permanent behaviour

Permanent deformation in UGMs is mainly related to resilient stresses or strains and number of load cycles. The effects of these two factors are often separated and the models for predicting permanent deformations can assume the form of:
                                                                                                               (1)
Amongst the numerous models that are available in the literature, Hornych (2004) shows that the Gidel model is particularly suitable to describe the variation of permanent deformations with stress level:
                                                                                              (2)
The link between permanent deformation and number of load cycles can instead be expressed by the simple Sweere model:
                                                                                                                                                (3)
or by the slightly more elaborated Hornych model:
                                                                                                                          (4)
All these models are derived empirically by fitting curves to laboratory data and do not take into account the effect that moisture content has on the permanent behaviour of the granular material. In theory, a different set of fitting parameters would need to be calculated for each value of moisture content considered. The ME-PDG, instead, uses a modified version of the Tseng and Lytton model that relates permanent strains to resilient strains, number of cycles and moisture content:
                                                                                                                      (5)
where β, ρ and ε₀/ε_r are functions of water content.

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Resilient Behaviour

In dealing with granular materials, the elastic properties are often defined by the resilient modulus (which replaces the modulus of elasticity to indicate the nonlinearity of the behaviour) and Poisson’s ratio. A common approach in dealing with the stress dependency of granular materials stiffness is the K-θ model, which expresses resilient modulus as a function of the sum of the principal stresses, or bulk stress.
                                                                                                                                (1)
Thanks to its simplicity, this model has been widely adopted for analysis of stress dependence of material stiffness. However, it has the drawbacks of assuming a constant Poisson’s ratio and of ignoring the effect of other stress parameters apart from the bulk stress. For this reason, several more elaborated versions of this model have been suggested such as, for instance, the so called “Universal Model”, first suggested by Uzan and then adopted by the ME-PDG:
                                                                                             (2)
This model includes the 3D effect of shear, or deviator, stress by means of the octahedral stress. Most of the models of this family have empirical origins and are mainly used in the US.
A more theoretical approach is taken by those models that characterise the non-linear behaviour by decomposing stresses and strains into volumetric and shear components. In this case, resilient modulus and Poisson’s ratio are replaced by bulk and shear moduli:
                                                                                                                                                  (3)
                                                                                                                                                (4)
where
                                                                                                                                 (5)
                                                                                                                                            (6)
                                                                                                                                 (7)
                                                                                                                            (8)
In 1980 Boyce used this type of approach to develop a theoretical nonlinear elastic model for the stress-strain relationship of granular materials:
                                                                                                                         (9)
                                                                                                                         (10)
this model has then been employed by many other researchers, particularly in mainland Europe. It is important to note that it is possible to express resilient modulus and Poisson’s ratio in terms of bulk and shear moduli:
                                                                                                                                   (11)
                                                                                                                                          (12)
This can be useful considering that our current approach to including the effect of moisture content in granular materials makes use of the resilient modulus.

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UGMs - Introduction

Over the years, many researchers have studied the complex behaviour of granular materials, using laboratory and in situ testing techniques. An extensive literature review was carried out by Lekarp et al. (2000) to collect findings from previous research and summarize in two companion papers the state of knowledge on resilient and plastic properties of granular materials. In this work the authors point out that these properties are affected by numerous factors such as stress, density, grading, moisture content, stress history, particle shape and load frequency, nonetheless the effect of stress parameters is certainly dominant.
If granular materials are simulated by means of Layered Elastic algorithms or using the Method of Equivalent Thickness, this stress dependency is usually dealt with through iterative processes (Figure 1):
Figure 1: Iterative determination of resilient modulus
The complexity of the problem meant that a large number of models can be found in the literature. An interesting review of available models for the prediction of permanent deformations in UGMs has also been conducted as part of the SAMARIS project by Hornych et al. in 2004.
Hereafter we present a selection of the main approaches to resilient and permanent behaviour of UGMs that might be implemented in the WLPPS.

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