Géotechnique

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Low-to-medium-density chalk can be destructured to soft putty by high-pressure compression, dynamic impact or large-strain repetitive shearing. These process all occur during pile driving and affect subsequent static and cyclic load-carrying capacities. This paper reports undrained triaxial experiments on destructured chalk, which show distinctly time-dependent behaviour as well as highly non-linear stiffness, well-defined phase transformation and stable ultimate critical states under monotonic loading. The chalk's response to high-level undrained cyclic loading invokes both contractive and dilative phases that lead to pore pressure build-up, leftward effective stress path drift, permanent strain accumulation, cyclic stiffness losses and increasing damping ratios that resemble those of silts. These outcomes are relatively insensitive to consolidation pressures and are distinctly different to those of the parent intact chalk. The maximum number of cycles that can be sustained under given combinations of mean and cyclic stresses are expressed in an interactive stress diagram which also identifies conditions under which cycling has no deleterious effect. Empirical correlations are proposed to predict the number of cycles to failure and mean effective stress drift trends under the most critical cyclic conditions. Specimens that survive long-term cycling present higher post-cyclic stiffnesses and shear strengths than equivalent ‘virgin’ specimens.

Chalk is a very soft biomicrite composed of silt-sized crushable calcium carbonate (CaCO3) aggregates. Vinck et al. (2022) demonstrate how low-to-medium-density chalks (with intact dry densities, IDD < 1·70 Mg/m3) develop stiff, brittle and ultimately dilative behaviour when sheared from in situ effective stress levels. However, their mechanical properties degrade markedly under dynamic, cyclic or high-pressure shearing, with important implications for problems such as the design of driven piles (Carrington et al., 2011; Diambra et al., 2014; Carotenuto et al., 2018; Buckley et al., 2020a).

Impact driving creates low-strength, destructured, chalk putty beneath the piles’ advancing tips, which spreads and further softens around their shafts. Buckley et al. (2018) and Vinck (2021) identified how destructuration varied with radial distance from the axes of open steel piles at shallow depths (above the water table), considering conditions after driving, and after long-term ageing and load testing. The thin annuli of putty formed around shafts on driving provided average driving resistances ≈ 20 kPa and reconsolidated over time to achieve notably lower water contents and significantly greater shear strengths. The response of the reconsolidated putty to monotonic and cyclic loading, as well as interface shear, is central to addressing axial capacity and cyclic loading performance for piles driven in chalk.

This paper explores the cyclic behaviour of reconsolidated destructured chalk. Stress-controlled cyclic triaxial tests are reported on material from the ‘axial–lateral pile analysis for chalk applying multi-scale field and laboratory testing’ (ALPACA) project's St Nicholas-at-Wade (SNW), UK pile research site, whose geotechnical profile and chalk properties are described by Vinck (2021). The destructured chalk's response to undrained cycling is interpreted with reference to those of saturated silts and silty sands, as reported by Carraro et al. (2003), Mao & Fahey (2003), Hyde et al. (2006), Sanin & Wijewickreme (2006), Sağlam & Bakır (2014) and Wei & Yang (2019). Ahmadi-Naghadeh et al. (2022) report parallel research into the intact chalk's cyclic response under similar cycling, identifying behaviour that differs starkly from that of unbonded soils and compares more closely with that of rocks, concretes or metals. Bialowas et al. (2018) and Alvarez-Borges et al. (2018, 2020) report earlier testing on reconstituted SNW chalk.

Laboratory dynamic compaction, applied at in situ water content, destructures low-to-medium-density chalk in an analogous way to pile driving (Doughty et al., 2018) and provides uniform batches for laboratory testing. Puttified specimens were formed for this study through compaction of block samples preserved from 1·4 m depth, whose unconfined compression strengths (UCS) exceeded 3 MPa, despite their 29–30% initial natural water contents. Up to 150 blows were applied at ≈ 2 s intervals with a 4·5 kg ram and 300 mm drop height to intact lumps contained in a 100 mm dia. mould to produce ≈ 0·3 litre batches of chalk putty. Mixing every 50 blows ensured uniformity, and the process, which took ≈ 10 min, involved slight drying with ≈ 1% water contents reductions. Index testing indicated 9 ± 3 kPa fall-cone undrained shear strengths, liquid and plastic limits of 30·6% and 24·2%, respectively, grain specific gravity Gs = 2·71 and median grain size D50 ≈ 3·0 μm.

Figure 1 presents constant-rate-of-strain (CRS; at 0·6%/h) oedometer compression curves for intact and destructured putty chalk samples. Also shown is a test on samples reconstituted by re-hydrating pulverised dried chalk to 1·4 times the liquid limit. The intact and reconstituted compression (NCL*) oedometer curves of natural clays reflect their different structures (Burland, 1990). Smith et al. (1992) employed the ratio of the intact soil's vertical effective stress at yield σvy to that projected onto the reconstituted curve at the same void ratio as a scalar ‘oedometer sensitivity’ measure of the clay's structure. Fig. 1 indicates an oedometer ‘sensitivity’ of ≈ 24 for the natural chalk. The intact chalk shows Su ≈ 1·2 MPa at this depth (Vinck et al., 2022), which suggests a higher undrained shear strength sensitivity ≈ 130. The intact CRS trace suggests that oedometer sensitivity declines towards unity as pressures increase post-yield and the eσv′ trace curves towards the NCL* whose compression index Cc* = 0·18. The fully destructured putty starts at a lower liquidity index than the reconstituted chalk, but follows a similar trend once σv′ > 50 kPa, falling far below the intact sample's curve, although exhibiting similar unloading curves and swelling indices Cs ≈ 0·01. The putty exhibited markedly time-dependent one-dimensional (1D) compression behaviour in parallel stage loaded oedometer tests that gave secondary compression coefficients Cαe = Δe/Δlog10(t) = 0·003 over the 100 < σv′ < 400 kPa range and a Cαe/Cc = 0·06 ratio, which is remarkably high for an inorganic soil (Mesri & Vardhanabhuti, 2006). As shown later, triaxial specimens prepared from the putty developed significant volumetric strains under relatively modest isotropic consolidation stresses and attained specific volume–mean effective stress (vp′) states well below (or ‘dryer than’) the destructured chalk followed in its CRS oedometer test. These findings and related features are discussed later in relation to the state parameter framework for sands (Been & Jefferies, 1985).

