Christophe Mobuchon

Pierre J Carreau

Marie-Claude Heuzey

Structural analysis of non-aqueous layered silicate suspensions subjected to shear flow

Non-aqueous layered silicate suspensions exhibit a complex rheological behavior due to a multiple length scale structure, which is sensitive to flow and flow history. In the present work, the nature of flow-induced non-equilibrium and metastable structures in non-aqueous layered silicate suspensions based on natural and organo-modified sodium montmorillonites was examined using rheometry and confocal laser scanning microscopy ͑CLSM͒. The scaling behavior of their linear and non-linear viscoelastic properties was investigated. Based on fractal scaling theories, the scaling laws of the solid-like properties were ascribed to the presence of space-filling percolating networks consisting of clusters with a mass-fractal dimensionality, D f ϳ 2. CLSM allowed us to detect the formation of aggregates under flow and to characterize their microscopic length scale. The shear-rate dependency of the microstructure characteristic length scale was attributed to a reversible shear-induced aggregation process. Upon cessation of flow, the observed thixotropic behavior of these suspensions was inferred from the CLSM observations to stem from local rearrangements at the nano-scale.

I. INTRODUCTION

Colloids and nanoparticles have found increasing uses as rheological modifiers in paints, inks, lubricating greases, cosmetics, and pharmaceutical formulations. More recently, layered silicates of the smectite group have attracted great interests as nanoscale fillers for reinforcement of polymers ͓polymer nanocomposites ͑PNCs͔͒ ͓Patel et al. ͑2006͔͒. Due to their particular nature, suspensions based on colloids and nanoparticles are quite complex since they undergo a fluid-to-solid transition with the formation of disordered solids defined by their dynamics and elasticity ͓Trappe and Sandkuhler ͑2004͔͒.

For attractive colloidal particles, Trappe et al. ͑2001͒ recently proposed a phase diagram that is a function of the particle volume fraction, energy of interparticle attraction, and applied stress. The non-equilibrium solid states of this diagram include two extremes: hard sphere colloidal glasses at high volume fraction, , and low attractive potential, U, and strongly attractive colloidal gels at low and high U. For the limiting case of hard-sphere colloidal particles ͑for which the interaction potential between spherical par-a͒ Author to whom correspondence should be addressed; electronic mail: pierre.carreau@polymtl.ca 1 ticles is determined solely by the excluded volume͒, the suspensions undergo a liquidto-glass transition at a volume fraction g ϳ 0.58 due to the crowding of single particles. The solid-like properties of colloidal glasses stem from the permanent kinetic trapping of the particles within cages formed by their nearest neighbors ͓Trappe and Sandkuhler ͑2004͔͒. The resistance to configuration changes of these cages provides a mechanism for energy storage ͑and viscous dissipation͒, which gives rise to a low-frequency plateau storage modulus ͓Crassous et al. ͑2005͒; Helgeson et al. ͑2007͒; Mason and Weitz ͑1995͔͒. Models based on the mode coupling theory predict a low-frequency plateau storage modulus ͓Mason and Weitz ͑1995͔͒, G 0 Ј, scaling with the volume fraction, , as G 0 Ј͑͒ ␣͑͒ n with n Ϸ 0.3-0.5 ͓Götze ͑1991͒; Zaccarelli et al. ͑2001͔͒. On the other hand, at low volume fractions, attractive spherical particles aggregate as they form self-similar clusters with a mass, M, scaling with their average size, c , as M͑ c ͒␣M 0 ͑ c / a ef f ͒ D f , where D f is the mass-fractal dimensionality, a ef f is a characteristic size, and M 0 is the mass of the cluster compact building blocks. Aggregation of clusters ultimately leads to a space-filling percolating network exhibiting macroscopic nonergodicity ͑i.e., breakdown in the equivalence of time and space averages͒ and, therefore, to a sol-to-gel transition. The solid-like behavior of colloidal gels originates from the stress bearing network made of close packed clusters. The dependency of the network elasticity on the volume fraction was shown to be stronger than predicted for colloidal glasses, with G 0 Ј͑͒ ␣͑͒ n and n Ϸ 4 ͑spherical colloidal particles͒ ͓Buscall et al. ͑1988͒; Khan and Zoeller ͑1993͒; Piau et al. ͑1999͒;Saint-Michel et al. ͑2003͒;Shih et al. ͑1990͒;Yziquel et al. ͑1999͔͒. Scaling models linking the fractal microscopic structure parameters to the macroscopic elastic properties of gels are presented in detail in the next section. Despite the structural differences between colloidal glasses and gels, their solidlike properties are similar in nature. Both result from the physical jamming or crowding of the characteristic particles within the suspensions: single particles in the case of glasses and clusters in the case of gels ͓Segrè et al. ͑2001͔͒.

Between the two limiting cases of glasses and gels, the structure and elasticity of colloidal suspensions in non-equilibrium solid states remain to be investigated. If suspensions of spherical colloidal particles have attracted a large amount of interest, attractive colloidal suspensions of anisotropic particles have been the subject of fewer investigations. The liquid-to-glass transition of suspensions containing anisotropic colloidal particles is expected to occur at volume fraction much lower than their maximum packing fraction since the excluded volume of anisotropic particles is larger than their actual volume ͓Jogun and Zukoski ͑1996, 1999͔͒. The particle anisotropy may also lead to differences associated with their orientation and consequently to a more complex picture of the non-equilibrium solid states ͓Lyatskaya and Balazs ͑1998͔͒.

Layered silicate particles are highly anisotropic. The most commonly used layered silicate is montmorillonite, in which the individual layers are about 1.2 nm thick, 320-400 nm long, and 250 nm wide for the larger ones ͓Cadene et al. ͑2005͔͒. Aqueous layered silicate suspensions undergo phase transitions controlled by repulsive or attractive interactions dependent on ionic strength and pH. The establishment of their state diagram has gradually emerged in recent years ͓Kroon et al. ͑1998͒; Michot et al. ͑2004͒;Mourchid et al. ͑1995͒;Pignon et al. ͑1996, 1997aPignon et al. ͑1996, , 1997b, 1998͒;, 1998͒;Ruzicka et al. ͑2006͒;Shalkevich et al. ͑2007͒;Tanaka et al. ͑2004, 2005͒;Tombacz and Szekeres ͑2004͔͒. With increasing ionic strength, clay layers become attractive and the state diagram is expected to gradually follow the jamming phase diagram for attractive particles proposed by Trappe and Sandkuhler ͑2004͒. In the same way, non-aqueous layered silicate suspensions belong to attractive colloidal suspensions as the repulsive electrostatic interactions are negligible and, therefore, should exhibit non-equilibrium fluid-to-solid transitions of the jamming phase diagram. Note that due to the high aspect ratio of the clay layers, the liquid-to-glass transition is not restricted to high volume fractions and is expected at low volume fractions where gelation can occur ͓see developments of King et al. ͑2007͒ andRen et al. ͑2000͒ for the calculation of g ͔. Actually, a fluid-to-solid transition has been observed for polymer layered silicate nanocomposites with the development of a lowfrequency plateau storage modulus increasing with volume fraction ͓see for instances Aubry et In the current paper, we investigate the nature of the non-equilibrium and metastable solid states of polar and non-polar non-aqueous layered silicate suspensions. At first, their viscoelastic properties are reported and their scaling behavior is explored according to both the volume fraction and the temperature ͑interaction energy͒. Afterwards, the rheological scaling laws observed are correlated with the clay microstructure. Confocal laser scanning microscopy ͑CLSM͒ is also used to verify the existence of a space-filling network and the observed length scales are compared to the predictions of fractal scaling theories. Finally, the flow history dependency of layered silicate suspensions reported in a previous publication ͓Mobuchon et al. ͑2007͔͒ is explained by the investigation of the network evolution under flow and at rest.

II. THEORETICAL BACKGROUND

Fractal scaling theories

In the framework of fractal scaling theories ͓Bremer et al

= a ef f 1/͑D f -d͒ , 1 Յ D f Յ 3, ͑1͒
where d is the Euclidian dimension of the system and a ef f is the characteristic size of the cluster compact building blocks. The cluster mass-fractal dimensionality, D f , can be interpreted as a measure of the compactness of the clusters and depends on the mechanism of particle aggregation. A value of 3 for D f corresponds to uniform dense ͑non-fractal͒ clusters, while diffusion-limited aggregation and reaction-limited cluster aggregation lead to values of 1.8 and 2.1, respectively ͓Bushell et al. ͑2002͔͒. As self-similarity is restricted to length scales below c , two levels of structure can be distinguished: fractal intra-microstructure of the clusters and inter-microstructure characterized by the cluster interactions.

Recently, Wu and Morbidelli ͑2001͒ developed a scaling model, based on the work of Shih et al. ͑1990͒, to relate the microscopic structural parameters to the macroscopic elastic properties. In their model, the intra-and inter-microscopic contributions to elasticity are considered in series to give the overall microscopic elasticity,

K ef f , 1/K ef f = 1/K c + 1/K l ,

͑2͒

where K c and K l are the intra-and inter-microscopic elastic constants, respectively. According to Shih et al. ͑1990͒, the elastic properties of a cluster are dominated by its backbone, which is approximated as a linear chain of springs with a fractal dimension, 1 Ͻ x Ͻ D f , different from that of the cluster. The intra-microscopic elastic constant K c can then be related to c using the following Khantor-Webman relation ͓Kantor and Webman ͑1984͒; Krall and Weitz ͑1998͔͒:

K c = K 0 ͑a ef f / c ͒ 2+x , ͑3͒
where K 0 is the spring constant of a bond between two cluster compact building blocks.

On the other hand, the inter-microscopic elastic constant, K l , refers to the interactions between the clusters and, therefore, is independent of the intra-microscopic structural parameters. The macroscopic elastic modulus, GЈ, can be related to K ef f as

GЈ = K ef f c = 1 c K c 1 + ͑K c /K l ͒ Х 1 c K c ͩ K l K c ͪ ␣ , ͑4͒
where ␣ is a constant in the range of ͓0, 1͔, which depends on the given range of K c / K l . The dependence of GЈ on is then derived from Eq. ͑5͒,

GЈ Х K a ef f n=␤/d-D f with ␤ = ͑d -2͒ + ͑2 + x͒͑1 -␣͒ and K = K l ␣ K 0 1-␣ . ͑5͒
This scaling model was also extended for the strain limit of linearity, ␥ 0 , following the original approach of Shih et al. ͑1990͒,

␥ 0 ϰ B with B = ͑d -␤ -1͒/͑d -D f ͒. ͑6͒
In the case of the strong-link regime, where links between the neighboring clusters are stronger than the intra-microscopic links within the clusters, ␣ = 0 and Eq. ͑6͒ predicts that the limit of linearity decreases with increasing . In contrast, ␥ 0 increases with increasing for the limiting condition of weak-link regime ͑␣ =1͒ for which the breaking of the network occurs at the intercluster links. Finally, the two preceding scaling laws ͓Eqs. ͑5͒ and ͑6͔͒ lead to the following scaling law for the yield stress, 0 : 0 = GЈ␥ 0 ϰ t=͑d-1͒/͑d-D f ͒ . ͑7͒

III. MATERIALS AND METHODS

A. Materials and preparation methods

Polar suspensions

The polar system used in this study consists of a natural layered silicate, a sodium montmorillonite, M x ͓Al 2-y Mg y ͔Si 4 O 10 ͓OH͔ 2 •nH 2 O, ͑Cloisite Na + , Southern Clay products, = 2.86 g / mL͒, dispersed in a Newtonian polyalkylene glycol, HO͓C 2 H 4 O͔ x ͓C 3 H 6 O͔ y -͓C 2 H 4 O͔ x H, ͑Plurasafe WT 90000, BASF͒ of molar mass around 1500 g/mol, viscosity of 59 Pa s and density of 1.09 g/mL at 25 °C. Clay volume fractions used are comprised between 0.001 and 0.005. The colloidal individual montmorillonite layers can be described as elliptical platelets 1.2 nm thick with lateral dimensions of 0.25-0.4 m ͓Cadene et al. ͑2005͔͒. The clay was added to the polyalkylene glycol via solution mixing as follows. First, natural clay platelets were exfoliated in de-ionized water using an ultrasound probe ͑Sonics & Material͒ ͓Ho et al. ͑2001͔͒. The pH was adjusted to approximately 9.5 in order to avoid dissolution of the silicate ͓Thompson and Butterworth ͑1992͔͒. Polyalkylene glycol was then added to the aqueous clay suspension and the entire mixture exposed to ultrasonication before extraction of the de-ionized water in a high vacuum oven at 40 °C. For the clay volume fractions used, the resulting suspensions were homogeneous translucid gels, stable over several months. Other methods aiming at directly incorporating the nanoclay in polyalkylene glycol, based on both sonication and high shear mixing, were observed not to induce gel formation. The mechanical energy involved in these two methods, conjugated with the good affinity between the clay and the polyalkylene glycol, does not suffice to overcome the energy of adhesion between the clay layers and hence hinders the polyalkylene glycol intercalation ͓Bousmina ͑2006͔͒.

For the confocal laser scanning microscopic studies described below, the natural clay was tagged with Safranine-O, C 20 H 19 ClN 4 , ͑Sigma-Aldrich͒, a fluorescent cationic dye, taking advantage of its cation exchange capacity ͑CEC͒. Fluorescent suspensions were prepared using the same procedure described above with the addition of an appropriate amount of Safranine-O ͓equivalent to 5% of the clay CEC ͑92.6 mmol/100 g clay͔͒ in de-ionized water. The low amount of adsorbed Safranine-O on the clay surface is later shown not to affect noticeably the rheological properties of the suspensions.

Non-polar suspensions

The non-polar system is based on a sodium montmorillonite ͑Cloisite ® 15A, Southern Clay products͒ organically modified with a dimethyl dihydrogenated quaternary ammonium salt, CH 3 CH 3 N + HTHT, where HT refers to the hydrogenated tallow of various carbon chain lengths: C 18 ͑65%͒, C 16 ͑30%͒, and C 14 ͑5%͒. The longest C 18 chains have an estimated length of 2.67 nm ͓Ho et al. ͑2001͔͒. A second organomodified montmorillonite ͑Cloisite ® 30B, Southern Clay products͒ with a methyl dihydroxyethyl, quaternary ammonium salt, CH 3 N + ͓CH 2 OH͔ 2 , was also used for comparison. The non-polar suspending fluid was a mixture of two Newtonian polybutenes, R͓CH 2 CCH 3 CH 3 ͔ x ͓CH 2 ͔ y R, of different molar mass, respectively, 910 g/mol ͑76.4 mass %, Indopol H100, BP͒ and 1300 g/mol ͑23.6 mass %, Indopol H300, BP͒ with an overall viscosity of 28.5 Pa s and a density of 0.89 g/mL at 25 °C. Clay volume fractions used were comprised between 0.0075 and 0.040. The non-polar suspensions were prepared following two different procedures, both resulting in stable homogeneous translucid gels. The first procedure consisted of incorporating directly the organomodified montmorillonite by ultrasonication. Details about the procedure and characterization of the corresponding dispersions are reported elsewhere ͓Mobuchon et al. ͑2007͔͒. The second procedure via solution mixing from toluene is equivalent to the procedure described before for the polar suspension. Toluene was selected, since the organomodified montmorillonite can be dispersed in it to yield a transparent gel ͓Ho et al. ͑2001͔͒. The fluorescent organoclay used for the CLSM studies was prepared separately from its suspensions. Safranine-O was dissolved in ethanol/de-ionized water ͑1:1͒ solution and stirred overnight after the addition of organoclay. The treated organoclay was filtered and washed with a hot solution of ethanol in de-ionized water ͑1:1 at first and then 4:1͒ until the filtrate became colorless. The fluorescent organoclay was then dried in a vacuum oven overnight and pulverized into fine powder. The non-polar suspensions based on the resulting fluorescent organoclay were prepared via the previously described ultrasonication and solution mixing methods.

B. Rheometry

The rheological characterization of the suspensions was performed using a stresscontrolled rheometer ͑Anton-Paar, Physica MCR 501͒. A cone-and-plate geometry with a radius of 50 mm and a cone angle of 0.04 rad was used for the tests performed at 25 °C, while a parallel plate geometry of 50 mm radius was employed for the investigation of the temperature effect. Actually, the true gap option of the MCR parallel plate geometry allows to monitor the gap and to keep it constant while the temperature of the instrument is changing. The steady-shear viscosity was measured by applying stepwise shear ratedecreasing ramps from 25 to 10 -3 s -1 . Linear viscoelastic measurements were conducted at a fixed strain of 0.005, which is below the limits of linearity, ␥ 0 , for all suspensions. The limits of linearity were determined, at a frequency of 6.28 rad s -1 , as the strain at which GЈ deviates from its plateau value by 10%. To ensure controlled initial microstructures ͓Mobuchon et al. ͑2007͔͒, all the dynamic measurements were preceded by a preshearing at a defined shear rate ␥ ˙i until steady state was reached and then followed by a rest time of 5400 s. For the stepwise shear rate measurements, data were considered after a period of 0.04 s, time required to reach the specified shear rate within an error of less than 2%. The standard deviation on the rheological data was estimated to be less than Ϯ9%. Investigations using various gaps and different roughness values for the parallel plates ruled out slip or wall effects for the two types of suspensions. However, flow instabilities with sample fracture localized in the middle of the gap were observed for clay volume fractions above 0.005 and 0.03 for the polar and non-polar suspensions, respectively, prepared by solution mixing ͑this phenomenon was never observed for the non-polar suspensions prepared by ultrasonication͒. The fracture was observed independently of the gap used and for the entire shear rate range investigated ͑from 25 to 10 -3 s -1 ͒. At low shear rates, the occurrence of fracture was inferred from the measured stresses, which were smaller than the value for the suspending fluid alone. The identification of the mechanisms leading to sample fracture is beyond the scope of this paper, but one may argue that the high elasticity of these suspensions drives this phenomenon. Hence, all results reported here for samples prepared by solution mixing are for volume fractions smaller than the critical values for sample fracture. For all the dynamic data reported in this paper, the standard deviation was estimated to be within Ϯ5%.

C. Optical measurements

CLSM

The micro-scale structures of the clay suspensions were directly observed using a confocal laser scanning microscope ͑Zeiss, Axioplan 2 & LSM 510 META͒ equipped with a commercial optical shear cell ͑Linkam Scientific Instruments, CSS450͒. The homogeneity of the imposed shear flow through the gap was established with a long working distance lens ͑Leica, 20ϫ / 0.33 numerical aperture, NA͒. A higher resolution oil immersion 40ϫ / 1.3 NA lens of 0.2 mm working distance ͑Carl-Zeiss, Plan-Neofluar͒ and a water immersion 40ϫ / 1.2 NA lens of 0.29 mm working distance ͑Carl-Zeiss, C-Apochromat͒ were used to observe the suspension microstructures. The original top plate of the shear cell is not compatible with a low working distance lens due to the thickness of its fixed parallel quartz disk. It was therefore necessary to replace it by a homemade lid with a glass cover of 160 m depth that acted as the fixed parallel plate. The Safranine-O was excited at 488 nm and its fluorescence was detected at wavelengths above 560 nm through a filter system. The lateral and focal plane resolutions were estimated to be about 0.17 ͑Ϸ0.4 / NA͒ and 0.46͑Ϸ1.4 / n.a. 2 ͒m ͓Pawley ͑2006͔͒, respectively. The spatial resolution of the images was chosen above the lateral resolution of the CLSM and the dynamic range of their intensity was 8 bits. The time resolution of the CLSM turned out to be enough to acquire images under flow at shear rates up to 0.1 s -1 and during relaxation. The real-space images were quantitatively analyzed to extract their characteristic length scale and to assess their isotropy ͓Kaufman et al. ͑2005͒; Lin et al. ͑2007͔͒. The image processing consisted first in a binarization of gray scale images with the threshold defined by their mean intensity and pixels with intensity above the threshold were considered to be part of the clay particles. A two-dimensional median filtering using two-by-two neighborhoods was then applied to remove "salt and pepper" noise ͑isolated pixels͒. The distances, , between nearest neighbors of pixels above the threshold were then determined within each row ͑x͒ and column ͑y͒. Their distribution, P͑͒, along these directions follows an exponential decay: P͑͒ ϰ exp͕-2 / 0 ͖, where 0 represents an average measure of the spatial correlation at the micro-scale. The same row and column distributions, P͑͒, and density distribution are expected in the case of an isotropic image. Finally, the ratio of the number of pixels above the threshold with the total number of pixels was used to estimate the amount, A, of surface occupied by the clay particles.

IV. RESULTS AND DISCUSSION

A. Flow induced non-equilibrium and metastable structures

In non-aqueous systems, the interaction potential between clay platelets includes short-range van der Waals attraction and forces induced by polymer chains and excluded volume. The potential of the van der Waals forces can be quantified by the Hamaker constant A H , which is a function of temperature and differences in refractive index and dielectric constants of the two phases. In the case of the non-polar suspensions, a steric barrier to the van der Waals forces is expected due to the compatibility between the surfactant hydrocarbon tails and the suspending fluid. A steric barrier is also expected in polar suspensions with the adsorption, during the solution mixing step, of the polyalkylene glycol chains on the clay surface. Depending on the extent of steric stabilization, the suspensions can belong to hard solid dispersions, where the van der Waals interactions are negligible relative to the Brownian motion and in other cases, to the weakly or strongly aggregated dispersions. Nevertheless, it will be shown later that the behavior and structure of both types of suspensions are fairly similar in nature, but for on a quite different range of clay concentration.

As previously reported ͓Mobuchon et al. ͑2007͔͒, the linear viscoelastic properties of layered silicate suspensions are strongly correlated with the flow history. This behavior is illustrated in Fig. 1͑a͒ for the non-polar suspension prepared by ultrasonication for = 0.01 and in Fig. 1͑b͒ for the polar suspension for = 0.003 ͑ = 126.5 rad s -1 ͒. Upon cessation of the steady-shear flow, the storage modulus, GЈ, is shown to grow from an initial value G i Ј to a time invariant value G ϱ Ј following an exponential form adapted by

GЈ͑t͒ = G i Ј + ͑G ϱ Ј -G i Ј͒ ͩ1-expͫ-ͩ t ͪ m ͬͪ ,

͑8͒

with m the stretching exponent and the characteristic time. The values of G i Ј and G ϱ Ј resulting from the non-polar suspension data of Fig. 1͑a͒ are shown in Fig. 2͑a͒ ͑unfilled and filled circles, respectively͒ to significantly decrease from high to low plateau values for increasing pre-shearing rate, ␥ ˙i. Therefore the non-equilibrium structures under flow, characterized by G i Ј, evolve at rest toward different metastable structures with larger elastic modulus G ϱ Ј values. In a similar fashion, the linear viscoelastic properties of the polar suspension ͓Fig. 2͑b͔͒ are sensitive to the flow history, but differ in the extent of the structural recovery that tends to vanish as the pre-shearing amplitude decreases. It is noteworthy to mention that a similar flow history dependency was also observed for organo modified suspensions at higher volume fraction ͑0.04, ͓Mobuchon et al. ͑2007͔͒͒ and higher temperature ͑40 °C, results not shown͒ and prepared by ultrasonication with synthetic layer silicates of different aspect ratios, p, ͑results not shown͒: a hectorite with p Ϸ 50 ͑Lucentite SANS, CBC Co.͒ and a fluoromica with p Յ 6000 ͑Somasif MAE, CBC Co.͒. This behavior appears to be a universal feature of strongly attractive colloidal suspensions.

