A new method to evaluate the mechanical behaviour of granular material with large particles: theory and validation

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A new method to evaluate the mechanical behaviour of

granular material with large particles: theory and

validation

Wei Hu, Etienne Frossard, Pierre Yves Hicher, Christophe Dano

To cite this version:

Wei Hu, Etienne Frossard, Pierre Yves Hicher, Christophe Dano. A new method to evaluate the me-chanical behaviour of granular material with large particles: theory and validation. 2nd International Conference on Long Term Behaviour of Dams, 2009, Graz, Austria. �hal-01007783�

A new method to evaluate the mechanical behaviour of granular

material with large particles: Theory and validation

W. Hu

1

, E. Frossard

2

, P-Y. Hicher

1

, C. Dano

1

1

Research Institute in Civil and Mechanical Engineering, UMR CNRS 6183, Ecole Centrale de Nantes, 44321 Nantes, France 2

Coyne et Bellier Bureau d’Ingénieurs Conseils, 9 Allée des Barbanniers, 92632 Gennevilliers, France E-mail: wei.hu@ec-nantes.fr

Abstract

Because of the lack of experimental devices for testing granular materials with large particles, the mechanical behaviour of rockfills is not very well known. This lack of knowledge can lead to improper design of rockfills dams, and as a consequence, to severe accidents including dam failures. A new method has been developed for evaluating the mechanical behaviour of rockfills based on the estimation of size effect in granular materials, affected by grain breakage. The size effect, resulting from Fracture Mechanics, can be expressed by the fact that the strength of individual particles decreases when their size increases. The aim of this new method is to determine the mechanical properties of an assembly made of large particles (grain size distribution G1) from the measured properties of a granular assembly made of smaller particles (grain size distribution G0), of the same mineral stock, taking into account the different breakage capacity of particles of different sizes. G0 and G1 should have parallel grain size distributions. Some triaxial and crushing tests were performed on granular assemblies made of homogeneous limestone particles with two parallel grain size distributions. Comparison between experimental results and numerical predictions shows that this method can estimate with good accuracy the strength of granular materials made of large particles.

Introduction

Because of the lack of experimental devices suitable for testing granular materials with large particles, an attractive way is to predict their mechanical behaviour from the behaviour of a finer fraction of the same material. An important outcome would be to improve the design of rockfills dams and, as a consequence, to prevent severe accidents of such hydraulic structures. Nevertheless, the influence of the particle size on the mechanical behaviour of granular assemblies is related to the amount of particle breakage during loading. We have developed a new method in order to evaluate the mechanical properties of assemblies

of large particles from the mechanical behaviour of assemblies of small particles. This method results from an in depth review of classical experimental results on rockfills in the light of present knowledge in mechanics of materials, during a recent research project about granular fill behaviour , Frossard (2005)[6].

Size effect in granular materials

Size effect is encountered in many fields of engineering, involving concrete, rock or soil. For granular materials, it has been found that with the increase of particle size, the strength and more particularly the internal friction angle decrease [1]– [2]–[9]

. In 1969, at the University of California, Marachi et al. [9]

performed a series of triaxial experiments on three kinds of materials. For each material, three specimens of different sizes were reconstituted with parallel grain size distributions chosen in proportion to the specimen sizes. The smallest specimen was about 70 mm in diameter and 178 mm in height, the medium specimen was about 305 mm in diameter and 1372 mm in height, the large specimen was about 915 mm in diameter and 2032 mm in height (Figure 1).

Figure 1: Different sizes of specimen ,Marachi et al[9] (1969) The experimental relationship between the internal friction angle and the confining pressures for the different sample

sizes is shown in Figure 2. The experimental relationship between the amount of grain breakage and the confining pressures for the different sample sizes is shown in Figure 3. Similar relationships were obtained with other granular materials, the only difference coming from the intensity of particle crushing. Particle breakage increases with particle size, as subsequently explained, and also induces a reduction of the dilatancy, causing the non linear decrease of the friction angle.

