Micromechanics of composites/Proof 11

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Identity 1[edit]

Let and be two second order tensors. Show that

Proof:

Using index notation,

Hence,

Identity 2[edit]

Let be a second order tensor and let and be two vectors. Show that

Proof:

It is convenient to use index notation for this. We have

Hence,

Identity 3[edit]

Let and be two second order tensors and let and be two vectors. Show that

Proof:

Using index notation,

Hence,

Identity 4[edit]

Let be a second order tensors and let and be two vectors. Show that

Proof:

For the first identity, using index notation, we have

Hence,

For the second identity, we have

Therefore,

Now, and . Hence,

Therefore,