Shear strength degradation of vibrated dry sand

Soil Dynamics and Earthquake Engineering 95 (2017) 106–117

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Shear strength degradation of vibrated dry sand

MARK

⁎

Nicolas Denies , Alain Holeyman Université catholique de Louvain, UCL, Civil and environmental engineering department (GCE), Place du Levant 1 boite L5.05.01, B-1348 Louvain-la-Neuve, Belgium

A R T I C L E I N F O

A BS T RAC T

Keywords: Vibrated dry sand Shear strength degradation Vibrations Vibrocompaction Vibrofluidization

The present paper first provides a state-of-the-art review of experimental researches characterizing the shear strength degradation of vibrated dry sand. Once subjected to vibrations, the shear strength of dry sand exponentially decreases with the acceleration amplitude of vibration to reach a particular state wherein dry sand behaves like a complex fluid: this is the vibrofluid behavior. In the present paper, fundamentals equations and criteria governing the shear strength degradation of vibrated dry sand are summarized. A revisited intrinsic Coulomb criterion, taking into account the absence of pore pressure to explain the shear strength degradation, is discussed and explained considering the existence of a corresponding “shaking” pressure. A “general” Critical State Soil Mechanics, including the effect of the acceleration, is introduced. Finally, a rapprochement with the fundamental researches of the physicists involved in the study of granular matter is proposed allowing the identification of the governing dimensionless parameters and the different dynamic regimes encountered by vibrated dry sands once subjected to shearing.

1. Introduction Because of its granular nature, cohesionless soil is difficult to categorize. It is a geometrically complex assemblage of grains of various sizes and shapes resulting in a particular grain-size distribution. Also the way the particles are in contact, the orientation and the distribution of these contacts, called the fabric of the assemblage, certainly play a role in its behavior. In addition, the sand is characterized by a presence of voids defining its porous character. This pore space can possibly be filled with some liquid according to its degree of saturation. Up to now, many researches have been dedicated to investigate the shear strength degradation of cohesionless soils in saturated condition, mainly under large cyclic strains resulting in the increase of the pore pressure (Seed and Lee [1] and [2]; Casagrande [3]; Ishihara and Li [4]; Castro [5]; Castro and Poulos [6]; Seed [7]; Dobry et al. [8]; Figueroa et al. [9]; Youd et al. [10]; Seed et al. [11]; De Alba and Ballestero [12]; Jefferies and Been [13]). As a result, the two phenomena of liquefaction flow and cyclic mobility have deeply been highlighted. If under saturated conditions, the “fluidization” of the sand can be explained by these two phenomena (liquefaction flow and cyclic mobility), it is not clear which parameters play a role in the shear strength degradation of dry sand when it is subjected to vibrations. An

interest is shown in this topic because of its importance in several industrial processes such as the vibrocompaction and vibrodriving. In spite of some research works performed in this field of geotechnical engineering, physical mechanisms in play in these processes remain poorly understood. In order to investigate this question, the authors have performed a large literature study to highlight the effects of the vibrations on the volume change and shear strength of dry sand on the basis of early experimental and numerical works. 2. Effects of vibrations on volume change of dry sand In the past, Mogami and Kubo [14], Barkan [15] and [16], Selig [17], Prakash and Gupta [18], D’Appolonia and D’Appolonia [19], Greenfield and Misiaszek [20], Ermolaev and Senin [21], D’Appolonia et al. [22], Kolmayer [23] and Dobry and Whitman [24] have conducted a variety of noteworthy experiments with the aim to study the effects of vibrations on the volume change of dry granular soils. They considered the influence of the following variables on the dynamic behavior of the dry sand: the direction of the vibrations, the harmonic vibration parameters (acceleration amplitude, a [m2/s], frequency, fr [Hz], and displacement amplitude, A [mm] of motion), the duration of the vibrations, the size of the container, the grain-size distribution of the sand and the static surcharge applied to the sample. Report on those results was performed in details by Denies et al. [25] whom

Abbreviations: EVR, vibratory equilibrium void ratio; CVR, critical void ratio; DEM, discrete element modeling ⁎ Corresponding author. Present address: Nicolas Denies, Belgian Building Research Institute, BBRI, 21, av. Pierre Holoffe, B-1342 Limelette, Belgium. E-mail address: [email protected] (N. Denies). http://dx.doi.org/10.1016/j.soildyn.2017.01.039 Received 2 November 2015; Received in revised form 3 November 2016; Accepted 26 January 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 95 (2017) 106–117

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In the densification range, the sand monotonously densifies, according to the same relationships as obtained for horizontal vibrations (Eqs. (1) and (2)) but with smaller values of αt and βt. Optimum acceleration amplitude close to 1g is then observed where a minimum void ratio can be approached. When the acceleration amplitude is increased beyond 1g, granular convection is observed and instability develops in the sand mass leading to the emergence of an inclined free surface. Finally, if the acceleration is further increased, the free surface progressively flattens. There is an impressive dilatation of the whole sample accompanied by the development of a bulge and grains saltation is noticed. The sand becomes fully vibrofluidized (Denies et al. [25]). The convection phenomenon will be later discussed in Section 6.1.

complete their review by the presentation of original experiments revealing that volume change arising during vibration cannot be explained without addressing related phenomena such as the motion pattern displayed by individual sand particles and vibrofluidization. According to the previous studies, the acceleration amplitude of the vibrations is the governing motion parameter of the vibrocompaction of dry granular soils for horizontal and vertical vibrations. For the sake of concision, in the continuation of this article, the term “acceleration amplitude” will be used as a substitute for “dimensionless acceleration amplitude”, defined as the ratio of the amplitude of the acceleration [m2/s] to the gravitational acceleration (g). Γ will symbolize that key variable, and will be indexed with h for horizontal vibrations and with v for vertical vibrations. Moreover, unless otherwise noted, all vibrations are considered to be sinusoidal.

