Cohesive Dynamics and Brittle Fracture

We formulate a nonlocal cohesive model for calculating the deformation state inside a cracking body. In this model a more complete set of physical properties including elastic and softening behavior are assigned to each point in the medium. We work within the small deformation setting and use the peridynamic formulation. Here strains are calculated as difference quotients. The constitutive relation is given by a nonlocal cohesive law relating force to strain. At each instant of the evolution we identify a process zone where strains lie above a threshold value. Perturbation analysis shows that jump discontinuities within the process zone can become unstable and grow. We derive an explicit inequality that shows that the size of the process zone is controlled by the ratio given by the length scale of nonlocal interaction divided by the characteristic dimension of the sample. The process zone is shown to concentrate on a set of zero volume in the limit where the length scale of nonlocal interaction vanishes with respect to the size of the domain. In this limit the dynamic evolution is seen to have bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement and the fracture set across which the displacement is discontinuous. We conclude illustrating how the approach developed here can be applied to limits of dynamics associated with other energies that $\Gamma$- converge to the Griffith fracture energy.Comment: 38 pages, 4 figures, typographical errors corrected, removed section 7 of previous version and added section 8 to the current version, changed title to Cohesive Dynamics and Brittle Fracture. arXiv admin note: text overlap with arXiv:1305.453

The Quantum Vlasov Equation and its Markov Limit

The adiabatic particle number in mean field theory obeys a quantum Vlasov equation which is nonlocal in time. For weak, slowly varying electric fields this particle number can be identified with the single particle distribution function in phase space, and its time rate of change is the appropriate effective source term for the Boltzmann-Vlasov equation. By analyzing the evolution of the particle number we exhibit the time structure of the particle creation process in a constant electric field, and derive the local form of the source term due to pair creation. In order to capture the secular Schwinger creation rate, the source term requires an asymptotic expansion which is uniform in time, and whose longitudinal momentum dependence can be approximated by a delta function only on long time scales. The local Vlasov source term amounts to a kind of Markov limit of field theory, where information about quantum phase correlations in the created pairs is ignored and a reversible Hamiltonian evolution is replaced by an irreversible kinetic one. This replacement has a precise counterpart in the density matrix description, where it corresponds to disregarding the rapidly varying off-diagonal terms in the adiabatic number basis and treating the more slowly varying diagonal elements as the probabilities of creating pairs in a stochastic process. A numerical comparison between the quantum and local kinetic approaches to the dynamical backreaction problem shows remarkably good agreement, even in quite strong electric fields, over a large range of times.Comment: 49 pages, RevTex/LaTeX2e, 8 .eps figures included in 404KB .gz file (~3MB total uncompressed). Replacement added \tightenpages command to reduce from 67 to 49 p

Convergence of nonlocal geometric flows to anisotropic mean curvature motion

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.Comment: 19 page

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamics model away from the fracture set. The nonlocal model treated here is characterized by a double well potential and is a smooth version of the peridynamic model introduced in n Silling (J Mech Phys Solids 48(1), 2000). The solutions of nonlinear peridynamics are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $\epsilon$ of non local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $\epsilon$. More generally for local Lagrange interpolation of order $p\geq 1$ the consistency error is of order $h^p/\epsilon$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics

Density-operator evolution: Complete positivity and the Keldysh real-time expansion

We study the reduced time-evolution of open quantum systems by combining quantum-information and statistical field theory. Inspired by prior work [EPL 102, 60001 (2013) and Phys. Rev. Lett. 111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling. This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage defines the dissipative timescales of the system --before having integrated out the environment completely-- the second stage sums up elementary physical processes described by CP superoperators. This allows us to establish the nontrivial relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit. Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by 'cutting' a group of Keldysh real-time diagrams 'in half'. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schr\"odinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion.Comment: Submission to SciPost Physics, 49 pages including 6 appendices, 13 figures. Significant improvement of introduction and conclusion, added discussions, fixed typos, no results change

On the evolution by fractional mean curvature

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.Comment: minor correction