Reappraisal of soil extrusion for geomechanical characterisation Open Access

A_f

die orifice area

A_o

bore cross-sectional area of the extrusion chamber

a, a′

regression coefficients for semi- and bi-logarithmic correlations, respectively

b, b′

regression exponents for semi- and bi-logarithmic correlations, respectively

c

extrusion coefficient

D_f

die orifice diameter

D_o

bore diameter of the extrusion chamber

d, d′

extrusion coefficients

F

applied force

F_e

force causing extrusion

I_L

liquidity index

I_P

plasticity index

I_RE

new consistency index parameter

K_RE

reverse-extrusion factor

k

shear yield stress

L

initial billet length

n

number of experimental measurements (data points)

PL₂₅

water content for s _u value 25 times greater than that mobilised at the fall-cone liquid limit

p_e

extrusion pressure

p_e(LL)

steady-state extrusion pressure for the liquid limit

p_e(PL)

steady-state extrusion pressure for the plastic limit water content

p_e(SL)

steady-state extrusion pressure for the shrinkage limit water content

R

extrusion ratio

R²

coefficient of determination

RE₂₅

water content corresponding to the steady-state extrusion pressure value of 25 × p _e(LL)

r_e

extrusion rate

s_u

saturated remoulded undrained shear strength

s_u(LL)

saturated remoulded undrained shear strength at the liquid limit

s_u(PL)

saturated remoulded undrained shear strength at the plastic limit

T

remoulding toughness

T_max

maximum remoulding toughness

t

elapsed time

v

die displacement rate

w

water content

w_L

water content at the liquid limit

w_L(cup)

water content at the Casagrande liquid limit

w_L(FC)

water content at the fall-cone liquid limit

w_n

natural water content of undisturbed test specimens

w_P

water content at the standard plastic limit

w_S

water content at the shrinkage limit

w_ST

water content at the stiffness transition

w_T

water content at the toughness limit

σ

standard deviation

Extrusion as a mechanical process involves the reduction of a billet’s cross-sectional area by forcing it to flow through a die orifice under pressure. The concept of extrusion has been applied to soil for brick manufacturing since the mid-nineteenth century (Whyte, 1982) and more recently for the rheological characterisation of various materials, such as food and ceramics (Cheyne et al., 2005; Göhlert and Uebel, 2009). The extrusion technique was first introduced for soil characterisation by Timár (1974) and subsequently promoted for determinations of undrained shear strength (Kayabali, 2011a; Kayabali and Ozdemir, 2013; Kayabali and Tufenkci, 2010a; Kayabali et al., 2015a, 2015b; Whyte, 1982) and Atterberg limits (Kayabali, 2011a, 2011b, 2012, 2013; Kayabali and Tufenkci, 2007, 2010a, 2010b; Kayabali et al., 2015c, 2016; Medhat and Whyte, 1986; Verástegui-Flores and Di Emidio, 2014, 2015). Extrusion has also been employed for three-dimensional clay printing, which allows the construction of physical heterogeneous models (Pua et al., 2018), as well as its promotion for possible effective-stress shear strength parameter determinations (Kayabali et al., 2015d).

Timár (1974) employed the direct-extrusion (DE) approach, whereby the soil test specimen (billet) is forced along a cylindrical chamber and extruded through a contoured orifice (Figure 1(a)). The material flowing through the orifice has a round bar shape, with a cross-section identical to that of the orifice. However, the method of data interpretation is complicated for DE due to the influence of the frictional resistance generated as the billet is forced along the chamber length to the die, with the value of the pressure causing extrusion (p_e: Equation 1) progressively reducing for the shortening billet (Figure 1(b)).

where F_e is the applied compressive force causing extrusion to occur and A_o is the extrusion chamber’s bore cross-sectional area.

Hence, investigations of soil extrusion after that of Timár (1974) generally adopted the indirect- or reverse-extrusion (RE) approach (Figure 2(a)). For this set-up, the billet is rigidly contained while the die advances towards it at a certain velocity, such that compared with the DE approach, the required value of p_e is significantly lower and attains a steady-state condition for the RE approach – that is, since no relative motion occurs between the billet and chamber wall for RE (resulting in much lower mobilisation of interface frictional resistance overall), the value of p_e remains approximately steady with increasing die displacement after the initial billet compression phase (Figure 2(b)). To maintain this steady-state condition, the soil billet must have a sufficient length, with an initial billet length of 1D_o to 1·5D_o typically employed, where D_o is the bore diameter of the extrusion chamber. The soil material yields when it enters the shear zone (fan), located in front of the ‘dead’ material zone localised between the chamber wall and die face. The initial approximately linear increasing portion of the extrusion pressure against die displacement plot signifies the work required in compressing the placed soil billet to reduce its air void content to practically zero. A small clearance (gap) between the die rim and chamber wall prevents metal friction between them, and the die orifice is also usually chamfered (e.g. see the die orifice shown in Figure 2(a)), so as to reduce the contact friction produced between the die and extruding soil worm to negligible levels.

Some principal physical dimensions of existing soil-extrusion apparatus and associated displacement rate (v) values employed, as reported in the literature, are summarised in Table 1. A 6·0 mm dia. orifice was used in the investigations by Whyte (1982) and subsequently in the many investigations by Kayabali and co-workers, since a standard thread-rolling plastic-limit (PL) test can be quickly performed on the extruded 6 mm dia. soil worm to confirm the soil’s consistency, if necessary.

Figure 3(a) presents the RE set-up employed in the studies by Verástegui-Flores and Di Emidio (2014, 2015), which is typical of other soil RE apparatus configurations. Using a uniaxial-compression test machine to provide the desired loading condition (Figure 3(b)), the die is set to travel against the confined soil specimen at a constant displacement rate v (i.e. displacement rate controlled), generally in the range 1–5 mm/min and typical of unconfined compression testing, while the die displacement and corresponding extrusion force are both recorded.

When the soil reaches the shear (plastic) zone within the extrusion chamber (zone labelled B in Figure 2(a)), it distorts and extrudes by way of the die orifice as a soil worm, with the value of the applied compressive force (and the value of p_e computed using Equation 1) assumed to relate to the undrained condition. As such, the soil’s resistance capacity is conventionally understood as controlled by its saturated remoulded undrained shear strength, s_u, the value of which is principally dependent for a given test soil on its water content (w) and to a lesser extent on the strain rate. The nature of the soil flow and redundant deformation defining the extent of the dead material zone within the soil billet are dependent on the apparatus extrusion ratio (R: Equation 2) and the die configuration (i.e. shape and geometrical layout of the die orifice(s)). Greater values of p_e are mobilised for higher R values since the associated energy for soil extrusion increases.

where A_o and A_f are the bore cross-sectional area of the extrusion cylinder and the die orifice area, respectively.

The dependence of the extrusion pressure on extrusion ratio and die displacement rate was investigated experimentally by Verástegui-Flores and Di Emidio (2014). They performed RE testing of a kaolin material at its fall-cone (FC) liquid limit (LL) water content (w_L(FC)) value, measuring the steady-state p_e value mobilised for R = 4, 16 and 64, with set values of v = 2, 4, 6 and 8 mm/min (Figure 4). As expected, they found that greater steady-state values of p_e were mobilised for higher R and v values. This dependency will be examined in greater detail later in the paper.

The extrusion rate (r_e: Equation 3), which is a function of the extrusion ratio, typically ranges 40–200 mm/min based on the reported RE soil testing set-ups summarised in Table 1.

