Numerical modeling of rock deformation 05 :: Continuum Mechanics

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Numerical modeling of rock deformation05
Continuum Mechanics

• www.structuralgeology.ethz.ch/education/teaching_m
  aterial/numerical_modeling
• Fallsemester 2013
• Thursdays 1015 1200
• NO D11 (lectures) NO CO1 (computer lab)
• Marcel Frehner
• marcel.frehner_at_erdw.ethz.ch, NO E3
• Assistant Marine Collignon, NO E1

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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
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The big picture Physical models

• Mechanical framework
• Continuum mechanics
• Quantum mechanics
• Relativity theory
• Molecular dynamics

• Solution technique
• Analytical solution
• Linear stability analysis
• Fourier transform
• Greens 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
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Goals of today

• Understand the concept of Taylor series expansion
• Derive the conservation equations for
• mass
• linear momentum
• angular momentum

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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.

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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
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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
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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
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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

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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

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Conservation of angular momentum (in 2D)

• Stress tensor is symmetric
• Conservation of linear momentum becomes
• Notation