Cyclic Hysteretic Behavior and Development of the Secant Shear Modulus of Sand under Drained and Undrained Conditions

In this study, cyclic triaxial and simple shear tests are utilized to evaluate the secant shear modulus of sand. In this section, the definitions of the stress and strain states in these tests are presented.

In the triaxial test, the axial and radial effective stress components can be described with σ₁ and σ₃, respectively (Fig. 2). The prime for expressing the effective stress is omitted in the following sections. With a cylindrical specimen, the stress and strain conditions are rotationally symmetric. The stress condition can be expressed with the Roscoe invariants mean stress p and deviatoric stress q as follows:

In the p–q plane, the stress state can be described with the stress ratio η = q/p (Fig. 3). For isotropic consolidation, η = 0. In the critical state, this stress ratio is η_c for compression and η_e for extension. The inclination of the critical state lines η_c and η_e can be calculated from the friction angle φ_c. After consolidation, a cyclic loading component q_ampl is applied to the specimen under average stress conditions p_av and q_av. The specimen deformation can be described with the axial strain ɛ₁ and the radial strain ɛ₃ (Fig. 2). The axial strain ɛ₁ is evaluated directly from the measurement of the axial displacement. The radial strain ɛ₃ is measured with a local displacement transducer (Wichtmann 2016; Goto et al. 1991) or calculated from the measured volumetric strain as ${\epsilon }_3=({\epsilon }_v\text{–}{\epsilon }_1)/2$. The common invariants of the strain components are the deviatoric strain ɛ_q and the shear strain γ with

The secant shear modulus G_s in the triaxial test can be defined as follows:with the increment in the deviatoric stress Δq = 2q_ampl = (q_max − q_min) and the increment in the deviatoric strain $\mathrm{\Delta }{\epsilon }_q=2{\epsilon }_{q_{\mathrm{ampl}}}=({\epsilon }_{q,\max }\text{–}{\epsilon }_{q,\min })$. In several studies, the increments Δq and Δɛ_q have also been referred to as double amplitudes. As an alternative to this calculation method, the secant shear modulus G_s can be estimated from Young's modulus E and Poisson's ratio ν, as follows:

This alternative method does not require measurement of the radial strain or volumetric strain. However, Poisson's ratio ν must be assumed. In undrained tests, Poisson's ratio is ν = 0.5, while in drained tests with sand, Poisson's ratio ν ranges from 0.20 to 0.40 (Wichtmann and Triantafyllidis 2009). Due to uncertainties in the estimation of Poisson's ratio, the authors prefer the calculation method according to Eq. (3). The difference between these two calculation methods is discussed in the Appendix.

Fig. 3 shows three possible initial stress states during the cyclic triaxial test: (I) compression; (II) isotropic conditions; and (III) extension. Under these conditions, the typical forms of the hysteresis loop are also presented in the figure. The definitions of the secant shear modulus in these cases differ and are explained in the following section.

Simple shear is defined as the vertical rectangular cross section deforming into a parallelogram during shearing, while the cross section in the horizontal plane remains unchanged. The stress and strain states of the simple shear test are shown in Fig. 2. In the most common type of simple shear device, only the normal and shear forces on the top surface of the specimen (σ_v and τ, respectively) can be measured. The complementary normal and shear stresses (σ_h and τ, respectively) on the side surface cannot be measured and are therefore unknown. The volumetric strain is identical to the axial strain ɛ_v = ɛ_a. Under this condition, the specimen deformation can be described with the axial strain ɛ_a and the shear strain γ, which is determined from the measured displacement of the top cap and the initial height of the specimen. The secant shear modulus G_s is calculated from the shear stress amplitude τ_ampl = (τ_max – τ_min)/2 and the shear strain amplitude γ_ampl = (γ_max – γ_min)/2 as follows:

Depending on the stress and loading conditions, a hysteresis loop can have different shapes. Therefore, a universal definition of secant shear modulus G_s does not exist. The problem of a nonclosed hysteresis loop is well known and has been mentioned in several studies (Wichtmann 2005; Glasenapp 2016; Wichtmann 2016; Le 2015). However, the determination of the secant shear modulus in these cases has not been clearly defined. In addition, in the current ASTM standards for simple shear and triaxial tests, a recommendation for the evaluation of G_s is missing, according to ASTM D8296-19 (ASTM 2019) ASTM D3999-1991 (Reapproved 2003) (ASTM 2003). In this section, the definitions of the secant shear moduli for the different forms of the hysteresis loop used in this study are presented.

