Numerical modeling of rock deformation 03 :: Continuum Mechanics

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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

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

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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

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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)

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Conservation of linear momentum

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

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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

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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