Related papers: The {\L}ojasiewicz-Simon inequality related to gra…
A curve shortening equation related to the evolution of grain boundaries is presented. This equation is derived from the grain boundary energy by applying the maximum dissipation principle. Gradient estimates and large time asymptotic…
In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the \L ojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove…
We study the motion of a grain boundary that separates two sets of mutually perpendicular rolls in Rayleigh-B\'enard convection above onset. The problem is treated either analytically from the corresponding amplitude equations, or…
We present a detailed analysis of the universal relationship between grain boundary (GB) free energy and GB self-diffusion coefficient derived by Borisov et al. (1964). This relationship was expressed by a simple equation that was used in…
A continuum grain boundary model is developed that uses experimentally measured grain boundary energy data as a function of misorientation to simulate idealized grain boundary evolution in a 1-D grain array. The model uses a continuum…
We provide sufficient conditions for the Lojasiewicz-Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. Our result provides a tool to study asymptotic properties of…
Recent grain growth experiments have revealed that the same type of grain boundary can have very different mobilities depending on its local microstructure. In this work, we use molecular dynamics simulations to quantify uncertainty in the…
A detailed theoretical and numerical investigation of the infinitesimal single-crystal gradient plasticity and grain-boundary theory of Gurtin (2008) "A theory of grain boundaries that accounts automatically for grain misorientation and…
In this paper, we present a continuum model for the dynamics of low angle grain boundaries in two dimensions based on the motion of constituent dislocations of the grain boundaries. The continuum model consists of an equation for the motion…
Based on the formula for the number density of vacancies in a solid under the stress or tension, the model of grain boundary diffusion in crystalline solids is developed. We obtain the activation energy of grain boundary diffusion…
We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg…
Models of grain boundary energy are essential for predicting the behavior of polycrystalline materials. Typical models represent the minimum boundary energy as a function of macroscopic boundary parameters. An energy model may allow for…
A threshold dynamics model of grain growth that accounts for the anisotropy in the grain boundary energy has been used to simulate experimentally observed grain growth of polycrystalline Ni. The simulation reproduces several aspects of the…
Cohesive laws are stress-strain curves used in finite element calculations to describe the debonding of interfaces such as grain boundaries. It would be convenient to describe grain boundary cohesive laws as a function of the parameters…
We state and prove a stabilisation result for solutions of abstract gradient systems associated with nonsmooth energy functions on infinite dimensional Hilbert spaces. One feature is that in this general setting the assumption on the range…
Grain boundaries affect properties of polycrystalline materials. The influence of a boundary is largely determined by its energy. Grain boundary energy is often portrayed as a function of macroscopic boundary parameters representing grain…
Granular systems present surprisingly complicated dynamics. In particular, nonlinear interactions and energy dissipation play important roles in these dynamics. Usually, constant coefficients of restitution are introduced phenomenologically…
Conventional grain growth is rate-limited by the mobility of grain boundary. To describe similar phenomena limited by the mobility of other grain junctions, we have developed a general theory allowing for size-dependent mobility and its…
The {\L}ojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this…
We define the elastic energy of smooth immersed closed curves in $\mathbb{R}^n$ as the sum of the length and the $L^2$-norm of the curvature, with respect to the length measure. We prove that the $L^2$-gradient flow of this energy smoothly…