03 - tensor calculus
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103 - tensor calculus
03 - tensor calculus -tensor analysis
2tensor calculus
tensor algebra - invariants
• (principal) invariants of second order tensor
• derivatives of invariants wrt second order tensor
3tensor calculus
tensor algebra - trace
• trace of second order tensor
• properties of traces of second order tensors
4tensor calculus
tensor algebra - determinant
• determinant of second order tensor
• properties of determinants of second order tensors
5tensor calculus
tensor algebra - determinant
• determinant defining vector product
• determinant defining scalar triple product
6tensor calculus
tensor algebra - inverse
• inverse of second order tensor
in particular
• properties of inverse
• adjoint and cofactor
7tensor calculus
tensor algebra - spectral decomposition• eigenvalue problem of second order tensor
• spectral decomposition
• characteristic equation
• cayleigh hamilton theorem
• solution in terms of scalar triple product
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
8tensor calculus
tensor algebra - sym/skw decomposition
• symmetric - skew-symmetric decomposition
• skew-symmetric tensor
• symmetric tensor
• symmetric and skew-symmetric tensor
9tensor calculus
tensor algebra - symmetric tensor
• symmetric second order tensor
• square root, inverse, exponent and log
• processes three real eigenvalues and corresp.eigenvectors
10tensor calculus
tensor algebra - skew-symmetric tensor
• skew-symmetric second order tensor
• invariants of skew-symmetric tensor
• processes three independent entries defining axial vector such that
11tensor calculus
tensor algebra - vol/dev decomposition
• volumetric - deviatoric decomposition
• deviatoric tensor
• volumetric tensor
• volumetric and deviatoric tensor
12tensor calculus
tensor algebra - orthogonal tensor
• orthogonal second order tensor
• proper orthogonal tensor has eigenvalue
• decomposition of second order tensor
such that and
interpretation: finite rotation around axis
with
13tensor calculus
tensor analysis - frechet derivative
• frechet derivative (tensor notation)
• consider smooth differentiable scalar field with
scalar argumentvector argumenttensor argument
scalar argument
vector argument
tensor argument
14tensor calculus
tensor analysis - gateaux derivative
• gateaux derivative,i.e.,frechet wrt direction (tensor notation)
• consider smooth differentiable scalar field with
scalar argumentvector argumenttensor argument
scalar argument
vector argument
tensor argument
15tensor calculus
tensor analysis - gradient
• gradient of scalar- and vector field
• consider scalar- and vector field in domain
renders vector- and 2nd order tensor field
16tensor calculus
tensor analysis - divergence
• divergence of vector- and 2nd order tensor field
• consider vector- and 2nd order tensor field in domain
renders scalar- and vector field
17tensor calculus
tensor analysis - laplace operator
• laplace operator acting on scalar- and vector field
• consider scalar- and vector field in domain
renders scalar- and vector field
18tensor calculus
tensor analysis - transformation formulae
• useful transformation formulae (tensor notation)
• consider scalar,vector and 2nd order tensor field on
19tensor calculus
tensor analysis - transformation formulae
• useful transformation formulae (index notation)
• consider scalar,vector and 2nd order tensor field on
20tensor calculus
tensor analysis - integral theorems
• integral theorems (tensor notation)
• consider scalar,vector and 2nd order tensor field on
greengauss gauss
21tensor calculus
tensor analysis - integral theorems
• integral theorems (tensor notation)
• consider scalar,vector and 2nd order tensor field on
greengauss gauss
22tensor calculus
voigt / matrix vector notation
• stress tensors as vectors in voigt notation
• strain tensors as vectors in voigt notation
• why are strain & stress different? check energy expression!
23tensor calculus
voigt / matrix vector notation
• fourth order material operators as matrix in voigt notation
• why are strain & stress different? check these expressions!
24example #1 - matlab
deformation gradient
• uniaxial tension (incompressible), simple shear, rotation
• given the deformation gradient, play with matlab tobecome familiar with basic tensor
operations!
25example #1 - matlab
second order tensors - scalar products
• (principal) invariants of second order tensor
• trace of second order tensor
• inverse of second order tensor
• right / left cauchy green and green lagrange strain tensor
26example #1 - matlab
fourth order tensors - scalar products
• symmetric fourth order unit tensor
• screw-symmetric fourth order unit tensor
• volumetric fourth order unit tensor
• deviatoric fourth order unit tensor
27example #1 - matlab
neo hooke‘ian elasticity
• 1st and 2nd piola kirchhoff stress and cauchy stress
• free energy
• 4th order tangent operators
28homework #2
matlab
• which of the following stress tensors is symmetric and
• play with the matlab routine to familiarize yourself with
• calculate the stresses for different deformation gradients!
tensor expressions!
could be represented in voigt notation?
• what would look like in the linear limit, for • what are the advantages of using the voigt notation?
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