Damping Ratio of Sand Containing Fine Particles in Cyclic Triaxial Liquefaction Tests

Abstract

Sand liquefaction triggered by earthquakes is a devastating geological disaster and has emerged as an engaging topic in earthquake engineering. With an enhanced understanding of pure sand liquefaction promoted by laboratory research, there is a growing concern, following filed investigations, over the influence of fine particles on the liquefaction potential of sand containing inclusions. Efforts have been devoted to clarifying the significance of certain physical indicators (e.g., plasticity index, particle shape and gradation characteristics), and fruitful conclusions can be found in the published literature. However, the relationship between the content of fine particles and the cyclic degradation in liquefaction process seems still unclear. To fill this knowledge gap, three sets of cyclic triaxial tests were performed on various sand–fines mixtures with the dry tamping method. The experimental results revealed that (i) fine particles provided a negative contribution to the global soil structure; (ii) however, the damping ratio measured from the obtained stress–strain loops manifested its independence from the fines content during cyclic degradation. In this paper, we propose a shearing mechanism on the microscopic scale to explain the above contrasting observations. For a given soil fabric, the fine particles around sand-to-sand contact points probably break strong force chains, intensifying the threat of liquefaction. By contrast, these fines play the same role in favouring relative sliding between sand grains during both the loading and unloading stages. As the maximum stored energy and the energy loss per cycle are amplified with the same scaling factor, the damping ratio, defined as the ratio between them, should display a macroscopic invariance in triaxial tests.

1. Introduction

Subjected to earthquake motions, sand liquefaction [1,2,3] can likely occur where an underground water table is pretty shallow. In such a case, the foundation soil might develop a “flow-type” behaviour due to the rapid accumulation of excess pore water pressure and experiences a sudden loss of shear resistance. Spectacular examples of engineering failure (e.g., civil buildings and underground lifelines) can be ascribed to the triggering of sand liquefaction, as shown in Figure 1. For instance, Kamisu city (Osaka province in Japan) was severely touched by sand liquefaction during the 2011 Tohoku earthquake (M_w = 9.1). According to the follow-up disaster report [4], a series of land cracks and some remains of sand boils associated with large permanent ground deformation were apparent on ground surfaces and walking streets, respectively.

Evidence from a number of field surveys has already established that sandy deposits containing a small amount of fine particles (i.e., fines content usually lower than 10.0%) are more likely to liquefy [5]. The similar conclusion has been repeatably drawn by others. For instance, in an experimental study including 17 worldwide earthquakes [6], more than half of the liquefied data fall within a small range of fines content less than 5.0%, as indicated in Figure 2. Hence, the influence of fine particles on the liquefaction potential has become one of the most significant current discussions and received increased attention in geotechnical earthquake engineering.

Fruitful studies reported in the literature that concentrate on this topic frequently adopt triaxial tests to assess the possible influence of fine particles in the light of certain physical indicators such as plastic index [7], particle shape [8] and gradation characteristics [9]. In an experimental study carried out by Xenaki [10], the relationship between the liquefaction potential and different void ratios have been carefully analysed: (i) for the same global void ratio, the liquefaction potential decreased with the addition of non-plastic fines from 0 to 44%; (ii) for the same intergranular void ratio with low fines contents, the increase in the fines content led to a higher liquefaction resistance, whereas a reversed tendency was found for the same interfine void ratio with higher fines contents. These references have mainly concentrated on homogeneous soil being the test specimens. With the purpose of embodying soil stratification, Jain [11] performed a series of cyclic triaxial tests directly with stratified sand–silt specimens and found that the increase in the number and thickness of silt layer significantly increased the corresponding liquefaction resistance. Recently, the initial soil fabric given by the specimen preparation method has been proven to be a predominant factor governing the liquefaction response [12]. Regarding sand–silt mixtures, Monkul [13] found that the angular nature of fines provided a negative contribution to shear strength since some fines can possibly located between coarse sand grains by breaking the global force chain yielding the shear resistance.

