Page 203 - Analysis and Design of Energy Geostructures

176 Analysis and Design of Energy Geostructures

in early studies by Melan (1938) and Prager (1949), and it has been employed for the
analysis of geomaterials such as soils by Drucker et al. (1957), for example.
Materials behaving according to the theory of hardening plasticity are characterised
by a size, location or shape of the yield surface that may change under the develop-
ment of plastic deformations. In these situations, the yield function depends on the
stress state and the stress history, and may explicitly depend also on temperature.
Materials characterised by hardening and an elastic limit that is assumed to be
insensitive to temperature (isothermal yield surface) are characterised by the following
yield function

~
f 5 f σ ij ; h i 5 0 ð4:92Þ
~
where σ ij is the relevant stress and h i are the internal variables that define the harden-
ing. Materials characterised by hardening and an elastic limit that is considered to be
sensitive to temperature are characterised by the following yield function

~
f 5 f σ ij ; h i ; T 5 0 ð4:93Þ
The hardening rule defines the modification of a yield surface during the process
of plastic flow. In most situations, it is assumed that the shape of the yield surface
remains unchanged while this surface can change in size, location or both. Reference
is made to isotropic hardening when upon the development of plastic strains changes in
size of the yield surface occur but the centre and orientation of the yield surface in the
stress space remain the same. Reference is made to kinematic hardening when upon the
development of plastic strains the size of the yield surface remains the same but its
location changes in the stress space. Reference is made to mixed hardening when the
size of the yield surface changes upon the development of plastic strains and translates
in the stress space.
When plastic deformation makes the yield surface expanding in size, the material is
said to have a hardening behaviour, which makes it more difficult to yield. In contrast,
when plastic deformation makes the yield surface contracting in size, the material is
said to have a softening behaviour, which makes it easier to yield. Hardening (and soften-
ing) can be linear or nonlinear.
The dependence of plastic deformation of hardening materials on the history of
the stress state can be appreciated through Fig. 4.21 with reference to the idealised
stress strain relation characterising a material with a linear elastic behaviour and a
nonlinear hardening behaviour. Points G and F are characterised by the same state of
strain but different states of stress.
The internal variables that define the hardening are commonly assumed to be a
function of plastic strains. In these situations, the material is characterised by a strain

~
~
hardening behaviour and h 5 h ε .
p
ij