Relationship of precompression stress with elasticity and plasticity indexes from uniaxial cyclic loading test

Soil & Tillage Research 180 (2018) 29–37

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Relationship of precompression stress with elasticity and plasticity indexes from uniaxial cyclic loading test

T

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Paulo Ivonir Gubiani , Patricia Pértile, José Miguel Reichert Soils Department, Federal University of Santa Maria (UFSM), Av. Roraima 1000, 97105-900 Santa Maria, RS, Brazil

A R T I C L E I N F O

A B S T R A C T

Keywords: Soil compressibility Soil elasticity Soil bearing capacity Compression curve

The precompression stress (σp) has been considered as the soil load bearing capacity, because σp theoretically separates predominantly elastic (loads lower than σp) from predominantly plastic (loads greater than σp) deformations. However, the relationship of σp and soil elasticity and plasticity indexes has not been widely investigated. In this study we determined σp and elasticity and plasticity indexes in cyclic loading tests (17 cycles per load) with a series of increasing σ (12.5, 25, 50, 100, 200, 400 and 800 kPa) of undisturbed samples of horizons A and B of eight soils (48 samples) from southern Brazil. The σp was also determined from static load test. The objective was to verify if σp match the load in the transition from predominantly elastic to predominantly plastic deformation. Two elasticity indexes were calculated from the logσ-ε diagram, which relates the void ratio (ε) to the log of the loads (σ): SI, as the slope of rebound line of the logσ-ε diagram; and SI’, being the ratio of recovered ε with soil decompression in relation to the change in ε by compression. Two plasticity indexes were also calculated: CI, as the slope of virgin compression line of log σ-ε diagram; and CI’, which is 1SI’. All these four indexes were calculated at 25, 50, 100, 200, 400 and 800 kPa loads. Both SI and SI' decreased during the repetitions of a given load, but neither SI nor SI’ have a same behavior throughout the entire range of loads. In some samples there was increase in SI with increased loading, but the opposite occurred in other samples. SI' also behaved like SI, but not at exactly for the same samples. Surprisingly, both SI and CI augmented with increased loading, suggesting that the soil becomes more elastic and more plastic simultaneously, which seems physically incoherent. This physical inconsistency does not result from SI’ and CI’ interpretation, because SI’ + CI’ = 1 is perfectly complementary. Therefore, SI’ and CI’ are more coherent indexes of soil elasticity and plasticity. Another critical finding of this study is that σp does not match the load where SI and CI or SI’ and CI’ change from predominantly elastic to predominantly plastic deformation. Furthermore, SI and CI or SI’ and CI’ do not support the assumption that σp define a transitions from elastic to plastic behavior.

1. Introduction

of soil compression of great environmental relevance. In the field of soil mechanics, static uniaxial compression is a technique that has been widely used to evaluate how soil resists the application of loads and recovers from deformations, as well as to understand the modifications caused in structural properties such as bulk density (ρ) and void ratio (ε). The relationship of these structural properties with applied loads (σ) is used to obtain the uniaxial compression curve (CC), from which soil mechanical parameters of resistance and recovery are derived. The CC of a soil sample can be described by the relationship of ε or ρ with logσ. Mechanical parameters are derived from the CC ε-logσ or ρlogσ, assuming it can be characterized by two approximately linear segments that can be described by regression lines. Studies by Dias and Pierce (1995), Gubiani et al. (2017) and Schäffer et al. (2010) are examples that illustrate the use of this approach. However, the terms

Soil compression by the application of an external load is a process that promotes the increase of soil density, reduction of porous space, exclusion of air and/or water from the pores, deformation of aggregates and rearrangement of particles (Gupta et al., 1989; Horn and Lebert, 1994; Reichert et al., 2016b). The structural changes caused by compression impair soil functional properties (Reichert et al., 2017, 2016b) related to air transport (air conductivity) (Braga et al., 2015; Dörner et al., 2012; Mentges et al., 2016; Mordhorst et al., 2012; Peth et al., 2010; Peth and Horn, 2006) and water transport (hydraulic conductivity) (Dörner et al., 2012; Reichert et al., 2016b), in addition to increasing the mechanical resistance to plant root growth (Reichert et al., 2016a, 2009a, 2009b; Reinert et al., 2008). All these changes affect the flow of matter and energy in the soil, which makes the study

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Corresponding author. E-mail address: [email protected] (P.I. Gubiani).

https://doi.org/10.1016/j.still.2018.02.004 Received 15 September 2017; Received in revised form 18 January 2018; Accepted 6 February 2018 0167-1987/ © 2018 Elsevier B.V. All rights reserved.

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recompression line (RCL) for the first segment and virgin compression line (VCL) for the second is a simplification of the mathematical linearization of often visibly non-linear segments, when using a continuous function to describe the experimental observations of the CC (e.g., Baumgartl and Köck (2004), Fritton (2001) and Gregory et al. (2006)). The RCL and VCL are separated by a visibly curved region, within which precompression stress (σp) is estimated by different graphical and mathematical techniques (e.g., Baumgartl and Köck (2004), Casagrande (1936), Dias and Pierce (1995), Fritton (2001), Gregory et al. (2006) and Gubiani et al. (2017)). The main assumption of Casagrande (1936) when proposing a graphical technique for calculating σp is that the RCL is a region of predominantly elastic deformations (recoverable deformations) and the VCL is a region of predominantly plastic deformations (unrecoverable deformations), and σp is the value of σ in the elastic-plastic transition of the soil. This assumption is also used for the proposition of several mathematical techniques subsequent to Casagrande (1936), such as those by Baumgartl and Köck (2004), Dias and Pierce (1995), Fritton (2001) and Gregory et al. (2006), as well as accepted by several studies, such as those by Holtz and Kovacs (1981) and Peth and Horn (2006). Another segment that is also mathematically described by regression lines is the rebound line (RBL), which can be generated in the CC by the gradual removal of loads from any point in the VCL segment (e.g., Braida et al. (2008), Gregory et al. (2006), Schäffer et al. (2010) and Reichert et al. (2018)). Similar to the RCL, the RBL is also considered a segment of elastic deformations. Therefore, the slope of the RBL is widely used as a soil elasticity index (SI), as opposed to the slope of the VCL that is widely used as a soil plasticity index also called compression index (CI) (Ajayi et al., 2009; Dias and Pierce, 1995; Keller et al., 2011). The rebound curve has a flatter slope than the virgin curve, and the recompression curve (the reloading curve) forms a hysteresis loop with the rebound curve; when the load is reapplied, the soil will re-consolidate along the recompression curve (Reichert et al., 2018). Another strategy to estimate elasticity and plasticity, dispensing regression lines, is using soil deformation and recovery. Considering points A, B, and C, respectively, at the beginning of loading, end of loading and end of unloading, where we calculate εA, εB and εC, then Δεt = εA − εB, Δεr = εC − εB and Δεp = εA − εC represent, respectively, total deformation, recoverable deformation, and unrecoverable deformation. With this information, Braida et al. (2008) proposed a percentage-based elasticity index (SI’% = 100Δεr/Δεt), obtained in a uniaxial compression test but with unloading-reloading. For the calculation of Δt, the authors defined εA as the ε of the beginning of the first loading. More recently and in cyclic compressibility tests, Peth et al. (2010) and Schäffer et al. (2010) calculated an elasticity index defined by SI’ = Δεr/Δεt. In addition to not presenting a percentagebased SI' as proposed by Braida et al. (2008), these authors defined εA as the ε of the beginning of each particular loading-unloading cycle. However, none of these authors calculated a corresponding plasticity index, which can be defined by CI’ = Δεp/Δεt, which is complementary to SI' in total deformation (CI’ + SI’ = 1). Cyclic compressibility was introduced to the study of mechanical soil changes when a given load is applied and removed repeatedly (Holthusen et al., 2018; Krümmelbein et al., 2008; Peth et al., 2010; Peth and Horn, 2006), which is known as loading-unloading cycles. With cyclic compressibility (also named as cyclic loading test), several loading and unloading cycles can be repeated with the same load (Peth et al. (2010) applied 100 cycles of 80 kPa) or for several loads (Mosaddeghi et al. (2007) applied ten cycles of 50, 100, 200, 400 and 800 kPa and Schäffer et al. (2010) applied one cycle of 50, 100, 200, 400 and 800 kPa) on a single soil sample. For each cycle of a given load, an RBL can be adjusted in the unloading segment, which contains its particular SI. In the segment that connects the ε of the last load of a set of cycles with the ε of the first load of the next set of cycles, a local VCL can be adjusted, providing its corresponding CI. The CI was calculated