The geometrical layout of the die can be reconfigured to reduce the size of the dead material zone but still maintaining the same R value, by using a die with multiple equally spaced smaller orifices rather than the customary single central orifice. This modification was adopted for the mud-press machine (MPM) extrusion device developed by Kayabali et al. (2015a, 2016) and which is described later in this section. Based on the presented p_e against die displacement traces and reported v values in the papers by Kayabali and Tufenkci (2007) and Kayabali and Ozdemir (2013), the steady-state p_e condition (i.e. soil worm extrudes at a steady rate) is achieved within ∼1–4 min from the start of the RE test.

An alternative soil-extrusion approach using an MPM device (see Figure 5(a)) was proposed in the papers by Kayabali et al. (2015a, 2016) – that is, a miniature DE machine which extrudes the soil billet contained in a 30 mm dia. chamber by way of 28 equally spaced 2·5 mm dia. orifices (i.e. R = 5·1). In this instance, the applied loading is rendered by way of a manually operated mechanical press device, with the value of F_e increasing steadily and quickly, becoming constant as soil worms extrude from the chamber orifices, as evident from the flat portions of the applied force against time traces in Figure 5(b). In other words, the extrusion velocity is not controlled for this experimental set-up, being dependent on the consistency of the test soil, the MPM’s R value and the applied force causing extrusion. Based on the extrusion force against time traces presented by Kayabali et al. (2015a) and reproduced as Figure 5(b) in this paper, the steady-state extrusion condition is achieved very quickly, typically within ∼3 s – that is, compared with the RE testing described earlier, the MPM device is portable and the test itself is significantly faster (completed within 1 min).

The RE approach has been used for undrained shear strength determinations on remoulded, compacted and undisturbed fine-grained soils (Kayabali, 2011a; Kayabali and Ozdemir, 2013; Kayabali and Tufenkci, 2010a; Kayabali et al., 2015a, 2015b; Whyte, 1982). Implicit in this approach are that, for given apparatus R and v values, the soil billet remains in an undrained condition during the course of the RE testing and, furthermore, the steady-state p_e/s_u ratio value is constant among different fine-grained soils – that is, if partial drainage (consolidation) of the soil billet were to occur, its shear strength value would increase, mobilising a higher steady-state p_e value, such that the steady-state p_e/s_u ratio value would increase accordingly. Both of these assumptions are examined in greater detail later in the paper.

In preparing remoulded RE test specimens (e.g. for consistency-limit determinations), excessive compactive effort should not be applied in forming the soil billet inside the extrusion chamber, with the static compaction method preferred. Otherwise, the billet’s compaction-induced overconsolidated state (O’Kelly, 2017; O’Kelly et al., 2019) would require higher applied extrusion pressures. Compared with specimens in the remoulded state, undisturbed test specimens invariably mobilise higher undrained shear strength in axial compression on account of their inherent fabrics (O’Kelly, 2006) and also possibly overconsolidated state, such that they too generally require higher extrusion pressures.

As described earlier, p_e = f_n(R, v, s_u), such that the steady-state extrusion pressure can be expressed in terms of the soil undrained shear strength for RE testing with given R and v values, as follows

where K_RE is the RE factor that accounts for the apparatus extrusion ratio, die displacement rate and also apparatus friction and inhomogeneous billet deformation.

Alternatively, knowing the value of the steady-state p_e/s_u ratio for specific experimental R and v values, the undrained shear strength value can be simply determined from the measured steady-state extrusion pressure value. As described earlier, implicit in these calculations is that the soil billet remains in a truly undrained condition throughout the RE testing.

Calibration of the RE device is performed experimentally, with its pertinent value of K_RE for specific R and v values producing a one-to-one correspondence between the correlations for measured pairs of steady-state extrusion pressure and undrained shear strength values determined for the reference test material prepared at various water contents. In principle, this approach is the same as that presented in the papers by O’Kelly (2014a, 2018) for the calibration of an FC device using unconsolidated-undrained (UU) triaxial-compression data obtained for test specimens prepared at various water contents. The described approach is particularly versatile since the RE device calibration can be performed for pertinent confinement pressure and shearing rate values that are controlled in performing the triaxial-compression tests. The MPM device can be calibrated in the same manner, but, because the extrusion velocity is not constant (depending on the MPM R value, soil undrained shear strength and maximum manually applied extrusion force), the MPM-derived strength values should be regarded as approximations only.

Provided that the soil billet remains in a truly undrained condition, following directly from Equation 4, one would expect for a given RE die configuration and set R and v values that the experimental steady-state p _e/s _u ratio would have the same value for various fine-grained soils investigated. In this section, the author investigates this central assumption of the RE approach for s _u measurement, making use of deduced steady-state p _e and measured s _u values presented in the paper by Kayabali and Ozdemir (2013), which is the only published research that reports s _u and associated steady-state p _e values at the same water contents for various fine-grained soils.

Specifically, Kayabali and Ozdemir (2013) reported pairs of s _u and steady-state p _e values corresponding to the PL water contents (i.e. s _u(PL) and p _e(PL), respectively) for 60 remoulded fine-grained soils (i.e. n = 60, where n is the number of samples). They also reported pairs of undrained shear strength and steady-state p _e values for 75 undisturbed natural clay soils with various natural water content (w _n) values. The steady-state p _e values were determined from RE testing for R and v values of 40·1 and 1 mm/min, respectively. Undrained shear strength was determined from unconfined compression testing of 38 mm dia. test specimens, with the remoulded specimens prepared by static compaction at the steady-state p _e values corresponding to their PL water content (w _P) values. Their reported tabulated data are plotted as the p _e(PL)/s _u(PL) ratio against both w _P and plasticity index (I _P) and also as the steady-state p _e/s _u ratio against w _n in Figures 6(a)–6(c). Consistency-limit values were reported by Kayabali and Ozdemir (2013) for the 60 remoulded soils, such that these could be categorised in the present investigation as either clay or silt materials according to the standard plasticity chart.

As evident from Figure 6, the value of the steady-state p _e/s _u ratio for the set R and v values varied widely for both the remoulded and undisturbed fine-grained soil specimens, with mean values of p _e(PL)/s _u(PL) = 13·9 (standard deviation, σ = 3·2 for n = 60) and steady-state p _e/s _u = 14·9 (σ = 4·7 for n = 75), respectively. The marginally higher p _e/s _u ratio value for the undisturbed test specimens is most likely explained by the effects of their inherent fabrics and strength anisotropy, considering the different shearing mechanisms occurring in extrusion compared with uniaxial compression. Further, the regression lines fitted to the p _e(PL)/s _u(PL) ratio values for the 60 remoulded specimens indicated an overall reducing trend with increasing w _P (Figure 6(a)) and also with increasing plasticity (Figure 6(b)), more so for the clay soils than the silt soils investigated. Assuming that the s _u measurements are sound, these observations point to a fundamental issue with extrusion testing that is soil dependent. As the constancy of the steady-state p _e/s _u ratio value is central to the RE s _u measurement approach, this aspect merits further in-depth investigation and is fully addressed later in the paper.

For extrusion of metals, the relationship between the steady-state p _e and R has been shown to be well fitted by Equation 5, which takes account of apparatus friction and inhomogeneous billet deformation.

where c and d are extrusion coefficients and k is the yield stress in shear.

Among the first soil-extrusion investigations, based on RE testing of two fine-grained soils, Whyte (1982) and Medhat and Whyte (1986) tentatively proposed two correlations of the form given by Equation 5 to relate the steady-state p _e with the soil resistance capacity – that is, with the s _u parameter replacing the k parameter in Equation 5. This relationship was further investigated by Kayabali (2011a), considering deduced p _e(PL) values and an assumed s _u(PL) = 160 kPa for 60 remoulded fine-grained soils. The deduced values of the coefficients c and d reported in these investigations are listed in Table 2. However, since the value of s _u(PL) is not unique, potentially varying over a very wide range between different soils (Haigh et al., 2013; Nagaraj et al., 2012; O’Kelly, 2013a), the c and d coefficient values reported for the Kayabali (2011a) data set can at best be regarded as mean approximations. Further, none of these investigations considered the dependence of the steady-state p _e/s _u ratio value on the die displacement rate in the case of soil-extrusion testing.

