Anisotropic nonlinear elasticity in a spherical bead pack: influence of the fabric anisotropy

Anisotropic nonlinear elasticity in a spherical bead pack: influence of the fabric anisotropy


Stress-strain measurements and ultrasound propagation experiments in glass bead packs have been simultaneously conducted to characterize the stress-induced anisotropy under uniaxial loading. These measurements, realized respectively with finite and incremental deformations of the granular assembly, are analyzed within the framework of the effective medium theory based on the Hertz-Mindlin contact theory. Our work shows that both compressional and shear wave velocities and consequently the incremental elastic moduli agree fairly well with the effective medium model by Johnson et al. [J. Appl. Mech. 65, 380 (1998)], but the anisotropic stress ratio resulting from finite deformation does not at all. As indicated by numerical simulations, the discrepancy may arise from the fact that the model doesn’t properly allow the grains to relax from the affine motion approximation. Here we find that the interaction nature at the grain contact could also play a crucial role for the relevant prediction by the model; indeed, such discrepancy can be significantly reduced if the frictional resistance between grains is removed. Another main experimental finding is the influence of the inherent anisotropy of granular packs, realized by different protocols of the sample preparation. Our results reveal that compressional waves are more sensitive to the stress-induced anisotropy, whereas the shear waves are more sensitive to the fabric anisotropy, not being accounted in analytical effective medium models.

45.70.-n, 43.35.+d, 81.05.Rm

I Introduction

The study of nonlinear elasticity in granular materials is not only of fundamental interest but also of practical importance in many fields, including soil mechanics and geophysics. Unlike consolidated porous materials such as sedimentary rocks, soils and sintered bead packs, dry granular materials acquire solely its elasticity as a result of the applied stress, forming a very inhomogeneous force network Goddard (1990); Jaeger et al. (1996); de Gennes (1999); Makse et al. (2004); Goldenberg and Goldhirsch (2005); Jiang and Liu (2007). The mechanical properties of such materials depend strongly on both the contact interactions between grains and the geometric arrangement (i.e. packing structure). The stress dependence of elasticity which originates with these contact forces at the grain level leads to a complex behaviour of the granular medium: nonlinear behaviour, loading-path dependence and stress-induced anisotropy Norris and Johnson (1997). The change of sound velocity with the applied stress (i.e. acousto-elastic effect) is the characteristic signature of this nonlinear elasticity Goddard (1990); Makse et al. (2004); Jia et al. (1999); Somfai et al. (2005). The hysteresis often found in the stress-strain experiments of granular media is also associated with the path-dependant nature of the contact forces Jiang and Liu (2007); Hoque and Tatsuoka (1998); García-Rojo et al. (2005). Another issue complicating the analysis of the mechanical properties of granular materials arises from the geometric or fabric anisotropy, which is basically related to the mode of grain deposition and the packing structure in a model system like the sphere pack Hoque and Tatsuoka (1998); Hicher and Chang (2006). This fabric anisotropy is important for the mechanical behaviour of granular media such as stress transmission Vanel et al. (1999), mechanical stability Grasselli and Herrmann (1997) and liquefaction resistance Ishibashi and Capar (2003).

Photoelastic visualizations Majmudar and Behringer (2005) and numerical simulations in 2D systems Radjaï and Roux (2001); Cambou et al. (2004); Luding (2004); Peyneau and Roux (2008) demonstrate that the anisotropy induced by an external load may have two distinct effects: one leads to an important change of the fabric anisotropy in the contact network, and the other develops an anisotropic force chain network. Understanding the nonlinear elastic responses and the associated stress-induced anisotropy represents a fundamental issue for granular mechanics. The problem may also be important for understanding the jamming phenomena and shear-induced yielding in a more general class of geomaterials formed from granular media Grasselli and Herrmann (1997); Ishibashi and Capar (2003); Majmudar and Behringer (2005). Development of non destructive methods of investigation is therefore desirable for monitoring the evolution of both geometric and mechanical anisotropies in real 3D granular materials, and allows us to gain a more comprehensive insight into their elastic properties along different stress paths.

