Sands Subjected to Repetitive Loading Cycles and Associated Granular Degradation

Park, Junghee; Santamarina, J. Carlos

doi:10.1061/JGGEFK.GTENG-11153

Open access

Technical Papers

Sep 14, 2023

Sands Subjected to Repetitive Loading Cycles and Associated Granular Degradation

Authors: Junghee Park https://orcid.org/0000-0001-7033-4653 [email protected] and J. Carlos Santamarina, A.M.ASCEAuthor Affiliations

Publication: Journal of Geotechnical and Geoenvironmental Engineering

Volume 149, Issue 11

https://doi.org/10.1061/JGGEFK.GTENG-11153

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Abstract

This study examines the load-deformation response of sands subjected to high- and low-stress cycles, i.e., both ends of the Wöhler’s fatigue curve. At high peak cyclic stress

σ_{f}

, the terminal void ratio decreases with

σ_{f}

due to crushing-dependent densification, and it can be smaller than

e_{\min}

when the peak stress approaches the yield stress

σ_{f} \to σ_{y}

. When

σ_{f} ≪ σ_{y}

, the soil retains memory of the initial fabric even after a very large number of cycles, and the terminal void ratio correlates with the initial void ratio

e_{o}

. Data show that the maximum change in relative density leads to simple strategies to estimate the maximum settlement for first-order engineering analyses. In agreement with Wöhler’s fatigue, tipping points in void ratio and stiffness trends occur at a small number of high-stress cycles or after a large number of small-stress cycles. During repetitive loading, sands stiffen with the number of cycles to reflect increased interparticle coordination following crushing, as well as contact flattening due to asperity breakage and fretting. The strong correlation between the resilient modulus

M_{r}

and the maximum shear modulus

G_{\max}

suggests the possible application of geophysical methods based on shear wave propagation to monitor geosystems subjected to repetitive loading cycles.

Introduction

Particles may crush when soils experience high effective stress such as beneath pile tips, steel rollers, and reservoirs during oil extraction. Particle crushing affects dilative tendencies, compressibility, and shear strength (Lade and Yamamuro 1996; Feda 2002; Uygar and Doven 2006; Sadrekarimi and Olson 2010). Furthermore, produced fines increase the soil specific surface area and reduce the hydraulic conductivity; they may migrate and clog the formation (Valdes and Santamarina 2006, 2008; Hafez et al. 2021) and could eventually cause undrained shear failures (e.g., railway ballast; Indraratna et al. 1997).

Repetitive loading cycles add the possibility of granular fatigue. Then, the load-deformation response may not be a single monotonic trend of the number of cycles and exhibit “tipping points” as observed in some datasets (Suiker et al. 2005; Indraratna et al. 2003, 2006; Werkmeister 2003; Wichtmann 2005; Werkmeister et al. 2005; da Fonseca et al. 2013; Indraratna and Nimbalkar 2013). In agreement with Wöhler’s S-N fatigue curve, particle degradation would be expected earlier under high-stress cycles than in low-stress cycles.

Granular degradation under cyclic loading affects the long-term performance of engineered geosystems. The purpose of this study is to advance our understanding of the load-deformation response and associated granular degradation when sands are subjected to repetitive vertical loading cycles under zero-lateral strain conditions. We explore both ends of Wöhler’s curve, i.e., high- and low-stress cycles, and monitor stiffness evolution and granular degradation.

Experimental Study

The experimental program consists of two end-member conditions: (1)

10^{2}

high-stress cycles (peak stress

σ_{f} = 2.5

, 5, 10, 15, and 25 MPa); and (2)

10^{6}

low-stress cycles (

σ_{f} = 167 kPa

).

Tested sand. The KAUST 20/30 sand used in all tests is a uniform quartzitic sand readily found in western Saudi Arabia. Table 1 summarizes its index properties including the coefficient of uniformity

C_{u}

, particle shape (i.e., roundness), extreme void ratios

e_{\max}

and

e_{\min}

, and yield stress

σ_{y}

under monotonic loading, i.e., the initiation of generalized crushing. (Note: there is no reliable criterion for selecting the yield stress; we fit the data using a hyperbolic function with asymptotically correct void ratios at low and high stresses, and adopt the point of maximum curvature as an indicator of yield stress.) For comparison and in order to confirm “tipping points,” we run a parallel set of low amplitude tests using Ottawa 20/30 another quartzitic sand, well studied in the literature; its index properties are included in Table 1 as well.

Table 1. Tested sands: properties

Property	Core study: KAUST 20/30 sand	Complementary study: Ottawa 20/30 sand
Particle size, $D$ (mm)	0.60–0.85	0.60–0.85
Mineralogy	Quartz 100%	Quartz 100%
Roundness, $R$	0.60	0.90
Coefficient of uniformity, $C_{u}$	1.20	1.20
Specific gravity, $G_{s}$	2.65	2.65
Maximum void ratio, $e_{\max}$	0.786	0.742
Minimum void ratio, $e_{\min}$	0.533	0.502
Critical state friction angle, $ϕ_{c s}$	31°	27°
Intercept (in $e - \log p^{'}$ ) $Γ$	0.845	0.802
Slope (in $e - \log p^{'}$ ) $λ$	0.074	0.047
Yield stress, $σ_{y}$ (monotonic loading)	10 MPa (at $D_{r} = 60 %$ )	28 to 48 MPa ( $D_{r} = 35 %$ -to-95%)

Instrumented oedometer cells. High- and low-stress tests use a floating ring oedometer cell configuration to minimize the effect of wall friction (Fig. 1). The tall rings guide the loading caps to prevent wall jamming and allow for thick specimens in order to avoid near-field effects in wave propagation. The wall thickness in both cases is designed to ensure a radial strain

ε_{r} \leq 5 \times 10^{- 5}

in order to maintain

k_{o}

conditions during repetitive loading (Okochi and Tatsuoka 1984). The resulting oedometer cell dimensions are summarized in Fig. 1.