In order to investigate the relationship between R, v and the steady-state p _e/s _u ratio the author has reinterpreted the data presented in Figure 4 for RE testing of kaolin specimens prepared at their FC LL water content (w _L(FC)) values, as determined using the British Standard (BS) 30°–80 g FC in accordance with BS 1377-2:1990 (BSI, 1990). Knowing that the corresponding saturated remoulded undrained shear strength (i.e. s _u(LL)) value is ∼1·7 kPa (O’Kelly et al., 2018), steady-state p _e/s _u ratio values have been computed from the steady-state p _e data values in Figure 4 and are replotted in Figure 7. Also, included in Figure 7 are

• reported p _e/s _u–R correlations in the papers by Whyte (1982) and Medhat and Whyte (1986) 

• representative correlation deduced by the author as part of the present investigation for the 60 remoulded fine-grained soils investigated in the paper by Kayabali (2011a) 

• the mean p _e(PL)/s _u(PL) ratio value of 13·9 deduced for the 60 remoulded fine-grained soils investigated in the paper by Kayabali and Ozdemir (2013) 

• steady-state extrusion pressure for the LL water content (p _e(LL) = 23·6 kPa (Verástegui-Flores and Di Emidio, 2015)) expressed as a ratio of FC s _u(LL) (1·7 kPa) for set R and v values of 16 and 4 mm/min, respectively – that is, p _e(LL)/s _u(LL) = 15·5.

As evident from Figure 7(a), even for the relatively narrow v data range 1–8 mm/min available, the value of the steady-state p _e/s _u ratio is strongly dependent on the values of both v and logarithm R, with its dependency on the former appearing to increase with increasing R.

Note that the deduced (reported) coefficient c values listed in Table 2 are generally very small. In other words, for RE testing and assuming no apparatus friction effect, one would expect a steady-state p _e value of zero for the quasi-static extrusion of a zero-shear-resistance material, so that Equation 5 would take the form of p _e/s _u = d′ ln R, which is analogous to stress equals a modulus coefficient times strain, with the true strain of the soil billet given by ln R.

To account for die displacement rate, the steady-state p _e/s _u data are plotted against extrusion rate r _e (= vR (mm/min)) in Figure 7(b) and also against r _e/D _o in Figure 7(c), with their best-fit regression lines given by Equations 6a–6c. Equation 6b is the result of the steady-state p _e/s _u–ln r _e regression analysis for the particular case with the value of coefficient c set as zero.

The extrusion rate (r _e) is the velocity of the soil worm. When normalised by a characteristic dimension, here taken as the bore diameter of the extrusion chamber, the r _e/D _o ratio (min⁻¹) gives the strain rate. Since the compiled data set considers only two D _o values (i.e. 38 and 40 mm), Equation 6c is only tentatively proposed, although if steady-state p _e/s _u data were available for a wide D _o range, it is suggested that this form of relationship would provide superior fitting for the enlarged data set. Further, this form of relationship would allow valid comparisons of s _u measurements obtained using different RE apparatus employing various R and v value combinations.

The strain-rate range 0·2–12·8 min⁻¹ for soil extrusion (see Figure 7(c)) is comparable with that usually employed for UU triaxial-compression testing but substantially lower than the shear strain rate of the order of 170 min⁻¹ typical of the range of penetration depths measured in practice for the BS FC LL test (Koumoto and Houlsby, 2001; Sivakumar et al., 2015). The very high strain rates for the FC test specimen are such that it undergoes undrained shearing for all practical purposes and triaxial specimens are fully enclosed by a sealing membrane to ensure a fully undrained condition. In contrast, the combination of relatively high confinement pressure acting on the soil billet and slow strain rate can potentially allow some consolidation to occur with drainage by way of the die orifice. This potential major shortcoming of the soil-extrusion approach, one that has received almost no attention in the published literature, is explored in detail later in the paper.

Since first introduced in soil mechanics approximately 45 years ago, the extrusion approach has been promoted for LL and PL determinations by a handful of research groups on the premise that, for given apparatus R and v values, the w_L and w_P values are associated with unique values of p_e(LL) and p_e(PL), respectively. Furthermore, Kayabali (2012, 2013) proposed that the RE approach could also be used for shrinkage-limit (SL) determinations, claiming that the SL water content (w_S) value was also associated with a unique steady-state extrusion pressure (i.e. p_e(SL)) value. Implicit in these approaches are that the steady-state p_e/s_u ratio value is constant among different fine-grained soils and the soil billet remains in a truly undrained condition throughout the RE test. Further, and, importantly, the determination of the consistency limits using the RE approach relies on them having strength-based definitions, and, while this holds true for w_L(FC) (O’Kelly et al., 2018), it is definitively not the case for the standard w_P and w_S (Barnes and O’Kelly, 2011; Haigh et al., 2013; Nagaraj et al., 2012; O’Kelly, 2013a; O’Kelly et al., 2018; Sivakumar et al., 2016). In other words, as elaborated in the following sections, it should not be unexpected if the experimental values of both p_e(PL) and p_e(SL) for various fine-grained soils were to vary over wide ranges.

Two approaches have been employed in previous RE investigations for the determination of the values of p _e(LL) and p _e(PL) associated with a given experimental set-up. In their investigations, Kayabali and co-workers employed regression and extrapolation of the steady-state p _e–w correlation obtained for a minimum of five specimens of the test soil prepared at various water contents within its liquidity index (I _L) range 0–0·5. As reported by Kayabali et al. (2015b), the values of water content that they investigated were limited to this I _L range because they found that RE testing became very difficult to conduct for I _L > 0·5, presumably because their experimental set-up did not have the required sensitivity for the p _e values mobilised by a lower-shear-strength material. Typical die force–displacement traces for high-plasticity soil are shown in Figure 8(a), from which data pairs of steady-state p _e and corresponding water content values were obtained and plotted in a semi-logarithmic chart (Figure 8(b)). The values of the y-intercept, a (i.e. for w = 0), and gradient −b were then determined for the best-fit correlation line (Equation 7) from regression analysis. Knowing the soil-dependent a and b values, the p _e(LL) and p _e(PL) values were computed by data extrapolation for the particular soil investigated on inputting, in turn, its conventionally measured w _L and w _P values in Equation 7. As discussed later in the paper, the same approach was used in the papers by Kayabali (2012, 2013) for the determination of the steady-state p _e values corresponding to the w _S values measured for various fine-grained soils.

A different experimental approach was adopted in the papers by Verástegui-Flores and Di Emidio (2014, 2015). The value of p _e(LL) was obtained from repeat RE tests performed on the reference kaolin clay material prepared at its w _L(FC) value. The p _e(PL) value was determined for the reference bentonite clay material from extrapolation of its best-fit steady-state p _e–w correlation to its measured thread-rolling w _P value, when analysed in a bi-logarithmic chart (Equation 8).

where a′ and b′ are the y-intercept and gradient, respectively, of the best-fit correlation line in a bi-logarithmic chart.

From the author’s perspective, the value of p_e(LL) would be best determined from RE testing of reference fine-grained soil prepared at its measured w_L(FC) value – that is, since the w_L(FC) value is strength based, with the BS FC LL (BS 1377-2:1990 (BSI, 1990)) condition corresponding to s_u(LL) ≈ 1·7 kPa, the RE approach is valid for w_L(FC) determinations provided that the steady-state p_e/s_u ratio remains constant, having the same value for different fine-grained soils. As explained earlier, this assumption is inherently dependent on the soil billet remaining in a truly undrained condition throughout the RE test.

