Pressure forces are independent of direction

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Reference no: EM131049785

Show that if the stress tensor Σ is diagonal (all off-diagonal elements zero) with respect to any choice of orthogonal axes, then it is in fact a multiple of the unit matrix. This gives an alternative and elegant proof that if there are no shearing stresses (in any coordinate system) then the pressure forces are independent of direction. To do this problem, you need to know how the elements of a tensor transform as we rotate our coordinate axes, as described in Section 15.17. Assume that with respect to one set of axes Σ is diagonal but that not all three diagonal elements are equal. (For example, σ11≠σ33.) It is not hard to come up with a rotation - that of Equation (15.36) will do - such that in the rotated system σ'13≠0.

Reference no: EM131049785

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