Finite-strain elastic-plastic torsion: analytical and FEM modeling for nonmonotonically hardening polymers

Polymeric materials, depending on the structure and chemical composition, exhibit various types of isotropic strain hardening. In particular, in the range of plastic deformation on the "true strain - true stress" curve, there may be a descending section of softening caused by the weakening of intermolecular bonds. This softening region is further replaced by a power-law hardening. The laws of deformation of materials can be established from simple experiments, one of which is often torsion. For torsion of thin-walled cylindrical specimens, the stress-strain state is practically uniform; therefore, such experiments are easy to interpret. However, stability problems arise at large deformations of thin-walled specimens. For solid cylindrical specimens, the stress state is inhomogeneous; interpretation of such experiments is possible on the basis of FEM modeling or using exact or approximate analytical solutions of the corresponding initial-boundary value problems of mechanics. In the present study, an exact analytical solution of the elastic-plastic problem of torsion of a cylindrical sample is presented, which is valid for an arbitrary law of isotropic hardening. The multiplicative decomposition is utilized as the kinematics of elastic-plastic deformation. The non-linear elastic properties of the material are described by the Mooney - Rivlin model. In the plasticity condition, the Tresca equivalent stress is used, which makes it possible to obtain a closed solution. The integral characteristics of the process (torque and axial force that represents a second order effect) are calculated. The analytical results are compared with the results of numerical simulations in MSC.Marc, as well as with the available experimental data. The analytical solution for torque corresponds very closely to the numerical solution obtained by the finite element method. Also, the curves of axial force coincide satisfactorily. At moderate strains, the analytical solution accurately describes the experimental results.