Page 206 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 179

Starting from an initial state (σ ij ; T) (for geomaterials, e.g. at a given void ratio)
and considering changes in stress and temperature occurring at rates dσ ij and dT, the
variation of f can be written as:

B
df 5 @f dσ ij 1 @f dh i 1 @f dT ð4:104Þ
B
@T
@σ ij
@h i
According to Hueckel and Borsetto (1990), the following situations are
conceivable:
1. The initial state lies inside the yield surface (f , 0) and dσ ij and dT are such that
the final state remains inside it (loading or unloading). In this case no plastic strain
~
is produced. Mathematically (Di Donna, 2014): dh i 5 0 and df can be positive
(loading), negative (unloading) or null (neutral loading).
2. The initial state lies on the yield surface (f 5 0) and dσ ij and dT are such that the
final state is inside it (unloading). In this case the plastic strain remains constant.
~
Mathematically (Di Donna, 2014): dh i 5 0 and df , 0. Thus:

dσ ij 1 dT , 0 ð4:105Þ
@f @f
@T
@σ ij
3. The initial state lies on the yield surface (f 5 0) and dσ ij and dT are such that the
final state remains on this setting. In this case, the condition of consistency for the
yield function sensitive to temperature of a material characterised by hardening
plasticity must stand true and reads:

~
df 5 @f dσ ij 1 @f dh i 1 @f dT 5 0 ð4:106Þ
~
@T
@σ ij
@h i
In this latter situation, two possible phenomena can occur (Di Donna, 2014):
~
a. No plastic strain is produced (dh i 5 0) and the increment of stress is balanced by
the increment of temperature (neutral loading):
dσ ij 1 dT 5 0 ð4:107Þ
@f @f
@T
@σ ij
~
b. Plastic strain is produced (dh i 6¼ 0) (loading):
dσ ij 1 dT . 0 ð4:108Þ
@f @f
@T
@σ ij
When f 5 0, the continuity conditions postulated by Prager (1949) guarantee the
possibility to have unloading (case 2), neutral loading (case 3.a) or loading (case 3.b)
(Di Donna, 2014).