Fig. 1. Instrumented oedometer cells and loading frames for (a) high-stress cycles ( $N = 100$ ); and (b) low-stress cycles ( $N = 10^{6}$ ). Both tests use floating ring oedometers, top and bottom caps instrumented with bender elements BE, and clamps for LVDTs. (a) Shows the peripheral electronics used to measure deformation and shear waves in both systems; and (b) shows the servo controlled pneumatic system (DAC = digital-to-analog converter; and PT = pressure transducer).

The repetitive loading system for high-stress cycles consists of a hydraulic jack (Enerpac, RSM-1000) connected to a hydraulic pump (ISCO–1000D) to produce a rectangular load cycle. The repetitive loading system for 1 million cycles involves a proportional-integral-derivative (PID) controller to generate the sinusoidal pressure cycle that drives the pneumatic cylinder (Fig. 1; details in Park and Santamarina 2019). LVDTs clamped to the top and bottom caps track changes in specimen height (TransTek DC 0242–Data logger Keysight 34970A).

Shear wave measurements for both high- and low-stress tests involve bender elements mounted on the top and bottom caps (Lee and Santamarina 2005). Both bender elements are parallel type to minimize crosstalk and for improved shielding, and cantilever 5 mm into the soil mass to ensure mechanical coupling. A function generator sends a 10 volt step signal every 50 ms to the source bender element (Keysight 33210A), and a filter amplifier connected to the receiver bender element conditions the received signal (500 Hz high-pass and 200 kHz low-pass window; Krohn-Hite 3364). We stack 1,024 signals to enhance the signal-to-noise ratio and store the resulting signal in the digital storage oscilloscope [Keysight DSOX 2014A; stacking design in Santamarina and Fratta (2005)].

Specimen preparation and test protocols. The specimen preparation method involves a predetermined tamping energy per layer to attain the target relative density

D_{r}

in each case (see details for dry tamping-based specimen preparation methods in Zhu et al. (2020, 2021). Specimens have a relative density

D_{r} \approx 75 %

for high-stress cycles, and three different relative densities

D_{r} = 30 %

, 50%, and 70% for low-stress cycles.

The loading sequence consists of monotonic loading to reach the initial vertical stress followed by the repetitive loading cycles:

•

High-stress (

N = 10^{2}

): We start the repetitive cycles from five different initial effective stress levels

σ_{o} = 0.5

, 1, 2, 3, and 5 MPa, and impose five different cyclic amplitudes

Δ σ = 2

, 4, 8, 12, and 20 MPa so that the ratio between the peak cyclic stress

σ_{f} = σ_{o} + Δ σ

and the initial stress

σ_{o}

is the same in all cases, i.e.,

σ_{f} / σ_{o} = 5

(Table 2). The loading period is

T = 60 s

for all specimens. Finally, we unload the specimens without applying additional static stress to avoid any additional grain crushing at these high-stress levels.

•

Low-stress (

N = 10^{6}

): The initial vertical stress is

σ_{o} = 67 kPa

and the repetitive cycles go from

σ_{o} = 67 kPa

to

σ_{f} = 167 kPa

; thus

Δ σ = 100 kPa

and the stress amplitude ratio is

(σ_{o} + Δ σ) / σ_{o} = 2.5

in all tests. The selected loading cycle period

T = 12 s

is a compromise between the sand relaxation time and the total test duration (139 days). After the stress cycles, we increase the static load to

σ_{\max} = 507 kPa

. Finally, we unload specimens.

Table 2. Test conditions: core study with KAUST 20/30 sand and complementary study with Ottawa sand

Test conditions	No.	$σ_{o}$ (MPa)	$Δ σ$ (MPa)	$σ_{f} = σ_{o} + Δ σ$ (MPa)	$σ_{f} / σ_{o}$	“As-prepared” void ratio $e_{initial}$	Void ratio at $i = 0$ $e_{o}$	Model parameters
Test conditions	No.	$σ_{o}$ (MPa)	$Δ σ$ (MPa)	$σ_{f} = σ_{o} + Δ σ$ (MPa)	$σ_{f} / σ_{o}$	“As-prepared” void ratio $e_{initial}$	Void ratio at $i = 0$ $e_{o}$	$e_{1}$	$e_{T}$	$m$	$N^{*}$
KAUST 20/30 sand
High stress $N = 100$ cycles	[1]	0.5	2	2.5	5	0.597	0.587	0.572	0.548	0.51	8
	[2]	1	4	5	5	0.589	0.577	0.559	0.533	0.54	8
	[3]	2	8	10	5	0.589	0.572	0.548	0.525	0.53	7
	[4]	3	12	15	5	0.590	0.568	0.533	0.487	0.50	9
	[5]	5	20	25	5	0.591	0.561	0.486	0.385	0.46	9
Low-stress $N = 10^{6}$ cycles	[6]	0.067	0.1	0.167	2.5	0.735	0.731	0.727	0.702	0.32	$4 \times 10^{5}$
	[7]	0.067	0.1	0.167	2.5	0.694	0.693	0.692	0.671	0.32	$3.5 \times 10^{5}$
	[8]	0.067	0.1	0.167	2.5	0.639	0.637	0.636	0.621	0.38	$3.3 \times 10^{5}$
Ottawa sand
One millime cycles	[9]	0.067	0.1	0.167	2.5	0.673	0.669	0.665	0.643	0.35	$5 \times 10^{5}$
	[10]	0.067	0.1	0.167	2.5	0.631	0.627	0.624	0.606	0.30	$4 \times 10^{5}$
	[11]	0.067	0.1	0.167	2.5	0.583	0.582	0.580	0.569	0.39	$3.5 \times 10^{5}$

