Soil Response to Repetitive Changes in Pore-Water Pressure under Deviatoric Loading

Soils often experience repetitive changes in pore water pressure. This study explores the volumetric and shear response of contractive and dilative sand specimens subjected to repetitive changes in pore water pressure, under constant deviatoric stress in a triaxial cell. The evolution towards a terminal void ratio eT characterizes the volumetric response. The terminal void ratio eT for pressure cycles falls below the critical state line, between emin < eT < ecs. Very dense specimens only dilate if they reach high stress obliquity ηmax during pressurization. The terminal void ratios for very dense and medium dense specimens do not converge to a single trend. The shear deformation may stabilize at shakedown, or continue in ratcheting mode. The maximum stress obliquity ηmax is the best predictor of the asymptotic state; shakedown prevails in all specimens subjected to stress obliquity ηmax < 0.95 · ηcs and ratcheting takes place when the maximum stress obliquity approaches or exceeds ηmax ≥ 0.95 · ηcs. Volumetric and shear strains can accumulate when the strain level during pressure cycles exceeds the volumetric threshold strain (about 5 × 10−4 in this study). A particle-level analysis of contact loss and published experimental data show that the threshold strain increases with confinement p 0 o. DOI: 10.1061/(ASCE)GT.1943-5606.0002229. This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/. Author keywords: Ratcheting; Pore water pressure cycle; Shakedown; Stress obliquity; Terminal void ratio. Introduction Soils experience repetitive changes in pore water pressure during groundwater level oscillations associated with tidal and river level fluctuations, and engineered structures such as docks and managed reservoirs (O’Reilly and Brown 1991; Chu et al. 2003; Orense et al. 2004; Leroueil et al. 2009; Page et al. 2010; Nakata et al. 2013; Shi et al. 2016). Coupled processes may also cause pore-fluid pressure oscillations, for example, in the case of a soft clay subjected to temperature cycles (Abuel-Naga et al. 2007). Pore-fluid pressure fluctuations affect a wide range of geotechnical systems from foundations and slope stability to pumpedstorage hydroelectric power stations, aquifer storage and recovery systems, compressed air energy storage, enhanced oil recovery by cyclic water flooding and cyclic steam injection, and repetitive CO2 injection (Premchitt et al. 1986; Olson et al. 2000; Gambolati and Teatini 2015; Huang 2016; Chang et al. 2017). Soils gradually deform in response to all kinds of repetitive excitations. Repetitive changes in water pressure imply effective stress cycles that can lead to the accumulation of plastic volumetric and shear strains. This study explores the volumetric and shear response of contractive and dilative sands subjected to repetitive changes in pore water pressure under constant deviatoric stress. The following section presents a detailed review of the state of the art and identifies salient gaps in knowledge. Previous Studies: Asymptotic States Pore-Fluid Pressure Oscillation Previous studies explored the effects of repetitive changes in pore-fluid pressure in the context of engineering needs, such as slope failures (Nakata et al. 2013) or aquifer oscillations (Hung et al. 2012). The selected test boundary conditions reflected field situations: triaxial stress (clays, Ohtsuka and Miyata 2001; Ohtsuka 2007), plane strain conditions (sand, Nakata et al. 2013), and Ko-conditions (sand-silt mixtures, Chang et al. 2017). In all cases, the strain accumulation induced by repetitive changes in pore-fluid pressure became more significant with an increasing pressure amplitude Δuw. However, previous studies did not separate the volumetric response from the shear response; these are analyzed next. Volumetric Asymptotic State: Terminal Void Ratio All soils evolve towards an asymptotic terminal void ratio during repetitive loading (Narsilio and Santamarina 2008—Refer to the p 0-e quadrant in Fig. 1). The tendency towards a terminal state is apparent in published data for all types of repetitive loads: porewater pressure cycles (Chang et al. 2017), Ko-loading and deviatoric stress cycles (Triantafyllidis et al. 2004; Wichtmann et al. 2005; Chong and Santamarina 2016), freeze-thaw (Viklander 1998), dry-wet (Albrecht and Benson 2001), and chemical cycles (Musso et al. 2003). There are irreversible structural changes during repetitive loading and soil properties adapt as the soil transitions towards the terminal void ratio; for example, the fabric changes of clays during dry-wet cycles (Croney and Coleman 1954), permeability increases in freeze-thaw cycles and dry-wet cycles (Chamberlain et al. 1990; Albrecht and Benson 2001), shear strength increases in freeze-thaw cycles (Ono and Mitachi 1997; Qi et al. 2006), and there is gradual stiffening in cyclic Ko loading (Park and Santamarina 2019). Postdoctoral Fellow, Earth Science and Engineering, King Abdullah Univ. of Science and Technology, Thuwal 23955-6900, Saudi Arabia (corresponding author). ORCID: https://orcid.org/0000-0001-7033-4653. Email: junghee.park@kaust.edu.sa Professor, Earth Science and Engineering, King Abdullah Univ. of Science and Technology, Thuwal 23955-6900, Saudi Arabia. Note. This manuscript was submitted on July 8, 2018; approved on October 30, 2019; published online on March 10, 2020. Discussion period open until August 10, 2020; separate discussions must be submitted for individual papers. This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, © ASCE, ISSN 1090-0241. © ASCE 04020023-1 J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 2020, 146(5): 04020023 D ow nl oa de d fr om a sc el ib ra ry .o rg b y K in g A bd ul la h U ni ve rs ity o f Sc ie nc e an d T ec hn ol og y L ib ra ry o n 06 /2 9/ 20 . C op yr ig ht A SC E . F or p er so na l u se o nl y; a ll ri gh ts r es er ve d.


