Numerical modeling of rock deformation 03 :: Kinematic models

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Numerical modeling of rock deformation03
Kinematic models Strain ellipses

• www.structuralgeology.ethz.ch/education/teaching_m
  aterial/numerical_modeling
• Fallsemester 2012
• Thursdays 1015 1200
• NO D11 (lectures) NO CO1 (computer lab)
• Marcel Frehner
• marcel.frehner_at_erdw.ethz.ch, NO E3
• Assistant Jonas Ruh, NO E69

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Goals of today

• Understand the displacement gradient tensor and
  the deformation gradient tensor
• Know what the left and right Cauchy-Green tensors
  are and how they are calculated
• Understand the concept of the strain ellipse and
  know how to calculate its principal axes
• Do some exercises!

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Displacement gradient tensor (H)Deformation
gradient tensor (F)

• Definitions
• Displacementgradient tensor
• Displacementat point P
• Deformed point P

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Displacement gradient tensor (H)Deformation
gradient tensor (F)

• Definitions
• Displacement at point Q
• Deformed point Q

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Right Cauchy-Green tensor

• Definition
• Ratio between newand old length
• Right Cauchy-Green tensor

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Left Cauchy-Green tensor

• Definition
• Ratio between oldand new length
• Left Cauchy-Green tensor

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Properties of the Cauchy-Green tensors

• rCG-tensor can be used to calculate the length of
  a vector after deformation from the length before
  deformation.
• lCG-tensor can be used to calculate the length of
  a vector before deformation from the length after
  deformation.
• Both CG-tensors are symmetric.
• Both CG-tensors contain information of the strain
  (change of lengths), but not of the rigid body
  rotation (rotation without change of length).
• The information about the total deformation is
  only provided by the displacement gradient tensor
  H or the displacement gradient tensor F.

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Principal strains

• The principal strain values can be calculated
  from the Eigenvalues of both CG-tensors
• The orientation of the principal strain values
  can be defined before or after deformation.
• Before deformationOrientation of a vector that
  will be deformed maximally or minimally is given
  by the Eigenvectors of the right CG-tensor.
• After deformationOrientation of a vector that
  was deformed maximally or minimally is given by
  the Eigenvectors of the left CG-tensor.

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Strain ellipse

• The axes of the strain ellipse are given by the
  principal strain values (lCG- or rCG-tensor) and
  the Eigenvectors of the left CG-tensor.

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Exercises 1 2

• Tips
• Use the command meshgrid to create the regular
  grid.
• Organize the coordinates as followswhere n
  nxny using the reshape-command.This should make
  it easier to deform the grid with the deformation
  gradient tensor.

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Exercises