Sand response to a large number of loading cycles under zero-lateral-strain conditions: evolution of void ratio and small-strain stiffness

Geotechnical structures often experience a large number of repetitive loading cycles. This research examines the quasi-static mechanical response of sands subjected to repetitive loads under zero-lateral-strain boundary conditions. The experimental study uses an automatic repetitive loading frame operated with pneumatic pistons. Both vertical deformation and shear wave velocity are continuously monitored during 10 000 repetitive loading cycles. The void ratio evolves towards the terminal void ratio eTas the number of load cycles increases. The terminal void ratio eT is a function of the initial void ratio e0 and the stress amplitude ratio Δσ/σ0. The number of cycles N* required to reach half of the final volume contraction ranges from N*!1 for densely packed sands (e0!emin) to N!10 for loosely packed sands (e0!emax). As the soil approaches terminal density at a large number of cycles, peak-to-peak strains are dominated by elastic deformations, and the minute plastic strains that remain in every cycle reflect local and sequential contact events. The shear wave velocity increases during cyclic loading with data suggesting a gradual increase in the coefficient of earth pressure K0 during repetitive loading. Changes in shear wave velocity track the evolution of the constrained modulus M; in fact, the constrained modulus can be estimated from the shear wave velocity to compute soil deformation in a given cycle. A simple procedure is suggested to estimate the potential settlement a layer may experience when subjected to repetitive mechanical loads.

The design of geo-structures needs to consider the influence of repetitive loads on long-term performance, serviceability and safety. For example, this applies to energy-related geosystems such as pumped hydro-storage, monopile-supported wind turbines, compressed air energy storage and energy piles (Peng et al., 2006;Andersen, 2009;Pasten & Santamarina, 2011;Sánchez et al., 2014;Loria et al., 2015). The number of mechanical loading cycles in these systems can be particularly high: monopilesupported wind turbines experience more than N = 10 7 cycles during a typical design life (Achmus et al., 2009;Li et al., 2015), while tidal cycles exceed N = 10 4 in a 30 year design life.
The present study explores the mechanical response of sands subjected to repetitive loading under zero-lateral-strain conditions. This paper extends a previous study by the authors (Chong & Santamarina, 2016) to address a large number of load cycles (i.e. N = 10 4 ), various stress amplitude ratios Δσ/σ 0 and the analysis of elastic cyclic deformation and the accumulation of plastic strain. The new repetitive loading system designed for this study facilitates automatic stresscontrolled repetitive loading cycles and provides small-strain data. Basic concepts are reviewed first, followed by the experimental programme and test results. The discussion provides a procedure for first-order cyclic deformation and cumulative settlement estimates that are valuable for engineering applications.

SOIL RESPONSE TO REPETITIVE LOADS: TERMS AND PROCESSES
The analysis and discussion of the long-term response of soils subjected to repetitive mechanical loads require terms and concepts that are not part of the standard geotechnical vocabulary. These concepts and associated observations are briefly introduced and defined next.

Threshold strains
The imposed strain level determines the underlying particle-scale deformation mechanisms. On the one hand, if the strain level is smaller than the elastic threshold strain ε th | el , elastic deformations occur at grain contacts and the fabric remains constant (Vucetic & Dobry, 1991;Vucetic, 1994;. On the other hand, strain accumulations through particle rearrangement and fabric changes take place above the volumetric threshold strain ε th | v (note: ε th | v % 30·ε th | el -Ishihara, 1996). A theoretical analysis based on the Hertzian contact predicts a lower threshold strain for stiffer particles and at lower confining stresses (Santamarina et al., 2001). It remains unclear whether the threshold strain levels obtained under quasi-static loading conditions apply to the soil response under repetitive loading.

Loading history
In this study, the effect of repetitive loading on a sand specimen under zero-lateral-strain conditions is investigated. More complex deformation mechanisms may arise in engineering situations where every point in a soil mass experiences different and evolving repetitive loading histories. For example, densification, convective flow cells and ratcheting develop in the complex boundary-valued problem of laterally loaded piles (Cuéllar et al., 2009(Cuéllar et al., , 2012.