Figure 2: Evolution of internal friction angle with the increase of cell pressure for different particle size, Marachi et

al (1969) [9]

Figure3: Increase of particle breakage with the increase of cell pressure for different specimen with different sizes,

Marachi et al (1969) [9]

Method for evaluating the maximum shear

strength envelope of coarse granular materials

In order to evaluate the mechanical behaviour of coarse granular materials, a method based on the size effect in granular materials has been developed by Frossard [6].

Feature of grain breakage

The relationship between the size and the maximum crushing force of a grain was investigated long ago by Marshal 1967 [10]

(Figure 4). The average experimental crushing load

P

_a

was fitted by a power function of the average diameter dm of the particles:

.

a m

P

=

η

d

λ

(1)

where η and λ are experimental constants.

Figure 4: Experimental results of the relationship between the average grain size and particle crushing force , Marsal (1967)

[10]

The breakage feature has also been described by a Weibull distribution considering the probability of survival within a population of fragile objects [11], exposed to loading conditions:

( )

0 0

exp

m s

V

P

V

V

σ

σ

⎡

_⎛

_⎞

⎤

⎢

⎥

=

−

_⎜

_⎟

⎢

⎝

⎠

⎥

⎣

⎦

(2)

modulus, V0 and σ0 are respectively the volume and the stress corresponding to a probability of 37 %.

The theories by Marsal (Equation.1) and Weibull (Equation.2) can be connected through a relationship between the parameter λ and Weibull modulus m. Indeed, as the grain volume V is proportional to the cube of its diameter dm, this approach leads for a given level of survival probability Ps to an average strength proportional to a power function of the diameter dm:

m m

d −3

∝

σ

(3)

If we consider now that the mean stress σ is proportional to the applied force during loading divided by the mean section of the grain, itself proportional to the square of the diameter, the bringing together of the two approaches gives a simple means to fix, for a given material, a representative Weibull distribution: 2 m a d P ∝

σ

(4)

From Equation 1, 3 and 4, we obtain:

3

2

m

λ

= −

or

3

2

m

λ

=

−

(5)

In the range of materials investigated by Marsal (1967)[10], the exponent λ was between 1.2 and 1.8 corresponding to a Weibull modulus m between 4 and 15, with a central value about 1.5 corresponding to a Weibull modulus m equal to 6.

Failure envelopes for rockfills

The shear strength of compacted rockfills can be determined by carrying out drained triaxial compression tests. It has been customary to express the shear strength measured in such a test at a particular confining pressure in terms of the internal friction angle.

(

)

(

)

' ' 1 3 1 ' ' 1 3

/

1

sin

/

1

f f

σ σ

φ

σ σ

−

⎡

_⎢

−

⎤

_⎥

=

⎢

+

⎥

⎣

⎦

(6)

where (σ’1/σ’3)f is the maximum principal stress ratio during the test. It has been generally found that when a series of tests is carried out on rockfills over a range of confining pressures, there is a marked decrease in the measured value of φ when the confining pressure σ’3 is increased. Therefore, as suggested by De Mello (1977) [4], a curved failure envelope expressed by the following formula is chosen:

( )

' b

f

A

τ

=

σ

(7)

Introduction of size effect in the expression of the failure envelope

Size effects provide an efficient rule, as derived by Frossard[6]–[7], to predict the strength failure envelope for rockfills with large particles of grading G1(D1 = Dmax), starting from the measured shear strength of rockfills with small particles of grading G0(D0 = Dmax), issued from the same homogeneous mineral stock, with parallel grading and same porosity (Figure 6). In the above frame, size effects can be integrated in the shear resistance formulations, like De Mello’s parabolic criterion (Equation.7), or shear resistance envelope through the Hoek and Brown’s criterion. If the general failure envelope for the material G0 is given by the following formula:

(

, 0

)