2.3. Effect of the normal pressure on the volume change during vibrations

2.1. Experimental observations for horizontal vibrations The application of horizontal vibrations leads to the densification of dry sand. For horizontal vibrations, Barkan [16] has early suggested a relationship describing the evolution of the void ratio of the dry sand in function of the duration of the vibrations:

e(t ) = eΓ + (einit − eΓ ) exp (−βt ) t

It is to note that the previous results have been obtained for dry sands vibrated with a free surface condition (without static surcharge applied to the sample). There are few experimental data characterizing the influence of the normal pressure applied to the sample on its behavior during vibration. As explained in Denies et al. [25], the application of a static surcharge to the surface of the sample or the increase of the depth of the sand deposit (corresponding to the application of an overburden pressure) could have the same effect: limiting densification of the sample. Additional vibration energy would then be required to overcome the resistance to density change as a result of the reinforcement of intergranular force chains in the sample, increasing in turn internal friction forces. These friction forces are increased by both static surcharge and added weight of sand (overburden pressure). These observations are valid for vertical (Ermolaev and Senin [26], D’Appolonia and D’Appolonia [19], and Kolmayer [23]) as well as for horizontal (Youd [27]) vibrations. The vibratory equilibrium void ratio (EVR) increases with an increase of the normal pressure applied to the sample. A formula is proposed in Kolmayer [23] to take into account the influence of the static surcharge applied to the sample on the parameters of Eq. (1) for vertical vibration experiments:

(1)

where e [-] is the void ratio, einit the initial void ratio resulting from dry pouring and t the vibration time [min]. βt [min−1] is an empirical coefficient depending on the nature of the soil and on the acceleration amplitude: it characterizes the rate of vibrocompaction. eΓ is the minimum void ratio that can eventually be reached under the imposed acceleration amplitude Γh. It can also be defined as the vibratory equilibrium void ratio (EVR): the stable void ratio for a sample being densified at particular acceleration amplitude, Γh. Considering the densification of dry sand under horizontal vibrations, one can consider a sand mass lying in a stable configuration (at rest, without vibration). Nevertheless, this initial configuration already corresponds to a vibratory equilibrium void ratio (EVR). Only the application of an acceleration amplitude larger than the threshold amplitude necessary to obtain this initial configuration will allow further densification. Barkan [15] has proposed the following equation to estimate the EVR:

eΓ = emin + (einit − emin ) exp (−αtΓh )

⎛ Γξ ⎞ βt = f ⎜⎜ v ⎟⎟ ⎝σ ⎠

(2)

(3)

where ξ is called the “static surcharge coefficient” by Kolmayer and σ is the static surcharge applied to the sample. The experiments of Kolmayer [23] are described in the Section 4 of the present paper. Ermolaev and Senin [26] illustrates the relationship between the index of vibratory compaction (αt), introduced in Eq. (2), and the static surcharge applied to the sample with the help of the following formula:

where emin is the minimum void ratio and αt [-] the index of vibratory compaction. Subjected to horizontal vibrations, a global settlement of the dry sand is always observed in spite of the level of the acceleration amplitude, Γh. The behavior of the sand mass can therefore be assessed with the help of Eqs. (1) and (2). No particular behavior or motion patterns (size-segregation, convection, particle recirculation…) are observed during horizontal shaking. For horizontal vibrations, no decompaction was reported by any authors in the literature (even for Γh values larger than 1) which is not the case with vertical direction of vibration.

αt = α0 exp(−K2σ )

(4)

where α0 is the index of vibratory compaction without static surcharge (for σ=0) and K2 [-] is an empirical coefficient dependent on the soil type according to the authors. Their experiments were conducted on some specimen of sandy loam, with water content close to 16.5%, subjected to vertical vibrations. Unfortunately, no precise description of their experimental set-up and procedure is available in the literature. Finally, Youd [27] studied the influence of the static surcharge applied to the sample on the parameter αt for horizontal vibration. The experiments of Youd [27] are described in the Section 5 of the present paper. On the basis of his experimental results, the following empirical relationship can be proposed:

2.2. Experimental observations for vertical vibrations When cohesionless soil, placed in a cylindrical container without static surcharge, was vertically vibrated under the gravitational field, experiments performed on dry Fontainebleau sand have allowed to Denies et al. [25] to distinguish three types of dynamic behaviors, depending on the acceleration amplitude, Γv:

αt = 0.174exp(−0.003σ )

– the densification behavior (Γv < 1), – the instability surface behavior (Γv≈1), and – the vibrofluid behavior (Γv > 1).

(5)

The previous relationship was established considering the experimental results of Youd [27] with a R-squared value close to 0.99 [-] and with σ in [kPa]. The similarity between Eqs. (4) and (5), respectively for 107

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vertical and horizontal vibrations, has to be underlined. As illustrated by these equations, the efficiency of the vibrocompaction process decreases with the increase of the static surcharge applied at the surface of the sand mass independently of the direction of the vibrations. 3. Effects of vibrations on internal friction of dry sand Considering the particular behaviors of vibrated dry sand already discussed in term of volume change, it is now interesting to investigate the effect of these vibrations on the shear strength of dry sand. The intrinsic Coulomb failure criterion of the dry cohesionless soil can initially be expressed as:

τds = σ tan φmob

Fig. 2. Decrease of the coefficient of internal friction of dry sand in function of the acceleration amplitude, Γh, for various displacement amplitudes and frequencies, data from Barkan [16].

(6)

where τds is the ultimate shear strength of dry sand [kPa], σ the effective normal stress [kPa] and ϕmob the mobilized angle of friction [°]. The index ds identifies dry sand. The pore pressure term is of course absent from the equation because of the dry condition. There are two different ways to consider the effect of the vibrations on the shear strength of vibrated dry sand. Either the researcher directly expresses the evolution of τds with the vibrations or he represents the variation of the coefficient of internal friction (f=tan ϕmob=τds/σ) as a function of the vibrations. The two approaches are similar from a scientific point of view. During several decades, many researches have been fulfilled, mainly in seismic regions, for a deeper understanding of the soil response to earthquakes with a view to protect structures and human lives. Simultaneously, others have still succeeded in harnessing soil degradation as a result of cyclic shearing. For example, Pavyluk, who studied the effect of vibrations maintained on soil, has introduced as soon as 1931 the concept of driving profiles into the ground using vibrations (as called vibrodriving process). This work was later reported by Barkan [15]. The results discussed in the present Section reflect both trends.

