numerical modeling of rock deformation: 04 continuum ...€¦ · numerical modeling of rock...

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich

Numerical modeling of rock deformation:

04 Continuum mechanics - Rheology

Stefan [email protected]

NO E 61

AS 2009, Thursday 10-12, NO D 11

mailto:[email protected]


Conservation equationsThe fundamental equations of continuum mechanics describe the conservation of• Mass • Linear momentum • Angular momentum and • Energy.

There exist several approaches to derive the conservation equations of continuum mechanics:• Variational methods (virtual work)• Derivations based on integro-differential equations (e.g., Stokes theorem)• Balance of forces and fluxes based on Taylor series.

We use in this lecture the balance of forces and fluxes in 2D, because it may be the simplest and most intuitive approach.


Taylor series

x0 x0 +xx

p 0 2

0 0

p xp x x p x x O x

x

0p x

0p x x

00

p xp x x

x


Conservation of mass in 2D

2pp x x p x x O xx

Taylor series

Net rate of mass increase

x y x yt t

must balance the net rate of flow of mass, e.g. vy, into the element

2 2

2 2

x xx x

y yy y

yx

v vx xv y v yx x

v vy yv x v xy y

vvx y y x

x y

2p xpx

x

y

2p ypy

2p ypy

x

y

2p xpx

p

2

1 kg kgx y mmt s sm

2

2

v =

v

kg mpsm

kg m kgy ms sm


Conservation of mass in 2D

0

0

yx

yx

vvx y x y y x

t x yvv

t x y

divt

v

Net mass increase in element balances net flow of mass into element

0

0

0

0

yx

yx

tvv

x yvv

x ydiv

v

If we assume the density to be constant then

2p xpx

x

y

2p ypy

2p ypy

x

y

2p xpx


Conservation of linear momentum

2xx

xxx

x

x

y

2yx

yxy

y

2yx

yxy

y

x

y

2xx

xxx

x

2 2

2 2

xx xxxx xx

yx yxyx yx

x x y zx x

y y x zy y

Force balance in the x-direction

0yxxx

x y

Force balance in the x-direction is fulfilled if

0

0

yxxx

xy yy

x y

x y

Force balance in two dimensions



0

0

yxxx

xy yy

x y

x y

General force balance in two dimensions

0

0

0

S

ji

jS V V

ji

j

dS

dS div dV dVx

x

T

σn σ

Derivation based on integro- differential equations

Gauss divergence theorem

0, 1, 2ji

j

jx

, 0, 1, 2ji j j

0div σ

Cauchy tensor



yxxx xx

xy yy yy

vF

x y tv

Fx y t

General force balance in two dimensions

yF g

Under gravity we use


Conservation of angular momentum

yx xy

Stress tensor is symmetric

This is the simplest version of the conservation of angular momentum and most common.Cosserat theory includes additional moments and the conservation equation becomes more complicated.


Conservation of energy

y yx xx y xx yx xy yy

v vv vDT T Tc k k QDt x x y y x y x y

Heat equation for two dimensions

Heat conduction-advection Heat source

Heat production due to shear heating

x yDT T T Tv vDt t x y

In Eulerian system the total time derivative is (material time derivative)


Ductile rheology1D viscous (Newtonian) rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Mostly incompressible

2 vx

The rheology is linear.Deviatoric stress is related to deviatoric strain rate.

2

1

vx


Ductile rheology1D power-law rheology

1

1 11

2

2

2

eff

n

n

nn

eff

vx

vx

v v vx x x

The rheology is nonlinear.The effective viscosity is a function of the strain rate.Iterations are usually necessary in numerical algorithms.

1 11expn n E VA

nRT

Typical structure of rock rheology.


Ductile rheology2D viscous rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Incompressible

2

2

122

xxx

yyy

yxyx

vp

xv

py

vvy x

p is pressure.

is total stress.


Ductile rheology2D non-Newtonian (power-law) rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Incompressible

1 1

1 1

1 1

2 2

2

2

122

1 14 4

xnxx II

ynyy II

yxnyx II

y yx xII

vp

xv

py

vvy x

v vv vx y y x


Closed sys. of eqns: incompressible fluid

yx xy

0

0

yxxx

xy yy

x y

x y

,,

,

,

,,

xx

yy

yx

xy

x

y

puu

Rheology,Three equations

Conservation of linear momentum,Force balance,Two equations

Conservation of angular momentum,One equation

Sevenunknowns

2

2

122

xxx

yyy

yxyx

vp

xv

py

vvy x

0yx vvx y

Conservation of mass,One equation


Elastic rheology

Elastic rheology•Time independent•Energy is conserved, no dissipation, no shear heating•In 2D different for plane strain and plane stress

11 1 2 1

11 1 2 1

2 1

yxxx

yxyy

yxyx

uE ux y

uE ux y

uuEy x

2

2

yxxx

yxyy

yxyx

uuL G L

x yuu

L L Gx y

uuG

y x

E = Young’s modulus

= Poisson ratioL = Lame parameterG = Shear modulus


Closed system of equations: solid

yx xy

0

0

yxxx

xy yy

x y

x y

11 1 2 1

11 1 2 1

2 1

yxxx

yxyy

yxyx

uE ux y

uE ux y

uuEy x

,,

,

,

,

xx

yy

yx

xy

x

y

uu

Rheology,Three equations

Conservation of linear momentum,Force balance,Two equations

Conservation of angular momentum,One equation

Sixunknowns


Rheology reformulated

Viscous rheology1 2 0 01 0 2 00 0 0 1

x

xxy

yy

yxyx

vxv

py

vvy x

01 2 0 01 0 2 0 00 0 0 1

xxx

yyy

yx

xv

pvy

y x

2

2

122

xxx

yyy

yxyx

vp

xv

py

vvy x

σ p DBu

Dσ B up


The constitutive equations

2

2

yxxx

yxyy

yxyx

uuL G L

x yuu

L L Gx y

uuG

y x

Constitutive equations for 2D plane strain elasticity

02 0

2 0 00 0

xxx

yyy

yx

xL G Lu

L L Guy

G

y x

σ D uB

σ DBu


Rheology – Force balance

0T B σ

σ DBu

0

0

x

y

y x

B

Elastic rheology

0

0

yxxx

yx yy

x y

x y

Force balance

TB DBu fK u f

Substitution of rheology in force balance equations

T B σ fVector f includes the boundary conditions if no physical external forces are present.

xx

yy

yx

σ

B(1,ii ) = DHDX(1,:);B(2,ii+1) = DHDX(2,:);B(3,ii ) = DHDX(2,:);B(3,ii+1) = DHDX(1,:);

E = MATPROP(1,Phase(iel));nu = MATPROP(2,Phase(iel));prefac = E/((1+nu)*(1-2*nu));D = prefac * [ 1-nu nu 0; nu 1-nu 0; 0 0 (1-2*nu)/2];

K = K +( B'*D*B )*wtx*detjacob;

Extract from finite element code


FEM Examples - linear viscous


FEM Examples – power law


FEM Examples – linear viscous & gravity


Next week: Matlab

• Next week we meet at 10:15 in HG E 27• Matlab scripts are on course web page

numerical modeling of rock deformation: 04 continuum ...€¦ · numerical modeling of rock...

Documents