节点梯度光滑有限元配点法

配点法构造简单、计算高效, 但需要用到数值离散形函数的高阶梯度,而传统有限元形函数的梯度在单元边界处通常仅具有C$^{0}$连续性,因此无法直接用于配点法分析. 本文通过引入有限元形函数的光滑梯度,提出了节点梯度光滑有限元配点法. 首先基于广义梯度光滑方法,定义了有限元形函数在节点处的一阶光滑梯度值,然后以有限元形函数为核函数构造了有限元形函数的一阶光滑梯度,进而对一阶光滑梯度直接求导并用一阶光滑梯度替换有限元形函数的标准梯度,即完成了有限元形函数二阶光滑梯度的构造.文中以线性有限元形函数为基础的理论分析表明,其光滑梯度不仅满足传统线性有限元形函数梯度对应的一阶一致性条件,而且在均布网格假定下满足更高一阶的二阶一致性条件.因此与传统线性有限元法相比,基于线性形函数的节点梯度光滑有限元法的$L_{2}$和$H_{1}$误差均具有二次精度,即其$H_{1}$误差收敛阶次比传统有限元法高一阶, 呈现超收敛特性.文中通过典型算例验证了节点梯度光滑有限元配点法的精度和收敛性,特别是其$H_{1}$或能量误差的精度和收敛率都明显高于传统有限元法.      

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.     

具有切应力梯度的平行平板流动腔的构造

在近年来关于流体切应力与细胞力学行为之间关系的研究中,流体切应力梯度被认为是诱发动脉粥样硬化和内膜增生的重要因素之一.本文探讨如何利用常用的离体细胞力学实验工具--平行平板流动腔模拟具有梯度的定常流切应力环境.结果表明,根据Hele-Shaw流的原理和常用复势W(Z)=AZn(n＞1)的特性,可构造出具有各种切应力梯度分布的流动腔.与其它模拟切应力梯度的方法比较,本文的方法更加简洁、可行.     

Plastic deformation modelling of tempered martensite steel block structure by a nonlocal crystal plasticity model

The plastic deformations of tempered martensite steel representative volume elements with different martensite block structures have been investigated by using a nonlocal crystal plasticity model which considers isotropic and kinematic hardening produced by plastic strain gradients. It was found that pronounced strain gradients occur in the grain boundary region even under homogeneous loading. The isotropic hardening of strain gradients strongly influences the global stress–strain diagram while the kinematic hardening of strain gradients influences the local deformation behaviour. It is found that the additional strain gradient hardening is not only dependent on the block width but also on the misorientations or the deformation incompatibilities in adjacent blocks.     

水下磁异常定位中两种矢量磁力仪配置方式的比较

在简述水下磁异常定位原理的基础上,提出了八单轴和十单轴两种矢量磁力仪配置方式,推导了两种配置方式下磁场大小及梯度的量测方程,对磁场大小及梯度的解算精度进行了比较分析。数值仿真了磁场大小及梯度、载体相对位置误差与测量基线长△X之间的变化关系,结果表明这些误差都随△X加大而增长。相比于八单轴磁力仪配置,十单轴磁力仪配置方式中磁场大小和梯度计算精确,定位误差很小,在水下磁定位实验中应选用十单轴磁力仪配置方式。     

ANCF curvature continuity: application to soft and fluid materials

The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shown that continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained using the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials.

     

Parametric processes due to plasma density gradients

Parametric processes due to density gradients in unmagnetized warm fluid plasmas are considered. Expressions for growth rate and threshold fields, which depend purely on density gradients, have been obtained for back and side scattering.     

The benefit of high-conductivity materials in film cooled turbine nozzles

This study presents an experimental and numerical investigation of the beneficial effect of higher conductivity materials in HP turbine nozzles. Most of the literature studies focus on the maximum temperature that a nozzle can withstand, whereas the effect of thermal gradients is often neglected. However thermal gradients have higher influence on the life of the components and they have to be given careful consideration. In this work it is shown that thermal gradients are reduced by using high conductivity materials and, as a consequence, the nozzles life is appreciably increased.A representative film cooled leading edge with an internal impingement plate was studied experimentally at Texas AM University. Two materials were used, namely polycarbonate and stainless steel, in order to highlight the impact of conduction on coolant effectiveness. Numerically conjugate heat transfer simulations have been carried out with an in house solver to analyse in detail the impact of conduction and internal convection. Both experimental and numerical results show that by increasing the conductivity in the solid region, the thermal gradients are strongly reduced. Numerically it is shown that using inserts of nickel-aluminide alloys in nozzles may reduce the thermal gradients from 3 to 4 times if compared to nowadays design.     

