Example Principal Stresses

The stress tensor is given at a point by u

determine the principal stress values and the corresponding directions. Solution:

From Eq.2.32 we have

Or upon expansion (and simplification) (A + 2)(A — 4)(A — 1) = 0, thus the roots are a(l) = 4, a(2) = 1 and a(3) = —2. We also note that those are the three eigenvalues of the stress tensor.

If we let xl axis be the one corresponding to the direction of a(3) and n3 be the direction cosines of this axis, then from Eq. 2.28 we have

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Similarly If we let x2 axis be the one corresponding to the direction of a(2) and n2 be the direction cosines of this axis,

Finally, if we let x3 axis be the one corresponding to the direction of a(l) and nl be the direction cosines of this axis,

—n\ + n2 + n3 = 0 nl — 4n2 +2n1 = 0 nl + 2n2 — 4n1 = 0

Finally, we can convince ourselves that the two stress tensors have the same invariants Ia, IIa and IIIa.

a jq

0 0

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