Tensor algebra - invariants 03 - tensor calculus - tensor ...

03 - tensor calculus tensor analysis

03 - tensor calculus

1

tensor algebra - trace

? trace of second order tensor

? properties of traces of second order tensors

tensor algebra - invariants

? (principal) invariants of second order tensor

? derivatives of invariants wrt second order tensor

tensor calculus

2

tensor algebra - determinant

? determinant of second order tensor

? properties of determinants of second order tensors

tensor calculus

3

tensor calculus

4

tensor algebra - determinant

? determinant defining vector product

? determinant defining scalar triple product

tensor calculus

5

tensor algebra - spectral decomposition

? eigenvalue problem of second order tensor

? solution

in terms of scalar triple product

? characteristic equation

? spectral decomposition ? cayleigh hamilton theorem

tensor calculus

7

tensor algebra - inverse

? inverse of second order tensor in particular

? adjoint and cofactor

? properties of inverse

tensor calculus

6

tensor algebra - sym/skw decomposition

? symmetric - skew-symmetric decomposition ? symmetric and skew-symmetric tensor ? symmetric tensor ? skew-symmetric tensor

tensor calculus

8

tensor algebra - symmetric tensor

? symmetric second order tensor ? processes three real eigenvalues and corresp.eigenvectors

? square root, inverse, exponent and log

tensor calculus

9

tensor algebra - vol/dev decomposition

? volumetric - deviatoric decomposition ? volumetric and deviatoric tensor ? volumetric tensor ? deviatoric tensor

tensor calculus

11

tensor algebra - skew-symmetric tensor

? skew-symmetric second order tensor

? processes three independent entries defining axial vector such that

? invariants of skew-symmetric tensor

tensor calculus

10

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 calculus

12

tensor analysis - frechet derivative

? consider smooth differentiable scalar field with

scalar argument vector argument tensor argument

? frechet derivative (tensor notation)

scalar argument

vector argument

tensor argument

tensor calculus

13

tensor analysis - gradient

? consider scalar- and vector field in domain

? gradient of scalar- and vector field

renders vector- and 2nd order tensor field

tensor calculus

15

tensor analysis - gateaux derivative

? consider smooth differentiable scalar field with

scalar argument vector argument tensor argument

? gateaux derivative,i.e.,frechet wrt direction (tensor notation)

scalar argument

vector argument

tensor argument

tensor calculus

14

tensor analysis - divergence

? consider vector- and 2nd order tensor field in domain

? divergence of vector- and 2nd order tensor field

renders scalar- and vector field

tensor calculus

16

tensor analysis - laplace operator

? consider scalar- and vector field in domain

? laplace operator acting on scalar- and vector field

renders scalar- and vector field

tensor calculus

17

tensor analysis - transformation formulae

? consider scalar,vector and 2nd order tensor field on

? useful transformation formulae (index notation)

tensor calculus

19

tensor analysis - transformation formulae

? consider scalar,vector and 2nd order tensor field on

? useful transformation formulae (tensor notation)

tensor calculus

18

tensor analysis - integral theorems

? consider scalar,vector and 2nd order tensor field on

? integral theorems (tensor notation)

tensor calculus

green gauss gauss

20

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