Introduction
Soils experience repetitive changes in pore water pressure during groundwater level oscillations associated with tidal and river level fluctuations, and engineered structures such as docks and managed reservoirs (O'Reilly and Brown 1991;Chu et al. 2003;Orense et al. 2004;Leroueil et al. 2009;Page et al. 2010;Nakata et al. 2013;Shi et al. 2016). Coupled processes may also cause pore-fluid pressure oscillations, for example, in the case of a soft clay subjected to temperature cycles (Abuel-Naga et al. 2007).
Pore-fluid pressure fluctuations affect a wide range of geotechnical systems from foundations and slope stability to pumpedstorage hydroelectric power stations, aquifer storage and recovery systems, compressed air energy storage, enhanced oil recovery by cyclic water flooding and cyclic steam injection, and repetitive CO 2 injection (Premchitt et al. 1986;Olson et al. 2000;Gambolati and Teatini 2015;Huang 2016;Chang et al. 2017).
Soils gradually deform in response to all kinds of repetitive excitations. Repetitive changes in water pressure imply effective stress cycles that can lead to the accumulation of plastic volumetric and shear strains. This study explores the volumetric and shear response of contractive and dilative sands subjected to repetitive changes in pore water pressure under constant deviatoric stress. The following section presents a detailed review of the state of the art and identifies salient gaps in knowledge.

Pore-Fluid Pressure Oscillation
Previous studies explored the effects of repetitive changes in pore-fluid pressure in the context of engineering needs, such as slope failures (Nakata et al. 2013) or aquifer oscillations (Hung et al. 2012). The selected test boundary conditions reflected field situations: triaxial stress (clays, Ohtsuka and Miyata 2001;Ohtsuka 2007), plane strain conditions (sand, Nakata et al. 2013), and K o -conditions (sand-silt mixtures, Chang et al. 2017). In all cases, the strain accumulation induced by repetitive changes in pore-fluid pressure became more significant with an increasing pressure amplitude Δu w . However, previous studies did not separate the volumetric response from the shear response; these are analyzed next.