Sound waves offer a sensitive and non invasive probe of both the structure and the mechanical properties of heterogeneous materials Toksöz and Johnston (1981); Winkler (1983); Guyer and Johnson (1999); Gilcrist et al. (2007). In a granular medium, sound propagation is controlled to a large extent by the properties of the contact force networks forming the solid frame of the material Makse et al. (2004); Jia et al. (1999); Liu and Nagel (1992); Jia (2004); Bonneau et al. (2007). The effective medium theory (EMT) has been commonly used to describe both the nonlinear elasticity Goddard (1990); Jiang and Liu (2007); Hicher and Chang (2006); Walton (1987); Johnson et al. (1998) and sound propagation in the long-wavelength limit Goddard (1990); Makse et al. (2004); Duffy and Mindin (1957); Digby (1981). These effective medium models are generally based on the Hertz-Mindlin theory at the grain contact and make use of the two main assumptions to obtain a mean field description of the granular elasticity: i) affine approximation or kinematic hypothesis, in which the motion of each grain follows the applied strain, and ii) statistically isotropic distribution of contacts around each grain. However, some acoustic measurements in random packs of glass beads under isotropic loading Makse et al. (2004); Domenico (1977) show that the ratio of bulk modulus to shear one is significantly larger than the value predicted by the EMT with bonded elastic spheres (no sliding). Numerical simulations reveal that the discrepancies can stem from the failure of the effective medium approximation: the bulk modulus is well described by the EMT but the shear modulus is not, principally because the EMT does not correctly allow the grains to relax collectively from the affine motion assumed by the theory Makse et al. (2004); Jenkins et al. (2004). In a recent work, the multiple scattering of shear acoustic waves through the stressed glass bead packs shows that sound propagation at small-amplitude (or incremental deformation) does not cause any significant rearrangement of the contact force network Jia (2004). This observation leads to a fundamental question as to whether overestimating the shear modulus by the EMT is related with the failure of the effective medium approximation for sound propagation or with the inadequate treatment of micromechanics at the grain contact level Norris and Johnson (1997); Winkler (1983).

In this paper, we present new results obtained from a granular model system, i.e. a glass bead pack in an œdometric test. We conducted simultaneously the stress-strain measurement and the ultrasound propagation experiment to characterize the stress-induced anisotropy by uniaxial loading. Both measurements of stress fields and elastic moduli via sound velocities parallel and perpendicular to the applied load are analyzed within the framework of the EMT developed by Johnson et al. specifically for stress induced anisotropy in the œdometric configuration Johnson et al. (1998). The aim of our work is twofold. Firstly, we examine the applicability of the EMT to small- and large-deformation mechanical tests, corresponding respectively to sound propagation and stress-strain measurements. The crucial role of the interaction nature at the grain contact for the appropriate prediction will be discussed. Then, we investigate the respective responses of compressional and shear waves to the elastic anisotropy in granular media. A particular attention is paid to the influence of the fabric anisotropy which is realized by the different sample preparation. Such inherent fabric anisotropy is expected to evolve little under œdometric loading compared to pure shear experiments.

Ii Experiments

In order to characterize the anisotropic elasticity of granular materials, we developed an apparatus coupling the mechanical test with the ultrasonic measurement. The schematic diagram of the experimental arrangement is illustrated in Fig. 1. Spherical glass beads of diameter between 300 m and 400 m are carefully filled in a duralumin cell up to 70 mm in height and closed by the fitted pistons at the top and bottom surfaces. A controlled vertical load is applied to the granular sample across the top piston. The forces re-directed in the horizontal direction are measured by the stiff force sensors placed at the middle of the cell to monitor the stress anisotropy evolution. This special compaction cell is presumed to be rigid enough that the sample experiences a macroscopic uniaxial strain with negligible lateral motion (œdometric test). The solid volume fraction of the granular assembly is determined by the axial displacement as a function of the applied load. To ensure the measurements of sound velocities in both the vertical (-axis) and horizontal (-axis) directions, a pair of compressional or shear piezoelectric transducers of diameter 30 mm in contact with the glass beads are placed at the top and bottom pistons, while the other ones are mounted at the lateral walls separated by a distance of 40 mm (see the inset of Fig. 1).