by Schäffer et al. (2010) for loads of 50, 100, 200, 400 and 800 kPa applied in a single cycle. These authors also calculated a SI' for each of the five loads. In contrast, Peth et al. (2010) calculated 100 SI' values with 100 cycles of the same load (80 kPa). The change in SI or SI' and CI or CI' resulting from cycles of the load sequence in a CC allows us to analyze changes in elasticity and plasticity with the deformation of the sample. With the 100 SI' values calculated in the 100 cycles of the 80 kPa, Peth et al. (2010) found that elasticity increases with the number of cycles, tending to full elastic before completing the 100 cycles (see Fig. 13 in Peth et al. (2010)). The change in elasticity across the full load range of the CC was not evaluated by Peth et al. (2010), but their findings indicate that soil elasticity can not be characterized by the slope of a single RBL obtained from a static compressibility test. As for the change in plasticity along the full load range of the CC, Schäffer et al. (2010) found that there may be an increase or decrease in CI, depending on the combination of the applied load (50, 100, 200, 400 and 800 kPa), the hydric condition of the sample (−1, −6 and −30 kPa of matric potential), and the occurrence of contraction and expansion (see Fig. 5 in Schäffer et al. (2010)). The σp can also be obtained in a cyclic compressibility test (Mosaddeghi et al., 2007; Peth et al., 2010; Schäffer et al., 2010). Thus, a soil sample can be characterized by a value of σp and a set of values of SI and CI, or SI' and CI'. The analysis of SI and CI (or SI' and CI') before and after σp makes it possible to verify if σp is a value of σ that clearly separates regions of elastic and plastic deformation. This is an underinvestigated research topic, but according to Keller et al. (2011) it would add relevant and urgent information to better understand what the σp determined in the laboratory means. To investigate the important aspects of this issue, this study aimed to evaluate changes in soil elasticity and plasticity indexes in cyclic loading test with a series of increasing loads and to verify if σp is a value of σ that clearly separates regions of elastic and plastic deformation. 2. Materials and methods 2.1. Soil sampling The study was conducted with samples of sixteen horizons (A and B horizons) of eight different soils from southern Brazil (region with humid subtropical climate), collected in previously described profiles. Soils were selected from the orders Latossolo/Oxisol, Argissolo/Ultisol, Planossolo/Alfisol and Vertissolo/Vertisol, as described in Table 1. Disturbed soil samples were collected in the middle of surface (A) and subsurface (B) horizons (the soils horizons will be referred simplified as A and B), air-dried and sieved with a 2-mm mesh. With this soil fraction, analyses were performed to determine textural class and total carbon content (TC) (Table 1). In the same position of the horizons, undisturbed soil samples were collected in metal rings of approximately 126 cm3 (9.8 cm in diameter and 3.0 cm in height) (6 replications per horizon, totaling 96 samples): three samples from each horizon were used for the static compression test (total of 48 soil samples), and the other three samples were used for the cyclic loading test and the determination of ρ (Table 1) (total of 48 soil samples). 2.2. Soil compression analysis For the static compression and the cyclic loading tests, the undisturbed samples were capillary-saturated for 48 h and remained on a sand column under water tension (wt) of 10 kPa for four days (Reinert and Reichert, 2006). Both tests were performed with a compression test device, with operation and data processing automated by Multistep software (Wazau, Germany). The loads were applied with a pneumatic piston (at a speed of 1.0 mm min−1) under confined lateral compression (sample contained within a metal ring), and the free water drainage was ensured by porous plates below and above the soil sample, as described by Krümmelbein et al. (2008). During the measuring process, a 30

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Table 1 Soils used in the study classified by the Brazilian System of Soil Classification - SiBCS (Santos et al., 2013) and Soil Taxonomy (Soil Survey Staff, 2014), symbology, horizons, depth, textural class, total carbon content (TC), and mean values ± standard deviation of bulk density (ρ), and initial degree of saturation before the cyclic loading test (Dsi) (see Pértile et al. (2016) for more information about these soils). SiBCSa

Soil Taxonomy

Symbology

Horizon

Depth cm

Textural classb

TCc g kg−1

ρd Mg m−3

Latossolo Vermelho Distrófico típico

Rhodic Hapludox

LVd1

Latossolo Vermelho Distrófico típico

Rhodic Kandiudox

LVd2

Latossolo Bruno Distroférrico rúbrico

Typic Hapludox

LBdf1

Latossolo Bruno Distroférrico típico

Typic Hapludox

LBdf2

Argissolo Vermelho Distrófico típico

Arenic Rhodic Paleudult

PVd1

Argissolo Vermelho Distrófico arênico

Arenic Hapludult

PVd2

Planossolo Háplico Eutrófico êndico

Arenic Albaqualf

SXe

Vertissolo Ebânico Órtico típico

Oxyaquic Hapluderts

VEo

Ap Bw1 A1 Bw1 Ap Bw1 A1 Bw1 A Bt A1 Bt1 A1 Btg A Biv

0–35 105–152 0–26 100–138 0–30 95–150 0–26 80–100 0–40 110–155 0–20 77–98 0–30 70–120 0–30 30–115

Sandy loam Sandy clay loam Clay Clay Clay Clay Clay Clay Sandy loam Clay Loamy sand Sandy clay Sandy loam Clay loam Silty clay Clay

11.0 4.2 23.2 6.4 34.4 5.0 33.6 11.4 5.7 5.4 6.2 5.0 9.5 4.5 44.1 32.1

1.31 1.48 0.81 1.21 0.93 1.04 0.98 1.12 1.62 1.55 1.68 1.59 1.44 1.41 0.92 1.03

a b c d

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

Dsi cm3 cm−3 0.04 0.05 0.04 0.05 0.04 0.02 0.02 0.03 0.02 0.04 0.03 0.04 0.07 0.04 0.02 0.02

0.43 0.56 0.44 0.80 0.59 0.82 0.72 0.85 0.48 0.75 0.56 0.89 0.54 0.83 0.92 0.96

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.05 0.02 0.02 0.04 0.03 0.01 0.04 0.02 0.03 0.05 0.13 0.06 0.06 0.07 0.02 0.02

Collected in profiles exposed on the edge of highways, except for VEo and PVd2 (under native pasture), LBdf2 (under implanted pasture), and LVd2 (under native forest). From the sand, silt and clay contents of the soil, determined according to Suzuki et al. (2015) with sodium hydroxide (or sodium hexametaphosphate -VEo) dispersant solution. Determination in autoanalyzer by dry combustion (with macerated soil samples). Determination according Blake and Hartge et al. (1986) with the samples used in the cyclic loading test.

potentiometric displacement sensor detected the vertical displacement of the soil sample. At the end of tests, all samples were oven dried at 105 °C to determine ρ and ε. 2.2.1. Static compression test Loads of 12.5, 25, 50, 100, 200, 400 and 800 kPa were successively applied, and each load was maintained for 10 min. The σp was calculated according to Casagrande (1936), mathematically operationalized with the equation of van Genuchten (1980), as proposed by Baumgartl and Köck (2004) (Eq. 1).

