The Thermodynamics of Porous Continua · The Thermodynamics of Porous Continua “Modern Trends in Geomechanics” Vienna, 25 June 2005 Prof. Guy Houlsby and Prof. Alexander Puzrin

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


The hyperelastic approach

( )ijg σ

ijij

gσ∂∂

−=ε

Gibbs free energy or(negative) complementary energy


The hyperplastic approach

( )ijijg ασ ,

( )ijijijd αασ &,,

ijij

gσ∂∂

−=ε

ijij

dgα∂∂

+α∂∂

=&

0

Gibbs free energy

Dissipation function



( )ijijg ασ ,


ijij

gσ∂∂

−=ε

Gibbs free energy


ijij

dgα∂∂

+α∂∂

=&

0



( )ijijg ασ ,


ijij

gσ∂∂

−=ε

ijij

dgα∂∂

+α∂∂

=&

0

Gibbs free energy


ijij

dα∂∂

=χ&

ijij

gα∂∂

−=χ

0=χ−χ ijij

⇒ {


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


An example of hyperplasticity: von Mises

ijijijijjjii GKg ασ−σ′σ′−σσ−=

41

181

ijijkd α′α′= &&2




41

181


Elasticity




41

181


Plasticity




41

181


von Mises


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?


Thermodynamics

• First Law

• Second Law

• Combine:

dQdWdU +=

θ≥

dQdSθ

=θ

−DdQdS

dSdWDdU θ+=+

⇒


Thermodynamics

• First Law

• Second Law

• Combine:

dQdWdU +=

θ≥

dQdSθ

=θ

−DdQdS

dSdWDdU θ+=+

Dissipation

⇒


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



( ) ( )( )( ) ( ) ( )∫∫∫

∫ ∫

−+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



( ) ( )( )( ) ( ) ( )∫∫∫

∫ ∫

−+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



( ) ( )( )( ) ( ) ( )∫∫∫

∫ ∫

−+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



( ) ( )( )( ) ( ) ( )∫∫∫

∫ ∫

−+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



( ) ( )( )( ) ( ) ( )∫∫∫

∫ ∫

−+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


Combined first + second laws

( )

( ) ( ) iiiwi

wi

wiii

wi

wwswsijijij

mpvsumpvg

wuswsvwpvpdpdu

,,,,,111

~~~~~1~

θηρ

−+θ−ρ

−−ρ

++θ+θ+−−δ+σρ

=+

dSdWDdU θ+=+


Functions of state

Internal energy is a function of state

( )( ) ( )wwwss

ijijs

wwssijij

svwusvu

svwsvuu

,,,,

,,,,,,

+α∆=

α∆=

Internal energyof skeleton

Internal energyof water


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∆= ,,~,,,,,,,


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

=


( )

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

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


( )

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


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.


( ) ( )( ) ( )

( ) 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 &&&&&


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Starting values


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Compression of particles


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Thermal expansion



( ) ( )( ) ( )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


η2222 &&&&&

Thermal capacity

( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Elasticity


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Thermal expansion


( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2


( ) ( )( ) ( )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


η2222 &&&&&

Plasticity



( ) ( )( ) ( )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


η2222 &&&&&

( ) ( )( ) ( )

( ) ijijkkijijjjii

sp

ss

s

sss

GK

cppK

ppv

spvg


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2

Pore water terms


( ) ( ) iiiim

w

ijijiiijijii kmm

kvkz ηη


η2222 &&&&&


( ) ( )( ) ( )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


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2

Drucker-Prager


( ) ( ) iiiim

w

ijijiiijijii kmm

kvkz ηη


η2222 &&&&&


( ) ( )( ) ( )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


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2

Dilation


( ) ( ) iiiim

w

ijijiiijijii kmm

kvkz ηη


η2222 &&&&&


( ) ( )( ) ( )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


−σσ

−

θθ−θ

−−θ−θα+−

−

θ−=

0

0

20

00

200

00

221

631

23

2

Fluxes (Darcy and Fourier)


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

Documents

The Thermodynamics of Porous Continua · The Thermodynamics of Porous Continua “Modern Trends in Geomechanics” Vienna, 25 June 2005 Prof. Guy Houlsby and Prof. Alexander Puzrin