Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.     

Conditions for Equality of Hulls in the Calculus of Variations

We simplify and sharpen several results by K. Zhang concerning properties of quasiconvex hulls of sets and quasiconvex envelopes of their distance functions. The approach emphasizes the underlying geometry and in particular we show that K ^pc=K ^c implies K ^rc=K ^c if and only if min{m,n}≤ 2 thus answering a question raised in [Z2]. (Accepted January 24, 2000)?Published online August 21, 2000     

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations,analysis and computation of the stabilization parameter

In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection‐diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear Coherence in Multivariate Research: Invariants and the Reconstruction of Attractors

First, some linear techniques in multivariate time-series analysis in EEG research are reviewed to highlight the problem of estimating the dimensionality of the state space (embedding dimension), the reconstruction of an attractor, and the evaluation of invariant properties of the attractor. The traditional linear techniques included the usual spectral and cospectral measures of power, phase, and coherence to which stepwise discriminant analysis was applied for canonical representation of the attractor. Then, some traditional nonlinear techniques of attractor reconstruction and dimensional analysis which use the time-lagged univariate approach of Ruelle and Takens (Takens, 1981) are reviewed. Next, updates and multivariate generalizations that use singular-value decomposition (Broomhead & King, 1986) are reviewed. Finally, Stewart's (1995, 1996) multivariate generalization of the method of false nearest neighbors (Abarbanel, Brown, Sidorowich, & Tsimring, 1993; Kennel, Brown, & Abarbanel, 1992) is reviewed. These are particularly relevant for evaluating multivariate coherence in research on the complex cooperative dynamical systems found in neuroscience, psychology, and social science when time series of sufficient length are investigated.     

Asymptotic Behavior of the Principal Eigenvalue for Cooperative Elliptic Systems and Applications

The asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.     

Attractors for Gradient Flows of Nonconvex Functionals and Applications

This paper addresses the long-time behaviour of gradient flows of nonconvex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of nonconvex evolution problems. In particular, we discuss the long-time behaviour of solutions of quasistationary phase field models and prove the existence of a global attractor.     

Fracture and Singularities of the Mass-Density Gradient Field

A continuum mechanical theory of fracture without singular fields is proposed. The primary contribution is the rationalization of the structure of a ‘law of motion’ for crack-tips, essentially as a kinematical consequence and involving topological characteristics. Questions of compatibility arising from the kinematics of the model are explored. The thermodynamic driving force for crack-tip motion in solids of arbitrary constitution is a natural consequence of the model. The governing equations represent a new class of pattern-forming equations.     

Principal Trajectories of Forced Vibrations for Discrete and Continuous Systems

Principal trajectories of forced vibration of linear and nonlinear continuous systems are introduced as such motions in which the system is equivalent to a Newtonian particle in the function space of the system configurations. The corresponding 'effective mass' of the particle gives physical characteristics of the system response, so that zero effective mass is associated with resonance. The methodology can be viewed as a complementary tool to the method of normal modes, when considering the class of forced vibrating systems, since the related basis accounts for the system physical properties as well as the external forcing factor. In particular, it is shown that a two degrees of freedom system can possess an infinite discrete set of in-phase and out-of-phase forced vibrations of the normal modes type. The corresponding forcing vector-functions obey the second Newton law due to the definition of principal trajectories.     

Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.     

判定广义Birkhoff系统稳定性的三重组合梯度方法

研究判定广义Birkhoff系统稳定性的三重组合梯度方法．首先,给出4类三重组合梯度系统的定义和微分方程;其次,得到广义Birkhoff系统成为三重组合梯度系统的条件,从而将广义Birkhoff系统化成三重组合梯度系统;最后,利用组合梯度系统的性质来研究系统的稳定性,举例说明结果的应用．     

判定非自治Birkhoff系统稳定性的广义组合梯度方法

研究判定非自治Birkhoff系统稳定性的广义组合梯度方法．首先,给出非自治Birkhoff系统和非自治广义Birkhoff系统的运动微分方程;其次,给出一类将广义梯度系统和广义斜梯度系统组合而成的广义组合梯度系统,并讨论广义组合梯度系统的一些性质;最后,将非自治Birkhoff系统和非自治广义Birkhoff系统在一定条件下表示成广义组合梯度系统,并用广义组合梯度系统的性质研究了这两类Birkhoff系统的稳定性．举例说明结果的应用．     

Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications

The current paper is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. It is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. Next we obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems. Finally we discuss the applications of the established principal eigenvalue theory to time periodic Fisher or KPP type equations with nonlocal dispersal and prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution which is globally asymptotically stable.     

梯度纳米结构金属力学性能、变形机理和多尺度计算研究进展

近年来,梯度纳米结构金属因其优越的力学性能和独特的塑性变形机理受到广泛关注,已成为材料与力学学科的热点和前沿.论文首先介绍梯度纳米结构金属的强度、塑性、加工硬化和抗疲劳等核心力学性能,以及晶粒长大、塑性应变梯度和几何必需位错等塑性变形机理及其力学研究.其次介绍梯度纳米结构金属的多尺度计算与模拟研究.最后讨论梯度纳米结构金属研究领域存在的挑战.     

混合光载波法分离动静态主应力

本文将等差载波位相调制原理与夹层全息术结合形成的混合光载波法,可同时获得等差载波条纹图与混合载波条纹图.对混合载波纹图进行光学付里叶变换,利用空间滤波技术可将等和载波条纹从中分离出来.利用图象处理技术实现了对两种载波条纹图从预处理到应力计算的自动化处理.将混合光载波法应用于动态全息光弹性,分离了动态主应力. 文中还提出了一个经济实用的大尺寸载波片制作技术.     

光弹塑性等倾线角度的判读一法

本文提出并证明了一套测定光弹塑性模型中任一点的等倾线角度的方法     

Remark on Single Exponential Bound of the Vorticity Gradient for the Two-Dimensional Euler Flow Around a Corner

In this paper, we consider the two–dimensional Euler flow under a simple symmetry condition, with hyperbolic structure in a unit square \({D = \{(x_1,x_2):0 < x_1+x_2 < \sqrt{2},0 < -x_1+x_2 < \sqrt{2}\}}\). It is shown that the Lipschitz estimate of the vorticity on the boundary is at most a single exponential growth near the stagnation point.     

ZrN单层、多层、梯度层及复合处理层对不锈钢固体粒子冲蚀行为的影响

对比研究了离子镀ZrN单层、多层、梯度层及其与离子渗氮复合处理对2Cr13马氏体不锈钢抗大小攻角固体粒子冲蚀性能的影响规律,通过硬度、韧性、结合强度、动静态承载能力等特性的综合测试与分析,探讨了表面处理层对不锈钢固体粒子冲蚀行为的作用机制.结果表明,上述4种表面改性层均显著提高了2Cr13不锈钢表面的硬度和抗微切削能力,因而有效改善了2Cr13钢抗30&#176;小攻角固体粒子冲蚀性能.经离子渗氮层上沉积ZrN梯度层的复合处理后,其硬度由表面至基材呈梯度分布,界面应力分布连续性好,承载能力和抗塑性变形能力高,多冲疲劳性能优,能够显著提高基材抗90&#176;大攻角固体粒子的冲蚀能力.然而,ZrN单层、多层、梯度层的抗多冲疲劳性能较差,其抗大攻角固体粒子的冲蚀能力反而不及2Cr13基材.