Flow analysis of a power-law fluid confined in an extrusion die

The one-dimensional equations in cylindrical coordinates governing flow in an arbitrary cross-sectional shape of a cavity and the slot are derived by accounting of the order of magnitude of terms by using scaling arguments and asymptotic techniques. The derived equation is based on an average momentum and mass balance within the cavity. The one-dimensional equations governing flow in a single-cavity die which can be used to predict the geometry of the cavity and the film thickness deviations for given geometry and operating points and thus yield a design strategy for extrusion die optimized for specific applications. The derived one-dimensional governing equations with the exception of coordinates system for extrusion dies are found to be identical to that of Leonard [Polym. Eng. Sci. 25 (9) (1985) 570] and Weinstein and Ruschak [AIChE J. 42 (9) (1996) 2401] who used the shape factor in the different manner and can be reduced to Miller’s [Ind. Eng. Chem. Fundam. 11 (4) (1972) 524] simple model under the certain restrictions.     

Similitude invariants and scaling laws for buckling experiments on anti-symmetrically laminated plates subjected to biaxial loading

Establishing the similitude between the model and prototype rigorously is a necessary step in designing an experiment efficiently. So far, to the best of the authors’ knowledge, no one has ever derived the similitude invariant for anti-symmetric cross- and angle-ply laminated plates subjected to biaxial loading before. This research paper is the first to establish the similitude invariant of anti-symmetric cross- and angle-ply laminated plates by applying the similitude transformation to the governing differential equations of buckling directly. Then the scaling laws for buckling loads of laminated plates subjected to biaxial loads are derived. But in reality, either due to the complexity of the scaling laws or to economize on costly experiments, it may not be feasible to construct the model conforming to the scaling laws completely, therefore partial similitude is investigated theoretically and approximate scaling laws are recommended. The buckling loads of the prototype predicted from the scaling laws are then compared with the available theoretical values. The complete similitude cases show exact agreement between results predicted from the scaling laws and the available analytical solutions. For partial similitude cases, the models distorted in stacking sequences, number of plies, and material properties are studied and the approximate scaling laws which yield good agreement are recommended.     

Use of scaled-down models for predicting vibration response of laminated plates

An analytical investigation was undertaken to show how and to what extent similitude theory can be applied in the design of scaled-down models. Through similitude the necessary similarity conditions, referred to herein as scaling laws, pertaining to free vibrations of laminated plates are derived. Establishment of scaling laws, based on the direct use of the governing equations is discussed and their use in the design of models is presented. These scaling laws provide the relationships between structural geometric parameters and frequency response parameters of a scaled-down model and its prototype. Later, these conditions are used to design a model, the experimental data of which can be projected in order to predict the behavior of the prototype. Attention is focused on the use of models for the analysis of free vibrations of multilayered composite rectangular plates. Angle-ply and cross-ply symmetric configurations are chosen for investigation. This analytical study indicates that distorted models with a different number of layers, material properties, and geometries than those of the prototype can predict the behavior of the prototype with good accuracy.     

Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition

In this paper, we study the free vibration of axially functionally graded (AFG) Timoshenko beams, with uniform cross-section and having fixed–fixed boundary condition. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, there exists a fundamental closed form solution to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions having simple polynomial variations, which share the same fundamental frequency. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of non-homogeneous Timoshenko beams. They can also be useful for designing fixed–fixed non-homogeneous Timoshenko beams which may be required to vibrate with a particular frequency.     

基于连续损伤的岩石渗流有限元分析

该文基于连续损伤力学,建立了含损伤的岩石渗流模型,研究岩石损伤破坏并分析其渗流特性。该文采用Weibull分布函数来模拟岩石弹性模量和强度的非均质特性,利用Biot本构关系建立含孔隙结构的弹性变形方程,结合Drucker-Prager强度准则、统计强度理论和连续损伤理论推导出岩石连续损伤演化控制方程及其定解条件,进而采用有限元法对其求解。该文给出的数值算例结果表明所提出的含损伤渗流模型适用于研究岩石的渗流特性,分析结果与实验数据吻合较好,且该求解方法可以对岩石体积压裂、岩层弱面进行可靠、有效的数值分析。     

Solutions of a twelfth order thick plate theory

Summary A system of equilibrium equations governing a twelfth-order theory for the bending of thick plates is shown to be equivalent to a biharmonic equation together with four Helmholtz equations. These equations are closely related to equations derived by Cheng for an elasticity based thick plate theory. Detailed comparisons between the solutions for the displacements and stresses predicted by the approximate plate theory and an exact theory give some basis for deciding the applicability of the plate theory. As an example of the application of the solution procedure presented here, some earlier results for the decay parameters for the end problem for finite width plates are extended to the present case of twelfth-order plate theory.With 5 Figures     