Volumetric Asymptotic State: Terminal Void Ratio
All soils evolve towards an asymptotic terminal void ratio during repetitive loading (Narsilio and Santamarina 2008-Refer to the p 0 -e quadrant in Fig. 1). The tendency towards a terminal state is apparent in published data for all types of repetitive loads: porewater pressure cycles (Chang et al. 2017), K o -loading and deviatoric stress cycles (Triantafyllidis et al. 2004;Wichtmann et al. 2005;Chong and Santamarina 2016), freeze-thaw (Viklander 1998), dry-wet (Albrecht and Benson 2001), and chemical cycles (Musso et al. 2003). There are irreversible structural changes during repetitive loading and soil properties adapt as the soil transitions towards the terminal void ratio; for example, the fabric changes of clays during dry-wet cycles (Croney and Coleman 1954), permeability increases in freeze-thaw cycles and dry-wet cycles (Chamberlain et al. 1990; Albrecht and Benson 2001), shear strength increases in freeze-thaw cycles (Ono and Mitachi 1997;Qi et al. 2006), and there is gradual stiffening in cyclic K o loading (Park and Santamarina 2019).

Shear Asymptotic State: Shakedown or Ratcheting?
The asymptotic condition in shear governs the response of all structures, from pavements (Sharp and Booker 1984) to metals (Johnson 1986). The shear response falls into one of three asymptotic regimes, as observed on the q-γ quadrant in Fig. 1 (Alonso- Marroquin and Herrmann 2004;Werkmeister et al. 2005) • Elastic shakedown: non-hysteretic, fully recoverable deformation in every cycle. • Plastic shakedown: hysteretic stress-strain response without permanent deformation at the end of each cycle. • Ratcheting: the stress strain response is hysteretic, and there is continued plastic strain accumulation in every cycle. The asymptotic condition depends on the stress amplitude ratio, the cyclic stress ratio Δσ amp z =2p 0 o , and the cyclic shear stress level Δτ amp zθ =Δσ amp z (Wu et al. 2017;Cai et al. 2018;Gu et al. 2018). Ratcheting should be expected at large stress amplitudes and high stress obliquity η ¼ q=p 0 . It may also develop when a large number of cycles reaches a fatigue-induced tipping point, or when the stress level causes particle crushing (see data in Werkmeister 2003;Alonso-Marroquin and Herrmann 2004;Werkmeister et al. 2005;Wichtmann et al. 2005;da Fonseca et al. 2013).

Experimental Study
Tested Sand: Properties Table 1 summarizes the main characteristics of the "KAUST 20/30 sand" used throughout this study and includes index properties such as the particle shape, the coefficient of uniformity C u , and the extreme void ratios e max and e min . Measured values are compared against predicted values from index properties for self-consistent verification (refer to Table 1 for details).
The critical state provides a "reference asymptotic state" for this study. Fig. 2 shows data for a set of conventional consolidatedundrained CU triaxial tests projected onto p 0 -q-ε z -e-u planes.

Experimental Devices and Configuration
The triaxial system used to conduct the repetitive pressure cycles consists of (1) a triaxial cell with an LVDT (Linear Variable Differential Transformer) to track the vertical displacement, (2) a loading frame to apply a constant deviatoric stress, and (3) a pressure panel that generates cyclic changes in pore water pressure and measures volume changes.

Sample Preparation
We prepare loose, medium dense, and dense specimens using a combination of raining and tamping techniques to obtain different initial relative densities between D r ¼ 15% and 70%.

