Geometrical Interpretation of Commutative-Symmetrical Stretch Tensor Products and the Decoupling of Material (An)isotropy From Deformation-Induced Anisotropy

Based on stretch tensor projections, the Lagrangean and Eulerian total, plastic/inelastic and elastic deformation ellipsoids, associated with Lagrangean or Eulerian commutative-symmetrical stretch tensor products [1, 2], can be interpreted geometrically as material deformations. The six internal degrees of freedom, reflected in three stretch eigenvalues and three principal directions of the symmetric stretch tensors, correspond to three semi-axes and three principal directions of the total, plastic/inelastic, elastic deformation ellipsoids. In the case of a plastic deformation ellipsoid, its six internal degrees of freedom can be controlled completely by a symmetric plastic-flow rule. And the Lagrangean and Eulerian total, plastic/inelastic, elastic deformation ellipsoids are related to each other by the Lagrangean and Eulerian commutative-symmetrical stretch tensor products, respectively. Since the multiplicative logarithmic strain space formulation [2] and the commutative-symmetrical stretch tensor products [1] represent both an Eulerian description and a Lagrangean description at the same time, their Eulerian characteristic may be employed to properly model the (overall) elastic isotropy and their Lagrangean characteristic may be utilized to decouple the material (an)isotropy from the deformation-induced anisotropy. This aspect is further discussed in particular with respect to material orthotropy. A new elasto-plastic/-inelastic interpretation of the continuum mechanics framework of the multiplicative logarithmic strain space formulation is presented in detail. This leads to new tensor definitions of the elastic and plastic/inelastic logarithmic Hoger stresses in addition to the well-known total logarithmic Hoger stress.

References

1. K. Heiduschke, A commutative-symmetrical multiplicative decomposition of left and right stretch tensors, International Journal of Solids and Structures, 144–145:59–65, 2018, https://doi.org/10.1016/j.ijsolstr.2018.04.013.
2. K. Heiduschke, On the multiplicative logarithmic strain space formulation, Technische Mechanik (ICMM5 Edition), 38:22–40, 2018, https://doi.org/10.24352/UB.OVGU-2018-004.