Page 213 - Analysis and Design of Energy Geostructures

186 Analysis and Design of Energy Geostructures

From Eq. (4.116), the plastic multiplier can be computed for a material charac-
terised by a thermoelastic, perfectly plastic behaviour, and reads

@f
D ij dε j 1 β dT
j
dλ 5 @σ i ð4:119Þ
p
@g
@f D ij @σ j
@σ i
From Eq. (4.117), the plastic multiplier can be computed for a material charac-
terised by a thermoelastic, plastic hardening behaviour, and reads

@f
D ij dε j 1 β dT
j
dλ 5 @σ i ð4:120Þ
p
2 p
@f @g @f @ ~ h k @g
D ij
@ ~ h k @ε @σ i
@σ j
@σ i
i
From Eq. (4.118), the plastic multiplier can be computed for a material charac-
terised by a thermoelastic, thermoplastic hardening behaviour, and reads

@f D ij dε j 1 β dT 1 @f
j @T dT
dλ 5 @σ i ð4:121Þ
p
2 p
@f @g @f @ ~ h k @g
D ij
@σ j
@ ~ h k @ε @σ i
@σ i
i
Knowledge of the plastic multiplier(s) allows obtaining the complete stress strain
relation for the modelled material.
4.10.9 Three-dimensional thermoelastic, plastic or thermoelastic,
thermoplastic modelling
The three-dimensional modelling of thermoelastic, plastic (or thermoelastic, thermo-
plastic) problems, assuming for simplicity the temperature distribution to be known, is
more complex compared to the modelling of thermoelastic problems treated in
Section 4.9.4. The reason for this is that additional functions and unknowns are associ-
ated with the mechanical behaviour of the modelled material(s).
Constitutive models accounting for the reversible and the irreversible behaviour of
materials necessitate the definition of four key components: (1) the relations governing
the reversible behaviour of the material (i.e. elastic deformation); (2) the criteria (yield
functions) characterising the limit for which an irreversible behaviour may occur (i.e.
plastic deformation); (3) the relations (plastic potential functions) characterising the
irreversible behaviour of the material, including the definition of the mechanisms gov-
erning the development of plastic deformation and the direction of its evolution and
potentially, (4) the expressions (hardening rules) describing the magnitude of plastic
deformation. Aspects (1) and (2 4) characterise the elastic and plastic constitutive
descriptions of the modelled material, respectively.