Abstract
The present study proposes a rigorous expression of a yield function for sand based on the linear elastic threshold strain concept and empirical expression for the maximum shear modulus. The new yield function was calibrated for Toyoura sand. The calibration results show that the proposed yield surfaces are nonlinear curves that depend on the void ratio of sand in the p′-q plane, whereas the linear lines have been adopted in the bounding surface modeling of sand. This study also found that elliptic yield surfaces are the best fitted with the proposed yield surface and they can be used as alternatives to the proposed yield surface under the undrained shearing where the void ratio (or density) of sand is fixed.
1. Introduction
Bounding surface models [1–7] have successfully described mechanical responses of sand including highly nonlinear stress-strain relationship [8–11], entrance to the critical state upon prolonged shearing [10–13], and dilatancy [10, 14–18], which is plastic volumetric deformation caused by plastic shear deformation. Figure 1 shows the bounding, critical-state, and yield surfaces that have been commonly applied in the bounding surface models [1–7] in the p-q′ (where p′ is mean effective stress (=σ′kk/3), q is von Mises stress =((3/2) sijsij), and σ′ij is the effective stress (=σij − uδij), u is pore-water pressure, and sij is deviatoric stress (=σij–(σkk/3) δij)) plane where those surfaces are expressed as linear lines of which slopes are Mb, Mc, and m, respectively, under the isotropic consolidation conditions. In Figure 1, the critical-state surface is the final destination of the stress upon shearing; the bounding surface corresponds to the peaks which depend on the density (or void ratio) of sand and confining pressure acting on sand; the yield surface is a region where sand shows elastic responses.
To calibrate the critical state surface, the bounding surface models [1–7] relies on the final stress state (e.g., final values of p′ and q) of sand upon shearing after full stabilization of stress was confirmed; then, they calculated the critical state stress ratio Mc (=(q/p′)c (where subscript c denotes the critical state)) using the stress at the critical state; thus, this calibration step is quite straightforward. For the calibration of the bounding surface (which corresponds to the peak), the pairs (e, p′) of the void ratio and mean effective stress at the peaks are used to calculate the values of the state parameter ψ [19] (which is defined as a vertical distance to the critical state line from the current (e, p′) state in the e-p′ space) for numerous shearing tests; after this step, the slope Mb of the bounding surface can be expressed as a function of the state parameter ψ; thus, the calibration of the bounding surface relies on the quantitative manner. For the yield surface, the previous research studies on the bounding surface modeling for sand [1–7] have tried to reflect the fact that the elastic region of sand is very small in the stress space. The authors in [1, 2, 4] set the m = 0.05 in their models, whereas Dafalias and Manzari [5] used a yield surface with m = 0.01; some bounding surface models [3, 6, 7] even relied on the yield line (so, the slope m of a yield surface is zero) so that there is no elastic region in their models. For these bounding surface models, however, the determination of the yield surface has not followed the calibration step using experimental data in a quantitative manner. From this background, the following concern arises: “Is the linear line enough to express the elastic range of sand in the stress space?”. This study deals with this concern using the elastic threshold strain concept.
Figure 2 shows the typical degradation curve of the ratio between the secant shear modulus Gs and maximum shear modulus G0 under the undrained cyclic shearing. In Figure 2, if the cyclic shear strain γ is less than the linear elastic threshold strain γtl, the sand shows the linear elastic response and there is no plastic dissipation; therefore, the linear elastic threshold strain represents the elastic range in the strain field and it should correspond to the yield surface in the stress space. Although Figure 2 represents the undrained response, this degradation is valid also under the drained conditions until the shear strain is less than the volumetric (or cyclic) threshold strain, which is much greater than the linear elastic threshold strain γtl, because there is no dilatancy before the strain reaches the volumetric threshold strain. Based on this, this study aims to propose the new yield surface (or yield function) using the concept of the elastic threshold strain.
