Numerical modeling of rock deformation 04 :: Continuum Mechanics

1. Numerical modeling of rock deformation04 :: Continuum Mechanics www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester2014 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. Goals of today • Understand the concept of Taylor series expansion • Derive the conservation equations for • mass • linear momentum • angular momentum

5. Conservation equations • The fundamental equations of continuum mechanics describe the conservation of • mass • linear momentum • angular momentum • energy • There exist several approaches to derive the conservation equations of continuum mechanics: • Variational methods (virtual work) • Based on integro-differential equations (e.g., Stokes theorem) • Balance of forces and fluxes based on Taylor terms • We use in this lecture the balance of forces and fluxes in 2D using Taylor terms, because it may be the simplest and most intuitive approach.

6. Conservation of mass (in 2D) • Taylor series expansion: • Mass flux at left boundary (in positive x-dir): • Mass flux at right boundary (in positive x-dir): • Mass flux at bottom boundary (in positive y-dir): • Mass flus at top boundary (in positive y-dir): y x

7. Conservation of mass (in 2D) • Net rate of mass increase must balance the net flux of mass into the element: • After some rearrangement: • For constant density (incompressible): y x

8. y x Conservation of linear momentum (force balance in 2D) Force balance in x-direction • Force at left boundary (in positive x-dir): • Force at right boundary (in positive x-dir): • Force at bottom boundary (in positive x-dir): • Force at top boundary (in positive x-dir): – Compression + Extension syx sxx sxx syx

9. y x Conservation of linear momentum (force balance in 2D) Force balance in x-direction • Force balance in x-direction (inertia force = sum of all other forces): • After some rearrangement: • Force balance in two dimensions:

10. y x Conservation of linear momentum (force balance in 2D) • General force balance in two dimensions (including body forces and inertial forces) • In a gravity field we use • In geodynamics, processes are often so slowthat we can ignore inertial forces

11. Conservation of angular momentum (in 2D) • Stress tensor is symmetric: • Conservation of linear momentum becomes: Notation