Page 211 - Analysis and Design of Energy Geostructures

184 Analysis and Design of Energy Geostructures

found subsequent applications to the modelling of geomaterials. Early applications of
the multisurface concept to geomaterials have been presented by Prévost (1977, 1978)
and Mrˇ oz et al. (1978, 1979, 1981), for example. Early applications of the bounding
surface concept to geomaterials have been proposed by Mrˇ oz et al. (1978, 1979),
Dafalias and Herrmann (1982), Al-Tabbaa and Wood (1989), Whittle (1993) and Li
and Meissner (2002), for example.
The theory of multisurface plasticity models the stress strain relation of materials
by various linear segments of constant tangential moduli. In the stress space, this con-
cept is associated with various yield surfaces f 1 ; f 2 ; f 3 ; .. . ; f i , with f 1 being the initial
yield surface and f 2 ; f 3 ; .. . ; f i defining regions of constant plastic work hardening mod-
uli (Yu, 2006). In this context, the yield surface moves (and may change in shape)
upon loading.
The theory of bounding surface (or two surface) plasticity models the stress strain
behaviour of materials through two yield surfaces, that is an inner yield surface f 5 0
and an outer yield surface F 5 0. The theory bounding surface plasticity coincides
with the multisurface theory proposed by Mrˇ oz (1967) with a continuum of interme-
diate loading surfaces where the distribution of these surfaces is analytically described
a priori Krieg (1975). The location of the inner and outer surfaces completely
describes the distribution of all intermediate surfaces so reference to them is not
necessary (Yu, 2006).
Both of the previous theories can be extended to account for the influence of tem-
perature on the irreversible behaviour of the material. Typically, these theories are
coupled with those characterising hardening plasticity and critical state plasticity.
A constitutive model resorting to the theories of bounding and multi surface plas-
ticity for geomaterials such as soils under nonisothermal conditions has been presented,
for example, by Laloui and François (2009). The considered model, which is reported
in Appendix C, has been based on successive developments presented by Laloui
(1993), Modaressi and Laloui (1997) and Laloui and Cekerevac (2008). Models of this
type may be employed in the analysis and design of energy geostructures to capture
detailed aspects of the mechanical behaviour of geomaterials under nonisothermal
conditions.

4.10.8 Thermoelastoplastic stress strain relations
The increment of elastic strain at each point of a material characterised by a thermoe-
lastic, plastic behaviour or by a thermoelastic, thermoplastic behaviour, can be written
as (e.g. considering a single plastic mechanism)

p 0
e
ð4:111Þ
dε 5 dε kl 2 dε 5 C klij dσ 2 β dT
kl
kl kl ij