties including stress and strength are affected by the specimen compaction level. It is thus expected that different compaction level produce different responses detected by the restrained ring method. Future work should investigate different compaction levels, solution properties, drying rates, and boundary conditions.
7. Summary and Conclusion
 The restrained ring method was presented as a means to understand and measure the internal and tensile stresses that develop in a soil when it is restrained from shrinking during drying. The stress development was estimated for experiments on silty clay loam (soil 1) and clay loam soils (soil 2). This method can provide key soil parameters for various soil applications, especially for unsaturated soil physics and mechanics as well as hydrology. It also provides an approach to physically characterize the shrinkage cracking behavior of soils with the point of cracking defined when stresses that develop owing to restrained exceed the tensile strength of the soil.
 Experimental results for the maximum tensile stress development of soil 1 and soil 2 using the restrained ring method were compared to ranges of similar soils reported in the literature, showing good correlation. Moreover, results provided the expected increased tensile strength as water content decreases (at least up to some low water contents). Finally, stresses measured by restrained ring method were compared to those predicted by water potential curve and the traditional effective stress defined in equation (8) and assuming for χ(θ) the empirical equation (9). Comparisons showed great discrepancies between σInternal and water potential and χ(ua − uw) toward the low water contents indicating that (1) the relationship between internal stress and water potential requires further investigation and (2) the relationship between the effective stress parameter (χ) and saturation is not appropriate, and χ(θ) must be related to soil structure.
pore water pressure.
volumetric water content (cm3 of water/cm3 of soil).
gravimetric water content (g of water/g of soil).
inside radius of ring.
outside radius of ring.
inside radius of soil (equal to RORing).
outside radius of soil.
strain measured by the ring at time t.
modulus of elasticity of the ring.
internal pressure of soil as measure by the ring.
circumferential stresses in the soil.
radial stresses in the soil.
maximum circumferential stress (tensile) in the soil, located in the inside of the soil ring.
effective stress parameter (Bishop equation).
soil total stress.
soil effective stress.
 This research was supported by a USDA grant 2005–03338 and a research grant from the French Embassy Office of Science and Technology in Chicago. The authors would like to acknowledge the Sensing Lab of the Department of Civil Engineering at Purdue University for providing the resources to perform the restrained ring tests. The authors also wish to acknowledge the valuable feedback received from the Editor, Associate Editor, and reviewers. Their feedback and comments significantly improved the quality of this contribution.