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Empirical Compression Models for Granular Media



2.3.5 Empirical Compression Models for Granular Media Influence of Saturation

Kjærnsli and Sande (1963) and Pigeon (1969) showed that, by flooding rockfill specimens, an increased and accelerated rate of compression was recorded, as presented in Figure 14. This behaviour was expected as the rate of dissipation of pore water pressure from rockfill specimens is much higher than that from clays and fine sands.

Also, this finding was in agreement with the field observation that repeated wetting of the fill during construction causes an accelerated compression under constant load, leading to a reduced compressibility with time. Fumagalli (1969) and Fumagalli et al.

(1970) reported that the stress-strain behaviour of the rockfill in dry-wet condition initially follows the dry state curve, and when subjected to saturation at elevated stress level, is transformed to follow the wet state curve (Fig. 15). Nevertheless, this could be attributed to the lubrication by water of the interparticle contact points. Later research by Bertacchi and Bellotti (1970), Marsal (1973), Donaghe and Cohen (1978), Frassoni et al. (1982), Hsu (1984) and Lee (1986) validated these findings for various types of material.

Chapter 2: Critical review of granular media with special reference to railway ballast



Adding water

Preload effect Short duration

Long duration + water added

Long duration

Figure 14. Effect of time and water on settlement of rockfill (after Kjærnsli, 1965)

0 4 8 12

Axial strain, %

0 1000 2000 3000

Axial stress, kPa




Initially dry

Initially saturated (Saturation)

Figure 15. Compressibility of rockfills in dry and saturated states (after Fumagalli, 1969)

(1961) found that there is a linear relationship between stress and strain in a double logarithmic scale. Janbu (1963) proved that the model could be applied to various

rockfills, the general form of the relationship is given by:

b v v =aσ′

ε (2.4)

where εv is the vertical or volumetric strain, σ´v is the applied vertical stress, and a and b are coefficients to be determined from test data, with b varying with initial density.

Schultze and Coesfeld (1961) applied the above model to data from 1-D repeated compression tests on basalt ballast and derived the following relationship between the tangent constrained modulus M and the applied vertical stress σ′v:

b -v1


M= ′σ′ (2.5)

where a′ and b are material constants that vary with the number of load applications and depend on the degree of compaction. In addition, it was found that irrespective of degree of compaction, the increase of M with the number of load applications was considerable in the initial stage of loading but its rate of variation diminished as the load was repeated (Fig. 16). These observations agreed with field observations, which showed that the stiffness of the railway track (ie. of ballast layer) is at its lowest value after construction or maintenance work. However, with each train passage the density of ballast layer increases, and the settlement rate decreases, ie. the stiffness of the track increases until a state of resilience is reached that could be modeled using the laboratory results.

Hansen (1968) attempted to account for the effect of the initial compaction state and he

Chapter 2: Critical review of granular media with special reference to railway ballast

1 10 100 1000

Number or load repetition, N

10 100 1000

Modulus of compressibility, M = 1/mv (MPa)

Loose (ρ < 1.5 t/m3 )

Medium dense (ρ = 1.5 t/m3 ) Dense (ρ > 1.8 t/m3 )

Loose (ρ = 1.4 t/m3 ) Unloading stage

1-D compression test

Plate loading test

Figure 16. Dependence of ballast compressibility on initial density and number of load repetitions (after Schultze and Coesfeld, 1961)

separated the initial void ratio in the stress-strain relationship:


c v 1 o

a a e d ⎟

⎜ ⎞

= ⎛ σ

ε (2.6)

where εa is the axial strain, σv is the applied vertical stress, eo is the initial void ratio and a1, b1, c and d are parameters to be obtained from experimental results. However, the difference in the particle size between sand and railway ballast may impede direct application of this model to the behaviour of ballast. Because there was no indication that this relationship hold upon unloading, it was unlikely to evaluate the elastic behaviour.

