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Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Obviously, there is a rotation of the main axes of the
response-envelopes. That means that the ratios of quasi-elastic
moduli depend on the stress ratio η.
This influence can be quantified. Figure 8 shows the ratio
Ev/Eh of the vertical stiffness Ev = Δσv/Δεv and the horizontal
stiffness Eh = Δσh/Δεh as a function of the stress-ratio η. For this
purpose data were analysed for stress-paths ασ = 90° and 270°,
(axial compression and extension) and ασ = 0° and 180° (radial
compression and extension).
Figure 8: Ratio Ev/Eh depending on the initial stress-ratio 
The dependence of the ratio Ev/Eh on the initial stress-ratio η
can be interpreted as a stress-induced anisotropy. Similar
observations are also made when investigating much smaller
stress- or strain-cycles, e.g. Ezaoui and Di Benedetto (2009) or
Hoque and Tatsuoka (1998). The coarser the sand, the more
distinctive is the difference between Ev and Eh, i.e. the ratio
Ev/Eh increases Hoque and Tatsuoka (1998). A detailed analysis
shows a stronger influence of the stress-ratio η on the vertical
than on the horizontal stiffness, see Bellotti, et al. (1996).
Figure 8 does not only show a stress induced anisotropy. At
the isotropic stress state with η = 0 the ratio Ev/Eh is ≠ 1. This
means, that there are no isotropic properties at an initial
isotropic stress state, i.e. there also is an inherent anisotropy.
Most authors come to similar conclusions. While Hoque and
Tatsuoka (1998) find out Ev/Eh ≥ 1 for all tested sands at
isotropic stress-states, Di Benedetto (2010) also finds ratios
Ev/Eh < 1 for the preparation-methods pluviation and vibration
and thus demonstrates a dependency of this ratio on the
specimen preparation-method. These discrepancies seem to be
due to several factors e.g. the grain-size distribution, the shape
of the specimen and the preparation-method.
4 SUMMARY AND FURTHER HINTS
Producing experimental or numerical response envelopes is a
convenient tool to investigate a soil’s incremental stress-strain
behaviour and to test or compare constitutive equations.
The investigation of the incremental stress-strain behaviour
of sand at low cycle loading procedures in triaxial testing
shows, that for stress-increments Δσ ≤ 50 kPa quasi-elastic
behaviour can occur after a low number of cycles. While the
influence of the sequence of the stress-paths on the quasi-elastic
strains seems to be negligible, a strong influence of the mean
pressure p on the size of the strain-response-envelopes is
observed.
For low number of cycles, the influence of an isotropic
prestress on the quasi-elastic strains seems to be negligible so
far. There is a stress-induced anisotropy, which can be shown
by the rotation of the axes of the ellipses depending on the
initial stress-ratio η.
Further triaxial tests are necessary in order to investigate e.g.
the influence of the void ratio and of a K0-preloading. Because
of the role of triaxial extension, further tests in extension region
will also be carried out. In addition plastic strains due to low
cycle loading will also be investigated; first results are already
available.
It is known, that some common constitutive models show
deficits when predicting deformations due to high and low cycle
loading processes, e.g. ratcheting in hypoplasticity, elastic
behaviour after the first un- and reloading in elastoplastic
constitutive models, missing anisotropy.
It is intended to use the results presented in this paper
together with future tests as a basis for calibrating and
validating more complex constitutive equations especially
developed for low cycle loading processes (Ehlers and Avci,
2011, Niemunis, et al., 2011).
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