In general, deformation characteristics of a soil element vary to a large extent decisively relying on the magnitude of shear strain. While assessing the liquefaction failure with a watchful eye toward large deformation, the effect of cyclic degradation [14] is an essential descriptor to assess the liquefaction process. The evolution of damping ratio in excess of the incipient deformation range [15,16] has been widely recognised as an intrinsic variable to describe the cyclic degradation. Moreover, a good understanding of the change in damping ratio due to the progression of shear strain can shed light on (i) the liquefaction response with the dynamic properties and (ii) the constitutive relationship to help simulate the liquefaction behaviour more accurately. Unfortunately, the aforementioned experimental studies so far provide very little information about the influence of fine particles on the evolution of damping ratio. To fill this knowledge gap, the primary objective of this study is to improve the understanding of the influence of fine particles on the damping ratio on account of sand liquefaction. With this aim, a standard silica powder was selected to create different binary sand–fines mixtures at three fines contents (i.e., 5.0%, 10.0% and 15.0%). Three sets of cyclic triaxial tests were then performed to establish cyclic shear resistance curves for clarifying the liquefaction potential with the variation in fines content. The damping ratio was then measured from the obtained stress–strain curves at each fines content to analyse its impact on cyclic degradation. With these experimental results, this paper presents a mechanism behind based on the microscopic perspective to rationally explain why the damping ratio and the fines content might be independent of each other.

2. Materials and Methods

2.1. Physical Properties of Test Materials

The base sand used in this study is a poorly graded French reference sand: Hostun sand 31 (termed “HN31” hereafter). The sand is characterised by uniform sub-angular grains and widely adopted in many European laboratories. A non-plastic silica powder C500 was selected as the added fines. The corresponding grain size distribution curves measured by the laser diffraction method are displayed in Figure 3, and the detailed index properties [17] are summarised in Table 1.

2.2. Experimental Programme

According to field investigations [5,6,18], liquefied soils are generally composed of a coarse sand matrix containing a few fine particles in relatively limited quantities. To get closer to this in situ condition, the density index of sand matrix I_Dmat was set as the controlled parameter to prepare all triaxial specimens:

(1)$I_{\mathrm{Dmat}}=\left(e_{\max }-e_{\mathrm{mat}}\right)/\left(e_{\max }-e_{\min }\right)$

where e_mat is the void ratio of sand matrix, e_max and e_min represent the maximum and minimum void ratio of the clean HN31 sand, respectively. The clear benefit of using such a concept is its ability to conserve the prevailing role of the coarse sand matrix, irrespective of the amount of fine particles [19]. A very loose state of I_Dmat = 0.00 was adopted to favour the triggering of sand liquefaction [20] since a high contractiveness was believed helpful in analysing the subsequent dynamic characteristics. Under the circumstance, the fines content F_c was defined as the dry mass of added fines divided by that of the base sand. Three relatively low values (5.0%, 10.0% and 15.0%) were used, more consistent with filed surveys. For further analysis, different intensities of shearing were applied by varying the cyclic shear ratio (CSR) so as to establish the relevant cyclic shear resistance curves. The detailed test programme is summarised in Table 2.

In the present work, the entire setup for cyclic triaxial tests consists of (i) a reaction frame, (ii) a confining chamber, (iii) CO₂, de-aired water tanks and (iv) an overhead crane. Figure 4a displays the schematic layout of the essential reaction frame and the confining chamber. A novel auto compensation system [21] was used to obtain the test specimen and the loading piston connected (Figure 4c). The primary function of the system was to avoid disturbing the fragile soil fabric as much as possible for improving the quality and repeatability. Cylindrical specimens with 100 mm in diameter and 200 mm (Figure 4b) in height were reconstituted with ten layers using the dry tamping technique [22]. After mixing base sand and fine particles through stirring and shaking in an enclosed container, all mixtures were divided into ten identical parts. Each part was carefully introduced into a split mould with a spoon, followed by slight compaction efforts to achieve the required thickness of 20 mm using a hand tamper. The confining pressure was then increased to 100 kPa to hold the test specimen properly, and the conventional three-step saturation was initiated. Carbon dioxide was first flushed (about 15 kPa) through the dry specimen for about 15 min with the purpose of replacing air, followed by the injection (about 9 kPa) of de-aired water. The final steps involved the application of a back pressure. Namely, the back pressure and confining pressure were alternatively increased in small steps of about 20 kPa until the Skempton’s B value was at least equal to or greater than 0.98. This criterion can be believed to be fair enough to achieve a full saturation for sandy deposits.

As for binary soil in laboratory research, one of the main obstacles is the likely segregation between coarse sand grains and fine particles, especially during the saturation and shearing stages. Hence, the assessment of repeatability has become an essential point to examine the adopted reconstitution method, as well as to validate both the entire experimental setup and protocol. In the companion paper of the present work [17], non-plastic (C500 with I_p < 10) and plastic fines (Speswhite Kaolinite with I_p = 30) were, respectively, added into HN31 sand in monotonic triaxial tests using the identical reconstitution and saturation methods described above. The obtained stress–strain curves exhibited high-level repeatability, demonstrating the objectivity of the obtained results.

