Mechanical properties of a carbonate sand from a dredged hydraulic fill Free

The ‘Mechanical properties of a carbonate sand from a dredged hydraulic fill’ (Giretti et al., 2018a) together with its companion paper ‘CPT calibration and analysis for a carbonate sand’ (Giretti et al., 2018b) are valuable because there are limited data in the literature on biogenic carbonate sands. ‘Compression of granular materials’ by Mesri & Vardhanabhuti (2009) includes some data on carbonate sands. In laterally constrained compression or isotropic compression, carbonate sands display a type C void ratio against effective stress behaviour for which significant level I and level II particle damage begin at low effective stresses and continue with or without gradual level III particle damage at higher stresses. Level I particle damage includes abrasion or grinding of particle surface asperities, level II particle damage includes breaking or crushing of particle surface protrusions and sharp particle corners and edges, and level III particle damage includes fracturing, splitting or shattering of particles.

The void ratio  plotted against effective stress relationships of most space-lattice silicate sands displays three distinct stages of compression. During the first stage, small particle movements further engage particle surface roughness and enhance interparticle locking. There is minor to small level I and level II particle damage; however, improved locking dominates over unlocking effects and M = Δσ′_v/Δε_v increases with the increase in σ′_v. The second compression stage begins with level III particle damage by the fracturing of the heavily loaded particles and collapse of the load-bearing aggregate framework. Particle fracturing unlocks the aggregate framework, allowing larger interparticle movements and M begins to decrease with an increase in σ′_v (see, for examples, Figs 2 and 22 in the paper by Mesri & Vardhanabhuti (2009)). The first inflection point in the e plotted against σ′_v relationship marks the beginning of the second stage at an effective vertical stress $(σ′v)Mmax$⁠, and a second inflection point at $(σ′v)Mmin$ marks the end at which major particle fracturing and splitting are substantially complete. In the void ratio plotted against logarithm of effective stress relationship of most sands, it is possible to identify a point of maximum curvature at $(σ′v)MC$ (see, for example, Figs 2 and 22 in the paper by Mesri & Vardhanabhuti (2009)).

Because in type C void ratio plotted against effective stress compression behaviour, significant level I and level II particle damage begin early at low effective stresses and continue with or without gradual level III particle damage at high stresses, the void ratio against effective stress relationship, such as that in Fig. 18(b) for M1 carbonate sand (from Fig. 3 of Giretti et al. (2018a)), does not display three distinct stages of compression. However, for some carbonate sands examined by Mesri & Vardhanabhuti (2009), it was possible to extract values of $(σ′v)MC$⁠, $(σ′v)Mmax$ and $(σ′v)Mmin$⁠, leading to $(σ′v)Mmax=0⋅67(σ′v)MC$ and $(σ′v)Mmin=2⋅0(σ′v)MC$⁠. In Fig. 18(a), a value of $(σ′v)MC=1MPa$ was selected for M1 carbonate sand, and the values of $(σ′v)Mmax$ and $(σ′v)Mmin$ were computed only for reference, as it is not possible to identify them on Fig. 18(b). A main difference between the compression behaviour of carbonate sands as compared to that of silicate sands, is that, for the data examined by Mesri & Vardhanabhuti (2009), for carbonate sands the values of $(σ′v)MC$ are in the range of 0·4–2 MPa, whereas for silicate sands, they are in the range of 4–40 MPa (Fig. 21 of Mesri & Vardhanabhuti (2009)).

According to cone penetration tests in calibration chambers, and interpretation by Houlsby & Hitchman (1988), for silicate sands, cone tip resistance, q_t, is governed primarily by effective horizontal stress, σ′_h, according to

in which p_a is atmospheric pressure, and parameter A had values of 50, 160 and 230, respectively, for loose, medium and dense silicate sands.

