多圆柱域上带间断系数的边值问题

本文研究多个复变数解析函数在多圆柱区域上带间断系数的Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ边值问题。文中给出了这个问题适定的变态提法，首先证明了相应变态问题解的存在唯一性。然后给出原边值问题可解的充要条件及解的积分表达式。     

On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.     

Nanoindentation and Microindentation of CuAlNi Shape Memory Alloy

Nanoindentation and microindentation studies were conducted within individual grains of a CuAlNi shape memory alloy. Linear surface features were observed near the indentations after unloading, many of which were responsive to heating. Crystallographic orientation information was obtained from electron backscattering diffraction in order to compare the orientation of observed surface features to predicted austenite–martensite interfaces, slip planes, and possible fracture planes in this alloy. Most of the features observed can be attributed to austenite–martensite interfaces, which remain in the material after unloading due to the constraints of the plastic deformation created by indentation. Due to the temperature dependence of the transformation stress in shape memory alloys, these stress-induced martensites are observed to diminish with heating and to reappear with cooling. Plastic deformation is observed in the form of pile-up near the indentation.     

Molecular-dynamics of a 2D Model of the Shape Memory Effect

This work investigates the thermodynamic properties of a qualitative atomistic model for austenite–martensite transitions. The model, still in 2D, employs Lennard-Jones potentials for the determination of the atomic interactions. By use of two atom species it is possible to identify three stable lattice structures in 2D, interpreted as austenite and two variants of martensite. The model is described in the first part of the work [6] in detail. The present work studies the thermodynamic properties of the model concerning a small, 2-dimensional test assembly consisting of 41 atoms. The phase stability is investigated by exploitation of the condition of minimal free energy. The free energy is calculated from the thermal equation of state, which is measured in numerical tensile tests. In the second part of this work a chain of eleven 41-atom assemblies is investigated. The chain is interpreted as an idealized larger body, where the individual crystallites represent crystallographic layers allowing for the creation of micro structure. By use of tensile tests at various temperature conditions we sketch how such chain may exhibit quasi-plasticity, pseudo-elasticity and the shape memory effect.     

Internal loops in pseudoelastic behaviour of Ti-Ni shape memory alloys: Experiment and modelling

The stress-strain isothermal hysteresis loops due to the incomplete martensitic transformation are analysed for Ti-Ni shape memory alloys. Experiments show the existence of two distinct yield lines for phase transition; one for the forward transformation austenitemartensite (AM), the other for the reverse transformation MA. The tensile behaviour of single crystals with only one yield line (AM) [1] can be considered as an ideal case. An extension of a thermodynamic model for pseudoelasticity [2] allows these two yield lines to be taken into account.

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Sommario Per leghe Ti-Ni con memoria di forma vengono analizzati i cicli di isteresi isotermici tensione-deformazione prodotti da una incompleta trasformazione martensitica. Gli esperimenti mostrano l'esistenza di due distinte linee di snervamento per la transizione di fase, una verso la trasformazione austenitemartensite (AM), l'altra per la trasformazione inversa MA. Il comportamento a trazione di un singolo cristallo con una sola linea di snervamento (AM) [1], può essere considerato un caso ideale. L'estensione ad un modello termodinamico pseudo-elastico [2] consente di analizzare queste due linee di snervamento.

     

Modeling the dynamic behavior of shape memory alloys

The paper studies the single degree of freedom vibration of a rigid mass suspended by a thin-walled shape memory alloy tube under torsional loading. The behavior is analyzed for the cases of quasiplasticity (low temperatures) and pseudoelasticity (high temperatures) on the basis of an improved version of the Müller–Achenbach model. To illustrate the strong hysteresis-induced damping capacity and the non-linear vibration characteristics, both, free and forced vibrations are considered in the first part of the paper. This is done on the basis of an isothermal version of the model, while the second part of the paper focuses on the effect of non-constant temperature caused by the rate-dependent release and absorption of latent heats.     

