On the overall elastic moduli of polymer-clay nanocomposite materials using a self-consistent approach. Part I: Theory

Although few investigations recently proposed to describe the overall elastic response of polymer-clay nanocomposite materials using micromechanical-based models, the applicability of such models for nanocomposites is far from being fully established. The main point of criticism to mention is the shelving of crucial physical phenomena, such as interactions and length scale effects, generally associated by material scientists, in addition to the nanofiller aspect ratio, to the remarkable mechanical property enhancement of polymer-clay nanocomposites. In this Part I of two-part paper, we present a micromechanical approach for the prediction of the overall moduli of polymer-clay nanocomposites using a self-consistent scheme based on the double-inclusion model. This approach is used to account for the inter-inclusion and inclusion-matrix interactions. Although neglected in the models presented in the literature, the active interaction between the nanofillers should play a key role in the reinforcing effect of nano-objects dispersed in a polymer matrix. The present micromechanical model incorporates the nanostructure of clay stacks, modeled as transversely isotropic spheroids, and the so-called constrained region, modeled as an interphase around reinforcements. This latter is linked to the interfacial interaction between matrix and reinforcements that forms a region where the polymer chain mobility is reduced. To account for length scale effects, interphase thickness and particle dimensions are taken as explicit model parameters. Instead of solving iteratively the basic homogenization equation of the self-consistent scheme, our formulation yields to a pair of equations that can be solved simultaneously for the overall elastic moduli of composite materials. When the interphase is disregarded for spheroids with zero aspect ratio, our formulation coincides with the Walpole solution (J Mech Phys Solids 1969;17:235-251). Using the proposed general form, a parametric study is presented to analyze the respective influence of aspect ratio, number of silicate layers, interlayer spacing and nanoscopic size of the transversely isotropic spheroids on the overall elastic moduli of nanocomposite materials.     

High-frequency torsional oscillations—I: Penny-shaped crack in an inhomogeneous medium

The problem of an inhomogeneous medium, whose shear modulus and density vary exponentially with radius, containing a penny-shaped crack undergoing high-frequency torsional oscillations is reduced asymptotically to Wiener-Hopf integral equation and solved by Carleman's method. Uniformly valid asymptotic results are obtained. Explicit expressions are derived for the normal displacement gradient outside the crack region, the stress-intensity factor and the energy of the crack.     

Thermoelastic composites with columnar microstructure and cylindrically anisotropic phases. Part II: One-parameter generalized self-consistent estimates

The effective transverse shear modulus of a transversely isotropic composite with columnar microstructure and with two cylindrically orthotropic phases is estimated by a one-parameter generalized self-consistent model. The equation characterizing the effective transverse shear modulus is a fourth-order polynomial equation instead of the quadratic one determined by the classical generalized self-consistent model. The result obtained in Part II of this work extends the relevant one of Hashin [Z. Hashin, Thermoelastic properties and conductivity of carbon/carbon fiber composites, Mech. Mater. 8 (1990) 293–308].     

On the overall elastic moduli of polymer-clay nanocomposite materials using a self-consistent approach. Part II: Experimental verification

Polyamide-6 (PA6) based nanocomposites were prepared using a modified montmorillonite (MMT) Cloisite 20A as nanofillers. The silicate weight fraction of the prepared nanocomposites, determined by burning off the PA6 matrix, was ranged from 0.2 wt% up to 7.5 wt%. The thermomechanical properties of both the neat PA6 and the PA6 filled with MMT nanoclay were measured by means of uniaxial tension tests and dynamic mechanical thermoanalysis, their crystallinity analyzed by differential scanning calorimetry and their morphology observed by transmission electron microscopy. The elastic stiffness of PA6-clay nanocomposites was examined under two moisture levels and was analyzed with the theory formulated in the Part I of this work. Predicted results are found in good agreement with our experiments. The model capabilities are also critically discussed by comparisons with both experiments issued from the literature and the Mori-Tanaka approach widely used in recent literature. It is demonstrated that the proposed micromechanical model is more efficient than the Mori-Tanaka approach. Moreover, the obtained results support the idea that the elastic stiffness of polymer-clay nanocomposites is governed by the same mechanisms as microcomposites, the effects of particle dimension or constrained region being of a second order.     