As a consequence of the flow history dependency, the polar and non-polar suspensions undergo a fluid-to-solid-like transition due to pre-shearing. The transition is depicted in Fig. 3 for the polar suspension ͑ = 0.003͒ where its storage modulus as a function of frequency is reported after different initial pre-shear rates ␥ ˙i. The low frequency elastic modulus measured after a 5400 s rest time is actually shown to evolve from a liquid-like behavior for high pre-shearing rates toward a non-terminal flow behavior ͑i.e., nearly independent of ͒ as the initial pre-shear rate is decreased.

B. Scaling behavior with volume fraction

Figure 4 shows the frequency dependence of the elastic modulus at various volume fractions, , ranging from 0.0075 to 0.040 for the non-polar suspensions prepared by ultrasonication. Prior to all measurements, the samples were systematically pre-sheared at a low pre-shearing rate ͑␥ ˙i =10 -3 s -1 ͒ and then allowed to rest during 5400 s to ensure similar flow history. As the volume fraction increases, the elastic modulus becomes significantly larger and displays a wider plateau at low frequencies, characterized by G 0 Ј. The dependence of the plateau value, G 0 Ј, on will be discussed further in relation to the nature of the clay network. An excellent superposition of the linear viscoelastic properties with the volume fraction is obtained based on the scaling factor as shown in Fig. 5. is defined as the ratio of G 0 Ј and a reference modulus, G 0r Ј , corresponding to a volume fraction, ref , arbitrarily chosen to be 0.020. This scaling is similar to that reported by Hobbie and Fry ͑2007͒ for concentrated nanotube suspensions. It is also in agreement with the universal scaling behavior of weakly attractive particles discussed by Trappe and Weitz ͑2000; 2001͒. Figure 5 shows that the scaling behavior applies for all data obtained under various pre-shearing rates, preparation methods, and different suspensions. Rather surprisingly, the reduced storage modulus of the polar suspensions shows a similar behavior than that of the non-polar suspensions, while the reference volume fraction is nearly ten times smaller ͑ ref = 0.003͒. Note that for the fluorescent suspensions used in CLSM, the low-frequency data of GЈ are unaffected by the presence of Safranine-O ͑semi-filled symbols in Fig. 5͒, suggesting that it does not perturb the interaction potential between the clay platelets.

Similarly, a scaling is proposed for the steady-state data ͑stress vs. shear rate͒, suggested by the Bingham equation and based on the apparent yield stress. Actually, the behavior of the layered silicate suspensions is typical of a yield stress material, as described by the Bingham equation: = 0 + 0 ␥ ˙, where , 0 , and 0 are the steady stress, the apparent yield stress, and the high-shear rate viscosity of the suspension, respectively. The shift factor, = 0 / 0͉ ref , where 0͉ ref is the apparent yield stress for the reference volume fraction, is applied to reduce the steady stress and the shear rate ␥ ˙. Figure 6 shows that the scaling effectively reduces the data of the non-polar suspensions prepared FIG. 7. ͑a͒ Variations of G 0 Ј ͑circles: ␥ ˙i =10 -3 s -1 and diamonds: ␥ ˙i = 0.3 s -1 ͒ and of 0 ͑triangles͒ as functions of the volume fraction ; ͑b͒ variation of ␥ 0 with for the non-polar suspensions prepared by ultrasonication ͑unfilled symbols͒ and the polar dispersion ͑filled symbols͒. by ultrasonication and by solution mixing in single master curves ͑ ref = 0.020͒. On the other hand, the yield stress for the polar suspensions was not well defined and it was deduced from the scaling procedure itself. A similar scaling behavior was observed for concentrated nanotube suspensions by Hobbie and Fry ͑2007͒ and is expected for other Bingham fluids ͓Trappe and Weitz ͑2000͔͒. The values of the plateau ͑low frequency͒ storage modulus G 0 Ј and of the yield stresses 0 used to calculate the shift factors, and , are plotted in Fig. 7͑a͒ as functions of the clay volume fraction, while Fig. 7͑b͒ presents the limit of linearity ␥ 0 of the two suspensions. The similarity within the behavior of these suspensions is discussed in the scope of the microstructure in Sec. IV.

C. Scaling behavior with temperature

The significant effect of temperature, T, on the non-equilibrium structures of the non-polar suspension prepared by ultrasonication ͑ = 0.040͒ and the polar suspension ͑ = 0.003͒ is illustrated in Fig. 8. The stress, , reduced by the contribution of the suspending fluid, m , is plotted at different temperatures as a function of the hydrodynamic stress, ␥ ˙, where is the viscosity of the suspension at high shear rate. Since the suspending fluids are Newtonian, the reduced stresses correspond to the contributions of the clay structures. The hydrodynamic contributions vary considerably over the temperature range covered experimentally. This is principally due to the viscosity changes of the suspending fluids with temperature ͑multiplied by ϳ8 and ϳ5 for the non-polar and polar suspensions, respectively, when the temperature is decreased from 65 to 15 °C͒. The non-superposition of the reduced stresses brings out the strong temperature dependency of the interaction potential between the clay platelets. In the same manner as that for the volume fraction, the shift factor based on yield stress allows to superpose the flow curves at different temperatures ͑filled symbols in Fig. 8 with T ref = 40 and 65 °C for the non-polar and polar suspensions, respectively͒. It will be shown below that, remarkably, the shift factors of the non-polar and polar suspensions follow the same exponential decay dependency with temperature.

To investigate the effect of temperature on the metastable structures, the frequency dependency of the elastic modulus was measured in the same temperature range as the stress measurements and the results are reported in Fig. 9. All the dynamic measurements were preceded by a pre-shearing followed by a 5400 rest time. For the non-polar suspension, two pre-shearing amplitudes were tested with initial hydrodynamic stresses, ␥ ˙i, equal to 0.95 and 95 Pa, while a pre-shearing at ␥ ˙i Ͻ 0.1 Pa with ␥ ˙i =10 -3 s -1 was used for the polar suspension. The variation of GЈ induced by the temperature dependency of the suspending fluid was taken into account by applying the time-temperature superposition. For that reason, the data were horizontally shifted with the respective factors a T of the suspending fluids, relative to the reference temperature, T ref , of 25 °C. The nonsuperposition of the plateau storage modulus, G 0 Ј, at low frequencies shown in Fig. 9͑a͒ for both suspensions reflects structure differences with temperature. However, in Fig. equilibrium structures ͑used in Fig. 8͒. Both shift factors are shown to follow the same exponential decay dependency with temperature within measurement accuracy, but they are not equal for all conditions investigated. Nevertheless, the temperature dependencies of the polar and non-polar suspensions as well as the temperature dependencies of the equilibrium and non-equilibrium structures are similar.

D. Correlation between rheological properties and microstructure

The power-law dependencies of G 0 Ј and of the apparent yield stress, 0 , on the volume fraction depicted in Fig. 7͑a͒ suggest that the microstructures of the polar and non-polar suspensions are fractal in nature. The scaling exponents n for G 0 Ј and t for 0 of the two suspensions do not differ much within their determination accuracy ͑see Table I͒. Note that the two different pre-shearing conditions ͑low and high ␥ ˙i͒ lead to similar exponents n. The limit of linearity of ␥ 0 for the two suspensions is also shown to follow a power-law relationship with the volume fraction ͓Fig. 7͑b͔͒ with a negative exponent B ͑Table I͒. As mentioned earlier, the limit of linearity was established according to the variation of GЈ with strain. Above ␥ 0 , while GЈ decreased continuously, GЉ exhibited a maximum ͑results not shown͒. Similar behaviors were reported by Yziquel et al. ͑1999͒ for fumed silica suspensions and by Page et al. ͑2002͒ for coating colors. The maximum ͑overshoot͒ was observed to strongly decrease with increasing frequency and decreasing volume fraction. It allows classifying the nonlinear viscoelastic behavior of these suspen- sions to type III or weak strain overshoot, according to Hyun et al. ͑2002͒. From the fractal scaling theory of Shih et al. ͑1990͒, a decrease of ␥ 0 with increasing volume fraction indicates that the elasticity of the system is ruled by the intra-microscopic elasticity K c , which is lower than the inter-microscopic elasticity K l . However, in the case of our suspensions, the volume fraction dependency of ␥ 0 is weak, which suggests mutual contributions of the two levels of elasticity. Actually, the theory of Shih et al. ͑1990͒, which only considers K c in the strong-link regime, fails by predicting an unphysical negative value for the fractal dimension x of the clusters in the case of our suspensions.

On the other hand, the scaling model of Wu and Morbidelli ͑2001͒ takes into account the two levels of elasticity with a parameter ␣ ranging from 0 to 1 for the strong-link and the opposite weak-link regimes, respectively. The determined power-law exponents n and B lead for the two types of suspensions to ␣ տ 0.4 if a realistic value of the fractal dimension x is considered ͑in the range of 1-1.3 for a colloidal gel͒. Consequently, the two types of suspensions belong to the transition regime for which both levels of elasticity contribute significantly to the overall elasticity. The values of the fractal dimensionalities D f for the two types of suspensions derived from Eqs. ͑5͒ and ͑6͒ are close to 2 ͑see Table I͒. This supports the fact that their structures are similar. They are also in good agreement with the predictions only based on yield stress ͓Eq. ͑7͒ and Table I͔.