Fig. 1. One-dimensional compression behaviour of destructured (puttified), reconstituted and intact chalk established from CRS (constant rate of strain; 0·6%/h) tests

Apparatus and procedures

Cyclic triaxial tests were performed with automated hydraulic stress-path apparatus. A suction cap and half-ball connection system helped to align the (initially soft) specimens with the load cells and minimise tilting and bedding. Layered latex discs and high-vacuum grease deployed at the specimen tops and bottoms reduced end constraint. Putty was placed in 5–10 g increments into a split mould, lined with a latex membrane, pre-set on the triaxial base platen. Care was taken to eliminate macro-voids and produce uniform 38 mm dia., 80 mm high, specimens with flat ends, topped with poly(methyl methacrylate) (PMMA) caps. The soft specimens’ ability to maintain regular shapes and resist disturbance during mould dismantling and instrumenting was improved through an ‘in-mould’ isotropic consolidation stage implemented by maintaining a triaxial cell-to-back-pressure difference of 70 kPa for 15 h under drained conditions, which led to volume strains of ≈ 10%. The resulting, relatively robust, specimens’ dimensions were then measured and sets of linear variable differential transducer (LVDT) local strain sensors were mounted, including a radial-strain belt.

The specimens were saturated by applying 300–400 kPa back-pressure, maintaining p′ = 20 kPa until B > 0·97, followed by isotropic consolidation at 1 kPa/min to reach the targeted mean effective stresses (p0), which led to average Cαe values (0·0034 and 0·0046) under p′ levels of 200 and 400 kPa, respectively. Creep periods of 8–12 days allowed residual axial straining to diminish to < 0·005%/day, 1000 times lower than the 5%/day applied in subsequent monotonic shearing stages. Samples consolidated to p0 = 200 kPa and 400 kPa had post-creep (pre-shearing) void ratios of 0·63 and 0·59, respectively, corresponding to water contents (23·3% and 21·8%) that, as noted in the putty zone around driven pile shafts, fell well below those of the undisturbed intact chalk and far below the oedometer curves shown in Fig. 1.

Test programme and code

Five monotonic ‘control’ tests characterised the putty's response to undrained shearing (at 5% axial strain/day) after isotropic consolidation to p0 = 70, 200 and 400 kPa followed by drained creep, that aimed to match the medium to high range of radial effective stresses (10 < σrf < 500 kPa) interpreted around the ALPACA pile shafts after full ageing (Buckley et al., 2020b). Specimen details and testing conditions are outlined in Table 1.

 Table 1. Summary of undrained monotonic triaxial tests: maximum Young's moduli, stress conditions and the corresponding axial strains (in brackets) at phase transformation (PT) and ultimate states

Table 1. Summary of undrained monotonic triaxial tests: maximum Young's moduli, stress conditions and the corresponding axial strains (in brackets) at phase transformation (PT) and ultimate states

Test e* p0: kPa
$Emaxu$
: MPa
$Emaxupref/p′0pref0⋅5$
q(PT): kPa (εa: %) p(PT): kPa q(ult): kPa (εa: %) p(ult): kPa (q/p′)ult ϕult′: degrees
DM-C1 0·714 70 643·1 7637·0 42·6 (0·6%) 39·9 309·7 (30·9%) 229·4 1·35 31·6
DM-C2 0·648 200 1195·3 8397·6 106·1 (1·4%) 100·3 606·1 (24·0%) 480·8 1·26
DM-C3 0·606 400 1472·5 7315·1 208·9 (1·2%) 198·3 1618·5 (25·0%) 1273·0 1·27
DM-E1 0·609 200 1114·1 7827·2 −132·1 (−1·1%) 149·1 −207·4 (−7·0%) 233·4 −0·89 30·1
DM-E2 0·580 400 1393·1 6920·7 −256·9 (−1·0%) 319·9 −364·2 (−7·0%) 453·9 −0·80

* Void ratio prior to undrained shearing; calculated based on post-test water content measurements.

pref, reference pressure (101·3 kPa).

The subsequent cyclic programme focused mainly on 11 tests at p0 = 200 kPa, supplemented by four experiments cycled from p0 = 400 kPa with, naturally, lower initial void ratios. Cell pressures were held constant, while deviator stresses varied sinusoidally about a fixed qmean by the amplitudes qcyc listed in Table 2. Note that q = (σv – σh) and p′ = (σv + 2σh)/3, and that qmean and qcyc are also shown as ratios of the putty chalk's p0 and 2Su values to aid interpretation. Relatively long periods of 300 s were adopted to enable full control, pore-pressure equalisation and detailed logging of all parameters. Recalling the material's time-dependent compression behaviour, the triaxial tests may overestimate the degree to which cycling affects offshore structures under typically shorter (perhaps 10 s) period cyclic loading. Each test's code is composed as

 (a) letter ‘D’ denotes destructured chalk (b) letter ‘M’ signifies monotonic and letters ‘Cy’ indicate cyclic loading (c) group letters A, B, C or D signify the level of maximum q applied, in ascending order (d) a numeral signifies the applied qcyc level in the A to D groups, in ascending order; letter ‘X’ represents the single case where negative qmean was applied (e) letter ‘E’ signifies the test series performed at the elevated p′0 of 400 kPa.

