Energy analysis of crack interaction with an elastic inclusion

This paper gives an energy analysis of an elastic solid with a crack which penetrates an elastic inclusion. The purpose of our work is to evaluate the energy release rates (ERR) associated with crack tip extension while the inclusion is stationary, and to evaluate the ERR due to inclusion translation, rotation and expansion with respect to the crack tip. Reduction and increase in the crack ERR caused by an inclusion (shielding and amplification effects of the inclusion) are expressed in terms of the inclusion elastic properties normalized by Young's modulus of the bulk material. The variation in ERR as a crack approaches and passes through a circular inclusion is also examined.     

Concentrated ring loading in a nonlocal elastic medium

Assuming an appropriate nonlocal modulus and using the Boussinesq–Galerkin vector representation of the nonlocal stress field the stress distribution in a nonlocal elastic medium has been found under the concentrated ring normal and shear loadings and force dipoles. The nonlocal modulus used in the paper is the Green function of the diffusion equation. To solve the corresponding boundary-value problem the Laplace transform with respect to the nonlocal parameter and the Hankel transform with respect to the radial coordinate are used. The Laplace transform is inverted analytically; inverting the Hankel transform the oscillatory integrals containing products of Bessel functions have been changed into integrands which decay exponentially, thus producing a solution more amenable to numerical quadrature. All classical singularities for stresses are eliminated.     

Finite element model updating using Hamiltonian Monte Carlo techniques

Bayesian techniques have been widely used in finite element model (FEM) updating. The attraction of these techniques is their ability to quantify and characterize the uncertainties associated with dynamic systems. In order to update an FEM, the Bayesian formulation requires the evaluation of the posterior distribution function. For large systems, this function is difficult to solve analytically. In such cases, the use of sampling techniques often provides a good approximation of this posterior distribution function. The hybrid Monte Carlo (HMC) method is a classic sampling method used to approximate high-dimensional complex problems. However, the acceptance rate of HMC is sensitive to the system size, as well as to the time step used to evaluate the molecular dynamics trajectory. The shadow HMC technique (SHMC), which is a modified version of the HMC method, was developed to improve sampling for large system sizes by drawing from a modified shadow Hamiltonian function. However, the SHMC algorithm performance is limited by the use of a non-separable modified Hamiltonian function. Moreover, two additional parameters are required for the sampling procedure, which could be computationally expensive. To overcome these weaknesses, the separable shadow HMC (S2HMC) method has been introduced. This method uses a transformation to a different parameter space to generate samples. In this paper, we analyse the application and performance of these algorithms, including the parameters used in each algorithm, their limitations and the effects on model updating. The accuracy and the efficiency of the algorithms are demonstrated by updating the finite element models of two real mechanical structures. It is observed that the S2HMC algorithm has a number of advantages over the other algorithms; for example, the S2HMC algorithm is able to efficiently sample at larger time steps while using fewer parameters than the other algorithms.     

Effect of Elastic Modulus on Biomechanical Properties of Lumbar Interbody Fusion Cage

This work focuses on the influence of elastic modulus on biomechanical properties of lumbar interbody fusion cages by selecting two titanium alloys with different elastic modulus.They were made by a new β type alloy with chemical composition of Ti-24Nb-4Zr-7.6Sn having low Young's modulus ～50 GPa and by a conventional biomedical alloy Ti-6Al-4V having Young's modulus ～110 GPa.The results showed that the designed cages with low modulus (LMC) and high modulus (HMC) can keep identical compression load ～9.8 kN and endure fatigue cycles higher than 5× 106 without functional or mechanical failure under 2.0 kN axial compression.The anti-subsidence ability of both group cages were examined by axial compression of thoracic spine specimens (T9～T10) dissected freshly from the calf with averaged age of 6 months.The results showed that the LMC has better anti-subsidence ability than the HMC (p<0.05).The above results suggest that the cage with low elastic modulus has great potential for clinical applications.     