This study refers to low-stress tests when stress conditions are significantly below the yield stress

σ ≪ σ_{y}

—in other words, when there will be no obvious contact damage during monotonic loading. Furthermore, we aim to extend monotonic loading beyond the stress at repetitive loading (i.e.,

σ_{\max} = 507 kPa

) to explore stiffness evolution past mechanical aging (without the effects of crushing). For example, the static-repetitive-static loading sequence allows us to examine the effect of repetitive loading on changes in

α

value and

β

exponent [see example in Park and Santamarina (2019)].

Shear wave signatures are gathered at the top of loading

σ_{f} = σ_{o} + Δ σ

at preselected cycles in both high- and low-stress studies. At the end of each test, we measure particle size distribution using sieve analysis and observe grains using optical and scanning electron microscopy. Given the relatively small fines fraction, we place special emphasis on fines separation (prolonged sieve shaking protocols without overloading the sieves), careful fines recovery, and precision weighing (= 0.0001g–ADB 200-4; KERN). Optical microscopy images confirm clean retained grains.

Experimental Results: Analyses

Particle Degradation

Fig. 2 presents particle size distributions before loading, and after

10^{2}

high-stress and

10^{6}

low-stress cycles. The amount of fines produced increases with the maximum stress level

σ_{f} = σ_{o} + Δ σ

. The repetitive load test that reaches

σ_{f} = 25 MPa

produces significantly more fines than the monotonic load that reaches

σ_{f} = 24 MPa

. The

10^{6}

low-stress cycles produce some fines as well even though the maximum stress is much lower than the yield stress

σ_{f} ≪ σ_{y}

(Fig. 2). The original particle size distribution before testing consists of 100% passing sieve #20 and retained on sieve #30; post-testing, the fines fraction passing sieve #30 is

\sim 7 %

after monotonic loading to 24 MPa, 0.6% to 30% after high-stress repetitive loading cycles (increases with stress amplitude

Δ σ

), and less than 0.8% for the low-stress repetitive loading cycles regardless of the initial packing density.

Fig. 2. Particle size distribution after test completion–KAUST 20/30 sand. Includes both high- and low-stress tests. Black circles show the original particle size distribution before testing. Numbers in square brackets [#] correspond to the test number (details in Table 2). Inset: schematic of the Wohler’s curve indicating the possible location of tests sequences in this study.

Based on our prior experience and published studies, we can anticipate higher fine production when more angular particles and higher initial void ratio specimens are involved (Lee and Farhoomand 1967; Sadrekarimi and Olson 2010). The particle size

d

and coordination number affect crushing. Smaller size particles tend to exhibit higher tensile strength when subjected to an axial force. On the other hand, smaller particles typically have a lower coordination number in a granular packing; therefore they are more prone to splitting (McDowell 1999; Coop et al. 2004; Guimaraes et al. 2007; Sadrekarimi and Olson 2010).

Optical and scanning electron microscopy (SEM) provide further insight into the evolution of particle degradation. Fig. 3 shows optical microscopy images of sand particles after repetitive loading to

σ_{f} = 25 MPa

(

Δ σ = 20 MPa

). The smaller particles produced during crushing have more angular shapes [see similar results in Guimaraes et al. (2007) and Sadrekarimi and Olson (2010)].

Fig. 3. Optical microscopy images of KAUST 20/30 sand after high-stress amplitude loading cycles: (a) original sand grains (0.60–0.85 mm) retained on different sieves after $10^{2}$ load cycles to $σ f = 25 MPa$ ( $σ_{o} = 5 MPa$ , $Δ σ = 20 MPa$ –Test 5 in Table 2); (b) 0.60–0.85 mm; (c) 0.25–0.425 mm; (d) 0.15–0.25 mm; (e) 0.15–0.075 mm; and (f) smaller than 0.075 mm. Note that the original particles are quite spherical and round, whereas the finer produced particles are platy and angular. (g) Shape chart modified after Krumbein and Sloss (1963) and Cho et al. (2006). Dotted lines indicate constant regularity = (Roundness + Sphericity)/2. R = “retained on”; P = “passing through”; and # = sieve number.

SEM images in Fig. 4 capture additional details, such as surface abrasion and fretting, asperity breakage, clean planar surface produced by tensile splitting, sharp blade-like fine particles, V-shaped pits, surface scratches, conchoidal fractures, preferential splitting of finer particles along cleavage planes, and sawtooth surfaces that result from interacting conchoidal fractures. (See Helland et al. 1997; Martinelli et al. 2020. Note: fretting refers to contact damage when the interacting bodies experience small repetitive movements tangential to the surface.) Clearly, surface abrasion, asperity breakage, and fretting prevail when sand grains experience a large number of low-stress cycles [Figs. 4(a and b)]. By contrast, sand particles subjected to high-stress cycles tend to split along cleavage planes to produce platy particles; the finer particles are flatter and more angular [Figs. 4(i–l)].

