Modal methods for the analysis of discrete systems

There has been much recent interest in the Distinct Element Method for analyzing soil and rock as a discontinuum composed of numerous individual deformable grains or blocks. The solution procedure for calculating element deformations relies on the superposition of element strain modes, where each strain mode is obtained by solving an independent dynamic equilibrium equation. It is shown here that this procedure sometimes applied incorrectly. For elements of arbitrary shape, the shear strain mode is not orthogonal to rotation and therefore cannot be superposed on rotation to give the true displaced shape and position.It is also shown that modal equations must be solved with respect to axes which translate and rotate with the element, and cannot be written with respect to fixed global coordinates as implemented in present distinct element codes.Finally it is shown that various alternative valid bases for modal decomposition can be derived from a single formulation and a number of examples are given for general element shapes.