Resilient modulus

From research carried out during the 1960’s, Hicks and Monismith (1971) summarised that the resilient response of granular materials under short-duration dynamic loading is significantly influenced by:

• Stress levels (confining pressures); • Degree of saturation;

• Dry density;

• Fines content (percentage passing 0.075mm sieve); and

• Aggregate properties (density, type, particle angularity, particle texture) Hicks and Monismith (1971) concluded that the resilient response of untreated granular materials is most significantly affected by the stress level and can therefore be related to the confining pressure, σ3, or to the bulk stress, θ = σ1 + σ2 + σ3, as follows, for as long as shear failure does not occur:

2 3 1 k r k M = σ Eq. 7 or 4 3 k r k M = θ Eq. 8

where Mr = Resilient modulus [MPa] σ3 = Confinement pressure [kPa]

θ = Bulk stress = σ1 + σ2 + σ3 [kPa] k1, k3 = model coefficients [MPa] k2, k4 = model coefficients [-]

The model shown in Equation 7 is used model for fine-grained soils, while the the model as shown in Equation 8 has been widely accepted to describe the stress-

σa ε εf σmax Esec Etan

dependent behaviour of a granular material, amongst others for its simplicity. The latter is often referred to as the Mr-θ model. On a log-log scale this model represents a linear function, whereby the k1-value is a measure of the intersection with the y- axis, while the k2-value indicates the slope of the line. The Mr-θ model as shown in Equation 8 will be used for analysis of the resilient modulus test results in this study. It is however not uncommon for a material to exhibit the following phenomena concurrently:

• An increase in resilient modulus (stiffening) with increasing confinement (σ3) at a constant deviator stress (σd); and

• A decrease in resilient modulus (softening) with increasing deviator stress (σd) at a constant confinement (σ3).

Both abovementioned changes in the stress conditions result in an increase of the bulk stress θ. The Mr-θ model can describe overall stiffening (positive k4-value) or overall softening (negative k4-value), but not stiffening due to increase in σ3 and softening due to increase in σd at the same time. Also, the Mr-θ model predicts an ever-increasing stiffness with increasing levels of bulk stress. The Mr-θ model is therefore fundamentally incorrect. It is however capable of accurately describing the stress-dependent behaviour of certain materials under certain stress conditions. May and Witczak (1981) found that the shear modulus of a granular material is not only dependent on the stress state but also on the level of shear strain. When the shear modulus is substituted for the resilient modulus using Poisson’s ratio, a similar dependency exists for the resilient modulus. This effect is neglected in Equations 7 and 8. Uzan (1985) concluded that these equations are therefore only valid in the range of low strain values and suggested the following extensions:

5 2 3 1 ak k r k M = σ ε for εa >10-5 Eq. 9 or 6 2 3 1 k d k r k M = σ σ for σd > 0.1σ3 Eq. 10

where εa = axial strain [-]

σd = deviator stress (σ1 – σ3) [kPa] k5, k6 = model coefficients [-]

other symbols same as Equation 8

Uzan (1985) concluded that Equation 10 appeared to be in good agreement with all aspects of granular material behaviour. Uzan et al. (1992) added that Equation 10 is valid for both granular and fine-grained materials. It is therefore also referred to as “the universal model”. As the material changes from granular to fine-grained, the k2- value approaches zero, the model degenerates into the model shown in Equation 7.

Whereas the Mr-θ model cannot describe stiffening due to increase in σ3 and softening due to increase in σd at the same time, the universal model (Equation 17) is capable of concurrently describing the opposite effects the confinement stress and deviator stress may have on the resilient modulus. The universal model is therefore physically more correct than the Mr-θ model.

It is however also common for granular materials to show stress-stiffening with increasing deviator stresses at a constant confinement pressure until a maximum stiffness is reached at a certain deviator stress. An increase in load levels beyond this deviator stress results in stress softening again. The universal model is not capable of, at a constant confinement stress, describing such initial stress-stiffening up until a certain deviator stress level and subsequent stress-softening beyond this deviator stress level.

In order to be able to describe both stress-stiffening and stress-softening at a constant confinement stress, van Niekerk (2002) adjusted the power-law relation between the resilient modulus and the deviator stress, as shown in Equation 10, to a parabolic relation as follows:

(

)

(

)

(

)

where S.R. = stress ratio (σd/σd,f) [-]

σd,f = deviator stress at failure [kPa] k7, k8, k9 = model coefficient [-]

other symbols same as in Equations 8 and 10

A negative value of the k7-coefficient would result in a parabola with a local maximum, which is capable of describing initial stress stiffening and subsequent stress softening. The opposite, a positive value of the k7-coefficient, would result in a parabola with a local minimum, i.e. initial stress softening and subsequent stress stiffening.

One has to be careful when predicting resilient modulus values outside the experimental range of deviator stress values. Unrealistic resilient modulus values may be predicted if the quadratic part of the parabola (k7(S.R.)2) becomes dominant and a small increase in deviator stress may result in a large increase or decrease in the resilient modulus. This is illustrated by an example shown in Figure 38.

The experimental data in Figure 38 shows stress softening with increasing deviator stress levels. When the parabolic model is fitted to the data a good fit is obtained, but unrealistic resilient modulus values are predicted for deviator stresses outside the experimental range.

Various researchers, amongst others Jenkins (2000) and van Niekerk (2002) have shown that the resilient modulus models developed for granular material can also be used to describe the stress-dependency of stabilised granular materials.

0 1000 2000 3000 4000 5000 6000 7000 8000 0 500 1000 1500 2000 2500

Bulk Stress θ [kPa]

R e s il ie nt M odul u s [ M P a ] 50 100 250 Model 50 kPa Model 100 kPa Model 250 kPa σ3 [kPa]: Figure 38: Measured resilient modulus of a foamed bitumen stabilised material

with fitted the parabolic model of Equation 11 (Jenkins and Ebels, 2004)