ε = εf + (ε0 −εf )[1 + (ασ )n]−m

(1)

−3

Where ε0 (m m ) is the void rate of the sample without load application; εf (m3 m−3) is the final void rate of the sample, for which Baumgartl and Köck (2004) suggested the value of 0.27, but in this study the εf was not fixed and was estimated as an adjustment parameter; α (kPa−1); n and m are parameters of adjustment, imposing the restriction m = 1 − 1/n. The Soil Compression Curve (SCC) Excel® Add-in developed by Gubiani et al. (2017) was used for fitting (Eq. 1) and calculating σp. In the fitting routine, SCC also converts the ε-σ relationship to the ε-log10σ relationship, which is the classic CC shape (ε-log10σ CC). From the εlog10σ CC, precompression stress (σp) was determined as the σ of the intersection point of the VCL (defined as the tangent line at the inflexion point) and the bisector between the horizontal and tangent lines through the point of maximum curvature. The point of maximum curvature was defined as the σ corresponding to the root of the third derivate of the ε-log10σ CC, numerically determined using the bisection root-finding method. 3

Fig. 1. Compression-test device configuration of cyclic loading applied in relation to time - 17 cycles per load, each cycle consisting of 36 s of loading and 36 s of unloading more the time for adjustment the loads by the device that worked at speed of 1.0 mm min−1.

interpreting the results. The Fig. 2A illustrates the results of a cyclic loading test. Fig. 2B shows the set of RBL segments between 10 and 200 kPa. In this set of segments, the points (ε, σ) of a RBL segment were highlighted (black circles) to illustrate how the regression analysis was used to obtain the SI coefficient of the lines corresponding to these segments. Applying the same calculation strategy of Keller et al. (2011), the equation Y = aX + b was adjusted in each RBL segment, where Y (cm3 cm−3) is the ε of RBL; X is the base-10 logarithm of the loads corresponding to ε; a (cm3 cm−3) and b (cm3 cm−3) are the slope and the intercept of the RBL, respectively. The RBL segments between 10 and 12.5 kPa were disregarded as the imprecise application of the loads in this small load range provided inconsistent RBL. For the adjustment of the regression lines described earlier, the matrix of points {σ, ε} of each RBL segment was first separated with an algorithm developed in VBA-Excel®. After obtaining each matrix of points {σ, ε}, the algorithm applied the function Application.WorksheetFunction.Slope(Y(), X())), which calculates the

2.2.2. Cyclic loading test The cyclic loading tests were performed by applying σ of 12.5, 25, 50, 100, 200, 400 and 800 kPa in loading-unloading cycles, which is the same load sequence used in the classical static compression test. Each cycle consists of 36 s loading and 36 s unloading, being applied 17 cycles per load, and totaling approximately 10 min of loading and 10 min of unloading, disregarding the load adjustment time by the device (Fig. 1). The unloading was carried out until about 10 kPa, because the device applies a minimum load of 10 kPa, but according to Peth et al. (2010) the maintenance of this minimum load is not problematic for 31

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Fig. 2. An example of a cyclic loading test output with void ratio (ε) and load (σ) (in logarithmic scale). A) The vertical arrow pointing downwards shows that there is variation around the lower load established in unloading (σmin = 10 kPa), due to non-adjustment precision equipment. The σmax values represent the programmed maximum load for each cycle. B) An example of the positioning of a RBL (belonging to the cycle with application of σmax = 200 kPa) and the VCL of the segment between the last σmax = 100 kPa and the last σmax = 200 kPa. C) An example of Δεt, Δεr and Δεp, which represent total deformation, recoverable deformation, and unrecoverable deformation, respectively.