Ab initio prediction of elastic and thermal properties of cubic TiO2

In this paper we focus on the novel solar material, namely cubic TiO₂. The full potential linearized augmented plane wave method in combination with the local density approximation (LDA) and the generalized gradient approximation (GGA) have been used. We calculated structural parameters, elastic constants, wave velocities and thermal properties of the material assuming the fluorite structure. The obtained values were in good agreement with the available theoretical and experimental data. Moreover, the pressure and temperature dependences of the bulk modulus, Debye temperature, Heat capacity and linear expansion coefficient have been addressed for the first time.     

Diameter-dependent elastic modulus supports the metastable-catalyst growth of carbon nanotubes

We measured the elastic modulus of individual multiwalled carbon nanotubes (MWCNTs) grown by catalytic chemical vapor deposition (CVD) over a broad diameter range (10-25 nm). Alternating current (ac) dielectrophoresis was used for efficient tube deposition, and atomic force microscope (AFM) force-displacement curve technique was used for stiffness measurements. The elastic modulus exhibits a strong diameter dependence, showing a difference of nearly 2 orders of magnitude in the 10-20 nm diameter range (thinner MWCNTs have higher elastic modulus). Our results support the metastable-catalyst model in which the catalyst's molten skin plays a key role.     

Sensitivity analysis for non-linear constrained elastostatic systems

Adjoint and direct differentiation methods are used to formulate design sensitivities for non-linear constrained elastostatic systems. Variations of a general response functional are expressed in explicit form with respect to all design field variations, including shape. The response functional is expressed over the undeformed configuration; however, the displacement gradient is included so that field quantities that are defined over the deformed configuration, such as Cauchy stress, may be modelled. The finite element method is used to evaluate the sensitivities in an example problem.     

含损伤复合材料剩余弹性模量预测的不确定分析

针对传统处理不确定问题概率统计方法的局限性, 提出两种非概率分析方法对具有不确定参数的含损伤复合材料剩余弹性模量问题进行研究。非概率方法将不确定变量描述为一区间数或凸集合, 再利用Taylor展开及区间四则运算, 便可得到含损伤复合材料剩余弹性模量的区间范围。非概率分析方法优点在于: 对于不确定参数数据信息依赖性较小, 计算方法简单、 实用, 并且精度可满足工程要求。通过一数值算例的两种情况对含损伤层合板的相对剩余弹性模量进行计算, 结果表明, 所提出的两种非概率方法在不确定信息较少时, 可以得到令人满意的结果。     

Planning repair effort in transfer lines

In this paper, we propose a methodology to optimally allocate the repair rates in transfer lines, which requires estimating the gradient vector of the output rate with respect to repair rates. Partial derivatives can be obtained exactly in the case of two-machine lines. The gradient estimation method for longer lines uses this result together with an algorithm developed to approximate the performance measures of the line. The method developed is used in an optimization context to find the optimal repair rate vector to maximize the throughput given a constraint on the sum of the individual repair rates. The results obtained by this method is compared to those obtained using the finite difference method in gradient estimation.     

Modeling the size-dependent elastic properties of polymeric nanofibers

We present a strain gradient (SG) theory to explain the strongly inverse size dependence between the elastic modulus and fiber diameter in polymeric nanofibers. For centrosymmetric and isotropic materials we showed that the three length-scale parameters can be combined into a single parameter that can be used to predict the onset of the size-dependent trend when the fiber diameter is reduced past its critical size. To address the issue of whether the SG offers a plausible explanation of the size-dependent behavior we conducted a series of uniaxial tensile and static bending tests involving polycaprolactone nanofibers. Since the elastic modulus is highly sensitive to the fiber diameter, it is necessary to correct the experimental data to account for the lack of circularity in the cross-section of the real fiber. Additionally, we applied the SG model to study the size-dependent elastic properties of polypyrrole nanotubes. By approaching the SG theory from a dynamics point of view, our model is able to capture size-dependent effects in the mechanics of fine-scale materials for both static and dynamic responses.     