Investigation of internal pressure gradients generated in electrokinetic flows with axial conductivity gradients

Field amplified sample stacking (FASS) is used to increase sample concentrations in electrokinetic flows. The technique uses conductivity gradients to establish a non-uniform electric field that accumulates ions within a conductivity gradient, and can be readily integrated with capillary electrophoresis. Conductivity gradients also cause gradients in near-wall electroosmotic flow velocities. These velocity gradients generate internal pressure gradients that drive secondary, dispersive flows. This dispersion leads to a significant reduction in the efficiency of sample stacking. This paper presents an experimental investigation of internally generated pressure gradients in FASS using micron-resolution particle image velocimetry (μPIV). We measure velocity fields of particles seeded into an electrokinetic FASS flow field in a glass microchannel with a single buffer–buffer interface. μPIV allows for the direct quantification of local, instantaneous pressure gradients by analyzing the curvature of velocity profiles. Measurements show internally generated pressure-driven velocities on the order of 1mm/s for a typical applied electric field of 100 V/cm and a conductivity ratio of 10. A one-dimensional (1D) analytical model for the temporal development of the internal pressure gradient generation is proposed which is useful in estimating general trends in flow dynamics.

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Laws of flow with a limiting gradient in anisotropic porous media

The equations of viscoplastic fluid flow through a porous medium are written for all types of anisotropy. It is shown that in anisotropic media the flows with a limiting gradient are characterized by two material tensors: the tensor of permeability (flow resistance) coefficients and the tensor of limiting gradients. A complex of laboratory measurements for determining the tensors of permeability coefficients and limiting gradients is considered for all types of anisotropic media. It is shown that the tensors of permeability coefficients and limiting gradients are coaxial. Conditions of flow onset and fluid flow laws are formulated for media with monoclinic and triclinic symmetries of flow characteristics.     

抽吸和压力梯度在层流边界层转捩过程中的作用

用空间模式的二次稳定性理论研究了抽吸和压力梯度对边界层三维亚谐扰动流动稳定性的影响．数值结果表明，固体边界上的抽吸有明显的层流控制作用，逆压梯度则有较强的不稳定作用．     

抽吸和压力梯度在层流边界层转捩过程中的作用

用空间模式的二次稳定性理论研究了抽吸和压力梯度对边界层三维亚谐扰动流动稳定性的影响．数值结果表明，固体边界上的抽吸有明显的层流控制作用，逆压梯度则有较强的不稳定作用．     

On the theory of mixtures of elastic solids

In this paper we derive a theory for binary mixtures of elastic solids in which the independent constitutive variables are the displacement gradients, displacement fields, volume fractions and volume fraction gradients. The theory is linearized and a uniqueness theorem with no definiteness assumption on the elasticities and no restriction on the initial stresses is presented.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

Detonation propagation in hydrogen–air mixtures with transverse concentration gradients

The influence of transverse concentration gradients on detonation propagation in \(\hbox {H}_2\)–air mixtures is investigated experimentally in a wide parameter range. Detonation fronts are characterized by means of high-speed shadowgraphy, OH* imaging, pressure measurements, and soot foils. Steep concentration gradients at low average \(\hbox {H}_2\) concentrations lead to single-headed detonations. A maximum velocity deficit compared to the Chapman–Jouguet velocity of 9 % is observed. Significant amounts of mixture seem to be consumed by turbulent deflagration behind the leading detonation. Wall pressure measurements show high local pressure peaks due to strong transverse waves caused by the concentration gradients. Higher average \(\hbox {H}_2\) concentrations or weaker gradients allow for multi-headed detonation propagation.     

On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fourier's law of heat conduction holds.     

Thermodynamics of non-simple elastic materials

Elastic materials whose local state depends upon the first and second order gradients of the deformation, the temperature, its gradient and the time rate of change of the temperature are studied according to an inequality proposed by Green and Laws. It is shown that in such materials either thermal disturbances can propagate with finite speed in the linear theory, and the constitutive quantities do not depend upon the second order gradients of the deformation or the constitutive quantities may depend upon the second order gradients of the deformation and in the linear theory thermal disturbances do not propagate with finite speed. In the latter case the entropy inequality reduces to the Clausius-Duhem inequality.     

Statistics of velocity gradients in turbulence at moderate reynolds numbers

A direct numerical simulation of the Navier-Stokes equation is performed in order to investigate the small scale structure of turbulence at moderately large microscale Reynolds numbers 40–140, using the spectral method with 340³ modes starting from a high-symmetric flow. It is shown that the small scale motion is statistically isotropic. The probability density distribution of the velocity is Gaussian, while those of the velocity gradients and the vorticity are not Gaussian but have long exponential tails. The moments of the velocity gradients are expressed in terms of the gamma function, and the ratio of the moments of the velocity gradients of successive orders increases linearly with the order. A comparison is made with a laboratory experiment.     

Shear banding analysis of geomaterials by strain gradient enhanced damage model

Shear band localization is investigated by a strain-gradient-enhanced damage model for quasi-brittle geomaterials. This model introduces the strain gradients and their higher-order conjugate stresses into the framework of continuum damage mechanics. The influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law. A weak-form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher-order stresses. Damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second-order characteristic surface. Analytical solutions for the distributions of strain rates and strain gradient rates, as well as the band width of localised damage are found. Numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of Poisson’s ratio and a parameter representing the shape of uniaxial stress–strain curve. It is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second-order discontinuous shear band boundary and the weak discontinuous bifurcation theory.