Loading Histories
We can simulate the effects of changes in water pressure through changes in either the back pressure or the confining pressure (Brenner et al. 1985;Anderson and Sitar 1995;Farooq et al. 2004;Orense et al. 2004). The two test procedures yield the same results if the Biot's coefficient χ ¼ 1 − B sk =B g remains close to χ ≈ 1.0, that is at a relatively low confining effective stress (note: B sk is the bulk modulus of the soil skeleton, and B g is the bulk modulus of the mineral that makes the grains, Skempton 1961; . In this study, we control the back pressure u w . Fig. 3 presents a subset of the stress paths explored in this study. Typical loading histories consist of five stages (1) isotropic consolidation, (2) drained deviatoric loading to stress obliquity η ¼ 0.33, (3) a decrease in back pressure u w to reach η ¼ 0.20, (4) repetitive changes in pore water pressure from η ¼ 0.20 to η max ¼ 0.50 for N ¼ 100 loading cycles (shown in red), and (5) strain-controlled undrained axial compression from η ¼ 0.20 to failure at a vertical strain rate of ε z ¼ 0.01=min. Table 2 summarizes the experimental study. Test parameters include the initial void ratio e o , cyclic pressure amplitude Δu w , and maximum stress obliquity η max ¼ q=p 0 min . Cyclic pressure amplitudes Δu w selected for this study represent various field conditions, such as tidal action (<170 kPa at Burntcoat Head and Leaf Basin in North America), seasonal fluctuations in ground water levels (70-100 kPa,- Hung et al. 2012;Huang 2016), injection pressures for injection wells and injection-recovery wells used in aquifer storage (<500 kPa,- Shi et al. 2016;Page et al. 2010), and some coupled processes (e.g., geothermallyinduced Δu w < 250 kPa, -Laloui 2001;Abuel-Naga et al. 2007).

Experimental Results
This section reports detailed experimental results for two sets of tests designed to explore the effects of maximum stress obliquity η max and initial confinement p 0 o . We analyze the complete dataset gathered in this study in the following section.  (Color) Conventional consolidated, undrained CU triaxial test data projected onto u w -ε z -q-p 0 -e planes (strain rate: ε z ¼ 0.01=min). Notation: αÞ, and stress obliquity η ¼ q=p 0 . Critical state parameters for the KAUST 20/30 sand: friction angle ϕ cs ¼ 31°, intercept of CSL at 1 kPa in e-log p 0 ¼ 0.845, and slope of CSL in e-log p 0 ¼ 0.074. For reference, the maximum and minimum void ratios are e max ¼ 0.786 and e min ¼ 0.533. Fig. 4 illustrates the load-deformation response of loose and medium dense sands initially loaded to the same p 0 ¼ 250 kPa and η min ¼ 0.20, and subjected to repetitive fluid pressure cycles to different maximum stress obliquities η max ¼ 0.33, 0.40, 0.45, and 0.50 (Fig. 3). The results show • Pre-loading. The void ratio decreases during isotropic confinement (p 0 ¼ 100 kPa, q ¼ 0) and deviatoric loading (p 0 ¼ 150 kPa, q ¼ 50 kPa). The vertical strain is very similar in all specimens as the stress obliquity reaches the initial value of η o ¼ 0.33 and during the first decrease in pore water pressure to reach η min ¼ 0.20. • Repetitive pressure cycles. All specimens exhibit volume dilation every time the pore pressure increases (the mean stress decreases, and obliquity increases from η min → η max ); however, there is residual contraction at the end of the cycle. The vertical strain increases during pressurization (η min → η max ) and accumulates at the end of every cycle. Volumetric contraction and vertical strain accumulation are more pronounced in specimens that reach a higher maximum stress obliquity η max during pressure cycles. Note that the initial void ratio e o of all specimens falls in the contractive zone just before repetitive loading; thereafter, the two specimens subjected to large pressure cycles (η max ¼ 0.50 and η max ¼ 0.45) become denser than at the critical state. • Undrained shear. All specimens reach the critical state line during the undrained deviatoric loading that followed the N ¼ 100 pressure cycles (p 0 -q-e space in Fig. 4). These results confirm that in the absence of overt localization the critical state line is not affected by the monotonic or cyclic loading history (Taylor 1948;Schofield and Wroth 1968;Castro et al. 1982;Mohamad and Dobry 1986).

