By L. G. Jaeger and B. G. Neal (Auth.)

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Any axis is a principal axis. From the above it follows that if a tensor has the form 0 0' 0 k 0 0 0 k "k with respect to a given triad of axes it has the same form with respect to any other triad. An elegant proof of this is possible using matrix theory as follows [Γ] = [λ] [Τ] = k[X] [I] [A]-i since [T] = k[I] = k[X] = k[I], [Τ'] = [T]. e. 6) The proof using suffices is even shorter (though the steps are, mathematically, identical) and is given later in this chapter after SYMMETRIC SECOND ORDER 45 TENSORS a discussion of a very frequent use of the Kronecker Delta.

The tensor is isotropic in the plane of the two equal components. 7. If all three principal components are equal, there is no uniquely defined principal axis, and any triad of mutually perpendicular axes is an acceptable set of principal axes. The tensor is then of the form kô , where k is a scalar, no matter what triad of axes is chosen. Such a tensor is an isotropic tensor. pq 8. The Kronecker Delta is a second order symmetric tensor. It is the unit isotropic tensor. Examples on Chapter 3 1. A plane area has axes χι,χ > mutually perpendicular, in the 2 plane.

XSpXtqXUr> · -Tstu. . 7. A scalar is a zeroth order tensor and a vector is afirstorder tensor. 8. In η dimensions, a Mi order tensor has n components. In η dimensions, a scalar, a vector and a second order tensor may be represented respectively as a 1 χ 1 matrix, an « χ 1 matrix, and a n « x « matrix. The transformation laws of vectors and second order tensors may be written in the following forms k Vector Suffix notation Matrix notation A\ = X^Aj and A = X^Aj [X] {A } and {Α} = [λ] {Α'} % {A'}= τ Second order tensor Suffix notation T' = Matrix notation [Γ] st XSpXtqTPq = [A] [T] and T = [X] and [Τ] pq T XSpXtqT'st = Τ [Χ] [Τ'] [A].