Numerical modeling of rock deformation 05 :: Rheology

1. Numerical modeling of rock deformation05 :: Rheology www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester 2014 Thursdays 10:15 – 12:00 NO D11 (lectures) & NO CO1 (computer lab) Marcel Frehner marcel.frehner@erdw.ethz.ch, NO E3

2. The big picture Indirect observations/interpretations from measured data Direct observations in nature Seismic velocities Thermal mantle structure Folds, Boudinage, Reaction rims, Fractures Conceptual Statistical • Model • Simplification • Generalization • Parameterization We want to understandwhat we observe Kinematical Physical/Mechanical Analogue

3. The big picture – Physical models • Mechanical framework • Continuum mechanics • Quantum mechanics • Relativity theory • Molecular dynamics • Solution technique • Analytical solution • Linear stability analysis • Fourier transform • Green’s function • Numerical solution • Finite difference method • Finite element method • Spectral methods • Boundary element method • Discrete element method • Constitutive • Equations • (Rheology, • Evolution • equation) • Elastic • Viscous • Plastic • Diffusion • Governing equations • Energy balance • Conservation laws • Differential equations • Integral equations • System of (linear) equations • Solution is valid • for the applied • Boundary conditions • Rheology • Mechanical framework • etc… • Closed system of equations • Boundary and initial conditions • Heat equation • (Navier-)Stokes equation • Wave equation Dimensional analysis

4. …from last week • Conservation of linear momentum (force balance) • 2 equations for 3 unknowns! • We need constitutive equations(i.e., rheological relationships).

5. Goals of today • Understand the constitutive equations in 1Dand 2D for • Linear elastic rheology • Incompressible linear viscous (Newtonian) rheology • Derive the closed system of equations

6. Elastic rheology – 1D • Model: Spring • Linear relationship betweenstress and strain • Instant deformationwhen stress is applied • Reversible deformationwhen stress is released

7. 2D plane strain • Note on strains:Engineering strain: • Note on stress: Compressive stresses are negative! • Separate stress and strain tensors into bulk (volumetric) and deviatoric (shear) parts. • Bulk: • Deviatoric: We will use:

8. Elastic rheology – 2D plane strain • Bulk: • Deviatoric: • Total in x-direction: • Total:

9. Viscous rheology – 1D • Model: Dashpot • Linear relationship betweenstress and strain rate • Increasing deformationwhen stress is applied • Irreversible deformationwhen stress is released

10. Viscous rheology – 2D plane strain • Bulk: No bulk viscosity (incompressible!): • Deviatoric: • Total in x-direction: • Total:

11. Matrix notation • Strain: • Elastic: • Viscous:

12. Force balance equation • Reminder: • Elastic: • Viscous:

13. Closed system of equations: Elastic • Conservation oflinear momentum:Two equations • Rheology:Threeequations • Closed system of equations:Two equations Fiveunknowns:

14. Closed system of equations: Viscous • Mass conservation One equation: • Conservation oflinear momentum:Two equations • Rheology:Threeequations • Closed system of equations:Two equations Sixunknowns: