Cartesian Tensors in Engineering Science by L. G. Jaeger and B. G. Neal (Auth.)

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

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Ans. 11 3 3 ~V6 7 VÏ8 3 5 7 VI8 3 3 29 V6 2 V12 V12 6 5. Notice that the answer to question 4 is symmetrical about the principal diagonal. g. question 3), then the components of this tensor with respect to any other set of axes are symmetrical. 6. Show that if the components of a second order tensor are symmetrical with respect to a given set of axes, the components are also symmetrical with respect to any other set of axes. 1 Mohr's Circle of Stress In a condition of plane stress, Fig. 12 shows the four com- FIG.

1 4. For a symmetric tensor there exist mutually perpendicular axes, with respect to which T = 0 when ρ and q are different. vq 47 SYMMETRIC SECOND ORDER TENSORS These are the principal axes. 5. If the three principal components of a symmetric tensor are all different, there is a uniquely defined triad of principal axes. 6. If two of the principal components of a symmetric tensor are equal and the other one is distinct, there exists a unique principal axis associated with the distinct component.

11 4. Axes x'i,x,2*3 are defi ned with respect to the (jci,jc2,X3) axes of Fig. 1 1 by m eans of the matrix of direction cosines given in question 1. Find the stress tensor components referred to these axes (use eqn. 20d)). Ans. 11 3 3 ~V6 7 VÏ8 3 5 7 VI8 3 3 29 V6 2 V12 V12 6 5. Notice that the answer to question 4 is symmetrical about the principal diagonal. g. question 3), then the components of this tensor with respect to any other set of axes are symmetrical. 6. Show that if the components of a second order tensor are symmetrical with respect to a given set of axes, the components are also symmetrical with respect to any other set of axes.

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