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1 03 - tensor calculus 03 - tensor calculus - tensor analysis 2 tensor calculus tensor algebra - invariants (principal) invariants of second order tensor derivatives of invariants wrt second order tensor 3 tensor calculus tensor algebra - trace trace of second order tensor properties of traces of second order tensors 4 tensor calculus tensor algebra - determinant determinant of second order tensor properties of determinants of second order tensors

Tensor Calculus. Stanford

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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

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 argument

vector argument

tensor 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 argument

vector argument

tensor 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?