Random marked nested tessellations applied to the modelling of deformation twinning in polycrystalline materials

Abstract: Stochastic geometry provides a powerful framework for modelling complex random structures, with applications in physics, materials science, biology, and other fields. The three-dimensional microstructure of polycrystalline materials is usually modeled by a randomly marked tessellation, where the marks correspond to crystallographic orientations. The purpose of this study is to extend the modelling approach to a finer scale, focusing on the subcells that emerge when a material specimen is exposed to mechanical loading. Specifically, the deformation twinning gives rise to nested tessellation, where the subcells are parallel twin lamellae and their complement is embedded within the original mother cells. The aim of this study is to develop a parametric mathematical model of marked nested tessellation and to realize it using stochastic simulations. We were able to deal with this model using computational tools. The sensitivity of the model to selected key parameters was investigated using statistical methods. As an application, a numerical simulation of the stress and strain fields resulting from deformation twinning is provided, and the contribution of the subcells to the total strain energy density under varying initial conditions was evaluated. This study highlights the dynamic capabilities of stochastic geometry in modelling a phenomenon that changes the microstructure.