At a point in a loaded component the stress tensor

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Invariance, Ch Index notation and Cartesian tensors

Problem

At a point in a loaded component the stress tensor, expressed in its principal axes, is σij=diag(100,50,−30) MPa\sigma_{ij} = \mathrm{diag}(100, 50, -30)\,\mathrm{MPa}. ((a)) Compute the traction vector ti=σijnjt_i = \sigma_{ij}n_j on the plane whose outward normal is ni=(0,1,0)n_i = (0, 1, 0), i.e. one of the principal planes. ((b)) Show that this traction has no component tangent to the plane (no shear on a principal plane) by computing ti−(tknk)nit_i - (t_kn_k)n_i.

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