Analysis of the micro-mechanics of shear wave propagation is shown in this work to be a powerful
method to study soil stiffness at small-strain levels. Discrete element modelling of triaxial
tests successfully captures the small-strain stiffness of granular materials. Various interpretative
methods are adopted to determine the travel time for shear waves that are transmitted through
assemblies of perfectly spherical particles with different densities, allowing small-strain stiffness
under anisotropic stress states to be measured. Along with the micro-scale data gathered from
true triaxial simulations, the stiffness anisotropy data are used to assess the effect of each principal
stress on the small-strain stiffness. In addition, these dynamic stiffnesses allow the void correction
function that quantifies the effect of sample density on stiffness to be studied. A notable conclusion
obtained is that the inclusions of micro-scale data (i.e. the coordination number) will likely give a
more accurate void ratio correction function when compared with the traditional function used in
the interpretation of laboratory test data where particle level information is unavailable.


Non-linear stiffness of soil is a long-standing topic of interest in geomechanics, with the degree
of nonlinearity being influenced by many factors including the stress path, the sample density
and the particle size distribution. These issues are extensively studied here using discrete method
simulations of triaxial tests in which samples were sheared along with different stress paths. Both
the macro-scale and the micro-scale data gathered from the true-triaxial DEM simulations allow
the non-linearity of stiffness and the interactions at the particle scale to be further understood.
The coordination number and the second-order fabric tensor provide a complementary insights to
further understanding of macro-scale response of granular materials. At a given strain level, both
the dynamic stiffness and the static stiffness are measured, allowing the degree of nonlinearity of
stiffness to be studied. The framework of kinematic modified yielding points proposed by Jardine
(1992) that identifies three main zones (i.e. linear elastic behaviour, non-linear elastic behaviour
and elasto-plastic behaviour) provides a benchmark for the gap between the dynamic stiffness and
the static stiffness to be studied in an effective manner. A key observation is that the dynamic
shear stiffness tends to increase to a peak value before experiencing of a decrease in magnitude.
Simulations of triaxial tests in which the samples were sheared along different stress paths allow
the yield surfaces to be captured, reconfirming that the sub-yield surfaces are dependant upon the
sample density and the confining effective stress.

The stress history and the particle size distribution (PSDs) have a large influence on the shape of
the non-linear stiffness degradation curve. In conjunction with the static stiffness, the dynamic stiffness was measured to quantify the effect of the over-consolidation ratio (OCR) on the shear
stiffness, resulting in the observation that the reductions in stiffness with increasing the OCR values
has strong link to drops in coordination number, while the input work required during shearing
triaxial tests is almost identical for samples with different OCR values. The stress history has a
measurable impact on the shear wave propagation, with higher travel time for samples with higher
values of OCR, indicating that both the dynamic stiffness and the static stiffness obtained from
DEM simulations reduce as OCR increases. The mathematical formulations used to capture the
non-linear stiffness degradation curves are examined in detail, the hyperbolic form is found to
closely capture the reductions in shear stiffness with increasing strain. Several DEM simulations
that consider effects of the coefficient of uniformity (Cu) on the non-linear behaviour of granular
materials are performed, arriving at some key observations that: (i) samples with lower values
of Cu have higher stiffness at small-strain levels; (ii) a higher amount of work should be input
for samples with coarse particles to attain a particular strain level; (iii) more energy dissipation is
observed for samples with a lower Cu value.