The Thermodynamics of Porous Continua “Modern Trends in Geomechanics” Vienna, 25 June 2005 Prof. Guy Houlsby and Prof. Alexander Puzrin (Oxford University and ETH Zurich)
The Thermodynamics of Porous Continua
“Modern Trends in Geomechanics”Vienna, 25 June 2005
Prof. Guy Houlsby and Prof. Alexander Puzrin(Oxford University and ETH Zurich)
Oxford University and ETH Zurich
Thermodynamics and plasticity theory
• Two possible approaches to make theories consistent with thermodynamics– Develop the theory and then check retrospectively that
it obeys the laws of thermodynamics
– Use a framework for the theory which automatically enforces the laws of thermodynamics
• Second method preferred
Oxford University and ETH Zurich
The hyperelastic approach
( )ijg σ
ijij
gσ∂∂
−=ε
Gibbs free energy or(negative) complementary energy
Oxford University and ETH Zurich
The hyperplastic approach
( )ijijg ασ ,
( )ijijijd αασ &,,
ijij
gσ∂∂
−=ε
ijij
dgα∂∂
+α∂∂
=&
0
Gibbs free energy
Dissipation function
Oxford University and ETH Zurich
The hyperplastic approach
( )ijijg ασ ,
( )ijijijd αασ &,,
ijij
gσ∂∂
−=ε
Gibbs free energy
Dissipation function
ijij
dgα∂∂
+α∂∂
=&
0
Oxford University and ETH Zurich
The hyperplastic approach
( )ijijg ασ ,
( )ijijijd αασ &,,
ijij
gσ∂∂
−=ε
ijij
dgα∂∂
+α∂∂
=&
0
Gibbs free energy
Dissipation function
ijij
dα∂∂
=χ&
ijij
gα∂∂
−=χ
0=χ−χ ijij
⇒ {
Oxford University and ETH Zurich
Advantages of hyperplasticity
• The entire model is specified through two scalar functions– Energy stored
– Energy dissipated
• The behaviour is then derived by applying “automatic”procedures to these functions
• The models automatically obey thermodynamics
• Models can be simply compared, developed and extended by examining/altering the two functions
Oxford University and ETH Zurich
An example of hyperplasticity: von Mises
ijijijijjjii GKg ασ−σ′σ′−σσ−=
41
181
ijijkd α′α′= &&2
Oxford University and ETH Zurich
An example of hyperplasticity: von Mises
ijijijijjjii GKg ασ−σ′σ′−σσ−=
41
181
ijijkd α′α′= &&2
Elasticity
Oxford University and ETH Zurich
An example of hyperplasticity: von Mises
ijijijijjjii GKg ασ−σ′σ′−σσ−=
41
181
ijijkd α′α′= &&2
Plasticity
Oxford University and ETH Zurich
An example of hyperplasticity: von Mises
ijijijijjjii GKg ασ−σ′σ′−σσ−=
41
181
ijijkd α′α′= &&2
von Mises
Oxford University and ETH Zurich
Generalisation of the hyperplasticity method
• Hyperplasticity is itself a special case of a more general method pioneered by Ziegler– but the work done on hyperplasticity helps to show the more
general work in a new light
• Dissipation arises from two types of process:– Changes with time of “internal variables” such as the plastic strain
– Flow processes which involve spatial gradients, e.g. seepage
• Both of the above are treated in thermodynamics texts... but any one text tends to treat either one or the other
• Why not examine both?
Oxford University and ETH Zurich
Thermodynamics
• First Law
• Second Law
• Combine:
dQdWdU +=
θ≥
dQdSθ
=θ
−DdQdS
dSdWDdU θ+=+
⇒
Oxford University and ETH Zurich
Thermodynamics
• First Law
• Second Law
• Combine:
dQdWdU +=
θ≥
dQdSθ
=θ
−DdQdS
dSdWDdU θ+=+
Dissipation
⇒
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
dU
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
dW
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
dQ
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
dS
Oxford University and ETH Zurich
First and second laws for a porous medium
( ) ( )( )( ) ( ) ( )∫∫∫
∫ ∫
−+ρ+ρ+−−
=ρ+ρ+ρ+ρ∂∂
Sii
Vi
wii
S
wiiii
V Si
wi
wi
swst
dSnqdVgwvvdSvnpnvtn
dSnvwuvudVwuu
1
( ) ( ) ∫∫ ∫ ⎟⎠
⎞⎜⎝
⎛θ
−≥ρ+ρ+ρ+ρ∂∂
Si
i
V Si
wi
wi
swst dSn
qdSnvwsvsdVwss
dQθ
Oxford University and ETH Zurich
Combined first + second laws
( )
( ) ( ) iiiwi
wi
wiii
wi
wwswsijijij
mpvsumpvg
wuswsvwpvpdpdu
,,,,,111
~~~~~1~
θηρ
−+θ−ρ
−−ρ
++θ+θ+−−δ+σρ
=+
dSdWDdU θ+=+
Oxford University and ETH Zurich
Functions of state
Internal energy is a function of state
( )( ) ( )wwwss
ijijs
wwssijij
svwusvu
svwsvuu
,,,,
