Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces. We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves. We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion

Active tuning of photonic device characteristics during operation by ferroelectric domain switching

Ferroelectrics have many unusual properties. Two properties that are often exploited are first, their complex, nonlinear optical response and second, their strong nonlinear coupling between electromagnetic and mechanical fields through the domain patterns or microstructure. The former has led to the use of ferroelectrics in optical devices and the latter is used in ferroelectric sensors and actuators. We show the feasibility of using these properties together in nanoscale photonic devices. The electromechanical coupling allows us to change the domain patterns or microstructure. This in turn changes the optical characteristics. Together, these could provide photonic devices with tunable properties. We present calculations for two model devices. First, in a photonic crystal consisting of a triangular lattice of air holes in barium titanate, we find the change in the band structure when the domains are switched. The change is significant compared to the frequency spread of currently available high-quality light sources and may provide a strategy for optical switching. Second, we show that periodically poled 90° domain patterns, despite their complex geometry, do not cause dispersion or have band gaps. Hence, they may provide an alternative to the antiparallel domains that are usually used in quasiphase matching and allow for tunable higher-harmonic generation

Existence of radial solution for a quasilinear equation with singular nonlinearity

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $\lambda>0$ sufficiently small, $0<\delta<1$ and $p-1<q<p^{*}-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.Comment: 16 page