Non linear constitutive models for lattice materials

We use a computational homogenisation approach to derive a non linear constitutive model for lattice materials. A representative volume element (RVE) of the lattice is modelled by means of discrete structural elements, and macroscopic stress–strain relationships are numerically evaluated after applying appropriate periodic boundary conditions to the RVE. The influence of the choice of the RVE on the predictions of the model is discussed. The model has been used for the analysis of the hexagonal and the triangulated lattices subjected to large strains. The fidelity of the model has been demonstrated by analysing a plate with a central hole under prescribed in plane compressive and tensile loads, and then comparing the results from the discrete and the homogenised models.     

Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries

Integral-type nonlocal damage models describe the fracture process zones by regular strain profiles insensitive to the size of finite elements, which is achieved by incorporating weighted spatial averages of certain state variables into the stress–strain equations. However, there is no consensus yet how the influence of boundaries should be taken into account by the averaging procedures. In the present study, nonlocal damage models with different averaging procedures are applied to the modelling of fracture in specimens with various boundary types. Firstly, the nonlocal models are calibrated by fitting load–displacement curves and dissipated energy profiles for direct tension to the results of mesoscale analyses performed using a discrete model. These analyses are set up so that the results are independent of boundaries. Then, the models are applied to two-dimensional simulations of three-point bending tests with a sharp notch, a V-type notch, and a smooth boundary without a notch. The performance of the nonlocal approaches in modelling of fracture near nonconvex boundaries is evaluated by comparison of load–displacement curves and dissipated energy profiles along the beam ligament with the results of meso-scale simulations. As an alternative approach, elastoplasticity combined with nonlocal and over-nonlocal damage is also included in the comparative study.     

The stiffness and strength of the gyroid lattice

Recently, a nanoscale lattice material, based upon the gyroid topology has been self-assembled by phase separation techniques (Scherer et al., 2012) and prototyped in thin film applications. The mechanical properties of the gyroid are reported here. It is a cubic lattice, with a connectivity of three struts per joint, and is bending-dominated in its elasto-plastic response to all loading states except for hydrostatic: under a hydrostatic stress it exhibits stretching-dominated behaviour. The three independent elastic constants of the lattice are determined through a unit cell analysis using the finite element method; it is found that the elastic and shear modulus scale quadratically with the relative density of the lattice, whereas the bulk modulus scales linearly. The plastic collapse response of a rigid, ideally plastic gyroid lattice is explored using the upper bound method, and is validated by finite element calculations for an elastic-ideally plastic lattice. The effect of geometrical imperfections, in the form of random perturbations to the joint positions, is investigated for both stiffness and strength. It is demonstrated that the hydrostatic modulus and strength are imperfection sensitive, in contrast to the deviatoric response. The macroscopic yield surface of the imperfect lattice is adequately described by a modified version of Hill’s anisotropic yield criterion. The article ends with a case study on the stress induced within a gyroid thin film, when the film and its substrate are subjected to a thermal expansion mismatch.     

High substitution of Fe for Zr in ZrV1.6P0.4O7 with small amount of FeV0.8P0.2O4 for low thermal expansion

Fe³⁺-doped ZrV_1.6P_0.4O₇ with Fe:Zr molar ratios of 1:9, 2:8, 3:7 and 4:6 was synthesized to reduce phase transition and obtain low thermal expansion. It is shown that the phase transition temperature of Fe-doped ZrV_1.6P_0.4O₇ is reduced obviously with increasing the content of Fe. Fe³⁺ ion with lower valence and smaller radius than that of Zr⁴⁺ favors to extend the bond angle of V–O–V(P) close to 180° in ZrV_1.6P_0.4O₇, which is considered responsible for the normal structure at room temperature and low thermal expansion. The thermal expansion coefficients of Fe-doped ZrV_1.6P_0.4O₇ for Fe:Zr molar ratios from 1:9 to 4:6 are calculated to be from −4.33×10⁻⁶$-4.33\times {10}^{-6}$ to 5.2×10⁻⁷ K⁻¹$5.2\times {10}^{-7}{\text{K}}^{-1}$ by linear thermal expansion measurement. The effect of small amount of FeV_0.8P_0.2O₄ formation with higher content of Fe³⁺ on thermal expansion coefficients of the samples is discussed.     