lower cylinder closed at its bottom and two rings inserted into a socket. The support (7) of the jack is rigidly attached to the vibrating platform (6) of the device. This latter can slide on rollers fixed to a support rigidly connected to the foundation of the installation. The horizontal vibrations are imposed by means of a vibrator (1) which is connected to the vibrating platform (6) with the help of a turnbuckle (8). The vibrator can slide on rollers (9) mounted on the guide frame (10) rigidly attached to the foundation of the installation. With this experimental set-up, the potential effects related to an oscillating normal load are avoided. Only the sand mass is vibrated during experiment. The influence of the horizontal vibrations on the coefficient of internal friction was analyzed comparing its value in static condition with values obtained for different acceleration amplitudes. In these experiments (as for all the experiments reported in the present paper), the shear strength is measured while the sand is vibrated. The evolution of the coefficient of internal friction with the acceleration amplitude is illustrated in Fig. 2. The experiments resulted in a decrease of the coefficient of internal friction of dry sand with increasing of the acceleration amplitude. The degradation of the coefficient of internal friction observed for these experiments can be described with the help of the following formula:

3.1. Experimental observations for horizontal vibrations

tan φmob = tan φres + (tan φst − tan φres ) exp (− ε Γh )

(7)

where tan ϕst is the value of the static coefficient of internal friction (thus without vibration), tan ϕres the asymptotical value of the coefficient of internal friction and ε [-] an empirical factor determining the intensity of the effect of the vibrations, known in the literature as the coefficient of frictional reduction (Youd [27]). Eq. (7) could already be introduced in Eq. (6) to build an intrinsic failure criterion for dry sand subjected to horizontal vibrations.

As reported in Barkan [16], Savchenko has characterized the decrease of the coefficient of internal friction of dry sand subjected to horizontal vibrations. In this aim, he had designed his own modified shear box, as illustrated in Fig. 1. It consists of a shear box which can be subjected to horizontal vibrations, a vibrator (1) and a monitoring device (2). The shearing device consists of three principal parts: the box (3) wherein dry sand is sheared, a spring jack (4) applying the vertical static surcharge on the sample, and the shearing system (5). This latter consists of a cable spanned over a pulley and a loading platform. For the purpose of preserving the static character of the shear load, the shearing system (5) was mounted on dampers. The shear box (3) is rigidly fixed to the vibratory platform (6). The shear box consists of a

3.2. Experimental observations for vertical vibrations Intrigued by the effects of vibrations on dry sand during earthquakes, the Japanese researchers Mogami and Kubo [14] have

Fig. 1. Experimental set-up of Savchenko [16].

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than the previous one: there is an exponential decrease of the shear strength with the acceleration amplitude until an asymptotical value is reached. The decreasing rate is in this case defined by two parameters: λ and Γ0. The analysis of their experimental data highlighted three stages of change in the shear strength, depending on the acceleration amplitude, Γv. At low acceleration (Γv < Γ0), the shear strength is close to its static value. Then, once this threshold is reached, there is a transthreshold region where the shear strength is swiftly decreased until a residual value. The distinction of these three dynamic behaviors as a function of the amplitude of acceleration (Γv) seems to correspond to the observations made by Denies et al. [25] for the volume change with a threshold amplitude (Γv=1) leading to a transition between different ranges of dynamic behavior. Considering this observation, the residual value of the shear strength is obtained when the dry sand is fully vibrofluidized. As a consequence, the “vibrofluidization” of a dry sand mass leads to the degradation of its shear strength. Nevertheless, it should be noted that in the experiments of Ermolaev and Senin [26], the material cannot be considered as fully tested in dry conditions. The appearance of a threshold acceleration could also be viewed by the detractors as the minimum level of acceleration necessary to break the apparent cohesion forces (due to the capillarity) between the grains of the assembly. It can finally be added that Ermolaev and Senin [26] are the only authors, investigating the shear strength degradation of dry sand under vertical vibrations, who have observed this threshold acceleration Γ0. Mogami and Kubo [14] have only observed a progressive degradation of the shear strength as a function of the increase of the acceleration amplitude. The experiments of Kolmayer [23], described in Section 4, have been conducted for vertical vibrations with acceleration amplitudes of Γv=2.5, 3.3 and 4.3 and under static conditions (Γv=0). It is therefore not possible to draw conclusions on the base of these experiments concerning a change of behavior around Γv=1, as no experiment was performed with values of Γv smaller or equal to 1.

Fig. 3. Experimental set-up of Mogami and Kubo [14].

4. Revisited intrinsic Coulomb failure criterion for vibrated dry sand As reported in Kolmayer [23], L’Hermite and Tournon have proposed to express the shear strength degradation of vibrated dry sand considering the development of a new intrinsic failure criterion. According to these authors, during the shearing of the vibrated dry sand, the vibrations applied to the sand sample would lead to the emergence of a “shaking pressure”, σS, which would decrease the normal pressure in a way similar to the pore pressure in the presence of an interstitial fluid. As a result, the intrinsic Coulomb failure criterion of the vibrated dry sand would become:

Fig. 4. Decrease of the shear strength of dry Soma sand in function of the acceleration amplitude for constant frequency and various static surcharges, data from Mogami and Kubo [14].

conducted some experiments to characterize the shear strength of dry Soma sand when subjected to vertical vibrations. In their experiments, dry Soma sand was filled into a metal box fixed on an electrical vibrating table as illustrated in Fig. 3. Fig. 4 illustrates the evolution of the shear strength of the dry sand in function of the acceleration amplitude, Γv. As a result, there is a progressive degradation of the shear strength of dry sand with the increase of the acceleration amplitude. As observed in Fig. 4, effect of degradation can be restricted (but not prevented) by the application of an increasing static surcharge to the sample. In the philosophy of harnessing soil degradation for vibrodriving, the Russian researchers Ermolaev and Senin [26] have subjected some specimen of sandy loam, with water content close to 16.5%, to shearing under the effect of vertical vibrations. As aforementioned, no description of their shear box is available in the literature. They proposed the following expression to describe the decrease of the shear strength with vertical vibrations:

τ = τres + (τst − τres ) exp (−λ(Γv − Γ0 ))