3. Results and discussion

slope of the regression line (coefficient a). In this function, Y () and X () are vectors ε and σ of the matrix of points {σ, ε}, respectively. For each sample, approximately 100 values of a were calculated, which correspond to the SI. For each CC, the segment in which VCL would be positioned was divided into six linear segments. Thus, it is possible to obtain discrete slope differences along this segment, as proposed by Schäffer et al. (2010). The end point of a loading cycle (logσi, εi) and that of a subsequent loading cycle (logσi+1, εi+1) constituted the beginning and end points of each segment, respectively (Fig. 2B). The CI (cm3 cm−3) of each segment was calculated by CI = |(εi+1 − εi)/log(σi+1/σi)|. For each loading and its subsequent unloading, we also calculated elasticity and plasticity indexes using only the deformation data. Fig. 2C shows total deformation (Δεt), recoverable deformation (Δεr) and unrecoverable deformation (Δεp) of a loading followed by unloading. Using this strategy, the elasticity index was calculated by SI' = Δεr/Δεt and the plasticity index by CI' = Δεp/Δεt, for each loading and its successive unloading. For each CC, the SI values of the RBL of each cycle were divided by the CI of the segment connecting the end of the previous cycle with the end of the cycle in question (Fig. 2B). The division of SI/CI was called the K1 index (K1 = SI/CI), which indicates the ratio (0 < K1 < 1, dimensionless) of recoverable deformation to unrecoverable deformation. Similarly, for each SI' and its corresponding CI', the ratio of SI'/CI' was called the K2 index (0 < K2 < 1, dimensionless) that also indicates the ratio of recoverable deformation to unrecoverable deformation. For each soil sample, the mean, median, percentiles, and confidence intervals of SI, CI, SI', CI' were shown graphically. For each CC, the relationship of SI, CI, SI', CI', K1 and K2 with σmax was analyzed using the Pearson correlation (P < 5%). For the total data set, the K1 and K2 indexes were still graphically related to the SI and SI' values, respectively. The pair (σmax, ε) of the first RBL of each set of RBL segments of each sample was used to form a matrix {σmax, ε} containing seven pairs (σmax, ε). A zero load and its corresponding ε was added to this matrix, resulting in {0, ε0; 12.5, ε12.5; 25, ε25; 50, ε50; 100, ε100; 200, ε200; 400, ε400; 800, ε800}. This new matrix was used to describe the CC, from which the σp was obtained. Both CC and σp were obtained as previously described in the static compression test (Section 2.2.1).

3.1. Evaluation of elasticity and plasticity indexes For the set of 48 soil samples, SI ranged from 0.002 to 0.089, CI ranged from 0.03 to 1.17, SI' ranged from 0.03 to 0.45 and CI' ranged from 0.49 to 0.97 (Fig. 3). Most SI values (Fig. 3A) are within the range observed in previous studies using static compressibility (SIst). For example, Mentges et al. (2013) observed SIst between 0.018 and 0.068, calculated for an unloading of 800–12.5 kPa, in Gleissolo Háplico (Endoaqualf) with wt of 10 kPa; Keller et al. (2011) found SIst between 0.005 and 0.021 for an unloading of 800–1 kPa in 48 samples of four Swedish soils with a variation in clay content of 10–62% and wt of 10 kPa; Braida et al. (2008) obtained SIst of 0.0056 in an Argissolo (Hapludalf) and of 0.0068 in a Nitossolo (Hapludox) for an unloading of 400–12.5 kPa and wt of 6 kPa. Water tension changes significantly during compression tests (Fazekas and Horn, 2005) and, thus, wt during unloading from 800 to 1 kPa was likely to be closer to saturation than wt during loading to 25 kPa (Keller et al., 2011). Comparing results is not a simple, straightforward approach since rebound lines are not parallel, as it will be shown later. Expressing the SI values in terms of structural soil properties provides a more comprehensive view of soil changes. Reichert et al. (2018) showed that the mean value of SI of 0.067, calculated from the loadingunloading-reloading loop of 0-300-0 kPa, implies a rebound of about 0.17 in ε, 0.12 Mg m−3 in ρ, and 14% in total porosity (n), calculated from the equations △ε = −0.067 log(300-0), n = ε/(1 + ε), and (ρ) = 2.65/(1 + ε). Braida et al. (2008), Keller and Arvidsson (2007) and Stone and Larson (1980) found somewhat lower rebound values, respectively, of ≤0.05, 0.06, and < 0.08 Mg m−3 in ρ. The CI variation of this study (0.03 and 1.17) (Fig. 3B) is much broader than the range 0.18 < I < 0.43 (considered mean ± standard error) plotted in the study by Schäffer et al. (2010) (see Fig. 5 in their study), because we used 16 horizons of eight soils with wide variation of density and composition (Table 1), whereas Schäffer et al. (2010) used only one soil with slight variation in density. As CI was calculated as the local slope of the CC (Fig. 2B) and as the slope of the CC changes a lot locally (Tang et al. (2009) found dε/dlogσ between 0.01 and 0.9 approximately), therefore the CI range contains values consistent with the condition of this study. The SI' range of this study (0.03–0.45) (Fig. 3C) is somewhat wider than those reported by Mentges et al. (2013) (0.11–0.35) and by Braida et al. (2008) (0.05–0.35), which (as is for CI) is also due to the higher variability of soils used in our study (Table 1). In Braida et al. (2008) and Mentges et al. (2013), as well as in our study, the SI' was calculated 32

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Fig. 4. Elasticity index (SI, dimensionless) with the repetition of loading/unloading cycles and increasing loads (following the order of 25, 50, 100, 200, 400 and 800 kPa) of a sample of each horizon A (A) and B (B) of the eight soils.

