Page 205 - Analysis and Design of Energy Geostructures

178 Analysis and Design of Energy Geostructures

Eq. (4.96) represents the condition at which the material stress state moves towards the
inside of the current yield surface. Eq. (4.97) corresponds to the condition in which the
material stress state remains or travels on the current yield surface. Eq. (4.98) represents
the condition in which the material condition has the tendency to go outside the yield
surface, but because this phenomenon is not possible (i.e. stress states cannot lie outside
the yield surface in the present framework) the current yield limit changes (i.e. it expands
to a larger one for a hardening material while it contracts for a softening material).
According to Prager (1949), when loading of a material characterised by a plastic
hardening behaviour happens, the material hardens while the stress state stays on a
new expanded yield surface. This is the condition of consistency for hardening materi-
als and indicates that loading from a plastic state leads to another plastic state.
According to this condition, starting from a plastic sate with:

~ ~
Current yield surface f σ ij ; h i 5 0 and df dσ ij ; dh i . 0 ð4:99Þ
the induced loading should result in a new plastic state where:

~ ~ ~
New yield surface f σ ij 1 dσ ij ; h i 1 dh i 5 0 and df dσ ij ; dh i 5 0 ð4:100Þ
Therefore the condition of consistency for the yield function insensitive to temper-
ature of a material characterised by hardening plasticity reads
~
df 5 @f dσ ij 1 @f dh i 5 0 ð4:101Þ
~
@σ ij
@h i
For the case of a material characterised by a strain hardening behaviour, the hard-
ening parameter may be the plastic strain. Accordingly, Eq. (4.101) can be written as

~
df 5 @f dσ ij 1 @f @h p dε 5 0 ð4:102Þ
p
~
@h @ε ij
@σ ij
ij
The consistency equation is used to determine the plastic multiplier and the magni-
tude of plastic strains. This aspect is addressed later.
The stress conditions characterising the reversible and irreversible response of materials
characterised by a thermoelastic, thermoplastic hardening behaviour are more complex.
According to Prager (1949) and Prager (1958), the only admissible states of stress
temperature are those that lie inside or on the yield surface after the loading: if the
material stress temperature state is inside theelastic domain theresponseispurely elastic;if
the material stress temperature state is on the yield surface it is elastoplastic. Mathematically,
the admissible states under nonisothermal conditions can be expressed as (Prager, 1958):
B

f 5 f σ ij ; h i ; T # 0 ð4:103Þ