Study 1: Maximum Stress Obliquity η max
Study 2: Confining Effective Stress p 0 Fig. 5 shows the p 0 -q-e-ε z load-deformation response of three medium dense specimens subjected to different initial mean stress values p 0 o . Initial conditions include specimens above and below the critical state line. Details of the loading history before repetitive loading is shown in Fig. 3. Pressure cycles cause changes in obliquity from η min ¼ 0.20 to η max ¼ 0.50 in all cases. The changes in void ratio and the vertical strain accumulation during the repetitive pressure cycles are more significant in the one specimen subjected to high initial mean stress p 0 o . Once again, all specimens shear and dilate as the pressure increases. However the overall void ratio trend is contractive at the end of every cycle. All three specimens land on the dilative side of the critical state at the end of cyclic loading and exhibit a dilative tendency during the final undrained shear.

Analysis of the Complete Dataset
This section analyzes the results of all tests conducted in this study (Table 2), with an emphasis on the shear strains and volume changes that occur during repetitive pressure cycles. Within triaxial boundary conditions, the shear strain γ ¼ ð3ε z − ε vol Þ=2 combines the vertical strain ε z and the volumetric strain ε vol . System compliance and inadequate saturation bias both the measured peak-topeak volumetric strain and the computed peak-to-peak shear strain. Therefore, figures and analyses in this section place emphasis on incremental and cumulative strains determined at the same pressure at the end of each cycle. Fig. 6 presents the shear strain accumulation as a function of pressure cycles. The initial mean stress is the same for all specimens, p 0 o ¼ 250 kPa, but pressure cycles reach different maximum stress obliquities η max . The shear strain accumulation model below fits data trends in all tests (modified from Chong and Santamarina 2016)

Shear Deformation
where a, b, c, and d are fitting parameters, and i is the number of loading cycles. The shakedown response corresponds to d ¼ 0, while d > 0 implies ratcheting. Table 2 summarizes the fitted model parameters for all tests. Results indicate that • The shear strain accumulation induced by pressure cycles is more pronounced in earlier cycles, in loose sands, in specimens that experience a higher maximum stress obliquity η max (for tests with the same initial p 0 o ), and in specimens subjected to a higher initial mean stress p 0 o (for tests that reach the same η max ).
• Shakedown is unmistakable for specimens with small η max . In general, all specimens subjected to stress obliquity η max ≤ 0.50  Table 1 for a complete description). The loading history consists of five stages: (1) isotropic consolidation, (2) drained deviatoric loading to stress obliquity η ¼ 0.33, (3) decrease back pressure u w to reach η ¼ 0.20, (4) repetitive change in pore water pressure from η ¼ 0.20 to η max ¼ 0.50 (shown in red), and (5) strain-controlled undrained axial compression from η ¼ 0.20 to failure at a strain rate of ε z ¼ 0.01=min. Notation:  a Shear strain accumulation model: [5] [4] [3] [1] [4] [5]  Fig. 4. (Color) Maximum stress obliquity: Loose and medium dense sands subjected to repetitive fluid pressure cycles to different maximum stress obliquities η max . In all four specimens, the pressure cycles begin at p 0 o ¼ 250 kPa and η min ¼ 0.20. Tests end with undrained axial compression from the same initial stress condition at η min ¼ 0.20. Notation:   Figure 2 shows all stress paths in detail. Notation: αÞ, and stress obliquity η ¼ q=p 0 . Numbers in square brackets [#] indicate the Test number in Table 2. exhibit a shakedown response regardless of their initial density. For reference, the obliquity at critical state is η cs ¼ 0.52.
• The dense specimen subjected to pressure cycles above the critical state (η max ¼ 0.56) shows a ratcheting response (d ¼ 4 × 10 −4 ). This specimen gradually dilates during pressure cycles. Hence the frictional resistance evolves from ϕ peak towards ϕ cs ; eventually, a pressure cycle above critical state obliquity will cause the soil to fail. • Overall, the initial packing density determines the different failure modes when soils are subjected to pressure fluctuations. Loose soil will contract. Dense soil will experience dilation when pressure cycles reach high stress obliquity (above values corresponding to ϕ cs ).
These observations indicate that shear strain accumulation is a function of the initial void ratio e o , the initial confinement p 0 o , and the maximum stress obliquity η max reached during pressure cycles. Fig. 7 presents the evolution of the void ratio with the number of cycles for all specimens where pressure cycles start at p 0 o ¼ 250 kPa. Specimens in Fig. 7 have distinct initial void ratios e o (from e o ¼ 0.59 to e o ¼ 0.71; for reference, e min ¼ 0.533 and e max ¼ 0.786) and reach different maximum stress obliquity values η max . The highest rate of change in void ratio occurs during earlier pressure cycles and is more pronounced as the maximum stress obliquity increases. The void ratio e i measured at the end of the i th cycle evolves towards an asymptotic terminal void ratio e T in all specimens. The following accumulation model properly fits all datasets (Park and Santamarina 2019):

