AN EXPERIMENTAL ANALYSIS ON THE RHEOLOGY OF A GRANULAR DEBRIS FLOW, PART II: QUASI STATIC STRESSES

Riccardo Martino, Carmine Sabatino and Lucio Taglialatela

Department of Hydraulic and Environmental Engineering “G. Ippolito”

University of Napoli (ITALY), Via Claudio, 21 – 80125 NAPOLI

E-mail: rmartino@unina.it - tel. +39 081 7683461 - fax: +39 081 5938936

Abstract: A rheological analysis concerning experiments carried out in the Hydraulics Laboratory of the University of Trent (ITALY) is presented. A free surface flow has been studied and velocity and concentration distributions have been obtained. Starting from the collisional rheological model, verified in a previous work, the quasi static component of the stress has been singled out.

The analysis showed a clear dependence between the extension of the static friction angle to the dynamic case, used modelling the shear quasi static stresses, and the local value of the solid concentration.

The extension of the friction angle rises when the solid concentration increases according to that happens with the static friction angle in the earth science.

Finally, for the material examined, the concentration at fluidity that many Authors use in their constitutive equations has been valuated 

Keywords:debris flow, granular system, rheology, quasi static stress, plastic stress, static friction angle

1    INTRODUCTION

The behaviour of a granular debris flow is typical of a granular mixture where for granular material we intend a collection of many individual solid particles, immerged in a interstitial fluid, that interact through short-range repulsive forces and frictional forces when in contact with other particles or boundaries (Seminara & Tubino, 1993).

A particular feature of a granular mixture is that a minimum value of the shear stress is necessary to the flowing of the granular mixture. This behaviour justifies that when grains are poured on a rough plane (with an inclination lower than a critical angle) heap up with a conic form. It is good described by Mohr Coulomb yield criterion: if the shear stress is lower than the critical value t_c

                                  _{( 1 )}

the granular material doesn’t flow. By the relationship (1) , valid only in static conditions, the critical value, t_c, depends on the normal stress s_P acting between the particles and on a static friction angle j_S. Moreover here cohesion is ignored because it is of significance when the granular material is wet but not saturated.

Another interesting feature is the analogy between the colliding macroscopic grains of a sheared granular material and the agitated molecules of a dense gas. When a granular material is sheared at a sufficiently high shear rate, the shear and normal stresses required to maintain its motion are observed to vary with the square of the shear rate (Bagnold, 1954; Savage & McKeown, 1983; Savage & Sayed, 1984; Martino et al., 2001). The interpretation of these results is that at right shear rates the dominant mechanism of momentum transfer is collisions between grains; on the contrary in these conditions the interstitial fluid (liquid or gas) plays a relatively minor role. This interpretation justifies the kinetic theory application to the flow of a granular material.

In the international literature it is traditional to think of two granular flow regimes, described above, like limiting regimes: the first is called the quasi static regime and the second the grain inertial (or collisional) regime. In the first one long-term contacts producing rubbing and sliding between particles occur and the shear stress does not depend on the rate; in the second one the contacts inter-particles are of short duration and the shear stress is depending on the rate.

In order to describe the transitional regime many rheological models for granular materials have been proposed and many authors (Johnson & Jackson, 1987; Chen, 1987; Johnson et al., 1990; Takahashi et al., 1997) suggest representing the total stress, , like the sum of a quasi-static stress,  and of a collisional stress  

                         (2)

With reference to the total normal stresses s_ACT,

                              (3)

where s_QS is the quasi-static normal stress, s_INT is pressure of the interstitial fluid and s_COLL is the collisional normal stress.

In literature the quasi static shear stress is modelled by the relationship similar to that of plastic stress, relationship (1 ), substituting the inter-particles stress s_P with the quasi-static normal stress s_QS obtaining, in the cohesion less case,

                                (4)

where j is an extension of the static friction angle j_S to the dynamic case.

Many authors assume j as constant and s_QS as fraction of the normal stress acting between the particles.

For a dry material (where s_INT is not present) Sayed and Savage (1983) assumed

                           (5)

In this way they represent the existence of a finite inter-particle quasi static stress that approaches zero when the solid concentration, c, approaches c₁. The value c₁ represents the concentration at fluidity that is the smallest concentration at which we can observe quasi static stresses. Obviously this relationship is not valid when the solid concentration is greater than maximum packing concentration c_¥.