In a perfectly closed hysteresis loop, the start and end points are identical (Fig. 1). The secant shear moduli calculated from the unloading and reloading branches are the same. However, under cyclic loading with large strains, the specimen is expected to undergo permanent deformation. The cyclic loop in these tests does not close. Depending on the loading conditions, a wide open hysteresis loop may occur (Fig. 3, Cases I and III). In these cases, the secant shear moduli for unloading and reloading must be separately evaluated.

In laboratory tests, cyclic loads are usually applied in the form of a combination of the static component and the harmonic oscillation component of the load. The static and oscillation components of the harmonic load are denoted by subscripts _av and _ampl, respectively. Under symmetric loading, the static load is zero (two-way loading). If the sign of the load remains unchanged during the test, it is referred to as one-way loading. The secant shear modulus for different forms of hysteresis loops is defined through example triaxial tests (Fig. 4). Figs. 4(a and d) show different oscillation components of the deviatoric stress q_ampl (Load cases I and II, respectively). In both cases, the deviatoric stress develops according to a sinusoidal curve with the same magnitude and frequency but with a π/2 phase shift. This suggests that the deviatoric stress q_ampl starts along the positive direction in Load case I [Fig. 4(a)] and along the negative direction in Load case II [Fig. 4(d)]. In Fig. 4, the first cycle is presented with a solid line, and the second cycle is presented with a dashed line. According to Load cases I and II, the hysteresis loops of the first and second cycles are presented under symmetrical two-way loading with q_av = 0 kPa [Figs. 4(b and e)] and under one-way loading with q_av = 100 kPa [Figs. 4(c and f)]. In Load case I, the secant modulus calculated from the extreme points of the loop is the value of the unloading branch, $G_{\mathrm{s}}^{\mathrm{U}}$ . In Load case II, the secant modulus between the extreme points is the value of the reloading branch, $G_{\mathrm{s}}^{\mathrm{R}}$. Considering this difference, the reloading modulus in Case I is defined based on the last quarter of the first cycle and the first quarter of the following cycle. $G_{\mathrm{s}}^{\mathrm{R}}$ is calculated as the inclination of the line between Extreme points B and C. Similarly, to evaluate the unloading modulus in Case II, Extreme points B and C are used. The turning point is not always clearly identifiable [Fig. 4(c), Point A]. In this case, the extreme point, defined as the tip of the loop according to Vucetic (1988), Mortezaie and Vucetic (2012), is used to calculate the modulus. In the simple shear tests, the evaluation procedures for the loading and unloading secant shear moduli are analogous.

To describe the reduction in the secant shear modulus with the variation in the shear strain amplitude γ_ampl, normalized values G_s/G_max are commonly used (Hardin and Drnevich 1972; Bai 2010; Wichtmann and Triantafyllidis 2013; Doroudian and Vucetic 1995; Kukusho 1980). Similarly, to quantify the development of G_s versus the number of cycles N during laboratory tests in large strain ranges, the concept of the stiffness index was proposed by Idriss et al. (1978). This index is defined as the normalization of the secant shear modulus with the modulus of the first cycle:

In cyclic tests within a large strain range, excess pore-water pressure or a change in density is expected. This can lead to a change in the secant shear modulus during the test. The stiffness index starts at δ = 1, expresses the variation in the shear modulus in relation to the initial value, and is called cyclic degradation.

The grain size distributions of the two sands tested in the present study are shown in Fig. 5. Sand A is a medium to coarse-grained sand originating from an excavation pit in northern Berlin. Sand B is also a medium to coarse-grained sand retrieved from another location in Berlin. Light microscopy images showed that these two sands exhibit a rounded grain shape (Glasenapp 2016). The soil mechanical characteristics of the tested sands are listed in Table 1. Further characteristics and standard test results for these sands can be found in Glasenapp (2016), Le (2015), and Bai (2010).

In the experimental investigations in this study, a simple shear device and two triaxial devices were used. The first cyclic triaxial device used a hydraulic driving system for axial load application and pneumatic controllers for cell and back pressure application. The volume was measured using a burette system, allowing an accurate measurement of the volume change during the test. This device was used to perform the tests under drained conditions with up to 100,000 cycles. More details on this triaxial device can be found in Le (2015). The second triaxial device used in this study was a device with an electromechanical driving system for cyclic axial load application. The back and cell pressures in this device were applied by two pressure controllers. These controllers were simultaneously used for volume measurement. The second triaxial device was more suitable for undrained tests with a significantly smaller number of cycles than for drained tests. The construction and technical details of this device were presented by Glasenapp (2016). All specimens for the triaxial tests exhibit the same diameter and height of D × H = 100 × 200 mm. The test materials were prepared by dry pluviation and then saturated under a back pressure of 300 kPa. The saturation degree was evaluated with the B value, according to Skempton (1954). All of the tests in this study yielded a B-value > 0.95.