The aforementioned approach was employed in the investigation by Verástegui-Flores and Di Emidio (2014), who reported a steady-state p_e(LL) value of 23·6 kPa for RE of a pure kaolin clay material prepared at its w_L(FC) value, as determined from three repeat RE tests with R = 16 and using v = 4 mm/min. As expected, good agreement was reported between the derived RE LL water content and the BS-measured w_L(FC) value for various artificial fine-grained soil mixtures composed of different percentages of kaolin and bentonite materials, thereby covering an extremely wide range of plasticity.

Should RE testing prove problematic at the LL water content owing to the soil’s very low undrained shear strength, the RE tests can instead be performed on the reference soil material for various lower water contents that produce values of steady-state p_e within the approximate range 10–50 kPa. The value of p_e(LL) is then determined from the steady-state p_e–w regression correlation as the extrapolated extrusion pressure value corresponding to the measured w_L(FC) value. Once the pertinent value of p_e(LL) has been established for the particular experimental RE set-up, the latter (extrapolation) approach can be employed for subsequent LL determinations on other fine-grained soils.

With a suitably large R value, it is probable that the MPM device used in combination with a rapid water content sensor device (e.g. see the papers by Caldwell et al. (2018) and Dettmann and Bechtold (2018)) could be used to obtain quick estimates of the steady-state p_e–w correlation for a given test soil. Hence, having previously established the MPM p_e(LL) value, the value of w_L(FC) could be estimated from extrapolation of the obtained correlation. This aspect may merit further research.

A significant shortcoming of the extrusion approach (and other strength-based methods, including FC (e.g. see the paper by O’Kelly et al. (2018)) is that it cannot demonstrate the significant change in behaviour, from ductile to brittle, obtained for rolling out of soil threads at water contents each side of the PL. The approach employed in previous RE investigations of soil consistency relied on correlations with the thread-rolling PL method to configure the RE apparatus set-up accordingly. As described earlier in the paper, when considering various fine-grained soils, however, any agreement between measured thread-rolling PL and RE-derived PL values is purely coincidental. In other words, the RE approach cannot consistently provide reliable w_P values for different soils investigated, and, as such, the author recommends that the RE approach not be used for PL determinations.

A new parameter, RE₂₅, termed the extrusion pressure parameter (analogous to the FC PL₂₅ parameter first proposed in the paper by O’Kelly et al. (2018)), is introduced at this stage and is defined as the water content value corresponding to a steady-state p_e value 25 times greater than the p_e(LL) value – that is, corresponding to s_u ≈ 42·5 kPa. Like PL₂₅, the water content corresponding to this new extrusion pressure parameter bears no relationship with the standard PL water content, any equivalence in their values for a given soil being purely coincidental. The extrusion pressure range considered is limited to a 25-fold increase (i.e. encompassing soft to medium-stiffness clays), since fine-grained soils are invariably in a plastic state over the associated water content range (O’Kelly et al., 2018) – that is, widening the range of steady-state p_e values considered in this analysis any further would incur the increasing probability of soil materials occurring in a brittle state near the associated lower-bound water content value.

Using the newly introduced RE₂₅ parameter allows significantly better correlations to be achieved between the steady-state p_e (and hence undrained shear strength) value and a new consistency index (I_RE: Equation 9) parameter than can be obtained with the conventional liquidity index parameter.

with I_RE being defined in logarithmic form since compared with the semi-logarithmic form, the bi-logarithmic steady-state p_e (s_u) against w correlation for a given soil material generally provides a regression coefficient value closer to unity.

Rearranging Equation 9 and assuming s_u(LL) = 1·7 kPa for I_RE = 0, the value of saturated remoulded undrained shear strength for water content values in the range w_L(FC) < w < RE₂₅ can be approximated in the proposed framework as (O’Kelly et al., 2018)

With R = 40·1 and v = 3–5 mm/min, respective mean values for p_e(LL) and p_e(PL) of approximately 20 and 2000 kPa for n = 120 (Kayabali, 2011a) and 100 (Kayabali, 2012), 30 and 2250 kPa for n = 20 (Kayabali and Tufenkci, 2007), 35 and 3000 kPa for n = 31 (Kayabali and Tufenkci, 2010b) and 40 and 3100 kPa for n = 30 (Kayabali and Tufenkci, 2010a) have been determined experimentally for different groups of fine-grained soils. Using the same RE apparatus (R = 40·1) but for a slower die displacement rate of 1 mm/min, Kayabali et al. (2015c) reported respective mean values for p_e(LL) and p_e(PL) of approximately 15 and 2300 kPa for n = 70, with the lower p_e(LL) value deduced for this experimental set-up consistent with the earlier presented theory that requires overall lower values of steady-state p_e for a slower die displacement rate.

Compared with p_e(PL), the mean p_e(LL) range 20–40 kPa deduced in the various studies by Kayabali and co-workers for R = 40·1 and v = 3–5 mm/min was much narrower. This is largely explained by the compatibility of the RE, FC LL and to a less extent the Casagrande LL measurement approaches, in that they are all essentially strength based, whereas the standard PL is not a strength-based phenomenon but a demarcation between ductile (or plastic) and brittle behaviour (Barnes and O’Kelly, 2011; Haigh et al., 2013; O’Kelly, 2013a; O’Kelly et al., 2018; Sivakumar et al., 2016).

Using an RE set-up with R = 16·0 and v = 4 mm/min, Verástegui-Flores and Di Emidio (2014, 2015) reported repeatable p_e(LL) = 23·6 kPa and p_e(PL) = 558·3 kPa (linked to the water contents at the BS FC LL (BS 1377-2:1990 (BSI, 1990)) and standard thread-rolling PL) for various soil samples prepared by mixing kaolin and bentonite clays in different proportions. These p_e(LL) and p_e(PL) values are significantly lower than the mean values of p_e(LL) and p_e(PL) ranging approximately 20–40 and 2000–3100 kPa, respectively, reported in the earlier-mentioned studies by Kayabali and co-workers, who investigated R = 40·1 and v = 3–5 mm/min. Again, this is consistent with the presented theory that requires overall lower values of steady-state p_e for a lower R value.

As expected, good agreement between the derived RE LL and BS-measured w_L(FC) values was reported by Verástegui-Flores and Di Emidio (2014) but not for the Casagrande-cup measured values (i.e. w_L(cup)), with strong divergence occurring between w_L(cup) and BS w_L(FC) for increasing bentonite content in the tested soil materials (Figure 9).

The systematic bias producing w_L(cup) > w_L(FC) for high-LL materials is well documented (e.g. see the papers by Haigh (2012), O’Kelly (2019) and Vardanega et al. (2018)). Specifically, BS w_L(FC) corresponds to an s_u(LL) value of ∼1·7 kPa (O’Kelly et al., 2018), whereas w_L(cup) corresponds to a specific strength (i.e. ratio of s_u to soil density) value of ∼1 m²/s² (Haigh, 2012). The latter equates to a relatively narrow Casagrande s_u(LL) range of ∼1–3 kPa (O’Kelly, 2019), reducing approximately linearly in value with increasing water content at the LL, when plotted in a semi-logarithmic chart (Youssef et al., 1965). As such, strong divergence between w_L(FC) and w_L(cup) values occurs for increasing LL beyond 120% water content (O’Kelly, 2013a; O’Kelly et al., 2018; Škopek and Ter-Stepanian, 1975; Wasti, 1987). Hence, along with differences in die displacement rates employed (v = 1–5 mm/min), the disparity in w_L values obtained for a given soil tested using the Casagrande-cup and FC LL methods may partly explain inconsistencies in the mean p_e(LL) values associated with BS w_L(FC) and w_L(cup) for a given extrusion ratio value.

