03 - Tensor Calculus - Tensor Analysis • (Principal) Invariants of Second Order Tensor

tensor algebra - invariants 03 - tensor calculus - tensor analysis • (principal) invariants of second order tensor

• derivatives of invariants wrt second order tensor

03 - tensor calculus 1 tensor calculus 2

tensor algebra - trace tensor algebra - determinant

• trace of second order tensor • determinant of second order tensor

• properties of traces of second order tensors • properties of determinants of second order tensors

tensor calculus 3 tensor calculus 4 tensor algebra - determinant tensor algebra - inverse

• determinant deﬁning vector product • inverse of second order tensor in particular • adjoint and cofactor

• determinant deﬁning scalar triple product • properties of inverse

tensor calculus 5 tensor calculus 6

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 7 tensor calculus 8 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 9 tensor calculus 10

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 11 tensor calculus 12 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 13 tensor calculus 14

tensor analysis - gradient tensor analysis - divergence • consider scalar- and vector ﬁeld in domain • consider vector- and 2nd order tensor ﬁeld in domain

• divergence of vector- and 2nd order tensor ﬁeld • gradient of scalar- and vector ﬁeld

renders vector- and 2nd order tensor ﬁeld renders scalar- and vector ﬁeld

tensor calculus 15 tensor calculus 16 tensor analysis - laplace operator tensor analysis - transformation formulae • consider scalar- and vector ﬁeld in domain • consider scalar,vector and 2nd order tensor ﬁeld on

• laplace operator acting on scalar- and vector ﬁeld • useful transformation formulae (tensor notation)

renders scalar- and vector ﬁeld

tensor calculus 17 tensor calculus 18

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

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

green gauss gauss

tensor calculus 19 tensor calculus 20 tensor analysis - integral theorems voigt / matrix vector notation • consider scalar,vector and 2nd order tensor ﬁeld on • strain tensors as vectors in voigt notation

• integral theorems (tensor notation) • stress tensors as vectors in voigt notation

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

tensor calculus 21 tensor calculus 22

voigt / matrix vector notation deformation gradient • given the deformation gradient, play with matlab to • fourth order material operators as matrix in voigt notation become familiar with basic tensor operations!

• uniaxial tension (incompressible), simple shear, rotation

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

tensor calculus 23 example #1 - matlab 24 second order tensors - scalar products fourth order tensors - scalar products • inverse of second order tensor • symmetric fourth order unit tensor • right / left cauchy green and green lagrange strain tensor • screw-symmetric fourth order unit tensor

• trace of second order tensor • volumetric fourth order unit tensor

• (principal) invariants of second order tensor • deviatoric fourth order unit tensor

example #1 - matlab 25 example #1 - matlab 26

neo hooke‘ian elasticity matlab

• free energy • play with the matlab routine to familiarize yourself with tensor expressions! • 1st and 2nd piola kirchhoff stress and cauchy stress • calculate the stresses for different deformation gradients! • which of the following stress tensors is symmetric and • 4th order tangent operators could be represented in voigt notation? • what would look like in the linear limit, for • what are the advantages of using the voigt notation?

example #1 - matlab 27 homework #2 28