EXPERIMENTAL STUDY: DEVICES AND TEST PROCEDURE
This experimental programme explores the evolution of the void ratio and the small-strain stiffness of sands subjected to repetitive mechanical loads under zero-lateral-strain conditions. The repetitive loading system consists of an instrumented floating-ring oedometer cell and an automatic stress-controlled frame (Fig. 1). The tall floating ring oedometer cell (inner diameter, ID = 52·5 mm, 3·9 mm wall thickness and 100 mm high) is designed to ensure the alignment of the top and bottom caps during repetitive loading (outer diameter, OD = 50·8 mm and 50·8 mm height).

Stress-controlled repetitive loading system
The loading system consists of a rigid reaction frame, pneumatic cylinders and peripheral control electronics. The computer sends the digitised command signal to the digital-to-analogue converter (DAC) (Labjack U3-LV). The controller (proportional-integral-derivative controller (PID) -Enfield Technologies C1) activates the pneumatic valve (Enfield Technologies LS-V25s) to match the analogue command signal from the DAC and the analogue signal received from the pressure transducer. The loading system can operate in a strain-controlled mode if the linear variable differential transducer (LVDT) response is used as the feedback signal instead of the pressure transducer.

Deformation monitoring
The LVDT (TransTek DC 0242) clamped to the top and bottom caps tracks the vertical deformation of specimen. LVDT data are saved by the data logger (Keysight 34970A, Fig. 1).

Shear waves: bender elements
The top and bottom caps include bender elements which are mounted inside removable nylon screws. Grounded parallel-type bender elements were used for both the source and receiver to minimise crosstalk (Lee & Santamarina, 2005). The bender elements are 12·7 mm Â 8 mm Â 0·7 mm in size, and are mounted with a 5 mm cantilevered length (7·7 mm anchored length). The function generator sends a 10 V step signal every 50 ms (Keysight 33210A). Received signals go through a filter-amplifier (Krohn-Hite 3364 -500 Hz high-pass and 200 kHz low-pass window) before they are averaged in the oscilloscope and stored. (Keysight DSOX 2014A -1024 stacked signalssee implications of signal stacking in Santamarina & Fratta (2005).)

Test procedure
The specimen preparation method involves different tamping energies to attain target relative densities between D r = 30 and 70%. The loading sequence consists of four stages: (a) static step loading to σ 0 ; (b) repetitive loading by N = 10 4 cycles with amplitude Δσ; (c) static loading to the maximum vertical stress; and (d ) unloading. Loading cycles have a period of T = 12 s to avoid dynamic effects ( f = 0·083 Hz). Previous studies have revealed that the loading frequency f does not affect the strain accumulation rate when f = 0·05 to 30 Hz (Youd, 1972;Shenton, 1978;Tatsuoka et al., 1986;Wichtmann, 2005).
Shear wave measurements during the repetitive loading stage take place at the same vertical effective stress σ 0 + Δσ at the top of cycles i = 1, 10, 100, 300, 1000, 3000 and 10 000. Table 1 summarises all test conditions.

EXPERIMENTAL RESULTS
All 33 tests in Table 1 exhibit similar trends. This section presents detailed experimental results for two specimens D r = 44% and D r = 86%, subjected to loading cycles with stress amplitude Δσ/σ 0 = 1·3 (Table 1). The complete dataset is then analysed in the subsequent section.
Effective stress σ′ z plotted against void ratio e Figure 2 presents the change in void ratio during the staticrepetitive-static loading history followed by the static unloading stage. The void ratio for the loosely and densely packed sands decreases monotonically during the initial static loading stage. The dotted circles indicate creep deformation at a constant load. Repetitive loading cycles cause higher contraction in the looser specimen.
Vertical effective stress σ′ z plotted against strain ε z Figure 3 presents the same repetitive load response shown in Fig. 2 in the stress-strain space. The recoverable and irrecoverable strains are clearly seen in the earlier loadingunloading cycles. The plastic strain accumulation becomes negligible as the number of cycles increases, but it remains hystereticthat is plastic shakedown. Figure 4 illustrates the void ratio evolution with the number of cycles for the two specimens reported above (D r = 44% and 86%; Δσ/σ 0 = 1·3). Once again, most changes in void ratio take place in earlier cycles and are more significant in the loosely packed sand. Figs 4(a) and 4(b) show experimental results fitted with a modified accumulation model (see Paute et al., 1993;Chong & Santamarina, 2016)