Role of structure and morphology in the elastic modulus of carbon nanotube composites

The nature of nanoscale reinforcements in the carbon nanotube composites indicates nanocomposite properties are heavily dependent on the micro/nano-structure and morphology. Macroscopic parameter-based properties estimation may lead to deviation as large as 30%. In this paper, a modified shear-lag model, combined with probability statistical theory and composites morphology, is established to investigate the elastic properties of single wall carbon nanotubes (SWNTs)-reinforced polymer composites. The computational results indicated that elastic modulus of nanocomposite was remarkably dependent on the micro/nano-structure, including diameter, length, and orientation of the dispersed SWNTs. Microstructure-dependent shape factor and orientation effect factor played a key role on achieving high-performance nanocomposites. Elastic modulus of nanocomposite with well-dispersed carbon nanotubes was more susceptible to the orientation. Similarly, nanocomposite modulus was more subject to the dispersion influence when SWNTs were well-aligned. The maximal modulus was located in the zone of small rope diameters and small orientation angles when adequate interfacial bonding was provided. The computational results were also compared with experimental outcome and demonstrated good consistence.     

Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity

The dispersive relations of Bloch waves in the periodic laminated structure formed by periodically repeating of two different gradient elastic solids are studied in this paper. First, the various wave modes in the gradient elastic solid, which are different from those in the classical elastic solid, are formulated. Apart from the dispersive P wave and SV wave, there are two evanescent waves, which become the P type and S type surface waves at the interface of two different gradient elastic solids. Next, the continuity conditions of displacement vector, the normal derivative of the displacement vector and the monopolar and dipolar tractions across the interface between two different gradient elastic solids are used to derive the transfer matrix of the state vector in a typical single cell. At last, the Bloch theorem of Bloch waves in the periodical structure is used to give the dispersive equation. The in-plane Bloch waves and the anti-plane Bloch waves are both considered in the present work. The oblique propagation situation and the normal propagation situation are also considered, respectively. The numerical results are obtained by solving the dispersive equation. The influences of two microstructure parameters of the gradient elastic solid and the microstructure parameter ratio of two different gradient elastic solids on the dispersive relation are discussed based on the numerical results.     

Effect of pressure on the structural and elastic properties of ZnS and MgS alloys in the B3 and B1 phases

From the first-principles calculations, we have investigated the elastic stiffness coefficients C₁₁, C₁₂, C₄₄ and the bulk modulus B of the II-VI semiconductors ZnS and MgS under hydrostatic pressure. The calculations are based on the density functional theory within the generalized gradient approximation (GGA) for exchange-correlation interaction. For the structural properties we have shown that ZnS adopt the rocksalt (NaCl or B1) structure over 11.87 GPa pressure, the same character is adopted by MgS over 0.8 GPa. The elastic coefficients have the same behavior for the different structures of alloys; they increase with increasing pressure values. Our results for the structural parameters and equilibrium phase elastic constants are in good agreement with the available theoretical and experimental data.     

Transient dynamic analysis of a fluid-saturated porous gradient elastic column

The dynamic response of a fluid-saturated porous gradient elastic column to a transient disturbance is determined analytically and numerically. The basic dynamic theory of a fluid-saturated poroelastic medium due to Biot is modified by replacing the classical linear elastic model of the solid skeleton by the simple gradient elastic model of Mindlin with just one elastic constant (internal length scale) in addition to the classical ones. Thus, the new theory, which is presently restricted to the one-dimensional case, can take into account the microstructural effects of the solid skeleton. After the establishment of appropriate boundary and initial conditions, the one-dimensional dynamic column problem is solved analytically with the aid of the Laplace transform with respect to time. The time domain response is finally obtained by a numerical inversion of the transformed solution. The effect of the solid microstructure on the response is assessed and discussed.     

Thermal expansion and elastic constants

We give a simple, useful relationship between thermal expansion, V/V, and elastic constants. The relationship permits estimation of thermal expansion from only elastic constants (second order and third order) and atomic volume. Elastic-constant temperature dependence is not required. We test the relationship for a variety of crystalline solids. Considering the 0-293 K region, measurement-calculation disagreement ranges from less than 1 to 15%. The model permits extrapolation of high-temperature (near-linear) thermal expansion to zero temperature.Paper presented at the Tenth International Thermal Expansion Symposium, June 6–7, 1989, Boulder, Colorado, U.S.A.     

