04 - tensor calculus - tensor analysis

04 - tensor calculus 1 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 2 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 3 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 4 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 5 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 6 tensor analysis - transformation formulae • consider scalar,vector and 2nd order tensor field on

• useful transformation formulae (tensor notation)

tensor calculus 7 tensor analysis - transformation formulae • consider scalar,vector and 2nd order tensor field on

• useful transformation formulae (index notation)

tensor calculus 8 tensor analysis - integral theorems • consider scalar,vector and 2nd order tensor field on

• integral theorems (tensor notation)

green gauss gauss

tensor calculus 9 tensor analysis - integral theorems • consider scalar,vector and 2nd order tensor field on

• integral theorems (tensor notation)

green gauss gauss

tensor calculus 10 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 11 voigt / matrix vector notation

• fourth order material operators as matrix in voigt notation

• why are strain & stress different? check these expressions!

tensor calculus 12