Evolution of void ratio
where the void ratio e i is measured after the ith cycle, and the void ratio e 1 corresponds to i = 1. The fitting parameters include the exponent m and the asymptotic terminal void ratio e T as i!∞; the number of cycles is (1 + N*) when the compaction is half of the asymptotic contraction (e 1 À e T )/2. 174 *Note: target value. The measured value is used in all plots and analyses.
Shear wave signals Figure 5 shows the shear wave signal cascades recorded during the 'static-repetitive-static' loading history followed by the unloading stage. The travel time decreases during the first quasi-static loading stage. Changes in the first arrival are less obvious during repetitive loading. The present authors use the time-stretched cross-correlation method in Coda wave analysis to assess these minor changes in travel time (Snieder, 2006;Dai et al., 2013). Then, the stretching factor θ allows minute changes in the shear wave velocity to be determined during repetitive loading cycles.

ANALYSES AND DISCUSSION
In this section, the complete dataset produced with the 33 tests run for this study (Table 1) is reported and analysed.

Terminal void ratio and maximum contraction
The terminal void ratio e T and the characteristic number of cycles N* describe the soil response to repetitive loads for engineering purposes (equation (1)). Fig. 6(a) shows that the asymptotic terminal void ratio e T scales with the initial void ratio e 0 . In other words, specimens do not evolve to a single soil fabric but retain memory of their initial fabric even after a very large number of cycles (see also López-Querol & Coop, 2012;Chong & Santamarina, 2016). However, the terminal void ratio e T decreases as the stress amplitude ratio Δσ/σ 0 increases (note: this statement applies to soils under zero-lateral-strain conditions, and should not be extended to dilative soils subjected to repetitive deviatoric loads under triaxial conditions). In fact, the dimensionless volume contraction λ = (e T Àe min )/(e 0 À e min ) is strongly correlated to the stress amplitude ratio Δσ/σ 0 as shown in Fig. 6 The characteristic number of cycles N* required to reach half of the asymptotic volume contraction Δe = (e 1 À e T )/2 increases with the increasing initial void ratio e 0 (Fig. 6(c)); in fact, N*!1 for densely packed sands (e 0 !e min ) and N*!10 3 for loosely packed sands (e 0 !e max ). Less obvious is the fact that N* also increases with stress ratio Δσ/σ 0 . This occurs because the terminal void ratio e T is lower for higher Δσ/σ 0 , hence, a larger number of cycles is required to attain half of the asymptotic contraction Δe = (e 1 À e T )/2. Fig. 6(d) shows that the m exponent is relatively constant, m = 0·45 ± 0·05. Finally, the number of cycles i η required to attain a certain η fraction of the ultimate compaction (e 1 À e i ) = η(e 0 À e T ) is computed from equation (1) Maximum change in relative density ΔD T . The modified accumulation model allows the asymptotic terminal void ratio e T to be estimated (equation (1) and Fig. 4). Then, the maximum change in relative density ΔD T = (e 0 À e T )/ (e max À e min ) from i = 0 to i!∞ is a function of the asymptotic contraction Δe = e 0 À e T and the attainable void ratio range e max -e min . The void ratio range e max -e min is selected to generalise these results to other coarse-grained soils subjected to repetitive loads. Figure 7 plots the maximum change in relative density ΔD T against the initial relative density D i=0 = (e max À e 0 )/ (e max À e min ) at i = 0 for all specimens and stress amplitude ratios Δσ/σ 0 = 0·8 to 4·0; data for Δσ/σ 0 = 0·10 extracted from Chong & Santamarina (2016) are superimposed on Fig. 7. Then, assuming no volume contraction when the initial void ratio e 0 = e min , the anticipated maximum change in relative density is a function of the stress amplitude ratio and the initial relative density as In this study, B % 20 to 25 and n = 0·7 for all cases.
Strain and particle-scale deformation mechanisms Strain levels. The plastic strain accumulation per cycle decreases and becomes negligible as the number of vertical K 0 load cycles increases. The first ten loading-unloading cycles illustrated by the dotted line in Fig. 3 highlight both recoverable and irrecoverable deformations. The upper and lower accumulation model trends fitted to test results e u and e l define the peak-to-peak strain in i cycle ε pp (i) (Fig. 4(b) and equation (1)) However, the permanent change in void ratio between two consecutive cycles defines the plastic strain ε pl (i) between the i and i + 1 cycles (Fig. 4(b))  For example, consider the data shown in Fig. 4. Equation (2) is used to compute the i η cycle for a target η = 90% of the compaction to terminal density, and equations (4) and (5) to estimate the peak-to-peak strain ε pp and the plastic strain ε pl in the i η cycle. The dense sand will reach 90% compaction after i η = 48 600 cycles, the peak-to-peak strain in that cycle will be ε pp = 3·1 Â 10 À4 and the plastic contraction will be ε pl = 1·6 Â 10 À9 ; corresponding values for the loose sand are i η = 85 000 cycles, ε pp = 2·9 Â 10 À4 and ε pl = 1·8 Â 10 À9 . Clearly, elastic deformations dominate the peak-to-peak strain as the specimen evolves towards terminal density at a large number of cycles; particle-scale deformation mechanisms are explored next.
Deformation mechanisms. Particle-scale deformation mechanisms in sands include elastic deformation at particle contacts, particle rotation (when the coordination number is low), contact slippage and crushing (Whitman, 1970;Omidvar et al., 2012). The release of stored strain energy during unloading may also cause contact slippage and yield; consequently elastic strains are not necessarily equal to recoverable strains (Rowe, 1954;Barden & Khayatt, 1966;Daramola, 1980). Monotonic one-dimensional (1D) compression test results for loose and dense Ottawa sand specimens show that the yield stress is σ y % 20 MPa (Roberts & de Souza, 1958;Valdes, 2002;Guimaraes et al., 2007). Therefore, the stress levels imposed during repetitive loading in this study are low, (σ + Δσ)/σ y , 0·03, and grain crushing is disregarded as a primary deformation mechanism. Still, asperity breakage and fretting damage are anticipated (Johnson, 1955(Johnson, , 1961SEM evidence in Park (2018)). Figure 8 presents peak-to-peak strains ε pp at η = 95% compaction. The corresponding plastic strains are between ε pl = 1 and 9 Â 10 À10 for all tests. For comparison, values of the volumetric threshold strain determined during cyclic loading are superimposed on Fig. 8. While the peak-to-peak strain levels are in the range of the threshold strain previously measured for cyclic loading, small plastic strains take place in every cycle. The plastic strain level per cycle is smaller than the strain level that would be required for atomic-scale displacement between contacts ε = 0·1 nm/D 50 % 10 À7 . Therefore, as the specimen approaches terminal density, the accumulation of plastic deformation consists of successive, individual contact events (slippage or abrasion) in every cycle, rather than evenly distributed plastic strains throughout the specimen.

