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Numerical modeling of rock deformation 02 :: Kinematic models – Strain ellipses

Numerical modeling of rock deformation 02 :: Kinematic models – Strain ellipses. www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester 2011 Thursdays 10:15 – 12:00 NO D11 & NO CO1 Marcel Frehner marcel.frehner@erdw.ethz.ch , NO E3

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Numerical modeling of rock deformation 02 :: Kinematic models – Strain ellipses

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  1. Numerical modeling of rock deformation02 :: Kinematic models – Strain ellipses www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling Fallsemester 2011 Thursdays 10:15 – 12:00 NO D11 & NO CO1 Marcel Frehner marcel.frehner@erdw.ethz.ch, NO E3 Assistant: Jonas Ruh, NO E69

  2. 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 a lot of exercises!

  3. Displacement gradient tensor (H)Deformation gradient tensor (F) • Definitions: • Displacementgradient tensor: • Displacementat point P: • Deformed point P’:

  4. Displacement gradient tensor (H)Deformation gradient tensor (F) • Definitions: • Displacement at point Q: • Deformed point Q’:

  5. Exercises 1 & 2 Tips: • Use the command meshgrid to create the regular grid. • Organize the coordinates as follows:where n= nx∙ny using the reshape-command.This should make it easier to deform the grid with the deformation gradient tensor.

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

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

  8. 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.

  9. 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 deformation:Orientation of a vector that will be deformed maximally or minimally is given by the Eigenvectors of the right CG-tensor. • After deformation:Orientation of a vector that was deformed maximally or minimally is given by the Eigenvectors of the left CG-tensor.

  10. 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.

  11. Exercises

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