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03 - Introduction Me338 - Syllabus to Vectors and Tensors 03 - introduction me338 - syllabus to vectors and tensors holzapfel nonlinear solid mechanics [2000], chapter 1.6-1.9, pages 32-55 03 - tensor calculus 1 introduction 2 tensor calculus tensor calculus tensor the word tensor was introduced • vector algebra in 1846 by william rowan hamilton. it was notation, euklidian vector space, scalar product, vector used in its current meaning by woldemar voigt in 1899. tensor calculus was deve- product, scalar triple product loped around 1890 by gregorio ricci-curba- • tensor algebra stro under the title absolute differential notation, scalar products, dyadic product, invariants, trace, calculus. in the 20th century, the subject determinant, inverse, spectral decomposition, sym-skew came to be known as tensor analysis, and achieved broader acceptance with the intro- decomposition, vol-dev decomposition, orthogonal tensor duction of einsteins's theory of general • tensor analysis relativity around 1915. tensors are used derivatives, gradient, divergence, laplace operator, integral also in other fields such as continuum transformations mechanics. tensor calculus 3 tensor calculus 4 vector algebra - scalar product vector algebra - vector product • euklidian norm enables definition of scalar (inner) product • vector product • properties of scalar product • properties of vector product • positive definiteness • orthogonality tensor calculus 5 tensor calculus 6 vector algebra - scalar triple product tensor algebra - scalar product • scalar triple product • scalar (inner) product area volume of second order tensor and vector • zero and identity • properties of scalar triple product • positive definiteness • properties of scalar product • linear independency tensor calculus 7 tensor calculus 8 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 9 tensor calculus 10 tensor algebra - dyadic product tensor algebra - invariants • dyadic (outer) product • (principal) invariants of second order tensor of two vectors introduces second order tensor • properties of dyadic product (tensor notation) • derivatives of invariants wrt second order tensor tensor calculus 11 tensor calculus 12 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 13 tensor calculus 14 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 15 tensor calculus 16 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 • cayleigh hamilton theorem • skew-symmetric tensor tensor calculus 17 tensor calculus 18 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 19 tensor calculus 20 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 with • deviatoric tensor interpretation: finite rotation around axis tensor calculus 21 tensor calculus 22 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 23 tensor calculus 24 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 25 tensor calculus 26 tensor analysis - laplace operator tensor analysis - transformations • 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 27 tensor calculus 28 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 29 tensor calculus 30 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 these expressions! • why are strain & stress different? check energy expression! tensor calculus 31 tensor calculus 32 .
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