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

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Page 1: 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)

Page 2: 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

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

Page 3: 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

Oxford University and ETH Zurich

The hyperelastic approach

( )ijg σ

ijij

gσ∂∂

−=ε

Gibbs free energy or(negative) complementary energy

Page 4: 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

Oxford University and ETH Zurich

The hyperplastic approach

( )ijijg ασ ,

( )ijijijd αασ &,,

ijij

gσ∂∂

−=ε

ijij

dgα∂∂

+α∂∂

=&

0

Gibbs free energy

Dissipation function

Page 5: 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

Oxford University and ETH Zurich

The hyperplastic approach

( )ijijg ασ ,

( )ijijijd αασ &,,

ijij

gσ∂∂

−=ε

Gibbs free energy

Dissipation function

ijij

dgα∂∂

+α∂∂

=&

0

Page 6: 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

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

⇒ {

Page 7: 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

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

Page 8: 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

Oxford University and ETH Zurich

An example of hyperplasticity: von Mises

ijijijijjjii GKg ασ−σ′σ′−σσ−=

41

181

ijijkd α′α′= &&2

Page 9: 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

Oxford University and ETH Zurich

An example of hyperplasticity: von Mises

ijijijijjjii GKg ασ−σ′σ′−σσ−=

41

181

ijijkd α′α′= &&2

Elasticity

Page 10: 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

Oxford University and ETH Zurich

An example of hyperplasticity: von Mises

ijijijijjjii GKg ασ−σ′σ′−σσ−=

41

181

ijijkd α′α′= &&2

Plasticity

Page 11: 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

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

Page 12: 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

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?

Page 13: 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

Oxford University and ETH Zurich

Thermodynamics

• First Law

• Second Law

• Combine:

dQdWdU +=

θ≥

dQdSθ

−DdQdS

dSdWDdU θ+=+

Page 14: 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

Oxford University and ETH Zurich

Thermodynamics

• First Law

• Second Law

• Combine:

dQdWdU +=

θ≥

dQdSθ

−DdQdS

dSdWDdU θ+=+

Dissipation

Page 15: 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

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

Page 16: 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

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

Page 17: 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

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

Page 18: 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

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

Page 19: 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

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

Page 20: 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

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

Page 21: 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

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+=+

Page 22: 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

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

Page 23: 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

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∆= ,,~,,,,,,,

Page 24: 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

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

=

Page 25: 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

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

Page 26: 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

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

∂∂

∂∂

=−

∂∂

∂∂

=−

∆∂∂

=πρ1

Page 27: 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

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

∂∂

∂∂

=−

∂∂

∂∂

=−

∆∂∂

=πρ1

Page 28: 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

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.

Page 29: 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

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 &&&&&

Page 30: 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

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

Page 31: 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

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

Page 32: 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

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

Page 33: 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

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

( ) ( )( ) ( )

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sp

ss

s

sss

GK

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ppv

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ασ−σθ−θα−σ′σ′

−σσ

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0

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20

00

200

00

221

631

23

2

Page 34: 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

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

Page 35: 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

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

Page 36: 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

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

Page 37: 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

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

Page 38: 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

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

Page 39: 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

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

Page 40: 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

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)

Page 41: 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

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