Polar decomposition and appropriate strains and stresses for nonlinear structural analyses

Work conjugacy, objectivity, vector expressions, directions and limitations of major pairs of stress and strain measures used in geometrically nonlinear and elastoplastic analyses of structures are investigated. Polar decomposition is examined in detail. For geometrically nonlinear analysis, Jaumann strains and stresses are the most appropriate pair of measures because Jaumann strains are proved to be objective (invariant under rigid-body rotations) corotated engineering strains. It is shown that Jaumann strains can be easily derived by using a new concept of local displacements without using the complex polar decomposition procedure. Moreover, the use of Jaumann stresses and strains results in a direct correlation between energy and Newtonian approaches and makes all structural energy terms interpretable in terms of vectors. For elastoplastic analysis, corotated Cauchy stresses and corotated Eulerian strain rates are shown to be the most appropriate pair of measures. Moreover, if the deformation follows the minimum work path, it is proved that corotated Eulerian strain rates correspond to the simultaneous variation of true stretches at a fixed ratio along the three fixed principal material lines. The derived corotated Cauchy stresses and Eulerian strain rates are useful in analyzing elastoplastic, viscoelastic and viscoplastic materials in deformation processing, such as metal forming.