Figure 1: Sketch of the experimental set-up. Inset: Top-view of the œdometric cell.

ii.1 Granular fabric and packing density

To study the influence of the granular fabric on the induced elastic anisotropy, the bead packs are realized by two different preparation protocols. The first one, known as rain deposition or air pluviation, consists in pouring the glass beads into the cell through two grids of mesh size 2 mm spaced by 2 cm and reaches a dense pack of . The second, termed as de-compaction protocol is to remove gently a horizontal grid through the bead pack from the bottom to the top after filling the beads in the cell; the rearrangements produced by this efficient shearing throughout the sample allow us to obtain a loose pack of . It is shown by numerical simulations Radjaï and Roux (2001); Emam et al. (2005) that the rain deposition protocol creates an anisotropic distribution of the contact angle with two preferred directions orientated roughly at 30 around the vertical (gravity) direction. However, the loose pack prepared by the de-compaction protocol is expected to produce a fairly isotropic distribution of the contact angle Reydellet (2002).

Before any measurement, a preloading up to kPa is applied to the sample in order to minimize the hysteretic effects related to the grain rearrangements and ensure a reproducible initial state. Then the stress-strain and ultrasonic velocity measurements are performed as a function of the applied stress ranging from 70 to 900 kPa. Figure 2 displays the evolution of the solid volume fraction versus in the dense and loose packs, obtained respectively by the two preparation protocols. These results show a good reproducibility of our sample preparation in terms of the initial packing density measured at kPa for 30 repeated measurements: and . The fact that the solid volume fraction in the dense packing sample is a little larger than the density of the random close packing (RCP) is due to the slight dispersion of the bead size. Furthermore, Fig. 2 shows that the packing density of our samples increases slightly, up to a resultant vertical deformation of at the applied kPa. Both the dense and loose bead packs keep clearly a signature of their preparation protocols. These results imply that for the applied stress in this work, much less than the value to produce the grain fracture of about 20 MPa for the glass beads McDowell and Humphreys (2002), there are few important rearrangements of grains in the œdometric test as compared to those in a pure shear experiment Majmudar and Behringer (2005) and the inherent fabric should not change much from the initial state Cambou et al. (2004).

Figure 2: Evolution of solid volume fraction of the loose and dense packs. Error bars illustrate the data dispersion over 30 measurements.

ii.2 Anisotropic stress field

Let us now investigate the stress-induced anisotropy under uniaxial loading in the œdometric configuration. By monitoring the resultant horizontal stress when increasing the vertical stress , we display in Fig. 3 the stress ratio measured as a function of the applied stress for the two granular samples, respectively. It is observed that both the bead packs evolve from their initial stress fields at low applied kPa, roughly isotropic, to anisotropic states at high . As mentioned in the previous experiment Johnson et al. (1998), the isotropic distribution of stress observed at low arises probably from a kind of tight wedging of grains produced during the preloading and unloading cycle. Such an interlock of grains may contribute to a residual stress isotropically distributed in the bead pack even when the top piston is removed and no stress is applied.

As the applied stress increases, the ratio is seen to increase to an asymptotic value at high . For the granular packs prepared by the two distinct protocols, the different asymptotic values of about reveal again a memory effect of the initial state of the sample as seen in the above density measurement. In the conventional œdometric test, the inverse of the asymptotic value is known as Jacky coefficient of the earth pressure at rest; it is empirically related to the internal friction angle of the medium by Michalowski (2005). These results thus suggest a possible correlation between the internal friction angle and the fabric of the medium. To examine this issue, we have measured the avalanche angle at the relative humidity for both the dense and loose packs, and obtained and for six avalanche experiments. The difference between the two measured avalanche angles is about , consistent with the difference of between the two internal friction angles deduced from the asymptotic values of in Fig. 3. Although the avalanche angle is not the direct measure of the internal friction angle, these stress measurements indicates again the significant effect of the preparation protocol on the granular fabric Vanel et al. (1999); Evesque et al. (1993).

