Many of the problems for Sections 2.1-2.4 require the determination of principal stresses and maximum shear stress, as well as the location of the planes on which they act. These quantities are required input in design and failure criteria for structural and mechanical systems.
Let (σx, σy, σz) be stress vectors on planes perpendicular, respectively, to the (x, y, z) axes. Show that the sum of the squares of the magnitudes of these stress vectors is an invariant, expressible in terms of the stress invariants I1 and I2.
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