 Table 2. Summary of cyclic triaxial test conditions and parameters

Table 2. Summary of cyclic triaxial test conditions and parameters

Test e* qmean: kPa qmean/(2Su) qcyc: kPa qcyc/(2Su) qmax: kPa qmax/(2Su) qmean/p0 qcyc/p0
DCy-A1 0·644 0 0 30 0·30 30 0·30 0 0·15
DCy-B1 0·607 0 0 45 0·45 45 0·45 0 0·23
DCy-C1 0·615 30 0·30 30 0·30 60 0·60 0·15 0·15
DCy-C2 0·621 15 0·15 45 0·45 60 0·60 0·08 0·23
DCy-C3 0·659 0 0 60 0·60 60 0·60 0 0·30
DCy-CX 0·616 −15 −0·15 45 0·45 −60 −0·60 −0·08 0·23
DCy-D1 0·621 79 0·79 17 0·17 96 0·96 0·40 0·09
DCy-D2 0·675 57 0·57 30 0·30 87 0·87 0·29 0·15
DCy-D3 0·648 44 0·44 44 0·44 88 0·88 0·22 0·22
DCy-D4 0·621 28 0·28 60 0·60 88 0·88 0·14 0·30
DCy-D5 0·591 0 0 75 0·75 75 0·75 0 0·38
DCy-A1-E 0·620 0 0 60 0·30 60 0·30 0 0·15
DCy-B1-E 0·587 0 0 90 0·45 90 0·45 0 0·23
DCy-C3-E 0·597 0 0 120 0·60 120 0·60 0 0·30
DCy-D5-E 0·575 0 0 150 0·75 150 0·75 0 0·38

* Void ratio prior to undrained monotonic pre-shearing or cyclic shearing; calculated based on post-test water content measurements.

The putty's response to undrained triaxial compression (TXC) and extension (TXE) is displayed in Fig. 2, plotting zoomed-in qp′ effective stress paths and deviatoric stress (q)–axial strain (εa) trends over the small to medium strain range. Specimens exhibited broadly linear elastic behaviour up to εa limits of 0·002% and 0·003% for the p0 = 200 and 400 kPa tests, respectively, corresponding to increments Δq ≈ 23·1 and 43·0 kPa with an average Δq/(2Su) ≈ 0·22. The qp′ effective stress paths rose nearly vertically upon compression and extension, suggesting that the re-consolidated (mildly aged) putty's initial stiffness response was largely isotropic (Vinck et al., 2022).

Fig. 2. Triaxial compression and extension behaviour of puttified chalk: (a) effective stress paths; (b) deviatoric stress–axial strain response (see also details in Table 1)

The effective stress paths rotated to follow leftward (contractive) stages after mobilising modest ‘peak’ resistances (after relatively small strains, εa < 0·2%) and showed strain softening as shearing continued up to phase transformation (PT) points at which their paths rotated abruptly and climbed towards ultimate (critical state) conditions (see Table 1). Continued straining led to markedly higher ultimate strengths as the specimens attempted to dilate from their states positioned well below the normal compression line indicated in Fig. 1.

As discussed later, the chalk putty's resistance to cyclic loading is dominated by its pre-PT behaviour. The peak pre-PT q(PT) points were taken as indicating the operational monotonic shear strengths (2Su), giving rounded Su values of 50 kPa and 100 kPa for the p0′ = 200 and 400 kPa tests, respectively, with Su/p0 = 0·25. Specimens undergoing extension developed similarly contractive pre-PT responses to shearing, followed by dilation after reaching PT, giving broadly similar, yet not fully symmetric stress paths and shear strengths to the compression tests, despite their different σ1 directions and b = (σ2 – σ3)/(σ1 – σ3) ratios (or Lode angles θ). While the isotropically consolidated putty did not manifest any significant combined effect of anisotropy or b ratio on its pre-PT shearing behaviour, the extension tests’ dilative post-PT stages were truncated prematurely by localised necking from εa ≈ −7·5% onwards that obscured any trend towards stable ultimate critical states.

Table 1 summarises the specimens’ linear elastic (maximum) Young's moduli (Emaxu), their PT stress points, large-strain ultimate (critical state) states and the corresponding strains. The q/p′ ratios at PT, critical state (in compression) and ultimate failure in extension in the (inherently more reliable) higher pressure tests were 1·05, 1·27 and 0·86, respectively. The latter two ratios both correspond to ϕcs ≈ 31°, matching the angle found in high-pressure tests on intact samples.

As listed in Table 2, the cyclic triaxial experiments investigated a range of one- and two-way qmean, qcyc and qmax conditions, including cycling into extension. The elevated pressure (p0 = 400 kPa) tests concentrated on symmetrical two-way conditions (with qmean = 0). The procedures mirrored the intact chalk testing by Ahmadi-Naghadeh et al. (2022). Target qmean values were applied by undrained strain-controlled loading at a rate of 5% axial strain/day, followed by pauses of 48–72 h in which the specimens sustained their qmean values without drainage until the local axial strain rates fell below 0·005%/day. These steps were critical for distinguishing the specimens’ subsequent cyclic straining from any creep provoked by applying the qmean component. Fig. 3 demonstrates how axial strains developed in DCy-D1 to DCy-D4 during their: (a) undrained monotonic pre-shearing; (b) extended creep pauses; and (c) first applied cycle. Also indicated are the corresponding average maximum Young's (Emaxu) and secant cyclic moduli (Esecu,cyc) developed over the first peak-to-trough half cycle. The creep strains are significant and increased with applied qmean to represent a large fraction of the overall straining. The creep pauses allowed the specimens to regain stiffness after pre-shearing and the subsequent cyclic moduli depended primarily on the stress amplitudes imposed. Doughty et al. (2018) and Vinck (2021) detail the abrupt stiffness degradation shown by chalk putty specimens; Emaxu moduli decay by 40% from initial values after shearing to 0·01% axial strain. They ascribe the rapid stiffness degradation to microstructural alteration and brittle re-cementation.

Fig. 3. Deviatoric stress–axial strain responses for tests DCy-D1 to DCy-D4 during monotonic pre-shearing, creep and first cycle, also indicating average maximum Young's modulus in pre-shearing and secant cyclic moduli for the first peak-to-trough half cycle

The tests which survived to 10 000 cycles extended for several weeks. All ‘surviving’ specimens were sheared to undrained monotonic failure; as shown later, stable cycling improved the puttified chalk's monotonic resistance and stiffness.

