Mohr's circle
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.
Ankit Jhille