τds = (σ − σS ) tan φmob

(9)

where σS would be the shaking pressure [kPa]. When the normal pressure is smaller than the shaking pressure, the shear strength of the sand gets close to zero. When the normal pressure becomes anew greater than the shaking pressure, the shear strength can increase. Hence, the shaking pressure would be an expanding pressure counteracting the normal pressure. In order to deal with this concept in depth, Kolmayer [23] has investigated the influence of vertical vibrations on the intrinsic failure criterion of dry sand. In this aim, he used a dry sand reconstituted in laboratory and presenting the following characteristics: an angular character, a d50 close to 0.85 mm and a coefficient of uniformity, Cu, of 1.8 [-]. The experimental set-up of Kolmayer is illustrated in Fig. 5. The test set-up consisted in a circular box fixed on a vibrating table. Dry sand deposit was 7 cm in diameter and 4 cm high. Before shearing, the sand was previously vibrated until it reaches a void ratio close to its vibratory equilibrium void ratio (EVR) in order to avoid volume change due to vibrations during shearing. Fig. 6 presents the influence of the acceleration amplitude on the intrinsic failure criterion of vibrated dry

(8)

where τst is the value of the static shear strength [kPa], τres the asymptotical value of the shear strength [kPa], λ an empirical factor [-], called vibro-shear exponent by the authors and Γ0 a relative threshold acceleration amplitude close to 1 [-]. This equation, only valid for Γv > Γ0, expresses the degradation of the shear strength in a similar way 109

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5. Critical state line for vibrated dry sands If the previous studies defined a set of Eqs. (7)–(9) expressing the shear strength degradation of vibrated dry sand in the (τ-σ) configuration, none of the aforementioned investigators has considered the effect of vibration on the void ratio during the shearing. While complete description of the sand behavior can be obtained only regarding the evolution of its full state not only characterized by the internal stress but also by its void ratio. The study of Kolmayer [23] actually consists in the first step of investigation. In his study, the sand is first vibrated until its vibratory equilibrium void ratio (EVR) is reached. Afterwards, the shearing is performed under vertical vibration without volume change. The following step would be to investigate the effect of the vibrations simultaneously on the variation of the void ratio and on the shear strength degradation of the vibrated dry sand. Fortunately, Youd ([27] and [28]) has studied this question. In his experiment, a direct shear apparatus was mounted on a horizontal vibrating table. The shear box was oriented on the table in such a manner that the horizontal shear displacement was horizontal and perpendicular to the direction of the vibrations. The normal load was applied vertically. Fig. 8 illustrates the experimental scheme of the Youd shear box. The shear box was 6.4 cm in diameter and the height of the sand specimen varied between 1.5 and 1.8 cm. Oven-dried Ottawa sand was used in these investigations. In the first test series of Youd [28], the sand was initially vibrated until it reaches its vibratory equilibrium void ratio (EVR) as in the experiments performed by Kolmayer [23]. A shear test was then conducted at the same frequency and displacement amplitude of vibration than during vibrocompaction. Afterwards, Youd [28] performed a second series of tests in which a frequency and a displacement amplitude, different from those used in the vibrocompaction phase, were used during the shearing of the sample. Fig. 9 illustrates the evolution of the coefficient of internal friction, for various acceleration amplitudes, in function of the initial void ratio of the sample just before shearing. All the tests were conducted with a constant frequency of 30 Hz and with a normal pressure (static surcharge) of 70 kPa. In the present case, Youd [28] defines tan ϕ as the ratio of the maximum shear force, Tmax, to the normal force, N, applied to the sample. As a result, the coefficient of internal friction decreases with increasing initial void ratio and acceleration amplitude. Constant amplitude lines were linear down to a void ratio of 0.57. Below this value, the coefficient of internal friction sharply increased for the static and lower amplitude tests. According to Youd [28], this sharp increase in tan ϕ with decreasing void ratio was thought to be due to the reinforced interlocking of the sand grains. Hence, the degradation of the shear strength would be related to the reduction of the grain interlocking when increasing acceleration amplitude. Youd [27,28] has also introduced the concept of critical void ratio

Fig. 5. Experimental scheme of Kolmayer [23] shear box tests.

Fig. 6. Influence of acceleration amplitude on the intrinsic failure criterion of dry sand,

Fig. 7. Influence of the acceleration amplitude on the coefficient of internal friction of dry sand and on the shaking pressure, data from Kolmayer [23].

sand. The vibrations decrease the shear strength of the material. The revisited intrinsic Coulomb failure criteria are straight lines but they do not go through the origin of the graph. They intersect the axis of normal pressure at a value corresponding to the shaking pressure, σs. As observed in the graph, the shaking pressure increases with the acceleration amplitude. Also the friction angle slightly decreases with acceleration amplitude. Fig. 7 presents the influence of acceleration amplitude on the coefficient of internal friction, tan ϕ, and on the shaking pressure, σS, for the test results of Fig. 6.

Fig. 8. Experimental scheme of Youd [28] shear box tests.

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Fig. 9. Decrease of the coefficient of internal friction, tan ϕ, in function of the initial void ratio (considered before shearing), data from Youd [28].

(CVR) in the field of the vibrated dry sand. In his work, the critical void ratio is reached when the sand can flows at a constant void ratio. His definition slightly differs from the definition introduced by Casagrande wherein the critical void ratio is reached when the soil can flow at constant void ratio, constant effective minor principle stress and constant shear stress. In his experiments, Youd [27,28] mainly focuses on the evolution of the void ratio during the shearing of the sand subjected to vibrations. The tests performed at constant volume state (hence without variation of the void ratio during the shearing) are identified by Youd [27,28] as the tests performed at the critical void ratio (CVR) as the sand flows at constant void ratio. Considering the previous work of Barkan [15] (see Section 2.1), this absence of volume change also means that the sand subjected to vibration has reached its equilibrium void ratio (EVR) as no variation of the void ratio is observed in spite of the application of the vibration. Youd [27,28] considers therefore that, for a vibrated dry sand, the critical void ratio (CVR) is also the vibratory equilibrium void ratio (EVR) as early defined by Barkan [15]. In order to illustrate his observation, Youd [28] illustrates the evolution of the EVR line (observed in the tests focusing on the vibrocompaction of dry sand) and the evolution of the CVR line (observed in the tests focusing on the shearing of vibrated dry sand) as a function of the acceleration amplitude, Γh (see Fig. 10). Within the limits of experimental accuracy, the equilibrium void ratio (EVR) and the critical void ratio (CVR) have the same value and the EVR line also corresponds to the CVR line. As a consequence, if a dry granular soil is compacted with a predetermined value of horizontal acceleration amplitude, Γh, until its vibratory equilibrium void ratio (EVR) and then sheared with the same vibration parameters, there is no volume change and the equilibrium void ratio is also the critical void ratio. That also means that if a granular material is compacted to a void ratio less than its critical void ratio, it may be sheared at that void ratio without dilating above it, or may even densify further (by increasing the acceleration amplitude) and be sheared, in consequence, at a smaller void ratio. Contrarily, if the sample was dilating to a void ratio above its critical void ratio, the vibratory energy would compact the granular material to its critical void ratio. The EVR line for Ottawa sand is also illustrated in Fig. 9. In this representation, the EVR line is the line passing through the values of tan ϕ obtained for the tests without volume change during the shearing phase. Two main conclusions can thus be drawn considering the work of Youd [27,28]. First, the critical void ratio (CVR) of a vibrated dry sand is also its vibratory equilibrium void ratio (EVR). Second, as observed in Fig. 10, the critical void ratio of dry sand can be reduced by the application of vibrations to the grain assemblage. The evolution of the EVR-CVR line in function of the acceleration