Undrained cyclic behaviour is often assessed in earthquake geotechnics through testing under symmetrical two-way loading. The failures that define the soils’ cyclic resistance ratios (Ishihara, 1996) are defined as occurring when specified double-amplitude (DA) axial (or shear) strain limits are met. Failure under non-symmetrical loading conditions is defined referring to either peak or accumulated cyclic strains (Yang & Sze, 2011). Cyclic failure criteria and strain limits are often tailored to reflect the geo-material's cyclic behaviour and the engineering problems addressed (Wijewickreme & Soysa, 2016).

Noting that stringent deformation tolerances are specified for offshore wind turbine design (Byrne et al., 2017), the cyclic strain limits were set lower than is routine in, for example, liquefaction assessment. Failure was defined by whichever of two criteria was satisfied first

 (a) criterion A: occurrence of 1% double-amplitude (DA = εa, peak−εa, trough) axial strain (b) criterion B: absolute peak or trough axial strain (|εa|) exceeding 1%.
The criteria reflect chalk putty's potentially marked stiffness degradation under cycling. As demonstrated later, they lead to outcomes that are compatible with other measures of cyclic failure, including trends for pore water pressures, shear strength reductions and damping ratios.

Table 3 summarises key outcomes from the cyclic experiments: the axial strains and ranges of cyclic stiffness (Esecu,cyc) and damping ratio (D) experienced up to the number of cycles (Nf) at which failure occurred, or the final cycle for tests that survived 10 000 cycles. Discussion on the mean effective stress drift trends follows later.

 Table 3. Strains, pore pressure changes, stiffness and damping ratio variations during cyclic loading, considering changes from the first cycle up to the Nf cycle, or final cycle in the unfailed tests

Table 3. Strains, pore pressure changes, stiffness and damping ratio variations during cyclic loading, considering changes from the first cycle up to the Nf cycle, or final cycle in the unfailed tests

Test qmean/(2Su) qcyc/(2Su) qmax/(2Su) Imposed cycles, Nmax Nf* εa at Nf: % εa/Nf: % ru: % Secant cyclic
$Esecu,cyc$
: MPa
Damping ratio, D: %
DCy-A1 0 0·30 0·30 10 024 Unfailed 0·078 7·78 × 10−6 0·10→34·5 903→1166 4·80→3·70
DCy-B1 0 0·45 0·45 717 486(A) −0·210 −4·32 × 10−4 1·91→85·3 951→14·0 6·51→21·2
DCy-C1 0·30 0·30 0·60 10 058 Unfailed 0·085 8·45 × 10−6 32·1→54·1 1067→1155 9·53→8·06
DCy-C2 0·15 0·45 0·60 676 645(B) 0·829 1·29 × 10−3 20·2→95·7 918→9·3 12·8→26·4
DCy-C3 0 0·60 0·60 181 65(A) −0·023 −3·54 × 10−4 3·86→77·2 841→23·2 13·0→21·3
DCy-CX −0·15 0·45 −0·60 361 357(A, B) −0·516 −1·45 × 10−3 10·1→80·3 1067→13·9 4·82→19·9
DCy-D1 0·79 0·17 0·96 9600 Unfailed 0·147 1·53 × 10−5 42·8→67·4 1334→1444 5·56→2·95
DCy-D2 0·57 0·30 0·87 10 614 5528(B) 0·984 1·78 × 10−4  40·1→82·7 1160→269·7 6·90→20·1
DCy-D3 0·44 0·44 0·88 240 183(B) 0·926 5·06 × 10−3 22·2→92·4 1045→35·6 10·9→25·9
DCy-D4 0·28 0·60 0·88 58 47(A, B) 0·523 0·011 30·3→90·2 955→21·6 11·1→25·7
DCy-D5 0 0·75 0·75 170 ≈18 −0·005 −2·78 × 10−4 6·45→76·9 508·0→46·9 17·9→23·9
DCy-A1-E 0 0·30 0·30 10 085 Unfailed 0·036 3·57 × 10−6 0·40→64·2 1428→1023 4·51→7·35
DCy-B1-E 0 0·45 0·45 149 142(A) −0·071 −5·0 × 10−4 1·81→89·0 1236→34·3 8·15→22·1
DCy-C3-E 0 0·60 0·60 57 50(A) −0·080 −1·6 × 10−3 2·70→82·5 1190→57·3 12·2→24·1
DCy-D5-E 0 0·75 0·75 18 16(A, B) −0·085 −5·31 × 10−3 6·56→67·3 815·0→80·9 17·0→23·3

* Superscripts (A) and (B) denote the applied cyclic failure criteria. Cycling control of test DCy-D5 deteriorated as specimen softened significantly after 13 cycles.

ru, pore water pressure ratio (%), defined as: ru = (p0′ − p′)/p0′ = Δu/p0′.

‡ Damping ratio calculated as: D = Aloop/(4πAelastic); Aloop – area enclosed by a stress–strain (qεa) loop for a complete sinusoidal stress cycle; Aelastic – unloading half-cycle elastic triangle area with height as qcyc ( = (qpeak − qtrough)/2) and width as cyclic strain ( = (εpeak − εtrough)/2).

Cycling from p0 = 200 kPa

Considering the tests performed from p0 = 200 kPa, Fig. 4 illustrates how Nf varies with the normalised loading parameters qcyc/(2Su)–qmean/(2Su) and qcyc/(p0)–qmean/(p0). The three unfailed cases are annotated as ‘> Nmax’ where Nmax is the number of stress cycles applied. Two nominal Nf = 1 contour lines are plotted from (qmean, qcyc) = (0, 2Su) to (2Su, 0) and (qmean, qcyc) = (0, 2Su) to (−2Su, 0), neglecting minimal variations in Su between compression and extension (see Fig. 2) and any possible rate dependency of shear strength between monotonic shearing at 5% per day and that developed over the final cycle of loading.