Flow and conduction of inhomogeneous media. II. Variation of the basic model of an inhomogeneous medium

On the basis of a basic model of an inhomogeneous medium, different variations are proposed. Analytical dependences are obtained for the conduction of the in homogeneous medium and the results of calculation and experiment are compared.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 46, No. 2, pp. 247–252, February, 1982.     

Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation, the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. In this Part I of a two-part investigation, the theory is extended to include propagation in anisotropic medium characterized by a generic wave-number profile for time-harmonic fields; in a companion paper [J. Opt. Soc. Am. A 22, 1208 (2005)], the theory is extended to time-dependent fields. The propagation characteristics of the beam propagators in a homogeneous anisotropic medium are considered. With use of Gaussian windows for the local processing of either ordinary or extraordinary electromagnetic field distributions, the field is represented by a phase-space spectral distribution in which the propagating elements are Gaussian beams that are formulated by using Gaussian plane-wave spectral distributions over the extended source plane. By applying saddle-point asymptotics, we extract the Gaussian beam phenomenology in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wave-number profile. The general results are applied to the special case of uniaxial crystal, and it is found that the asymptotics for the Gaussian beam propagators, as well as the physical phenomenology attached, perform remarkably well.     

A partially debonded ellipsoidal inclusion in an elastic medium. Part I: Stress and displacement fields

The paper calculates the elastic field of an ellipsoidal inclusion which has undergone an internal deformation and has debonded over a part of its boundary from the surrounding medium. The problem is reduced to the solution of a (singular) integral equation for the displacement discontinuity across the debond. The essential steps of a method of solving this equation are outlined. The elastic field is used in the companion paper (Part II) to calculate the stress intensity factors along the edge of debond.     

The collinear cracks in an inhomogeneous medium subjected to transient load

Summary A laminate model is employed to solve the elastodynamic problem of a collinear crack in an inhomogeneous material. The inhomogeneous material is treated as a series of thinner layer. The Laplace and Fourier transforms are used to reduce the problem to a set of singular integral equations that is solved numerically. Numerical results of two collinear cracks in a functionally graded material strip are obtained to show the influence of material inhomogeneity and crack position on crack tip field intensities.     

Two-sided estimates for thermal resistance of an inhomogeneous solid body

A mathematical model of heat transfer is constructed for a composite inhomogeneous body of arbitrary shape, which includes elements of anisotropic materials. Thermal contact between these elements can generally be imperfect. A dual formulation of the variational problem of stationary heat transfer is used in obtaining two-sided estimates for the thermal resistance of the body.     

内设加劲肋空间DKT形相贯节点轴压承载力公式

规范认为T形和十字形圆钢管节点的平面内抗弯承载力相同,但两类节点的受力特性不同。为了全面对比T形节点和十字形节点的平面内受弯性能,进行了6个节点(4个十字形和2个T形)平面内往复弯矩加载试验,并进行了受力机理分析和有限元分析,其中十字形节点进行了两侧支管同向受弯、反向受弯两种加载模式。结果表明：所有节点均为节点域主管管壁延性撕裂破坏,表现出较好的抗震性能;增加支主管直径比β能明显提高节点的抗弯承载力、延性、耗能能力;T形节点的抗弯承载力高于十字形节点,但延性不如十字形节点,T形节点的耗能能力介于十字形节点在两种加载模式下的耗能能力之间;对比反向受弯加载,同向受弯加载对β较小(接近及小于0.7)十字形节点的性能(耗能、延性和承载力)不利,但对β较大(接近及大于0.9)十字形节点的性能有利。     

非均匀可激发介质中一对螺旋波的共存

基于Barkley模型研究了非均匀性可激发介质中螺旋波的共存性态.该模型中各参数与可激发介质的属性密切相关,数值模拟表明,可激发介质的非均匀性对螺旋波波纹的粗细、疏密程度以及螺旋波的共存有较大的影响.存在于空间中的一对螺旋波由于参数的不同将导致共存或漂移现象.     