Void Ratio
where the m-exponent varies between m ¼ 0.8 to 1.0. The model parameter N Ã is the number of cycles required for a given specimen to reach half of the asymptotic volume change ðe o − e T Þ=2. Table 2 lists fitted model parameters for all tests.

Particle-Scale Deformation Mechanisms: Threshold Strain
In the absence of grain crushing, particle-scale deformation mechanisms relate to the strain level γ the soil experiences. There are two threshold strains under monotonic loading conditions 1. all deformations take place at contacts until the elastic threshold strain γ ≤ γ th j el that is selected at G=G max ≈ 0.99, and 2. there are minimal fabric changes until the volumetric threshold strain γ ≤ γ th j v . Typically, γ th j v ≈30 · γ th j el (Sands: Vucetic 1994, Ishihara 1996 Table 2. Slip-down, grain roll-over, and high frictional losses take place at strains above the volumetric threshold (Ishihara 1996;Mueth et al. 2000). Let's consider three spherical particles arranged in a triangular configuration and subjected to a normal force N [ Fig. 8(a), inset]. The shear force T increases until the contact force F 13 between particles ① and ③ becomes F 13 ¼ 0, which indicates contact loss. The extension of the 1 and 3 contact and the contraction of the 2 and 3 contact follow Hertzian behavior. Then, the horizontal displacement δ Ã of the top particle ③ relative to the interlayer height d · cos 30°yields the equivalent shear strain for contact loss γ th j loss as a function of the mineral shear modulus G g and the applied confining stress σ 0 estimated from the applied force as σ 0 ∝ N=d 2 This analysis anticipates that the threshold strain at contact loss increases with confining stress σ 0 in agreement with experimental evidence (Dyvik et al. 1984;Kim et al. 1991;Vucetic 1994). The threshold strain estimated using Eq. (3) is γ th j loss ≈ 5 × 10 −4 at p 0 o ¼ 250 kPa (Table 1; see data in Silver and Seed 1971;Dobry et al. 1982;Vucetic 1994;Santamarina and Shin 2009).
Clearly, there can be no volumetric strain accumulation when the cyclic strain level is too low for contact loss and fabric change. But, what is the threshold strain for repetitive pressure cycles? Let us compute the incremental volumetric strain in a given cycle Δε vol j i as a function of the change in void ratio between two consecutive cycles i and i þ 1 (taken at the same fluid pressure at the end of each cycle) Fig. 8(a) shows the absolute value of the incremental volumetric strain Δε vol j i for contractive and dilative specimens plotted against the peak-to-peak vertical strain ε pp z for all cycles. Data trends show that (1) volumetric changes diminish as the number of pressure cycles increases, and (2) volumetric changes vanish Δε vol j i → 0 as the peak-to-peak vertical strain ε pp z → 2-to−5 × 10 −4 .

Shakedown or Ratcheting?
The initial state of stress (p 0 o , q o ) and void ratio e o together with the amplitude of pressure cycles Δu w and the maximum stress obliquity η max determine the shear strain response of a soil subjected to repetitive changes in pore water pressure under constant deviatoric stress. The incremental shear strain Δγ i between two consecutive cycles i and i þ 1 scales with the maximum stress obliquity when η max < 0.95 · η cs , and gradually diminishes towards shakedown [Figs. 6(a) and 8(b)]. Ratcheting takes place when the maximum stress obliquity approaches or exceeds η max → η cs [Figs. 6(b) and 8(b)]. Note that Wu et al. 2017 report the onset of ratcheting behavior at η ¼ 0.50, i.e., close to failure.