Fig. 4. Cyclic interaction diagram expressed in normalised qcyc/(2Su)–qmean/(2Su) and qcyc/p0′–qmean/p0′ stress space, also indicating the interpreted contours of number of cycles (Nf) to failure for p0′ = 200 kPa and p0′ = 400 kPa tests series as summarised in Table 3

A tentative family of curved Nf ( = 10, 30, 100, 300, 1000, 3000 and 10 000) contours that extends to qmean/(2Su) = −0·15 is included to illustrate stress interaction patterns in the one- and two-way cycling regions of the 200 kPa p0 tests. Nominal contours are shown as dashed straight lines over the unpopulated extension region and link the contours towards a putative lower qcyc/(2Su) limit of 0·15. The lower level, high Nf, contours show less curvature and tighter spacings than those representing high-level cycling. The interactive stress diagram region below the Nf = 10 000 contour represents the stable area within which, although strains could accumulate slowly and effective stresses reduce, cyclic failure did not occur.

As shown in Fig. 4, the contours applying in the one-way compressive cyclic region (qmean ≥ qcyc) situated above the Nf = 10 000 contour, curve downwards rapidly towards the right-hand corner (qcyc/(2Su) = 0). The interpreted two-way contours (where qmean ≤ qcyc) confirm that chalk putty is more susceptible to compression–extension loading than one-way compression. Similar interactive failure schemes were established from axial cyclic loading field tests on piles driven in chalk by Buckley et al. (2018).

The test outcomes for the 400 kPa p0 tests are broadly compatible with the contours in Fig. 4, although the higher pressure tests developed excess pore pressures at higher rates and failed at a significantly earlier stage in DCy-B1-E, as detailed later. The chalk's response to high-level cyclic loading is demonstrated by two unstable tests, DCy-C3 and DCy-D4, which were pre-sheared to different qmean but cycled with identical qcyc. Figs 5 and 6 plot their stress–strain response, the overall effective stress paths as well as zoomed-in illustrations of six illustrative cycles prior to and shortly after their nominal failure at N = Nf. Fig. 7 shows how the secant undrained cyclic Young's modulus (Esecu,cyc) and damping ratio (D) evolved in test DCy-D4, showing the following.

 (a) Axial strain accumulation accelerates markedly as cyclic failure develops. Straining tended towards a positive εa (bulging) pattern in DCy-D4, while negative εa (and necking) developed in DCy-C3. While DCy-D4 satisfied failure criteria A and B simultaneously (at Nf = 47), DCy-C3 met criterion B one cycle after matching criterion A at Nf = 65. (b) Stress–strain curves fall initially in tight bands, fanning out as failure approached. ‘Kinks’ (as termed by Wijewickreme & Soysa (2016)) were evident in the post-Nf stress–strain loops, where strain hardening occurred and (tangent) stiffnesses increased as deviatoric stresses cycled towards their peaks and troughs. (c) Effective stress paths drift leftward invariably as pore water pressures grow. The paths traversed the PT points and slopes defined by monotonic loading (see Fig. 2), as well as the critical state slopes M. (d) Both contractive and dilative behaviour occurs during individual cycles, as shown in Fig. 6. Best-fit lines drawn through the cyclic PT points identified from the effective stress path loops of two-way cyclic tests with qmean = 0 indicate (q/p′)PTcyc gradients of ≈ 0·54 and 0·38 in compression and extension, respectively, that fall well below the monotonic PT stress ratios. Mao & Fahey (2003) report similar findings for calcareous silts as do Porcino et al. (2008) for an uncemented carbonate sand under cyclic simple shear. (e) There is systematic variation in the damping ratio and cyclic stiffness trends. Damping ratios show maxima near failure, followed by marked post-failure reductions as cyclic strains increase. Doygun & Brandes (2020) reported similar, although less abrupt, post-peak decreases for sands.

Fig. 5. Unstable cyclic response in tests DCy-C3 and DCy-D4: stress–strain behaviour

Fig. 6. Stress paths for unstable tests DCy-C3 and DCy-D4 and the identified cyclic PT lines

Fig. 7. Damping ratio and cyclic stiffness evolution in unstable test DCy-D4

Equivalent traces are shown in Fig. 8 for two typical ‘stable’ (Nf > 10 000) tests, DCy-A1 and DCy-D1, with the latter being pre-sheared to the highest qmean and both being cycled with the lowest qcyc in their sub-group. The specimens accumulated only small axial strains over ≈ 10 000 cycles, developing εaN patterns that can be matched by power-law functions (εa = a × Nb) with b = 0·502 for both cases. Their stress–strain loops evolved steadily, with moderately increasing cyclic stiffnesses and decreasing damping ratios, as listed in Table 3. Applying a large number of such low-level cycles resulted in a stable, non-linear, but principally reversible response that enhanced the destructured chalk's cyclic resistances. Similar outcomes were reported for silica sands (by Aghakouchak et al. (2015)) and stiff glacial till (see Ushev & Jardine (2022)). The pore water pressure ratio ru ( = (p0 − p′)/p0 = Δu/p0, p0 = 200 kPa) tended to stabilise after ≈ 2000 cycles and eventually reached 34·5% and 67·4% in DCy-A1 and DCy-D1, respectively. The specimens’ cyclic stress path orientations correlated directly with their position relative to the (q/p′)PTcyc lines indicated in Fig. 8.

Fig. 8. Unfailed (Nf > 10 000) tests DCy-A1 and DCy-D1: (a) axial strain; (b) effective stress path

A ‘transitional’ response, between the above ‘stable’ and ‘unstable’ styles, was observed in DCy-D2 under qmean/(2Su) = 0·57 and qcyc/(2Su) = 0·3. Fig. 9 shows how the specimen's axial strain accumulated almost linearly up to N = 4000 and accelerated rapidly up to N ≈ 7000, followed by a far slower ‘near-stable’ trend towards N = 10 000, leading to a large ultimate strain of 5·5%. Strain criterion A was met when peak axial strain reached 1% at Nf = 5528, while the strain amplitude remained far below the 1% criterion B limit throughout. The specimen's resistance to loading (from qmin to qmax) was maintained by its tendency to dilate, which kept ru largely constant at 82% and the effective stress loops settled to a stable pattern after N > 7000.