Based on the K₀ versus σ′_v measurements in Fig. 12, the authors selected an average value of K₀ = 0·5 for the M1 carbonate sand with void ratio in the range of 0·58–0·89. Because σ′_h, and thus K₀ as a function of void ratio, is a significant factor in cone tip resistance according to equation (6), in Fig. 19 the K₀ values from Fig. 12 have been plotted for the M1 carbonate sand against void ratio. The extrapolation, in Fig. 19, of the data to initial void ratio of M1 carbonate sand (Fig. 3, M1 small oedometer) leads to a K₀ = 0·36. According to Mesri & Hayat (1993), the Jaky (1948) equation for K_0p is

which together with $ϕ′cv=40⋅3deg$ leads to K_0p = 0·36. Note that for Mexico City clay with poriferous diatom shells, $ϕ′cv=43deg$ and measured K_0p = 0·30 (Mesri et al., 1975; Mesri & Hayat, 1993). For comparison with behaviour of the M1 carbonate sand, Fig. 20 shows K₀ plotted against void ratio for three silicate sands which were densified by vibration (Mesri & Vardhanabhuti, 2007).

The discussers have interpreted, and shown in Table 3, the cone tip resistance, q_t, in dry M1 sand in Fig. 12, of Giretti et al. (2018b) using equation (6) together with K₀ in Fig. 19. The computed values of parameter A, ranging from an average value 110 at e = 0·823 to 259 at e = 0·666, are plotted in Fig. 19. Values of $ϕ′tc$ for the M1 carbonate sand have also been computed, as listed in Table 3, by means of the empirical equation reported by Houlsby & Hitchman (1988) for silicate sands

where N_h = q_t/σ′_h. Most of the computed magnitudes of $ϕ′tc$⁠, only several degrees higher than $ϕ′cv=40⋅3deg$⁠, are reasonable because the highly crushable calcium carbonate particles cannot mobilise significant dilatant geometrical interference beyond $ϕ′cv=40⋅3deg$⁠. However, for one q_t measurement at e = 0·727 and four q_t measurements at e = 0·823, the computed values of $ϕ′tc$ are actually smaller than $ϕ′cv=40⋅3deg$⁠. Apparently, for these combinations of void ratio and effective stress condition in M1 carbonate sand with crushable particles, the $ϕ′p$ component of friction angle resulting from interparticle interference during shear is smaller than the magnitude mobilised at a constant volume shearing condition (Terzaghi et al., 1996).

The authors thank the discussion contributors for their very interesting observations, which provided the possibility of further analysing some aspects of the mechanical behaviour of carbonate sands.

The authors acknowledge the observation that M1 sand shows a type C void ratio–effective stress behaviour, with a progressive increase of the constrained modulus M.

As to the interpretation of the centrifuge cone penetration tests (CPTs) on M1, the basic concept of the authors’ analysis is that both the stress-dilatancy behaviour and the normalised cone resistance of M1 sand depend on the state parameter ψ.

This is shown

• (a)

  in Fig. 8 of Giretti et al. (2018a) where the peak shearing resistance angle is plotted against ψ; the trend ϕ′_tc–ψ can be expressed with a linear function a

  $ϕ′tc=ϕ′cs−aψ$

  9

  where ϕ′_cs = 40·3° is M1 shearing resistance angle at critical state and a = 25·5 is the function slope.
• (b)

  in Figs 16–18 of Giretti et al. (2018b), where the normalised cone resistance Q_p = (q_t − p)/p′ is plotted against ψ; an exponential function has been adopted to interpret the experimental trend

  $Qp=kexp(−mψ)$

  10

  where k and m are calibration coefficients whose values are 42 and 5·1 for dry M1 sand and 35 and 5·1 for saturated M1 sand.

From the data in Table 3, ϕ′_tc has been recomputed according to equations (9) and (10). The recomputed values are listed in Table 4; they are slightly larger than ϕ′_cs in the looser sample, up to 9° higher than ϕ′_cs in the denser sample.

The difference between the ϕ′_tc values computed by the contributors and the authors’ estimation could be due to the value of the constants Q and R of the stress-dilatancy model of Bolton (1986), on which the Houlsby & Hitchman (1988) q_t/σ′_h − ϕ′_tc equation is based – constants which were calibrated for silica sand. According to the Bolton (1986) model, Q is a particle strength parameter equal to 10 for quartz sands and R is normally assumed as equal to 1.

In Fig. 21 the difference ϕ′_tc − ϕ′_cs computed for M1 sand is represented as a function of the relative density D_r (data from Fig. 8 in the paper by Giretti et al. (2018a)); in the figure, the Bolton equation, reported below, is also represented as a solid line.

A very poor correlation for M1 sand can be observed.