Fig. 3. Elasticity SI (A) and SI' (C), and plasticity CI (B) and CI' (D) indexes for the 48 soil samples (three samples per horizon). For each soil sample, N = 100 for SI, SI' and CI' and N = 6 for CI. In the box plot, the lower limit of the box indicates the 25th percentile, and the upper limit indicates the 75th percentile; within the box the black line marks the median, and the blue line marks the mean; error lines above and below the box indicate the 90th and 10th percentiles; and the black points mark outlying points.

by the Δεr/Δεt relationship, where Δεt is always defined with the same initial ε (i.e., ε0) (Fig. 2C). Therefore, the SI' did not exceed 45% of the maximum value (1). A SI' value similar to 1 was obtained by Peth et al. (2010). However, they defined Δεt for each particular loading-unloading cycle. Thus, the SI' calculated by Peth et al. (2010) represents the recovery of the deformation of a particular cycle, whereas the SI' of our study represents the recovery of the total deformation up to the cycle under analysis. As CI' is complementary to SI' in total deformation (CI' + SI' = 1), the strategy to define Δεt in our study resulted in high CI' values (0.49–0.97) (Fig. 3D), indicating almost completely plastic deformations, whereas the strategy of Peth et al. (2010) resulted in high SI' values, which indicated almost completely elastic deformations. However, the use of an index that indicates greater plasticity with the load sequence was intentional in our study, as will be explained later in the discussion of the relationship of SI' with σp. Since six CI values were calculated for each CC, and as CI' complements SI' in total deformation (CI' + SI' = 1), only results of SI and SI' for one sample of each horizon of the eight soils were presented (Figs. 4 and 5). Additionally, in each CC there are 100 calculated values of SI and SI'. The samples in Fig. 4 show that magnitude of SI is affected by the soil and the horizon, however, SI always decreases with the repeated loading of a given load, but it increases with increasing loads, except for samples LBdf1 A and LBdf2 A. The samples in Fig. 5 show that the magnitude of SI' is also affected by the soil and the horizon,

Fig. 5. Elasticity index (SI', dimensionless) with the repetition of loading/unloading cycles and increasing loads (following the order of 25, 50, 100, 200, 400 and 800 kPa) of a sample of each horizon A (A) and B (B) of the eight soils.

however, SI' also always decreases with the repeated loading of a given load, but there are fewer samples in which increasing loads cause an increase in SI'. The low negative correlation between SI and SI' (r = −0.43) (data not shown) shows the low degree of agreement between them in 33

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refer to Fig. 2B, which shows a RBL and a segment of the VCL. If all the deformation is recovered in the segment to which the RBL was fitted, the soil would be completely elastic. In this case, SI (slope of RBL) should numerically represent total elasticity and CI (slope of VCL) should represent zero plasticity. However, a VCL with slope equal to zero (CI = 0) would be a geometric representation of a sample without deformation, for which there would also be no recovery (SI = 0). This mathematical independence between SI and CI is the reason we can experimentally verify that both may increase with increasing σmax (Fig. 7A) and misleadingly indicate that the soil can become more elastic and plastic simultaneously. The inconsistency in the complementarity of SI and CI does not occur with SI' and CI', because SI' + CI' = 1 (Fig. 7B). This mathematical approach does not result in any physical incoherence, because if the soil is completely elastic (SI' = 1) it will not be plastic at all (CI' = 0) and vice versa. With the use of SI' and CI', there is no possibility of the soil being more elastic and more plastic simultaneously, because the relationship between SI' and CI' is reverse (Fig. 7B).

indicating elasticity, and also that the load sequence would modify elasticity in the opposite direction. Furthermore, both SI and SI' would not be well correlated with the SI' calculated by Peth et al. (2010) (see Fig. 13 in their study). Their SI’ value shows that repeating the application of 80 kPa induces the soil to rebound as a full elastic behavior before 100 cycles. In contrast, repeating each load decreased SI and SI' for most of the samples in our study (Figs. 4 and 5), indicating a decrease in the elastic behavior of the soil. These inconsistencies are of concern because both SI and SI' were used in previous studies to assess soil elasticity. Because our study is the first to produce a large set of SI and SI' resulting from 17 cycles at six different loads and in different soils, there is no other set of results with this range to compare with our data. Therefore, the following interpretation of our results should be reassessed in future studies. The change in SI with the increased loads emphasizes the violation of the assumption that RBL are parallel, as observed by Keller et al. (2011) using SI calculated at 25 and 800 kPa. Also, RBL are not generally linear curves in a log σ – ε representation (Holtz and Kovacs, 1981). Therefore, the characterization of elasticity with a single RBL of a static compressibility test (Braida et al., 2008; Mentges et al., 2013) can characterize only a value of soil elasticity of a particular compression-decompression cycle. The increase in SI with increasing loads can be confirmed by the correlation between SI and σmax (Fig. 6). For the 48 soil samples, the correlation between SI and σmax is greater than 0.75 in 37 samples (approximately 77%) and less than 0.5 in six samples (approximately 12%). For CI, the percentages were approximately 63% and 19%, respectively, for r > 0.75 and r < 0.5. As for SI' and CI', their relationship with σmax was weaker and more distributed in both directions (positive and negative). As shown in Fig. 4, SI predominantly increases with increasing σmax. The correlations (Fig. 6) indicate that CI also increases with increasing σmax, resulting in a positive correlation between SI and CI (Fig. 7A). Based on these relationships, the interpretation of elasticity and plasticity with the use of SI and CI suggests that the soil becomes more elastic and more plastic simultaneously with increasing loads, which seems physically incoherent. A hypothesis to explain this supposed conflict is to consider that SI and CI contain only a portion of the total elasticity and plasticity of the sample, so that both can increase up to a certain load even though total elasticity decreases with increasing total plasticity. Another assumption is that SI and CI geometrically describe better the CC segments than the elasticity and plasticity of the soil. To explain this point of view, we will