Real space observations of the microstructures of the non-polar and polar suspensions were done using CLSM. Figure 11 shows microstructural images from CLSM when no pre-shearing, except a slight squeezing flow to insert the sample within the CLSM holder gap, was imposed prior to their acquisition. The areas rich in fluorescent layered silicate appear bright in the micrographs. The dense regions in layered silicate within the nonpolar suspensions prepared either by ultrasonication ͓Fig. 11͑a͒, = 0.020͔ or by solution mixing ͓Fig. 11͑b͒, = 0.020͔, and the polar suspension ͓Fig. 11͑e͒, = 0.003͔, form a continuous network with an overall appearance similar to those reported for polymer gels ͓Bremer et al. ͑1993͒; Crompton et al. ͑2005͒;Firoozmand et al. ͑2007͒;Hirokawa et al. ͑1999͔͒. The space-filling networks are consistent with the fractal structures inferred previously. At the micro-scale, these networks were shown to be static over the time scale of observation ͑ϳ60 min͒, indicating the dominance of prevailing attraction forces over Brownian forces. It is also noteworthy to mention that the non-polar suspension based on the more polar organomodified montmorillonite ͑Cloisite 30B͒ did not exhibit such networks ͓Fig. 11͑d͔͒. For a volume fraction of = 0.020, the microstructure of the nonpolar suspension prepared by ultrasonication ͓Fig. 11͑a͔͒ is shown to contain some large agglomerates in contrast to the suspension prepared by solution mixing ͓Fig. 11͑b͔͒. The clay platelets contained within these agglomerates do not contribute to the percolation of the network, which explains the weaker rheological properties reported previously for this suspension ͑Figs. 5 and 6͒. By considering Eq. ͑5͒ with the low plateau value of the storage modulus G 0 Ј given in Fig. 5 ͑with ␥ ˙i =10 -3 s -1 ͒ and assuming the same fractal dimensionality D f and similar ␤ ͑Table I͒ for both suspensions, the network induced by ultrasonication is expected to include only approximately one half of the total clay volume fraction. Consequently, and due to its fractal nature, it should have a characteristic length scale, 0 , larger than that of the network prepared by solution mixing ͑Eq. ͑1͒, assuming 0 is proportional to c ͓Gefen et al. ͑1981͔͒͒. Its coarser microstructure observed by CLSM ͓Fig. 11͑a͒ vs. Fig. 11͑b͔͒ qualitatively confirmed this trend, but this should be considered with care since the two preparation methods involve different flow histories. In the next paragraph, the microstructure characteristic size 0 is actually observed to depend on the pre-shearing rate. Similarly, a more open network is seen when the volume fraction of the non-polar suspension is decreased by half to 0.010 ͓Fig. 11͑c͒ compared to Fig. 11͑b͔͒. In contrast, the polar suspension exhibits the same behavior when the volume fraction is decreased from 0.003 ͓Fig. 11͑e͔͒ to 0.0015 ͓Fig. 11͑f͔͒. Figure 12 presents the distance distribution function, P͑͒, used to quantify the network characteristic length scale 0 of the CLSM images of Fig. 11. The observable isotropy of the quiescent microstructures is confirmed by the similarity of the distributions along the x and y directions. The corresponding characteristic length scales, 0 ͓shown in the legends of Figs. 12͑a͒ and12͑b͔͒, are in agreement with the main concept of fractal scaling theories described by Eq. ͑1͒. Actually, according to this equation and the massfractal dimensionalities, D f , given in Table I, an increase by a factor of 2 of the volume fraction would result in a decrease of 0 by factors of 0.5 and 0.6 for the polar and non-polar suspensions prepared by solution mixing, respectively. These theoretical values are in close agreement with their corresponding experimental factors equal to 0.5Ϯ 0.05 for the two suspensions. In addition, the order of magnitude of the characteristic length scales, 0 , of the non-polar and polar suspensions are similar while the volume fraction range for the polar suspensions is one decade smaller. The differences in the range of volume fractions result in different sizes a ef f of the compact building blocks of the clusters, as calculated from Eq. ͑1͒. This equation predicts that the compact building blocks of the non-polar suspensions are almost three times larger than those of the polar suspensions ͓ratio of Eq. ͑1͒for the non-polar and polar suspensions estimated from 0 of Fig. 12 and the fractal dimensions from Table I͔. This is attributed to differences in the nanoscale interactions between the particles during the solution mixing step in preparing the suspensions. Actually, it has been shown that the organomodified clay in toluene forms tactoids of 3-6 platelets whereas the natural clay is fully exfoliated in de-ionized water ͓Ho et al. ͑2001,2002͒; Ho and Glinka ͑2003͔͒. Finally, the comparable rheological properties of these suspensions observed on different volume fraction ranges are explained by differences in the elastic constant K. From Eq. ͑5͒ and using the scaling exponents and fractal dimensions given in Table I, one obtains the elastic constant K for the polar suspensions nearly 100 times larger than for the non-polar suspensions. Assuming that the elasticity is governed by the interaction potential between particles and clusters ͓Pantina and Furst ͑2006͔͒, the interaction potential level within the polar suspensions is therefore much higher than that within the non-polar ones.

Figure 13 reports micrographs for the metastable microstructures of the non-polar suspension prepared by solution mixing ͑ = 0.020͒ and polar suspension ͑ = 0.003͒ following various pre-shearing rates ␥ ˙i from 10 to 10 -2 s -1 . Their elasticity is given by G ϱ Ј in Fig. 2. These metastable microstructures reflect their corresponding non- equilibrium microstructures, since no overall structural recovery at the micro-scale was observed in CLSM upon cessation of steady shear flow. Therefore, the structure build-up at rest resulting in the increase of GЈ from G i Ј to G ϱ Ј depicted in Fig. 1 is due to local rearrangements at smaller length scales. The lack of micro-structural recovery toward a single equilibrium structure outlines the high interparticle attraction energy ͑Ͼ20 k B T͒, a characteristic of strongly aggregated systems. On the shear rate range investigated, from 10 -2 to 10 s -1 , the metastable microstructures of the polar suspensions display no apparent anisotropy, while for the non-polar suspensions a slight one along the vorticity direction is distinguishable, with the difference in the distributions along the x and y directions ͑Fig. 12͒. A constant surface coverage by the structure was also observed on the shear rate range explored, with A ͑described in the optical measurement section͒ equal to 0.48Ϯ 0.02 and 0.39Ϯ 0.02 for the non-polar and polar suspensions, respectively. The metastable structures and, hence, the non-equilibrium structures differ essentially from each other by their characteristic length scale, 0 , which tends to decrease for increasing pre-shearing rate amplitude ␥ ˙i as revealed quantitatively in Fig. 14. A comparable aggre- gation process was recently reported by Rahatekar et al. ͑2006͒ for a carbon nanotube suspension in epoxy. The dependency of the rheological properties with the pre-shearing and shearing amplitudes ͑Figs. 2, 3, and 8͒ results consequently from the shear-controlled characteristic length scale of the microstructure, as supported by the strong correlation between G ϱ Ј ͑Fig. 2͒ and 0 ͑Fig. 14͒. This structural dependency is attributed to a reversible shear-induced aggregation. As a result, 0 follows a power-law dependency at intermediate shear rates with a power exponent m = -0.22 for the non-polar suspensions and m = -0.42 for the polar suspensions. Sonntag and Russel ͑1986, 1987͒ reported a value of m = -0.35 for their dilute flocculated suspensions ͑polystyrene lattices͒. These values are within the limits of Ϫ0.23 and Ϫ0.5 determined by Potanin ͑1993͒ using computer simulations. The low limit is related to the non-central components of the particle attraction potential ͑function of the angle between adjacent bonds͒, while the high limit belongs to the purely central component ͑function of the distance between the centers of neighboring particles͒.

Finally, the shear-induced aggregation has been directly visualized during stepwise reductions in shear rate from 10 to 10 -2 s -1 . Figure 15 suspension ͑ = 0.003͒. The first decrease in -at short times, corresponding to the viscoelastic response of the suspension ͑stress relaxation͒, is followed by a monotonic increase as 0 starts to increase due to the shear-induced aggregation. The transient stress response at large deformations is therefore shown to be governed by a structural evolution at the micro-scale. Similar observations have been made for the non-polar suspensions.

V. CONCLUDING REMARKS

In this paper, the nature of flow-induced non-equilibrium and metastable structures of polar and non-polar nanoclay suspensions has been probed by studying the scaling behavior of their linear and non-linear viscoelastic properties and by quantifying real space observations obtained with CLSM. The power-law dependencies of the viscoelastic properties with the volume fraction have been interpreted in light of the fractal nature of the suspensions. Fractal dimensions close to 2 were estimated using the scaling model of Wu and Morbidelli ͑2001͒. Direct observations of the microstructures using CLSM confirmed the presence of fractal networks with fractal dimensions ϳ2 for both types of suspensions. As far as we are aware, such CLSM observations for colloidal suspensions under shear flow have never been reported. They demonstrate the potential of CLSM to correlate the complex rheological behavior of colloidal suspensions with their underlying structures.

The similar rheological properties of the non-polar and polar suspensions were explained to arise from differences in particle interaction potential, while the volume fraction for the polar suspension was ten times smaller. The attraction potential responsible for the fractal networks was clarified owing to its temperature dependency. The exponential decay of the shift factors with temperature was shown to be similar for the two suspensions.

By increasing the shearing amplitude, the characteristic length scale of the nonequilibrium microstructures was shown to decrease from high to low plateau values. This structural variation with the shearing amplitude was attributed to a reversible shearinduced aggregation. Upon cessation of flow, no overall structural recovery at the microscale was observed in CLSM, leading to metastable microstructures with the same characteristic length scale than their non-equilibrium counterparts. Consequently, the thixotropic behavior of these suspensions is due to local rearrangements at the nano-scale and its physical interpretation should be elucidated using wide-angle scattering techniques. Based on these findings, the rheological dependency on the flow history is attributed to the shear-controlled characteristic length scale of the microstructure.

  al. ͑2005͒, Lim and Park ͑2001͒, and Ren et al. ͑2000͔͒. The non-terminal flow behavior is usually attributed to the formation of a percolated three-dimensional network of clay layers or tactoids and, thus, to a fluid-to-glass transition ͓Galgali et al. ͑2001͒; Mitchell and Krishnamoorti ͑2002͒; Ren et al. ͑2000͔͒. A sol-to-gel transition through the formation of fractal networks has been lately considered ͓Durmus et al. ͑2007͒; Vermant et al. ͑2007͔͒. Until now, the literature reveals a lack of studies of colloidal attractive suspensions in light of jamming phase diagrams including structural and rheological descriptions of their non-equilibrium states.

FIG. 1 .

 1 FIG. 1. GЈ as a function of time measured at a frequency of 125.6 rad s -1 upon cessation of steady shear flow. T = 25 °C. Symbols indicate the values of the initial pre-shear rate ␥ ˙i; solid lines represent best fits using Eq. ͑8͒. ͑a͒ The non-polar suspension prepared by ultrasonication for = 0.01 and ͑b͒ the polar suspension for = 0.003.

FIG. 6 .

 6 FIG. 6. Scaling behavior of as a function of scaled shear rate ␥ ˙/ as is varied for the polar suspensions ͑ ref = 0.003; filled symbols͒ and the two non-polar suspensions ͑unfilled symbols͒ prepared by solution mixing and ultrasonication ͑ ref = 0.020͒. Solid lines represent the shear stress vs shear rate of the suspending fluids.

FIG. 8 .

 8 FIG. 8.Temperature dependency ofm ͑unfilled and semi-filled symbols͒ and its scaling behavior ͑filled symbols͒ for the non-polar suspension prepared by ultrasonication ͑ = 0.04͒ and the polar suspension ͑ = 0.003͒; data for the polar suspension have been multiplied by 0.1.

  FIG. 9. ͑a͒ Temperature dependency of GЈ and ͑b͒ its scaling behavior for the non-polar suspension prepared by ultrasonication ͑ = 0.040͒ and the polar suspension ͑ = 0.003͒; data for the polar suspension have been multiplied by 0.1.

FIG. 10 .

 10 FIG. 10. Temperature dependency of the shift factors ͑circles͒ and ͑diamonds͒ for the non-polar suspension prepared by ultrasonication ͑ = 0.040, unfilled symbols: ␥ ˙i = 0.95 Pa; semi-filled symbols: ␥ ˙i = 0.95 Pa͒ and for the polar suspension ͑filled symbols = 0.003͒. Solid line represents the best fit using an exponential decay:, =6673ϫ e -0.03T .

FIG. 11 .

