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

Independent Candidates in a Parliamentary Election in India: A Poisson Regression Model

The paper attempts to explain the number of independent candidates in Indian parliamentary election in the year 2004. The statistical models developed are applications and generalizations of Poisson and Negative Binomial distributions. Our results suggest that the distribution of independent candidates can be explained well with a negative binomial probability model or its generalizations. Our results also help to identify three major factors behind the variations in the number of independent candidates. First, a major determinant of the number of independent candidates is political fractionalization. Results suggest that the number of non-independent candidates would typically lead to more independent candidates in the fray. Interestingly, our analysis points out that the major determinant appears to be political fractionalization at the State level rather than at the constituency itself. Second, we find some indirect evidence of presence of free riders. Free riders typically stand in urban constituencies and against the so called VIP candidates. Third, our results suggest that SC and ST constituencies would have typically lower number of independent candidates due to lack of potential candidates as compared to general constituencies.Independent Candidates, Election, Poisson, Negative Binomial

Demand and Supply of Currencies of Small Denominations: A Theoretical Framework

The paper presents theoretical framework of demand and supply of currencies of small denominations. In our framework both demand and supply equations emerge out of an optimization framework. Demand functions for small denominations are obtained from a linear expenditure system. Our main contention is that economic agents would like to hold a fixed number of small changes, independent of their respective total cash holdings. However, in our model the fixed quantity is influenced by the probability that in a currency transaction, the counterparty would be able to provide the small change if needed. The supply function is derived from an optimization problem where the central bank balances its operational cost with the probability that an individual would be able to carry out “small” transactions independently, without the help of counterparty. In this demand-supply framework, the probability that a randomly chosen individual in an economy would hold certain currency combinations is interpreted as “price”. We attempt to show that in a dynamic environment, such interaction could be understood by specifying a cob-web type model where expectations are formed based on previous period’s experience. As an operational rule, it is proposed that the central bank should increase the supply of small denominations at a rate marginally above the growth rate of economically active population and stop minting as soon as some of the small denominations start return in the currency chest. We also suggest how demand for “small change” could be estimated from the “lifetime” of the “smallest” denomination.Small Change, Denomination, Currency Management, Poisson Distribution

A micromechanics-inspired constitutive model for shape-memory alloys

This paper presents a three-dimensional constitutive model for shape-memory alloys that generalizes the one-dimensional model presented earlier (Sadjadpour and Bhattacharya 2007 Smart Mater. Struct. 16 S51–62). These models build on recent micromechanical studies of the underlying microstructure of shape-memory alloys, and a key idea is that of an effective transformation strain of the martensitic microstructure. This paper explains the thermodynamic setting of the model, demonstrates it through examples involving proportional and non-proportional loading, and shows that the model can be fitted to incorporate the effect of texture in polycrystalline shape-memory alloys

Influence of thermomechanical loads on the energetics of precipitation in magnesium aluminum alloys

We use first principles calculations to study the influence of thermomechanical loads on the energetics of precipitation in magnesium-aluminum alloys. Using Density Functional Theory simulations, we present expressions of the energy of magnesium-aluminum binary solid solutions as a function of concentration, strain and temperature. Additionally, from these calculations, we observe an increase in equilibrium volume (and hence the equilibrium lattice constants) with the increase in concentration of magnesium. We also observe an increase in the cohesive energy of solutions with increase in concentration, and also present their dependence on strain. Calculations also show that the formation enthalpy of β phase solutions to be strongly influenced by hydrostatic stress, however the formation enthalpy of α phase solutions remain unaffected by hydrostatic stress. We present an expression of the free energy of any magnesium aluminum solid solution, that takes into account the contributions of strain and temperature. We note that these expressions can serve as input to process models of magnesium-aluminum alloys. We use these expressions to report the influence of strains and temperature on the solubility limits and equilibrium chemical potential in Mg-Al alloys. Finally, we report the influence of thermomechanical loads on the growth of precipitates, where we observe compressive strains along the c axis promotes growth of precipiates with a (0001)_α habit plane, whereas strains along the a and b directions do not influence the growth of precipitates

Stress-Induced Phase Transformations in Shape-Memory Polycrystals

Shape-memory alloys undergo a solid-to-solid phase transformation involving a change of crystal structure. We examine model problems in the scalar setting motivated by the situation when this transformation is induced by the application of stress in a polycrystalline material made of numerous grains of the same crystalline solid with varying orientations. We show that the onset of transformation in a granular polycrystal with homogeneous elasticity is in fact predicted accurately by the so-called Sachs bound based on the ansatz of uniform stress. We also present a simple example where the onset of phase transformation is given by the Sachs bound, and the extent of phase transformation is given by the constant strain Taylor bound. Finally we discuss the stress–strain relations of the general problem using Milton–Serkov bounds

A Mesoscopic Electromechanical Theory of Ferroelectric Films and Ceramics

We present a multi-scale modelling framework to predict the effective electromechanical behavior of ferroelectric ceramics and thin films. This paper specifically focuses on the mesoscopic scale and models the effects of domains and domain switching taking into account intergranular constraints. Starting from the properties of the single crystal and the pre-poling granular texture, the theory predicts the domain patterns, the post-poling texture, the saturation polarization, saturation strain and the electromechanical moduli. We demonstrate remarkable agreement with experimental data. The theory also explains the superior electromechanical property of PZT at the morphotropic phase boundary. The paper concludes with the application of the theory to predict the optimal texture for enhanced electromechanical coupling factors and high-strain actuation in selected materials