**5. Conclusions**

We have analyzed the most popular relations of crystal plasticity models in the context of geometric nonlinearity description and fulfillment of thermodynamic constraints in case of elastic deformation. These models are focused on the description of significant inelastic deformations and material structure evolution caused by these deformations. That is why the formulations in the rate form written in terms of the current configuration seem preferable to constructing advanced constitutive models able to take into account different deformation mechanisms and their complex interactions. The main advantages of these formulations over the formulation in the finite form are as follows:


The use of a corotational derivative in such a formulation means that the model is based on the decomposition of motion into the deformation motion and the rigid motion of a moving coordinate system, the rate of stress change relative to which is associated with the strain rate.

We have obtained the relations of the mesolevel model with an explicit separation of the motion of the moving coordinate system and elastic distortion of crystallites with respect to this system in the deformation gradient. The MCS is related to the elements of the material symmetry of the anisotropic crystallite. The proposed formalism with an explicit consideration of the MCS allows us to reasonably pass from the formulation in terms of the unloaded lattice configuration in the finite form (where the thermodynamic constraints for purely elastic deformation are strictly fulfilled) to the close formulation in rate form written in terms of the current configuration.

We have compared these relations with a popular crystal plasticity formulation, which makes it possible to establish their proximity to one another. The results of the performed analytical calculations show the equivalence or similarity (in the sense of the response determined under the same loads) of the formulations under consideration. These conclusions were supported by the results of the numerical calculations. It should be noted that the proposed approach aimed to determine the spin of the MCS also could include other physically justified rotation models, e.g., those taking into account the interaction of defects of neighboring crystallites or the contribution from the mechanism of grain boundary sliding. Such models cannot be introduced into other known approaches to constructing constitutive relations via the use of kinematic variables without explicit consideration of the material symmetric properties.

**Author Contributions:** Conceptualization, P.T. and A.S.; methodology, P.T. and A.S.; software, A.S.; validation, A.S., P.T., and N.K.; formal analysis, A.S. and N.K.; investigation, A.S., P.T., and N.K.; data curation, A.S. and N.K.; writing—original draft preparation, A.S.; writing—review and editing, P.T., A.S., and N.K.; visualization, A.S. and N.K.; supervision, P.T.; funding acquisition, N.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The study was carried out with financial support from the Ministry of Education and Science of the Russian Federation as part of the implementation of the national project "Science and Universities" (the state task fulfillment in the Laboratory of Multilevel Structural and Functional Materials Modeling).

**Data Availability Statement:** Not available.

**Conflicts of Interest:** The authors declare no conflict of interest.
