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03 - tensor calculus - tensor analysis. 03 - tensor calculus. tensor algebra - invariants. • (principal) invariants of second order tensor. • derivatives of invariants wrt second order tensor. tensor calculus. tensor algebra - trace. • trace of second order tensor.

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tensor calculus

tensor algebra - invariants

• (principal) invariants of second order tensor

• derivatives of invariants wrt second order tensor

tensor calculus
tensor calculus1

tensor algebra - trace

• trace of second order tensor

• properties of traces of second order tensors

tensor calculus
tensor calculus2

tensor algebra - determinant

• determinant of second order tensor

• properties of determinants of second order tensors

tensor calculus
tensor calculus3

tensor algebra - determinant

• determinant defining vector product

• determinant defining scalar triple product

tensor calculus
tensor calculus4

tensor algebra - inverse

• inverse of second order tensor

in particular

• adjoint and cofactor

• properties of inverse

tensor calculus
tensor calculus5

tensor algebra - spectral decomposition

• eigenvalue problem of second order tensor

• solution in terms of scalar triple product

• characteristic equation

• spectral decomposition

• cayleigh hamilton theorem

tensor calculus
tensor calculus6

tensor algebra - sym/skw decomposition

• symmetric - skew-symmetric decomposition

• symmetric and skew-symmetric tensor

• symmetric tensor

• skew-symmetric tensor

tensor calculus
tensor calculus7

tensor algebra - symmetric tensor

• symmetric second order tensor

• processes three real eigenvalues and corresp.eigenvectors

• square root, inverse, exponent and log

tensor calculus
tensor calculus8

tensor algebra - skew-symmetric tensor

• skew-symmetric second order tensor

• processes three independent entries defining axial vector

such that

• invariants of skew-symmetric tensor

tensor calculus
tensor calculus9

tensor algebra - vol/dev decomposition

• volumetric - deviatoric decomposition

• volumetric and deviatoric tensor

• volumetric tensor

• deviatoric tensor

tensor calculus
tensor calculus10

tensor algebra - orthogonal tensor

• orthogonal second order tensor

• decomposition of second order tensor

such that and

• proper orthogonal tensor has eigenvalue

with

interpretation: finite rotation around axis

tensor calculus
tensor calculus11

tensor analysis - frechet derivative

• consider smooth differentiable scalar field with

scalar argument

vector argument

tensor argument

• frechet derivative (tensor notation)

scalar argument

vector argument

tensor argument

tensor calculus
tensor calculus12

tensor analysis - gateaux derivative

• consider smooth differentiable scalar field with

scalar argument

vector argument

tensor argument

• gateaux derivative,i.e.,frechet wrt direction (tensor notation)

scalar argument

vector argument

tensor argument

tensor calculus
tensor calculus13

tensor analysis - gradient

• consider scalar- and vector field in domain

• gradient of scalar- and vector field

renders vector- and 2nd order tensor field

tensor calculus
tensor calculus14

tensor analysis - divergence

• consider vector- and 2nd order tensor field in domain

• divergence of vector- and 2nd order tensor field

renders scalar- and vector field

tensor calculus
tensor calculus15

tensor analysis - laplace operator

• consider scalar- and vector field in domain

• laplace operator acting on scalar- and vector field

renders scalar- and vector field

tensor calculus
tensor calculus16

tensor analysis - transformation formulae

• consider scalar,vector and 2nd order tensor field on

• useful transformation formulae (tensor notation)

tensor calculus
tensor calculus17

tensor analysis - transformation formulae

• consider scalar,vector and 2nd order tensor field on

• useful transformation formulae (index notation)

tensor calculus
tensor calculus18

tensor analysis - integral theorems

• consider scalar,vector and 2nd order tensor field on

• integral theorems (tensor notation)

green

gauss

gauss

tensor calculus
tensor calculus19

tensor analysis - integral theorems

• consider scalar,vector and 2nd order tensor field on

• integral theorems (tensor notation)

green

gauss

gauss

tensor calculus
tensor calculus20

voigt / matrix vector notation

• strain tensors as vectors in voigt notation

• stress tensors as vectors in voigt notation

• why are strain & stress different? check energy expression!

tensor calculus
tensor calculus21

voigt / matrix vector notation

• fourth order material operators as matrix in voigt notation

• why are strain & stress different? check these expressions!

tensor calculus
example 1 matlab

deformation gradient

• given the deformation gradient, play with matlab to

become familiar with basic tensor operations!

• uniaxial tension (incompressible), simple shear, rotation

example #1 - matlab
example 1 matlab1

second order tensors - scalar products

• inverse of second order tensor

• right / left cauchy green and green lagrange strain tensor

• trace of second order tensor

• (principal) invariants of second order tensor

example #1 - matlab
example 1 matlab2

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

example #1 - matlab
example 1 matlab3

neo hooke‘ian elasticity

• free energy

• 1st and 2nd piola kirchhoff stress and cauchy stress

• 4th order tangent operators

example #1 - matlab
homework 2

matlab

• play with the matlab routine to familiarize yourself with

tensor expressions!

• calculate the stresses for different deformation gradients!

• which of the following stress tensors is symmetric and

could be represented in voigt notation?

• what would look like in the linear limit, for

• what are the advantages of using the voigt notation?

homework #2