Grioli's theorem: rotation minimizing deformation

Abstract

In this paper, the celebrated theorem of G. Grioli is considered according to which the rotation factor in the polar decomposition of the deformation gradient minimizes Biot's strain tensor. The theorem is demonstrated by applications to some cases in large displacement theory: simple shear, plane deformation, Euler-Bernoulli and Timoshenko beam theories, and bar element in space. An interpretation could be that the material behaves economically: first occurs the part of deformation which does not induce any stresses and then the material starts to resist the deformation.