Cartesian Tensors in Engineering Science, Edition: 1St by L. G. Jaeger and B. G. Neal (Auth.)

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

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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]. 30 CARTESIAN TENSORS IN E N G I N E E R I N G SCIENCE Examples on Chapter 2 1. A matrix of direction cosines is 1 1 1 V3 V3 V3 1 1 0 V2 'V2 1 1 2 V6 V 6 V6 (a) Verify that this is a valid array of direction cosines which satisfies all the six requirements of eqns.

Some further discussion of the principal axes of a symmetric tensor will be found in Chapter 4 under the discussion of eigenvectors. Summary of Chapter 3 1. In a plane the components of a symmetric tensor with respect to various pairs of perpendicular axes in the plane may be displayed by Mohr's circle. 2. In a Mohr's circle the abscissa is a component Τ[ of the tensor and the ordinate is — 7\' . λ 2 3. The transform [Τ'] = [λ] [Τ] [λ]- involves rocking from one set of axes to the other, and then back again to the first.

For any given diameter of bar, there is one particular length for which h = h = h \ in this one case the moment of inertia tensor is isotropic with respect to all three axes, and the xi direction is no longer uniquely defined. In fact any axis through G is then a principal axis. Some further discussion of the principal axes of a symmetric tensor will be found in Chapter 4 under the discussion of eigenvectors. Summary of Chapter 3 1. In a plane the components of a symmetric tensor with respect to various pairs of perpendicular axes in the plane may be displayed by Mohr's circle.

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