Fig. 9. Axial strain accumulation and double-amplitude trends for the ‘transitional’ test DCy-D2

Cycling from p0′ = 400 kPa

Cycling from p0 = 400 kPa and p0 = 200 kPa (with qmean = 0) provoked broadly compatible cyclic patterns. Although the higher pressure tests developed lower (absolute) strains and higher cyclic stiffness under similar normalised loading levels (see Table 3), higher pore water pressure ratios were observed. The unfailed test DCy-A1-E developed a final ru twice that of its low-pressure equivalent DCy-A1 after 10 000 cycles, while unstable test DCy-B1-E developed pronouncedly more rapid ru growth than DCy-B1 (see Fig. 10). Cyclic failure was accompanied by marked and simultaneous changes in axial strains, pore water pressures, cyclic stiffness and damping (see Fig. 7). Fig. 10 also gives further details on how the stress path loops evolved in tests DCy-B1 and DCy-B1-E with reference to the (q/p′)PTcyc lines indicated by the two-way cyclic tests. The cyclic PT lines appear largely independent of p0′ level in both triaxial compression or extension.

Fig. 10. Tests DCy-B1 (p0′ = 200 kPa) and DCy-B1-E (p0′ = 400 kPa): (a) pore water pressure ratio; (b) selected stress path loops near cyclic failure

Cyclic strain accumulation and stiffness trends

The specimens’ (permanent) strain (captured at the end of each full-stress cycle) accumulation, cyclic stiffness degradation and mean effective stress drifting patterns can be categorised into three broad groups. Accumulated cyclic strain trends are plotted in Fig. 11(a) for the unstable cyclic resistance ratio ‘CRR’ (qmean = 0) group. The specimens developed minimal (≤ 0·025%) straining over their initial stable stages, before trending abruptly towards negative (extension), as cyclic failure approached, and peak-to-trough amplitudes exceeded 1% (criterion A). Most tests developed positive strains over their initial cycles, followed by a trend to reverse towards extension failure. Tests DCy-D5 and DCy-D5-E which cycled with the highest (qcyc/(2Su)) ratio of 0·75 accumulated negative strains throughout. The overall straining patterns were consistent between the p0 = 200 and 400 kPa tests, while their cyclic stiffnesses were markedly dependent on the levels of p0 and cyclic stress ratio, as shown in Fig. 12.

Fig. 11. Trends for accumulated cyclic strain against number of cycles: (a) unstable group, qmean = 0; (b) unstable group, qmean ≠ 0; (c) ‘stable’ (unfailed within 10 000 cycles) group

Fig. 12. Cyclic stiffness degradation trends for unstable tests: (a) qmean = 0; (b) qmean ≠ 0

Figures 11(b) and 12 plot the corresponding trends for the unstable qmean ≠ 0 group tests. Axial strains accumulated in the direction of pre-shearing qmean and the straining was generally more abrupt in tests with higher qcyc/(2Su) or negative qmean (test DCy-CX). Test DCy-D2 developed ‘transitional’ cycling behaviour (as discussed previously) with its stiffness degrading significantly over its initial 6000 cycles but recovering subsequently. As revealed in Fig. 7, damping ratios generally increased over the initial cycles with the applied qcyc/(2Su) ratio, but reached similar peaks (on average 23·4%, see Table 3) as failure approached.

Specimens in the unfailed group (Nf > 10 000) accumulated moderate (< 0·16%) permanent strains, as demonstrated in Fig. 11(c). The cyclic stiffness decreased slightly but gained ultimately by on average 15% in the p0 = 200 kPa tests, as listed in Table 3, while the elevated stress test (DCy-A1-E) lost 30% stiffness, correlating with its much greater proportional reduction in p′ (see Fig. 14(a) later).

Figure 8 gives examples of how equation (1) power-law functions match the evolution of permanent cyclic strains with N. A similar approach was applied to other unfailed tests, and to the initial stages of unstable tests prior to strain reversal or significant acceleration. Parameters a and b were controlled predominantly by qcyc and were relatively insensitive to qmean. The empirical curve-fitting equations (2) and (3) provide a means of estimating permanent strains developed prior to cyclic failure and can be employed when developing and calibrating analyses of overall pile response to axial cycling through global or local tz approaches, as described by Jardine (2020).
$εa=a×(N)b$
1
$a=−0⋅0011×lnqcycp′0−0⋅0012(in%)$
2
$b=0⋅265×e[5⋅328×(qcyc/p′0)]$
3
Cyclic resistance ratios
Drawing on the symmetrical cycling tests (with qmean = 0), Fig. 13 demonstrates trends for cyclic stress (qcyc/(2Su)) and resistance (CRR: qcyc/(2p0)) ratios against Nf or Nmax. The trends are compatible with those commonly observed for carbonate sands and silts (Sanin & Wijewickreme, 2006; Porcino et al., 2008) and can be matched by the function given as equation (4), which is plotted as a dashed line in Fig. 13 through both the p0 = 200 and 400 kPa test points.
$qcyc(2Su)=0⋅24+11⋅32+0⋅35×[log10(Nf)]2⋅5$
4

Fig. 13. Trends for cyclic stress ratio (qcyc/(2Su)) and cyclic resistance ratio (CRR: qcyc/(2p0′)) against number of cycles to failure (Nf)

The qcyc/(2p0)–Nf trend can be derived by noting the correlation of 2Su ≈ triaxial compression (q)PT ≈ p0/2 found in the monotonic tests. The function implies a qcyc/(2Su) ratio of 0·32 at Nf = 10 000 and a lower limit of 0·24 below which regular symmetric two-way loading can be applied indefinitely. The qcyc/(2Su) limit of 0·24 is lower than in tests DCy-A1 and DCy-A1-E, which did not show fully stable trends for mean effective stress (discussed below). The lower limit exceeds the linear elastic (Y1) threshold with Δq/(2Su) ≈ 0·22 developed under monotonic loading (as discussed previously) and could be regarded as the limit to the outer kinematic Y2 surface within which specimens develop hysteretic closed stress–strain loops with non-linear yet recoverable straining (Smith et al., 1992; Kuwano & Jardine, 2007; Ushev & Jardine, 2022).

