Multiscale Models of Growth: Microstructure Evolution & Mechanics of Surface Growth

The process of growth, which involves the change in shape and structure of a body, holds significant importance across various physical phenomena such as solidification, biological tissue growth, environmental processes, additive manufacturing, and annealing heat treatment. The modeling approach for studying growth is highly influenced by the underlying physics involved, the scale of growth, and whether it occurs throughout the volume or concentrated at boundaries and interfaces. Within the scope of this thesis, our focus is specifically on two aspects of growth: grain growth in polycrystalline materials, as well as the mechanics and thermomechanics of surface growth. 

Grain growth in polycrystalline materials. The evolution of the microstructure in polycrystalline materials, such as metals and rocks, occurs through the movement of grain boundaries, which is facilitated by heat treatment processes like annealing in order to enhance material properties. Despite the experimental evidence highlighting the significance of grain boundary energy anisotropy in grain boundary motion, many existing microstructure simulations overlook this dependency and assume isotropic uniform grain boundary energy. In this thesis, we employ a threshold dynamics algorithm and a spherically non-symmetric anisotropic convolution kernel to investigate the evolution of microstructures in nickel. The associated convolution kernel is computed using a five-parameter energy function evaluated from molecular dynamics simulations. The simulation results are then compared with experimental data obtained from near-field high-energy X-ray diffraction microscopy of a nickel (Ni) polycrystal sample consisting of approximately 2700 grains evolved through five different annealing stages. Our findings demonstrate that the anisotropic simulation accurately predicts the increase in the area fraction of low energy boundaries and the decrease in the area fraction of high energy density boundaries, consistent with the observed trends in the experimental data. Conversely, the isotropic simulation predicts the opposite evolution.

Mechanics of surface growth. To understand failure and instabilities in various processes involving surface growth, such as biological tissue growth, melting, solidification, and additive manufacturing, realistic models are required to study the interaction between mass addition and stress. Modeling the mechanical behavior of surface growth, where mass is added at the body’s boundary, poses challenges for standard Lagrangian formulations designed for a constant set of material particles. In this thesis, an Eulerian approach is proposed to overcome the need for constructing the reference configuration. However, this introduces the challenge of determining the solid’s stress response, which typically relies on the deformation gradient not readily available in the Eulerian framework. To resolve this, additional kinematic descriptors, namely the relaxed zero-stress deformation and the elastic deformation, are introduced, which do not need to satisfy kinematic compatibility. The proposed method is applied to simplified examples, showcasing non-normal growth, Actin network assembly and disassembly, and additive manufacturing, with closed-form solutions obtained in the Eulerian framework. 

The model is coupled with the phase field method to develop an Eulerian numerical approach that solves the equations in a fixed computational domain larger than the growing body, with a fixed discretization. This coupling allows for a straightforward extension of the mechanical formulation to incorporate other physics. Utilizing a free energy-based approach offers a unified and thermodynamically-consistent framework to investigate the thermomechanics of a growing solid. This approach combines the developed mechanical model with a thermal model. The unified formulation is implemented in the open-source finite element framework Firedrake, enabling the investigation of the role of thermomechanical stress in the pressure melting of glaciers.