Energy loss and stiffness
Damping ratio D and constrained modulus M. The damping ratio D = ΔW/(4πW) relates the consumed energy ΔW to the stored energy W in a cycle. However, the constrained modulus in a cycle relates the applied cyclic stress to the peak-to-peak strain M = Δσ/ε pp . From data in Silver & Seed (1971) From data in Dobry et al. (1982) Vucetic (1994 Fig. 8. Peak-to-peak strain ε pp values after η = 95% compaction. Data show the effect of initial void ratio e 0 and stress amplitude ratio Δσ/σ 0 . The corresponding plastic strains are between ε pl = 1 × 10 −10 and 9 × 10 −10 for all tests a selected number of cycles i = 1, 2, 10, 100, 1000 and 10 000. The damping ratio decreases while the constrained modulus increases with the number of cycles in all 33 tests. Most of the reduction in the damping ratio takes place in the first few cycles (see also Stokoe et al. (1999)), whereas the increase in the constrained modulus continues even after a large number of cycles (see resilient modulus data in Monismith, 2004;da Fonseca et al., 2013). The stress amplitude ratio Δσ/σ 0 affects the variation of the constrained modulus as the number of cycles increases. Overall, repetitive mechanical loads result in denser, stiffer and less attenuating sand fabrics under zero-lateral-strain conditions.
Shear wave velocity during static loading. The shear wave velocity is a power function of the vertical σ′ z and horizontal σ′ x effective stresses (vertical propagation, horizontally polarised - Yu & Richard, 1984;Santamarina et al., 2001) where the α factor is the shear wave velocity at effective stress σ′ mean = 1 kPa and the β exponent represents the stress sensitivity of the shear wave velocity. The shear wave velocity-stress relation captures both contact behaviour during wave propagation and fabric changes associated with stress changes (Cha et al., 2014). Figure 10(a) summarises the evolution of the shear wave velocity measured during the static (labelled 1), repetitive (2) and static (3) loading stages for a loose and a dense sand. Fig. 10(b) presents the α factors and β exponents computed by fitting equation (6) to the static load stages before and after repetitive loading for all tests. The data trend is consistent with the relationship between α and β values reported by Cha et al. (2014). Overall, the sand becomes stiffer (lower C c ), the α factor increases, the β exponent decreases and the soil fabric becomes less sensitive to stress changes after repetitive loading.
Shear wave velocity during repetitive loading. Small-strain stiffness measurements can be used to estimate the lower bound for the repetitive elastic deformation. Previous studies showed that the small-strain stiffness evolves during repetitive loading cycles (e.g. resonant column: Drnevich & Richart (1970) and Triantafyllidis et al. (2004); V p and V s : Wichtmann (2005) and Papadopoulos (2014)). Fig. 10(c) presents shear wave velocity data gathered in this study against the number of loading cycles for three loosely packed sands subjected to different stress amplitude ratios Δσ/σ 0 . The shear wave velocity increases with log(i) for all stress amplitude ratios Δσ/σ 0 (Fig. 10(c)) The d exponent is a function of the shear wave velocity for the first cycle V s | 1 (Fig. 10(d)). Data points cluster along linear trends for each stress amplitude ratio Δσ/σ 0 . It is concluded that the rate of increase in small-strain stiffness during repetitive loading is more pronounced in looser and  softer sands subjected to high stress cycles; that is, when more significant fabric changes are expected.
Evolution of K 0 . The coefficient of earth pressure K 0 at rest relates the in situ horizontal effective stress to the vertical effective stress under zero-lateral-strain conditions (Mayne & Kulhawy, 1982;Talesnick, 2012). The velocity-stress data before and after the repetitive loading stage will now be analysed to estimate the evolution of the coefficient of earth pressure K 0 during repetitive loading. From equation (6) V Assuming a nominal initial value K 0 | 1 = 0·42, the α and β values fitted before and after repetitive loading allow a stress ratio K 0 | N % 0·48 ± 0·02 to be anticipated immediately after repetitive loading with stress amplitude ratio Δσ/σ 0 = 1·3. Hence, indirect shear wave velocity measurements suggest a gradual increase in the horizontal stress during repetitive loading under zero lateral strains (see also data in Sawicki & Swidzinski (1995)). From an activation energy perspective, repetitive loads facilitate the development of processes that must overcome an energy barrier, such as friction and particle rearrangement. Related phenomena include time and ageing, heating (in creep), and vibration (rheopexy). During repetitive loading, horizontal stresses are gradually locked in by structural changes and friction, and K 0 increases proportionally to the stress amplitude ratio Δσ/σ 0 .
Estimation of the constrained modulus from shear wave velocity. The 'mechanistic' method for pavement design uses the resilient modulus to compute elastic deformations (Monismith, 2004;NCHRP, 2004). The resilient modulus E r = σ d /ε r is measured in a triaxial cell and is defined as the ratio between the applied vertical cyclic stress Δσ and the recoverable axial strain ε r after a predetermined number of load cycles (Davich et al., 2004). The resilient modulus can be estimated from elastic wave propagation to circumvent measurement difficulties and for the assessment of existing pavements. Published results show that the ratio between the measured resilient modulus and the small-strain Young's modulus computed from the wave velocity is in the range E r /E max = 0·4 to 0·8 (Davich et al., 2004;Schuettpelz et al., 2010).
In this study, the ratio M/G max is explored, between the constrained modulus M preferred for 1D deformation analysis and the maximum shear stiffness G max obtained from shear wave velocity (note: M and G max involve different strain levels and they are not related through a single value of Poisson ratio). Fig. 11 presents the M/G max ratio plotted against the number of cycles for different stress amplitudes Δσ/σ 0 . The stiffness ratio ranges between 1·2 M/G max 2·3; lower values apply to denser specimens, M/G max increases with the number of cycles and for lower stress amplitude ratios Δσ/σ 0 .