 11 FIG. 11. CLSM images of the quiescent microstructures of the layered silicate suspensions. White areas are rich in layered silicate. Insets show the corresponding image part after binarization. The micrographs were taken at room temperature ͑about 23 °C͒. ͑a͒ 2.0% vol. of non-polar suspension ͑Cloisite 15A͒ prepared by ultrasonication; ͑b͒ 2.0% vol. of non-polar suspension ͑Cloisite 15A͒ prepared by solution mixing; ͑c͒ 1.0% vol. of non-polar suspension ͑Cloisite 15A͒ prepared by solution mixing; ͑d͒ 2.0% vol. of non-polar suspension of ͑Cloisite 30B͒ prepared by solution mixing; ͑e͒ 0.30% vol. of polar suspension ͑natural clay͒ prepared by solution mixing; ͑f͒ 0.15% vol. of polar suspension ͑natural clay͒ prepared by solution mixing.

FIG. 12 .

 12 FIG.12. distributions along x ͑unfilled symbols͒ and y ͑filled symbols͒ directions for the ͑a͒ non-polar suspensions and ͑b͒ polar suspensions. Dashed lines represent log-linear regressions.

FIG. 13 .

 13 FIG. 13.Effect of the pre-shearing amplitude on the flow-induced metastable microstructures of the layered silicate suspensions. Right: 2 vol. % non-polar and left: 0.3 vol. % polar suspensions prepared by solution mixing ͑a͒ pre-shear rate of 10, ͑b͒ 1, ͑c͒ 0.1, and ͑d͒ 0.01 s -1 . The pre-shearing flow direction was from bottom to top. Insets show the corresponding image parts after binarization. The micrographs were taken at room temperature ͑about 23 °C͒.

  FIG. 14.Dependence of the characteristic length scale 0 with the pre-shearing amplitude ␥ ˙i for the non-polar ͑unfilled symbols͒ and polar suspension ͑filled symbols͒ metastable structures.

  . ͑1989͒; Buscall et al. ͑1988͒; Shih et al. ͑1990͒; Wu and Morbidelli ͑2001͔͒, the network structure is considered to result from the close packing of mass-fractal clusters with an average size c following a scaling relation with the volume fraction ͓Mellema et al. ͑2000͔͒

	c

  Mobuchon et al. ͑2007͒ from Dullaert and Mewis ͑2005͒

TABLE I .

 I Values of the scaling exponents n, B, ␤, and t and fractal dimensions D f of non-polar and polar suspensions.

	Sample		n	B	D f	␤	t ͑D f ͒ b
	Non-polar suspensions a	␥ ˙ = 0.95 Pa	2.2	Ϫ0.7	1.7	2.8	1.7 ͑1.8͒
		␥ ˙ = 95 Pa	2.6
	Polar suspensions	␥ ˙ Ͻ 0.10 Pa	2.6	Ϫ0.6	2	2.6	2.3 ͑2.1͒

a Non-polar suspensions prepared by ultrasonication. b D f derived from Eq. ͑7͒.

ACKNOWLEDGMENTS

The authors are most grateful to Dr. N. Tran-Khanh and Dr. M. Thibault for their precious help with CLSM experiments. They also thank Professor M. Bousmina for the use of the Linkham shear cell. One of the authors ͑C.M.͒ is thankful to Alcan Inc. for a scholarship. Southern Clay Products kindly provided the nanoclays used in this study. Finally, financial support from NSERC ͑Natural Science and Engineering Research Council of Canada͒ and from FQRNT ͑Fonds Québécois de Recherche en Nature et Technologies͒ is gratefully acknowledged.


A.-H Soubra

Static and seismic passive earth pressure coefficients on rigid retaining structures

Keywords: limit analysis, passive pressure, earthquake analyse limite, butée des terres, séisme. Notes 478

Static and seismic passive earth pressure coefficients on rigid retaining structures

Introduction

Earthquakes have the unfavorable effects of increasing active lateral earth pressures and reducing passive lateral earth pressures. Hence, the assessment of seismic lateral earth pressures or changes in lateral earth pressures as the result of an earthquake is of practical significance in most seismic designs of retaining walls. The traditional method for evaluating the effect of an earthquake on the lateral earth pressures is the so-called "pseudo-static method." This method continues to be used by consulting geotechnical engineers because it is required by the building codes; it is easy to apply and gives satisfactory results. Quasi-static analysis using the seismic coefficient concept is therefore of great practical value in many cases, although the assessment of the seismic coefficient still relies highly on past experience.

The well-known Mononobe-Okabe analysis of seismic lateral earth pressures proposed by [START_REF] Mononobe | On determination of earth pressure during earthquakes[END_REF] and [START_REF] Okabe | General theory on earth pressure and seismic stability of retaining walls and dams[END_REF] is a direct modification of the Coulomb wedge method where the earthquake effects are replaced by a quasi-static inertia force whose magnitude is computed on the basis of the seismic coefficient concept. As in the Coulomb analysis, the failure surface is assumed to be planar in the Mononobe-Okabe method, regardless of the fact that the most critical sliding surface may be curved. Similar to the Coulomb analysis, the Mononobe-Okabe analysis may underestimate the active earth pressure and overestimate the passive earth pressure. Note, however, that the Mononobe-Okabe analysis has been experimentally proved by [START_REF] Mononobe | On determination of earth pressure during earthquakes[END_REF] and [START_REF] Ishii | Lateral earth pressure in an earthquake[END_REF] to be effective in assessing the seismic active earth pressure; it is generally adopted in current practice for seismic design of rigid retaining walls. The Mononobe-Okabe solution is therefore practically acceptable at least for the active pressure case, although its applicability to the passive pressure case is somewhat in doubt.

Recent research conducted by [START_REF] Chang | Lateral earth pressures on rigid retaining walls subjected to earthquake forces[END_REF] (cf. Chen and Liu 1990) using a log-sandwich mechanism within the framework of the kinematical method in limit analysis has shown that the upper-bound solutions they obtained were practically identical to those given by the Mononobe-Okabe method for the active case. However, the passive earth pressure coefficients are seriously overestimated by the Mononobe-Okabe method; they are in most cases higher than those obtained by the upper-bound method for a logsandwich mechanism.

In this paper, the static and seismic passive earth pressure problems are investigated by the upper-bound method of limit analysis using a translational failure mechanism. This mechanism allows the slip surface to develop more freely in comparison with the log-sandwich mechanism presented by [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF] (cf. Chen 1975 and[START_REF] Chen | Limit analysis and soil plasticity[END_REF][START_REF] Chen | Limit analysis in soil mechanics[END_REF] in the static case and by [START_REF] Chang | Lateral earth pressures on rigid retaining walls subjected to earthquake forces[END_REF] A.-H. Soubra. École Nationale Supérieure des Arts et Industries de Strasbourg, 24, Boulevard de la Victoire, 67084 Strasbourg CEDEX, France.

(cf. [START_REF] Chen | Limit analysis in soil mechanics[END_REF]) in the seismic case; hence it leads to smaller upper-bound solutions of the passive earth pressure problem.

The upper-and lower-bound theorems of limit analysis

The upper-bound theorem, which assumes a perfectly plastic soil model with an associated flow rule, states that the rate of energy dissipation in any kinematically admissible velocity field can be equated to the rate of work done by the external forces, and so enables a strict upper-bound on the true limit load to be deduced (see [START_REF] Drucker | Extended limit design theorems for continuous media[END_REF][START_REF] Chen | Limit analysis and soil plasticity[END_REF][START_REF] Salençon | An introduction to the yield design theory and its application to soil mechanics[END_REF]. A kinematically admissible velocity field is one which satisfies compatibility, the flow rule, and the velocity boundary conditions. To provide solutions that are useful in practice, the upper-bound theorem is often used in tandem with the lower-bound theorem. The latter also assumes a perfectly plastic soil model with an associated flow rule and states that any statically admissible stress field (which satisfies equilibrium and the stress boundary conditions, and nowhere violates the yield criterion) will provide a lower-bound estimate of the true limit load (see [START_REF] Drucker | Extended limit design theorems for continuous media[END_REF][START_REF] Chen | Limit analysis and soil plasticity[END_REF][START_REF] Salençon | An introduction to the yield design theory and its application to soil mechanics[END_REF]. By using these two theorems, the exact limit load can often be bracketed with an accuracy which is sufficient for design purposes.

In this paper, only the upper-bound theorem of limit analysis is applied to the static and seismic passive earth pressure problem using a kinematically admissible velocity field. It should be noted here that the upper-bound theorem gives an unsafe estimate of the passive failure load. The aim of this work is to improve the best available upper-bound solutions given by [START_REF] Chen | Limit analysis in soil mechanics[END_REF] in both the static and seismic cases.

Theoretical analysis of the seismic passive earth pressure problem

An earthquake has two possible effects on a soil-wall system. One is to increase the driving forces, and the other is to decrease the shearing resistance of the soil. The reduction in the shearing resistance of a soil during an earthquake is in effect only when the magnitude of the earthquake exceeds a certain limit and the ground conditions are favorable for such a reduction. Research conducted by [START_REF] Okamoto | Bearing capacity of sandy soil and lateral earth pressure during earthquakes[END_REF] indicated that when the average ground acceleration is larger than 0.3g, there is considerable reduction in strength for most soils. However, he claimed that in many cases the ground acceleration is less than 0.3g, and the mechanical properties of most soils do not change significantly in these cases.

The assumptions made in the analysis can be summarized as follows:

(1) Only the reduction of the passive pressures due to the increase in driving forces is investigated under seismic loading conditions. The shear strength of the soil is assumed unaffected as the result of the seismic loading. This hypothesis is currently made by consulting geotechnical engineers (see, for instance, Commission of the European Communities 1994) and it has been adopted by many investigators in the seismic stability analysis of geotechnical problems (see, for instance, [START_REF] Sarma | Seismic bearing capacity factors of shallow strip footings[END_REF][START_REF] Richards | Seismic bearing capacity and settlement of foundations[END_REF][START_REF] Budhu | Seismic bearing capacity of soils[END_REF][START_REF] Paolucci | Seismic bearing capacity of shallow strip foundations on dry soils[END_REF]andSoubra 1997, 1999).

(2) A constant seismic coefficient is assumed for the entire soil mass involved. Only the horizontal seismic coefficient K h is considered, the vertical seismic coefficient often being disregarded.

(3) A translational multiblock failure mechanism is assumed. This mechanism is a generalization of the one-block failure mechanism considered in the Mononobe-Okabe method.

(4) The soil is assumed to be an associated flow rule Coulomb material obeying the maximal work principle of Hill. However, real soils do not obey the associative flow rule, since frictional soils are found experimentally to dilate at increments considerably less than those predicted by the normality condition, that is, dilatancy angle (ψ) < angle of internal friction (φ). Recent theoretical considerations made on translational failure mechanisms [START_REF] Drescher | Limit load in translational failure mechanisms for associative and non-associative materials[END_REF][START_REF] Michalowski | Closure on 'Stability of uniformly reinforced slopes[END_REF]Shi 1995, 1996) allow one to conclude that for a nonassociative material the limit load can be obtained using the flow rule associated with a new yield condition in which φ and cohesion c are replaced by φ* and c* as follows:

[1] tan * cos sin sin sin

φ φ φ = - Ψ Ψ 1 [2] c c * cos cos sin sin = - Ψ Ψ φ φ 1
Hence, the results presented in this paper can be used for nonassociative material provided φ and c are replaced with φ* and c* calculated from eqs. [1] and [2], respectively.

