Page 212 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 185

The previous relations can be written as
dσ 5 D ijkl dε kl 1 β dT 2 dε p ð4:112Þ

0
kl
ij kl
The increment of plastic deformation is given by the flow rule. Therefore
Eq. (4.112) can be rewritten as

dσ 5 D ijkl dε kl 1 β dT 2 dλ p @g ð4:113Þ
0
ij kl @σ kl
Once an expression of the form introduced through Eq. (4.113) is established, the
plastic multiplier must be defined. In this context, Eq. (4.113) must fulfil the unique-
ness condition of the stress strain solution formulated by Prager (1949). The value of
the plastic multiplier can be computed by substituting Eq. (4.113) in either among the
consistency Eqs (4.91), (4.101) and (4.106) written for a perfectly plastic material
whose yield function is insensitive to temperature, for a hardening material whose
yield function is insensitive to temperature, and for a hardening material whose yield
function is sensitive to temperature, respectively. In this context, the vectorial notation
instead of the tensorial one can be used for the strain and the stress tensors
(Timoshenko and Goodier, 1951):

T
ð4:114Þ
ε yz
dε i 5 ε x
ε y
ε z ε xy
ε xz

T
ð4:115Þ
σ z σ xy
σ xz
σ yz
σ y
dσ i 5 σ x
The consistency equation for a material characterised by a thermoelastic, perfectly
plastic behaviour reads

D ij dε j 1 β dT 2 λ p 5 0 ð4:116Þ
@f @g
j @σ j
@σ i
The consistency equation for a material characterised by a thermoelastic, plastic
hardening behaviour reads
~

D ij dε j 1 β dT 2 λ p 1 p λ p 5 0 ð4:117Þ
@f @g @f @h k @g
~ @ε
j
@σ i
@σ j
@h k
i @σ i
The consistency equation for a material characterised by a thermoelastic, thermo-
plastic hardening behaviour reads
B

D ij dε j 1 β dT 2 λ p 1 dT 1 p λ p 5 0 ð4:118Þ
@f @g @f @f @h k @g
j B @ε
@T @σ i
@σ j
@σ i
i
@h k