Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

The classical overlapping Schwarz algorithm is here extended to the spectral element discretization of linear elastic problems, for both homogeneous and heterogeneous compressible materials. The algorithm solves iteratively the resulting preconditioned system of linear equations by the conjugate gradient or GMRES methods. The overlapping Schwarz preconditioned technique is then applied to the numerical approximation of elastic waves with spectral elements methods in space and implicit Newmark time advancing schemes. The results of several numerical experiments, for both elastostatic and elastodynamic problems, show that the convergence rate of the proposed preconditioning algorithm is independent of the number of spectral elements (scalability), is independent of the spectral degree in case of generous overlap, otherwise it depends inversely on the overlap size. Some results on the convergence properties of the spectral element approximation combined with Newmark schemes for elastic waves are also presented.     

微桥法镍膜的弹性模量和残余应力测量

利用MEMS技术制作了不同尺寸的镍(Ni)膜微桥结构样品。采用纳米压痕仪XP系统测量了微桥载荷与位移的关系,并结合微桥力学理论模型得到了两种不同尺寸的Ni膜的弹性模量和残余应力。结果表明,两种不同尺寸的Ni膜的弹性模量结果一致,为190 GPa左右,但是残余应力变化较大。与采用纳米压痕仪直接测得的带有硅(Si)基底的Ni膜弹性模量186.8 7.5 GPa相比较,两者符合较好。     

Influence of thermal treatments on the elastic parameters in iron rich metallic glasses

The change of magnetoelastic properties after thermal treatments has been investigated for two groups of metallic glasses. (Fe₇₉Co₂₁)_75+xSi_15−1.4xB_10+0.4x (x (at.%)=0, 2, 4, 6, 8, 10) has been studied both in the as-prepared state and after thermal annealing in an applied magnetic field, to achieve a particular domain structure, at temperatures well below the crystallization temperatures. Changes in the ΔE effect, magnetomechanical coupling (k) and internal friction coefficient (Q⁻¹) are reported, reaching values of about 60% of the saturation value E_S. Fe₆₄Ni₁₀Nb₃Cu₁Si₁₃B₉ alloys annealed in vacuum for 1 h in the temperature range 350–550 °C showed maximum values of the ΔE effect and k of 61% and 0.85, respectively, accompanied by a minimum value of Q of around 2 for the sample annealed at 460 °C. These variations are related to the progress of nanocrystalization. The properties achieved are among the best reported for magnetomechanical applications.     

Materials modelling: A bridge from atoms to bulk properties

The use of molecular-level materials modelling techniques in the development of advanced performance polymers is discussed, with particular emphasis upon bridging the large difference in the scales of dimensions between atomic structure and fabricated parts. The advantages and disadvantages of bulk quantitative structure-property relations and atomistic modelling are assessed, and the method of group interaction modelling is suggested as a means of bridging the dimensional scales.After a brief introduction to the concept of group interaction modelling, examples of modelling the engineering properties of polymers are presented which are difficult to model quantitatively by any other means. The important phase transitions from the crystal and glassy states of matter to those of rubber- and liquidlike states are shown quantitatively to be due to the same isoenergetic condition. The viscoelastic properties of a polymer are critical for many applications and expressions are derived for the loss and storage components of the complex modulus, with reference to failure initiation conditions. The effect of crosslinking in thermosets upon the glass transition temperature and viscoelastic properties is outlined. Finally, the scaling of time from atomic vibrations to the years involved in creep and ageing effects are discussed.     

