03 - tensor calculus tensor analysis
03 - tensor calculus
1
tensor algebra - invariants
• (principal) invariants of second order tensor
• derivatives of invariants wrt second order tensor
tensor calculus
2
tensor algebra - trace
• trace of second order tensor
• properties of traces of second order tensors
tensor calculus
3
tensor algebra - determinant
• determinant of second order tensor
• properties of determinants of second order tensors
tensor calculus
4
tensor algebra - determinant
• determinant defining vector product
• determinant defining scalar triple product
tensor calculus
5
tensor algebra - inverse
• inverse of second order tensor
in
particular
• adjoint and cofactor
• properties of inverse
tensor calculus
6
tensor algebra - spectral decomposition
• eigenvalue problem of second order tensor
• solution
product
in terms of scalar triple
• characteristic equation
• spectral decomposition
• cayleigh hamilton theorem
tensor calculus
Qu ickT ime™ and a
TIF F (U ncom pres sed) deco mpre ssor
are nee ded t o see this pictu re.
7
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 - 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 - vol/dev decomposition
• volumetric - deviatoric decomposition
• volumetric and deviatoric tensor
• volumetric tensor
• deviatoric tensor
tensor calculus
11
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 - 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 - 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 - 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 (tensor notation)
tensor calculus
18
tensor analysis - transformation formulae
• consider scalar,vector and 2nd order tensor field on
• useful transformation formulae (index notation)
tensor calculus
19
tensor analysis - integral theorems
• consider scalar,vector and 2nd order tensor field on
• integral theorems (tensor notation)
green
gauss
gauss
tensor calculus
20
tensor analysis - integral theorems
• consider scalar,vector and 2nd order tensor field on
• integral theorems (tensor notation)
green
gauss
gauss
tensor calculus
21
voigt / matrix vector notation
• strain tensors as vectors in voigt notation
• stress tensors as vectors in voigt notation
• why are strain & stress different? check energy
expression!
tensor calculus
22
voigt / matrix vector notation
• fourth order material operators as matrix in voigt
notation
• why are strain & stress different? check these
expressions!
tensor calculus
23
deformation gradient
• given the deformation gradient, play with matlab
tobecome familiar with basic tensor operations!
• uniaxial tension (incompressible), simple shear,
rotation
example #1 - matlab
24
second order tensors - scalar products
• inverse of second order tensor
• right / left cauchy green and green lagrange strain
tensor
• trace of second order tensor
• (principal) invariants of second order tensor
example #1 - matlab
25
fourth order tensors - scalar products
• symmetric fourth order unit tensor
• screw-symmetric fourth order unit tensor
• volumetric fourth order unit tensor
• deviatoric fourth order unit tensor
example #1 - matlab
26
neo hooke‘ian elasticity
• free energy
• 1st and 2nd piola kirchhoff stress and cauchy stress
• 4th order tangent operators
example #1 - matlab
27
matlab
• play with the matlab routine to familiarize yourself
tensor expressions!
with
• calculate the stresses for different deformation
gradients!
• which of the following stress tensors is symmetric
and
could be represented in voigt notation?
• what would
look like in the linear limit, for
• what are the advantages of using the voigt
notation?
homework #2
28