Fig. 4. SEM images: (a–c) abrasion and fretting on the particle surface; (d) asperity breakage; (e) planar surface from tensile splitting; (f) micro fracture; (g) V-shaped pits; (h) clean cross-sectional area; (i) conchoidal fracture; and (j–l) preferential splitting of finer particles along cleavage planes. Blade-shaped particles prevail among the finer produced particles.

Void Ratio Evolution

Fig. 5 presents the change in void ratio versus the vertical effective stress for the five KAUST 20/30 sand specimens subjected to high-stress amplitude cycles. We superimpose a complementary monotonic oedometer test data gathered at the same initial relative density

D_{r} = 75 %

and loaded to a maximum vertical stress

σ_{f} = 24 MPa

. The yield stress

σ_{y} \approx 10 MPa

corresponds to the point of maximum curvature. Repetitive loading cycles—all with the same

Δ σ / σ_{o}

ratio—cause more significant volume contraction as the peak cyclic stress increases from

σ_{f} = σ_{o} + Δ σ = 2.5 MPa

(Test 1) to 25 MPa (Test 5). The void ratio for specimens that experience

σ_{f} = 15 MPa

and

σ_{f} = 25 MPa

reaches values smaller than the minimum void ratio

e_{\min} = 0.533

. On the other hand, the gray square in Fig. 5 illustrates the change in void ratio for a specimen subjected to

10^{6}

low-stress cycles (Test 8 in Table 2). Fig. 6 presents the void ratio evolution with the number of cycles in both

e - i

and

e - \log (i)

scales to highlight trends. Marked changes in the void ratio occur in the first 10 cycles, and the volume contraction becomes more pronounced as the maximum stress

σ_{f} = σ_{o} + Δ σ

increases from 2.5 to 25 MPa.

Fig. 5. High-stress amplitude repetitive loading tests—void ratio evolution ( $N = 100$ cycles, KAUST 20/30 sand). Numbers in square brackets [#] correspond to the test number (details in Table 2). The orange line shows data gathered in a monotonic test run to 24 MPa vertical stress. For reference, the minimum void ratio is $e_{\min} = 0.533$ , and the gray plot indicates the change in the void ratio for KAUST 20/30 sand specimen (Test 8 in Table 2) during 1 million loading cycles.

Fig. 6. Evolution of void ratio against number of cycles $i$ during high stress amplitude loading cycles ( $N = 100$ cycles–KAUST 20/30 sand): (a) linear scale; and (b) logarithmic scale. Test conditions: stress amplitude ratio $σ_{f} / σ_{o} = 5$ for all specimens (refer to Table 2). Numbers in square brackets [#] correspond to the test number (details in Table 2).

Fig. 7 shows the change in the void ratio

e

versus the number of cycles

i

for the three KAUST 20/30 sand specimens subjected to

10^{6}

low-stress cycles. The complementary analogous study conducted with Ottawa 20/30 sands exhibits very similar trends (Fig. S1).

Fig. 7. Low-stress amplitude repetitive loading tests—void ratio evolution ( $N = 10^{6}$ cycles, KAUST 20/30 sand). “As-prepared” void ratio $e_{initial}$ and void ratio $e_{o}$ at $i = 0$ : (a) Test 6, $e_{initial} = 0.735$ , $e_{o} = 0.731$ and $e_{1} = 0.727$ ; (b) Test 7, $e_{initial} = 0.694$ , $e_{o} = 0.693$ and $e_{1} = 0.692$ ; and (c) Test 8, $e_{initial} = 0.639$ , $e_{o} = 0638$ and $e_{1} = 0.636$ . The three tests start at different void ratios $e_{o}$ but experience the same stress history: initial vertical stress $σ_{o} = 67 kPa$ , and vertical stress cycles from $Δ σ = 100 kPa$ (from $σ = 67$ -to-167 kPa). For reference, the extreme void ratios are $e_{\min} = 0.533$ and $e_{\max} = 0.786$ .

The complete dataset for high- and low-stress cycles is plotted in terms of the cumulative volumetric strain

ε_{v o l} = Δ e / (1 + e_{1})

in Figs. 8(a and b); for completeness, volumetric strain data extracted from published studies are superimposed on Fig. 8(b). Both high- and low-stress data deviate from the initial linear trends (in

\log i

scale) once the number of cycles exceeds a “tipping point.” A change in trends hints to the emergence of a new deformation mechanism apparently associated with contact fatigue.

Fig. 8. Cumulative volumetric strain $ε_{vol}$ with number of cycles during high stress cycles ( $N = 100$ , KAUST 20/30 sand) and low-stress cycles ( $N = 10^{6}$ , KAUST 20/30 and Ottawa 20/30 sands): (a) volumetric strain from 0 to 0.1; and (b) volumetric strain from $- 0.002$ to 0.016. Numbers in square brackets [#] correspond to the test number (details in Table 2).

