Nonlinear finite elements/Homework 7/Hints



Hints 1: Index notationEdit

Index notation:

 

If  

 
 

Dummy indices are replaceable.

Hint 2: Index notationEdit

Index notation:

 

Multiply by  :

 

Multiplication by   leads to replacement of one index.

 

Hint 3: Index notationEdit

Index notation:

 

From the definition of dyadic product, we can show

 

Contraction gives:

 

Hint 4: Tensor productEdit

Index notation:

 

Definition of dyadics products:

 

We can show that

 

Contraction gives:

 

Hint 5 : Tensor productEdit

Tensor Product of two tensors:

 

Tensor product:

 

Hint 6: Vector transformationsEdit

Change of basis: Vector transformation rule

 

  are the direction cosines.

 

In matrix form

 

Other common form: Vector transformation rule

 
 

In matrix form

 

Hint 7: Tensor transformationsEdit

Change of basis: Tensor transformation rule

 

where   are the direction cosines.

In matrix form,

 

Other common form

 

In matrix form,