0 f n D

G

σ

τ

= (8)

the failure envelope for material G1 is given as follows:

3 3 1 1 1 0 0 0

,

m m G n

D

D

f

D

D

D

τ

σ

−

_⎧

_⎫

⎛

⎞

_⎪

⎛

⎞

_⎪

=

_⎜

_⎟

⋅

_⎨

⋅

_⎜

_⎟

_⎬

⎝

⎠

⎪

_⎩

⎝

⎠

⎪

_⎭

(9)

Substituting Equation 7 into Equation 9, we can get the failure envelope for the material G1:

( )

( )

b n m b G G m b G D D A D D

_τ

_σ

τ

_⎟⎟ ⋅ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ = ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − − − − 31 0 1 1 3 0 1 0 0 1 (10)

Validation of the method

In order to validate the method, we have chosen the experimental results provided by Lee (1992) [8].

Description of the material

Two kinds of limestone subsequently named MOLS and POLS, with similar gradations (Figure 5) and different particle sizes have been chosen for the experiments. MOLS is the limestone with comparatively small particles and POLS is the material with larger particles. The ratio between the respective mean diameters of MOLS and POLS grains is about 10.

Crushing test

Crushing tests similar to those presented in Figure 7 were performed on different particle sizes [8]. With the increase of

the average particle size, the crushing strength of the particles decreases (Figure 8). According to fracture mechanics, this is due to the distribution of micro-cracks with are statistically more present or initially larger in large particles. According to Marsal formula (Equation. 1), the crushing function for the limestone is:

P

a

=

4.508

d

m1.65

(11)

The values of the two parameters can also be calculated λ = 1.65 and m = 8.57.

Sieve size (mm)

Figure 5 : Similar gradation for material MOLS and POLS, Lee (1992) [8]

Figure 6: General size effect relationship

Figure 7 : Crushing test on limestone particles, Marsal (1967) [10]

Figure 8: Relationship between average particle size and crushing stress Lee (1992) [8]

Triaxial tests

In order to obtain the failure envelope of the materials MOLS and POLS, several triaxial tests were performed by Lee (1992) [8]. For the material MOLS, confining pressures equal to 40kPa, 80kPa, 160kPa and 320kPa are selected. For the material POLS, confining pressures equal to 40kPa, 80kPa, 120kPa and 160kPa are chosen. The results are presented in Figure 9 (deviatoric stress vs axial strain) and Figure 10 (volumetric strain vs axial strain). The gradations before and after loading are compared in Figure 11.

From Figures10 to 12, the following conclusions can be drawn:

- The specimens with comparatively smaller particles (MOLS) exhibit a greater shear strength than specimens with larger particles (POLS). This difference is caused by size effect in granular materials;

- The POLS specimens with larger particles have more ruptures than the MOLS specimens, which is very clear by comparing the gradation curves before and after the experiments (Figure 11) ;

MOLS

- The POLS specimens are more contractant than the MOLS specimens (Figure 10);

Figure 9 : Comparison of stress-strain relationship between POLS and MOLS materials, Lee (1992) [8]

Figure 10: Comparison of volumetric evolution between POLS and MOLS materials Lee (1992) [8]

Calculation of the failure envelope

The aim of the method is to estimate the failure envelope of the specimen with larger particles (POLS) from that of the specimen with smaller particles (MOLS). From triaxial experiments and Equation 7, the failure envelope for the MOLS material can be easily fitted:

0.89

(

)

0.71

M n n

τ σ

=

σ

(12) Then the failure envelope of the POLS material should have the following form, according to Equation 10:

( )

3(1 ) 3(1 )

(

)

b b m m b P P P n M n M n M M

D

D

A

D

D

τ σ

τ σ

σ

− − − −

⎛

⎞

⎛

⎞

=

_⎜

_⎟

⋅

=

⋅

_⎜

_⎟

⋅

⎝

⎠

⎝

⎠

(13)