3.2. Precompression stress as elastic-plastic transition of soil To facilitate comparing the evolution of SI, CI, SI' and CI' with increasing loads and also verifying graphically if σp is a value of σ that clearly separates regions of elastic deformation from plastic deformation, the means of SI, CI, SI' and CI' in each σmax were normalized with respect to their amplitude in the CC [Yi = (Xi − Xmin)/ (Xmax − Xmin)]. Thus, normalized SI, CI, SI' and CI' (SIn, CIn, SI'n and CI'n, respectively) ranged from zero to 1. The normalized SIn and CIn of the 48 samples are represented in Fig. 8 and the normalized SI'n and CI'n of the 48 samples are represented in Fig. 9. Comparing the SIn and CIn indexes, there is no region of predominantly elastic deformations followed by a region of predominantly plastic deformations (Fig. 8). Therefore, in the graphs of SIn and CIn, it is not possible to fit the concept of σp as a value of σ that clearly separates regions of elastic deformation from plastic deformation. Comparing the SI'n and CI'n indexes, we found a decrease in SI'n accompanied by an increase in CI'n in 15 of the 48 samples (Fig. 9). However, in these 15 samples there are cases where the region of decreased SI'n is preceded and succeeded by regions of increased SI'n, which implies that the region of increased CI'n is preceded and succeeded by regions of decreased SI'n. Furthermore, in 12 of the 48 samples, SI'n always increases and CI'n always decreases with the increase of σmax. Consequently, also in the graphs of SI'n and CI'n, there is no regularity in the relationship of SI'n and CI'n to reliably fit the concept of σp as a value of σ that clearly separates regions of elastic deformation from plastic deformation, even though the manner in which we calculated SI' favored the increase in plasticity with the increase of the loads, as previously explained. If we had used the calculation strategy of Peth et al. (2010), which favors the increase in elasticity with the load sequence, the lack of experimental proof of the assumptions of the concept of σp would be even more evident. The acceptance that the σp is a value of σ that clearly separates regions of elastic deformation from plastic deformation becomes unsupported when positioning the values of σp found in the 48 samples in the respective graphs of SIn and CIn or SI'n and CI'n. Most of the average of σp were less than 100 kPa and in only two samples they exceeded 150 kPa (Fig. 10). The σp for each compression curve had little error attributed to fitting Eq. (1), since R2 was greater than 0.95 for all fitted curves. Relating σp with SIn and CIn or SI'n and CI'n, we found that there is no reliable evidence in the graphs of SIn and CIn or SI'n and CI'n that, below 200 kPa, there is a point that separates regions of elastic deformation from plastic deformation. Consequently, the σp of this study does not contain this characteristic, regardless of whether it was obtained by static compression test or cyclic loading test. The differences in the σp between the two tests (Fig. 10) also do not modify this interpretation. Furthermore, the acceptance of the values of σp also

Fig. 6. Frequency of Pearson correlation coefficient (r) (P < 5%) for elasticity (SI and SI') and plasticity (CI and CI') indexes with σmax. For each of the 48 r (48 soil samples), N = 100 for SI, SI' and CI' and N = 6 for CI.

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Fig. 7. Relationship of the plasticity index CI with the elasticity index SI (A); and of the plasticity index CI' with the elasticity index SI' (B) for the 48 soil samples.

Fig. 9. Relationship of normalized plasticity and elasticity indexes, respectively, SI' and CI' (SI'n and CI'n, dimensionless) with the maximum load of each cycle for the 48 soil samples.

Fig. 8. Relationship of normalized plasticity and elasticity indexes, respectively, SI and CI (SIn and CIn, dimensionless) with the maximum load of each cycle for the 48 soil samples.

implies that the elastic deformations after σp may be greater than before σp (and plastic deformations before σp may be greater than after σp), in most cases (Figs. 8 and 9). The set of results of this study does not corroborate the definition of σp as a load value in the transition from elastic to plastic deformation. On the contrary, it corroborates the evidence that σp is a product from a mathematical artifact, the semi-logarithmic (log σ-ε) diagram (Keller

et al., 2011). This lack of evidence from the experimental observations with the theoretical assumptions of σp is added to another lack of experimental-theoretical support. The laboratory studies of Dastjerdi and Hemmat (2015) and Somavilla et al. (2017) highlight the weak relationship of σp with the maximum load previously received by the soil, but a strong relationship would be expected according to the theoretical 35

