Lamé's Ellipsoid and Mohr's Circles (Part 3: Meridians)
For a continuous body, the stress tensor is a symmetric matrix representing the stresses at a point. The traction vector on a plane defined by its unit normal vector is the matrix product of the stress tensor with . The stress tensor has three real eigenvalues (the principal stresses, , , and ) and three associated eigenvectors (the principal directions). In the coordinate system defined by these eigenvectors, the Lamé's ellipsoid represents the locus of the traction vector heads.[more]
This Demonstration shows ellipses, called meridians, obtained by cutting Lamé's ellipsoid by sheaves of planes through the , , and axes, with equations
In the first two cases, is the angle between the plane and the - plane (); in the third case is the angle between the plane and the - plane ().
The points of the meridians are endpoints of the traction vectors, whose intrinsic components (normal and tangential) are represented by green loci within the Mohr circles.[less]