,,,,,,
+α∆=
α∆=
Internal energyof skeleton
Internal energyof water
Oxford University and ETH Zurich
Dissipation
• Dissipation is a function of:– state– rate of change of internal variables– flow
• Assumed to be a “quasi-homogeneous” function
( )iiijwwss
ijij msvwsvdd ηαα∆= ,,~,,,,,,,
ii
ii
ijij
zmmzzd η
η∂∂
+∂∂
+αα∂∂
= ~~
( )iiijwwss
ijij msvwsvzz ηαα∆= ,,~,,,,,,,
Oxford University and ETH Zurich
Combine the equations …
( )
( ) iwi
wi
wi
ii
iii
iw
i
ijijij
s
ww
ww
w
w
ss
ss
s
s
ijij
s
ijo
mpvsu
zmmzpvg
zu
ssuwv
vupw
ssuv
vupu
,,,
,,
1
11
~~
~~
~~~10
+θ−ρ
−
η⎟⎟⎠
⎞⎜⎜⎝
⎛η∂∂
−θρ
−+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−ρ
+
α⎟⎟⎠
⎞⎜⎜⎝
⎛
α∂∂
−α∂∂
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−+
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−+∆⎟⎟⎠
⎞⎜⎜⎝
⎛
∆∂∂
−πρ
=
Oxford University and ETH Zurich
( )
ii
i
ii
iw
i
ijijij
s
ww
w
ww
w
ss
s
ss
s
ijij
s
ijo
z
mmzpvg
zu
ssu
vvup
ssu
vvup
u
η⎟⎟⎠
⎞⎜⎜⎝
⎛η∂∂
−θρ
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−ρ
=
α⎟⎟⎠
⎞⎜⎜⎝
⎛
α∂∂
−α∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
∆⎟⎟⎠
⎞⎜⎜⎝
⎛
∆∂∂
−πρ
=
,
,
10
10
~~0
~0
~0
~0
~0
~10
Oxford University and ETH Zurich
( )
ii
i
ii
iw
i
ijijij
s
ww
w
ww
w
ss
s
ss
s
ijij
s
ijo
z
mmzpvg
zu
ssu
vvup
ssu
vvup
u
η⎟⎟⎠
⎞⎜⎜⎝
⎛η∂∂
−θρ
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−ρ
=
α⎟⎟⎠
⎞⎜⎜⎝
⎛
α∂∂
−α∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
∆⎟⎟⎠
⎞⎜⎜⎝
⎛
∆∂∂
−πρ
=
,
,
10
10
~~0
~0
~0
~0
~0
~10
w
w
w
w
s
s
s
s
ij
s
ijo
suvup
suvup
u
∂∂
=θ
∂∂
=−
∂∂
=θ
∂∂
=−
∆∂∂
=πρ1
Oxford University and ETH Zurich
( )
ii
i
ii
iw
i
ijijij
s
ww
w
ww
w
ss
s
ss
s
ijij
s
ijo
z
mmzpvg
zu
ssu
vvup
ssu
vvup
u
η⎟⎟⎠
⎞⎜⎜⎝
⎛η∂∂
−θρ
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−ρ
=
α⎟⎟⎠
⎞⎜⎜⎝
⎛
α∂∂
−α∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−θ=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=
∆⎟⎟⎠
⎞⎜⎜⎝
⎛
∆∂∂
−πρ
=
,
,
10
10
~~0
~0
~0
~0
~0
~10
( )
ii
ii
wi
ijij
s
zmzpvg
zu
η∂∂
=θρ
−
∂∂
=−ρ
α∂∂
=α∂∂
−
,
,
1
1
~
w
w
w
w
s
s
s
s
ij
s
ijo
suvup
suvup
u
∂∂
=θ
∂∂
=−
∂∂
=θ
∂∂
=−
∆∂∂
=πρ1
Oxford University and ETH Zurich
Variables and their conjugates
Strain
Internal variable
Density of solids
Entropy of solids
Density of water
Entropy of water
Mass flux of water
Entropy flux
( )ρθ−⇔η
ρρ−⇔
θ⇔
⇔ρ
θ⇔
⇔ρ
ρχ=ρχ⇔α
ρπ⇔∆
ii
wiii
w
w
s
s
oijoijij
oijij
pgms
ps
p
,
,
1
1
Effective stress
“Generalised stress”
Pore pressure
Temperature
Pore pressure
Temperature
Excess head grad.
Temperature grad.
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Starting values
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Compression of particles
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Thermal expansion
Oxford University and ETH Zurich
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Thermal capacity
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Elasticity
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Thermal expansion
Oxford University and ETH Zurich
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
Plasticity
Oxford University and ETH Zurich
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
Pore water terms
Oxford University and ETH Zurich
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
Drucker-Prager
Oxford University and ETH Zurich
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
Dilation
Oxford University and ETH Zurich
( ) ( ) iiiim
w
ijijiiijijii kmm
kvkz ηη
θ++α′α′β+αΛ+α′α′µσ+=
η2222 &&&&&
A porous-thermo-elastic-plastic material
( ) ( )( ) ( )0
20
00
200
00
23
2 θθ−θ
−−θ−θα+−
−
θ−=
wp
ww
w
www
cppK
ppv
spvg
( ) ( )( ) ( )
( ) ijijkkijijjjii
sp
ss
s
sss
GK
cppK
ppv
spvg
ασ−σθ−θα−σ′σ′
−σσ
−
θθ−θ
−−θ−θα+−
−
θ−=
0
0
20
00
200
00
221
631
23
2
Fluxes (Darcy and Fourier)
Oxford University and ETH Zurich
Conclusions
• Plasticity models have been successfully formulated
using just two potential functions
• This concept has been extended to include flow
processes
• What is new about this approach?– Flow and internal variables in same formulation
– The z-potential