Figure 3: Evolution of the vertical-to-horizontal stress ratio versus the applied stress in the dense and loose packs.

ii.3 Anisotropic elastic moduli

Different transports of sound waves in granular media have been detailed in the previous work Jia and Mills (2001). When the wavelength is much larger than the bead size, the coherent waves propagate as through an effectively homogeneous medium. Measuring the elastic wave velocity allows one to access to the elastic modulus of the granular medium by with the material density Makse et al. (2004); Jia and Mills (2001); Brunet et al. (2008a). In our apparatus, the compressional P-waves and shear S-waves are excited and detected, respectively, by longitudinal or transversal piezoelectric transducers of diameter 30 mm. Figure 4 displays a typical pulsed ultrasound transmission along the vertical direction. The arrivals of P- and S-wave pulses are well separated, which allows us to clearly identify the different modes and measure adequately the wave velocities by the time-of-flight.

Figure 4: Typical ultrasound transmission along the vertical direction (500 kPa), excited and dectected by shear transducers.

For the granular materials under uniaxial loading and the geometry of the cell considered here, the elastic tensor relating the incremental stress and strain , , would have a symmetry belonging to the orthotropic class. In such a case, there are nine independent constants as shown by the matrix Royer and Dieulesaint (2000):

By means of P- and S-waves of different polarization propagating along the horizontal and vertical directions (i.e. principal axes), respectively, we can infer the diagonal elastic components () from the velocity measurement . For determining the off-diagonal components, velocity measurements of waves propagating along a direction inclined to the principal axes should be necessary. We display in Fig. 5a and 5b the wave velocities of several elastic waves measured a function of the applied stress in our loose pack and dense one, respectively. Here and correspond to the velocities of P-waves propagating along the horizontal and the vertical (- and -axis) directions. denotes to the velocity of the S-wave propagating vertically, whereas and correspond to those of the S-waves propagating horizontally and being polarized along the vertical and perpendicular (-axis) directions. Each data presented in Fig. 5 results from an average of six experimental runs and the data dispersion or error-bar is lower than in the case of the loose pack and in the dense sample, illustrating thus a good reproducibility of these acoustic measurements.

As shown in Fig. 5a for the loose pack, there is a clear difference of P-wave velocity between and () going up to when the stress is increased to 900 kPa. For the S-wave, the velocity difference is less important than that of P-wave: indeed, at kPa is about and greater than and , respectively. In contrast to the previous works under load superior to a few MPa Makse et al. (2004); Johnson et al. (1998); Domenico (1977), our acoustic measurements are realized at much lower stress, ranging from 70 kPa to 900 kPa. In such range of loading, it is expected that both the solid volume fraction (Fig. 2) and the coordination number would vary little Makse et al. (2004). As the de-compaction protocol tends to create an isotropic fabric in the loose granular sample, our measurements indicate that the elastic anisotropy observed here shall stem from the induced stress anisotropy shown in Fig. 3.

For the dense pack prepared by rain deposition, Fig. 5b displays again a significant difference between the P-wave velocities and . When the anisotropy of the stress field is developed, such elastic anisotropy rises to at kPa. Unlike the P-wave velocity, the S-wave velocity appears to be sensitive to the fabric anisotropy produced by rain deposition in the dense granular pack. Contrary to the measurement in the loose packing, we observe a significant difference between m/s, m/s and m/s at the lowest stress kPa where the stress field is nearly isotropic (see Fig. 3). Moreover, when is increased the S-wave velocities evolve differently, but recover a similar behaviour at high load: a difference of is found again between and at kPa. These results imply a particular sensitivity of S-wave to the fabric anisotropy which affects the elastic anisotropy together with the stress anisotropy.

Figure 5: Wave velocity versus applied stress in the loose pack (a) and the dense one (b). Experimental data (, , , and ) are compared to the EMT predictions (lines) .

