03 - introduction to vectors and tensors. me338 - syllabus. introduction tensor calculus. tensor calculus. tensor calculus.

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1 03 - introduction to vectors and tensors me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter , pages introduction 2 tensor the word tensor was introduced in 1846 by william rowan hamilton. it was used in its current meaning by woldemar voigt in was developed around 1890 by gregorio ricci-curbastro 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 introduction of einsteins's theory of general relativity around tensors are used also in other fields such as continuum mechanics. vector algebra 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 decomposition, vol-dev decomposition, orthogonal tensor tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations 3 4

2 vector algebra - scalar product euklidian norm enables definition of scalar (inner) product vector algebra - vector product vector product properties of scalar product properties of vector product positive definiteness orthogonality 5 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 7 8

3 tensor algebra - scalar product tensor algebra - scalar product scalar (inner) product scalar (inner) product of two second order tensors and zero and identity of two second order tensors scalar (inner) product properties of scalar product of fourth order tensors and second order tensor zero and identity 9 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 11 12

4 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 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 15 16

5 tensor algebra - spectral decomposition eigenvalue problem of second order tensor tensor algebra - sym/skw decomposition symmetric - skew-symmetric decomposition solution characteristic equation spectral decomposition cayleigh hamilton theorem in terms of scalar triple product symmetric and skew-symmetric tensor symmetric tensor skew-symmetric tensor tensor algebra - symmetric tensor tensor algebra - skew-symmetric tensor symmetric second order tensor processes three real eigenvalues and corresp.eigenvectors square root, inverse, exponent and log skew-symmetric second order tensor processes three independent entries defining axial vector such that invariants of skew-symmetric tensor 19 20

6 tensor algebra - vol/dev decomposition volumetric - deviatoric decomposition volumetric and deviatoric tensor volumetric tensor deviatoric tensor 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 analysis - frechet derivative tensor analysis - gateaux derivative consider smooth differentiable scalar field scalar argument vector argument tensor argument frechet derivative (tensor notation) scalar argument vector argument tensor argument with consider smooth differentiable scalar field scalar argument vector argument tensor argument gateaux derivative,i.e.,frechet wrt direction (tensor notation) scalar argument vector argument tensor argument with 23 24

7 tensor analysis - gradient tensor analysis - divergence consider scalar- and vector field in domain consider vector- and 2nd order tensor field in domain gradient of scalar- and vector field divergence of vector- and 2nd order tensor field renders vector- and 2nd order tensor field renders scalar- and vector field 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 27 28

8 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 gauss gauss green gauss gauss 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! 31 32