Fig. 10. Dependence of the EVR and CVR values on the acceleration amplitude of the vibration, data from Youd [28].

amplitude could have a huge impact on the Critical State Soil Mechanics (as defined in Roscoe et al. [29]; Schofield and Wroth [30] and Atkinson and Bransby [31]). Actually, it means that for different acceleration amplitudes, new dynamic complete state boundary surfaces (such as illustrated in Fig. 11) developing in the (q′: p′: e) space could be build. Conducting additional experiments in deviatoric conditions with various effective stresses applied on the sample, new critical state lines would be discovered for each level of acceleration amplitude. Hence, the application of vibrations to the sand mass would theoretically lead to the emergence of an ensemble of complete state boundary surfaces in the (q′: p′: e) space depending on the level of acceleration amplitude, Γ. In physics, the special theory of relativity remains valid until the acceleration amplitude of motion is considered. To include the effect of the acceleration amplitude on the behavior of a system, a more general theory was developed: the general relativity. As for the relativity, to include the influence of the acceleration amplitude on the dynamic 111

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is increased at the surface of the sample:

ε = 0.267exp(−0.003σ )

(10)

Eq. (10) is established considering the experimental results of Youd [27] with a R-squared value close to 0.87 [-] and with σ in [kPa]. The empirical factor ε and therefore the efficiency of the degradation process decrease when the static surcharge is increased. 6. Numerical study of sheared and vibrated granular material in dry condition With regard to the review of the experimental researches characterizing the shear strength degradation of vibrated dry sand, few references have been found in the literature after 1970. This could be related to the advent of the numerical methods which have more and more replaced experimental researches in most of the universities around the world. This trend was accentuated by the possibility to simulate grain assemblage in particular with the help of the discrete element modeling (DEM) approach (Cundall and Strack [32]). Nevertheless, with regard to the present engineering issue, the consideration of an important number of particles necessary to obtain a representative volume of material is really time-consuming. Moreover the correct (rheological and physical) modeling of the grain characteristics, the inertial and the damping effects due to the dynamics of the problem remain a real challenge for the computational engineers (Denies [33]). If a decrease of the experimental works was observed in the field of soil dynamics with time, the study of the behavior of vibrated granular material seems to become progressively the core business of the physicists and the engineers especially specialized in powder technology and granular matters. A large body of the literature focusing on the vibrated granular matters concerns the granular convection and the granular sizesegregation. Concerning the shearing behavior of granular material, the physics of dense granular flows was also largely studied by the physicists. The purpose of the following paragraphs is to give to the reader a short overview of the main information coming from the field of granular matter and relevant for the understanding of the present topic: the shear strength degradation of vibrated dry sand.

Fig. 11. Complete state boundary surfaces according to the Critical State Soil Mechanics (as defined in Roscoe et al. [29]; Schofield and Wroth [30] and Atkinson and Bransby [31]).

behavior of dry sand, it would be necessary to consider a “general” Critical State Soil Mechanics which would be a generalized version of the well-known theory including the effect of the acceleration amplitude of the vibration imposed to the sand mass. In the present paper, only the influence of the vibration on the volume change and on the shear strength of dry sand is considered. Nevertheless, as demonstrated in Barkan [15] and Youd [27], the shear modulus of dry sand is also reduced by vibration. As a conclusion, vibrations not only modify the failure criterion and the state boundary surface of dry sand but also its field of elastic strains. As a research perspective, the realization of additional experiments in deviatoric conditions will be not only justified by the need to build a new set of complete state boundary surfaces but also by the fact that the experimental results presented in this paper come from direct shear test devices. In this test configuration, the sand mass is not uniformly strained and most of the deformation occurs within a restrained shear band. Actually, the observation of Youd [27,28], highlighting the fact that the critical void ratio of the sand mass corresponds to its vibratory equilibrium void ratio, assumes that the void ratio was nearly homogeneous throughout the sample during the experiments. In his paper, Youd [27] introduced another interesting concept to consider the role of the vibrations with regard to the state of a sand mass: the “energy barriers”. According to his analysis, each phenomenon (densification or dilatation of the sample) corresponds to an amount of potential and frictional energies mobilized during the motion of the grains. The modification of the sand mass will occur only with a sufficient input of work into the system. Youd [27] then defines the energy barrier as the amount of external work required to cause the motion of sand grains by sliding from one stable position to another. Considering his approach, the role of the vibrations would be therefore to contribute to the reduction of these energy barriers and to promote the motion of the grains in the sand mass. Finally, such as demonstrated in the study of Kolmayer [23] for vertical vibrations, Youd [27] observed during his experiments that an increase of the normal pressure applied to the sand sample (the static surcharge) resulted in an increase of the shear strength for a given acceleration amplitude. On the basis of his experimental results, the following empirical relationship can be proposed to illustrate the decrease of the empirical factor ε of Eq. (7) when the static surcharge

