Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives

The positivity of quadratic integrals involving variable coefficients and derivatives of any order is studied. The result is determined by the solution of an initial value problem for a system of first order nonlinear differential equations. The system is identified as the matrix Riccati differential equation in control theory. A complete conclusion is reached by considering the cases when the solution is bounded and when the solution is unbounded.     

Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

In 1985 Franz Rothe [J. Reine Angew Math. 355 (1985) 129–138] found, by means of the thermodynamical equilibrium theory, an asymptotic estimate of the period of solutions of ordinary differential equations originated by predator–prey Volterra–Lotka model. We extend some of the Rothe's ideas to more general systems:

迭代估计法单幅条纹图的相位解调

基于相位曲面的光滑连续性,提出一种新的自适应相位解调方法,该方法通过迭代估计,能有效从干涉条纹图获得较准确的条纹相位分布,文中通过实例说明了其处理过程。     

Cosserat Operators of Higher Order

A higher order version of Cosserat Operators is introduced. With a compactness result (the proof of which we can only sketch here, for details see Riedl in Cosserat operators of higher order and applications, PhD thesis, University of Bayreuth, 2010) based on a regularization property of these operators we gain insight to invertibility of the operator div : ${\underline{H}^{m,q}_0 (G) \rightarrow H^{m-1,q}_{0,0} (G)}$ , where ${m \in \mathbb{N}, 1 < q < \infty}$ and ${G \subset \mathbb{R}^n}$ is a bounded domain with sufficiently smooth boundary. As an application, we get a very simple and effective method of treating higher order generalizations of Stokes’ system.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Algebra of Transversely Isotropic Sixth Order Tensors and Solution to Higher Order Inhomogeneity Problems

In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.     

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.      

Complex Dynamics in a ODE Model Related to Phase Transition

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation

$$\begin{aligned} -\ddot{x} + \left( 1 + \varepsilon ^{-1} A(t)\right) G'(x) = 0, \end{aligned}$$

where A(t) is a nonnegative T-periodic function and \(\varepsilon > 0\) is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima \(x_0\) and \(x_1\) of G(x). Such solutions stay close to \(x_0\) or \(x_1\) in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case \(x_0 =0\) and \(x_1 = 1\).

     

单晶硅滑动磨损性能及其相变研究

考察了单晶硅在室温和低接触应力条件下的摩擦磨损行为随速度的变化情况．结果表明，单晶硅的摩擦系数和磨损率随滑动速度的提高而呈现降低趋势；单品硅在低速、短滑动时间下的磨损表面形貌特征以微断裂为主，并伴有一定程度的塑性变形;随着滑动时间的延长，塑性变形特征逐渐减弱，塑性变形同具有金属延性特征的p—Sn(简称si—II)相密切相关，Si—II在滑动过程中可转变为体心立方结构(简称Si-III)、斜方六面体结构(简称Si—XII)和非晶硅相;在高速条件下，单晶硅磨损表面呈现微断裂和较弱的塑性变形特征；尽管通过Raman分析证实磨损表面存在si—III相，但其对磨损机制的影响有待于进一步研究．     

Analytical Solutions for Free Vibration and Buckling of Composite Beams Using a Higher Order Beam Theory

To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton's principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy's. A parametric study is also performed to investigate the effects of geometry and material parameters.     

分数阶粘弹性地基上薄板波动和振动特性分析

本文研究了粘弹性地基上薄板的波动和振动问题.主要讨论了基于分数导数理论的粘弹性地基模型上薄板弯曲波的传播特性以及固有频率对地基的依赖特性.推导了三种经典粘弹性地基模型的复模量.并利用分数导数的性质得到分数阶粘弹性地基上Kirchhoff板中弯曲波的传播速度、衰减系数以及自由振动的复固有频率.数值算例表明粘弹性地基对弯曲波传播特性存在显著影响,不同粘弹性模型所对应的色散和衰减特性也存在较大差别.分数阶导数可以实现相邻整数阶导数之间的光滑过渡.利用分数导数的本构关系可以更加真实地描述粘弹性地基的历史依赖行为,更准确地表现出粘弹性地基板中弯曲波的色散和衰减特性.     

润滑介质对单晶硅相变与脆塑行为的影响

研究了分别在十五烷、无水乙醇和蒸馏水润滑下单晶硅的摩擦磨损行为及其相变和脆塑行为.结果表明:在十五烷润滑下单晶硅的摩擦系数和磨损体积损失最低,而在蒸馏水润滑下的摩擦系数和磨损体积损失最大;单晶硅在非极性溶剂十五烷润滑下发生明显的Si-I→a-Si相变,磨损表面光滑并呈现明显的金属塑性特征;单晶硅在无水乙醇润滑下发生轻度Si-I→a-Si相变,磨损表面特征为微弱的塑性变形和微断裂共存;在蒸馏水润滑下,单晶硅发生轻度的Si-I→Si-III相变,磨损表面变得粗糙并伴有大量微断裂;润滑介质的极性是影响单晶硅磨损表面相变和脆塑行为的主要因素之一.     