In addition to the triaxial devices, an electromechanical cyclic simple shear device of the modified NGI type was used in this study. This test device could perform monotonic and cyclic loading with different boundary conditions: constant normal stress (drained condition) and constant volume (undrained condition). Furthermore, specimens of different sizes could be tested. In this study, a specimen diameter of 9 cm with a height of 2 cm was preferred. The construction and technical details of this device were presented by Le and Rackwitz (2020). As in the triaxial tests, the materials were prepared via dry pluviation. All simple shear tests were conducted with dry specimens. Undrained conditions were ensured via dry specimens with a constant-volume controller. The simple shear test procedure in this study was in accordance with ASTM D8296-19 (ASTM 2019).

The main focus of the study is the investigation of the characteristics and evolution of the secant shear modulus during cyclic loading considering the drainage condition and form of the hysteresis loop. Regarding these aims, cyclic triaxial and simple shear tests were conducted employing two sands under different stress and drainage conditions. The test programs are presented in Tables 2–4.

In several tests, the secant shear moduli for unloading $G_{\mathrm{s}}^{\mathrm{U}}$ and reloading $G_{\mathrm{s}}^{\mathrm{R}}$ differ. Fig. 6 shows a series of simple shear tests (Table 3: SS1‒SS8), in which the secant shear modulus is investigated depending on the average shear stress τ_av. All tests were conducted under constant normal stress conditions (drained) with σ_v = 200 kPa and an amplitude of τ_ampl = 30 kPa (stress-controlled). The average shear stress τ_av (also denoted as the static shear stress) varies between 0 and 70 kPa. The diagrams show the development of the secant moduli $G_{\mathrm{s}}^{\mathrm{U}}$ and $G_{\mathrm{s}}^{\mathrm{R}}$ depending on the cycle number up to N = 10,000. In all tests, except for symmetric loading (τ_av = 0 kPa), a significant gap between $G_{\mathrm{s}}^{\mathrm{U}}$ and $G_{\mathrm{s}}^{\mathrm{R}}$ is observed. This discrepancy between the secant shear modulus increases with increasing average shear stress. After a certain number of cycles, the gap between $G_{\mathrm{s}}^{\mathrm{U}}$ and $G_{\mathrm{s}}^{\mathrm{R}}$ vanishes. In the tests with higher τ_av, more cycles are needed to close the gap. The unloading secant shear modulus $G_{\mathrm{s}}^{\mathrm{U}}$ is unaffected by the variation in the average stress τ_av, while the reloading modulus $G_{\mathrm{s}}^{\mathrm{R}}$ decreases with increasing τ_av.

In Fig. 7, the evolution of the unloading and reloading secant shear modulus is presented exemplarily for drained and undrained triaxial tests. These two tests experienced identical initial stress conditions and the same cyclic loading (two-way loading with the same average deviatoric stress and amplitude). In the drained test [Fig. 7(a)], the unloading modulus $G_{\mathrm{s}}^{\mathrm{U}}$ does not indicate a clear tendency. During the first 100 cycles, the reloading modulus $G_{\mathrm{s}}^{\mathrm{R}}$ is lower than $G_{\mathrm{s}}^{\mathrm{U}}$ and continuously increases. Afterward, it reaches $G_{\mathrm{s}}^{\mathrm{U}}$ and remains with the same magnitude range. In the undrained test [Fig. 7(b)], the difference between $G_{\mathrm{s}}^{\mathrm{U}}$ and $G_{\mathrm{s}}^{\mathrm{R}}$ is much larger than that in the drained test. Due to the large deformation under undrained conditions, $G_{\mathrm{s}}^{\mathrm{R}}$ is significantly lower than $G_{\mathrm{s}}^{\mathrm{U}}$. The modulus for reloading $G_{\mathrm{s}}^{\mathrm{R}}$ rapidly increases throughout the whole test and approaches the curve of $G_{\mathrm{s}}^{\mathrm{U}}$. With a reversal tendency, $G_{\mathrm{s}}^{\mathrm{U}}$ continuously decreases during the test. These tests clearly show that the larger accumulation deformation in the undrained test leads to a larger difference between $G_{\mathrm{s}}^{\mathrm{U}}$ and $G_{\mathrm{s}}^{\mathrm{R}}$.