Kayabali (2012) and Kayabali and Tufenkci (2010a) argued that, because the consistency limits are threshold water contents, they should also correspond to threshold undrained shear strengths and hence fixed steady-state p_e values for a given RE set-up and die displacement rate. As explained earlier, while this statement holds true for w_L(FC), it has been definitively established that the value of s_u(PL) is not unique but can vary over a wide range in considering different fine-grained soils (Haigh et al., 2013; Nagaraj et al., 2012; O’Kelly, 2013a). A major factor explaining the very wide mean p_e(PL) range 2000–3100 kPa reported by Kayabali and co-workers for their RE set-up with R = 40·1 and using v = 3–5 mm/min is, therefore, the expected variation in s_u(PL) values for different fine-grained soils. Other contributing factors include differences in die displacement rates employed and also the fact that different methods were adopted for PL determinations, with Kayabali and Tufenkci (2010a, 2010b) employing the standard thread-rolling method, whereas Kayabali (2012) and Kayabali et al. (2015b, 2015c) employed the PL rolling device described in ASTM D 4318-17e1 (ASTM, 2017). Specifically, PLs obtained using this rolling device generally underestimate the standard thread-rolling PLs (Bobrowski and Griekspoor, 1992), most likely because the paper covering the two rolling plates tends to cause inhomogeneity of the soil thread, the outside becoming drier than its centre, during the rolling-out procedure (O’Kelly et al., 2018). As such, compared with standard thread-rolling, values of s_u(PL) and hence p_e(PL) for a given fine-grained soil would be expected to be higher overall for PLs obtained using the rolling device.

As mentioned earlier, the RE approach has also been proposed for the determination of the values of w_S for fine-grained soils. For instance, with R = 40·1, p_e(SL) values of 12 000 kPa for v = 5 mm/min (n = 100 (Kayabali, 2012)) and 15 000 kPa (n = 120 (Kayabali, 2013)) were reported as providing reasonably good estimates of the values of w_S, as determined using the mercury method in accordance with ASTM D 427-98 (ASTM, 1998). As such, the RE approach may appear as an attractive proposition for SL determination. However, since the SL is the transition water content below which no further reduction in the soil volume occurs (i.e. not a strength-based definition), the RE approach cannot consistently provide reliable w_S values. Consequently, the author does not recommend the RE approach for SL determination.

In this section, the extensive data sets presented in the papers by Kayabali (2011a, 2012, 2013), Kayabali and Ozdemir (2013), Kayabali and Tufenkci (2007, 2010b) and Kayabali et al. (2015b, 2015c), which together report original pairs of p_e(LL) and p_e(PL) values deduced for the consistency limits of 457 different fine-grained soils, are combined and reassessed as part of the present investigation. These data sets relate to an RE apparatus with an R value of 40·1 and employed die displacement rates of 1 (Kayabali and Ozdemir, 2013; Kayabali et al., 2015b, 2015c), 3 (Kayabali, 2011a) and 5 mm/min (Kayabali, 2012; Kayabali and Tufenkci, 2007, 2010b) to investigate a very wide range of soil plasticity; w_L = 26·4–105·0%, w_P = 15·4–46·9% and I_P = 8·3–75·9% (see Table 3 and Figure 10). Using this large data set, potential relationships between reported consistency limits and their steady-state p_e values are investigated, including the effects of changes in soil plasticity categories (levels) and die displacement rate. Further, the p_e(LL) and p_e(PL) values deduced for the consistency limits of another 275 different fine-grained soils using the MPM device approach and reported in the paper by Kayabali et al. (2016) are separately reassessed as part of the present investigation for the same purposes.

As already mentioned, different methods were adopted in these studies for the determination of the consistency limits – that is, the Casagrande-cup or BS FC methods for LL determinations and standard thread-rolling or rolling-device methods for PL determinations. Multiple repeat tests were performed in these studies towards ensuring good reliability of reported LL and PL values. For instance, the w_L(FC) and w_P values reported for each soil sample investigated in the paper by Kayabali and Tufenkci (2010b) are the computed mean of 15 or 22 pairs of LL and PL test results obtained for each soil sample by various experienced operators in eight different soil mechanics laboratories. However, the nuances of the experimental methods themselves and their derived results must be considered when interpreting the various data trends for the combined data sets.

Considering the entire combined RE data sets for v = 1–5 mm/min, p _e(LL) had a mean value of 22·3 kPa (σ = 16·0 kPa) and demonstrated general overall strong trends of reducing exponentially from approximately 90 to 2 kPa for increasing w _L from 35 to 95% (Figure 11(a)) and also for increasing I _P (Figure 11(b)). The effects of adopting different LL determination methods were unexpected and counterintuitive. For a given RE set-up, since w _L(FC) corresponds to a particular reference s _u value, the associated p _e(LL) values should, in theory, remain approximately constant for different LL water content values. However, this was not the case; instead, they exhibited a strong exponentially reducing trend with increasing w _L(FC), as evident for the Kayabali et al. (2015b, 2015c) data sets presented in Figure 11(a). Further, the Kayabali et al. (2015c) data set was obtained for the lowest die displacement rate investigated (1 mm/min) and generally plotted as a lower bound for the entire combined data set, indicating a significant extrusion (strain) rate effect. Also unexpected was the values of p _e(LL) corresponding to the w _L(cup) values for the Kayabali (2012) and Kayabali and Tufenkci (2010b) data sets generally plotting above the Kayabali et al. (2015c) data set – that is, w _L(cup) > w _L(FC) for higher LL water contents implies lower values of p _e(LL) expected for w _L(cup). The relationship between p _e(LL) and w _P was also investigated, but no consistent overall trend was apparent.

Considering the entire combined data sets for v = 1–5 mm/min, p _e(PL) had a mean value of 2240 kPa (σ = 839 kPa), with 900 and 5300 kPa appearing as approximate tentative lower- and upper-bound values (Figure 12). Apart from the Kayabali and Tufenkci (2007, 2010b) data sets, the overall trend suggests that the values of p _e(PL) appear to reduce exponentially from approximately 4000 to 1100 kPa when w _P increases from 17 to 37% (Figure 12(a)). The apparent opposing trend of increasing p _e(PL) values with increasing w _P observed for the Kayabali and Tufenkci (2007, 2010b) data sets may be related to the fact that they were determined for the highest die displacement rate of v = 5 mm/min investigated – that is, overall higher steady-state p _e values could be anticipated on account of the greater strain-rate effect compared with, for instance, the Kayabali et al. (2015c) data set obtained for v = 1 mm/min, which clearly exhibited a decreasing trend with increasing value of w _P.

In relation to the plasticity index, the overall trend considering the entire combined data sets is that the value of p _e(PL) marginally increases with increasing I _P (Figure 12(b)) – that is, apart from the Kayabali et al. (2015c) data set, which again may be explained by the lower die displacement rate employed for that investigation. When the entire combined data sets were plotted in a bi-logarithmic chart, the value of p _e(PL) was found to increase approximately with increasing p _e(LL) according to a power-law relationship (see Figure 12(c)). The relationship between p _e(PL) and w _L was also investigated, although no consistent overall trend was apparent.

The ratio of steady-state extrusion pressures corresponding to the consistency limits (i.e. p _e(PL)/p _e(LL)) had a mean value of 204 but with an extremely wide range, showing a general strong trend of increasing exponentially from approximately 30 to 2000 when w _L increases from 35 to 100% (Figure 13(a)) and also when I _P increases from 10 to 75% (Figure 13(b)). In relation to PL, regression of the entire combined data sets indicated that the p _e(PL)/p _e(LL) ratio reduces approximately exponentially in value with increasing w _P, although the Kayabali and Tufenkci (2007, 2010b) and Kayabali et al. (2015c) data sets tended to suggest that the opposite occurs.

