1 03 - tensor calculus 03 - introduction to vectors and tensors holzapfel nonlinear solid mechanics [2000], chapter 1.6-1.9, pages 32-55 2 introduction me338 - syllabus 3 tensor calculus tensor the word tensor was introduced in 1846 by william rowan hamilton. it was 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. tensor calculus 4 tensor calculus • tensor analysis • vector algebra notation, euklidian vector space, scalar product, vector product, scalar triple product notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew decomposition, vol-dev decomposition, orthogonal tensor derivatives, gradient, divergence, laplace operator, integral transformations • tensor algebra tensor calculus
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1 03 - tensor calculus
03 - introduction to vectors and tensors
holzapfel �nonlinear solid mechanics� [2000], chapter 1.6-1.9, pages 32-55
2 introduction
me338 - syllabus
3 tensor calculus
tensor the word tensor was introduced in 1846 by william rowan hamilton. it was 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.
tensor calculus
4 tensor calculus
• tensor analysis
• vector algebra notation, euklidian vector space, scalar product, vector product, scalar triple product
notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew decomposition, vol-dev decomposition, orthogonal tensor
derivatives, gradient, divergence, laplace operator, integral transformations
• tensor algebra
tensor calculus
5 tensor calculus
• euklidian norm enables definition of scalar (inner) product
• properties of scalar product
• positive definiteness • orthogonality
vector algebra - scalar product
6 tensor calculus
• vector product
• properties of vector product
vector algebra - vector product
7 tensor calculus
• scalar triple product
• properties of scalar triple product
area volume
• linear independency
vector algebra - scalar triple product
8 tensor calculus
• scalar (inner) product
• properties of scalar product
of second order tensor and vector
• zero and identity • positive definiteness
tensor algebra - scalar product
9 tensor calculus
• scalar (inner) product
• properties of scalar product
of two second order tensors and
• zero and identity
tensor algebra - scalar product
·
10 tensor calculus
• scalar (inner) product
of fourth order tensors and second order tensor • zero and identity
• scalar (inner) product of two second order tensors
tensor algebra - scalar product
11 tensor calculus
• dyadic (outer) product
• properties of dyadic product (tensor notation) of two vectors introduces second order tensor
tensor algebra - dyadic product
12 tensor calculus
• (principal) invariants of second order tensor
• derivatives of invariants wrt second order tensor
tensor algebra - invariants
13 tensor calculus
• trace of second order tensor
• properties of traces of second order tensors
tensor algebra - trace
14 tensor calculus
• determinant of second order tensor
• properties of determinants of second order tensors
tensor algebra - determinant
15 tensor calculus
• determinant defining vector product
• determinant defining scalar triple product
tensor algebra - determinant
16 tensor calculus
• inverse of second order tensor in particular
• properties of inverse
• adjoint and cofactor
tensor algebra - inverse
17 tensor calculus
• eigenvalue problem of second order tensor
• spectral decomposition
• characteristic equation
• cayleigh hamilton theorem
• solution in terms of scalar triple product
tensor algebra - spectral decomposition
18 tensor calculus
• symmetric - skew-symmetric decomposition
• skew-symmetric tensor
• symmetric tensor
• symmetric and skew-symmetric tensor
tensor algebra - sym/skw decomposition
19 tensor calculus
• symmetric second order tensor
• square root, inverse, exponent and log
• processes three real eigenvalues and corresp.eigenvectors
tensor algebra - symmetric tensor
20 tensor calculus
• skew-symmetric second order tensor
• invariants of skew-symmetric tensor
• processes three independent entries defining axial vector such that
tensor algebra - skew-symmetric tensor
21 tensor calculus
• volumetric - deviatoric decomposition
• deviatoric tensor
• volumetric tensor
• volumetric and deviatoric tensor
tensor algebra - vol/dev decomposition
22 tensor calculus
• orthogonal second order tensor
• proper orthogonal tensor has eigenvalue
• decomposition of second order tensor
such that and
interpretation: finite rotation around axis
with
tensor algebra - orthogonal tensor
23 tensor calculus
• frechet derivative (tensor notation)
• consider smooth differentiable scalar field with scalar argument vector argument tensor argument
scalar argument vector argument tensor argument
tensor analysis - frechet derivative
24 tensor calculus
• 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
tensor analysis - gateaux derivative
25 tensor calculus
• gradient of scalar- and vector field
• consider scalar- and vector field in domain
renders vector- and 2nd order tensor field
tensor analysis - gradient
26 tensor calculus
• divergence of vector- and 2nd order tensor field
• consider vector- and 2nd order tensor field in domain
renders scalar- and vector field
tensor analysis - divergence
27 tensor calculus
• laplace operator acting on scalar- and vector field
• consider scalar- and vector field in domain
renders scalar- and vector field
tensor analysis - laplace operator
28 tensor calculus
• useful transformation formulae (tensor notation)
• consider scalar,vector and 2nd order tensor field on
tensor analysis - transformations
29 tensor calculus
• integral theorems (tensor notation)
• consider scalar,vector and 2nd order tensor field on
green gauss gauss
tensor analysis - integral theorems
30 tensor calculus
• integral theorems (tensor notation)
• consider scalar,vector and 2nd order tensor field on
green gauss gauss
tensor analysis - integral theorems
31 tensor calculus
• stress tensors as vectors in voigt notation
• strain tensors as vectors in voigt notation
• why are strain & stress different? check energy expression!
voigt / matrix vector notation
32 tensor calculus
• fourth order material operators as matrix in voigt notation
• why are strain & stress different? check these expressions!
voigt / matrix vector notation
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