6.1. Convective motion in vibrated granular matter As previously introduced in Section 2.2, granular convection phenomenon only arises when a granular material is vertically vibrated in a container with a sufficient energy or acceleration amplitude (Clément et al. [34], Gallas et al. [35], Grossman [36], Jaeger et al. [37,38], Knight et al. [39] and Lee [40]). As reported in Denies et al. [25], convective motion can be observed in dry sand when it is vertically vibrated in a cylindrical container with an acceleration amplitude larger than 1g. With increasing the acceleration beyond 1g, there is progressively an upward granular flow developing in the center of the container and a downward granular flow within a thin veneer along the sides (see Fig. 12). Grossman [36] has attributed this phenomenon to the inelastic frictional interactions between the grains and the walls of the container. Convective motion can be understood by following the sand grains– container interactions during their cycling history such as described in Denies [33] and Denies et al. [25]. During the upward movement of the sand grains subjected to vertical vibrations, the gravitational acceleration is opposed to the displacement of the sand grains. There are then few relative movements between the sand particles. Contrarily, during the downward movement of the container, there is a relative acceleration between the sand grains and the container walls (Γ−1)g. The sand grains are also momentarily separated from the bottom of the container. There are more relative movements between sand particles. Considering the study of Lee [40], it is possible to explain the 112

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such as density, inelasticity and friction can play an important role. The nature of the energy input, boundary conditions and interstitial air are also significant factors governing size-segregation. The presence of convection can enhance mixing or lead to size separation depending on the experimental conditions (Kudrolli [49]). Within the present context, considering vibrated dry sand, no size-segregation between sand grains was still observed in the experiments of Denies et al. [25] regardless of the encountered dynamic behavior (densification, instability surface or vibrofluid behaviors). After vibrations, similar grain size distribution was observed in the different parts of the vibrated sand mass independently of the dynamic behavior reached during the test. No size-segregation phenomena were observed by any authors in the other geotechnical studies reported in the previous Sections. Fig. 12. Illustration of the convective motion developing when a granular matter is vertically vibrated in a container increasing the acceleration amplitude of the vibrations beyond 1g.

6.3. Shearing of granular matter – dense granular flow regime Concerning the shearing behavior of granular material (without application of vibration to the sample), one can consider the study of dense granular flows. As highlighted by the GDR MiDi [50], the behavior of dense assemblies of dry grains submitted to continuous shear deformation has been the subject of many experimental investigations and DEM simulations (Brown and Richards [51], Duran [52], Herrmann et al. [53], Chevoir and Roux [54], da Cruz et al. [55], De Gennes [56], Savage [57] and Rajchenbach [58]). As explained in GDR MiDi [50], granular flows can be divided in three different regimes depending on the flow velocity. First, there is a “quasi-static” regime wherein the inertia of the particles is negligible. The granular material is then treated considering the quasi-static theories of the Critical State Soil Mechanics as introduced in Schofield and Wroth [30] (=typical soil mechanics approach). Secondly, there is the “gaseous” regime wherein the granular material is strongly agitated and the grains are far apart one from another. In this regime, particles interact through binary collisions and a kinetic theory close to the kinetic theory of gases can be used to describe the behavior of the particle flow (Campbell [59] and Goldhirsch [60]). Between these two regimes, there is a “dense flow regime” where the grain inertia and the contact network developing between the grains simultaneously play a role in the physics of the flow (Pouliquen and Chevoir [61]). In GDR MiDi [50], dense granular flows are studied considering different experimental set-ups. Within the framework of the present paper, only the plane shear configuration is regarded as it is close to the experimental conditions of the experiments encountered in the previous Sections focusing on the “pure” geotechnical review. In the plane shear configuration, the flow is obtained between two parallel rough walls (separated by a distance L) and moving at the relative velocity Vw. In his paper, GDR MiDi [50] clearly identifies the dimensionless parameters governing the dynamic behavior of the granular material once subjected to shearing. Apart from the contact laws, the results of the simulations depend on two parameters: the velocity of the upper wall imposing the shearing, Vw, and the normal stress, σ, or the density, ρ of the granular assembly. On the basis of this consideration, GDR MiDi [50] proposes a single dimensionless parameter describing the relative importance of inertia and confining stresses:

Fig. 13. Convective motion of the grains along the container sides during vertical vibrations. a) During relative upward motion of the grains relative to the side, b) during relative downward motion.

progressive downward motion of the sand grains along the walls of the container (see Fig. 12) considering the effect of the density change over the course of a cycle on the shear forces exerted by the container sides on the grains. During the upward motion of the sand grains relative to the sides of the container (see Fig. 13a), sand grains are closely packed and display high density. Contrarily, during the relative downward motion, sand mass expands to a lower density (see Fig. 13b). At higher densities, during the upward relative motion (Fig. 13a), the side of the container exerts a larger downward drag force on the particles than during the downward relative motion (Fig. 13b). That relative behavior results therefore in a downward shear force exerted by the container walls on the particles over the course of a cycle explaining the resulting downward motion of the grains along the sides of the container, such as illustrated in Fig. 12. The grey arrows in Fig. 13a and b represent the absolute displacements of the sand grains during each part of the cycle. Within the present context (shear strength degradation of vibrated dry sand), the understanding of the convection phenomenon is important because, in particular experimental configurations, it can take an active part in the onset of the vibrofluid behavior of dry sand which results in the degradation of its shear strength as explained in Section 3.2 of the present article.

I= 6.2. Granular size-segregation in vibrated granular matter

γwd σ /ρ

(11)

where γw is the mean shear rate (=Vw/L) and d is the grain diameter. It is to note that if slight differences exist in the scientific literature concerning the definition of the dimensionless ratio I, the meaning of this parameter still remain always the same: it expresses the relative importance of inertia with regard to the confining stresses developing inside the granular material. As it was already the case with the geotechnical studies, the shearing behavior of the granular material can be expressed considering the ratio of the shear stress, τ, to the pressure inside the material, σ,

Another topic related to the physics of vibrated granular matter is the granular size-segregation (Williams [41,42], Ahmad and Smalley [43], Rosato et al. [44], Knight et al. [45], Ottino and Khakhar [46], Hong et al. [47], Breu et al. [48] and Kudrolli [49]). As deeply explained in Kudrolli [49], large particles in a vibrated granular system normally rise to the top. However, they may also sink to the bottom or show other motion patterns, depending on the experimental conditions. While size ratio is a dominant factor, particle-specific properties 113

