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

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

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Extending these ideas to three dimensions the reader will readily verify, using eqn. e. that symmetry with respect to any given triad of axes implies symmetry with respect to any other triad. e. one canfinddirection cosines such that pq 2 Γ In 0 0 0 ^22 0 0 ' 0 = 733 . ^11 λΐ2 λΐ3 ' λ21 λ22 ^23 _ ^31 λζ2 ^33 _ Τη Τ\2 Τ\ζ Τ\2 Τ22 Τ2Ζ ^12 ^22 ^32 Γ ΐ 3 Γ23 733 _ /ίΐ3 ^23 ^33 ^21 ^31 SYMMETRIC SECOND ORDER TENSORS 41 To see that this is so, let the nine direction cosines be treated as unknowns. Then insistence that T[ 7\' and T \ shall be zero gives three equations between these nine direction cosines.

7>0>~ T F 3. Figure 11 shows a cube of metal which is subjected to direct stresses ση = 6 ton/in , σ 2 = 4 ton/in , σ33 = 1 ton/in . The shear stresses on these planes are zero. Write down the stress tensor. 2 2 2 Ans. 0 4 0 2 P R O P E R T I E S OF D I R E C T I O N C O S I N E A R R A Y S 31 χ> 1 Σ-, 2= 4 ton/irr • Fig. 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.

3) — SpqTvq. On changing axes, the right-hand side becomes (λαρλ^Ξ'αυ) x Applying X\>qXdq — St>d this reduces to X pX SbdSabTcd and on using "change b into d and drop

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