On Estimation of Temperature Uncertainty Using the Second Order Adjoint Problem

The uncertainty of temperature prediction from the heat flux error is estimated using first and second order adjoint equations. The adjoint codes developed for the inverse heat transfer problems provide the uncertainty estimation for the corresponding forward problems. Numerical tests corroborate the feasibility of fast uncertainty estimation using Hessian maximum eigenvalue obtained via second order adjoint equations.     

Perturbation Solutions for Flow in a Slowly Varying Fracture and the Estimation of Its Transmissivity

The spatial distribution of hydrogeological properties is essentially heterogeneous. Heterogeneity can be characterized quantitatively using geostatistics, which conventionally assumes that the stochastic process is stationary. However, growing evidence indicates that the spatial variability has the multiscale self-similarity characteristics and can be better characterized using non-stationary model but with statistically homogeneous increments. A general framework is developed in this work to conduct the uncertainty quantification analysis by using truncated power variogram model, which can explicitly account for measurement scale, observation scale, and window scale. The effect of the multiscale characteristics of the hydrogeological properties on the uncertainty and the consequential risk associated with the groundwater flow process is investigated. A synthetic two-dimensional saturated steady-state groundwater flow problem is used to evaluate the performance to predict the flow field distribution. For comparative purposes, the evaluation is based on both the truncated power and the traditional variogram models when the underlying porous medium is a random fractal field. The results show that the truncated power variogram model can perform the uncertainty quantification more accurately, and the adoption of traditional variogram model tends to result in a smoother estimation on the flow field and underestimate the uncertainty associated with the hydraulic head prediction. Upscaling is generally inevitable to avoid predictive uncertainty underestimation when the underlying random field exhibits multiscale characteristics.

     

液晶聚合物/柔性聚合物相分离的方向序与流动效应分析

本文用数值方法模拟液晶聚合物(LCP)／柔性链聚合物(FP)的相分离动力学过程，并用有序度S和长主轴倾斜角θ构成向量图，用来分析方向序和剪切流动对相分离形态的影响。结果表明(液晶富)基元的长主轴有沿条状界面平行排列的趋向，在弯曲条纹转角剧烈处或T字口处，有序度明显恶化；另外发现，对系统加剪切流动会引起邻近(液晶富)区域的凝聚，使得条纹变粗变长，同时，剪切率的大小对条纹的粗细和指向矢的分布有影响，表现为剪切率越大条纹拉伸越长，而指向矢的有序度越差。     

苯并三氮唑及其衍生物在菜籽油中的摩擦学性能研究

利用四球试验机考察了苯并三氮唑及其衍生物在菜籽油中的摩擦学性能,并用X射线光电子能谱和扫描电子显微镜分析观察磨斑表面的化学组成和形貌。结果表明,苯并三氮唑在菜籽油中有良好的抗磨作用,在其分子中引入长链烷基虽然提高了其在菜籽油中的溶解度,但却降低了其抗磨减摩性能,这主要是由于苯并三氮唑有效成分的减少以及添加剂和基础油之间的竞争吸附所致。     

Phase Transition upon Solidification from a Liquid State

A criterion that allows one to predict a priori a mechanism (equilibrium or nonequilibrium) according to which the phase transition upon solidification of a melt from a liquid state will occur is obtained. The fundamental possibility of experimental determination of the kinetic constant and the energy of activation is shown on the basis of the phase–transition criterion obtained and the solution of the problem of sequential nonequilibrium crystallization.     

Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.     

单峰接触研究及其在分形表面接触中的应用

基于有限元方法,建立了弹塑性单峰的接触模型.粗糙峰为理想的弹塑性材料,为了考虑不同的材料特性对微凸体变形的影响,分别对9种不同的材料进行了分析.根据有限元计算结果,分析了接触面积,平均接触压力和接触力与变形干涉量之间的关系,并进行了经验公式的拟合.单峰接触所经历的4个不同的阶段,以及不同阶段之间的转化点均作了明确的表达.然后,根据分形理论,将单峰接触模型扩展到了三维的粗糙表面的接触,并提出了一个计算接触表面法向刚度的模型.通过与实验数据和以往模型的结果对比,证明本文中所提出的模型具有较高的精度.     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.