The tests’ detailed, cycle-by-cycle measurements enable further interpretation and application in the laboratory test-based predictive framework for axial cyclic pile loading assessment described by Jardine et al. (2012), Rattley et al. (2017) and Jardine (2020). Figs 14(a)–14(c) plot the ratios of mean effective stress changes (Δp′ = p′N=ip′N=1) for all tests by reference to specimens’ pre-shearing p0 of 200 or 400 kPa, in ascending qcyc/p0 sequences. The following observations apply.

 (a) All tests showed Δp′/p′0 decreasing continuously against N. It is possible that cycling at lower levels would identify conditions under which no reduction occurred. (b) Steeper rates of Δp′/p′0 drift were observed in specimens cycled from higher pressures (p′0 = 400 kPa) under qcyc/p′0 ratios < 0·3 than in equivalent p′0 = 200 kPa experiments, but the influence became less discernible at higher qcyc/p′0. (c) The rates of Δp′/p′0 degradation depended principally on the cyclic stress ratio (qcyc/p′0). The influence of qmean/p0′ was modest over the central portion of the interactive stress diagram and became more significant as (qcyc + qmean)/p′0 exceeded 0·4 in the p′0 = 200 kPa tests, causing the Nf contours to curve down markedly (see Fig. 4).

Fig. 14. Drifting trends for mean effective stress during cycling: (a) qcyc/p0′ ≤ 0·15; (b) qcyc/p0′ ≈ 0·23; (c) qcyc/p0′ ≥ 0·30; (d) fitted trends for tests with p0′ = 200 kPa and mostly (qcyc + qmean)/p0′ < 0·4

Tests on dense sands and stiff clays show that pre-cycling with relatively high stress ratios (as occurs during pile driving) reduces and, at low qcyc/p0 can even reverse, the Δp′/p0 drift rates observed on renewed cycling at lower qcyc/p0 levels (Aghakouchak, 2015; Aghakouchak et al., 2015; Rattley et al., 2017). It remains to be established whether such trends apply to destructured chalk. The above authors demonstrated how the Δp′/p0N (or Δσz/σz0N) relationships from cyclic triaxial, hollow cylinder apparatus or simple shear tests could be expressed by the power-law form in equation (5) and applied to generate shaft capacity degradation trends that match trends from cyclic pile tests (Jardine & Standing, 2012).
$Δp′p′0=A×B+(qcyc/p′0)×NC$
5
where A, B and C are parameters defining the rate of p′ degradation and the maximum cyclic stress ratio that could lead to beneficial, null, or deleterious cycling effects. The p0 = 200 kPa tests with (qcyc + qmean)/p0 < 0·4 indicate A = −0·05 and B = −0·12, regardless of the applied qcyc/p0 ratio, with the following best-fitting linear correlation for parameter C.
$C=3⋅48×qcycp′0$
6

The chalk putty tests indicated a far wider 0·3 < C < 1·3 range than reported for dense sands or stiff clays. Fig. 14(d) demonstrates how the above correlations and parameters provide generally good matches (shown as dashed lines) with the (mainly p0 = 200 kPa) experiments. Note that fully stable responses are expected when qcyc/p0 < |B| = 0·12 (or qcyc/(2Su) = 0·24), in keeping with the qcyc/(2Su) lower limit implicit in equation (4).

Table 4 summarises key outcomes of post-cyclic monotonic undrained shearing stages conducted after the five tests that sustained 10 000 cycles, listing the initial stress conditions and pore pressure ratios (ru) along with the initial maximum and normalised Young's moduli (Emaxu) and the ultimate stresses and strains attained. When sheared monotonically from initial conditions with an average ru = 62%, the pre-cycled specimens exhibited maximum Young's moduli that were broadly comparable to those of ‘virgin’ specimens sheared from ru = 0 (see Table 1). More pronounced changes were found in the putty's normalised stiffness ratios, (Emaxu/pref)/(p0/pref)0·5, where pref = 101·3 kPa, which were on average 58% higher than for the monotonic tests conducted without pre-cycling. Long-term (≈ 35 days) undrained low-amplitude stress cycling enhanced stiffness and led to slightly higher shear strengths. The pre-cycled specimens manifested principally dilative responses and trended towards ultimate shear strengths with ultimate stress ratios M = (q/p′)ult rising moderately from 1·27 to 1·34.

 Table 4. Summary of maximum undrained Young's moduli ($Emaxu$), pore water pressure ratio, ultimate stress states and the corresponding axial strains (in brackets) of the post-cyclic monotonic tests

Table 4. Summary of maximum undrained Young's moduli (

$Emaxu$
), pore water pressure ratio, ultimate stress states and the corresponding axial strains (in brackets) of the post-cyclic monotonic tests

Test p′: kPa q: kPa ru0: %
$Emaxu$
: MPa
$Emaxupref/p0′pref0⋅5$
*
q(ult): kPa (εa: %) p(ult): kPa (q/p′)ult ϕult: degrees
DCy-A1 120·2 0 39·9 1265·8 11 471 837·1 (31·5%) 640·4 1·31 33·2
DCy-C1 103·6 30 52·7 1159·4 11 319 1002 (27·8%) 744·9 1·35
DCy-D1 94·8 77·3 65·3 1110·4 11 330 919·5 (22·3%) 705·3 1·30
DCy-D2 47·6 67·2 87·4 936·0 13 483 645·2 (19·5%) 465·9 1·38
DCy-A1-E 143·4 0 64·2 1492·4 12 384 1205·3 (18·8%) 881·0 1·37

* pref, reference pressure (101·3 kPa).

Figure 15 considers the state paths followed in the v–log(p′) plane by the destructured chalk during undrained triaxial compression tests from p0′ of 70, 200 and 400 kPa. This treatment enables further interpretation and modelling of chalk's mechanical response and destructuration behaviour with critical-based approaches, such as the Lagioia & Nova (1995) framework.