(5) The angle of friction δ at the soil-structure interface is assumed to be constant. This hypothesis is in conformity with the kinematics assumed in this paper.

(6) An adhesive force P a is assumed to act at the soilstructure interface. The intensity of this force is cl (tan δ)/(tan φ), where l is the length of the structure.

(7) The velocity at the soil-structure interface is assumed tangential to the wall (see [START_REF] Chen | Limit analysis and soil plasticity[END_REF]. Other investigators (see [START_REF] Drescher | Limit load in translational failure mechanisms for associative and non-associative materials[END_REF][START_REF] Michalowski | Closure on 'Stability of uniformly reinforced slopes[END_REF] assumed that the interfacial velocity is inclined at δ to the wall to respect the normality condition. Both hypotheses lead to the same result of the limit load (see Appendix 1).

Failure mechanism

The failure mechanism is shown in Fig. 1. It is composed of a radial shear zone including n triangular rigid blocks. The angles α i and β i (i = 1, …, n) are as yet unspecified.

As shown in Fig. 2a, the wall is translating horizontally and all the triangles move as rigid bodies in directions which make an angle φ with the discontinuity lines d i (i = 1, …, n).

The velocity of each triangle is determined by the condition that the relative velocity between the triangles in contact should have a direction which forms an angle φ with the contact surface. The velocity hodographs are shown in Fig. 2b. The velocities so determined constitute a kinematically admissible velocity field.

In the present analysis, the work equation is applied to the soil mass in motion. Two another alternative approaches considering the whole soil-structure system can also be used and lead to an identical limit load (see Appendix 1).

Calculations of rate of external work

As shown in Fig. 3, the external forces contributing to the rate of external work consist of the passive earth force P pE , the adhesive force P a , the weight of the soil mass W i (i = 1,..., n) the surcharge q on the ground surface, and the different inertia forces. These inertia forces concern the soil mass and the surcharge loading.

The rate of external work for the different external forces can be easily obtained; the calculations are presented in Appendix 2.

Calculations of rate of internal energy dissipation

Since no general plastic deformation of the soil mass is permitted to occur, the energy is dissipated solely at the dis- continuity surfaces d i (i = 1, …, n) between the material at rest and the material in motion and at the discontinuity surfaces l i (i = 1, …, n -1) within the radial shear zone. The rate of energy dissipation per unit length along such velocity discontinuities can be expressed as

[3] & cos D cV = φ
where V is the velocity that makes an angle φ with the velocity discontinuity according to the associated flow rule of perfect plasticity. Calculations of the rate of energy dissipation along the different velocity discontinuities are given in Appendix 2.

Work equation

By equating the total rate of external work (eq. [B14], Appendix 2) to the total rate of internal energy dissipation (eq. [B19], Appendix 2), we have

[4] P pE = K l K q l K c l i i i i i i p E pqE pcE γ α β γ α β α β ( , ) ( , ) ( , ) 2 2 +
+ in which the seismic passive earth pressure coefficients K p γ E , K pqE , and K pcE can be expressed in terms of the 2n -1 as yet unspecified angles α i and β i . They are given as follows:

[5] K f K f p E h γ β φ δ = - + -- 1 2 1 sin( ) [6] K f K f pqE h = - + -- 3 4 1 sin( ) β φ δ [7] K f f f pcE = + + -- 5 6 7 1 sin( ) β φ δ

Numerical results

The most critical passive earth pressure coefficients can be obtained by minimization of these coefficients (eqs. [5]- [7]) with regard to the mechanism parameters. A computer program has been developed for assessing the critical static and seismic passive earth pressure coefficients. In the following sections, we present and discuss in succession (i) the static passive earth pressure coefficients K pγ , K pq , and K pc given by the present failure mechanism for K h = 0; and (ii) the seismic passive earth pressure coefficients K p γ E , K pqE , and K pcE for various values of the seismic coefficient K h .

Static passive earth pressure coefficients

Table 1 presents the K pγ factor obtained from the computer program for φ = 45°, δ / φ = 1, β / φ = 1, λ / φ = 0, K h = 0, and various values of n (the number of triangular rigid blocks).

The upper-bound solution can be improved by increasing the number of rigid blocks. The reduction in the K p γ value decreases with an increase in n and attains 0.4% for n = 14. The same trend has been observed for the coefficients K pq and K pc .

On the other hand, the results obtained from the computer program indicate that the coefficient K pc is related to the coefficient K pq0 by the following relationship (cf. theorem of corresponding states of [START_REF] Caquot | Tables de poussée et de butée[END_REF]):

[8] K K pc pq0 = - 1 cos tan δ φ
φ = 45°, δ / φ = 1, β / φ = 1, and λ / φ = 0.
where K pq0 is the coefficient of passive earth pressures due to a surcharge loading acting normally to the ground surface. This result is to be expected, since in the present analysis an adhesive force P a = cl (tan δ)/(tan φ) is assumed to act along the soil-wall interface.

Figure 4 shows the critical slip surfaces obtained from the numerical minimization of the coefficient K pγ for φ = 45°, δ / φ = 2/3, β / φ = 1/3, λ / φ = 0, and three values of n (3, 7, and 14).

For n = 14, the critical failure mechanism is composed of a radial shear zone sandwiched between two triangular rigid wedges. The shear zone is not bounded by a log-spiral slip surface as is the case of the log-sandwich mechanism proposed by [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF]. Thus, the present mechanism leads to smaller upper-bound solutions of the passive earth pressure coefficient K pγ . However, the coefficients K pq and K pc are identical to those given by the logsandwich mechanism, since the multiblock mechanism approaches the log-sandwich mechanism as the number of rigid blocks increases. The coefficients K pq and K pc are also almost identical to those given by L'herminier and Absi (1962) (cf. Kérisel and Absi 1990) using the slip line method, and the maximum error does not exceed 0.5%. It should be noted that the critical angular parameters (α i , β i ) obtained from the minimization of both K pq and K pc give exactly the same critical geometry.

Table 2 presents the coefficients K pγ and K pq obtained from the computer program for φ ranging from 10°to 45°a nd five values of δ / φ when β / φ = λ / φ = 0; the coefficient K pc can be calculated using eq. [8]. In Table 2 and hereafter, the results are given for n = 14, which means that the minimization procedure is made with regard to 27 angular parameters.

Comparison with Rankine's solution

For the general case of an inclined wall with a sloped backfill ( β / φ ≠ 0, λ / φ ≠ 0), the Rankine passive earth pressure coefficient K p γ is given as follows (cf. [START_REF] Costet | Cours pratique de mécanique des sols, plasticité et calcul des tassements[END_REF]):

[9] K pγ β β β λ β ω α ω -β) φ ω + β -2 λ )] = - + cos ( ) sin cos sin ( [ sin cos( 1 where [10] sin sin sin ω β φ β =
As is well known, this pressure is inclined at an angle α with the normal to the wall (cf. Fig. 5) irrespective of the angle of friction at the soil-wall interface, where To validate the results of the present analysis, one considers a soil-wall friction angle δ equal to the α value given by Rankine. The numerical solutions obtained from the computer program have shown that in this special case the present results are similar to the exact solutions given by Rankine (eqs.

[9] and [12]). [START_REF] Caquot | Tables de poussée et de butée[END_REF] and [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF] As mentioned earlier, the log-sandwich mechanism proposed by [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF] gives the exact solution solely in the case of a weightless soil. As a result, the comparison of the present solutions with solutions of other authors will be limited to the passive earth pressure coefficient K pγ .

Comparison with

There are many solutions for K pγ in the literature based on (i) the limit equilibrium method [START_REF] Janbu | Earth pressure and bearing capacity calculations by generalised procedure of slices[END_REF][START_REF] Rowe | Stress-dilatancy, earth pressures, and slopes[END_REF][START_REF] Lee | Stability analysis. Application to slopes, rigid and flexible retaining structures[END_REF][START_REF] Packshaw | Earth pressure and earth resistance[END_REF]Shields andTolunay 1972, 1973;[START_REF] Spencer | Forces on retaining walls using the method of slices[END_REF][START_REF] Rahardjo | General limit equilibrium method for lateral earth forces[END_REF][START_REF] Bilz | Earth pressure of soils with friction and cohesion[END_REF], (ii) the slip line method [START_REF] Caquot | Tables de poussée et de butée[END_REF][START_REF] Sokolovski | Statics of soil media[END_REF][START_REF] Sokolovski | Statics of soil media[END_REF][START_REF] Graham | Calculation of passive pressure in sand[END_REF][START_REF] Hettiaratchi | Boundary wedges in two-dimensional passive soil failure[END_REF]; and (iii) the limit analysis theory [START_REF] Lysmer | Limit analysis of plane problems in soil mechanics[END_REF][START_REF] Lee | A theoretical study of the pressures acting on a rigid wall by a sloping earth on rockfill[END_REF][START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF][START_REF] Basudhar | Optimal lower bound of passive earth pressure using finite elements and non-linear programming[END_REF][START_REF] Soubra | Application de la méthode variationnelle au problème de détermination des pressions passives des terres. Influence des forces d'écoulement[END_REF][START_REF] Soubra | Étude de la butée des terres en présence d'écoulement[END_REF], 1999;[START_REF] Chen | Limit analysis in soil mechanics[END_REF]. The tendency today in practice is to use the values given by [START_REF] Caquot | Tables de poussée et de butée[END_REF] (cf. Tables of [START_REF] Kérisel | Tables de poussée et de butée des terres[END_REF].

Figure 6 shows the comparison of the present solutions with those of [START_REF] Caquot | Tables de poussée et de butée[END_REF] (cf. [START_REF] Kérisel | Tables de poussée et de butée des terres[END_REF] in the case of a vertical wall and horizontal backfill (

β / φ = λ / φ = 0).
The values from the present analysis are greater than those of Caquot and Kérisel, with the maximum difference being less than 12%.

Figure 7 shows the comparison of the present solutions with those of [START_REF] Caquot | Tables de poussée et de butée[END_REF] for different values of β and λ when φ = 45°and δ / φ = 1.

As in the case of a vertical wall and a horizontal backfill, the values from the present analysis are greater than those of [START_REF] Caquot | Tables de poussée et de butée[END_REF]. The difference is not significant for small values of β / φ (≤ -0.4) and for large values of λ ( ≥ 25°); however, it attains 38% when φ = 45°, δ / φ = 1, β/φ = 1, and λ/φ = -1/3. Note that for practical configu-rations ( φ ≤ 45°, 0 ≤ δ/ φ ≤ 2/3, λ / φ = 0, β / φ ≤ 1/3) the maximum difference does not exceed 12.2%. On the other hand, rigorous upper-bound solutions are proposed in the literature by [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF]. These authors considered six failure mechanisms and showed that the log-sandwich mechanism gives in most cases the least upper-bound solutions. The results given by the present failure mechanism and those given by the logsandwich mechanism proposed by Chen and Rosenfarb are presented in Fig. 8 in the case of φ = 45°and δ / φ = 1.

K pγ K pq φ (°) 0 1/3 1/2 2/3 1 0 1/3 1/2 2/
It is clear that the present upper-bound solutions are better than those of [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF]; the improvement attains 21% when φ = 45°, δ / φ = 1, β / φ = 1, and λ / φ = -1/3.

