一类多孔固体的等效偶应力动力学梁模型

一维多孔固体结构可采用等效连续介质梁模型来研究其动力学行为. 当类梁结构的高度尺寸和多孔固体单胞结构尺寸相近时,等效模型的力学行为会产生尺寸效应现象. 等效经典模型由于不包含尺度参数而无法描述尺寸相关特点,而广义连续介质力学模型则可以准确地考虑尺寸效应的影响. 基于偶应力理论,对一类单胞含有圆形孔洞的周期性多孔固体类梁结构,给出了分析其横向自由振动的等效连续介质铁木辛柯梁模型. 通过对单胞分析,在应变能等价和几何平均的意义下,定义了等效偶应力介质的材料常数. 利用已有的材料常数,推导了等效铁木辛柯梁的动力学微分方程. 将实际多孔固体结构进行完全的动力学有限元离散计算,所获得的解作为精确解以检验等效梁模型所获得的频率和振型的精度. 振型的比较借助于模态置信准则矩阵方法. 大量算例表明,等效偶应力铁木辛柯梁模型在频率和振型两方面均具有较高的计算精度. 重点研究了单胞孔径的相对大小、类梁结构高度与单胞尺寸比以及类梁结构长高比对等效梁模型精度的影响. 在此基础上,偏保守地建议了多孔固体类梁结构自振分析方法.      

基于偶应力模型的连续体结构拓扑优化设计

经典连续介质理论不包含材料尺度参数,因而基于经典理论的结构拓扑优化无法显现尺度效应.本文在偶应力理论的框架下,构造了四节点四边形离散偶应力单元,将传统的SIMP方法推广至偶应力介质.结果表明,在以结构的最大刚度为目标的设计中,偶应力介质的最优结果取决于宏观结构尺寸与材料微结构尺寸(或者特征长度)的比值,最优结果具有明显的尺度效应,具体为,二者比值较大将产生与传统理论相似的结构,而二者比值相当则产生独特的偶应力主导的结构.     

含弱界面的微纳尺度双层梁的热弹性分析

众多微尺度实验已经证实了一些材料在微纳尺度下的力学行为具有尺寸效应.这种现象采用经典的弹性理论无法得到合理的解释,因而需要新的理论,修正偶应力理论就是其中一种.采用修正偶应力理论研究微纳尺度下两端自由铁木辛柯双层梁受热载荷后的弯曲响应,考虑两层之间存在弱界面.获得了梁的挠度、曲率以及界面剪力等表达式,并与经典弹性力学的结果进行了比较.通过分析计算可知,采用修正偶应力理论可预测微纳尺度下双层梁的尺寸效应,而当梁的特征尺寸远大于其材料的内禀尺度时,则与经典理论的结果一致.     

偶应力理论的等效量定义

从应力和偶应力、外加面力和体力、外加面力偶矩和体力偶矩满足的平衡方程及其边界条件出发,讨论应力和偶应力场的静力学性质.由此导出代表性体积上等效应力和等效偶应力的定义,给出它们与相应的体积平均值间的差异.用本文给出的等效应力和等效偶应力的定义,单位体积上虚功的表达式与均匀介质具有同一性.如果用体积平均值作为等效应力和等效偶应力的定义,则虚功的表达式与均匀介质不具有同一形式.     

考虑应力梯度效应的金属微梁屈服模型研究

一系列微加载测试结果表明,金属微梁的弯曲强度会随着材料外部几何特征尺寸的减小而显著升高,呈现出明显的尺寸相关性.基于位错塞积模型,探讨了纯金属单晶微梁的初始屈服应力,并提出了描述其尺寸相关性行为的关键内禀特征长度.通过综合分析现有的微梁弯曲实验及其离散位错动力学数值模拟结果,并考虑到位错-自由表面交互作用的影响,提出了一种仅涉及位错源的位错塞积构型.在此构型下,对线性应力梯度作用下的位错塞积行为进行了连续性分析,并建立了一个由位错源主导的应力梯度屈服模型.该模型有效地解释了微梁初始屈服应力的尺寸相关性,并与实验结果一致.研究结果表明,针对外部几何特征尺寸在数微米及以下的纯金属单晶微梁,位错塞积行为是其尺寸相关性行为的主导机制,而且刻画这种行为需要两个内禀特征长度参数,即位错源长度和位错塞积长度.为解释非均匀加载条件下微尺度晶体材料屈服应力的尺寸相关性行为,特别是纯金属单晶微梁,提供了新的视角.     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论,建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件,并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明,在材料几何尺寸较小时,本文模型预测到的屈曲载荷明显大于传统理论的结果,有效地反映了尺度效应。几何尺寸较大时,尺度效应消失,本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论，建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件，并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明，在材料几何尺寸较小时，本文模型预测到的屈曲载荷明显大于传统理论的结果，有效地反映了尺度效应。几何尺寸较大时，尺度效应消失，本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

微极介质混合物的等效本构方程

对微极介质混合物引入代表性体积的等效均匀体,用代表性体积边界上的面力和面力偶定义等效应力和等效偶应力,提出了建立微极介质混合物的等效本构方程的一般原理和方法.讨论了以十字形框架为胞元的多胞材料面内变形问题的等效本构方程,给出合理的解析结果.     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