Minimum Volumetric Strain
The volumetric strain ε vol ¼ Δu=B max computed using the smallstrain maximum skeletal bulk stiffness B max provides a lower bound estimate of the volumetric strain the soil will experience during a given pressure cycle Δu w . The maximum skeletal bulk stiffness can be computed from the in situ shear wave velocity B max ¼ 2 · ðV 2 s ρÞð1 þ νÞ=½3 · ð1 − 2νÞ, where ν is the small-strain Poisson's ratio. For example, consider a KAUST 20/30 specimen subjected to p 0 o ¼ 250 kPa and Δu w ¼ 100 kPa where the shear wave velocity for KAUST 20/30 sand increases with confining stress as V s ¼ 89 m=sðp 0 o =1 kPaÞ 0.21 and the small-strain Poisson's ratio is ν ≈ 0.15 (Note e o ≈ 0.65 in- Table 1). Then, the minimum peak-to-peak volumetric strain is ε vol ≈ 6 × 10 −4 .

Maximum Volumetric Strain
Terminal Void Ratio Fig. 9(a) compares the initial void ratio e o and the terminal void ratio e T for specimens with different e o , p 0 o , and η max (Note: p 0 o ¼ 250 kPa for the eight specimens in the dotted box, but symbols are p 0 -shifted to facilitate the visualization). Previous studies have suggested that there is a characteristic "terminal void ratio" for each loading condition (Narsilio and Santamarina 2008). Note that the critical state CS is the terminal state for monotonic shear. Results reported in this study show that loose to medium dense specimens contract to reach terminal void ratios that are denser than CS.   Table 2.
However, very dense specimens only dilate if pressurization causes high stress obliquity η max , and may rapidly evolve to failure without reaching a unique terminal state.

Potential Volume Change: Obliquity
Let us define the normalized asymptotic volume change ðe o − e T Þ=ðe o − e min Þ in terms of the initial void ratio e o at the beginning of pressure cycles (i ¼ 0), the terminal void ratio e T ð→ ∞Þ, and the minimum void ratio e min . Results discussed above suggest that the normalized volume change caused by fluid pressure cycles depends on the maximum stress obliquity η max [ Fig. 9(b)]. Contractive specimens experience volume change when obliquity exceeds η max > 0.3, and it is proportional to η max thereafter. On the other hand, significant volumetric dilation in dense specimens requires a stress obliquity η max greater than the critical state stress obliquity η max ≥ η cs ¼ 0.52. The minimum void ratio e min , the void ratio at critical state e cs , and the terminal void ratio for pressure cycles e T are all "asymptotic states" for a given sand (where e cs and e T are initial stress dependent). The preceding results show that terminal void ratios fall below the critical state line between e min < e T < e cs . Together, Figs. 7-9 suggest that the balance between internal deformation mechanisms depends on initial stress conditions p 0 o and q o , the maximum obliquity η max reached in pressure cycles and the initial void ratio e o .

Design Guidelines
The volumetric strain ε T associated with the maximum asymptotic change in void ratio Δe ¼ e o − e T induced by pressure cycles as i → ∞ is We cannot propose a definitive approach to estimate the terminal volumetric strain ε T for pressure cycles due to the limited dataset available at this time. However, the results in Fig. 9(b) suggest • The terminal change in void ratio for loose and medium dense sands is a μ-fraction of the void ratio difference e o − e min . In other words, • The μ-fraction is relatively low (i.e., μ ≤ 0.3) and is a function of the maximum stress obliquity η max .