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distinguished between them (Fig. 11). This indicates that an increase in the elasticity/plasticity ratio (K1 = SI/CI) is logical if the soil becomes more elastic (higher SI). However, SI always increased with increasing compaction (Fig. 8). Consequently, the positive relationship of K1 to SI is also positive with increasing compression. This means that the more compacted the soil is, the greater the elasticity/plasticity ratio. We also found that K1 is soil dependent. In general, the greater the slope of the K1-SI relationship, the lower the maximum elasticity (SI) of the soil, but the elasticity/plasticity ratio increases. The SI and CI can incorrectly indicate elasticity and plasticity, as previously discussed. Therefore, there is a risk of error in the interpretation of soil elasticity and plasticity when analyzing K1. The relationship of K2 to SI' was also positive (Fig. 12), but it is not dependent on the soil, as K2 is only a function of SI', K2 = SI'/(1 − SI'). In the calculation of K2, the logical increase in the elasticity/plasticity ratio (K2) is clear if the soil becomes more elastic (higher SI'), which is not evident in the calculation of K1. There were several cases where SI' also increased with increasing compression (Fig. 9), although in fewer instances than in the case of SI. However, even if the course of SI' changes from decreasing to increasing during deformation, there is no physical inconsistency in the SI' and CI' relationship, as highlighted previously. Therefore, K2 incorporates no physical inconsistency and is more reliable than K1 in evaluating soil elasticity/plasticity relationships.

Fig. 10. Mean (bars) and standard deviation (error lines) of precompression stress (σp) determined by static compression test and cyclic loading test for the 16 horizons.

4. Conclusion The magnitude of SI, CI, SI', CI' and σp on 48 soil samples from A and B horizons were similar to those observed in previous studies. Soil SI and CI were positively correlated with σ, indicating that the soil becomes simultaneously more plastic and elastic as the σ increase. This sounds physically incoherent, and this inconsistency does not arise mathematically from SI’ and CI’, because CI’ + SI’ = 1, i.e. ε_recovered + ε_unrecovered = ε_total deformation. Hence, SI’ and CI’ are more coherent indexes of soil elasticity and plasticity. When relating σp with the elastic and plastic index, there was no evidence that σp is the σ where SI and CI or SI’ and CI’ indicates a transition from predominantly elastic deformation to predominantly plastic deformation. Therefore, SI and CI or SI’ and CI’ do not endorse the assumption that σp represent a transition from elastic to plastic soil behavior.

Fig. 11. Ratio of elasticity that is manifested in relation to plasticity (K1, dimensionless), considering the elasticity (SI, dimensionless) and plasticity (CI, dimensionless) indexes for the 48 soil samples.

Acknowledgements The authors are grateful to National Council for Scientific and Technological Development (CNPq) and Coordination for the Improvement of Higher Education Personnel (Capes) from Brazil. References Ajayi, A.E., Dias Junior, M. de S., Curi, N., Araujo Junior, C.F., Souza, T.T.T., Inda Junior, A.V., 2009. Strength attributes and compaction susceptibility of Brazilian latosols. Soil Tillage Res. 105, 122–127. http://dx.doi.org/10.1016/j.still.2009.06.004. Baumgartl, T., Köck, B., 2004. Modeling volume change and mechanical properties with hydraulic models. Soil Sci. Soc. Am. J. 68, 57–65. http://dx.doi.org/10.2136/ sssaj2004.5700. Blake, G.R., Hartge, K.H., 1986. Bulk density. In: Klute, A. (Ed.), Methods of Soil Analysis. Part 1-Physical and Mineralogical Methods. SSSA Book Ser. 5.1. SSSA, ASA, Madison, pp. 363–375. http://dx.doi.org/10.2136/sssabookser5.1.2ed.c13. Braga, F. de V.A., Reichert, J.M., Mentges, M.I., Vogelmann, E.S., Padrón, R.A.R., 2015. Propriedades mecânicas e permeabilidade ao ar em topossequência ArgissoloGleissolo: Variação no perfil e efeito de compressão. Rev. Bras. Cienc. do Solo 39, 1025–1035. http://dx.doi.org/10.1590/01000683rbcs20140724. Braida, J.A., Reichert, J.M., Reinert, D.J., Sequinatto, L., 2008. Elasticidade do solo em função da umidade e do teor de carbono orgânico. Rev. Bras. Cienc. do Solo 32, 477–485. http://dx.doi.org/10.1590/S0100-06832008000200002. Casagrande, A., 1936. The determination of pre-consolidation load and its practical significance. In: Proceedings 1st International Conference on Soil Mechanics and Foundation Engineering. Cambridge. pp. 60–64. Dastjerdi, M.S., Hemmat, A., 2015. Evaluation of load support capacity of remoulded fine and coarse textured soils as affected by wetting and drying cycles. Soil Res. 53, 512–521. http://dx.doi.org/10.1071/SR14209.

Fig. 12. Ratio of elasticity that is manifested in relation to plasticity (K2, dimensionless), considering the elasticity (SI', dimensionless) and plasticity (CI', dimensionless) indexes for the 48 soil samples.

definition of σp. These observations suggest that the theoretical concept of σp could not be accessed in agricultural soils or that laboratory techniques are not capable of accessing it. As the approach based on plastic and elastic indexes failed to show a predominantly-elastic followed by a predominantly-plastic deformation region in the compression curve, new methodological approaches to find these two regions or another soil physical behavior are necessary for the σp not be considered a mathematical artifact. The relationship of K1 with SI was positive for all soils and well36

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