Non-elliptic deformation field near the tip of a mixed-mode crack in a compressible hyperelastic material

This paper gives an asymptotic analysis of the deformation field near the tip of an arbitrary mixed-mode crack in a compressible hyperelastic harmonic material which loses ellipticity at sufficiently large deformations. It is found that the near-tip deformation field is characterized by a localized non-elliptic deformation band issuing from the crack-tip and bounded by two curves of discontinuous deformation gradient. Explicit expression for the near-tip deformation field is obtained both inside and outside the localized deformation band. In particular, a simple relation is derived that determines the orientation of the deformation band in terms of two complex governing parameters of the near-tip fields inside and outside the deformation band, respectively.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems. Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany.     

三色圆偏振激光组合脉冲驱动氦原子产生椭圆偏振的阿秒脉冲

我们提出利用频率为ω,3ω和强度较弱的2ω组成的三色圆偏振激光组合脉冲驱动氦原子能够得到椭偏率较大的阿秒脉冲链的一种方法.通过强场近似方法,计算了氦原子在两色和三色圆偏振激光组合脉冲驱动下发射高次谐波谱及其合成阿秒脉冲链,比较了氦原子(初态为s态)在这两种情况下发射高次谐波谱的特点及其合成阿秒脉冲链椭偏率的大小,结果发现,在反旋的两色ω,3ω激光脉冲基础上加入了频率为2ω的第三色激光脉冲联合作用到氦原子上,所得到的阿秒脉冲链的椭偏率相对于双圆场情况下有所增加,通过调整ω,3ω激光的强度比,并且选择适当的第三色激光的强度,对初态为s态的原子,仍能够得到具有较大椭偏率的阿秒脉冲链.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

Poly(ethylene oxide)–poly(propylene oxide)–poly(ethylene)oxide triblock copolymers at the water/air interface and in foam films

The behavior of commercial poly(ethylene oxide)(PEO)–poly(propylene oxide)(PPO)–PEO triblock copolymers at the water/air interface and in microscopic foam films is studied. In aqueous solution these amphiphilic nonionic substances exhibit a surfactant-like aggregation and adsorption behavior. Even below the critical micelle concentration (cmc) the surface concentration is so high that the PEO chains are squeezed and protrude into the solution in order to accommodate to the situation at the interface. As evidenced by measurements of the ellipticity of light reflected from the free surface of the solution a PEO brush is created at the fluid interface. The microscopic foam film is used as a tool for investigating the normal interaction between two PEO brushes facing each other. Stable foam films are obtained at concentrations below the cmc and steric repulsion predominates (in 0.1 M NaCl). A brush-to-brush contact is established only at higher capillary pressures and the disjoining pressure isotherm follows de Gennes' scaling prediction. At lower pressure a softer steric repulsion occurs. It is governed by the bulk copolymer concentration and hence is fundamentally different from the brush-to-brush repellency. On the whole PEO–PPO–PEO copolymers behave as nonionic surfactants, but the large size of their molecules exemplifies the excluded-volume features. Received: 13 July 1999/Accepted: 27 July 1999     

基于多视场星点椭圆率的望远镜主动准直方法

光学元件的准直失调会引起天文望远镜在观测过程中的像质退化,该问题在大口径快焦比的天文光学系统中更为突出。针对此问题,本文提出一种用于望远镜日常观测过程中的主动准直方法,通过星像解算实时校正副镜位置及姿态达到维持望远镜像质的目的。该方法基于多视场星点椭圆率,利用粒子群优化算法迭代求解望远镜光学元件失调量,从而校正由光学元件失调引起的低阶像差。利用1.6 m多通道测光巡天望远镜光学系统进行模拟仿真,求解副镜失调量残余误差小于1%,在系统设计公差范围以内。利用南极巡天望远镜AST3-3模拟及实验验证,表明该方法可高精度求解望远镜光学元件的失调误差。     

Andreev reflection and bound states in topological insulator based planar and step Josephson junctions

A superconductor-topological insulator-superconductor (S/TI/S) junction having normal region at angle θ is studied theoretically to investigate the junction angle dependency of the Andreev reflection and the formation of the Andreev bound states in the step and planar S/TI/S structures. It is found that the Andreev reflection becomes θ dependent only in the presence of the potential barrier at the TI/S interface. In particular, the step and planar TI/S junction have totally different conductive behavior with bias voltage and potential barrier in the regime of retro and specular Andreev reflection. Interestingly, we find that the elliptical cross section of Dirac cone, an important feature of topological insulator with step surface defect, affects the Fabry-Perot resonance of the Andreev reflection induced Andreev bound states (which become Majorana zero energy states at low chemical potential) in the step S/TI/S structure. Unlike the usual planar S/TI/S structures, we find these ellipticity affected Andreev bound states lead to non-monotonic Josephson super-current in the step S/TI/S structure whose non-monotonicity can be controlled with the use of the potential barrier, which may find applications in nanoelectronics.     

Loading restrictions for the Blatz-Ko hyperelastic model—application to a finite element analysis

In this paper, we consider the Blatz-Ko constitutive model for compressible elastic solids. It is shown that the Cauchy stress tensor leads to a normal loading limitation. This limitation induces slow convergence of the Newton-Raphson algorithm near the maximum authorized normal loading value and divergence if this value is exceeded. In addition, convergence of the Newton-Raphson scheme also depends on ellipticity and strong ellipticity conditions. These various points are discussed in the case of a rectangular specimen subjected to a tensile load and modeled with finite elements.