Nonlinear finite elements/Homework 7/Hints

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Hints 1: Index notation[edit]

Index notation:

If

Dummy indices are replaceable.

Hint 2: Index notation[edit]

Index notation:

Multiply by :

Multiplication by leads to replacement of one index.

Hint 3: Index notation[edit]

Index notation:

From the definition of dyadic product, we can show

Contraction gives:

Hint 4: Tensor product[edit]

Index notation:

Definition of dyadics products:

We can show that

Contraction gives:

Hint 5 : Tensor product[edit]

Tensor Product of two tensors:

Tensor product:

Hint 6: Vector transformations[edit]

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 transformations[edit]

Change of basis: Tensor transformation rule

where are the direction cosines.

In matrix form,

Other common form

In matrix form,