2. Formulation
According to Hardin and Richart [20], the maximum shear modulus G0 is a function of void ratio e and mean effective stress p′:where pA is the reference pressure (=100 kPa) and are positive material parameters. Equation (1) implies that G0 has the greater value as e is the less or p′ is the greater. If the shear strain is less than the elastic threshold strain, the sand shows the linear elastic response (Figure 2); in this case, under the pure shear loading conditions after the isotropic consolidation, the relationship between shear stress τ and (engineering) shear strain γ is written by
When shear strain γ exceeds the linear elastic threshold strain γtl, the plastic deformation starts to happen; thus, the yield shear stress τy can be written bywhich represents the yield condition of sand under the pure shear loading conditions after the isotropic consolidation. Substitution of equation (1) into equation (3) leads
Under the pure shear conditions (or the simple shear conditions with very small strain) after isotropic consolidation, the relationship between shear stress τ and von Mises stress q is
After substitution of equation (5) into equation (4),where qy is the yield von Mises stress; thus, under the pure shear loading after the isotropic consolidation, the yield function of sand can be written by
The bounding surface models [1–7] have considered the stress anisotropy for the critical-state and bounding surfaces; however, they have assumed the isotropic yield surface so far; for the sake of simplicity, this study also relies on the isotropic yield surface for sand and there is no dependency of the yield function on loading directions. Considering the kinematic hardening rule, the yield function in equation (7) can be extended towhere a is the back-stress ratio which represents the middle line of the yield surface. The gradients ∂f/∂q and ∂f/∂p′ of the yield function f to q and p′ arewhere variable s has 1 and −1 if q > ap′ and q < ap′, respectively. The gradient ∂f/∂q (equation (9)) of f to q has the same form with the linear line yield functions [1, 2]; however, the gradient ∂f/∂p′ (equation (10)) of f to p′ has an additional term (the second term in equation (10)), which makes the yield surface gradually parallel to the back-stress ratio as p′ increases (as is generally less than 1).
3. Calibration Yield Surface for Toyoura Sand
The construction of the yield function (equation (8)) requires the calibration of , and γtl for sand. According to Bolton and Oztoprak [21], the lower bound, upper bound, and mean value of the linear elastic threshold strain γtl are 0, 3 × 10−5, and 7 × 10−6 from 750 test data for various sands; in this study, the average value 7 × 10−6 of γtl is used for the yield surface. For Toyoura sand (clean uniform sand of which maximum and minimum void ratios are approximately 1.0 and 0.6, respectively), Woo et al. [22] and Woo and Salgado [2] calibrated as 850, 2.17, and 0.45, respectively, using a number of the resonance column, torsional shear, and bender element test results. Table 1 listed the calibrated parameters of the proposed yield surface for Toyoura sand.
Figure 3 plots the calibrated yield surfaces with the back-stress ratio a = 0 for Toyoura sand when void ratio e = 0.6, 0.7, 0.8, 0.9, and 1.0; it also draws linear yield surfaces with m = 0.005, 0.01, 0.02, and 0.05. In Figure 3, the proposed yield surface (developed from the linear elastic threshold strain concept and based on the experimental data) depends on the void ratio; the denser sand has the greater yield surface. Although the linear yield surfaces have been commonly used for the bounding surface models [1, 2, 4, 5], Figure 3 shows that the proposed yield surfaces are nonlinear curves in the p′-q plane. Focusing on low confinement situations (p′ < 100 kPa), a yield surface [1, 2, 4] with m = 0.05 overestimates the elastic region of Toyoura sand, whereas yield surfaces [3, 5–7] with m < 0.01 generally underestimates the elastic region of Toyoura sand in the stress space.
