Kinematic hardening and size effects in elastoplastic nonlinear Timoshenko beams

Architected materials, like lattice structures composed of beams, have attracted increasing interest due to their unique properties. For instance, auxetic materials, which exhibit a negative Poisson’s ratio, offer potential advantages for impact protection, including increased indentation resistance, fracture toughness, and energy adsorption. These properties appear promising in the search for lighter materials for impact protection. In literature, numerous architectures have been proposed to achieve auxetic properties, however with limited insight into nonlinear effects. Efficient numerical models are required to capture geometric and material nonlinearities and to explore the behavior of different architectures.
Nonlinear Timoshenko beams can be used to model lattice materials. These beams commonly account only for linear material behavior.
Recently Herrnböck et al. [1,2] have developed a framework to determine the yield surface and hardening tensor in the full sixdimensional cross-sectional force and moment space. This extension to include plasticity in the modelling of beams allows the efficient simulation of elastoplastic lattice structures under large deformation in impact scenarios. In their framework, they describe the scaling of the yield surface in relation to the macroscopic geometric size and the scaling of the hardening tensor in relation to the microscopic hardening properties. For the study of different lattice architectures under finite deformation, the scaling of the hardening tensor with respect to the macroscopic geometric size is of further interest and has not been discussed so far.
The objective of this research is to examine macroscopic geometrical scaling effects of the hardening tensor in the full six-dimensional cross-sectional force and moment space. A numerical framework is established for conducting elastic analysis of single non-linear Timoshenko beams and multi-beam structures. The framework is then extended to include the yield surface of Herrnböck et al. [1] for ideal plasticity. Careful consideration is given to meshing of the beams and to load-stepping in relation to the explicit return mapping scheme. Kinematic hardening, as described by Herrnböck et al. [2], is subsequently added to the analyses. The effects of geometric scale on the hardening tensor are explained and a method to adapt the hardening tensor in order to account for scale effects is presented. The investigations are conducted on both single cantilever beams and lattice architectures for auxetic metamaterials.
[1] Herrnböck et al., Comput Mech. 67 (2021), pp. 723–742.
[2] Herrnböck et al., Comput Mech. 71 (2022), pp. 1–24.