Cyclic deformation and maximum settlement
Finally, an estimate will be made of the deformation a layer will experience in a given cycle, as well as the cumulative settlement of the layer when it reaches the terminal void ratio.
The strain in the ith cycle ε i = Δσ/M is the ratio between the cyclic stress amplitude Δσ and the constrained modulus M, which can be estimated from shear wave velocity measurements in the field, as indicated earlier (Fig. 11). Then, a layer of thickness H will deform δ = HΔσ/M.
The maximum cumulative strain ε T as i!∞ is estimated from the initial and the terminal void ratios Δe = e 0 À e T (refer to results in Figs 6(a) and 6(b)) where a = 0·05 for the sand tested in this study ( Fig. 6(b)). Then, the settlement S T of a layer thickness H subjected to repetitive mechanical loading under zero-lateral-strain conditions is These equations highlight the importance of the initial e 0 and terminal e T void ratios. Furthermore, the terminal void ratio is readily estimated from e min and the stress amplitude ratio Δσ/σ 0 . For example, consider a large footing on a homogeneous sandy sediment (e max = 0·74, e min = 0·50, e 0 = 0·60, a = 0·05). The footing applies σ 0 (z = 0) = 100 kN/m 2 and Δσ = 50 kN/m 2 . From Figs 6(a) and 6(b) and equation (11), the settlement expected for an H = 5 m thick layer is S T = 5·7 mm.