Comparison with other theoretical and experimental solutions

Recently, [START_REF] Kobayashi | Laboratory experiments with an oblique passive wall and rigid plasticity solutions[END_REF] performed laboratory tests to compare the experimental passive earth pressure coefficients obtained in the case of a large wall oblique angle with those given by theoretical predictions based on the rigid plasticity theory. The passive wall was pushed into a sand mass using two different methods. Method A translates the passive wall normally with a slight shear force, whereas method B sinks the passive wall vertically with a full shear force. The relationship between the observed K pγ and tan δ is demonstrated in Fig. 9. Passive earth pressure coefficients given by the present analysis, by [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF] (cf. Chen 1975) using the upper-bound method in limit analysis, and by [START_REF] Sokolovski | Statics of soil media[END_REF] using the slip line method are also indicated in Fig. 9.

Observed values of K pγ are smaller than the theoretical predictions for large values of δ and K pγ decreases considerably from the peak to the residual. This may be explained by the progressive failure observed along a shear band. On the other hand, the comparison with the theoretical results given by [START_REF] Chen | Limit analysis and soil plasticity[END_REF] indicates that our solution improves the best upper-bound solution given by this author. The improvement attains 20.4% when φ = 42°, tan δ = 0.9, λ = -60°, and β = 0°. The present solutions are greater than those given by [START_REF] Sokolovski | Statics of soil media[END_REF], and the difference attains 19% when φ = 42°, tan δ = 0.9, λ = -60°, and β = 0°. [START_REF] Fang | Earth pressures with sloping backfill[END_REF] presented experimental data of earth pressure acting against a vertical rigid wall which moves toward a mass of dry sand with an inclined surface. The instrumented retaining-wall facility was used to investigate the variation of earth pressure induced by the translational wall movement. Based on experimental data, it has been found that the earth pressure distributions are essentially linear at each stage of wall movement and that the wall movement S/H (where S is the horizontal wall movement and H is the wall height) required for the backfill to reach a passive state increases with an increasing backfill inclination β. The relationship between the passive earth pressure coefficient K pγ,h and backfill inclination β at a different stage of wall movement S/H is demonstrated in Fig. 10.

Figure 10 shows that K pγ,h increases with increasing sloping angle β. For large values of β, the experimental K pγ,h values occur at a large wall displacement. As an example, for β = 20°, the required wall movement is 0.46H. For an arbitrarily assumed displacement criterion (S/H = 0.2) chosen by the authors, the present results and those given by the log-spiral method proposed by [START_REF] Terzaghi | Theoretical soil mechanics[END_REF] are in fairly good agreement with the experimental data.

Seismic passive earth pressure coefficients

Earthquakes have the unfavorable effect of reducing passive earth pressures. Equation [7] shows that the coefficient K pcE is unaffected by the seismic loading. To investigate how the passive earth pressure coefficients K pγE and K pqE are affected by earthquakes, extensive numerical results based on the present failure mechanism are presented in Tables 3 and4. The passive earth pressure coefficients K pγE and K pqE decrease with an increase in K h . The reduction is more significant for looser soils with lower φ values than for denser soils with higher φ values. For example, for δ / φ = 1, the reduction of the coefficient K pγE is 16% for φ = 45°and 35% for φ = 20°when K h increases from 0 to 0.3. Finally, it should be mentioned that the seismic acceleration generated by earthquakes not only imposes extra loading on a soil mass but also shifts the sliding surface to less favorable positions.

Figure 11 shows that the critical slip surface becomes more extended as the acceleration intensity increases.

Comparison of results with existing solutions

To see the validity of the present upper-bound solution, the seismic passive earth pressure coefficients are calculated and compared with solutions given by other authors.

Chang and Chen (1982) (cf. Chen and Liu 1990) considered a log-sandwich failure mechanism and gave rigorous upper-bound solutions for the coefficient K pγE . The upperbound solutions given by the present mechanism and those given by [START_REF] Chen | Limit analysis in soil mechanics[END_REF] are presented in Fig. 12.

As in the static case, the present upper-bound solutions are better than those of [START_REF] Chen | Limit analysis in soil mechanics[END_REF]; the improvement attains 18.2% when φ = 45°, δ / φ = 1, β / φ = 1, λ / φ = 0, and K h = 0.3. Therefore, the present failure mechanism gives interesting solutions for the seismic passive earth pressure coefficients for translational wall movement. Further investi-

(β / φ = λ / φ = 0).
gation remains necessary for the elaboration of adequate rotational failure mechanism.

Conclusions

A translational, kinematically admissible failure mechanism has been considered for the calculation of the static and seismic passive earth pressures using the upper-bound method of the limit analysis theory. The solutions presented are rigorous upper-bound solutions in the framework of the limit analysis theory. The numerical results obtained lead to the following conclusions.

For the static case, the present results for the coefficient K pq are almost identical to those given by Kérisel and Absi (1990) using the slip line method and those given by Chen and Liu (1990) using the upper-bound method in limit analysis for a log-sandwich mechanism. For the coefficient K pc , the present analysis has shown that the traditional formula given by the theorem of corresponding states [START_REF] Caquot | Tables de poussée et de butée[END_REF]) is also valid in the present analysis using the upper-bound method of the limit analysis theory. For the coefficient K pγ , the present upper-bound solutions are better than those of [START_REF] Chen | Limit analysis in soil mechanics[END_REF], since one obtains smaller upper-bound solutions; the improvement is 21% for φ = 45°, δ / φ = 1, β / φ = 1, and λ / φ = -1/3. On the other hand, the comparison between the present solutions and the currently accepted values of [START_REF] Caquot | Tables de poussée et de butée[END_REF] has shown that the maximum difference is less than 12.2% for the practical configurations (φ ≤ 45°, 0 ≤ δ/ φ ≤ 2/3, λ / φ = 0, β / φ ≤ 1/3).

For the seismic case, the present multiblock failure mechanism continues to give smaller upper-bound solutions for the coefficient K pγE than the log-sandwich mechanism; the improvement is 18.2% for φ = 45°, δ / φ = 1, β / φ = 1, λ / φ = 0, and K h = 0.3. Extensive numerical results for the present seismic passive earth pressure coefficients K pγE and K pqE are presented for practical use in geotechnical engineering. [START_REF] Chen | Limit analysis in soil mechanics[END_REF] for φ = 45°, δ / φ = 1, λ / φ = 0, and K h = 0.3. K p γ ,h horizontal component of static passive earth pressure coefficient Kp γ K pq0 static passive earth pressure coefficient due to a surcharge loading acting normally to the ground surface l wall length n number of rigid blocks in the failure mechanism P a adhesive force P p static passive force P pE seismic passive force q surcharge loading S horizontal wall movement S i area of block i V velocity along velocity discontinuity V 0 wall velocity V 0,1 relative velocity at the soil-structure interface V i velocity of block i V i,i+1 relative velocity between blocks i and i + 1

V i+1 velocity of block i + 1 W i weight of block i & , W P P

Appendix 1

In the case of a vertical rough wall, [START_REF] Drescher | Limit load in translational failure mechanisms for associative and non-associative materials[END_REF] and more recently [START_REF] Michalowski | Closure on 'Stability of uniformly reinforced slopes[END_REF] have indicated that two different ways of incorporating the wall friction in the energy-balance equation can be found in the literature (cf. [START_REF] Chen | Limit analysis and soil plasticity[END_REF][START_REF] Mroz | Limit plasticity approach to some cases of flow of bulk solids[END_REF][START_REF] Collins | The upper-bound theorem for rigid/plastic solids to include Coulomb friction[END_REF][START_REF] Collins | A note on the interpretation of Coulomb's analysis of the thrust on a rough retaining wall in terms of the limit theorems of plasticity theory[END_REF], and that both approaches lead to an identical force P p . The aim of this appendix is to present and discuss the two approaches in the general case of an inclined rough wall. A third approach is also presented and discussed. The following three approaches consider the simple case of a single rigid block (Fig. A1) ; however, the results remain valid for a multiblock mechanism.

First approach (Chen 1975)

In the approach of Chen (1975), the velocity jump vector is assumed to be tangential to the wall (Fig. A2). The wall moves horizontally with velocity V 0 and the wedge moves with velocity V 1 . Thus, sliding on the wall surface occurs with sliding vector magnitude V 0,1 = V 1 cos ( )/ cos β λ φ λ --(see velocity hodograph), and the rate of work dissipation associated with this sliding is

[A1] & ( sin ) , D P P V 1 0 1 = + = p a δ ( sin ) cos ( ) cos P P V p a δ β λ φ λ + -- 1
The energy dissipation rate along the failure surface is given by

[A2] & ( cos ) cos cos ( D c V l 2 1 = - φ λ β λ)
The work rates of forces P p and P a and the soil weight are

-- - + - + --- -       P a cos ( ) cos sin sin( ) cos β λ φ λ λ β φ λ
Second approach [START_REF] Mroz | Limit plasticity approach to some cases of flow of bulk solids[END_REF][START_REF] Collins | The upper-bound theorem for rigid/plastic solids to include Coulomb friction[END_REF][START_REF] Collins | A note on the interpretation of Coulomb's analysis of the thrust on a rough retaining wall in terms of the limit theorems of plasticity theory[END_REF] In this approach, the inclination of the velocity jump V 0,1 across the wall is taken as δ (Fig. A3). The wall moves horizontally with velocity V 0 and the wedge moves with velocity V 1 (see velocity hodograph).

The rate of work dissipation along the wall is given by

[A6] & cos cos cos ( ) cos ( ) , D PV P V 3 0 1 1 = = -- - a a δ δ β λ φ λ δ
λ φ β λ + -- - - -- -   P a cos cos ( ) cos ( ) sin sin ( ) cos ( ) δ β λ φ λ δ λ β δ φ λ δ    

Third approach (present analysis)

In this approach, the passive force P p and the adhesive force P a are considered as two external forces acting on the soil wedge which moves with velocity V 1 (Fig. A4).

The rate of work dissipation along the failure surface is given by eq. [A2]. The work rates of forces P p and P a are )

β φ δ γ λ β β φ λ β λ -- = -- - 2 2 + - + - cl P cos cos cos ( ) [cos ( )] λ φ β λ β φ a

Discussion

By simple trigonometric manipulations or by numerical calculations one can easily see that the three approaches lead strictly to the same result of P p . Hence, in the present paper the third approach is adopted in which the passive and adhesive forces are considered external forces acting on the soil wedge which moves with velocity V 1 . In this analysis, no indication is made with regard to the soil-wall velocity V 0,1 . This confirms the conclusion of Drescher and Detournay [START_REF] Mroz | Limit plasticity approach to some cases of flow of bulk solids[END_REF] and [START_REF] Collins | The upper-bound theorem for rigid/plastic solids to include Coulomb friction[END_REF][START_REF] Collins | A note on the interpretation of Coulomb's analysis of the thrust on a rough retaining wall in terms of the limit theorems of plasticity theory[END_REF]. (1993) and [START_REF] Michalowski | Closure on 'Stability of uniformly reinforced slopes[END_REF] that the hypothesis of associativeness or nonassociativeness along the soil-structure interface has no influence on the limit load.

Appendix 2

This appendix presents the rate of external work and the rate of internal energy dissipation along the different velocity discontinuities.