Iii Comparison with the effective medium theory

The theory of elasticity a granular pack is primarily based on the Hertz-Mindlin model of contact between grains Johnson (1985). The macroscopic stress-strain relations are commonly derived using the effective medium description, where it is assumed that the motion of grains is affine with the applied macroscopic strain at least on average, and the distribution of contacts is statistically isotropic and homogeneous. Because the presence of the tangential forces at the contacts gives rise to load-displacement relations which are not only nonlinear but also inelastic, the mechanical response of the medium, namely stress-strain relations, must be expected to depend on the entire past history of loading. However, it is shown that the incremental response of the medium, i.e. the second-order effective elastic constants and consequently sound velocities are path-independent of loading Norris and Johnson (1997). Within the framework of the effective medium approach, Walton Walton (1987) analyzed the mechanical responses of a random pack of identical elastic spheres under isotropic strain and purely uniaxial compression, respectively. Furthermore the incremental elastic moduli were derived for these specific initial deformed states. For simplicity the spheres were assumed to be either infinitely rough or perfectly smooth in the calculations; nevertheless these results provide a physical insight of the effect of friction in real grain contacts.

As noted above, our measurements were made in the compaction cell with rigid walls in which the granular sample is subjected neither to purely isotropic nor uniaxial compression. To compare with the experimental data obtained in such widely used œdometric tests, Johnson et al. proposed an analytical model based on the effective medium theory Johnson et al. (1998). This model combines isotropic and uniaxial strains to describe the stress-induced elastic anisotropy in transversely isotropic granular materials: with if , otherwise . Here the internal strain stems from the residual stress distribution, namely an initial isotropic stress state produced by the preloading-unloading process as mentioned above, whereas the axial strain results from the applied stress . As differs from only by (Fig. 5), the approximation of the transverse isotropy used in this model would be adequate for interpreting our measurements. The predictions of such a model reduce to the results for purely isotropic compression in the limit and to those for purely uniaxial compression in the opposite limit , respectively Walton (1987).

Fig. 3 presents the ratio computed for the packs of both bonded (“rough”) and frictionless (“smooth”) spheres, as a function of the applied stress . The latter case is obtained by canceling the tangential contact stiffness in the model, i.e. . As described in eqs. 21 and 22 of Johnson et al. (1998), the stress tensor is a function of several parameters, where is the coordination number, (= 24MPa) is the shear modulus and (= 0.2) is the Poisson ratio of the glass bead. For a given the ratio versus the applied stress is only parameterized by the glass bead property and (Fig. 3). The axial strain is measured from the displacement of the top piston versus and the residual isotropic strain is deduced from the isotropic residual stress 70 kPa measured at , yielding .

We observe from Fig. 3, that the theoretical curve calculated with the frictionless spheres () is faithful to the qualitative trend observed in our experiments and is overall in agreement with the measured data; at the high load the stress ratio approaches to the limit value of uniaxial loading, i.e. Walton (1987); Johnson et al. (1998). In contrast, Fig. 3 and the inset show that the no-slip assumption with bonded spheres gives rise to a huge overestimation of this stress ratio, indicating the crucial role of the contact interaction law in the effective medium model for describing adequately the evolution of the stress field anisotropy. We may understand the discrepancy found with the bonded spheres by the following picture. At large deformation applied here of the order of , the grains need to rearrange in order to relax the tangential stress accumulated at the contacts. The no-slip assumption with infinitely frictional spheres forbidden such a process and hence fail, whereas the effective medium analysis with frictionless spheres allow the stress relaxation and provide thus an adequate description of the experiments.

