The behaviour of unbound granular materials in a pavement structure is stress — dependent. For that reason the linear elastic model is not very suitable. A non-linear elastic model, with an elastic modulus varying with the stress and strain level is, therefore, needed.
For isotropic materials, moduli depend only on two stress invariants1: the mean stress level, p, and the deviatoric stress, q, which are given in the general, as well as the axi-symmetric case (cylindrical state of stress with o1 = oaxial and o2 = o3 =
oradial as:
General
P = T
q = 2 0ij0ij with 0ij = 0ij pSij
In a similar way strain invariants can be introduced. The volumetric strain ev and the deviatoric strain eq, are defined as:
An invariant has the same value regardless of the orientation at which it is measured.
(9.3)
The stresses and strains are interconnected through the material properties as stated in Eq. 9.1 and the elastic (resilient) response of the material can be expressed according to Hooke’s law as a diagonal matrix:
|
Sv |
1 |
3 (1 — 2v) 0 2 |
p |
|
Sq_ |
= E |
_ 0 — (1 + v)_ |
q |
where E and v are the material stiffness modulus (or the resilient modulus, usually denoted Mr) and Poisson’s ratio respectively, defined as:
Дq Дє3
Mr or E = and v = — (9.5)
Дє1 Дє1
|
. * Д p and Дє„ = Kr |
where Д stands for the incremental change during the loading. An alternative formulation is:
where Kr and Gr are the bulk and shear moduli of the material. The bulk and the shear moduli are connected to the resilient modulus and the Poisson’s ratio through:
for isotropic materials.
The resilient modulus for most unbound pavement materials and soils is stress — dependent but the Poisson’s ratio is not, or at least to a much smaller extent. Biarez (1961) described the stress-dependent stress-strain behaviour of granular materials subjected to repeated loading. Independently, similar work was performed in the United States (Hicks and Monismith, 1971). Both results presented the к-0 model, which is written with dimensionless coefficients like:
3 p k2
Mr = ki pa and v = constant (9.8)
PaJ
where Mr is the resilient modulus, p is the mean stress, pa is the reference pressure (pa = 100 kPa) and k1, k2 are coefficients from a regression analyses usually based on repeated load triaxial test results. This model has been very popular for
describing non-linear resilient response of unbound granular materials. It assumes a constant Poisson’s ratio and that the resilient modulus is independent of the devia — toric stress. To address this latter limitation the Uzan-Witczak model — often called the “Universal” model — has become widely promoted, especially, in recent years, by authors in North America, e. g. Pan et al. (2006). It takes the form:
Mr = kpa 1 + and v = constant (9.9)
Pa PaJ
Subgrade soils are also stress-dependent and can also be modelled by one of the k-0 approaches. The principle difference between granular materials and many soils is that the former exhibit a strain-hardening stiffness whereas the latter, typically, exhibit strain-softening behaviour under transient stress loadings. In practice, the incorporation of non-linearity into the stiffness computations for subgrade soils is often less important than for granular materials as the stress pulses due to traffic loading will be a far smaller part of the full stress experienced by the subgrade than is the case for the unbound granular layer. Thus the error introduced by ignoring subgrade non-linearity will be correspondingly smaller.
In 1980, Boyce presented some basis for subsequent work on the stress-dependent modelling of the resilient response of cyclically loaded unbound granular material. The Boyce model takes into account both the mean stress and the deviatoric stress, with the bulk and shear moduli, K and G, of the material calculated as:
where Ka, Ga, and n are material parameters determined from curve fitting of repeated load triaxial tests results.
|
and q * = уо-j — ст3 |
Anisotropy of pavement materials is increasingly being recognised as a property that must be modelled if the pavement is to be adequately described (e. g. Seyhan et al., 2005). The Boyce model was modified to include anisotropy in the early 1990’s (Elhannani, 1991; Hornychetal., 1998). Hornych and co-workers ntroduced anisotropy by multiplying the principal vertical stress, cti in the expression of the elastic potential by a coefficient of anisotropy у so that p and q are redefined as follows (c. f. axi-symmetric part of Eq. 9.2):
and the stress-strain relationships are defined as:
Дєq* = 2 (ДЄ1* — Де/) = — and
q 3 1 1 ! 3Gr
Д n*
Дє„ * = Дє1*+ 2Де3* = — .
Kr
Kr and Gr, the bulk and shear moduli, respectively as:
The k— model, Universal model, Boyce model, and the modified Boyce model must be considered in pavement modelling to ensure a valid stress, strain, and deflection evaluation in pavements. When subjected to repeated loading, two types of deformations are exhibited, linear or non-linear elastic (or resilient) and plastic deformations. Models based on non-linear elasticity deal with resilient deformations only. Their biggest disadvantage is that permanent deformations cannot be modelled.