3. Results

Regarding cyclic liquefaction tests, it is customary to take either (i) the excess pore water pressure ratio r_u (r_u = ∆u/σ′_c) or (ii) axial strain ε_a as the standard to recognise the outset of sand liquefaction [7,23,24]. Between these two criteria, the latter [15] is more preferred since the excess pore water pressure ∆u of sandy specimens containing fine particles was experimentally found not to completely develop till the consolidation stress but rather stop building up when it reached a value equal to approximately 90–95% of the consolidation stress. For this reason, the achievement of 5% axial strain in double amplitude (D.A.) during one cycle was thought to estimate the attainment of liquefaction failure.

Figure 5 presents the experimental results of HN31-C500 mixtures with the fines content F_c equal to 5.0%. For better viewing, two partial enlarged views are provided on the right. The excess pore water pressure ∆u displayed in Figure 5a suggests that the initial increment in CSR from 0.075 to 0.090 could significantly accelerate the development of sand liquefaction. Figure 5b displays the relationship between the axial strain ε_a and the number of cycles N_cyc. In the initial loading stage, ε_a accumulated in a granular manner. After r_u attained a higher value of about 80%, the accumulation rate of ε_a was sharply amplified and the specimen finally liquefied with the transient axial strain on the extension side. With the increase in CSR from 0.075 to 0.090, the number of cycles to liquefaction (i.e., 5% in D.A.) N_L decreased from 269 to 56. In contrast, the next increment in CSR was relatively inefficient to favour the build-up of liquefaction. The effective stress paths shown in Figure 5c were mainly governed by the instability line [25] referring to the flow liquefaction during which the test specimens passed from the solid to liquid state and behaved almost like a liquid. Close to the failure stage, a tiny dilatant response (as outlined by a blue arrow in Figure 5c) was indeed observed for CSR = 0.075. This puzzling phenomenon can be attributed to the fact that the coupled hydraulic actuator was incapable of perfectly managing the transient axial displacement during the liquefaction stage.

Given the limited space, only the evolutions of axial strain against the number of cycles are provided with partial enlarged views in Figure 6 for F_c = 10.0% and 15.0%, respectively. In the graph, the axial strain first developed in a gradual manner followed by a rapid increase with the attainment of liquefaction triggering, very similar to what has been previously observed in Figure 5. These curves are two key ingredients in constructing the subsequent cyclic shear resistance curves for further analysis.

Figure 7 displays all cyclic shear resistance curves for HN31-C500 mixtures as a function of fines content F_c. With the increase in F_c, the curve clearly moves towards the left. This indicates that the gradual addition of fine particles provides a quite negative contribution to the global soil structure because N_cyc required for the attainment of the given failure criterion (5.0% in D.A.) decreases to be much smaller for specimens containing more inclusions. This adverse effect due to the addition of fine particles is consistent with many experimental investigations documented in the published literature [27,28,29]. However, the above phenomenon is contrary to the experimental data reported by Xenaki [10] as discussed in the first section. A possible reason for this discrepancy might be the fact that different specimen preparation methods were employed creating different initial soil fabrics in which the influence of fines content might not necessarily be the same.

As already known, the mechanical response of a soil element is highly nonlinear and strain-dependent. Subjected to cyclic loading, the strain-dependent property of sand can be manifested by the variation of energy loss per cycle ∆W with the development of shear strain on the stress–strain curve. Figure 8 presents the stress–strain curves of HN31-C500 mixtures according to the shearing intensity and the fines content. The stress–strain curves at the initial loading stage primarily develop around the zero point, indicating a slight energy loss (i.e., quasi-elasticity) due to the external loading. However, after triggering the instability state with liquefaction failure, huge transient axial strains suddenly occur on the extension side, referring to a “flow-type” behaviour.

As a measure of dynamic characteristic, the damping ratio, defined as the energy loss per cycle ∆W divided by the maximum stored energy W, is more frequently employed as it is an indicative quantity describing the evolution of material states. A common manner of computing W is to assume that it is proportional to the area of the triangle bonded by a straight line defining the secant modulus on the stress–strain curve. However, this method is not suitable for this study as the axial deformations of the test specimens develops in an asymmetrical manner with an accumulative trend towards the extension side (ε_a < 0), as shown in Figure 6 and Figure 8. From a microscopic viewpoint, this phenomenon can be explained by the following factors. Under gravity, the long axes of coarse sand grains preferentially orient along the horizontal direction to achieve a naturally stable state. Subjected to compressive shearing, sand grains tend to come into better contact with one another [30,31]. In addition, near the liquefaction failure with the development of transient deformation, the inertia force that characterises the dynamic behaviour as distinguished from the static one appears in the form of enhancing the lateral confinement against external loading [23]. Therefore, for cyclic triaxial tests involving loading reversals (from triaxial compression to extension and vice versa), both the above reasons imply a higher axial resistance to deformation on the compression side than that on the extension side.