Fumagalli (1969) showed that the load-deformation relationship for rockfill during 1-D compression was in good agreement with a power law. Interpreting data from results by Sowers et al. (1965), Tombs (1969) and Marsal (1973), Parkin (1977) proposed the following relationship for deformation of rockfill with pressure variation:


H= σ′v

Δ (2.7)

where ΔH is the settlement of the rockfill specimen, σv is the applied vertical stress, and A and B are material parameters to be determined from test data. The shortfall of this

equation is that it does not identify the elastic component of the settlement, ie. no indication on how Eq. (2.7) changes for the unloading branch of the graph to account for the rebound that rockfill may exhibit. Also the initial density is not separated from the other effects.

Hardin (1987) identified the inherent deficiencies of the previous models as follows:

• the e-log σ′v model predicts e →

as σ′v → 0 and at high stress levels it predicts negative void ratios;

• the power model ignores progressive development of plastic deformation during first loading;

• the power model predicts εv > 1 for very large stresses for a and b positive.

Making use of results from published studies (Schultze and Moussa, 1961; Kjærnsli and Sande, 1963; Roberts, 1964, Leslie, 1975; Schmertmann, 1986), Hardin presented a more comprehensive semi-empirical equation to describe the void ratio variation over a wide range of vertical stress:

Chapter 2: Critical review of granular media with special reference to railway ballast


a v o S1D p

+ 1 e

1 e



⎜⎜ ⎞

= ⎛ σ′ , and (2.8)

( )


CR v b 2 o

1D 1D

1D 1D

+ e 1


+ S

S max min




⎛ σ′


− σ′

= − (2.9)

where 1/e is the reciprocal of void ratio, 1/eo is the intercept at σ´v = 0, S1D is the dimensionless stiffness coefficient for one-dimensional strain, σ´v is the applied vertical stress, pa is the atmospheric pressure, S1Dmin, and S1Dmax are the limits of S1D, σ´CR is the crushing reference stress, σ´b is the “break point” at which the crushing mechanism begins showing its effect, and b´ and c´ are coefficients to be determined from test data.

It was shown that b´=0.5 fits a very large amount of data for granular media available in the literature, and that the crushability characteristic of the material is defined by three parameter σ´b, σ´CR and c´.

Although the equation accounts for the effect of initial density, particle size and shape and for the crushing effect on the stiffness of granular media, it is restricted to the first loading condition and when applied to rockfill like materials the results are not as consistent as for sands. Moreover, some of the input parameters (S1Dmin, σ´CR) are not well defined and/or lack physical correlation to materials behaviour.

Pestana and Whittle (1995, 1999) proposed a simple four-parameter elasto-plastic model that describes the non-linear volumetric behaviour of cohesionless soils in hydrostatic and one-dimensional compression. The relationship can simulate the accumulation of plastic deformations in loading-unloading-reloading cycles and the

hysteresis effect.

Two main assumptions were made for this model:

• the incremental volumetric strains (εv) can be subdivided into elastic and plastic components, where plastic strains occur throughout first loading of freshly deposited cohesionless soils;

• the tangent bulk modulus (K) for loading can be written as a separable function of the current void ratio e and mean effective stress p′.

During the loading stage the model is described by the dimensionless relationship:


a a

a p

p n C p



⎜⎜ ⎞

= ⎛ ′ , (2.10)

and during the unloading stage by:

3 1

a b a E

p p n C p K



= ⎛ ′ (2.11)

where, K and KE represent the tangent bulk modulus for loading and unloading stage respectively, pa the atmospheric pressure (an arbitrary reference stress proposed by Janbu, 1963), p′ is the mean effective stress, n is the instantaneous porosity of soil, and Ca, Cb and d′ are constants which together with n infer the effects of particle mineralogy, size, shape, texture and grading on compressibility. The model is valid for a very large range of stress and by differentiating Eqs. (2.10) and (2.11), the incremental volumetric strains can be obtained. Although, this model seems to be very simple it was developed for sand subjected to high level of stress to allow full collapse of sand skeleton. Also, the crushing of grains is not defined by a separate parameter.

Consequently, its applicability to the behaviour of railway ballast (having coarse grains

Chapter 2: Critical review of granular media with special reference to railway ballast

with uniform gradation ie. particle fragmentation is expected even at a lower level of stress) may require extensive modifications.