Lamé's Ellipsoid and Mohr's Circles (Part 4: Spheres)
Given a continuum body, the stress tensor is a symmetric matrix that contains the stresses at a point. The traction vector on a plane defined by its perpendicular unitary vector, , is obtained multiplying the stress tensor by . The stress tensor has three real eigenvalues (the principal stresses: , , and ) and three associated eigenvectors (the principal directions). On the coordinate system defined by these eigenvectors Lamé's ellipsoid represents where the heads of the traction vectors lie.
This Demonstration shows the curves obtained by cutting Lamé's ellipsoid with spheres of different radius; the points of these curves are the extremes of the traction vectors with the same norm. Their intrinsic components (normal and tangential) are shown as the green locus over Mohr's circles.