Powers of Second Order Tensors Hamilton Cayley Equations

64 When expressed in term of the principal axes, the tensor array can be written in matrix form as

A

(l)

0

0

0

A(2)

0

0

0

65 By direct matrix multiplication, the quare of the tensor T.j is given by the inner product TikTkj, the cube as TikTkmTmn. Therefore the nth power of T.j can be written as n=

A

n (l)

0

0

0

n A(2)

0

0

0

Since each of the principal values satisfies Eq. 1.84 and because the diagonal matrix form of t given above, then the tensor itself will satisfy Eq. 1.84.

where I is the identity matrix. This equation is called the Hamilton-Cayley equation.

0 0

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