Renormalized Energy and Peach-K\"ohler Forces for Screw Dislocations with Antiplane Shear

We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by Cermelli and Gurtin, methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-K\"ohler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in a forthcoming work.Comment: 22 page

On the Stability of Stochastic Parametrically Forced Equations with Rank One Forcing

We derive simplified formulas for analyzing the stability of stochastic parametrically forced linear systems. This extends the results in [T. Blass and L.A. Romero, SIAM J. Control Optim. 51(2):1099--1127, 2013] where, assuming the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this leads to an enormous simplification.Comment: 16 page

Dynamics for Systems of Screw Dislocations

The goal of this paper is the analytical validation of a model of Cermelli and Gurtin for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a "maximal dissipation criterion", leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved.Comment: 35 pages, 5 figure

Sediment accumulation rates in subarctic lakes: Insights into age-depth modeling from 22 dated lake records from the Northwest Territories, Canada

Age-depth modeling using Bayesian statistics requires well-informed prior information about the behavior of sediment accumulation. Here we present average sediment accumulation rates (represented as deposition times, DT, in yr/cm) for lakes in an Arctic setting, and we examine the variability across space (intra- and inter-lake) and time (late Holocene). The dataset includes over 100 radiocarbon dates, primarily on bulk sediment, from 22 sediment cores obtained from 18 lakes spanning the boreal to tundra ecotone gradients in subarctic Canada. There are four to twenty-five radiocarbon dates per core, depending on the length and character of the sediment records. Deposition times were calculated at 100-year intervals from age-depth models constructed using the 'classical' age-depth modeling software Clam. Lakes in boreal settings have the most rapid accumulation (mean DT 20±10 yr/cm), whereas lakes in tundra settings accumulate at moderate (mean DT 70±10 yr/cm) to very slow rates, (>100yr/cm). Many of the age-depth models demonstrate fluctuations in accumulation that coincide with lake evolution and post-glacial climate change. Ten of our sediment cores yielded sediments as old as c. 9000cal BP (BP=years before AD 1950). From between c. 9000cal BP and c. 6000cal BP, sediment accumulation was relatively rapid (DT of 20-60yr/cm). Accumulation slowed between c. 5500 and c. 4000cal BP as vegetation expanded northward in response to warming. A short period of rapid accumulation occurred near 1200cal BP at three lakes. Our research will help inform priors in Bayesian age modeling

On the Aubry-Mather theory for partial differential equations and the stability of stochastically forced ordinary differential equations

textThis dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three separate papers. In Chapter 2 we consider gradient descent equations for energy functionals of the type [mathematical equation] where A is a second-order uniformly elliptic operator with smooth coefficients. We consider the gradient descent equation for S, where the gradient is an element of the Sobolev space H[superscipt beta], [beta is an element of](0, 1), with a metric that depends on A and a positive number [gamma] > sup |V₂₂|. The main result of Chapter 2 is a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator, and we provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional. In Chapter 3 we investigate the differentiability of the minimal average energy associated to the functionals [mathematical equation] using numerical and perturbation methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the minimal average energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter [epsilon], and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series. In Chapter 4 we present a method for determining the stability of a class of stochastically forced ordinary differential equations, where the forcing term can be obtained by passing white noise through a filter of arbitrarily high degree. We use the Fokker-Planck equation to write a partial differential equation for the second moments, which we turn into an eigenvalue problem for a second-order differential operator. We develop ladder operators to determine analytic expressions for the eigenvalues and eigenfunctions of this differential operator, and thus determine the stability.Mathematic

Renormalized energy and Peach-Köhler forces for screw dislocations with antiplane shear

We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by Cermelli and Gurtin [6], methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-K¨ohler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in [4]