A second point to be considered is that normally consolidated (NC) M1 sand has shown a clear dependence of the cone tip resistance q_t on the vertical effective stress σ′_v, as shown by Giretti et al. (2018b) in Figs 11–15. Similar behaviour has been observed by Fioravante & Giretti (2016) from centrifuge CPTs carried out in NC Ticino and Toyoura silica sands.

Jamiolkowski et al. (2003), analysing a very large dataset of calibration chamber CPTs in Ticino, Toyoura and Hokksund silica sands, observed that the cone penetration resistance depends on the vertical σ′_v in the case of NC sands, and the horizontal stress σ′_h plays a major role in the case of overconsolidated sands.

Recognising the importance of the horizontal stress on the cone resistance, the authors have selected the stress invariant p′ for the computation of a normalised q_t.

To check and verify whether the stress-dilatancy behaviour observed for M1 is consistent with the mechanical behaviour of other carbonate sands, the authors have interpreted a series of unpublished laboratory and centrifuge tests available for Quiou sand (QS), performed at Istituto Sperimentale Modelli e Strutture (ISMES; Italy) in the 1990s.

QS is a skeletal carbonatic sand of biogenic origin (Fioravante et al., 1994, 1998; Porcino et al., 2008; Mesri & Vardhanabhuti, 2009).

Among the triaxial tests available for QS, Fig. 22 shows the results of selected tests which reached the critical state and allowed the definition of the QS critical state line in e–p′ and q–p′ planes. QS has the same ϕ′_cs as M1 and quite similar compressibility characteristics.

Figure 23 shows the peak shear angle ϕ′_tc achieved in drained tests on QS as a function of the state parameter ψ at the end of consolidation. As for M1 sand, a linear relationship (equation (9)) can be used to interpolate the experimental data and the slope of the function is a = 49·5.

Seven q_t−σ′_v profiles measured on normally consolidated centrifuge models of dry QS are reported on Fig. 24. The samples have an average void ratio e = 0·96 and e = 0·87 and have been tested at a centrifugal acceleration of 30g and 80g. As for M1 sand, the q_t profiles have been normalised against p′ and plotted against ψ in Fig. 25. Also for QS an exponential function can be adopted to fit the experimental trends and the m and k parameters of equation (10) are 4·1 and 61, respectively.

Following the same procedure as M1, equations (9) and (10) have been combined to derive the peak resistance angles as a function of Q_p (see Fig. 26).

In conclusion, the state parameter ψ appears to be a useful independent state indicator to interpret the mechanical behaviour of carbonate sands.

A

parameter in equation (6)

a

fitting parameter of equation (9)

C_c

compression index, Δe/Δlog σ′_v

C_u

uniformity coefficient, D₆₀/D₁₀

D₁₀

grain size at which 10% is finer

D₅₀

mean grain size

D₆₀

grain size at which 60% is finer

D_r

relative density

e

void ratio

K₀

coefficient of earth pressure at rest

K_0p

coefficient of earth pressure at rest in normally consolidated young loose sands

k

fitting parameter of equation (10)

M

tangent constrained modulus, Δσ′_v/Δε_v

M_max

tangent constrained modulus at the first inflection point of σ′_v plotted against ε_v

M_min

tangent constrained modulus at the second inflection point of σ′_v plotted against ε_v

m

fitting parameter of equation (10)

N_h

q_t/σ′_h, used in equation (8).

P_a

atmospheric pressure

p

mean total stress

p′

mean effective stress

Q

fitting parameter of equation (11)

Q_p

normalised cone resistance

q_t

cone penetration test cone resistance

R

fitting parameter of equation (11)

ε_v

vertical strain; volumetric strain

σ′_h

horizontal effective stress

σ′_v

vertical effective stress

$(σ′v)Mmax$

effective vertical stress at the yield point, defined at the first inflection point of e plotted against σ′_v M_max

$(σ′v)Mmin$

effective vertical stress at the yield point, defined at the second inflection point of e plotted against σ′_v, defining the end the second stage of compression.

$(σ′v)MC$

effective vertical stress at the yield point, defined at the point of maximum curvature of e plotted against σ′_v

ϕ′_cs

shearing resistance angle at critical state

ϕ′_cv

constant volume friction angle

ϕ′_p

component of friction angle resulting from interparticle interference during shear

ϕ′_tc

effective stress friction angle mobilised in triaxial compression

ψ

state parameter