Traditionally, Mohr’s circle has been used as a graphical method for performing
coordinate transformations for stress, but the technique applies equally well to any
tensor matrix. Mohr’s circle also provides rapid graphical estimations for
eigenvalues and eigenvectors, which is extremely useful for verifying analytical
results. Mohr’s circle is not just for stress tensors, but it is typically taught in only that
context in introductory materials mechanics courses. For stress tensors, Mohr’s circle
can be used to visualize and to determine graphically the normal and shear stresses
acting on a plane of any given orientation. For symmetric tensors, Mohr’s circle can
be generalized to matrices for a graphical depiction of the set of all possible
normal and shear components. The traditional definition of Mohr’s circle for
symmetric matrices is presented with numerous examples for performing coordinate
transformations, finding the plane(s) of maximum shear, and identifying eigenvalues
and eigenvectors. An important but little-known enhancement to Mohr’s circle
(called the Pole Point) is shown to rectify counter-intuitive factors of 2 when
converting angles in physical space to angles in the Mohr diagram. The extension of
Mohr’s circle to matrices is presented with application to the Mohr-Coulomb
theory of material failure. The basic construction of Mohr’s circle is shown to apply
with only minor modification to nonsymmetric matrices, in which case the circle no
longer remains symmetric about the normal axis. Applications of nonsymmetric
Mohr’s circle include rapid eigenvalue/eigenvector determination and fast polar
decompositions for deformation matrices. A numerical exploration is presented that
suggests there is no simple extension of Mohr’s circle to nonsymmetric
matrices.