tensor calculus 02 - basics and maths - tensor the word tensor was introduced in 1846 by william rowan hamilton. it was notation and tensors used in its current meaning by woldemar voigt in 1899. tensor calculus was deve- loped around 1890 by gregorio ricci-curba- stro under the title absolute differential calculus. in the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the intro- duction of einsteins's theory of general relativity around 1915. tensors are used also in other fields such as continuum mechanics.

02 - tensor calculus 1 tensor calculus 2

tensor calculus - repetition vector algebra - notation • vector algebra • einstein‘s summation convention notation, euklidian vector space, scalar product, vector product, scalar triple product • tensor algebra notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew • summation over any indices that appear twice in a term decomposition, vol-dev decomposition, orthogonal tensor • tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations

tensor calculus 3 tensor calculus 4 example - position vector / displacement vector vector algebra - notation

• kronecker symbol

• permutation symbol

tensor calculus 5 tensor calculus 6

vector algebra - euklidian vector space vector algebra - euklidian vector space • euklidian vector space • euklidian vector space equipped with norm

• is defined through the following axioms • norm defined through the following axioms

• zero element and identity

• linear independence of if is the only (trivial) solution to

tensor calculus 7 tensor calculus 8 vector algebra - euklidian vector space vector algebra - scalar product • euklidian vector space equipped with • euklidian norm enables definition of scalar (inner) product euklidian norm

• representation of 3d vector • properties of scalar product

with coordinates (components) of relative to the basis • positive definiteness • orthogonality

tensor calculus 9 tensor calculus 10

example - radial displacement vector algebra - vector product • vector product

• properties of vector product u = x - X ur = u · r

tensor calculus 11 tensor calculus 12 vector algebra - scalar triple product tensor algebra - second order tensors

• scalar triple product • second order tensor

area volume with coordinates (components) of relative to • properties of scalar triple product the basis • transpose of second order tensor

• linear independency

tensor calculus 13 tensor calculus 14

tensor algebra - second order tensors example - scaled identity tensor / pressure • second order unit tensor in terms of kronecker symbol

with coordinates (components) of relative to the basis • matrix representation of coordinates

• identity

tensor calculus 15 tensor calculus 16 tensor algebra - third order tensors tensor algebra - fourth order tensors

• third order tensor • fourth order tensor

with coordinates (components) of with coordinates (components) of relative relative to the basis to the basis • fourth order unit tensor • third order permutation tensor in terms of permutation symbol • transpose of fourth order unit tensor

tensor calculus 17 tensor calculus 18

tensor algebra - fourth order tensors tensor algebra - scalar product • scalar (inner) product • symmetric fourth order unit tensor

• screw-symmetric fourth order unit tensor of second order tensor and vector • zero and identity • volumetric fourth order unit tensor • positive definiteness • properties of scalar product • deviatoric fourth order unit tensor

tensor calculus 19 tensor calculus 20 tensor algebra - scalar product tensor algebra - scalar product • scalar (inner) product • scalar (inner) product

of two second order tensors of two second order tensors and • scalar (inner) product • zero and identity • properties of scalar product of fourth order tensors and second order tensor • zero and identity

tensor calculus 21 tensor calculus 22

tensor algebra - dyadic product tensor algebra - dyadic product • dyadic (outer) product • dyadic (outer) product

of two vectors introduces second order tensor of two vectors introduces second order tensor • properties of dyadic product (tensor notation) • properties of dyadic product (index notation)

tensor calculus 23 tensor calculus 24 tensor algebra - invariants tensor algebra - trace • trace of second order tensor • (principal) invariants of second order tensor

• properties of traces of second order tensors • derivatives of invariants wrt second order tensor

tensor calculus 25 tensor calculus 26

tensor algebra - determinant tensor algebra - determinant

• determinant of second order tensor • determinant defining vector product

• properties of determinants of second order tensors • determinant defining scalar triple product

tensor calculus 27 tensor calculus 28 tensor algebra - inverse eigenvalue problem - maximum principal value

• inverse of second order tensor in particular • adjoint and cofactor

• properties of inverse

tensor calculus 29 tensor calculus 30

tensor algebra - spectral decomposition tensor algebra - sym/skw decomposition

• eigenvalue problem of second order tensor • symmetric - skew-symmetric decomposition

• solution in terms of scalar triple product • symmetric and skew-symmetric tensor • characteristic equation • symmetric tensor • spectral decomposition • skew-symmetric tensor • cayleigh hamilton theorem

tensor calculus 31 tensor calculus 32 tensor algebra - symmetric tensor tensor algebra - skew-symmetric tensor

• symmetric second order tensor • skew-symmetric second order tensor

• processes three real eigenvalues and corresp.eigenvectors • processes three independent entries defining axial vector such that • square root, inverse, exponent and log • invariants of skew-symmetric tensor

tensor calculus 33 tensor calculus 34

tensor algebra - vol/dev decomposition tensor algebra - orthogonal tensor

• volumetric - deviatoric decomposition • orthogonal second order tensor

• decomposition of second order tensor • volumetric and deviatoric tensor such that and • volumetric tensor • proper orthogonal tensor has eigenvalue

• deviatoric tensor with interpretation: finite rotation around axis

tensor calculus 35 tensor calculus 36 tensor analysis - frechet derivative tensor analysis - gateaux derivative • consider smooth differentiable scalar field with • consider smooth differentiable scalar field with

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

scalar argument scalar argument vector argument vector argument tensor argument tensor argument

tensor calculus 37 tensor calculus 38

tensor analysis - gradient example - displacement gradient / strain • consider scalar- and vector field in domain

• gradient of scalar- and vector field

! = !u (x)

renders vector- and 2nd order tensor field !tt = t ·!t · t

tensor calculus 39 tensor calculus 40 tensor analysis - divergence tensor analysis - laplace operator • consider vector- and 2nd order tensor field in domain • consider scalar- and vector field in domain

• divergence of vector- and 2nd order tensor field • laplace operator acting on scalar- and vector field

renders scalar- and vector field renders scalar- and vector field

tensor calculus 41 tensor calculus 42

tensor analysis - transformation formulae tensor analysis - transformation formulae • consider scalar,vector and 2nd order tensor field on • consider scalar,vector and 2nd order tensor field on

• useful transformation formulae (tensor notation) • useful transformation formulae (index notation)

tensor calculus 43 tensor calculus 44 tensor analysis - integral theorems tensor analysis - integral theorems • consider scalar,vector and 2nd order tensor field on • consider scalar,vector and 2nd order tensor field on

• integral theorems (tensor notation) • integral theorems (tensor notation)

green green gauss gauss gauss gauss

tensor calculus 45 tensor calculus 46

voigt / matrix vector notation voigt / matrix vector notation • strain tensors as vectors in voigt notation • fourth order material operators as matrix in voigt notation

• stress tensors as vectors in voigt notation

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

tensor calculus 47 tensor calculus 48