Terminal void ratio. A soil will evolve towards a stable asymptotic terminal void ratio

e_{T}

as

i \to \infty

when subjected to repetitive loading (D’Appolonia and D’Appolonia 1967; Narsilio and Santamarina 2008). The terminal void ratio can be inferred by fitting void ratio trends using asymptotically correct functions. Previous studies show that a modified hyperbolic function properly matches volumetric changes under

k_{o}

-repetitive loading [modified from Chong and Santamarina (2016)]:

e_{i} = e_{T} + (e_{1} - e_{T}) {[1 + {(\frac{i - 1}{N^{*}})}^{m}]}^{- 1} for i \geq 1 and m > 0

(1)

where the void ratio

e_{i}

after the

i

th cycle is a function of the void ratio

e_{1}

after the first cycle and the terminal void ratio

e_{T}

as

i \to \infty

. The

m

exponent is

m < 1

. Half of the asymptotic contraction

(e_{1} - e_{T}) / 2

takes place when

i = N^{*} + 1

, i.e., immediately after the characteristic cycle number

N^{*}

. Eq. (1) fits both high- and low-stress data well. Model parameters listed in Table 2 show that both the terminal void ratio

e_{T}

and the characteristic cycle number

N^{*}

decrease as the peak stress

σ_{f}

increases.

Fig. 9 presents the terminal void ratio

e_{T}

against the initial void ratio

e_{o}

at

i = 0

for all specimens tested in this study. The terminal void ratio for low-stress cycles varies with the initial void ratio

e_{o}

(red circles and triangles,

Δ σ / σ_{o} = 1.50

). The proportionality between

e_{T}

and

e_{o}

for low-stress cycles suggests that sand specimens retain memory of their initial fabric even after a very large number of cycles. On the other hand, the terminal void ratio for high-stress cycles depends on the peak stress

σ_{f} = σ_{o} + Δ σ

due to crushing-dependent densification.

Fig. 9. Terminal void ratio $e_{T}$ versus void ratio $e_{o}$ at $i = 0$ . Test conditions with KAUST 20/30 sand for high-stress amplitude loading cycles: $σ_{o} = 0.5$ to 5 MPa, stress amplitude ratio $Δ σ / σ_{o} = 4$ , and total number of cycles $N = 100$ . Test conditions with both KAUST 20/30 and Ottawa 20/30 sands for low-stress million cycles tests: $σ_{o} = 67 kPa$ , stress amplitude ratio $Δ σ / σ_{o} = 1.50$ (from $Δ ς = 67$ -to-167 kPa). Numbers in square brackets [#] correspond to the test number (details in Table 2).

Time-stepping numerical simulations are expensive and subject to error accumulation. Instead, robust empirical functions that satisfy asymptotic conditions—such as Eq. (1)—can be combined with constitutive models to ensure equilibrium and compatibility in efficient hybrid numerical simulators (Suiker and de Borst 2003; Niemunis et al. 2005; François et al. 2010; Pasten et al. 2014; Masin 2021).

Maximum change in relative density

Δ D_{T}

. The ratio between the asymptotic contraction

Δ e = e_{o} - e_{T}

and the extreme void ratio range

e_{\max} - e_{\min}

for a given sand can help us estimate the maximum change in relative density

Δ D_{T}

at

i \to \infty

with respect to the initial relative density

D_{o}

at

i = 0

during repetitive loading:

Δ D_{T} = \frac{e_{o} - e_{T}}{e_{\max} - e_{\min}}

(2)

D_{o} = \frac{e_{\max} - e_{o}}{e_{\max} - e_{\min}}

(3)

Data normalization with respect to index properties

e_{\max} - e_{\min}

allows us to superimpose data for different sands and test conditions. Fig. 10 presents

Δ D_{T}

versus

D_{o}

data: (1) all specimens tested in this study under low- and high-stress cycles; and (2) data gathered under low-stress cycles in this and previous studies. As the terminal densities for high-stress cycles

σ_{f} = 15 MPa

and 25 MPa result in

e_{T} < e_{\min}

, the predicted asymptotic relative density will exceed

D_{r} > 100 %

. The maximum possible change in relative density corresponds to a very loose specimen subjected to

σ_{f} \to \infty

and reaching

e_{T} \to 0

, so the maximum possible change in relative density is

Δ D_{T} = 1 / (1 - e_{\min} / e_{\max}) \approx 3

[cutoff in Fig. 10(a)].

Fig. 10. Estimation of maximum settlement due to the repetitive loading: (a) maximum change in relative density $Δ D_{T}$ for both $N = 10^{2}$ high-stress and $10^{6}$ low-stress cycles versus initial relative density $D_{i = 0}$ at $I = 0$ (Sand: KAUST 20/30 and Ottawa 20/30 sands). Table 1 summarizes details for test conditions for each test such as initial stress $σ_{o}$ , stress amplitude Δσ, and final stress $σ_{f}$ . The model parameter is $n = 0.7$ in both low and high stress loading cycles [Eq. (4)]. Assuming the void ratio $e \to 0$ , the cut off for the maximum change in relative density is $1 / (1 - e_{\min} / e_{\max}) \approx 3$ ; and (b) $Δ D_{T}$ versus $D_{i = 0}$ at $i = 0$ for comparison to other published works where model parameter $n = 0.7$ in all cases Open symbols = data extracted from Chong and Santamarina (2016) and Park and Santamarina (2019). Numbers in square brackets [#] correspond to the test number (details in Table 2).

The following curve-fitting function satisfies the data and allows us to predict the maximum change in relative density

Δ D_{T}

as a function of the stress amplitude

Δ σ

and the initial relative density

D_{o}

:

Δ D_{T} = \frac{Δ σ \cdot {[1 - D_{o}]}^{n}}{B}

(4)

The

n

exponent determines the function curvature; we adopt

n = 0.7

in all cases to limit the number of degrees of freedom during curve fitting. The fitted

B

parameter increases with peak cyclic stress

σ_{f}

from

B \approx 0.7

for

σ_{f} ≪ σ_{y}

to

B \approx 12

for

σ_{f} > σ_{y}

. Fig. S2 shows a trend between fitted

B

values and the maximum stress

σ_{f}

. The data spread is in part due to the emergence of tipping points and underlying differences in grain degradation with

σ_{f}

.

