stress tensor4

# stress tensor4 - 12.005 Lecture Notes 4 Stress...

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12.005 Lecture Notes 4 Stress Tensor (continued) Before getting bogged down in rotation and notation, let’s consider a special case: rotation of the coordinate system by 45 o about x 3 . In parallel, consider the tractions on a plane with its normal in the x 1 – x 2 plane at 45 o to x 1 and x 2 . ˆ n = (2 /2, 2 /2,0) T = 2 /2 2 0 11 12 12 22 13 23 2 2 Tn σσ σ + ⎛⎞ ⎜⎟ =⋅= + + ⎝⎠ ± ±± ± x 2 x 1 x 3 , x 3 x 2 ' x 1 45 o ' ' Figure by MIT OCW. Figure 4.1

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Notes: Case 1: ± T (on plane at 45 o ) is only σ n = p , no τ . Case 2: is still only ± T 45 o n = 1 . Case 3: ± T on faces 1 and 2 only n , no . is pure ± T 45 o , no n . Case 4: ± T on faces 1 and 2 all , no n . σ 3 τ τ σ 1 σ 1 x 2 P CASE 1 CASE 3 CASE 4 "Hydrostatic" "Uniaxial" "Plane Stress" CASE 2 P P x 3 x 1 x 2 x 3 x 1 +S -S Figure by MIT OCW. Figure 4.2 is all T 45 o ± n , no .
Evidently, there are “special” directions, at least for these cases, where τ = 0 ! Is this true in general? Can we find the “principal” frame where σ ij p = 11 p 00 0 22 p 0 33 p

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stress tensor4 - 12.005 Lecture Notes 4 Stress...

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