Numerical modeling of rock deformation:
04 Continuum mechanics - Rheology
Stefan Schmalholz [email protected] NO E 61
AS 2009, Thursday 10-12, NO D 11
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation equations
The 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.
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Taylor series
p px 0 px x px x O x2 00x
px 0 x px px 0 x 0 x px 0
x x0 x0+ x
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of mass in 2D
Taylor series p y p p pxxpx xOx 2 y 2 x
Net rate of mass increase 1 kg kg xy mm 2 tssm x yxy y tt p x p p x p y x p x 2 x 2 must balance the net rate of flow of kg m x mass, e.g. vy, into the element p v= m2 s kg m kg vym vvxxxx2 vyvyxx s sm p y xx22 p y 2 vvyyyy vxvxyy yy22 v v x xyy yx xy
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of mass in 2D
Net mass increase in element balances p y net flow of mass into element p y 2 v v x yxyyx x y txy v v x y 0 tx y y p x p x p y p divv 0 x t x 2 x 2 x If we assume the density to be constant then 0 t p y v v p x y 0 y 2 xy v v x y 0 xy divv 0
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum
Force balance in the x-direction y yx xx yx xx xx y 2 xx xx yz xx22
yx yy yx yx yx x z yy22 y x x xx y xx xx x xx Force balance in the x-direction is x 2 x 2 fulfilled if x xx yx 0 xy y yx yx y 2 Force balance in two dimensions xx yx 0 xy xy yy 0 xy
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum
General force balance in two dimensions Derivation based on integro- differential equations xx yx 0 TdS 0 xy Cauchy tensor S σndS divσ dVji dV 0 xy yy x 0 SV Vj xy ji 0 x j Gauss divergence theorem ji 0,j 1,2 x j
ji, j 0,j 1,2
divσ 0
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum
General force balance in two dimensions
v xxxyx F xyx t v xy yy F y xyy t
Under gravity we use
Fgy
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of angular momentum
Stress tensor is symmetric
yx xy
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.
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of energy
Heat equation for two dimensions
DT T T vvxxvvyy ckx kQ y xx yx xy yy Dt x x y y x y x y
Heat conduction-advection Heat Heat production due to source shear heating
In Eulerian system the total time derivative is (material time derivative) DT T T T vv Dt txy x y
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 1D viscous (Newtonian) rheology •Time dependent •Energy is not conserved, dissipation, shear heating •Mostly incompressible v 2 2 x 1 The rheology is linear. Deviatoric stress is related to deviatoric strain rate.
v x
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 1D power-law rheology v n 2 x 1 v n 2 x 1 1 1 vvn v n 2 eff x xx eff
The rheology is nonlinear. 11 The effective viscosity is a function of the strain rate. 1 E V nnA exp Iterations are usually necessary in numerical algorithms. nRT Typical structure of rock rheology.
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 2D viscous rheology •Time dependent •Energy is not conserved, dissipation, shear heating •Incompressible
v p 2 x xx x p is pressure. is total stress. vy p 2 yy y
1 vx vy yx 2 2 yx
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 2D non-Newtonian (power-law) rheology •Time dependent •Energy is not conserved, dissipation, shear heating •Incompressible
1 1 v p 2 n x xx II x 1 1 v p 2 n y yy II y 1 1 v n 1 vx y yx2 II 2 yx 22 11vvxxvvyy II 44xy yx
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Closed sys. of eqns: incompressible fluid
Conservation of xx yx 0 linear momentum, xy
Force balance, xy yy Seven 0 Two equations xy unknowns , xx Conservation of
yy , angular momentum, yx xy One equation yx , v v xy , Conservation of mass, x y 0 One equation xy p, v ux , x xx p 2 u x y v Rheology, y yy p 2 Three equations y
1 vx vy yx 2 2 yx Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Elastic rheology
Elastic rheology E = Young’s modulus •Time independent = Poisson ratio •Energy is conserved, no L = Lame parameter dissipation, no shear heating G = Shear modulus •In 2D different for plane strain and plane stress
u E 1 u u ux y x y LG2 L xx xx 112 x 1 y x y
ux uy E 1 ux uy LLG2 yy yy x y 1121 x y ux uy E ux uy yx G yx yx 21 yx
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Closed system of equations: solid
Conservation of xx yx 0 Six linear momentum, xy unknowns Force balance, Two equations xy yy 0 xy xx , , yy Conservation of yx xy yx , angular momentum, , One equation xy E 1 u u x y u , xx x 112 xy 1 u y E 1 u u Rheology, x y yy Three equations 1121 xy
E ux uy yx 21 yx Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Rheology reformulated
vx x 1 200 Viscous rheology xx v p 1020 y yy y yx 0001 v v x y vx yx xx p 2 x 0 1200x v xx y vx p 2 yy p 10200 yy y v y 0001 y yx yx 1 vx vy yx 2 2 yx σ p D B u
σ pDBu
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich The constitutive equations
Constitutive equations for 2D plane strain elasticity 0 x ux uy xx LG 20 L xx LG2 L ux x y yy LLG200 y uy u u yx 00G LLGx 2 y yy x y yx
ux uy yx G yx σ D B u
σ DBu
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Rheology – Force balance
Elastic rheology 0 Substitution of rheology in force x xx balance equations σ B 0 σ DBu yy y yx yx BDBufT Ku f Force balance Extract from finite element code
xx yx B(1,ii ) = DHDX(1,:); 0 B(2,ii+1) = DHDX(2,:); BT σ 0 xy B(3,ii ) = DHDX(2,:); B(3,ii+1) = DHDX(1,:); yx yy 0 xy E = MATPROP(1,Phase(iel)); T nu = MATPROP(2,Phase(iel)); prefac = E/((1+nu)*(1-2*nu)); B σ f D = prefac * [ 1-nu nu 0; nu 1-nu 0; 0 0 (1-2*nu)/2]; Vector f includes the boundary conditions if no physical external forces are present. K = K +( B'*D*B )*wtx*detjacob;
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples - linear viscous
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples – power law
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples – linear viscous & gravity
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich 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: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich