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Continuum Mechanics - Rheology 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 2 px00 x px x O x x px 0 x px 0 px 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 2 p xxpx xOx 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 uy ux y x 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 uy x u , xx x 112xy 1 uy E 1 u uy Rheology, x 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 uy yx 00G LLGx 2 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.
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