The settlement caused by repetitive loads is the depth integral of vertical strain

ε_{z} = (e_{o} - e_{T}) / (1 + e_{o}) = Δ D_{T} \cdot (e_{\max} - e_{\min}) / (1 + e_{o})

. Then, the anticipated maximum change in relative density

Δ D_{T}

can be used to obtain first-order estimates of the maximum settlement a geosystem may experience during repetitive loading.

Shear Wave Velocity

Fig. 11 presents shear wave signal cascades recorded throughout the loading history for high- and low-stress tests. Trends in shear wave velocity as a function of stress capture both contact behavior and fabric changes. The first arrival times decrease during the initial static loading stage in both cases. We pick first arrivals from amplified signals and disregard near-field effects [Criteria in Lee and Santamarina (2005)]. There are minor changes in travel time and wave form during repetitive loading cycles; thus, we use CODA wave analysis to determine accurate changes in travel times for different cycles in comparison with the signal at

i = 1

(Snieder 2006; Dai et al. 2013).

Fig. 11. Cascade of shear wave signals for KAUST 20/30 sand: (a) high-stress cycle; and (b) low-stress cycle. Test conditions for (a) $σ_{o} = 5 MPa$ , $Δ ς = 20 MPa$ ( $σ = 5$ -to-25 MPa), $σ_{f} / σ_{o} = (σ_{o} + Δ σ) / σ_{o} = 5$ , and total number of cycles $N = 100$ (Test 5 in Table 2). Test conditions for (b) $σ_{o} = 67 kPa$ , $Δ ς = 100 kPa$ ( $σ = 67$ -to-167 kPa), $σ_{f} / σ_{o} = (σ_{o} + Δ σ) / σ_{o} = 2.5$ , total number of cycles $N = 10^{6}$ , and void ratio $e_{o} = 0.693$ at $i = 0$ (Test 7 in Table 2).

The shear wave velocity is a function of the mean effective stress in the polarization plane (Roesler 1979; Yu and Richart 1984; Santamarina et al. 2001). Values in Fig. 12 measured during static and repetitive loading stages exhibit a linear trend with stress in log–log scale; therefore

V_{S} = α^{*} {(\frac{σ_{z}^{'} + σ_{x}^{'}}{2 kPa})}^{β} = α^{*} {(\frac{σ_{z}^{'}}{1 kPa})}^{β} {(\frac{1 + K_{0}}{2})}^{β} \approx α {(\frac{σ_{z}^{'}}{1 kPa})}^{β}

(5)

where the

α

factor is the shear wave velocity at 1 kPa, and the

β

exponent represents the stress sensitivity of the shear wave velocity. The

α

and

β

values fitted to static loading trends in this study satisfy the inverse relationship

β = 0.70 - 0.25 \cdot \log [α / (1 m / s)]

observed for all soils (Cha et al. 2014).

Fig. 12. Shear wave velocity captured throughout the loading history for KAUST 20/30 sand specimens: (a) $N = 100$ high stress cycles; and (b) $N = 10^{6}$ low-stress cycles. Vertical arrows indicate the stress at the top of the loading cycle $σ_{o} + Δ σ$ . The colored squares indicate the evolution of shear wave velocity during the $N = 100$ high stress cycles. Numbers in square brackets [#] correspond to test number (details in Table 2).

High-stress data in Fig. 12(a) show a higher

α

factor and a lower

β

exponent when the stress level exceeds the yield stress

σ_{y} = 10 MPa

. Furthermore, data in Fig. 12(b) show that sand specimens become stiffer, and the soil fabric becomes less sensitive to stress changes after

10^{6}

low-stress cycles (higher

α

factor and a lower

β

exponent). Apparently, particle crushing and contact fretting enhance contacts, interparticle coordination, and fabric stability.

Finally, the shear wave velocity evolves linearly with

\log (i)

during early loading cycles for both high- and low-stress tests (Fig. 13). However, shear wave velocity trends rise above the linear trend either due to high-stress [Fig. 13(a); Test 5 for

σ_{f} = 25 MPa

after

i > 40

] or a high number of cycles [Fig. 13(b) for

i > \sim 10^{4}

]. These observations suggest the emergence of grain degradation and pore filling associated with grain crushing, enhanced particle contacts, contact fretting, and asperity breakage as observed in terms of grain size (Fig. 2) and shape (Figs. 3 and 4), and in agreement with volume changes (Fig. 8).

Fig. 13. Evolution of the shear wave velocity with number of cycles: (a) high-stress cycles ( $N = 100$ , KAUST 20/30 sand); and (b) low-stress cycles ( $N = 10^{6}$ , KAUST 20/30 sand). Numbers in square brackets [#] correspond to the test number (details in Table 2).

Small Strain Shear Modulus versus Resilient Constrained Modulus

Pavement deformation and rutting reflects the evolution of vertical strains versus the number of load applications (Monismith 2004). The applied cyclic stress amplitude

Δ σ

and the recoverable vertical strain

ε_{r}

define the resilient modulus

E_{r} = Δ σ / ε_{r}

(NCHRP 2004). In view of nondestructive pavement characterization, one may wonder about the relationship between the resilient modulus

E_{r}

and the maximum stiffness

E_{\max}

inferred from elastic wave velocity measurements. Previous studies show that the stiffness ratio is about

E_{r} / E_{\max} \approx 0.4

to 0.8 (Davich et al. 2004; Schuettpelz et al. 2010).