The numerical solution of two-dimensional,steady flow problems by the finite element method

Finite element equivalents of the equations governing shearing and buoyancy driven flows are derived, and reduced to upwind forms suitable for the solution of problems in which the Reynolds and Rayleigh numbers are large. A modification to the central difference iterative method is studied which increases the Reynolds and Rayleigh numbers for which a central difference form may be used. A comparison is made between the results obtained using the central and upwind forms of the finite element method and those predicted by finite difference methods in the case of flow in a cavity. A mesh refinement study is made. The upwind forms of the finite element equations are applied to the solution of a complex flow problem involving the flow of glass in a throated furnace in the case of constant- and temperature- dependent viscosity and conductivity.     

The influence of metal vapors resulting from electrode evaporation in a thermal puffer-type circuit breaker

The present work is a computational investigation of thermal plasmas in SF₆ switchgear, which burns in a mixture of working gas and PTFE and metal vapors resulting from nozzle ablation and electrode erosion, respectively. In order to solve the concentration of PTFE and electrode vapors conservation equations for the PTFE vapor and the metal vapor are solved together with the governing equations for mass, momentum, and energy of the gas mixture under the assumptions of local thermodynamic equilibrium (LTE) and local chemical equilibrium (LCE). The influence of metal vapor due to electrode evaporation on the arc behavior in SF₆ switchgear can be explained with some important parameters indicating the characteristics of SF₆ arc plasma. First, temperature in arc column is considerably different from that of the case without contact evaporation. The contamination by metal vapors broadens the arc radius and lowers the core temperature when compared to the case without evaporation. Second, the concentration of PTFE vapor from nozzle surfaces can more easily move to the expansion volume during high current period due to the existence of metal vapors near electrodes. Finally, the electrical conductivity in the arcing zone can be higher and may negatively affect the interruption limit of switchgears due to the increasing rate of breakdown by the metal vapors between the electrodes.     

Numerical simulation of variable polarity vertical-up plasma arc welding process

Three-dimensional transient governing equations were developed based on conservation laws of energy, momentum and mass. These equations described physical phenomena of convection in weld pool and heat transfer in workpiece during variable polarity vertical-up plasma arc welding process. Boundary conditions for the developed governing equations were given. Welding energy input for variable polarity vertical-up plasma arc welding process was quantitatively expressed. Free surface deformation of the keyhole molten pool was coupled into calculation. Effect of wire filling on the geometry of molten pool and weld reinforcement was considered in the simulation. Correlations of temperature and thermophysical properties for aluminum alloy 2219 were quantitatively established. A control volume based finite difference method was used to solve the discrete governing equations. Moreover, dynamic evolutions of geometrical profile, dimension and fluid flow for the molten pool and keyhole were simulated through the developed computational routines, which achieved transient solution of fluid flow field coupling with thermophysical properties, temperature field and weld pool free surface deformation. Besides, the effect of the workpiece thickness on the moments of keyhole formation and stable keyhole establishment was analyzed, and thermal cycles for the main welding stage were calculated. In addition, experiments via variable polarity vertical-up plasma arc welding technique were conducted, and the established models were experimentally verified through weld cross-section profiles.     

结霜过程的变密度分析

采用分子扩散模型对结霜过程进行数学描述。在支配方程推导及求解过程中，采用了霜层变密度分析并了据试验数据拟合的传质系数。计算结果表明，霜层厚度及霜层密度，与试验数吻合较好。     

Study of wave propagation in nanowires with surface effects by using a high-order continuum theory

A high-order continuum model is developed to study wave propagation in nanowires. By using the model, heterogeneous nanostructure effects can be captured especially for high wave frequency cases. Surface stress effects are also included by using the incremental deformation approach. The governing equations of motion in the nanowire are derived including both the strain-independent and strain-dependent surface stresses. For simplicity and clarity, specific attention will be paid to the effects of strain-independent surface stress in this study. The accuracy of the proposed model is validated by comparing dispersion curves of longitudinal wave propagation from the current model with those from the exact solution. By conducting a reduced formulation, the results predicted by the current model will be compared with those based on existed high-order models to show capability of the current model. Numerical simulations are then conducted to study both longitudinal and flexural wave propagation in nanowires. The surface stress effects upon both longitudinal and flexural wave propagation in nanowires are demonstrated, from which the size dependent wave information in nanowires can be observed. Some new physical wave phenomena related to the surface stress effects are discussed.     

