Lecture 1. How to model: physical grounds

Outline

• Physical grounds: parameters and basic laws

• Physical grounds: rheology

Physical grounds: displacement, strain and stress tensors

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

r

r´

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

Lagrangian

r

r´

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

xi= xi (ξ1,ξ2,ξ3,t) i=1,2,3

Lagrangian

r

r´

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

xi= xi (ξ1,ξ2,ξ3,t) i=1,2,3

Lagrangian

r

r´u

u= r´-r -displacement vector

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

xi= xi (ξ1,ξ2,ξ3,t) i=1,2,3

Lagrangian

r

r´u

ui= ξi -xi= ui (x1,x2,x3,t) i=1,2,3

u= r´-r -displacement vector

Strain

x1

x2

x3

t

t´

x1,x2,x3

ξ1,ξ2,ξ3

ξi= ξi (x1,x2,x3,t) i=1,2,3

xi= xi (ξ1,ξ2,ξ3,t) i=1,2,3

Lagrangian

r

r´u

ui= ξi -xi= ui (x1,x2,x3,t) i=1,2,3

ui= ξi -xi= ui (ξ1,ξ2,ξ3,t) i=1,2,3u= r´-r -displacement vector

Eulerian

Strain tensor

1

2

3

Strain tensor1 2

3

Strain tensor

Strain and strain rate tensors

Geometric meaning of strain tensor

Changing length

Changing angle

Changing volume

Stress tensor

is the i component of the force acting at the unit surface which is orthogonal to direction j

Tensor invariants, deviators

Stress deviator

P is pressure

Stress norm

Principal stress axes

Physical grounds: conservation laws

Mass conservation equation

Material flux

0)(

i

i

x

v

t

0

i

i

x

v

Dt

D

ii xv

tDt

D

Substantive time derivative

Mass conservation equation

01

i

i

x

v

Dt

DT

Dt

DP

K

Dt

DT

Dt

DP

KDt

DT

TDt

DP

PDt

D

Momentum conservation equation

3,..1,

ix

vv

t

v

Dt

Dvg

x j

ij

iii

j

ij

3,..1,

iDt

Dvg

xx

P ii

j

ij

i

Substantive time derivative

Energy conservation equation

Substantive time derivative

chemijijii

p HAx

T

xDt

DTC

)(

Heat flux

Shear heating

)(2

1

i

j

j

iij x

v

x

v

Physical grounds: constitutive laws

RheologyRheology – the relationship between stress and strain

• Rheology for ideal bodies– Elastic– Viscous– Plastic

An isotropic, homogeneous elastic material follows Hooke's Law

Hooke's Law: = E

For an ideal Newtonian fluid:differential stress = viscosity x strain rate

viscosity: measure of resistance to flow

Ideal plastic behavior

Constitutive laws

Dt

D

Gij

elasticij

2

1 Elastic strain rate

ijviscij

2

1 Viscous flow strain rate

ijII

plasticij

Plastic flow strain rate

ijijII

plasticijviscijelasticijij

Superposition of constitutive laws

ijII

ijij Dt

D

G

)2

1(

2

1

ijIIeff

ijij TPDt

D

G

),,(2

1

2

1

ijijII ;

chemijijii

p HAx

T

xDt

DTC

)(

ijIIeff

ij

i

j

j

iij TPDt

D

Gx

v

x

v

),,(2

1

2

1)(

2

1

Dt

Dvg

xx

P ii

j

ij

i

Full set of equations

mass

momentum

energy

01

i

i

x

v

Dt

DT

Dt

DP

K

How Does Rock Flow?

• Brittle – Low T, low P

• Ductile – as P, T increase

Brittle

Brittle-ductile Ductile

Modes of deformation

Deformation Mechanisms

• Brittle– Cataclasis

– Frictional grain-boundary sliding

• Ductile– Diffusion creep

– Power-law creep

– Peierls creep

Cataclasis

• Fine-scale fracturing and movement along fractures.

• Favoured by low-confining pressures

Frictional grain-boundary sliding

• Involves the sliding of grains past each other - enhanced by films on the grain boundaries.

PcII

Diffusion Creep• Involves the transport of

material from one site to another within a rock body

• The material transfer may be intracrystalline, intercrystalline, or both.

• Newtionian rheology:

n ~

~ exp

LL L II

HB

RT

Dislocation Creep• Involves processes internal to

grains and crystals of rocks. • In any crystalline lattice there

are always single atoms (i.e. point defects) or rows of atoms (i.e. line defects or dislocations) missing. These defects can migrate or diffuse within the crystalline lattice.

• Motion of these defects that allows a crystalline lattice to strain or change shape without brittle fracture.

• Power-law rheology :

n ~ exp

n NN N II

HB

RT

Peierls Creep

For high differential stresses (about 0.002 times Young’s modulus as a threshold), mixed dislocation glide and climb occurs. This mechanism has a finite yield stress (Peierls stress, ) at absolute zero temperature conditions.

))1(exp(P

IIPPP RT

HB

Dislocation

Diffusion

Peierls

Three creep processes

( Kameyama et al. 1999)

Diffusion creep

Dislocation creep

Peierls creep

chemijijii

p HAx

T

xDt

DTC

)(

ijeff

ij

i

j

j

iij Dt

D

Gx

v

x

v

2

1

2

1)(

2

1

Dt

Dvg

xx

P ii

j

ij

i

Full set of equations

mass

momentum

energy

01

i

i

x

v

Dt

DT

Dt

DP

K

Final effective viscosity

IIPNLeff

/)(2

1

exp

LL L II

HB

RT

exp

n NN N II

HB

RT

))1(exp(P

IIPPP RT

HB

Plastic strain

IIPNLeff

/)(2

1

PcII 0 if

0 PcII if

if 0,0 c cII Von Mises yield criterion

Drucker-Prager yield criterion

bac, softening

Final effective viscosity

IIPNLeff

/)(2

1

exp

LL L II

HB

RT

exp

n NN N II

HB

RT

))1(exp(P

IIPPP RT

HB

PcII 0 if

0 PcII if

Strength envelope

0

50

100

StrengthD

epth

(km

)

watercrustMoho

Mantle

0

50

100

Strength

Moho

Mantle

Crust