Effective couple-stress continuum model of cellular solids and size effects analysis

The purpose of this paper is to develop a homogeneous, orthotropic couple-stress continuum model to take the place of the periodic heterogeneous cellular solids. Through generalizing the definition of the characteristic length for isotropic couple-stress continuum, four characteristic lengths are introduced as material engineering constants for such kind of continuum. In order to determine the effective moduli and the characteristic lengths of the effective couple-stress continuum, a Representative Volume Element (RVE) method is constructed. The effective properties are obtained based on the response of the RVE under prescribed boundary conditions, and our results agree with the analytical solutions in literature. In addition, the influences of the relative density, the topology, the size, and the properties of the solid material of cellular materials on the effective moduli as well as the characteristic lengths are discussed, respectively. Furthermore, the size effects in cellular solid beams are investigated using our effective couple-stress continuum model. The results show that the developed continuum model in this paper can precisely capture the size effects in cellular solids.     

Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

Cellular solids are usually treated as homogeneous continuums with effective properties. Nevertheless, these mechanical properties depend strongly on the ratio of the specimen size to the cell size. These size effects may be accounted for according to preliminary static analysis of effective continuums based on couple-stress theory. In this paper an effective dynamic continuum model, based on couple-stress theory, is proposed to analyze the behavior of free vibrations of periodic cellular solids. In this continuum model, the effective mechanical constants of the effective continuum are deduced by an equivalent energy method. The cellular solid structure is then replaced with the equivalent couple-stress continuum with same overall dimension and shape. Moreover, the finite element formulation of the couple-stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis. The eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method. A conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution. Several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model. Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

A micromechanically based couple-stress model of an elastic orthotropic two-phase composite

We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin's (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couple-stress properties from the response of a unit cell under several types of boundary conditions: displacement, displacement-periodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions' volume fraction of 18%.     

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

Elastic constants for granular materials modeled as first-order strain-gradient continua

Formulation of a stress–strain relationship is presented for a granular medium, which is modeled as a first-order strain-gradient continuum. The elastic constants used in the stress–strain relationship are derived as an explicit function of inter-particle stiffness, particle size, and packing density. It can be demonstrated that couple-stress continuum is a subclass of strain-gradient continua. The derived stress–strain relationship is simplified to obtain the expressions of elastic constants for a couple-stress continuum. The derived stress–strain relationship is compared with that of existing theories on strain- gradient models. The effects of inter-particle stiffness and particle size on material constants are discussed.     

A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds

We show that the mean wall-shear stresses in wall-modeled large-eddy simulations (WMLES) of high-speed flows can be off by up to \(\approx 100\%\) with respect to a DNS benchmark when using the van-Driest-based damping function, i.e., the conventional damping function. Errors in the WMLES-predicted wall-shear stresses are often attributed to the so-called log-layer mismatch, which, albeit also an error in wall-shear stresses \(\tau _\mathrm{w}\), is an error of about \(15\%\). The larger error identified here cannot be removed using the previously developed remedies for the log-layer mismatch. This error may be removed by using the semi-local scaling, i.e., \(l_\nu =\mu /\sqrt{\rho \tau _\mathrm{w}}\), in the damping function, where \(\mu \) and \(\rho \) are the local mean dynamic viscosity and density, respectively.     

Measurement and scaling analysis of critical energy for direct initiation of gaseous detonations

In this paper, the critical energies required for direct initiation of spherical detonations in four gaseous fuels (C₂H₂, C₂H₄, C₃H₈ and H₂)–oxygen mixtures at different initial pressures, equivalence ratios and with different amounts of argon dilution are reported. Using these data, a scaling analysis is performed based on two main parameters of the problem: the explosion length R _o that characterizes the blast wave and a characteristic chemical length that characterizes the detonation. For all the undiluted mixtures considered in this study, it is found that the relationship is closely given by R_o ? 26 l{R_{\rm o} \approx 26 \lambda} , where λ is the characteristic detonation cell size of the explosive mixture. While for C₂H₂–2.5O₂ mixtures highly diluted with argon, in which cellular instabilities are shown to play a minor role on the detonation propagation, the proportionality factor increases to 37.3, 47 and 54.8 for 50, 65 and 70% argon dilution, respectively. Using the ZND induction length Δ _I as the characteristic chemical length scale for argon diluted or ‘stable’ mixtures, the explosion length is also found to scale adequately with R_o ? 2320 D_I{R_{\rm o} \approx 2320 \Delta_I} .     

Measurement of apparent young's modulus in the bending of cantilever beam by heterodyne holographic interferometry

Detection of micromechanical phenomena in material requires a sufficiently high-accuracy measurement. This paper shows the possibility of applying heterodyne holographic interferometry to such experimental verification. The effects of the ratios of thickness to grain size on the apparent Young's modulus are evaluated based on the deflection patterns of the cantilever beam. The characteristic length ℓ of couple-stress theory is calculated by applying the analytical results of Koiter to our experimental results. This value is about one fifth of the grain size.