基于一种新修正偶应力理论的平面正交各向异性功能梯度梁静弯曲模型及尺度效应

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁模型。模型中包含两个材料尺度参数,因此能够分别描述在两个正交方向上由尺度效应带来的不同大小弯曲刚度增强。基于最小势能原理推导了平衡方程和边界条件,并以自由端受集中载荷作用的悬臂梁为例给出了弯曲问题的解析解。该梁模型的控制方程以及解的形式和经典梁模型是一致的,只是在刚度项中增加了一项和尺度效应有关的项。算例结果表明:采用本文模型所预测的梁挠度总是小于经典理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的减小而增大,并在梁的几何尺寸远大于尺度参数时逐渐消失。     

Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.     

Refined theory for the analysis of laminated orthotropic structures

A new higher-order theory for the analysis of laminated orthotropic plates and shells subject to both mechanical and thermal loads is developed. Using the variational approach the system of governing differential equations and corresponding boundary conditions are derived. Two refined models of the stress and strain state are considered, their application and accuracy are discussed. The analytical solution is obtained for plates and shells with the Navier boundary conditions on the side surfaces. The results of calculations are given and compared with an exact three-dimensional solution available in the literature. The influence of the laminated structure upon the exactness of results and the characteristics of stress–strain state is studied and discussed.     

剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。该方程与相应的边界条件构成了微分特征值问题。直接求解该问题,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

Numerical manifold method based on the method of weighted residuals

Usually, the governing equations of the numerical manifold method (NMM) are derived from the minimum potential energy principle. For many applied problems it is difficult to derive in general outset the functional forms of the governing equations. This obviously strongly restricts the implementation of the minimum potential energy principle or other variational principles in NMM. In fact, the governing equations of NMM can be derived from a more general method of weighted residuals. By choosing suitable weight functions, the derivation of the governing equations of the NMM from the weighted residual method leads to the same result as that derived from the minimum potential energy principle. This is demonstrated in the paper by deriving the governing equations of the NMM for linear elasticity problems, and also for Laplaces equation for which the governing equations of the NMM cannot be derived from the minimum potential energy principle. The performance of the method is illustrated by three numerical examples.     

Trefftz-type boundary elements for the evaluation of symmetric coefficient matrices

The classical Trefftz-method can be generalized such that different types of finite elements and boundary elements are obtained. In a Trefftz-type approach we utilize functions which a priori satisfy the governing differential equations. In this paper the systematic construction of singular Trefftz-trial functions for elasticity problems is discussed. For convenience a list of solution representations and particular solutions is given which did not appear together elsewhere. The Trefftz-trial functions with singular expressions on the boundary are constructed such that the physical components (stresses, strains, displacements) remain finite in the solution domain and on the boundary. The unknown coefficients of the linearly independent Trefftz-trial functions for the physical components can be obtained by using a variational formulation. The symmetric coefficient matrix in the discussed procedure can be obtained from the evaluation of boundary integrals. As an application of the proposed boundary element algorithm, the symmetric stiffness matrices of subdomains (finite element domains) are calculated. For the numerical example the solution domain is decomposed into triangular subdomains so that a standard finite element program could be used to assemble the system of equations. The chosen example is meant as a simple test for the proposed algorithm and should not be understood as a proposal for a new triangular finite element. Using the proposed boundary element techniques, symmetric stiffness matrices for irregular shaped subdomains (finite elements) can be derived. However, in order to use the method in a finite element package for the coupling of irregular shaped subdomains some program modifications will be necessary.     

The steady state response of geometrically non-linear shallow arches

The non-linear steady state response of structures with curvature is investigated through the expository example of a shallow circular arch. A consistent mass finite element formulation is employed to derive the governing non-linear differential equations. These equations are solved by assuming a single mode expansion reducing the governing equations to the single degree-of-freedom Duffing's equation with a quadratic term. The non-symmetric amplitude-frequency curve is derived and compared with results previously obtained by direct integration of the equations of motion.     

A new formulation and computation of the triphasic model for mechano-electrochemical mixtures

From the generalized first law of thermodynamics for an irreversible thermodynamical system, a new set of governing equations for the mixture theory is derived based on the triphasic model for mechano-electrochemical mixtures. It is shown that, in the case of electroneutral solution, a new biphasic mixture theory including the electrochemical effects can be derived from the new governing equations. The chemical-expansion stress representing both the influences of deformation on the fixed charge density and the electric potential of fixed charge field is given. For comparison and verification purposes, the numerical solution for a confined compression problem of a charged hydrated soft tissue is computed using the multiquadric method. Received: 1 November 1998     

On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.