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Chapter 10: Q2MP (page 535)
Let bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct .
It has been shown that the scalar triple product is .
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If P and S are -rank tensors, show that coefficients are needed to write each component of P as a linear combination of the components of S. Show that is the number of components in a -rank tensor. If the components of the -rank tensor are , then equation gives the components of P in terms of the components of S. If P and S are both symmetric, show that we need only 36different non-zero components in . Hint: Consider the number of different components in P and S when they are symmetric. Comment: The stress and strain tensors can both be shown to be symmetric. Further symmetry reduces the 36components of C in (7.5)to 21or less.
Show that the transformation equation for a -rank Cartesian tensor is equivalent to a similarity transformation. Warning hint: Note that the matrix C in Chapter 3 , Section 11 , is the inverse of the matrix A we are using in Chapter 10 (compare). Thus a similarity transformation of the matrix T with tensor components is. Also, see 鈥淭ensors and Matrices鈥� in Section 3 and remember that A is orthogonal.
Verify Hints: In Figure , consider the projection of the slanted face of area onto the three unprimed coordinate planes. In each case, show that the projection angle is equal to an angle between the axis and one of the unprimed axes. Find the cosine of the angle from the matrix A in .
Bipolar.
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