Characterizing the non-proportional and out-of-phase extent of tensor paths

Scalar measures are developed for characterizing the non-proportional and o ut-of-phase extent of any symmetric second-order tensor path. These scalar measures are constructed by introducing the concept of the principal axes o f the tensor path. These measures are independent of the parameterization o f the tensor path. As required these measures are invariant under coordinat e transformations. For an example, these measures are used to characterize the stress history in the combustion dome of an internal-combustion automot ive engine.