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Micromechanics of composites/Proof 2

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Tensor-vector identity 2

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Let be a vector field and let be a second-order tensor field. Let and be two arbitrary vectors. Show that

Proof:

Using the identity we have

From the identity , we have .

Since is constant, , and we have

From the relation we have

Using the relation , we get

Therefore, the final form of the first term is

For the second term, from the identity we get, .

Since is constant, , and we have

From the definition , we get

Therefore, the final form of the second term is

Adding the two terms, we get

Therefore,