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0.25

which is actually the coefficient of internal friction, f, of the material. This coefficient is sometimes called the effective-friction coefficient by the physicists and they note this dimensionless ratio µeff. As observed in da Cruz et al. [55], when a granular material is sheared in a shear plane configuration without gravitational acceleration, the effectivefriction coefficient µeff increases approximately linearly with I, starting from a minimum value µeff,min and finally reaches a threshold value for I≥0.2:

0.2

Vertical vibrations

0.15 f = µeff (-)

(12)

where µeff, min≈0.25 and b≈1.1 (for inelastic frictional rigid grains characterized by a inter-particle friction coefficient of 0.4 and a restitution coefficient varying between 0.1 and 0.9). As underlined in da Cruz et al. [55], small values of I correspond to the “quasi-static” regime governed by the theory of the Critical State Soil Mechanics of Schofield and Wroth [30] while large values of I correspond to the “gaseous” collisional regime governed by the kinetic theory of gases. For intermediate values of I, corresponding to the dense granular flow regime, the effective-friction coefficient linearly increases with I, indicating a shear-rate-dependent regime. da Cruz et al. [55] propose boundary values between the three different regimes: the transition between the quasi-static regime and the dense granular flow regime is observed for a I-value close to 10-3 and the transition between the dense granular flow regime and the collisional flow regime (=gaseous regime) is observed for a I-value close to 10-1. For I-values larger than 0.2, the effective-friction coefficient µeff reaches a threshold value expressing the fact that, at that I-values, the density of the sample becomes too reduced to allow the development of an extra shear strength in the sample. In presence of I-values larger than 0.2, the behavior of the granular sample is governed by the collisions between the particles and not more by the confining pressure applied to the sample. Considering the result of the study of da Cruz et al. [55], it is tempting to conclude that the shear state of the granular material is only determined by a single dimensionless parameter, I, everywhere in the granular flow. This assumption, called the local-rheology assumption by GDR MiDi [50], is still related to the plane shear configuration and to the zero gravity conditions considered by the authors in their simulations to obtain linear velocities profiles and a uniform stress distribution in the sheared granular material. For more complex shearing configurations, a non-local rheology approach has to be considered (GDR MiDi [50]).

0.1

0.05 Without vibration 0 0.01

0.1 I (-)

1

0.25 I = 0.8 I = 0.4

0.2

0.15 f = µeff (-)

I = 0.23

Increase of I

f = μeff ≅ μeff , min + b I

Volume fraction 50% Interparticle friction coefficient = 0.5 Particle diameters randomly distributed between 0.9 and 1.1 mm

0.1

I = 0.15

I = 0.13 I = 0.09 I = 0.065

0.05

0 0

1

2

3

4

5

6 (-)

7

8

9

10

11

12

v

Fig. 14. Coupled influence of the dimensionless parameters Γ and I on the development of the coefficient of internal friction. Data from numerical simulations performed by Baran and Kondic [62] on 2000 spherical particles confined and sheared in an angular Couette cell.

the material is subjected to a gravitational field. In a second set of simulations, they considered the effect of vertical vibrations applied at the bottom of the shear cell. The authors analyzed the variation of µeff as a function of I varying the value of the dimensionless acceleration, Γ, in their simulations. As a result (see Fig. 14), they observed that the vertical vibrations significantly modify the value of µeff for small and intermediate values of I. The increase of µeff as I increases is slower as Γ simultaneously increases or in other words, the increase of µeff with I is mitigated by the application of vertical vibrations (larger than 1g). For similar values of I, smaller values of µeff were observed with increasing Γ. It has to be noted that this trend (mitigation of the increase of µeff with increasing I by the application of vertical vibrations) was only observed for values of Γ larger than 1 (corresponding to the aforementioned vibrofluid regime). For values of Γ < 1, similar increase of µeff with increasing I was observed than for the simulations conducted without oscillations confirming the existence of the transition between the densification regime and the vibrofluid regime as experimentally observed by Denies et al. [25] and Ermolaev and Senin [26] for dry sand. In addition, Baran and Kondic [62] confirmed the observation of L’Hermite and Tournon (see Section 4 of the present paper) postulating the existence of a shaking pressure. In their numerical simulations, the normal stress inside the material clearly decreases with increasing the value of Γ. As a conclusion, it is possible to postulate that the shearing behavior of vibrated dry sand can fully be understood considering the evolution of the two dimensionless parameters I and Γ. But that remains a postulate because the nature and the properties of the sand grains (their shape, their angularity…) differ from the typical spherical

6.4. Shearing of vibrated granular matter The urge to connect the dense granular flow regime and the vibrofluid behavior as discussed in the first part of the present article is large. Nevertheless, the definition of the two problems - dense granular shear flow (observed without vibration but with the application of a shear rate to the sample) and vibrated dry sand subjected to shearing – differs. A first connection is still established by Baran and Kondic [62] in their numerical study of the influence of the gravitational acceleration and external vibrations on the stresses developing in a granular material sheared in an angular Couette cell. In their threedimensional DEM simulations, a uniform granular material is sheared in an annular configuration and it is simultaneously subjected to the gravity and to vertical sinusoidal vibrations. This configuration is therefore close to the experimental conditions considered in the geotechnical review described in the previous Sections 2–5. In a first set of simulations, Baran and Kondic [62] studied the behavior of a dense granular material under a gravitational field and subjected to shearing without application of vibrations. For that conditions of simulation, they confirmed the existence of the relationship (12) proposed by da Cruz et al. [55]. As a result, the effectivefriction coefficient, µeff, increases approximately linearly with I even if 114