Fig. 15. Critical state relationships established for destructured and intact chalk

As noted earlier, the pre-shearing states achieved by the destructured specimens after their initial ‘in-mould’ and later isotropic consolidation and creep stages fell (unexpectedly) well below the CRS 1D compression path, when projected assuming K0 = 1 − sin(ϕcs) = 0·48 from Fig. 1. The tests’ PT points are shown and a critical state relationship is drawn through the tests’ ultimate shearing points, which fall, as expected, below the 1D compression curve, but also below the critical state relationship established for intact chalk through high-pressure triaxial tests. It is interesting to speculate whether the intact and destructured critical state line (CSL) relationships would fall closer if it had been possible to compress beyond the ≈ 30% maximum axial strain that can be imposed in triaxial tests.

The destructured samples’ tendency to dilate significantly post-PT is compatible with their ‘dry’ position relative to their CSL. In terms of the Been & Jefferies (1985) framework, the specimen's state parameters (ψ = e − ecs) decrease systematically as p0′ grows, a trend that is reflected in the high-pressure tests’ tendency to produce higher cyclic excess pore pressures.

As noted in the introduction, the cyclic response of piles driven in chalk is affected by both the putty zone around the shaft and the surrounding intact mass. It is therefore interesting to contrast the undrained cyclic triaxial behaviour of chalk in both conditions, drawing on representative paired tests that applied identical mean and cyclic stresses. Fig. 16 offers such a comparison for tests with qcyc/(2Su) = qmean/(2Su) ≈ 0·44. Cyclic failure in the putty involves pore pressure build-up, leftward drifting of the effective stress path, cyclic stiffness loss and growing damping ratios (see also Figs 5–7 and 10–12). In contrast, the intact chalk shows a very stiff cyclic response, with little or no sign of pore pressure change or impending instability. The cyclic effective stress paths remain within a tight band beforehand and until shortly before abrupt brittle failure, which leads to markedly dilative pore pressure trends. Ahmadi-Naghadeh et al. (2022) conclude that the intact response is closer to that of harder rocks and solids, such as concretes and metals, whose cyclic or fatigue failure is dominated by their inherent microstructures and triggered by local stress concentrations that promote progressive wear and shearing.

Fig. 16. Effective stress path of intact (Ahmadi-Naghadeh et al., 2022) and destructured chalk (samples with qcyc/(2Su) = qmean/(2Su) ≈ 0·44; plotted up to Nf + 3 cycles)

Heavily destructured soft putty forms in the field under intense compression or large-strain, repetitive shearing. Understanding the putty's behaviour after reconsolidation is crucial to advancing driven pile design in chalk. This paper reports 20 high-resolution triaxial experiments and other tests on putty formed by dynamic compaction followed by re-consolidation and explores their responses to both monotonic and one- and two-way sinusoidal deviatoric undrained loading. The experiments, which ran in parallel alongside an intact chalk programme, led to the following main conclusions.

Overall, the interactive cyclic triaxial stress diagrams, effective stress drift, permanent displacement and stiffness trends provide key information to aid both the interpretation of the ALPACA field tests and the cyclic design of piles driven in other comparable chalks.

ACKNOWLEDGEMENTS

The experimental study was undertaken as part of the ALPACA and ALPACA Plus Projects funded by the Engineering and Physical Science Research Council (EPSRC) grant EP/P033091/1, Royal Society Newton Advanced Fellowship NA160438 and Supergen ORE Hub 2018 (EPSRC EP/S000747/1). Byrne is supported by the Royal Academy of Engineering under the Research Chairs and Senior Research Fellowships scheme. The authors acknowledge the provision of additional financial and technical support by steering committee members and partners: Atkins, Cathie Associates, Equinor, Fugro, Geotechnical Consulting Group (GCG), Iberdrola, Innogy, LEMS, Ørsted, Parkwind, Siemens, TATA Steel and Vattenfall. Imperial College EPSRC Centre for Doctoral Training (CDT) in Sustainable Civil Engineering and the DEME Group (Belgium) are acknowledged for supporting the doctoral study of Ken Vinck. Invaluable technical support was provided by Steve Ackerley, Graham Keefe, Prash Hirani, Stef Karapanagiotidis at the Department of Civil and Environmental Engineering of Imperial College London is acknowledged gratefully.

NOTATION
 A, B, C fitting parameters for mean effective stress degradation Aelastic unloading half-cycle elastic triangle area with height as qcyc( = (qpeak − qtrough)/2) and width as cyclic strain ( = (εpeak − εtrough)/2) Aloop area enclosed by a stress–strain (q–εa) loop for a complete sinusoidal stress cycle a, b fitting parameters for permanent cyclic strain Cc compression index of intact specimen Cc* compression index of reconstituted specimen Cs swelling index of intact specimen Cαe rate of secondary compression of intact specimen c′ soil cohesion D damping ratio ( = Aloop/(4πAelastic)) D50 mean particle diameter $Emaxu$ maximum undrained Young's modulus $Esecu$ undrained secant vertical Young's modulus $Esecu,cyc$ cyclic undrained secant vertical Young's modulus e0 specimen initial void ratio Gs specific gravity K0 earth pressure coefficient at rest M critical state q/p′ stress ratio N number of cycles Nf number of cycles to failure p′ mean effective stress p0′ initial mean effective stress q deviatoric stress qcyc cyclic deviatoric stress amplitude ( = (qpeak − qtrough)/2) qf deviatoric stress at failure qmax maximum q applied in stress cycle ( = qmean + qcyc) qmean mean q applied in stress cycle qPT deviatoric stress at the phase transformation state qpeak peak q applied in stress cycle ( = qmax) qtrough minimum q applied in stress cycle ( = qmean − qcyc) ru pore water pressure ratio ( = (p′0 − p′)/p′0 = Δu/p0′) Sr saturation degree Su undrained shear strength Δu excess pore water pressure εa axial (vertical) strain εpeak axial strain at qpeak εr radial (horizontal) strain εs shear strain ( = εa for undrained triaxial condition) εtrough axial strain at qtrough ϕ′cs critical state shear resistance angle ϕ′peak shear resistance angle at peak

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• Tingfa Liu , Reza Ahmadi-Naghadeh , Ken Vinck , Richard J. Jardine , Stavroula Kontoe , Róisín M. Buckley and Byron W. Byrne