Similar concepts can be extended to all foundation systems subjected to repetitive loads. We investigate the ratio

M_{r} / G_{\max}

between the resilient constrained modulus

M

extracted from 1D oedometric data and the maximum shear modulus

G_{\max} = ρ \cdot V_{s}^{2}

computed from density

ρ

and shear wave velocity

V_{s}

. We compute the resilient constrained modulus

M_{r}

as the ratio between the stress amplitude

Δ σ

and the peak-to-peak strain

ε_{p p}

in the

i

th cycle

ε_{p p} = (e_{i}^{U} - e_{i}^{L}) / (1 + e_{i}^{L})

where superscripts correspond to

L

= loading when

σ = σ_{o} + Δ σ

and

U

= unloading at

σ = σ_{o}

.

Fig. 14 shows the evolution of the stiffness ratio

M_{r} / G_{\max}

versus the number of cycles for

10^{2}

high-stress and

10^{6}

low-stress tests; for comparison, Fig. 14(b) presents previously published data. The stiffness ratio

M_{r} / G_{\max}

ranges between

0.85 \leq M_{r} / G_{\max} \leq 2.25

, and increases with the number of cycles, and higher ratios correspond to higher peak cyclic stress

σ_{f}

and lower densities.

Fig. 14. Stiffness ratio $M_{r} / G_{\max}$ versus number of cycles. The midstrain resilient constrained modulus $M_{r}$ and the maximum shear modulus measured at small strains $G_{\max}$ : (a) this study: $10^{2}$ high-stress and $10^{6}$ low-stress cycles; numbers in square brackets [#] correspond to the test number (details in Table 2); and (b) $10^{4}$ intermediate stress cycles from Park and Santamarina (2019); open symbols = loose packings; and filled symbols = dense packings.

The peak-to-peak strain level is

ε_{p p} \sim 3

-to-

8 \times 10^{- 3}

in high-stress cycles,

ε_{p p} \sim 2

-to-

4 \times 10^{- 4}

in low-stress-stress cycle tests, and about

ε_{p p} \sim 10^{- 8}

in wave propagation measurements. Therefore, the resilient constrained modulus

M_{r}

and the small-strain shear modulus

G_{\max}

involve different values of Poisson’s ratio. Similarly, the relationship between published

E_{r} / E_{\max}

values reported above and

M_{r} / G_{\max}

measured in this study require an intermediate-strain Poisson value

ν_{r}

to relate

E_{r}

and

M_{r}

, and a small-strain Poisson value

ν_{o}

to link

E_{\max}

to

G_{\max}

:

\frac{M_{r}}{G_{\max}} = \frac{\frac{E_{r} (1 - υ_{r})}{(1 + υ_{r}) (1 - 2 υ_{r})}}{\frac{E_{\max}}{2 (1 + υ_{o})}} = \frac{E_{r}}{E_{\max}} \cdot \frac{2 (1 - υ_{r}) (1 + υ_{o})}{(1 + υ_{r}) (1 - 2 υ_{r})}

(6)

For

ν_{r} = 0.25

and

ν_{o} = 0.10

, the ratio

(M_{r} / G_{\max}) / (E_{r} / E_{\max}) \approx 2.6

is in agreement with the data.

Tipping Points: Deformation Mechanisms

While the modified hyperbolic function in Eq. (1) fits volumetric data well, “tipping points” are apparent in volumetric strain (Fig. 8) and shear wave velocity trends (Fig. 13) versus the number of cycles in logarithmic scale

\log (i)

. Deviations become apparent within the first 20–30 cycles when the maximum stress exceeds the yield stress

σ_{f} > σ_{y}

, or after

\sim 10^{4}

cycles when the maximum cyclic stress

σ_{f} \approx σ_{y} / 60

(Refer to Wöhler’s curve; inset in Fig. 2).

Grain size distributions, optical photographs, and SEM images suggest different underlying mechanisms. High-stress cycles can produce fines that fill voids and participate in subsequent crushing events to produce even finer particles (Level III damage; Mesri and Vardhanabhuti 2009). On the other hand, sand contraction under a large number of low-stress cycles starts with the buckling of unstable granular columns together with particle sliding and rolling (Lambe and Whitman 1969; Omidvar et al. 2012); then, stress concentration at interparticle contacts leads to fatigue-induced asperity breakage and contact fretting (Djordjevic and Morrison 2006; Sadrekarimi and Olson 2010; Radjai et al. 1998).

Conclusions

This study explored the response of sands subjected to repetitive

k_{o}

loading at both ends of the Wöhler’s curve, i.e., at a small number of high-stress cycles and for large number of low-stress cycles. Salient conclusions follows:

•

Higher stress cycles produce more fines. Particle degradation mechanisms vary with the maximum stress level. Sand particles subjected to high-stress cycles tend to split along cleavage planes to produce platy particles; the finer particles are flatter and more angular. On the other hand, surface abrasion, asperity breakage, and fretting prevail when sand grains experience a large number of low-stress cycles.

•

Sands subjected to repetitive loading reach asymptotic terminal void ratios

e_{T}

. Its value is recovered by fitting the void ratio evolution versus the number of cycles using asymptotically correct functions. A modified hyperbolic function fits both high- and low-stress data well.