In order to overcome the above asymmetrical property creating troubles in properly measuring the damping ratio D, the maximum stored energy W is supposed to be proportional to the triangular area [32] (as shaded in Figure 8a, linking two extreme points belonging to a single hysteresis loop and one point sharing the same coordinates with them). Recently, such a definition has been successfully applied to establish the relationship between the damping ratio D and the amplitude of shear strain γ_a for the same HN31 sand for highlighting the influence of different waveforms on the liquefaction response [20]. Figure 9 presents the evolutions of D due to the progression of cycles for HN31-C500 mixtures with three different percentages. For each content, D measured with different CSR is provided together in the graph and the amplitude of shear strain γ_a is obtained by converting ε_a through the following relation:

(2)${\gamma }_a=\left(1+\upsilon \right)·{\epsilon }_a$

where ν denotes Poisson’s ratio, which is supposed to be 0.50 [15] since the test specimens were fully saturated and verified by the Skempton’s B value, as mentioned previously. What is striking in the graph is that a rough unification is achieved for a given F_c and all data points seem to fall within the same narrow confidence band despite the sharp variation in CSR. This fact conforms well to the intrinsic strain-dependent behaviour of granular materials subjected to shearing. Moreover, the three trend lines obtained from the regression analysis are characterised by the control parameters very close to one another, as displayed in the lower right corner. This observation furthermore reveals that the development of D for sandy specimens containing a few fines is independent of the fines content.

4. Discussion

On the question of liquefaction potential with fine particles, one interesting issue that emerges from the experimental results is that the fine particles provided a quite negative contribution to the global soil skeleton since the undrained shear strength of the HN31-C500 mixtures continuously decreased with the increase in fines content F_c. By keeping the quantity of the main sand matrix constant, the gradual introduction of fine particles (i.e., F_c from 5.0% to 15.0%) indeed densified the triaxial specimens. Corresponding to this densification effect, the basic soil mechanics regularly suggests an enhanced mechanical response contrary to our experimental outcomes. The other unanticipated result from the experimental results is the independence of the damping ratio D with the variation in F_c. In the following sections, these two phenomena will be further discussed from a microscopic point of view.

As for granular assemblies, previous studies [33,34,35,36,37,38] have already underlined the close relationship between the microscopic soil fabric and the macroscopic mechanical response in laboratory tests. For instance, a thin section of a sand specimen containing 20.0% silica fines was examined through the scanning electron microscopy (SEM) [38]. Based on the observation of the relative positions of coarse sand grains and fine particles on the microscale, it was thus inferred that three kinds of force chains might exist in a sand–fines mixture, including (i) sand-to-sand force chain, (ii) fine-to-fine force chain and (iii) fine-to-sand force chain. Regarding the clean sand specimen shown in Figure 10a, the straight line represents the strong force chain constructed by two coarse sand grains in contact with each other. Due to the influence of confining pressure, strong force chains can form an interlocking system and thus prevent the sand grains from displacing each other. Upon shearing, these force chains sustain the main stress transfer and yield the overall soil resistance to deformation. Regarding the sand–fines specimen shown in Figure 10b, two other force chains can then be defined according to the location of fine particles. Since the fine particles are much smaller than the coarse sand, they certainly have a greater level of mobility. A large proportion of them is thus accommodated into the coarse sand matrix, forming the fine-to-fine force chain (highlighted by the red crosses in the graph). Subjected to external loading, these fines are almost free to shift and move without truly participating in the global stress transfer. As a result, these inactive fine particles might be assumed to behave as “voids” and make no actual contribution to the overall resistance. The other part of fine particles is located nearby the sand-to-sand contact points by separating coarse grains in contact, which forms the fine-to-sand force chain (highlighted by the discontinuous red lines in the graph). In this case, some strong force chains (sand-to-sand) yielding the global soil resistance disappear, further substituted by unstable fine-to-sand force chains. Moreover, this kind of fines can serve as minor ball bearings during shearing and thus facilitate the relative sliding between coarse sand grains. As a consequence, the contractiveness of the global soil matrix might be essentially favoured. With the increase in fines content F_c, the added fines are more likely to destroy the strong force chains. This rationale behind can explain why the decrease in the liquefaction resistance is interestingly accompanied with the densification process.

