Microcontinuum field theories, including Micromorphic theory, Microstructure theory, Micropolar theory, Cosserat theory, nonlocal theory and couple stress theory, are the extensions of the classical field theories for the applications in microscopic space and time scales. They have been expected to overlap atomic model at microscale and encompass classical continuum mechanics at macroscale. Thi...

The double bracket dissipation approach is applied to the Vlasov kinetic equation. The Vlasov equation is then transformed by a Poisson map to moment dynamics, leading to a nonlocal form of Darcy’s law. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the double bracket dissipation form. The moment dynamics for these double bracket kinetic equ...

Energy-decreasing continuum flows of geometric order parameters are considered. The dynamics of pattern formation for an arbitrary geometric quantity is derived using a generalization of Darcy’s law based on the geometric nature of the order parameter. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanic...

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct can be made for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including...

Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel f(xc) that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting f(xc) accounts for both self-energy corrections and the electron-hole interaction. It is static, nonlocal, and has a long-range Coulomb tail. Taking the example of bulk silicon, we sh...

Abstract. Usually smeared crack techniques are based on the following features: the fracture is represented by means of a band of finite elements and by a softening constitutive law of damage type. Often these methods are implemented with nonlocal operators which control the localization effects and reduce the mesh bias. We consider a non-local smeared crack energy defined for a finite element ...

We propose that sharp stability estimates are essential for evaluating the predictive capability of atomistic-to-continuum coupling methods up to the limit load for atomistic instabilities such as fracture, dislocation movement, or crack tip propagation. Using rigorous analysis, asymptotic methods, and numerical experiments, we obtain such sharp stability estimates for the basic conservative qu...

We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of sp...

We establish the existence and uniqueness of a solution to master equation for mean field game controls with absorption. The arises as continuum limit dynamic exhaustible resources modeling Cournot competition between producers. proof relies on an analysis forward-backward system nonlocal Hamilton-Jacobi/Fokker-Planck equations Dirichlet boundary conditions. In particular, we new priori estimat...

Center for Concrete and Geomaterials, Northwestern University, Tech-241 0, Evanston, I L 60208 A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal, In contrast to the original imbricate no...