Network topology and the theory of rubber elasticity

The earliest investigations on rubber elasticity, commencing in the 19th century, were necessarily limited to phenomenological interpretations. The realisation that polymers consist of very long molecular chains. commencing c. 1930, gave impetus to the molecular theory of rubber elasticity (1932-). according to which the high deformability of an elastomer, and the elastic force generated by deformation, stem from the configurations accessible to long molecular chains. Theories of rubber elasticity put forward from 1934-1946 relied on the assumption that the junctions of the rubber network undergo displacements that are affine in macroscopic strain. The theory of James and Guth (1947) dispensed with this premise, and demonstrated instead that the mean positions of the junctions of a ‘phantom’ network consisting of Gaussian chains devoid of material properties are affine in the strain. The vital significance of the distinction between the actual distribution of chain vectors in a network and their distribution if the junctions would be fixed at their mean positions went unnoticed for nearly 30 years. Experimental investigations, commencing with the incisive work of Gee in 1946. revealed large departures from the relationship of stress to strain predicted by the theories cited. This discrepancy prompted extensive studies, theoretical and experimental, during succeeding years. Inquiry into the fundamentals of polymer networks, formed for example by interlinking very long polymer molecules, exposed the need to take account of network imperfections, typically consisting of chains attached at only one end to a network junction. Various means were advocated to make corrections for these imperfections. The cycle rank ζ of the network has been shown (1976) to be the fundamental measure of its connectivity, regardless of the junction functionality and pattern of imperfections. Often overlooked is the copious interpenetration of the chains comprising typical elastomeric networks. Theories that attempt to represent such networks on a lattice are incompatible with this universal feature. Moreover, the dense interpenetration of chains may limit the ability of junctions in real networks to accommodate the fluctuations envisaged in the theory of phantom networks. It was suggested in 1975 that departures from the form predicted for the elastic equation of state are due to constraints on the fluctuations of junctions whose effect diminishes with deformation and with dilation. Formulation of a self-consistent theory based on this suggestion required recognition of the non-affine connection between the chain vector distribution function and the macroscopic strain in a real network, which may partake of characteristics of a phantom network in some degree. Implementation of the idea was achieved through postulation of domains of constraint affecting the equilibrium distribution of fluctuations of network junctions from their mean positions. This led in due course to a theory that accounts for the relationship of stress to strain virtually throughout the ranges of strain accessible to measurement. The theory establishes connections between structure and elastic properties. This is achieved with utmost frugality in arbitrary parameters.     

纵横波测井资料在储层评价中的应用

将弹性波基础理论，Gasman理论，Biot理论和实验研究结果相结合，探讨了沉积岩地层中速度和弹性模量，弹性模量和孔隙度，剪切模量和孔隙度，速度和孔隙度，孔隙度和Biot系数之间的关系，利用从各种声学测井资料中提取出的P波和S波时差（或速度）定量评价地层孔隙度和骨架物质，辅以泥质指示测井曲线，可以计算地层孔隙度，饱和度，岩性等参数，在M，Krief方法的基础上，采用优化算法寻求适合于任何岩性地层的孔隙度与Biot系数之间的关系式，提高了Biot系数的计算精度，该评价方法有坚实的实验基础和理论基础，评价结果与实验室岩心分析数据吻合。     

退火对GaAs窗口晶体断裂模数的影响

LEC GaAs晶片经高温退火后，残余应力得以部分释放；从而减小残余应力诱生断裂的可能性，提高了GaAs晶体的断裂模数。原生GaAs晶片加工的样品的断裂模数平均值约为135MPa，经退火的GaAs晶片加工样品的断裂模数平均值更高，约为150MPa，断裂模数最高值达163MPa。     

Axial vibrations of U-shaped bellows with elastically restrained end conditions

Previous work by Li et al. in the area of axial vibrations of bellows dealt with fixed end conditions. However, it is seen on several occasions that bellow ends are welded to a small pipe spool that has a lumped mass such as a valve or an instrument. Hence, the present paper aims at finding out the effect of elastically restrained ends on the axial natural frequencies. The analysis considers finite stiffness axial restraints on the bellows, i.e. solving the set of equations with non-homogeneous boundary conditions. Two bellow specimens are considered for comparison having the same dimensions as taken by Li in his analysis. The transcendental frequency equation deduced is accurate as the first, second and third mode frequencies computed are in close agreement to the ones obtained by Li.     

A direct stiffness analysis of a composite beam with partial interaction

The use of the conventional semi-analytical stiffness method in finite element analysis, in which interpolation polynomials are used to develop the stiffness relationships, leads to problems of curvature locking when beam-type elements are developed for composite members with partial interaction between the materials of which it is comprised. The curvature locking phenomenon that occurs for composite steel–concrete members is quite well reported, and the general approach to minimizing the undesirable ramifications of curvature locking has been to use higher-order polynomials with increasing numbers of internal nodes. This paper presents an alternate formulation based on a direct stiffness approach rather than starting from pre-defined interpolation polynomials, and which does not possess the undesirable locking characteristics. The formulation is based on a more general approach for a bi-material composite flexural member, whose constituent materials are joined by elastic shear connection so as to provide partial interaction. The stiffness relationships are derived, and these are applied to a simply supported and a continuous steel–concrete composite beam to demonstrate the efficacy of the method, and in particular its ability to model accurately both very flexible and very stiff shear connection that causes difficulties when implemented in competitive semi-analytical algorithms. Copyright © 2004 John Wiley & Sons, Ltd.