In the following sections, only the behavior of the unloading secant shear modulus is discussed. A detailed designation with superscript letter ^U is thus omitted.

The principle of the deformation behavior of soils differs under drained and undrained conditions. In simple shear tests with dry sand, these conditions can be realized with a constant volume (undrained) or a constant normal stress (drained). In this section, the difference in the development of the secant shear modulus between the drained and undrained tests is presented. Fig. 8 shows the results of the test program comprising 16 simple shear tests. The program consists of two series with different initial relative densities, I_D0: dense sand with an average relative density of approximately I_D0 = 60% [Figs. 8(a–d)] and medium sand with an average density of approximately I_D0 = 40% [Figs. 8(e–h)]. All tests were conducted with the same consolidation stress σ_v = 150 kPa under symmetric two-way loading (τ_av = 0 kPa). In the first test series, the amplitude was adjusted from 7.5 to 20 kPa. Considering the quicker liquefaction for medium sand, the amplitude in the second series was reduced to a value between 5 and 15 kPa. At each amplitude, a pair of drained and undrained tests was conducted. It can be observed, as expected, that the development of the secant shear modulus fundamentally differs. While under cyclic loading in drained conditions, the specimens become denser and stiffer. The secant shear modulus increases during the loading process and remains high. In the undrained tests, the secant shear modulus decreases in general and reaches zero upon liquefaction. The stiffness of the specimens is lost. The drainage condition only slightly affects the secant shear modulus of the first cycle. Except for a small discrepancy, the drained and undrained tests initially yield quite similar shear moduli. This statement can be confirmed by a comparison of the drained and undrained triaxial tests to analog test pairs (Glasenapp 2016).

In several studies, the stiffness index normalized with the modulus of the first cycle is used. However, the deformation measurement from the first cycle is not always objective. The first cycle in laboratory tests often exhibits an atypical deformation behavior relative to the further test cycles. The first cycle is called the irregular cycle, as reported in several studies (Wichtmann 2005; Le 2015; Glasenapp 2016; Niemunis et al. 2005). Due to several unavoidable issues with specimen preparation (e.g., the setting problem of the top cap and flattening of the surface or orientation of the soil particles during preparation), the first cycle usually indicates a significantly larger strain. Due to this problem, the use of this value in the evaluation of the stiffness index can lead to several uncertainties in the test results. The stiffness index shows the behavior of the specimen in the test devices but not always the natural characteristics of the soil. To investigate this problem in detail, an alternative stiffness index with the value of the second cycle G_{s,N = 2} was used to express the discrepancy when the first cycle was not considered. Furthermore, an additional triaxial test with a precycle (TX6, Table 2) for the elimination of the influence of the irregular cycle was conducted.

Fig. 9 shows demonstrative results of the undrained triaxial test with different definitions of the stiffness index. Fig. 9(a) shows the first three cycles of the tests with q_ampl = 20 and 40 kPa. The first cycle (dashed curve) indicates a visibly larger shear strain. The second and third cycles (solid line) exhibit a very similar hysteresis loop. In Fig. 9(b), the normalized secant shear modulus with G_{s,N = 1} is shown. The secant shear moduli in the test with q_ampl = 20 and 40 kPa increase at the beginning of the test and then decrease during cyclic loading. The maximum growth at q_ampl = 20 is more than 15%, and that at q_ampl = 40 kPa exceeds 10%. This behavior agrees with the simple shear test results reported in Vucetic et al. (2021) and Vucetic and Mortezaie (2015). Fig. 9(c) presents the normalized modulus using G_s,N=2. Here, the curves start at N = 2, and the value of the first cycle is not considered. The development of the degradation curves G_s,N/G_s,N=2 follows a similar trend to that of the G_s,N/G_s,N=1 curves. However, for q_ampl = 40 kPa, the degradation curve does not show an increase at the beginning of the test. For q_ampl = 20 kPa, the growth in the secant shear modulus at the beginning is significantly smaller (5% versus 15%).

To demonstrate the influence of the irregular cycle, the triaxial test with q_ampl = 40 kPa (TX4) was repeated with an additional precycle. In this repeated test, the influence of the first cycle was eliminated during the precycle. The precycle was applied under drained conditions and exhibited the same amplitude. After the precycle, the density condition of the test changed only minimally. The loading scheme of the two tests is presented in Fig. 10(a). At the start of the tests (undrained conditions), both tests indicate the same stress conditions and similar densities. The development of the secant shear modulus and stiffness index fundamentally differ [Figs. 10(b and c)]. In the test with the precycle, the degradation curve does not show an increase at the beginning of the test. In other words, the increase in the degradation curves in the test without the precycle generates due to the atypically lower modulus G_N=1 of the irregular cycle. The concept of the precycle to eliminate the effect of the irregular first cycle of undrained tests has been used in several studies (Wichtmann 2016; Wichtmann et al. 2010).