Let us now examine the applicability of the effective medium approach to the wave velocity measurements. Given a stress state and corresponding macroscopic deformation , the elastic moduli can be derived from the incremental stress of the medium subject to an incremental strain by . Following the work of Johnson et al. (1998), yields


where , and denotes the Kronecker symbol. The brackets represent an average over all unit vector uniformly distributed since the distribution of contact angle (i.e. granular fabric) is assumed to be isotropic. To determine the wave velocities from the elastic moduli in eq. 1, we perform the calculations by two steps. Firstly, we compute the stress state of a frictionless spheres packing () created by the large axial deformation, which provides an adequate description of the experimental stress field shown in Fig. 3. Then, we consider the appropriate value of () according the Hert-Mindlin theory to calculate the elastic moduli by eq. 1. Note that this procedure of computation is also used in many numerical simulations where the friction between the particles is turned off during the packing preparation and is turned on to measure the incremental response Makse et al. (2004). Fig. 5a and 5b illustrate the calculated wave velocities in both the loose packing and the dense one. The agreement between the theoretical predictions and experimental results remains fairly good (better than ), for the present effective medium model employing only one adjustable parameter, i.e. in the dense pack and in the loose pack.

Iv Discussions

iv.1 Stress dependence of P-wave velocities

Based on the Hertz-Mindlin theory of contact and the assumption of isotropic fabric in the sphere pack, the effective medium theory (EMT) predicts a power-law dependence of the elastic modulus (or ) on isotropic compression Goddard (1990); Makse et al. (2004); Duffy and Mindin (1957); Digby (1981). The same power law scaling is found for the pure uniaxial loading (i.e. ) as or since the stress ratio is constant Walton (1987). Numerous acoustic velocity measurements in sands and glass bead packs showed however that the exponent in the power-law scaling may not be constant, varying from at low pressure to at high pressure Goddard (1990); Jia et al. (1999); Richart et al. (1970); Gilles and Coste (2003). Several mechanisms have been proposed to explain this discrepancy, including the contact recruitment by buckling of particle chains or the conical contact due to irregular surfaces Goddard (1990), the effect of soft shell on coated spheres de Gennes (1996) and the fluctuation of the stress field Velický and Caroli (2002). However, for the preloaded glass bead packs and the range of the applied stress considered here, the contact recruitment is expected to have a negligible effect, as that suggested by the above density measurement (Fig. 2).

We investigate here the influence of the stress anisotropy on the scaling behaviour of the P-wave velocity. To do this, we displays in Fig. 6 the rescaled P-wave velocities and as a function of the applied axial stress , obtained in the loose and dense packs, respectively. For kPa, is seen to follows adequately the scaling of , as expected within the framework of the Hertz-Mindlin theory. By contrast, deviates drastically from the scaling, which is due to the evolution of stress-field anisotropy illustrated in Fig. 3. Indeed, if rescaling with the stress parallel to the propagation direction, we recover a scaling behaviour (insets to Figs. 6a and 6b), similar to those for in both the loose and the dense packs having the different granular fabric.

Figure 6: Stress dependence of P-wave velocities propagating vertically and horizontally in the loose pack (a) and the dense one (b). The dashed lines correspond to the predictions by Hertz-Mindlin theory of contact.

iv.2 Stress and fabric dependences of S-wave velocities

Compared to the P-wave velocity, the stress dependence of the S-wave velocity is more complicated and strongly influenced by the fabric of the pack. As mentioned above (in sec. II), S-waves are not only sensitive to the stress field anisotropy but also to the fabric anisotropy. Let us first examine the scaling behaviour of the shear velocities versus in the loose bead pack with isotropic fabric. As shown in Fig. 7a, both and scale roughly as a power law i.e. for kPa, which agree fairly well with the stress-dependence predicted by the Hertz-Mindlin theory of contact. Compared to the P-wave velocity , the scaling behaviours of the shear velocity is somewhat surprising, however, it may be understood by a heuristic picture which accounts for both the propagation direction and the polarization. Indeed, unlike the P-wave propagating in the horizontal direction (), the S-wave travelling along the -axis is polarized vertically along the -axis () and it would also be affected by the stress component . This scenario is consistent with the preceding observation showing that the S-wave velocity is less sensitive to the stress-field anisotropy than the P-wave velocity is (shown in Fig. 5). However, for the S-wave travelling along the -axis but polarized horizontally along the -axis (), Fig. 7a shows a scaling behaviour of deviated from but being consistent with the power law (inset to Fig. 7a), a situation reminiscent to that of the P-wave ().

