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Title: Numerical modeling of rock deformation 03 :: Continuum Mechanics


1
Numerical modeling of rock deformation03
Continuum Mechanics
  • www.structuralgeology.ethz.ch/education/teaching_m
    aterial/numerical_modeling
  • Fallsemester 2011
  • Thursdays 1015 1200
  • NO D11 NO CO1
  • Marcel Frehner
  • marcel.frehner_at_erdw.ethz.ch, NO E3
  • Assistant Jonas Ruh, NO E69

2
Goals of today
  • Understand the concept of Taylor series expansion
  • Derive the conservation equations for
  • mass
  • linear momentum
  • angular momentum

3
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, because it may be the simplest and
    most intuitive approach.

4
Conservation of mass
  • Taylor series expansion
  • Mass flux at left boundary
  • Mass flux at right boundary
  • Mass flux at bottom boundary
  • Mass flus at top boundary

5
Conservation of mass
  • Net rate of mass increase must balance the net
    flux of mass into the element
  • After some rearrangement
  • For constant density (incompressible)

6
Conservation of linear momentum
  • Force balance in the x-direction (without body
    forces and inertial forces)
  • After some rearrangement
  • Force balance in two dimensions

7
Conservation of linear momentum
  • 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

8
Conservation of angular momentum
  • Stress tensor is symmetricThis is the simplest
    version of the conservation of angular momentum
    and most common.
  • Conservation of linear momentum becomes
  • Notation
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