CONCLUSIONS
The void ratio, the state of stress, and the small-strain stiffness of sands change during quasi-static repetitive mechanical loads under zero-lateral-strain conditions. Notable conclusions from this study are listed below.
(a) The void ratio evolves towards the terminal void ratio e T as the number of load cycles i!∞. The terminal void ratio e T is a function of the initial void ratio e 0 when repetitive loads take place under zero-lateral-strain conditions. In other words, the soil retains a memory of its initial fabric. (b) The characteristic number of cycles N* required to reach half of the asymptotic volume contraction Δe = (e 1 À e T )/2 increases with an increasing initial void ratio e 0 ; in fact, N*!1 for densely packed sands (e 0 !e min ), and N!10 3 for loosely packed sands (e 0 !e max ). (c) As the soil approaches terminal density at a large number of cycles, peak-to-peak strains are dominated by elastic deformations, and the plastic strains are much smaller than the strain level required for atomic-scale displacement between contacts ε = 0·1 nm/D 50 . Therefore, the accumulation of plastic deformation near the terminal density consists of successive, individual contact events rather than evenly distributed plastic strains throughout the specimen. (d ) The changes in void ratio, damping ratio and constrained modulus reveal that repetitive K 0 loading results in denser, stiffer and less attenuating sand fabrics. All these changes are a function of the initial void ratio e 0 and the stress amplitude ratio Δσ/σ 0 . (e) Shear wave velocity-stress trends V s = α(σ′ mean /kPa) β capture both contact behaviour and fabric changes. Data show that sands become stiffer (lower C c ), the α factor increases, the β exponent decreases and the soil fabric becomes less sensitive to stress changes after repetitive loading. The increase in shear wave velocity indicates a gradual increase in horizontal stress during repetitive loading under zero lateral strains. ( f ) Changes in shear wave velocity and G max track the evolution of the constrained modulus M during repetitive loading. The stiffness ratio ranges between 1·2 M/G max 2·3 (lower values apply to denser specimens). Then, the maximum stiffness G max computed from the shear wave velocity can be effectively used to estimate the elastic cyclic deformation in a given cycle. (g) The maximum settlement a layer may experience when subjected to repetitive mechanical loads is a function of the initial and terminal void ratios. Data gathered in this experimental programme suggest a simple procedure to estimate the maximum settlement based on the initial and the minimum void ratios (e 0 À e min ) and the cyclic stress ratio Δσ/σ 0 .