Provided that the presence of these fine particles in a sand matrix is capable of facilitating the relative sliding between coarse sand grains, more macroscopic deformation can then be imagined in triaxial tests. Because of this, the maximum stored energy W induced by the external loading (as shown in Figure 11a) is boosted, corresponding to a higher liquefaction tendency. Following the same logic, fine particles can also contribute positively to the relative sliding after the loading reversal (as shown in Figure 11b). During the unloading stage, both the reversible and irreversible deformations are enhanced with the same scaling factor as that in the loading stage. In other words, the energy loss ∆W per cycle associated with the irreversible deformation increases proportionally as the maximum stored energy W. By combining the above two factors, the numerator and denominator determining the damping ratio D are amplified with the same factor. This can logically explain why this descriptor can remain unchanged in triaxial tests on sandy specimens with different fines contents. It should be noted that the above reasoning can be only established with the same density index of sand matrix as the comparison basis. In other words, the mass of sand matrix remains always constant, which is furthermore irrespective of the addition of fine particles. On the contrary, as for the experimental data based on the same value of global void ratio while preparing test specimens, the initial introduction of fines certainly reduces the quantity of sand grains forming the global skeleton structure. In such a case, the weakened soil skeleton might probably develop a higher mobility and thus manifest a higher damping ratio with the increase in fines content [39].

5. Conclusions

In this study, the influence of fine particles on the evolution of damping ratio during the liquefaction process was investigated by performing three sets of undrained cyclic triaxial tests on various sand–fines mixtures with the density index of sand matrix equal to 0.00. The obtained results can be used to draw the following conclusions:

• For test specimens containing fine particles reconstituted with the dry tamping method, the addition of fines provides a negative contribution to the global soil structure; consequently, the liquefaction potential rises significantly. From the microscopic viewpoint, this weakening effect can be attributed to the fact that the presence of fine particles can possibly break a part of strong force chains which bear the main external loading.

• Although the liquefaction potential is indeed affected by the presence of fine particles, the evolution of the damping ratio remains shear strain-dependent, regardless of the fines contents. This is due to the similar positive influence of fine particles on the relative sliding between coarse sand grains during both the loading and unloading stages. Thus, the damping ratio manifests itself invariant on the macroscopic scale in triaxial tests.

Author Contributions

Methodology, Z.Z.; formal analysis, J.L.; writing-original draft, Z.Z., J.Z. and H.Z.; supervision, Z.Z. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to restrictions and privacy.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Schematic of liquefaction-induced failures under seismic loading.

Figure 2. Triangular classification chart for liquefied soil including 17 worldwide earthquakes [6].

Figure 3. Grain size distribution curves of the test materials [17].

Figure 4. (a) Schematic layout of the reaction frame, (b) test specimen and (c) auto compensation system [21].

Figure 5. Evolutions of (a) excess pore water pressure, (b) axial strain and (c) effective stress path for HN31-C500 mixtures with Fc = 5.0% [26].

Figure 6. Evolution of axial strain with (a) Fc = 10.0% and (b) Fc = 15.0%.

Figure 7. Cyclic shear resistance curves of HN31-C500 specimens with different fines contents.

Figure 8. Stress–strain curves for HN31-C500 specimens subjected to different levels of shearing with (a) Fc = 5.0%, (b) Fc = 10.0% and (c) Fc = 15.0%.

Figure 9. Relationships between the damping ratio and amplitude of shear strain of HN31-C500 mixtures with (a) Fc = 5.0%, (b) Fc = 10.0% and (c) Fc = 15.0%.

Figure 10. Schematic of force chains for (a) clean sand specimen and (b) sand–fines mixture.

Figure 11. Schematic of soil fabrics in the process of (a) loading and (b) unloading.

Table 1

Physical properties of test materials.

Material	Mineral Composition	D50 (1) (mm)	emin (2)	emax (3)	Gs (4)	Ip (5)
HN31	Sand	0.35	0.656	1	2.65	NP
C500	Silica	0.006	-	-

Note: ⁽¹⁾ mean grain size; ⁽²⁾ minimum void ratio; ⁽³⁾ maximum void ratio; ⁽⁴⁾ special gravity; ⁽⁵⁾ plasticity index.

Table 2

Cyclic triaxial test programme.

Ref.	Fc (%)	CSR (1)	f (Hz)	σ′c (kPa)
CTT01	5.0	0.120	0.10	200
CTT02	5.0	0.105
CTT03	5.0	0.090
CTT04	5.0	0.075
CTT05	10.0	0.120
CTT06	10.0	0.105
CTT07	10.0	0.100
CTT08	10.0	0.090
CTT09	10.0	0.075
CTT10	15.0	0.120
CTT11	15.0	0.105
CTT12	15.0	0.090
CTT13	15.0	0.075

Note: ⁽¹⁾ CSR = τ_cyc/σ′_c.

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