The linewidth of infrared light transitions between the Landau levels in graphene

We theoretically propose a structure that the population inversion between the Landau levels (LLs) of the graphene can be achieved by the electrical injection. This structure may be used for the Landau level-laser and related infrared and terahertz emitters. We mainly study the linewidth of the optical transitions between LLs in graphene due to the electron–acoustic phonon scattering. Within the Huang–Rhysʼs lattice relaxation model, we improve the effective single-phonon mode (ESM) for the acoustic phonon to calculate the linewidth of the optical transition and compare the obtained results with that of in the low and high-temperature limit. We find that the ESM provides a very good approximation for the temperature dependence of linewidth, which covers the dominating features of the low and high-temperature limit.     

纳米二氧化钛的制备及Eu3＋掺杂发光研究

以钛酸四丁酯为前驱物,采用溶胶-凝胶法制备了四种不同配方Eu3+掺杂的TiO2纳米晶.利用扫描电镜（SEM)、EDS能谱、光致发光光谱对样品的形貌、成份及性能进行了表征.研究了退火温度、稀土Eu3+离子掺杂摩尔分数、溶剂乙醇量等对发光性能的影响,并对其发光机理进行了探讨.结果表明:稀土Eu3+掺杂TiO2纳米晶样品,掺杂均匀、颗粒大约在30~80 nm|从EDS能谱分析可得Ti:O原子个数比并不是按化学计量TiO2满足1:2,这是因为在TiO2中形成的是Ti-O-Ti键,Eu3+离子很可能取代了Ti4+离子,同时又形成了氧空位,表明稀土Eu3+离子进入TiO2晶格中|样品的主发射峰在614 nm(5D0→7F2)处发光最强,且在593 nm(5D0→7F1)处出现了属于磁偶极跃迁的发射峰,制备Eu3+∶TiO2纳米晶的组分、退火温度、溶剂乙醇的量不同,发射光谱的强度也不同.     

格点量子Zakharov方程组紧致核截面的存在性与熵的估计

讨论一类非自治带量子纠正的修正Zakharov方程组在无穷格点上解的渐近行为,应用截断函数的技巧证明该格点方程组紧致核截面的存在性,并给出核截面的Kolmogorov-ε熵的上界估计.     

Efficient Lagrangian scalar tracking method for reactive local mass transport simulation through porous media

Mass transfer in the presence of chemical reactions for flows through porous media is of interest to many disciplines. The Lattice Boltzmann method (LBM) is particularly attractive in such cases due to the ease with which it handles complicated boundary conditions. However, useful Lagrangian information (such as solute survival distance, effective diffusivity, collision frequency) is challenging to obtain from the LBM. In this paper, we present a straightforward and efficient Lagrangian methodology (Lagrangian scalar tracking, LST) for performing solute transport simulations in the presence of heterogeneous, first‐order, irreversible reactions, based on a velocity field obtained from LBM. The hybrid LST/LBM technique tracks passive mass markers that have two contributions to their movement: convective (obtained through interpolation of a previously obtained velocity field) and Brownian. Various Schmidt number solutes and different solute release modes can be modeled with a single solvent flow field using this method. Moreover, the mass markers can have a range of reaction rate coefficients. This allows for the exploration of the whole spectrum of first‐order heterogeneous reaction rates with just a single simulation. In order to show the applicability of the LST/LBM scheme, results from a case study are presented in which the consumption of oxygen and/or nutrients within a porous bone tissue engineering scaffold is modeled under flow perfusion culturing conditions. Although the reactive LST methodology described in this paper compliments the LBM, it can also be used with any other flow simulation that can generate the velocity field. Copyright © 2010 John Wiley & Sons, Ltd.