A method based on small amplitude homogenization for detecting defects using elastic waves

This work is concerned with the resolution of inverse problems for the detection of defects inside a medium using the propagation of elastic waves, under the assumption of small contrast on the value of the stiffness between the matrix material and that of the defect. This is the so-called small amplitude, small contrast or small aspect ratio assumption. Following the framework developed for optimal design problems, we consider a formal second order asymptotic expansion with respect to the aspect ratio, which allows us to simplify the inverse problem considering it as an optimization problem. According to this and through solving the wave equation in the time domain, we can develop a gradient type algorithm that reduces, in the time interval being considered, the difference between the boundary values obtained from a problem with certain defect distribution that is numerically solved and those values obtained from an assumption on the distribution of the defect. An adaptive procedure is presented for locating the wave source in order to improve the results.     

System reliability analysis of spatial variance frames based on random field and stochastic elastic modulus reduction method

This paper presents the stochastic elastic modulus reduction method for system reliability analysis of spatial variance frames based on the perturbation stochastic finite element method (PSFEM) and the local average of a random field. The stochastic responses and reliability index of each element of a structural frame are characterized by the PSFEM and the first-order second-moment method, to properly handle the correlation structures and scale of fluctuation of random fields. A strategy of elastic modulus adjustment for the estimation of system reliability is developed to determine the range and magnitude of elastic modulus reduction, by taking the element reliability index as a governing parameter. The collapse mechanism and system reliability index of a stochastic framed structure are determined through iterative computations of the PSFEM. Compared with the failure mode approaches in traditional system reliability analysis, the proposed method avoids two major difficulties, namely the identification of significant failure modes and estimation of the joint probability of failure modes. The influences of the correlation structure and scale of fluctuation of the random field upon system reliability are investigated to demonstrate the accuracy and computational efficiency of the proposed methodology in system reliability analysis of spatial variance frames.     

非连续弹性圆柱实心桩屏障对弹性波的隔离

采用波函数展开法和Graf加法定理,根据桩土界面处位移和应力连续的边界条件,得到了非连续弹性圆柱实心桩屏障对入射平面P波和SH波散射系数的理论解,通过引入无量纲位移,即屏障后某处土体内由入射波和散射波引起的总的纵向位移和水平位移与未设屏障时由入射平面P波和SH波在该处土体内单独引起的位移幅值的比值,根据求得的散射系数的理论解,绘制了平面P波和SH波单独入射情况下,弹性圆柱实心桩屏障后不同位置的无量纲位移、不同剪切模量的弹性圆柱实心桩屏障后的无量纲位移等值线和透射系数随桩间距的变化曲线,通过研究这些图件进而分析了屏障后不同位置、剪切模量、桩间距和入射频率对非连续弹性圆柱实心桩屏障的隔离效果的影响,为非连续弹性屏障的隔振设计提供了理论依据.     

Mapping the elastic properties of granular Au films by contact resonance atomic force microscopy

Endowed with nanoscale spatial resolution, contact resonance atomic force microscopy (CR-AFM) provides extremely localized elastic property measurements. We advance here the applicability of CR-AFM on surfaces with nanosize features by considering the topography contribution to the CR-AFM signal. On nanosize granular Au films, the elastic modulus at the grain scale has been mapped out by considering a self-consistent deconvolution of the contact geometry effect in the CR-AFM image. Significant variation in the contact area over granular topography arises as the probe is either in single-?or multiple-asperity contact with the surface. Consequently, in extracting the elastic modulus from CR-AFM measurements on granular surfaces we considered both the normal and lateral couplings established through multiple-asperity contacts between the tip and the surface. Thus, by appropriately considering the change in the contact mechanics during CR-AFM imaging, variations in the elastic modulus have been revealed in the intergrain regions as well as across individual grains.