4. Comparision with the Elliptic Yield Surface
Taiebat and Dafalias [23] assessed the availability of the elliptical, lemniscate, distorted lemniscate, and eight-curve functions as yield functions for soil constitutive modeling; according to Taiebat and Dafalias [23], the lemniscate, distorted lemniscate, and eight-curve yield surfaces have a sharp tip at the origin in the p′-q plane, whereas the elliptical yield surface has a smooth tip at the origin; as the proposed yield surface (Figure 3) has a smooth tip at the origin in the p′-q plane, this study selects the elliptic yield function to compare it with the proposed yield surface. In Taiebat and Dafalias [23], one of the elliptic yield functions can be written bywhere a is the back-stress ratio and m and p′0 are material parameters. Figure 4 illustrates the effect of parameters m and p0 on the elliptic yield surface in the p′-q plane. Figure 4(a) implies that when p0 is fixed, an increase of m inflates the elliptic yield surface. In Figure 4(b), the elliptic yield surface can be defined until p′ = p0 and the greater p′0 makes the elliptic yield surface have the greater vertical size at the same p′ value in the p′-q plane.
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(b)
The gradients of the elliptic function fe (equation (11)) to q and p′ are written by
The gradient of fe to q (equation (12)) is identical to the gradient of f (the proposed yield function) to q (equation (9)), whereas the gradient of fe to p′ (equation (13)) has a different second term from the gradient of f to p′ (equation (10)). In equation (10), ∂f/∂p′ is −∞ at the origin (p′ = 0) in the p′-q plane; as p′ increases, ∂f/∂p′ increases and it approaches to −sa as p′ goes to infinity; ∂fe/∂p′ is also −∞ at the origin (p′ = 0) in the p′-q plane; ∂fe/∂p′ approaches to −sa as p′ evolves to (1/2)p0 from the origin; beyond this point (p′ = (1/2)p0), ∂fe/∂p′ starts to increases; at p′ = p0, ∂fe/∂p′ gets ∞; thus, for the elliptic yield surface to describe the proposed yield surface properly, the parameter p0 should have great value so that the elliptic surface within p′ < p0/2 is close to the proposed yield surface.
Figure 5 shows how to determine p0 of the elliptic yield surface corresponding to the proposed yield surface for Toyoura sand within the typical range (0 to 3000 kPa) of p′. In Figure 5, the grey lines are elliptic yield surfaces when p0 = 5,000, 10,000, and 20,000 with m values that make the elliptic yield surfaces as close to the proposed yield surface as possible. Figure 5 shows that the greater p0 enables the elliptic yield surface more closed to the proposed yield surface with a wide range of p′; this study sets p0 as 20,000 for Toyoura sand. Figure 6 illustrates the proposed yield surfaces for Toyoura sand with void ratio e = 0.6, 0.8, and 1.0 overlapped the elliptical yield surfaces with various values of m and fixed value of p0 (=20,000). According to Figure 6, for Toyoura sand specimens with e = 0.6, 0.8, and 1.0, the best matched elliptic yield surfaces have m = 1.03e−3, 0.7e−3, and 4.5e−4, respectively. For the undrained shearing, where the density (or void ratio) does not change, these calibrated elliptic yield surfaces (which have a simpler form than the proposed yield surface) can be adopted instead after proper determination of p0 and m.
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(b)
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5. Conclusions
The present study proposes a rigorous mathematical expression of a yield function for sand based on the linear elastic threshold strain concept for the advanced constitutive modeling of sand. Conceptually, the linear elastic threshold strain in the strain space should correspond to the yield surface in the stress space. The new yield surface is formulated based on the linear elastic threshold strain that locates the yield points and empirical expression of the maximum shear modulus G0, which represents the stress-strain relationship within the elastic range. The proposed yield function was calibrated for Toyoura sand. The calibration results show the following: (1) the size of yield surface depends on the void ratio of sand; the denser sand has the greater yield surface and (2) the proposed yield surfaces are nonlinear curves in the p′-q plane, whereas the linear lines have been adopted in the bounding surface modeling of sand.
This study also compared the elliptic yield surface with the proposed yield surface. The elliptic yield surface can describe the proposed yield surface accurately with adjustment of parameters and it can be used alternatives to the proposed yield surface under the undrained shearing where the void ratio (or density) of sand is fixed.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The author gratefully acknowledges the financial support from the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (no. NRF-2018R1A2B2002869).