As evident from Figure 14, the same general trends in terms of the steady-state DE pressures for the consistency limits are found from reanalysis of the Kayabali et al. (2016) data set obtained using the MPM device with R = 5·1 – that is, a significant approximately exponential decrease in the value of p _e(LL) occurs for increasing w _L(FC), whereas the value of p _e(PL) does not exhibit this trend with increasing I _P (appearing independent of one another for I _P > ∼50%), such that the value of the p _e(PL)/p _e(LL) ratio exhibits a general strong exponentially increasing trend for increasing I _P. Referring to Figure 14, the ranges of the MPM p _e(LL) and p _e(PL) values are different from those presented for RE testing in Figures 11–13 since the die configurations and values of the R and v parameters were significantly different between these two experimental approaches.

The regression correlations included in Figures 11 and 12 would suggest that the values of the steady-state RE pressures corresponding to the consistency limits are not unique but instead are strongly dependent on the soil plasticity, with the values of p_e(PL) and, in particular, p_e(LL) both reducing exponentially for increasing values of w_P and w_L, respectively (Figures 11(a) and 12(a)). However, there are a number of major anomalies between these suggested behaviours and the actual soil behaviour.

In relation to LL, within the limits of experimental error and following directly from Equation 4, the value of p_e(LL) should remain constant for increasing value of w_L(FC), since the associated s_u(LL) value remains approximately constant – for example, at ∼1·7 kPa in the case of the BS FC LL. However, the p_e(LL)-against-w_L(FC) data set from Kayabali et al. (2015c) presented in Figure 11(a) clearly shows that their deduced values of p_e(LL) substantially decrease for increasing value of w_L(FC), with values of p_e(LL) as low as 0·8–2 kPa for w_L(FC) > 80%. In other words, the trend in the Kayabali et al. (2015c) data set would imply that, for given R and v values, the value of p_e(LL) corresponding to the w_L(FC) is not uniquely related to the soil s_u(LL) value, but it is also significantly dependent on its plasticity characteristics. If this statement were true, the non-uniqueness in the value of p_e(LL) associated with w_L(FC) would invalidate the extrusion approach for LL determination and undrained shear strength measurement. Further, for increasing values of w_L(cup), it is well documented that the associated values of s_u(LL) decrease exponentially over a relatively narrow range of ∼1–3 kPa (O’Kelly, 2019; Youssef et al., 1965). Hence, with p_e(LL) solely dependent on the soil billet’s s_u value for given R and v values according to Equation 4, the value of p_e(LL) would also be expected to exhibit a proportionate exponential decrease with increasing value of w_L(cup). However, as evident in Figure 11(a), an approximate two-orders-of-magnitude decrease in the value of p_e(LL) (reducing from approximately 90 to 2 kPa) occurred for increasing w_L(cup) from 35 to 95%. Clearly, such an enormous reduction in the value of p_e(LL) cannot be reconciled with the more modest reduction occurring in the value of Casagrande s_u(LL) for increasing w_L(cup). Taken together, these observations indicate a major issue with the values of p_e(LL) reported by Kayabali and co-workers, particularly for w_L > ∼60% and I_P > ∼35%, as tentatively deduced from the general pattern of the data spread evident in Figures 11 and 13.

In relation to PL, consistent with the general overall trend expected for s_u(PL) with increasing soil plasticity, one would anticipate that the values of p_e(PL) and the p_e(PL)/p_e(LL) ratio would decrease overall with increasing value of I_P. However, the data trends in Figures 12(b) and 13(b) seem to suggest that this was not the case, with the values of p_e(PL) and particularly the p_e(PL)/p_e(LL) ratio increasing exponentially with increasing I_P for the various fine-grained soils investigated. For instance, the value of the p_e(PL)/p_e(LL) ratio increased from approximately 30 to 2000 for increasing I_P from 10 to 75%. Further, with Equation 4 implying that the steady-state p_e is analogous to s_u, this approximately 67-fold increase in the value of the p_e(PL)/p_e(LL) ratio (and hence the s_u(PL)/s_u(LL) ratio) for increasing I_P of fine-grained soil is simply not credible – see the discussion on typical strength gain over plastic range presented in the paper by O’Kelly (2013a). Based on all of the aforementioned observations, it is evident that other influencing factors are at play, with three significant ones identified by the author investigated in the following sections – that is, it is the author’s contention that inconsistencies in the reported p_e(LL) and p_e(PL) values and their inferred correlations with the consistency limits largely occur for the following reasons.

• Some consolidation of the soil billet may occur under the compressive force applied by the slowly advancing die, particularly for stiffer soils that require very high extrusion pressures and particularly those with higher hydraulic conductivity values. In these instances, the value of shear strength mobilised in the billet’s yielding zone would correspond to a partially drained condition (i.e. greater than the assumed undrained shear strength value).

• For the water content range of w_P ≤ w < w_T (where w_T is the transition water content below which the soil has measurable toughness), the property governing extrusion resistance capacity is the soil remoulding toughness (T), rather than simply its undrained shear strength.

• When considering a wide range of water contents, the general piecewise-linear nature of the steady-state p_e–w relationship impacts on the reliability of extrusion pressure values deduced from regression analyses and data extrapolation.

Relative to the consistency of the soil billet, the extrusion pressures required for plastic soils are very high, with their values significantly increasing for soil with greater shear strength and for higher experimental R and v values. For instance, p _e(LL) and p _e(PL) values ranging approximately 20–40 and 2000–3100 kPa, respectively, were reported for the RE apparatus with an R value of 40·1 employed in the investigations of various fine-grained soils by Kayabali (2011a, 2012) and Kayabali and Tufenkci (2007, 2010a, 2010b). For the slow die displacement rates of typically 1–5 mm/min employed, it is entirely plausible that some localised consolidation of the soil billet may occur, caused by drainage of the highly pressurised pore water from the die end of the billet, by way of the extruding soil worm which is under no confinement pressure. More pronounced localised consolidation would be expected for stiffer soils that require higher extrusion pressures, those with greater hydraulic conductivity and also for lower die displacement rates, such that the drainage time increases. In this event, the shear strength mobilised in the billet’s yielding zone would correspond to a partially drained condition, representing a significant departure from the extrusion theory that was adopted from metallurgy. If substantiated, this anomaly could adversely impact or possibly even invalidate the RE approach for the determination of the true undrained shear strength for fine-grained soil and by association its consistency-limit values. A full review by the author of the pertinent literature produced only two experimental evidences, which are elaborated below, concerning this particular aspect of RE soil testing.

The most compelling evidence supporting the billet consolidation hypothesis occurs in connection with RE testing of pure kaolin clay (w _L = 53·2%; w _P = 31·0%) prepared at water contents slightly higher than its PL, as reported in the papers by Verástegui-Flores and Di Emidio (2014, 2015). Rather than reaching a steady-state p _e value, the mobilised extrusion pressure continuously increased with increasing die displacement, from which Verástegui-Flores and Di Emidio (2014, 2015) concluded that not only did the kaolin clay billets undergo extrusion but they were also undergoing consolidation within the RE chamber. This assessment was confirmed by the extruded soil worm having an increased value of water content compared with the remaining compressed billet. In other words, consolidation of the soil billet (with drainage by way of the die orifice) under the very high applied extrusion pressures progressively increased the shear strength value of the kaolin clay billet within the extrusion chamber, such that a steady-state p _e value could not be achieved for large die displacements. These researchers reported no similar issues for RE testing of the same kaolin clay at its w _L value, or for natural Boom Clay containing illite (w _L = 51·0%; w _P = 32·4%) and pure bentonite clay (w _L = 374·7%; w _P = 62·9%) prepared at water contents slightly higher than their w _P values. As such, it would appear that fine-grained soil with higher shear strength can be susceptible to billet consolidation and that the other major determining factors are clay mineralogy and gradation. The dependence of shear resistance and compressibility on clay mineralogy is well documented (O’Kelly, 2014b) – that is, among kaolinite, illite and smectite clay minerals, kaolinite has a smaller specific surface area and shows higher hydraulic conductivity, so it is more prone to releasing water when subjected to confinement stress.