Solution of the unsteady heat-conduction equation in an inhomogeneous medium

The solution of an unsteady two-dimensional heat-conduction problem in an inhomogeneous medium is investigated by using differential operators.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 2, pp. 357–360, August, 1978.     

Friction drive of an SAW Motor. Part I: measurements

The surface acoustic wave motor in this study utilized transparent lithium niobate for a stator. We then measured the normal and tangential displacements of the frictional surface of the slider via the transparent stator by means of 2 laser Doppler vibrometers. We thoroughly inspected the measurement conditions and indicated that the measured data were reliable and usable for subsequent precise analyses of the friction drive. The driving conditions for the measurements were a driving frequency of 9.61 MHz and a wave vibration amplitude of 20 nm. The start-up transients of the motor for a duration of 10.4 micros were measured. The measurements showed that the frictional surface of the slider displaced in both the normal and tangential directions followed each wave vibration. The displacements increased with the wave's vibration amplitude: they increased to 10 nm in both directions, in response to the transient increase of the wave's vibration amplitude to 20 nm, under the 15 N preload condition. Moreover, the slider surface rotated in the same direction as the wave surface and its trajectories were a tilted elliptical orbit. Since the surface of the wave rotated in an upright elliptical orbit, the result indicated that the tangential displacement of the slider surface was delayed in relation to that of the wave. The delay was in the range from 30 degrees to 60 degrees under the 15 N preload condition.     

The overall thermoelastic moduli of transversely isotropic composites according to the self-consistent method

We discuss the problem of determining the overall thermoelastic moduli of some transversely isotropic composites. The self-consistent method is used to obtain precise formulae for the overall moduli of certain fibre reinforced and platelet reinforced composites.     

Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement

It is shown that the absorption field inside an inhomogeneous, rotationally symmetric medium with a spatially variable refractive index can be reconstructed by means of a tomographic technique. The classic Abel transform is extended to non-Euclidean optical media. The optical behavior of such a medium is described and, provided that the product of the refractive index with the radial distance is a monotonic function, an exact inverse formula is found. Both a numerical and an analytical test on a phantom function is carried out to prove the exactness of this formula. In contrast, when the assumption of a monotonic function is not true, it is shown that the reconstruction problem becomes subdeterminate because of the presence of annular regions, known as blind areas, inside of which no curved path reaches an extremum. The spatial localization and the size of these regions are related to the extrema of the index of refraction times the radial distance.     

Choice of sink for air conditioning systems. Part 2

The technical superiority of water-cooled air conditioning systems for different applications in terms of overall power rating have been established by the authors in an earlier publication. This paper deals with the generalized approach for evaluating the techno-economic choice of sink for an air conditioning system. A single parameter of unit cooling cost in fils/kWh_c (100 fils = 1 Kuwaiti dinar = US$3.25), combining the system's technical performance and economic aspects, is an effective cost-benefit criterion for making the choice for the type of sink. The results of an analysis, applied to example case studies in Kuwait, favour the water-cooled system for both the direct expansion (mostly used in the residential sector) and the chilled water (commonly used for non-residential applications) air conditioning systems.     

Scattering of shear waves by a rigid cylinder in an inhomogeneous medium

We consider the problem of scattering of shear (SV) waves by a rigid circular cylinder embedded in an inhomogeneous medium, in which density and shear modulus vary as the power of the distance from the center of the cylinder. The problem is investigated by the method of dual integral transformation. The resulting integrals are evaluated asymptotically to obtain the time solution. Numerical computations are done to investigate the behaviour of reflection co-efficients and divergence factors for shear and compressional waves respectively. The results show that the expectations in case of a homogeneous earth are also valid in the heterogeneous case.     

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h⁴) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L₂ projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L₂ projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.