For a saturated debris flow Takahashi et al. (1997) assumed

                   (6)

With reference to the collisional stresses Bagnold (1954) found the relationships

                        (7)

and

                          (8)

where du/dy is the shear rate, r_S is the particles density, d_S is the particles diameter, F_D is an experimental value, called dynamic friction angle and equal to 17.75°, a is an other experimental value equal to 0.042, and l is the linear concentration (function of the solid concentration c and of maximum packing concentration c_¥)

A local rheological analysis (Martino et al., 2001) has been worked out to check the existence of collisional-dilatant regime as provided by a Bagnold type relationship. The results have been carried out on a uniform flow in an open channel of solid (particles of PVC) and water mixture measuring the velocity and concentration. Typical experimental results are showed in Fig. . In these conditions we valuated the tangential and normal stresses, t_ACT and s_ACT, by the following relationships

                       (9)

                            (10)

where q is the slope of the sediment layer deposited on the flume bed, h is the depth of the flow and r is the interstitial fluid density (function of the point considered).

Fig. 1    Typical concentration and velocity profiles (Martino, 1999).

The Bagnold relationships have been found satisfactory for large values of the Bagnold number (greater than 1500). The experimental values of the parameters a and F_D, for the material used in the experiments are: a=0.0258 and F_D=33.7°. On the other hand the quasi static stresses seem to be predominant for Bagnold values lower than 1500 (Martino et al., 2001).

2    QUASI STATIC STRESSES

Valuing the total stresses t_ACT and s_ACT through the relationships (9) and (10), and using the relationships (7) and (8) in order to model the collisional stresses, t_COLL and s_COLL, we can calculate, in correspondence of a generic value of y, the quasi-static shear stress

                        (11)

and the quasi static normal stress, acting between the particles

                        (12)

In order to model the collisional stresses we used the values of a and of F_D found previously for the granular mixture that we are examining.

The pressure of the interstitial fluid, s_INT, has been modelled with a hydrostatic distribution

                           (13)

In Fig. we can see the acting shear stress (whole curve) the shear collisional (hatched curve) and shear quasi static stresses distribution in a section.

The quasi static stresses are predominant near to the bottom. Obviously in correspondence of the point where the velocity is zero, the collisional stress is not present and the entire stress is a quasi static stress.

In the case of the test of Fig. and Fig. the collisional stress is prevalent at the top of the flow even if we can see little differences that we attribute to the technique of discretization in order to calculate the collisional stresses.

Fig. 2    Shear stresses distribution in the section.

If we consider the terms that appear in the relationship (4) they are all known except the extension of the static friction angle, j. The resolution of this equation permits to obtain j in correspondence of the generic value of y (obviously where the granular material is present) and so for each solid concentration value.

The results are showed in Fig. where the friction angle j against the solid concentration c, is reported. The dependence is very clear: when the concentration c grows the friction angle j increases because with high concentration the inter particles spaces are limited and the collisions can’t occur.

We can see that when the solid concentration, c, is lower than the value about 0.54 the extension of the static friction is null. We can justify this behaviour because when the solid concentration is not high the inter-granular space is so big that the long-term contacts producing rubbing and sliding can’t occur and the shear stress arises only from the collisions between the particles (pure grain-inertial regime).

When the concentration is very high (very close to the maximum value) the entire stress is a quasi static stress as for example near to the bed. In these conditions is reasonable to affirm the extension of the friction angle, j, tends to the value of the static friction angle.

According to the considerations above, for our material we can consider the value of 0.54 as the concentration at fluidity used by Sayed and Savage (1983) and by Takahashi et al. (1997).

Between the point with the concentration equal to 0.54 and the other experimental points a jump of j is present: this behaviour can be explained from the great gradient and from the small availability of the experimental results in this range.

For this reason it would be very interesting to develop other tests with local values of the concentration straddling the concentration at fluidity.

Fig. 3    Extension of the static friction angle j against solid concentration.

3    CONCLUSIONS

A rheological analysis concerning the quasi static stresses has been showed.

The analysis permitted to affirm that the extension of the static friction angle to the dynamic case (used in order to model the shear quasi static stresses) depends on the local value of the solid concentration.

This angle rises when the solid concentration increases according to that happens with a soil in static situation but near to the breaking conditions.

Finally the concentration at fluidity has been calculated for the granular mixture used.

To confirm the dependence found between the extension of the friction angle and the solid concentration is one of the main research tasks in the near future. 

Acknowledgements

We thank prof. Aronne Armanini who permitted to dr Riccardo Martino, during his PhD studies, to develop the experiments in the hydraulic laboratory of University of Trent.

References

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