On Euler’s stretching formula in continuum mechanics

Abstract

The Lie time-derivative of the material metric tensor field along the motion is the proper mathematical definition of the physical notion of strain rate or stretching. Its expression, as symmetric part of the velocity gradient in Euclid space, is provided by a celebrated formula conceived by the genius of Leonhard Euler around the middle of the eighteenth century and since then reproduced in articles, books and treatises on continuum mechanics. We present here a formulation, in the proper geometric context of the four-dimensional space-time manifold endowed with an arbitrary linear connection and referring to a material body of arbitrary dimensionality. The expression involves the material time-derivative of the metric field and torsion-form and gradient of the velocity field, according to the connection induced on the trajectory. As an application, the expressions of the Gram matrix of the stretching in natural and in normalized (or engineering) reference systems induced by orthogonal polar coordinates are provided.