Crack growth resistance of shape memory alloys by means of a cohesive zone model

Crack growth resistance of shape memory alloys (SMAs) is dominated by the transformation zone in the vicinity of the crack tip. In this study, the transformation toughening behavior of a slowly propagating crack in an SMA under plane strain conditions and mode I deformation is numerically investigated. A small-scale transformation zone is assumed. A cohesive zone model is implemented to simulate crack growth within a finite element scheme. Resistance curves are obtained for a range of parameters that specify the cohesive traction-separation constitutive law. It is found that the choice of the cohesive strength t₀ has a great influence on the toughening behavior of the material. Moreover, the reversibility of the transformation can significantly reduce the toughening of the alloy. The shape of the initial transformation zone, as well as that of a growing crack is determined. The effect of the Young's moduli ratio of the martensite and austenite phases is examined.     

一类新的(2n-1)点二重动态逼近细分

利用正弦函数构造了一类新的带有形状参数ω的(2n-1)点二重动态逼近细分格式.从理论上分析了随n值变化时这类细分格式的C~k连续性和支集长度;算法的一个特色是随着细分格式中参数ω的取值不同,相应生成的极限曲线的表现张力也有所不同,而且这一类算法所对应的静态算法涵盖了Chaikin,Hormann,Dyn,Daniel和Hassan的算法.文末附出大量数值实例,在给定相同的初始控制顶点,且极限曲线达到同一连续性的前提下和现有几种算法做了比较,数值实例表明这类算法生成的极限曲线更加饱满,表现力更强.     

Effects of Periodic Excitation on the Flow Around a D-Shaped Cylinder at Low Reynolds Numbers

The baseline and forced flow around a bluff body with semi-elliptical D-shape was investigated by solving the 2D Navier–Stokes equations at low Reynolds numbers. A D-shape rather than the canonic circular-cylinder was selected due to the fixed separation points in the latter, enabling to study a pure wake rather than boundary-layer control. The correlation between Strouhal and Reynolds numbers, the mean drag, the lift and drag oscillations vs. the Reynolds number and wake structure were investigated and compared to experimental and numerical data. Effects of open-loop forcing, resulting from the influence of zero-mass-flux actuators located at the fixed separation points, were studied at a Reynolds number of 150. Fluidic rather than body motion or volume forcing was selected due to applicability considerations. The motivation for the study was to quantify the changes in the flow field features, as captured by Proper Orthogonal Decomposition (POD) analysis, due to open-loop forcing, inside and outside the “lock-in” regime. This is done in order to evaluate the suitability of low-order-models based on POD modes of this changing flow field, for future feed-back flow control studies. The evolution of the natural and the excited vortices in the Kármán wake were also investigated. The formation and convection regions of the vortex evolution were documented. It was found that the forcing causes an earlier detachment of the vortices from the boundary-layers, but does not affect their circulation or convection speeds. The results of the POD analysis of the near-wake flow show that the influence of the bluff body shape (“D”-shaped versus circular cylinder) on the baseline POD wake modes is small. It was found that the eigenfunctions (mode-shapes) of the POD velocity modes are less sensitive to slot excitation than the vorticity modes. As a result of the open-loop excitation, two types of mode-shape-change were observed: a mode can be exchanged with a lower-energy mode or shifted to a low energy level. In the latter case, the most energetic mode becomes the “actuator” mode. The evolution of one-slot excitation on still fluid (“Synthetic jet”) was studied and compared to published data and to “actuator” modes with external flow present. Based on the current findings, it is hypothesized that the cross-flow velocity POD modes are suitable for feedback control of wake flow using periodic excitation, due to their low sensitivity to the excitation as compared to the streamwise velocity or vorticity modes.     

Models for one-variant shape memory materials based on dissipation functions

Some simple models for the macroscopic behavior of shape memory materials whose microstructure can be described as a mixture of two phases are derived on the basis of a free energy and a dissipation function. Keeping a common expression for the free energy, each model is based on a different expression for the dissipation function. Temperature-induced as well as isothermal, adiabatic and convective stress-induced transformations are studied. Attention is paid to closed form solutions, comparison among the models and parameter identification.