Figure 11 : Comparison of gradations after different triaxial tests, Lee (1992) [8]

(1) Experimental shear strength envelope for MOLS (2) Experimental shear strength envelope for POLS (3) Shear strength for large size material extrapolated

from MOLS, through the size effect rule

Figure 12: Comparison of experimental and predicted shear strength envelopes POLS MOLS

1

3

2

The two parameters m and b can be fitted: m = 8.57 and b = 0.89. DM = 0.3 mm and DP = 2.5 mm are the maximum particle diameters of materials MOLS and POLS, respectively. Substituting Equation 12 into Equation 13, the failure envelope for POLS can be predicted:

0.89

(

)

0.657

P n n

τ σ

=

σ

(14) The failure envelopes obtained by this method and from triaxial experiments have been compared, as shown in Figure 12. The prediction given by the theory fits very well the experimental results, although the size effect is not so much marked, due to the particularly high values of both parameters m and b (in Equation. 13, it may be shown that when b tends towards value of 1, the size effect vanishes as the shear strength envelope tends towards the Coulomb failure criteria straight line) and because the gap between the maximum grain sizes is not so large.

Conclusion

A new method for the evaluation of the mechanical behaviour of rockfills materials based on size effects has been presented. In this method, the Weibull theory and experimental observations by Marsal about the rupture of granular particles have been connected through the relationship between Weibull’s modulus m and the parameter

λ in Marsal’s formula. Basically, these two parameters give

an indication on the “rupture capability” of particles. Size effects have also been introduced and used in the method for connecting the strength of specimens with comparatively large particles and that of specimens with small particles. A series of crushing tests and triaxial experiments on limestone granular assemblies have been used to validate the method. Comparison of model prediction and experimental results demonstrates that the method is able to predict with very good accuracy the maximum strength of coarse granular materials.

Acknowledgements

This work is a part of the French research Project ECHO (Scale effects in fill works in Civil Engineering), sponsored by the French National Agency for Research, targeted to validate the method exposed above and explore its limitations, through a wide experimental testing program, including 1m diameter triaxial tests on granular fills.

References

[1] Barton N., Kjaernsli B. (1981). Shear Strength of Rockfills. Journal of Geotechnical Engineering Vol. 107, pp. 873-891.

[2] Charles J., A., Watts K. S. (1980). The influence of confining pressure

on the shear strength of compacted rockfills. Géotechnique Vol.30, pp.

353-367.

[3] Charles J.A., Soares M.M. (1984). Stability of compacted rockfills

slopes. Géotechnique Vol.34, pp.61-67.

[4] De Mello V.F.B. (1977). Reflections on design decisions of practical

significance to embankment dams. Géotechnique Vol. 27, pp. 281-355.

[5] Frossard E. (1979). Effect of sand grain shape on interparticle friction :

indirect measurements by Rowe’s stress-dilatancy theory.

Géotechnique Vol. 29, pp.341-350.

[6] Frossard E. (2005). Macroscopic behaviour of granular materials used

in dam construction – Coyne et Bellier Report on Investigation Program on Micromechanics in Rockfills Dams (sponsored by France Ministere de la Recherche) –

[7] Frossard, E. (2009) On the Structural safety of large Roockfill Dams –

Proceedings XXIII° International Conference on Large Dams, Brasilia, May 2009

[8] Lee D.M. (1992). The angles of friction of granular fills. Doctor degree thesis, Cambridge university

[9] Marachi N.D., Chan C.K., Seed H.B., Duncan J.M. (1969) . Strength

and deformation characteristics of rockfills materials. Report No.

TE-69-5, Department of civil engineering, University Of California, Berkeley.

[10] Marsal, R.J. (1967). Large-scale testing of rockfills materials. Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 93, pp. 27-44.

[11] Weibull W. (1951). Statistical distribution function of wide

applicability, Journal of Applied Mechanics, ASCE, Vol.18, pp.