On the other hand, based on <1% difference between the measured water contents of the soil billet and worm parts for 12 specimens of a single high-plasticity clay soil (w _L = 57·2%; w _P = 23·0%) reverse-extruded at various water contents, Kayabali and Tufenkci (2010b) concluded that the soil material in the billet’s yielding zone remained in a fully undrained condition. However, a water volume balance requires that pore water draining out of the extrusion chamber by way of the die orifice brings about recovery for the extruded soil worm of any reduction in its water content value that occurred earlier within the yielding zone to (approximately) the billet’s initial water content value. As such, this experimental evidence presented by Kayabali and Tufenkci (2010b) does not, in itself, conclusively support their assertion of a truly undrained billet condition.

Compared with the high-plasticity clay soils investigated by Kayabali and Tufenkci (2010b) and Verástegui-Flores and Di Emidio (2014, 2015), any billet consolidation effect would be significantly more pronounced for fine-grained soils with substantially higher hydraulic conductivity (e.g. silty fine sand material), which is dependent on the soil gradation. Further, in terms of the experimental set-up, a higher degree of localised consolidation and hence greater gain in shear resistance capacity of these soils could be expected for higher applied extrusion pressures and slower die displacement rates. In other words, under these circumstances, the soil material within the billet’s yielding zone could be partially drained and would therefore mobilise a higher shear resistance capacity compared with its initial s _u value. Further, different degrees of localised billet consolidation for various fine-grained soils with the same starting s _u value would produce different levels of strength increase, thereby mobilising different values of the steady-state p _e for these soils. Consequently, the values of the steady-state p _e/s _u ratio for these soils would be different.

It is the author’s opinion that billet consolidation, combined with data analysis and extrapolation issues discussed in the following sections, largely explains the very wide p _e(LL) and p _e(PL) value ranges reported for various fine-grained soils investigated using the RE apparatus with set R and v values – that is, consistent with the data trends evident in Figures 11, 12(a) and 14(a), fine-grained soils with lower plasticity, which typically have higher hydraulic conductivity, generally mobilise significantly greater values of p _e(LL) and p _e(PL). As such, the extrusion approach may not be a reliable method for undrained shear strength determinations on low-plasticity soils, producing unconservative (high) shear strength values if some localised billet consolidation occurs during the testing. Counteraction measures include reducing the apparatus R value, thereby reducing the pressure values required to cause extrusion, and (or) increasing the die displacement rate, thereby reducing the time period over which billet consolidation could occur. Other potential shortcomings of the RE experimental set-up include required maintenance of the extrusion chamber and the risk of the die jamming, particularly considering the high extrusion pressures required as the PL water content is approached.

Compared with the predefined shear surfaces for direct shear and shear vane strength-measurement methods and the shearing plane or general plastic deformation of the test specimen for triaxial compression (e.g. see the papers by O’Kelly (2013b, 2013c, 2015)), the extrusion process is distinctly different in that it involves plastic flow of the soil element through the die orifice. At the PL, clayey silt and clayey silty sand materials have slight toughness, whereas low- to medium-plasticity soils have medium toughness and high-plasticity clay soils have high toughness, toughness here referring to the effort required in (re)moulding. As such, soil with greater toughness would be expected to provide higher resistance to extrusion – that is, greater energy would be required to cause extrusion, noting that the extrusion pressure against die displacement plot can be regarded as the work or power diagram of extrusion.

Typical bi-linear T–w relationships determined for clay soils using a PL thread-rolling device developed by Barnes (2013a, 2013b) are presented in Figure 15. In this figure, the plastic range is subdivided based on the deduced toughness relationship into an adhesive (sticky) plastic region and workable soft- and stiff-plastic regions, as delineated by the reducing water content values of w _L, w _T, the stiffness transition (w _ST) and w _P. For water contents within the adhesive-plastic region (w _T ≤ w < w _L), fine-grained soil exhibits zero toughness. In this instance, for given apparatus R and v values and a truly undrained condition, the steady-state extrusion pressure value is entirely controlled by the soil’s undrained shear strength. Based on experimental data for rolling out of soil threads presented by Barnes (2013b), measurable toughness for fine-grained soil is typically first evident at I _L ≈ 0·4.

For the water content range of w _P ≤ w < w _T, however, it is the author’s contention that the steady-state extrusion pressure value relates to the soil toughness. Compared with low-plasticity fine-grained soil, greater toughness for higher-plasticity soil would be expected to provide greater resistance capacity to extrusion for water contents in the workable soft-plastic and, in particular, stiff-plastic regions. This interpretation is consistent with the earlier observation in connection with Figure 12(b) of overall increasing value of p _e(PL) with increasing I _P. Further, as discussed in the next section, whether plotted in normal, semi- or double-logarithmic charts, the s _u–w (and hence steady-state p _e–w) correlation can generally be expected to be piecewise linear. This can have major implications for the reliability of regression analyses and extrapolation of steady-state p _e–w data.

An approach commonly employed for quantifying relative remoulding toughness is to compute the area enclosed by the experimental shear stress–shear strain curve. For the RE test, under the action of the steady-state extrusion pressure, the die displacement rate and hence extrusion velocity of the soil worm remain constant, meaning that the associated work rate (=p _e A _o v) is also constant. As such, for the workable soft- and stiff-plastic regions, the steady-state p _e value mobilised for a given apparatus die configuration and R and v values provides an assessment (measure) of the test soil’s remoulding toughness. The maximum remoulding toughness, T _max, occurs for the PL water content. In assessing the relative value of T _max using the RE approach, the extrusion testing should be limited to soil water contents within the workable stiff-plastic region, given the characteristic approximately bi-linear nature of the T (steady-state p _e)–w correlation (see Figure 15(a)). Having determined the value of w _P from standard thread-rolling of the extruded 6 mm dia. soil worm (RE apparatuses with 6 mm dia. orifices are the norm), the associated value of p _e(PL) can then be deduced from regression of the obtained experimental steady-state p _e–w correlation, thereby providing a relative measure of the test soil’s T _max value.

The piecewise-linear or curved nature of the steady-state p _e–w correlation covering the plastic range for a given fine-grained soil impacts to varying degrees on the results obtained from regression analyses and data extrapolation, such that deduced values of p _e(PL) and, in particular, p _e(LL) may not be reliable. For example, the tabulated values of p _e(LL) and p _e(PL) reported in the studies by Kayabali and co-workers were determined from data extrapolation using Equation 7, inputting the values of the a and b fitting parameters obtained from regression analysis of typically five p _e and w data pairs confined to the I _L range 0–0·5 – that is, in this instance, significant extrapolation of the experimental data was required to determine the value of p _e(LL) corresponding to I _L = 1·0.

The principal reasons for the piecewise-linear or curved nature of the steady-state p _e–w correlation are the following.

• As described earlier, depending on the consistency of the soil billet, different mechanical properties can control its resistance capacity – that is, remoulding toughness for w _P ≤ w < w _T and shear strength for w ≥ w _T.