To quantify the cyclic degradation in the stiffness and the development of the secant shear modulus, the stiffness indexes δ_N (calculated with G_{s,N = 1}) dependent on the number of cycles are evaluated via simple shear tests (Tests SS9–SS24, Table 3). In the tests under constant-volume conditions [Figs. 11(a and c)], the stiffness index rapidly decreases to zero immediately after the specimen attains liquefaction. With small amplitudes, δ_N increases at the beginning of the test and reaches a peak (e.g., 1.11–1.15 for τ_ampl = 5 and 7.5 kPa). Under a constant normal stress, the specimen is compacted under cyclic loading. The density continuously increases during the test. In comparison to the behavior under constant-volume conditions, the specimen under constant normal stress in drained condition shows a completely different trend. The stiffness under constant normal stress conditions significantly increases [Figs. 11(b and d)]. In most of the tests, δ_N grows in the first 100 cycles and remains relatively constant at a high level under further loading. For similar amplitudes, the tests with looser sand indicate a larger increase in the stiffness index.

The stress level has a large influence on the stiffness of the soil. In this study, the dependency of the secant shear modulus on the stress level is investigated by means of a series of drained simple shear tests under strain-controlled cyclic loading. All tests presented in the previous sections are performed with stress-controlled cycles. In these tests, the shear strain amplitude changes continuously during cyclic loading. Therefore, to quantify the influence of the stress level, a series of strain-controlled simple shear tests with the same shear strain amplitude of γ_ampl = 0.03% and at different consolidation stresses in the range of σ_v = 25‒300 kPa were carried out. Fig. 12 shows the evolution of the secant shear modulus with the number of cycles [Fig. 12(a)] and the stiffness index [Fig. 12(b)]. Under cyclic loading in drained conditions, the specimens become denser and stiffer. The secant shear modulus increases during the loading process. The stiffness index expresses the percentage growth of the secant shear modulus compared with the modulus at the beginning of the test (first cycle). It can be observed that the evolution of the stiffness index is greater in the test with lower stress levels. In the test with σ_v = 25 kPa, the secant shear modulus doubles after 1,000 cycles, while at σ_v = 300 kPa, the increase is 25%. The significant change is observed primarily in the first 100 cycles.

To quantify the effect of the stress level on the stiffness, the secant shear modulus measured in the 1st, 10th, and 1,000th cycles is utilized. The mean stress levels in the simple shear test cannot be directly evaluated due to the unknown horizontal stress. Therefore, the mean confining stress is assumed to be p = σ_v(1 + 2K₀)/3 according to Silver and Seed (1971) with the coefficient K₀ = 1–sin(φ_c). The parameter φ_c is the critical friction angle (Table 1). The results in Fig. 13 show that the secant shear modulus increases disproportionately with the mean confining stress. The pressure dependence of G_s can be described here with the exponential relationship G_s ∼ pⁿ. Fitting to the test results gives an exponent of n = 0.78 for the modulus in the first cycle and n = 0.58 for the 1,000th cycle. The exponent n for describing G_s depends on the shear strain amplitude. For the same sand, an exponent of n = 0.52 is evaluated for G_max based on the resonant column test results (Le 2015). According to Hardin and Black (1966) and Wichtmann and Triantafyllidis (2009), the average value for the exponent to describe G_max is n = 0.5.

The influence of anisotropy in the loading condition can be investigated using triaxial tests. The development of the secant shear modulus in undrained triaxial tests under anisotropic and isotropic consolidation is presented in Fig. 14. All tests were conducted with the same mean confining stress p_av = 200 kPa and similar densities. The amplitude q_ampl was adjusted from 20 to 80 kPa. Under undrained conditions, the test with q_ampl = 80 kPa liquefies within two cycles and is not presented in Fig. 14(b). All anisotropic tests involve 1,000 cycles [Fig. 14(a)]. However, due to the small range of the local displacement transducer and the large deformation of the specimens, the secant shear modulus here cannot be evaluated over the whole test. The tests show cyclic mobility under anisotropic conditions, and the cycle indicated a butterfly shape, while under isotropic conditions, the specimens experience liquefaction. The results show that the modulus in the anisotropic tests approaches a constant value, while the modulus is zero upon liquefaction under isotropic conditions.