Geometry

For the triangular block i, the lengths l i and d i and the surface S i are given as follows:

Velocities

From the velocity hodographs (cf. Fig. 2b), the blocks and interblocks velocities are given as follows:

[B4] V V i j j j i j= i = - - + - ∏ sin ( + 2 ) sin ( 2 ) α β φ β φ 1 1 1 [B5] V i i i i i i j j , sin( ) sin( )
+ + + = - + - - 1 1 1 2 β β α β φ α β φ sin ( + 2 ) sin ( 2 ) β φ j j= i V + - - ∏ 1 1 1 1

Rate of external work

The rate of external work for the failure mechanism can be calculated as follows:

(1) Rate of external work due to self-weights and inertia forces of the n triangular rigid blocks: The rate of external work due to self-weight in a rigid block is the vertical component of the velocity in that block multiplied by the weight of the block. The rate of external work due to the seismic force in a block can be obtained by the multiplication of the horizontal inertia force of that block and the corresponding horizontal velocity. By summing this rate of external work due to self-weights and inertia forces over the n triangular blocks, one obtains

[B6] & [ ( , ) ( , )] W l f K f V i i i i soil h = + γ α β α β

Fig. 1 .

 1 Fig. 1. Failure mechanism for static and seismic passive earth pressure analyses.

Fig. 2 .

 2 Fig. 2. (a) Velocity field of the failure mechanism. (b) Velocity hodographs.

Fig. 3 .

 3 Fig. 3. Free body diagram of the failure mechanism.

  of the slip surface with the horizontal direction is given as follows:

Fig. 4 .

 4 Fig. 4. Critical slip surfaces for φ = 45°, δ / φ = 2/3, β / φ = 1/3, λ / φ = 0, and three values of n (3, 7, and 14).

Fig. 5 .

 5 Fig. 5. Rankine solution in the case of a general soil-wall system.

Fig. 6 .

 6 Fig. 6. Comparison of the coefficient K pγ from the present solution with that of Caquot and Kérisel (1948) (β / φ = λ / φ = 0).

Fig. 7 .

 7 Fig. 7. Comparison of the coefficient K pγ from the present solution with that of[START_REF] Caquot | Tables de poussée et de butée[END_REF] for different β and λ values.

Fig. 8 .

 8 Fig. 8. Comparison of the coefficient K pγ from the present solution with that of Chen and Rosenfarb (1973) for different β and λ values.

Fig. 9 .

 9 Fig. 9. Comparison of results from the present solution with other theoretical and experimental results[START_REF] Chen | Limit analysis and soil plasticity[END_REF][START_REF] Sokolovski | Statics of soil media[END_REF] Kobayashi 1998 "Methods A and B").

Fig. 10 .

 10 Fig. 10. Comparison of results from the present solution with other theoretical and experimental results (Terzaghi 1943; Fang et al. 1997) ( φ = 30.9°, δ = 19.2°, and λ = 0°).

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Critical slip surfaces for φ = 20°, δ / φ = 1, β / φ = λ / φ = 0, and K h = 0 and 0.2.

  due to forces P p and P a & , W P P p E a rate of external work due to forces P pE and P a & W q rate of external work due to surcharge loading and corresponding inertia force & W soil rate of work due to soil weights and the corresponding inertia forces in the multiblock mechanism & W γ rate of work due to soil weight in the one-block mechanism α inclination of the Rankine passive earth pressures α i , β i angular parameters of failure mechanism β backfill inclination ψ dilatancy angle δ angle of friction at the soil-wall interface φ angle of internal friction of the soil φ * residual friction angle due to nonassociativeness γ unit weight of the soil λ angle between the soil-wall interface and the vertical direction θ inclination of the Rankine slip surface with the horizontal direction ω β intermediate angular parameter

  rate of external work (eqs. [A3] and [A4]) to the rate of internal energy dissipation (eqs. [A1] and [

Fig. A1 .

 A1 Fig. A1. One block failure mechanism.

  of soil weight is given by eq. [A4]. By equating the rate of external work (eqs. [A4], [A9], and [A10]) to the rate of internal energy dissipation (eq. [A2]), one obtains [

Fig. A2 .

 A2 Fig. A2. Velocity field according to Chen (1975).

Fig. A3 .

 A3 Fig. A3. Velocity field according to[START_REF] Mroz | Limit plasticity approach to some cases of flow of bulk solids[END_REF] and[START_REF] Collins | The upper-bound theorem for rigid/plastic solids to include Coulomb friction[END_REF][START_REF] Collins | A note on the interpretation of Coulomb's analysis of the thrust on a rough retaining wall in terms of the limit theorems of plasticity theory[END_REF].

Fig. A4 .

 A4 Fig. A4. Velocity field according to the present analysis.

[

  Rate of external work due to the passive force P pE and the adhesive force P a : The rate of external work due to these forces is Rate of external work due to the surcharge loading and the corresponding inertia force: This rate of external work is given of external work is the summation of these three contributions, that is, eqs. [B6], [B9], and [B11]:Rate of internal energy dissipation(1) Along linesd i (i = 1, …, n) Along the radial lines l i (i = 1, …, n-1)The total rate of energy dissipation is the summation of these two parts, that is, eqs. [B15] and [B17]:

Table 1 .

 1 K pγ values versus number of rigid blocks n for

	n	K pγ	% reduction
	2	2362.66
	3	673.26	71.50
	4	499.92	25.75
	5	441.87	11.61
	6	414.50	6.19
	7	399.18	3.70
	8	389.68	2.38
	9	383.34	1.63
	10	378.90	1.16
	11	375.65	0.86
	12	373.21	0.65
	13	371.32	0.51
	14	369.83	0.40

Table 2 .

 2 K pγ and K pq values for φ ranging from 10°to 45°and δ / φ of 0, 1/3, 1/2, 2/3, and 1 (β / φ = λ / φ = 0).

	3	1

Table 3 .

 3 Seismic

			φ (°)
	δ / φ	K h	15	20	25	30	35	40	45
	0	0	1.70	2.04	2.46	3.00	3.69	4.60	5.83
		0.05	1.63	1.97	2.38	2.91	3.59	4.49	5.71
		0.10	1.56	1.89	2.30	2.82	3.49	4.38	5.58
		0.15	1.47	1.80	2.21	2.73	3.39	4.27	5.46
		0.20	1.37	1.71	2.12	2.63	3.29	4.15	5.33
		0.25	1.22	1.61	2.02	2.53	3.18	4.03	5.20
		0.30	-	1.48	1.91	2.42	3.06	3.91	5.07
	1/3	0	1.89	2.39	3.08	4.05	5.48	7.70	11.35
		0.05	1.81	2.30	2.97	3.91	5.31	7.48	11.07
		0.10	1.71	2.19	2.84	3.77	5.14	7.27	10.78
		0.15	1.60	2.08	2.72	3.62	4.96	7.04	10.49
		0.20	1.47	1.95	2.58	3.47	4.78	6.81	10.19
		0.25	1.28	1.81	2.44	3.31	4.59	6.58	9.88
		0.30	-	1.64	2.28	3.13	4.39	6.33	9.57
	1/2	0	1.99	2.58	3.43	4.69	6.67	9.99	15.98
		0.05	1.89	2.47	3.29	4.53	6.46	9.70	15.57
		0.10	1.79	2.35	3.15	4.35	6.24	9.40	15.15
		0.15	1.67	2.22	3.01	4.18	6.01	9.10	14.73
		0.20	1.53	2.08	2.85	3.99	5.78	8.79	14.29
		0.25	1.32	1.92	2.68	3.79	5.54	8.48	13.85
		0.30	-	1.73	2.50	3.59	5.29	8.15	13.40
	2/3	0	2.08	2.77	3.79	5.40	8.06	12.83	22.22
		0.05	1.98	2.65	3.64	5.20	7.80	12.46	21.65
		0.10	1.87	2.52	3.48	5.00	7.53	12.08	21.06
		0.15	1.74	2.37	3.32	4.79	7.25	11.68	20.47
		0.20	1.58	2.22	3.14	4.57	6.96	11.28	19.86
		0.25	1.36	2.04	2.95	4.34	6.66	10.87	19.24
		0.30	-	1.83	2.74	4.10	6.35	10.44	18.61
	1	0	2.25	3.12	4.51	6.86	11.13	19.62	38.61
		0.05	2.13	2.98	4.33	6.61	10.76	19.05	37.61
		0.10	2.01	2.83	4.13	6.35	10.39	18.46	36.60
		0.15	1.86	2.66	3.93	6.07	9.99	17.85	35.56
		0.20	1.69	2.48	3.71	5.79	9.59	17.23	34.51
		0.25	1.44	2.28	3.47	5.49	9.17	16.60	33.43
		0.30	-	2.03	3.22	5.17	8.74	15.94	32.33

passive earth pressure coefficient K p γ E (β / φ = λ / φ = 0).

Table 4 .

 4 Seismic passive earth pressure coefficient K pqE

			φ (°)
	δ / φ	K h	15	20	25	30	35	40	45
	0	0	1.70	2.04	2.46	3.00	3.69	4.60	5.83
		0.05	1.63	1.97	2.38	2.91	3.59	4.49	5.71
		0.10	1.56	1.89	2.30	2.82	3.49	4.38	5.58
		0.15	1.47	1.80	2.21	2.73	3.39	4.27	5.46
		0.20	1.37	1.71	2.12	2.63	3.29	4.15	5.33
		0.25	1.22	1.61	2.02	2.53	3.18	4.03	5.20
		0.30	-	1.48	1.91	2.42	3.06	3.91	5.07
	1/3	0	1.88	2.37	3.03	3.95	5.28	7.29	10.49
		0.05	1.80	2.28	2.93	3.83	5.14	7.11	10.27
		0.10	1.71	2.18	2.82	3.71	4.99	6.93	10.04
		0.15	1.60	2.07	2.70	3.57	4.84	6.75	9.81
		0.20	1.47	1.95	2.57	3.43	4.67	6.55	9.56
		0.25	1.28	1.81	2.43	3.28	4.50	6.35	9.32
		0.30	-	1.64	2.28	3.12	4.32	6.14	9.06
	1/2	0	1.96	2.52	3.30	4.44	6.17	8.92	13.62
		0.05	1.88	2.42	3.19	4.31	6.00	8.71	13.33
		0.10	1.78	2.32	3.07	4.17	5.83	8.49	13.03
		0.15	1.67	2.20	2.95	4.02	5.65	8.26	12.73
		0.20	1.53	2.07	2.81	3.86	5.46	8.02	12.42
		0.25	1.32	1.92	2.65	3.69	5.26	7.78	12.09
		0.30	-	1.73	2.48	3.51	5.05	7.52	11.76
	2/3	0	2.03	2.65	3.56	4.93	7.09	10.72	17.29
		0.05	1.94	2.56	3.45	4.78	6.90	10.47	16.93
		0.10	1.84	2.45	3.32	4.63	6.71	10.20	16.55
		0.15	1.73	2.33	3.18	4.46	6.50	9.93	16.17
		0.20	1.58	2.19	3.03	4.29	6.28	9.64	15.77
		0.25	1.36	2.03	2.86	4.10	6.05	9.35	15.36
		0.30	-	1.83	2.68	3.90	5.81	9.04	14.93
	1	0	2.13	2.87	4.00	5.81	8.86	14.42	25.55
		0.05	2.04	2.77	3.87	5.64	8.62	14.08	25.01
		0.10	1.94	2.65	3.73	5.45	8.38	13.72	24.45
		0.15	1.82	2.52	3.57	5.26	8.12	13.35	23.88
		0.20	1.66	2.37	3.40	5.05	7.85	12.97	23.29
		0.25	1.43	2.19	3.22	4.83	7.56	12.57	22.69
		0.30	-	1.98	3.01	4.59	7.26	12.16	22.06

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