Comparison between Pressure Cycles versus K o -Loading Cycles
The terminal void ratio evolves to its asymptotic state when the sand is subjected to repetitive vertical loading under zero lateral strain (previously reported in Park and Santamarina 2019). While boundary conditions are very different, both studies show that • There is a minimum strain required for plastic strain accumulation. The vertical threshold strain in the K o cell varies in the range of 2 to 7 × 10 −4 , which is similar to estimated values in this study. • All specimens contract in K o -loading cycles, but not in the pore-water pressure cycles with deviatoric loads (Fig. 7). Yet, the terminal void ratio falls between e o > e T > ð0.7 · e o þ 0.3 · e min Þ in both K o -loading and pressure cycle studies.

Ratio between Horizontal-to-Vertical Plastic Strains
The shear strain accumulation model γ i [Eq.
(1)] and the void ratio evolution model e i [Eq.
(2)] allow us to compute the incremental plastic vertical strain Δε pl z and plastic volumetric strain Δε pl vol between two consecutive cycles i and i þ 1. This approach avoids the inherent error magnification in incremental computations using experimental data For small strains, the ratio ν Ã between the incremental horizontalto-vertical plastic strains in axisymmetric conditions is   Table 2.
A ratio ν Ã ¼ 0.5 implies vertical deformation at constant volume (i.e., accumulation of vertical deformation at the terminal density). A ratio ν Ã → 0 corresponds to volume contraction under zero-lateral strain. A negative ratio ν Ã < 0 indicates that both vertical and horizontal contraction take place during repetitive loading; in fact, ν Ã ¼ −1 implies isotropic volume contraction. Finally, a positive ratio ν Ã > 0 indicates Δε pl vol < Δε pl z . Fig. 10 shows the evolution of the plastic strain ratio ν Ã with the number of cycles for loose, medium dense, and dense specimens. All trends exhibit an early dip into lower values of the plastic strain ratio (i.e., towards global contraction), followed by a gradual evolution to asymptotic trends.

Conclusions
Repetitive changes in pore water pressure can lead to the accumulation of plastic volumetric and shear strains. The initial state of stress and void ratio (p 0 o , q o , e o ), the amplitude of pressure cycles Δu w , and the maximum stress obliquity η max determine the volumetric and shear strain response.

Volumetric Response
The void ratio evolves towards an asymptotic terminal void ratio e T as the number of pressure cycles increases; the rate of change is more pronounced for high stress obliquity η max .
• The terminal void ratio for pressure cycles e T falls below the critical state line. The void ratio at critical state e cs for the same initial stress p 0 o and the minimum void ratio e min of the sand "bound" the terminal void ratio for pressure cycles e min < e T < e cs .
• The terminal void ratios for dilative and contractive specimens do not converge to a single trend.
• The terminal change in the void ratio (e o − e T ) in loose and medium dense sands increases with stress obliquity η max and is a fraction of (e o − e min ); for reference, ðe o − e T Þ ≤ 0.3 · ðe o − e min Þ in this study. • Dense dilative sands experience minimal void ratio changes and only dilate when η max approaches the critical state, η max ≥ 0.95 · η cs . Consequently, the frictional resistance evolves from ϕ peak towards ϕ cs and soils may fail in shear during subsequent pressure cycles.

Shear Response
The shear strain accumulation is more pronounced in earlier cycles, in loose sands, in specimens subjected to higher initial mean stress p 0 o and in specimens that experience a higher maximum stress obliquity η max .
• The shear deformation may stabilize at shakedown, or continue in ratcheting mode. The maximum stress obliquity η max is the best predictor of shakedown or ratcheting. • Shakedown should be expected as long as pressure amplitudes keep the stress obliquity below η max < 0.95 · η cs . Conversely, ratcheting takes place when the maximum stress obliquity approaches or exceeds η max ≥ 0.95 · η cs . Volumetric and shear strain accumulation during repetitive pressure cycles requires a minimum threshold strain which is estimated to be γ ≈ 5 × 10 −4 in this study. A particle-level analysis of contact loss and published experimental data show that the threshold strain increases with confinement p 0 . [7] max= 0.50 (p' o=125kPa)