Non-Coaxiality of Principal Normal Stresses and the ‘Strain’ Ellipsoid in the Classical Theory on Infinitesimal Deformation

Abstract

A FUNDAMENTAL assumption in the classical theory on infinitesimal deformation in isotropic substances is that such an equation as (6) reduces to principal normal ‘strains’ η₁₁, η₂₂ only when the co-ordinate axes are in the principal normal stress directions^1,2. This hypothesis is shown in this communication to be untrue, in general, since this ‘strain’ ellipse is defined by means of gradients of ‘spatial-displacement’ U. Because the reference axes as used in the classical theory are fixed in space and independent of the deformed body, U contains a component due to ‘rotation-of-the-body-as-a-whole’³ even when this rotation is only infinitesimal. Thus, while all workers agree on the meaning of physical principal normal strain and stress directions, and that they are coaxial in an isotropic substance, the use of ‘spatial-displacement’ gradients to define strain leads to mathematical non-coaxiality with stress.