• When presented in semi- or bi-logarithmic charts, compared with intermediate- and high-plasticity fine-grained soils which exhibit approximately linear relationships for w _P ≤ w > w _L (Figure 16(a)), some extremely-high-plasticity soils were found to exhibit a piecewise-linear behaviour at near their w _L values. The latter behaviour is most evident for the FC s _u–w power correlations presented for two pure bentonite clays in Figure 16(b)). Compared with intermediate- and high-plasticity soils, this distinctly different behaviour is rationalised as the LL of kaolinitic soils mainly due to their fabric, as governed by interparticle attractive forces (Sridharan et al., 1988), whereas that of montmorillonitic soils is largely due to the diffuse double-layer-held water (Sridharan et al., 1986).

• When investigating a wide range of water contents, experimental steady-state p _e (s _u)–w data often exhibit some curvature when plotted in a semi-logarithmic chart. For this reason, bi-logarithmic presentations of the data are usually preferred (Butterfield, 1979; Koumoto and Houlsby, 2001; O’Kelly, 2014a, 2016; Sharma and Bora, 2003). Experimental supporting evidence in terms of RE testing is provided in the papers by Verástegui-Flores and Di Emidio (2014, 2015), with the bi-logarithmic steady-state p _e–w plot of their data covering the I _L range 0·05–0·44 reported as producing a very well-defined linear correlation, whereas the semi-logarithmic presentation of the same data was non-linear.

In the studies by Kayabali and co-workers, their experimental steady-state p _e–w correlations presented in a semi-logarithmic chart were obtained for 0 < I _L < 0·5 – that is, the various fine-grained soils tested had measurable toughness. In view of the issues highlighted earlier, it is evident that the significant extrapolation of the steady-state p _e–w correlation from I _L < 0·5 to an I _L value of unity in order to establish the value of p _e(LL) will likely result in an underestimation of its true value. As explained earlier using the soil behaviours shown in Figure 16(a) as examples, the underestimation may be substantial for extremely-high-plasticity soils investigated (w _L > 90%) and possibly also for some high-plasticity soils (w _L = 70–90%). This interpretation is consistent with the extremely low values of p _e(LL) reported for these soil plasticity categories – that is, with R = 40·1 and using v = 1–5 mm/min, Kayabali and co-workers deduced p _e(LL) values ranging 0·8–10 and 0·8–30 kPa for extremely high- and high-plasticity soils, respectively (refer to Figure 11(a)) based on the described data extrapolation approach. Although these issues also affect the deduced values of p _e(PL), the resulting errors are generally likely to be smaller since the required data extrapolation is not as great. However, consideration of the impact of the reported bi-linear T–w relationship on the reliability of the described regression and extrapolation approaches merits further investigation.

Computed values of p _e(LL) and p _e(PL) can also be particularly sensitive to reduced precision (rounding) of the deduced b or b′ fitting parameter values in Equations 7 and 8. For instance, Figure 17 compares the values of p _e(LL) and p _e(PL) reported for 31 fine-grained soils in the paper by Kayabali and Tufenkci (2010b) with those computed by the author as part of the present investigation from tabulated p _e and w data pairs reported in their paper. The author performed the same computation as used in the original study – that is, a and b values were deduced for each soil from regression analysis of its reported steady-state p _e–w data set, from which the values of p _e(LL) and p _e(PL) were computed using Equation 7 for the reported values of w _L and w _P, respectively. The only difference between these analyses was that the author’s deduced b exponent values had an enhanced precision of four decimal places. Not surprisingly, as evident from Figure 17, the deduced values of p _e(LL) and p _e(PL) can be significantly affected by rounding errors. As evident from the regression lines presented in Figure 17 for the reported and author-determined data values, this can have knock-on effects for inferred steady-state p _e–w relationships. For instance, the author-derived p _e(LL)–w _L(cup) correlation presented in Figure 17(a) is consistent with the trend of reducing value of Casagrande s _u(LL) for increasing LL water content (O’Kelly, 2019; Youssef et al., 1965), whereas the Kayabali and Tufenkci (2010b) regression correlation is ambiguous regarding this accepted trend.

Finally, as mentioned earlier, the RE approach was proposed by Kayabali (2012, 2013) for the determination of the SL water content, with the value of w _S supposedly associated with a particular steady-state p _e value (i.e. p _e(SL)). As evident from Figure 18 and with s _u–I _L as a proxy for the steady-state p _e–w, the pattern of relationships undergoes a noticeable transition at the PL (I _L = 0). As such, the proposed back-extrapolation of the logarithmic steady-state p _e–w correlation determined for the plastic range to water contents in the semi-solid region does not seem a valid approach for the determination of the w _S value.

The steady-state p_e value for RE of fine-grained soil is strongly dependent on its water content and the logarithm of extrusion rate, r_e (= Rv). The following are concluded from reassessment of the combined RE data sets presented in the literature for many hundreds of remoulded fine-grained soils covering an extremely wide plasticity range.

• The value of the steady-state p_e/s_u ratio and hence K_RE can vary widely between soils, appearing to decrease overall with increasing plasticity. For instance, a standard deviation of 3·2 occurred for the mean p_e(PL)/s_u(PL) = 13·9 deduced for 60 remoulded fine-grained soils investigated in the paper by Kayabali and Ozdemir (2013).

• The values of both p_e(LL) and p_e(PL) exhibited substantial exponentially reducing trends with increasing values of w_L and w_P, respectively. This behaviour was also strongly apparent for the value of the steady-state p_e(LL) corresponding to the strength-based BS w_L(FC) deduced for s_u(LL) ≈ 1·7 kPa.

• The values of p_e(PL) and the p_e(PL)/p_e(LL) ratio increased overall with increasing plasticity, although one would have anticipated the contrary, given the general expected trend of s_u(PL) reducing in value with increasing plasticity.

Based on these points, it is concluded that, for given apparatus R and v values, the steady-state p_e/s_u ratio value does not remain constant among different remoulded fine-grained soils. Depending on mineralogy and gradation, some localised billet consolidation may occur for stiff low-plasticity soils under the very high extrusion pressures required and low strain-rate range of 0·2–12·8 min⁻¹ typically employed, particularly for those soils with higher hydraulic conductivity values. Evidence of this effect was presented in the papers by Verástegui-Flores and Di Emidio (2014, 2015) for pure kaolin clay near its PL water content. With drainage to atmosphere allowed by way of the die orifice, some consolidation of soil material within the billet’s highly stressed shear zone would have the overall effect of increasing the value of the steady-state p_e/s_u ratio.

Further, compared with standard strength-measurement approaches (e.g. vane shear, direct shear and triaxial compression), extrusion is distinctly different in that it involves plastic flow of the soil, such that remoulding toughness (rather than simply undrained shear strength) is the mechanical property governing resistance capacity for water contents in the range of w_P ≤ w < w_T. This interpretation is consistent with the overall observed trend of the experimental p_e(PL) value increasing with increasing soil plasticity, given that higher-plasticity soils have greater toughness at the PL.

Caution is urged in interpolating values of extrusion pressure and (/or) water content from regression analysis of steady-state p_e–w correlations and particularly for their extrapolation over a wider water content range on account of the probable piecewise-linear nature of p_e–w relationships when considering the full plastic range and beyond.

Since the standard thread-rolling PL relates to the ductile–brittle transition, as essentially a strength-based approach, the extrusion technique cannot be reliably employed for its determination. Further, the standard SL relates to the water content below which no further reduction in soil volume occurs, which again cannot be reliably determined using the strength-based extrusion approach. The standard FC and Casagrande LL are strength based, but, as mentioned already, the various data sets presented by Kayabali and co-workers indicate that the value of p_e(LL) generally reduces significantly with increasing soil plasticity. Therefore, in conclusion, the substantial experimental evidence is such that the present extrusion approach is not recommended for undrained shear strength measurement or for Atterberg-limit determinations.