•

The empirical hyperbolic function used in this study adequately captures the evolving volumetric and stiffness response with the number of cycles. The use of such empirical functions within a hybrid numerical approach reduces numerical error and allows for efficient simulations that track the plastic strain accumulation during repetitive loading cycles.

•

The terminal void ratio for high-stress cycles depends on the peak cyclic stress

σ_{f} = σ_{o} + Δ σ

due to crushing-dependent densification, and it can be smaller than

e_{\min}

when the peak cyclic stress approaches or exceeds the sand yield stress

σ_{f} \to σ_{y}

(measured in monotonic loading). On the other hand, the soil retains memory of the initial fabric even after a very large number of low-stress cycles, and the terminal void ratio for low-stress cycles correlates with the initial void ratio

e_{o}

.

•

The change in void ratio from the initial state to the terminal void ratio (

e_{o} - e_{T}

) can be normalized by (

e_{\max} - e_{\min}

) to compare data for different sands and test conditions. Data show that the maximum change in relative density

Δ D_{T} = (e_{o} - e_{T}) / (e_{\max} - e_{\min})

is a function of the sand initial relative density

D = (e_{o} - e_{\min}) / (e_{\max} - e_{\min})

, the peak cyclic stress

σ_{f}

, and the stress amplitude

Δ σ

. These results lead to simple yet robust strategies to estimate the maximum settlement for first-order engineering analyses.

•

Detailed analyses of volumetric and stiffness trends reveal tipping points and deviations from initial trends. In agreement with Wöhler’s fatigue, tipping points occur at a small number of high-stress cycles (as small as

\sim 20

cycles when

σ_{f} \sim σ_{y}

) or after a large number of small-stress cycles (

> 10^{4}

cycles when

σ_{f} ≪ σ_{y}

). Loosely packed sands experience tipping points at a lower number of cycles than dense sands due to higher contact forces for lower interparticle coordination.

•

The small-strain stiffness in monotonic loading increases with confinement and density. During repetitive loading, sands stiffen with the number of cycles to reflect increased interparticle coordination following crushing, as well as contact flattening due to asperity breakage and fretting.

•

The resilient constrained modulus

M_{r}

evolves more rapidly than the shear modulus

G_{\max}

particularly during the earlier cycles when fabric changes are most pronounced; consequently, the stiffness ratio

M_{r} / G_{\max}

increases with the number of cycles. The stiffness ratio varies between

0.85 \leq M_{r} / G_{\max} \leq 2.25

for high-stress cycles and between

1.6 \leq M / G_{\max} \leq 2.3

for low-stress cycles. These

M_{r} / G_{\max}

values reflect the different strain levels involved in resilient modulus and wave velocity measurements and cannot be anticipated from theory of elasticity using a single Poisson’s ratio. The strong correlation between resilient modulus

M_{r}

and shear modulus

G_{\max}

suggest the possible application of geophysical methods based on shear wave propagation to monitor geosystems subjected to repetitive loading cycles.

•

Repetitive loading studies under zero-lateral strain conditions allow us to anticipate the maximum change in relative density and the potential settlement a layer may experience when subjected to repetitive mechanical loads. Finally, 1D loading test results can be used to estimate the resilient constrained modulus for pavement design guideline.

Supplemental Materials

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Data Availability Statement

All data, models, and code generated or used during the study appear in the published article.

Acknowledgments

Support for this research was provided by the KAUST Endowment at King Abdullah University of Science and Technology. Gabrielle E. Abelskamp edited the manuscript.

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Information & Authors

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Published In

Journal of Geotechnical and Geoenvironmental Engineering

Volume 149 • Issue 11 • November 2023

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Jul 3, 2022

Accepted: Mar 31, 2023

Published online: Sep 14, 2023

Published in print: Nov 1, 2023

Discussion open until: Feb 14, 2024

Authors

Affiliations

Junghee Park https://orcid.org/0000-0001-7033-4653 [email protected]

Assistant Professor, Dept. of Civil and Environmental Engineering, Incheon National Univ., 119 Academy-ro, Yeonsu-gu, Incheon 22012, Republic of Korea; formerly, King Abdullah Univ. of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (corresponding author). ORCID: https://orcid.org/0000-0001-7033-4653. Email: [email protected]

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J. Carlos Santamarina, A.M.ASCE

Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332; formerly, King Abdullah Univ. of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia.

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Carlos D. Rodriguez-Hernandez, Junghee Park, Heriberto Echezuria, J. Carlos Santamarina, Volume Contraction and Reduction in Hydraulic Conductivity during Particle Crushing, Journal of Geotechnical and Geoenvironmental Engineering, 10.1061/JGGEFK.GTENG-12234, 150, 6, (2024).
Abstract

Sands Subjected to Repetitive Loading Cycles and Associated Granular Degradation

Abstract

Introduction

Experimental Study

Experimental Results: Analyses

Particle Degradation

Void Ratio Evolution

Shear Wave Velocity

Small Strain Shear Modulus versus Resilient Constrained Modulus

Tipping Points: Deformation Mechanisms

Conclusions

Supplemental Materials

Data Availability Statement

Acknowledgments

References

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Abstract

Introduction

Experimental Study

Experimental Results: Analyses

Particle Degradation

Void Ratio Evolution

Shear Wave Velocity

Small Strain Shear Modulus versus Resilient Constrained Modulus

Tipping Points: Deformation Mechanisms

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