We now turn on the scaling behaviour of the S-wave velocities in the dense pack presenting the fabric anisotropy obtained by rain deposition. Contrary to the behaviours observed in the loose packing with isotropic fabric, there is no clear stress dependence of the shear velocities , , and in the range of the applied stress (Fig. 7b and the inset). This observation reveals the extreme sensitivity of the shear wave velocity to the fabric of the granular pack. However, at the present stage, it is difficult to speculate the scaling behaviour of the S-wave velocity on stress, due to the interplay between the evolution of the stress anisotropy and the inherent fabric anisotropy. Indeed, the latter parameter is not included in most of analytical models within the framework of the effective medium theory.

Figure 7: Stress dependence of S-wave velocities propagating vertically and horizontally in the loose pack (a) and the dense one (b). The dashed lines correspond to the predictions by the Hertz-Mindlin contact theory.

iv.3 Correlation between induced elastic anisotropy and stress-field anisotropy

In the above sections, we have observed that the elastic moduli and consequently sound velocities (P- and S-waves) depend on both the induced stress-field anisotropy and the inherent fabric of the granular sample. The protocols of the sample preparation used here, i.e. de-compaction and rain deposition, allow us to obtain two distinct granular fabrics which also differ in the packing density as shown in Fig. 2 and probably in the coordination number . As suggested by eq. (1), we may investigate the correlation between the induced stress anisotropy and the elastic anisotropy via the ratio of elastic components dropping thus the parameters and .

Fig. 8a displays the ratio of the compressional

Figure 8: Correlation between the induced elastic anisotropy and stress-field anisotropy in the granular packs of different fabric.

moduli versus the induced stress anisotropy for two granular samples with different fabric (loose and dense). The experimental data agree well with the effective medium model (eq. 1), demonstrating that the anisotropy of compressional moduli is principally determined by the stress field anisotropy and is much less sensitive to the fabric of the granular pack. Also, we depict the evolution of the shear modulus ratio as a function of in Fig. 8b. These results confirm again the correlation of the elastic anisotropy with the stress-field anisotropy, though the anisotropy is less pronounced for the shear modulus than for the compressional one. In contrast to the latter, the shear moduli are however very sensitive to the granular fabric. For the loose pack with isotropic fabric, the ratio of the shear moduli is in good agreement with the effective medium theory model based on the assumption of the isotropic distribution of contact angle.

V Conclusion

In summary, we have studied the anisotropic elasticity of dry glass bead packs using ultrasonic measurements. Both the influences of the fabric anisotropy and the stress anisotropy are investigated thanks to the granular samples of different inherent fabric under uniaxial loading in an œdometric cell. The stress anisotropy is characterized from the horizontal and vertical force measurements, whereas the elastic moduli are determined from velocity measurements of P- and S-waves propagating along the vertical and horizontal directions, respectively. For the range of the applied stress in this work, our results show clearly that the compressional modulus anisotropy is more sensitive to the stress field anisotropy while the shear modulus anisotropy is more sensitive to the fabric anisotropy.

We have tested the applicability of the effective medium theory to sound propagation associated with the incremental deformation. The predictions by the analytical model based on the affine approximation agree well with the measurements of sound velocity. Moreover, our velocity measurements of compressional waves confirm the scaling behaviour on stress, , where is the normal component of the stress along the propagation direction. As for the shear waves, the stress dependences of velocities are more complex, being also strongly influenced by the fabric anisotropy which is not considered in existing analytical EMT models.

The present effective medium model is also applied to analyze the finite deformation of the granular pack. Our stress-strain measurements show the breakdown of this model to describe adequately the deformed state at large deformation. However, such discrepancy between theory and experiment can significantly be reduced when the friction between grains is switched off. As indicated previously Makse et al. (2004); Jenkins et al. (2004), our observation confirms that the model based on the affine approximation fails because it prevents the relaxation of the tangential stress at the grain contact, especially for large deformation. Other effective medium models are required for describing the large deformations of granular packs Jiang and Liu (2007); Hicher and Chang (2006).


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