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sands and “vibrofluidization” phenomenon in dry sands seem to present common denominator. In saturated sand, the shear strength is degraded by the increase of the pore pressure while in dry conditions, the shear strength degradation is related to the increase of the shaking pressure. The urge to build an analogy between both types of processes is large. But in saturated conditions, the degradation of the granular skeleton is due to the increase of the pore pressure (because of the no volume change condition), while in dry conditions; the degradation of the granular skeleton is related to the application of vibrations which could lead to volume change. There is therefore a fundamental difference between the two processes but a strong connection is still made between them considering the results of Youd [27,28] who demonstrated that “vibrofluidization” can occur without volume change when the initial void ratio is also the critical void ratio. In the present paper, the role of the static normal pressure applied to the sample (also called the static surcharge) has also been considered. Regardless the direction of the vibration, the application of a static surcharge to the sand mass reinforces the intergranular force chains in the sample, increasing in turn internal friction forces or energy barriers according to the reasoning of Youd [27]. As a result, the increase of the normal pressure is opposed to the densification of the sand mass and to the reduction of the shear strength by vibration. The influence of the initial void ratio is also highlighted by Youd [28]. The coefficient of internal friction, f, increases with decreasing the initial void ratio due to the reinforced interlocking of the sand grains. The study of Youd [27,28] presents major conclusions with regard to the dynamic behavior of dry sand. First, Youd [28] observed that the critical void ratio (CVR) was also the vibratory equilibrium void ratio (EVR) and second, he showed that the two void ratios exponentially vary with the acceleration amplitude of the vibration, Γ. Such as discussed in Section 5, the evolution of the EVR-CVR line in function of the acceleration amplitude could have a huge impact on the Critical State Soil Mechanics with the definition of a “general” Critical State Soil Mechanics which would be a generalized version of the wellknown theory including the effect of the acceleration amplitude of the vibration imposed to the sand mass. A new set of complete state boundary surfaces in the (q′: p′: e) space could be build depending on the level of acceleration amplitude, Γ. In the Section 6, the authors comment several recent numerical studies from the field of granular matters. As a result of this review, it can be concluded that the dynamic behavior of a dry granular material, once simultaneously vibrated and sheared under the gravitational field, can be described by two distinct dimensionless parameters: Γ and I where I represents the relative importance of inertia with regard to the confining stresses inside the granular material. As demonstrated in da Cruz et al. [55], the transition between the different shearing regimes

particles used in the numerical simulations considered in the present Section. 7. Discussion and conclusions The present paper discusses the results of previous researches investigating the influence of vibrations on the shear strength of dry sand. In spite of the variety of all these studies, the important role of the acceleration amplitude on the behavior of the vibrated dry sand has been highlighted. The vibratory equilibrium void ratio (EVR), the critical void ratio (CVR) and the shear strength of vibrated dry sand (uttered with the help of f=tan ϕ), all vary exponentially with the dimensionless acceleration amplitude of the vibration, Γ. It is still necessary to nuance this conclusion especially with regard to the direction of the vibration which seems to lead to different dynamic behaviors in dry sand once vertically vibrated (see Sections 2.2 and 3.2). For horizontal vibrations, shear strength degradation is always observed independently of the value of Γh. For vertical vibrations, shear strength degradation seems to be initiated once Γv > 1. The results of all the experimental studies investigating the reduction of the shear strength of dry sand by vibration are summarized in Fig. 15. As highlighted in Eqs. (7) and (8), the shear strength of dry sand exponentially decreases with the acceleration amplitude. If similar trends are observed, the results of Mogami and Kubo [14] should be considered with caution not only with regard to the very weak normal pressure applied to the sample during shearing but also considering the fact that the normal load applied to the sample oscillated during the tests which is not the case for the other experimental works analyzed in this paper. Moreover, these results must also be compared with attention due to the different procedures followed by the aforesaid authors and regarding the field of vibration used for their experiments. Fig. 16 illustrates in a bi-logarithmic diagram (Velocity vs. Period) the domains of investigation covered by the experimental researches reviewed in this paper. Inclined axes of this diagram correspond to constant displacement or acceleration amplitudes. The advantage of this presentation is to provide alternate features of a harmonic motion. For the sake of information, researches concerning the volume change reported in Denies et al. [25] are also included in the figure. Considering the physics of the problem, the vibrations applied to the sand mass seem to play alternately the role of lubricant of the contacts between sand grains or can be seen as a “shaking pressure” counteracting the effect of the normal load applying on the sand mass. In this sense, a revisited intrinsic Coulomb failure criterion has been proposed (see Eq. (9)) to take into account the presence of this “shaking pressure”. Considering the expression of the intrinsic Coulomb failure criterion, liquefaction or cyclic mobility in saturated

Fig. 15. Influence of the acceleration amplitude on the internal friction coefficient of dry sand: summary of the results of previous experimental studies.

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Fig. 16. Domains of investigation of previous experimental studies characterizing the influence of vibrations on the volume change and the shear strength of dry sand.

the transition between the vibration states is governed by Γ. If this cross-over can be criticized (with regard to the reduced number of vibrated particles and the spherical shape of the grains in the numerical simulations), a first analyse has still allowed a rapprochement between the two worlds (geotechnical engineers and physicists) and certainly constitutes a first step for a deeper understanding of the dynamic behavior of vibrated dry sand.

(the quasi-static, the dense granular flow and the gaseous-collisional regimes) can be controlled considering the value of the parameter I. Moreover, during shearing, the coefficient of internal friction can be related to this parameter according to Eq. (12). f (or µeff) increases approximately linearly with I (in the dense granular flow regime), starting from a minimum value µeff,min (in the quasi-static regime) and finally reaches a threshold value (in the collisional regime). The results of Baran and Kondic [62] investigating the dynamic behavior of a granular material simultaneously subjected to shearing and to vertical vibrations under a gravitational field highlight the coupled influence of Γ and I on the shear strength of the granular material. The increase of f (or µeff) with increasing I is slower as Γ simultaneously increases. For similar values of I, smaller values of µeff were observed with increasing Γ. There is therefore a degradation of the shear strength with increasing Γ, as previously demonstrated in the geotechnical experiments for vibrated dry sand. Nevertheless, this degradation was only observed for values of Γ larger than 1 confirming the existence of the aforementioned vibrofluid regime. For values of Γ < 1, no significant influence of Γ was observed in the simulations. Crossing the results of experimental studies obtained from geotechnical engineers and the results of numerical simulations of physicists of the granular matters, it was possible to identify different regimes, respectively associated with the shearing (quasi-static, dense granular flow and gaseous regimes) and with the vibration state (densification, instability surface and vibrofluid regimes), governing the dynamic behavior of vibrated dry sand. The transition between the shearing regimes is controlled by the dimensionless parameter I and

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