Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.Comment: 22 pages, 4 figure

Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of non-periodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed non-periodically. Using the methods of locally periodic two-scale convergence (l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary unfolding operator, we are able to analyze differential equations defined on boundaries of non-periodic microstructures and consider non-homogeneous Neumann conditions on the boundaries of perforations, distributed non-periodically

Interior Regularity Estimates in High Conductivity Homogenization and Application

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance epsilon^{1+tau} (for some tau>0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in L^p as soon as p>2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres direction, and weighted by the high-contrast conductivity. The dependence on powers of epsilon of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures. We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance epsilon^{1+tau} away from the fibres

Effective macroscopic dynamics of stochastic partial differential equations in perforated domains

An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domain without holes. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of \emph{convergence in probability distribution}, and the effectivity of the macroscopic system in the sense of \emph{convergence in energy} are also proved

Bounds on strong field magneto-transport in three-dimensional composites

This paper deals with bounds satisfied by the effective non-symmetric conductivity of three-dimensional composites in the presence of a strong magnetic field. On the one hand, it is shown that for general composites the antisymmetric part of the effective conductivity cannot be bounded solely in terms of the antisymmetric part of the local conductivity, contrary to the columnar case. So, a suitable rank-two laminate the conductivity of which has a bounded antisymmetric part together with a high-contrast symmetric part, may generate an arbitrarily large antisymmetric part of the effective conductivity. On the other hand, bounds are provided which show that the antisymmetric part of the effective conductivity must go to zero if the upper bound on the antisymmetric part of the local conductivity goes to zero, and the symmetric part of the local conductivity remains bounded below and above. Elementary bounds on the effective moduli are derived assuming the local conductivity and effective conductivity have transverse isotropy in the plane orthogonal to the magnetic field. New Hashin-Shtrikman type bounds for two-phase three-dimensional composites with a non-symmetric conductivity are provided under geometric isotropy of the microstructure. The derivation of the bounds is based on a particular variational principle symmetrizing the problem, and the use of Y-tensors involving the averages of the fields in each phase.Comment: 21 page

Brownfield Action: Dissemination of a SENCER Model Curriculum and the Creation of a Collaborative STEM Education Network

Brownfield Action (BA) is a web-based environmental site assessment (ESA) simulation in which students form geotechnical consulting companies and work together to solve problems in environmental forensics. Developed at Barnard College with the Columbia Center for New Media Teaching and Learning, BA has been disseminated to ten colleges, universities, and high schools, resulting in a collaborative network of educators. The experiences of current users are presented describing how they have incorporated the BA curriculum into their courses, as well as how BA affected teaching and learning. The experiences demonstrate that BA can be used in whole or in part, is applicable to a wide range of student capabilities and has been successfully adapted to a variety of learning goals, from introducing non-science-literate students to basic concepts of environmental science and civic issues of environmental contamination to providing advanced training in ESA and modeling groundwater contamination to future environmental professionals

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

Radiation Combined With Thermal Injury Induces Immature Myeloid Cells

The continued development of nuclear weapons and the potential for thermonuclear injury necessitates the further understanding of the immune consequences after radiation combined with injury (RCI). We hypothesized that sub-lethal ionization radiation exposure combined with a full thickness thermal injury would result in the production of immature myeloid cells. Mice underwent either a 20% total body surface area (TBSA) full-thickness contact burn or sham procedure followed by a single whole body dose of 5-Gy radiation. Serum, spleen and peripheral lymph nodes were harvested at 3 and 14 days post-injury. Flow cytometry was performed to identify and characterize adaptive and innate cell compartments. Elevated pro- and anti-inflammatory serum cytokines and profound leukopenia were observed after RCI. A population of cells with dual expression of the cell surface markers Gr-1 and CD11b were identified in all experimental groups, but was significantly elevated after burn alone and RCI at 14 days post-injury. In contrast to the T-cell suppressive nature of myeloid-derived suppressor cells (MDSC) found after trauma and sepsis, myeloid cells after RCI augmented T-cell proliferation and were associated with a weak but significant increase in IFN-γ and a decrease in IL-10. This is consistent with previous work in burn injury indicating that a MDSC-like population increases innate immunity. RCI results in the increase of distinct populations of Gr-1+ CD11b+cells within the secondary lymphoid organs, and we propose these immature inflammatory myeloid cells provide innate immunity to the severely injured and immunocompromised host